terça-feira, 3 de abril de 2018

## FREGE'S PHILOSOPHY OF LANGUAGE: A CRITICAL STUDY

DRAFT OF A CHAPTER FOR THE BOOK 'PHILOSOPHICAL SEMANTICS'


An Extravagant Reading
of Fregean Semantics



Wenn es eine Aufgabe der Philosophie ist, die Herrschaft des Wortes über den menschlichen Geist zu brechen, indem die Täuschungen aufdeckt, die durch den Sprachgebrauch über die Beziehungen der Begriffe oft fast unvermeidlich entstehen (…) so wird meine Begriffschrift, für diese Zwecke weiter ausgebildet, den Philosophen ein brauchbares Werkzeug werden können.
[If it is a task of philosophy to break the power of the word over the human spirit by laying bare the misconceptions that through the use of language often almost unavoidable arise … then my ideography, further developed for these purposes, can become a useful tool for the philosopher.]
—Gottlob Frege

…might the time not have come to reflect about the very foundations of analytic philosophy, and to see it as one task of philosophy to break the power of the mathematical sign over the philosophical mind?
—Edward Kanterian

The importance of Fregean semantics for the philosophy of language derives from its unique blend of theoretical simplicity, explanatory scope and philosophical relevance. In this chapter, I want to revise and reconstruct the essentials of Fregean semantics. I intend to make it clear that his basic concept of sense can be paraphrased in terms of semantic-cognitive rules and that his concept of existence can be reinterpreted in terms of the effective applicability of semantic-cognitive rules, leading to some unexpected consequences regarding the explanation of the concepts of verification, fact and truth. With the identification of senses with rules, I intend to show the real link between Wittgenstein’s semantics – that is, the way I understood him in the last chapter – and Frege’s semantics. This link was already noted by Michael Dummett, though he still offered no pragmatic exploration. Anyway, my aim here is not to produce a work of Fregean scholarship. My aim is instead to reconstruct Frege’s semantic work with him, against him, and beyond him, in order to provide a more rigorous framework for the rather vague semantic insights gained in the first chapters.
     As is general knowledge, Frege explains reference (Bedeutung) using a semantic intermediary link, which he called sense (Sinn) (1891:14). The schema below shows how Frege deals with these two main levels, (1) sense and (2) reference in the case of a predicative singular assertive sentence (Satz) of the form Fa:

singular term: a                             general term: F                    sentence: Fa
1. sense                                            sense                                      thought
2. reference                                     concept (> object)               truth-value

Although Fregean semantics was a development of unparalleled importance for contemporary philosophy of language, it is not free from well-known oddities. My intuitively natural reading of its main semantic elements in terms of conceptual rules will also show how to purge Frege’s semantics of its most puzzling eccentricities.
1. Reference of the singular term
Let’s start with singular terms. The reference of a singular term is, for Frege, the object itself, taken in an enlarged sense. The reference of the name ‘Moon’, according to him, is the Moon itself with its craters. To designate the reference, he uses the German word ‘Bedeutung,’ whose literal translation in English is ‘meaning.’ Most English translators have chosen words like ‘reference,’ ‘denotation,’ and ‘nominatum,’ in this way making clear what Frege really had in mind. There are also other terms, like ‘semantic value,’ ‘semantic role’ and ‘truth-value potential.’ These terms underline the contributions of the references of a sentence’s components to the truth-value of the sentence as a whole. Although the literal translation of ‘Bedeutung’ as ‘meaning’ remains the correct one, for the sake of clarity I will use the word ‘reference.’[1]
     There is also an interpreter’s discussion of the reason why Frege would have chosen the unexpected word ‘Bedeutung’ for the reference of a nominal term. A widespread interpretation is that one of the meanings of ‘Bedeutung’ (as well as of ‘meaning’ or ‘signification’) is relevance or importance, since reference is what matters most for truth (Tugendhat 1992: 231). While this may be the case, it seems clear to me that the strongest reason, at least with regard to the reference of natural language terms, is that by introducing the term ‘Bedeutung’ Frege substantivated the verb ‘bedeuten.’ In this way, the word no longer expresses the act of pointing at (deuten) or of designating (bezeichnen), but rather what is pointed at (die Bedeutung), what is designated (das Bezeichnete), that is, the reference itself.[2] In German these derivations could be diagrammed as follows:

Bedeutet... deutet...      bezeichnet...      →     was gedeutet, bezeichnet wird/
(means)                          (indicate... designates)  (what is denoted, designated)
                                                                                                                                                                                                          
                                                                                           ↓
                                                                                                                                                                           
                                                                                 die Bedeutung
                                                                            (meaning = reference)

This would have been the small semantic twist with which Frege turned the word ‘Bedeutung’ into a technical term – a twist that seems to betray some semantic-referentialist influence.
2. Sense of the singular term
Now we come to what Frege understands as the sense of a singular term. To introduce it, compare the following two sentences:

1.     The morning star has a dense atmosphere of CO2.
2.     The evening star has a dense atmosphere of CO2.

Sentences (1) and (2) concern to the same thing regarding the planet Venus. But in spite of this, a person can know the truth of (1) without knowing the truth of (2) and vice versa. Frege’s explanation for this is that although the two singular terms ‘the morning star’ and ‘the evening star’ refer to the same planet Venus, they convey different informative contents, that is, they have different senses (Sinne).[3] The word ‘sense’ is defined by Frege as an object’s way of being given (die Art des Gegebenseins des Gegenstandes), which is usually translated as a mode of presentation. The senses of the singular terms ‘the morning star’ and ‘the evening star’ are different, because ‘the morning star’ presents Venus as the brightest celestial body usually seen just before sunrise, while ‘the evening star’ presents the same planet Venus as the brightest celestial body usually seen shortly after sunset…
     Frege writes that words express their senses (drücken ihre Sinnen aus), while senses determine (bestimmen) their reference, since the mode of presentation should show us how to find the reference. Even in cases where the reference does not exist, this determination of reference through sense is given as a possibility, since even in this case the words preserve their senses. This fact points to a flaw in Frege’s idea that sense is the way an object presents itself to us, for in the case of empty terms there is no object to be presented to us. This is why sense can be better understood as the intended mode of presentation instead of as a mode of presentation given by the object (Textor 2010: 134); sense is the way we intentionally present an object or reference to ourselves, whether it exists or not. At any rate, for Frege an expression can have a sense without a reference, but cannot have a reference without its determination by means of a sense.
     Frege extended his notion of sense to other terms and to sentences. In the case of the senses of (declarative) sentences, he calls it cognitive or (more literally) epistemic value (Erkenntniswert). The last term is also appropriate. The Fregean concept of sense has epistemological interest, for it constitutes the proper informative content of the linguistic expression. It is what makes ‘the evening star’ and other expressions informative. Or, using Dummett’s words, ‘sense is what we understand when we understand an expression’ (1990: 92). The philosophical importance of Fregean semantics is largely due to the epistemological and ontological imports of the concept of sense (this is what distinguishes it from a more exclusively linguistic semantics like that of Ferdinand de Saussure.)
     Frege is a Platonist about sense. For this reason he conceives senses as abstract entities which can only be analyzed in terms of constituents that are also senses. A consequence of his Platonism of senses is that it prevents him from analyzing senses in terms of other concepts. However, it is just this task that naturally imposes itself. For it seems very plausible to understand senses as semantic-cognitive criterial rules. We see here a fundamental difference between Fregean semantics and the semantic considerations of the later Wittgenstein, who regarded senses or meanings as depending on episodic uses of expressions determined by rules. Dummett was perhaps the first to defend the idea that senses are rules as the most natural reading of Frege’s use of the term senses. As he wrote in his book on Frege’s philosophy of language:
The sense of a word consists in a rule which, taken together with the rules constitutive of the senses of the other words, determines the condition for the truth of a sentence in which the word occurs. (Dummett 1981b: 194; my italics)
And concerning the singular sentences in Frege, understanding with the term ‘criterion’ the condition of satisfaction of a semantic rule, he wrote:
To know the sense of a proper name is to have a criterion for recognizing, for any given object, whether or not it is the bearer (referent) of that name; to know the sense of a predicate is to have a criterion for deciding, for any given object, whether or not the predicate applies to that object; and to know the sense of a relational expression is to have a criterion for deciding, given any two objects taken in a particular order, whether or not the relation it stands for holds between the first object and the second. (Dummett 1981b: 229)[4]
The identification between senses and rules proves particularly compelling when we take numerical expressions as examples. Consider the following expressions:

1 + 1,
 6/3,
(7 + 3) – 8,
(874 – 870)/2
5 – 3

All these numerical expressions have the same reference: the number 2. But their senses or modes of presentation are in each case different. At the same time, they are expressions of procedures, methods, semantic-cognitive rules or, in most cases, combinations of such rules by means of which we reach the identification of the same number 2 as a result (See Runggaldier 1985: 91 f.).
     By treating senses as semantic-cognitive rules and these rules in the primary case as shared conventions, we contrast them with what Frege called colorations and illuminations (Färbungen and Beleuchtungen), which are feelings often associated with image representations (Vorstellungen) and sensory-perceptions (Anschauungen), as such all belonging to an intrinsically subjective level (Frege 1892: 31). These ‘colorations’ and ‘illuminations’ are names for what we would more often call expressive meanings, that is, sensory-emotional states that we normally and customarily associate with expressions. Thus, for example, the words ‘love,’ ‘dog’ and ‘hell’ in the sentence ‘Love is a dog from hell’ (Bukowski) contrastively associate words linked with strong specific emotions in order to create an epigrammatic effect.
     As Frege realized, the kind of appeal or lack of appeal that the colorations associated with words have for different persons depends correspondingly on similarities and differences in their human natures. Because of this, they do not require conventions to be communicated, as in the case of senses. This is why some people are emotionally moved by a certain poem, while others are not. Consequently, it is very difficult to translate poetry, which depends so much on colorations acquired by expressions in a particular language and form of life. Hence, colorations are not results of conventional rules; they are rather regularities originating from shared aspects of human nature within a historically developed cultural context. If my understanding of Wittgenstein’s argument against private language is correct, then his attempt to explain phenomenological language as a simple replacement of public behavioral criteria like uttering ‘ouch!’ under conditions that would cause pain with a sentence like ‘I feel pain’ is insufficient (1984d, sec. 244). It is an attempt to assimilate the referential meaning of the phenomenal language to its expressive meaning (I suppose that both can be legitimated).
     If in opposition to Frege we accept the view that sense is usually only something with the form of a rule (etwas Regelartiges), namely, a convention or a combination of conventions, we can easily solve the problem of the com­municability of senses that has long tormented philosophers like him. This is because the reason can easily be found for the objectivity (interpersonal accessibility) of senses, as well as for their consequent communicability. This reason is that Fregean senses are cognitive senses typically reducible to conventional semantic-cognitive rules or associations of them, and such conventions are interpersonally established and agreed upon in a pre-reflexive manner. Indeed, accepting the conclusions reached through our discussion of Wittgenstein’s views, senses typically result either from the direct application of interpersonally established semantic conventions or, more importantly, from associations or combinations of these conventions.
     Accepting that the sense of a singular term is the same thing as a rule understood as a conventional or conventionally grounded procedure that plays a decisive role in the identification of the object, it is easy to go further and accept that this rule can be typically expressed by means of definite descriptions. Hence, the sense or mode of presentation expressed by the singular term ‘the morning star’ is a conventional rule that can be understood as requiring as a criterial condition for the cognitive identification of the morning star that we see as the brightest celestial body not too far from the Sun just before or after the Sun rises. Concisely stated, this rule can be expressed by the definite description ‘the brightest celestial body that is seen close to where the Sun is about to rise.’ Without assuming that definite descriptions are expressions of rules, Frege also approached this in a note on the name ‘Aristotle’ (Frege 1892: 28). For him the proper name ‘Aristotle’ abbreviates a cluster of modes of presentation of the object that can be expressed by descriptions, which may include (i) ‘the disciple of Plato,’ (ii) ‘the teacher of Alexander the Great,’ and (iii) ‘a person born in Stagira.’ If this is the case, then (i), (ii) and (iii) express different senses, different rules that in one way or another help us to determine the reference of the proper name ‘Aristotle’ (Cf. also Frege 1918-19: 63).[5]
     Of course, there is a controversy about this issue, which arose from Kripke’s arguments against descriptivist views of proper names like Frege’s. However, it seems to me out of question that Kripke’s arguments are successfully countered by the kind of meta-descriptivist bundle theory suggested in the Appendix to Chapter I of the present book.[6]
3. Reference of a predicative expression
Frege has something to say about the reference of a predicative expression, which he calls a concept (Begriff) and which may include relations. This is odd, because it seems natural to call a concept something like the sense of a conceptual expression – the mode of presentation of its designata – while the reference itself should be called a property (ex.: a red patch) or some combination of properties (ex.: a bird’s colorful feathers).
     A traditional philosopher like Kant understood the concept as immediately related to a schema, which, as I understand him, is a rule able to lead to the formation of a manifold variety of sensory patterns that are satisfied by those things to which the concept applies (Cf. Kant 1988, B 180). Although Kant’s text on schematism is terminologically impenetrable, it is easy to paraphrase his intuition using the terminology we have developed based on Wittgenstein by saying that a concept is a semantic-cognitive rule or procedure that requires the satisfaction of criteria by particularized properties (p-properties) or tropes, what is also consonant with Dummett’s and Tugendhat’s analyses of singular statements. Consequently, we have good reasons to suspect that a concept should be the sense of the predicative expression, its mode of presentation, and not its reference, as in Frege’s strange use of the term.
     To be fair to Frege, he also says that when an object falls under a concept, the concept may be called a property (Eigenschaft) of the object (1892: 201), seemingly acknowledging that ‘property’ is the right term for the reference of a predicative expression. However, for him the criterion of identity for two concepts is the sameness of their value-range (Wertverlauf), or of their extension, which means that predicative expressions with different senses but the same extension must refer to the same concept (2001: 31). So, for instance, ‘…animal with a kidney’ and ‘…animal with a heart’ must be predicative expressions referring to the same concept, since they have the same extension. But it is intuitively obvious that kidneys and hearts are very different concepts.
     In addition to belonging to the realm of reference, Frege also sees his concepts as functions. The mathematical concept of function can be defined as a rule that has as its input arguments and as its output values (for example: ‘3 + x = y’ is a function by means of which when we give as input the number 2 as the argument for x, we get as an output the number 5 as the value of y). For Frege, a concept is a function whose argument is the object that ‘falls under it’ (fällt unter etwas) or does not and whose value is a truth-value, which can be alternatively two abstract objects: ‘The True’ (das Wahre) when the object falls under the given concept and ‘The False’ (das Falsche) when it does not. For example, the concept designated by the conceptual term ‘...is a satellite of the earth’ has the value true for the object Moon and the value false for the object Jupiter.
     For Frege, concepts cannot be objects, neither collections of objects, nor extensions (2001: 26). The reason is that objects, collections of objects and extensions are complete (vollständig) entities. That is, they do not require anything to complete them. A concept, by contrast, being a function, is seen by Frege as necessarily open: he calls it an incomplete (unvollständig) or unsaturated (ungesättigt) entity, needing to be completed by those arguments represented by the objects falling under the concept. In contrast, objects referred to by proper names are complete (vollständig), saturated (gesättigt) or independent (unabhängig).
     One could say that the saturated-unsaturated distinction can be found on three distinct levels: linguistic, semantic and referential. For instance: the predicate ‘…is a horse’ could be called an unsaturated linguistic expression (the unsaturatedness is shown by the gap ‘…’), expressing a supposedly unsaturated sense, which refers to an unsaturated concept (property) as the ultimate unsaturated ground. This unsaturated concept, for its part, becomes saturated when some object falls under it, for instance, the object named ‘Bucephalus’ referred to by the predicative sentence ‘Bucephalus is a horse.’
     With metaphors like those of ‘unsaturation’ and ‘incompleteness,’ Frege hoped to open the way to the solution of the problem of the logical distinction between the subject and predicate of a sentence. After all, the subject (the singular term) would refer to the saturated object, which would complete the unsaturated concept referred to by the predicate (general term).
      Unsaturated predicative expressions and saturated singular terms combine to form saturated singular sentences like ‘Bucephalus is a horse,’ which being complete must also be the name of an object. For Frege this object is the truth-value of the sentence. It was already noted that this would be confirmed by the possibility we have of nominalizing sentences in the form of definite descriptions, since the latter are also singular terms. Thus, the sentence ‘Bucephalus is a horse’ can be transformed in the description ‘the horse named Bucephalus,’ which appears in the sentence as ‘The horse named Bucephalus was black.’ The problem with this argument is that the same can also be done with general terms: ‘…is a horse’ can be nominalized as ‘the horse’, as found in sentences like ‘The horse is an herbivorous animal.’ Hence, the argument isn’t persuasive.
4. Ontological level
Discussing the unsaturated nature of the references of predicative expressions leads us to the question of the ontological nature of what Frege meant by a concept. If a concept is an unsaturated entity, what kind of entity is it? If it is an abstract entity, it seems that we should also have concepts as referred-to abstract entities of empty predicates, like ‘…is a yeti,’ which seems to be an ontologically abusive admission.
     Anyway, it is by now clear that Frege uses the word ‘concept’ as a technical term that contrasts too strongly with the word’s ordinary use. For our ordinary language intuition there is surely an empty concept expressed by the predicate ‘…is a yeti,’ but this concept should be called empty because it is nothing but the sense of a predicate that has no reference at all! It is no wonder that Frege has nothing to say about the sense of predicative expressions, since he has beforehand emptied them by absorbing the semantic level into the ontological one.
     My final conclusion is that it is better to drop the Fregean technical notion of a ‘concept.’ This is a problematic remnant of ontological realism that does nothing to explain predication. Instead, we will understand the word ‘concept’ here in an intuitive way as the sense of the predicative expression: its mode of presentation of something. It is counter-intuitive to assume that ‘...is a yeti’ must have any reference; but this predicate clearly has a sense intuitively expressing what we ordinarily understand by a concept, namely, that of the abominable snowman of the Himalayas. Thus, it seems that the best way to give a legitimate role to the word ‘concept’ is to see it as the sense of a predicative expression understood as its cognitive meaning, that is, its ascription rule.
5. Referring to particularized properties: trope theory
But if we drop Frege’s technical notion of concept, what is the reference of a predicative expression? I think that nowadays the most reasonable answer to this question consists in an appeal to the ontology of tropes as proposed in the Appendix of Chapter III of this book. Thus, I propose to replace Frege’s reference of predicative expressions with what we now call a trope, which I characterize simply as any spatio-temporally individualizable property.[7]
     There are many examples of tropes that are genetically primary and directly accessible to experience: the white color I see when I look at newly fallen snow on a sunny day, and which is there in my visual field, the smooth surface of this couch, the rectangular shape of my computer screen, its hardness or my headache. All these are tropes – spatio-temporally particularized properties or simply p-properties – that may range from simple objective or subjective qualities to complex homogeneous or heterogeneous tropes, like the music I listen to in the former case and the personality of a human being or a country’s political system or a social upheaval for the latter, though such things are in a less specific way also spatio-temporally localizable. Also very indirectly experienceable things like physical forces could be derivatively constructed from perceived tropes, since they are spatio-temporally localizable, and it is not unconceivable that even space and time, together with formal properties could be eventually reducible to tropes, as I tried to show in the Appendix of Chapter III.
     Moreover, it is easy to suggest a particularistic construction of universals built on the basis of particularized properties or tropes. In my view, a universal can be disjunctively defined as:

Any chosen trope model Tm or any other trope strictly similar[8] to model Tm.

I suggest this assuming that the trope we take as the model Tm is at our discretion and may vary according to the epistemic subject and even concerning the same epistemic subject on different occasions.[9] In this case, tropes T1, T2… Tn are identified as instantiations of the universal only because they are strictly similar (qualitatively identical) to an arbitrarily chosen trope model Tm. An additional point is that usually the trope-model needs to be intermediated by memory: we (usually) don’t bring with us physical patterns to compare things with, but have a memory of them. The memory-trope cannot be the primary trope we intend to consider, since it must stand for the experienced one.
     A material object could be constructed as a cluster of tropes. It can in principle be understood as a cluster of tropes displaying at least compresence, that is, it must consist of a co-located & co-temporal cluster of tightly connected varied tropes. Moreover, there are some general characterizing property-tropes like unity, displaceability, volume, solidity, resistance to pressure – scientifically explained as inertial mass – that typically comprise material objects.
     Concerning tropes I usually avoid using the word ‘property,’ not because it isn’t the best one, but because the philosophical tradition has too often hypostasized this word as referring to some scarcely intelligible non-empirical entity, vitiating our philosophical language. This tradition has stubbornly ignored the fact that in ordinary language the word ‘property’ has always been used to refer to simple or complex tropes. Anyway, I intend to use the word trope exactly as the word ‘property’ is ordinarily used. Thus, I explicitly include among the tropes complex tropes made up of different kinds of tropes, these complex tropes possibly being designated by composite predicates like ‘…a black horse of the best Thessalonian strain’ in the sentence ‘Bucephalus was a black horse of the best Thessalonian Strain.’ This does not make this complex trope (complex property) a singular material object, mainly because, as we will see later, a singular material object can exist independently if compared with the complex trope to which it is tied (in a different possible world Alexander’s beloved horse, Bucephalus, could still exist even if he were just a tired old nag).
     According to the understanding of the reference of predicative terms that I am proposing, a predicative expression like ‘... is white’ in the sentence ‘The moon is white’ does not refer to any Fregean concept. It primarily ascribes, denotes, designates (or refers to) a particularized property, namely, a trope, which is the whiteness of the Moon as normally seen by observers on the Earth. Secondarily but distinctively, however, the predicate ‘…is white’ also alludes to (or connotes) the fact that this trope exemplifies the universal property of whiteness, here understood in the already explained particularist way as this same model trope that is being considered, or any other trope that is like it. Summarizing, a predicative expression has mainly a twofold function:

(A) An ascriptive function: that of ascribing or denoting the trope (property) belonging to the object referred to by the subject term,
(B) An allusive function: that of alluding to or connoting the denoted trope or any other tropes that would be strictly similar to the model-trope that could be considered by the speaker as designated by the predicative expression, building what might be called the universal, here understood in an ontologically unobjectionable particularist way.

The allusive function is subsidiary to the ascriptive function: to identify a trope you do not necessarily need to grasp its role as an instance of a universal.[10] Better said, as particularized properties tropes have not only ontological, but also epistemic priority if compared with their role in the identification of universals.
     Furthermore – opposing the overwhelming influence of the logicist tradition – we have, as a still more subsidiary element: (C) the extension. Although relevant, differently from (A) and (B), extension isn’t primarily associated with predication. Extension doesn’t even need to be implicitly considered in the act of predication! However, it can be derived from the application of the allusive function of the predicate plus additional knowledge, allowing us to infer or even find: (C1) an extension of tropes as the set of tropes strictly similar to the trope in question and (C2) an extension of objects as a set of objects having tropes strictly similar to the trope in question. However, in both cases the extension is a further element that is usually an only vaguely inferred set.[11] As a rule, you do not need to take it into consideration to use a predicate ascriptively.
6. Difficulty with the concept of unsaturation
The main objection to the idea of incompleteness or unsaturation is that it fails to serve its main purpose, which is that of distinguishing a predicative expression from a nominative or singular term. Between the object referred to by the subject and the property designated by the predicate, there is a well-known asymmetry: the nominative term always refers to its object and cannot properly take the place of a predicate; on the other hand, we can easily turn a predicate into a subject by means of nominalization.[12] For instance, ‘Socrates’ in ‘Socrates is wise’ always refers to its object and cannot properly take the place of a predicate, while ‘… is wise’ can be nominalized as ‘wisdom’ in a statement like ‘Wisdom is a virtue.’ To make the point more convincing, consider the following sentences:

1.       <A man who lived in Antiquity> was called Socrates.
2.       <Wisdom> is a property of Socrates.
3.       <Xantippe’s husband> is Socrates.
4.       <There> is Socrates!

In these sentences, the name ‘Socrates’ at least seems to occupy a predicative position. However, this name clearly continues to be used logically as a proper name, since the logical form of these sentences can be easily expressed, respectively by:

1.       <Socrates> was a man who lived in Antiquity.
2.       <Socrates> has the property of being wise.
3.       <Socrates> is the husband of <Xantippe>.
4.       <Socrates> is in < that place>![13]

One cannot effectively transform a singular term as such into a predicate, while predicates seem to be easily transformed by nominalization into singular terms. However, we can show that the nominalized predicate is in fact a disguised universal predication: the sentence ‘Wisdom is a virtue,’ for instance, could be analyzed as, ‘For any x, if x has wisdom then x is virtuous.’ However, the asymmetry returns at this deeper level, since we cannot analyze a proper nominal term (like ‘Socrates’) in the same way. The asymmetry suggests that subjects and predicates play different logical roles in sentences, which requires explanation. The question is: can the Fregean distinction between saturation and unsaturation really do anything to explain the difference?
     At first glance, the answer is in the negative. Frege’s distinction does not explain the difference between subject and predicate in a logical sense, because it is also possible to suggest that a singular term and, therefore, its sense and reference, is unsaturated or incomplete! After all, what is the difference between:

[Bucephalus, Silver, Black Beauty, Fury… Pegasus] …is a horse.

And

Bucephalus is... [black, strong, restless, swift… of the best Thessalonian strain]?

In the first case, the concept ‘…is a horse’ is a function that according to Frege may have as an argument any object and as a value a resulting truth-value, which for the object Bucephalus is ‘The True’ and for the object Alexander is ‘The False.’ However, it makes just as much sense to apply the same reasoning to the second case. One can suggest that the nominal expression ‘Bucephalus is…’ refers to an object that is a function that may have as its argument any property designated by any predicative expression. If it is the property white, it has as a value ‘The False,’ and if it is the property black, it has ‘The True’ as its value, since we know that Bucephalus was a black horse. The undesirable conclusion is that in a singular predicative sentence both the general and the singular terms can be viewed as unsaturated in the sense of denoting functions that can be supplemented by a myriad of arguments able to bring in ‘The True’ or ‘The False’ as the resulting values!
7. Unsaturation as ontological dependence
Notwithstanding, I think that the metaphor of unsaturation is not exhausted so easily. In chemistry, a carbon compound is said to be unsaturated when it contains carbon-carbon bonds that can be broken by the addition of hydrogen atoms, which make it a saturated compound. The hydrogen atoms aren’t said to be unsaturated. Isn’t there a hint in the metaphor of an answer that was not sufficiently explored by Frege?
     In what follows, I offer a reading of the reference of a predicative expression in terms of tropes that enables us to make a useful paraphrase of the Fregean distinction between saturation and unsaturation. This paraphrase is inspired by the Aristotelian independence definition of primary substance as the individual:

All the other things are either said of the primary substances as subjects or in them as subjects. For example, animal is predicated of man and therefore also of the individual man; for were it predicated of none of the individual men it would not be predicated of man at all… Thus, all the other things are either said of the primary substances as subjects or in them as subjects. So, if the primary substances did not exit it would be impossible for any of the other things to exist. (1984, vol. 1, Categories, sec. 5)

That is, some things can exist apart, and some cannot, and the former are substances.
     I am not here interested in questioning if there are substances, what they are and if they are ultimately able to exist apart. However, applied to individuals or material objects understood as (at least) clusters of tropes displaying compresence, the independence definition suggests that the objects typified by material things exist in a manner relatively independent of their tropes in the composition of facts understood as tropical arrangements in the world.[14] Moreover, I hold that the individual referred to as a subject is only independent relatively to the predicated trope-property, because the relation of existential independence/dependence is here understood in a way restricted to the internal context of the fact represented by the statement.
     In other words, my thesis is that the true dichotomy distinguishing subject from predicate is between independence and dependence, terms only rarely used by Frege. Thus, what distinguishes the designatum of a predicative expression in the fundamental case of a predicative or relational sentence is that this reference is a trope (simple or complex, homogeneous or heterogeneous) whose existence as part of the fact depends on a cluster of selected compresent tropes constituting the individual referred to by the singular term, which is independent relatively to that trope. In my view, this fragile principium individuationis is the only thing that really distinguishes the references of logical subjects. Here are some clear examples supporting this view:[15]

Mary’s smile depends on Mary’s existence.
The car’s skidding depends on the car’s existence.
The snubness of Socrates’ nose depends on Socrates’ existence.
Amundsen’s expedition to the South Pole depended on the existence of both Amundsen and the South Pole.

These examples also make it clear that we do not mean that the dependent tropes (like those of smile, skidding, snubness, expedition to South Pole…) could not exist independently of other individuals as clusters of compresent tropes, but that they could not exist as they are independently of the individual or individuals belonging to the fact represented by the respective statements. – Qualitatively identical tropes of smile, skidding, snubness… could obviously exist in the dependence of other individuals.
Concerning singular statements, my thesis can be summarized as follows:

In the constitution of a fact represented by a true singular (predicative or relational) statement, the trope ascribed by the predicative expression exists as a part of the whole fact in the dependence on the existence of the compresent trope-cluster constitutive of the object(s) referred to by the nominal term(s).

Hence, it is important to see that the considered existential tropical dependence is relative to the fact it is a constituent (See section 23).
     In trying to explore this view in more detail, we can begin by remembering Peter Simons’ nuclear trope theory of material objects. According to this theory, individuals are in the standard case formed by an essential nucleus or core of mutually founding tropes, which is necessarily surrounded by a looser cluster of accidental peripheral tropes, so that these peripheral tropes require the nucleus of essential tropes for their existence (See Appendix to Chapter III, sec. 3). To this we should add, as already noted for the relevant case of material objects, that belonging to the nucleus are typically tropes like those of hardness, form, volume and resistance to pressure or solidity, a trope that in physics was elaborated under the label of inertial mass, all of them related by the dependent trope of compresence.
     Unfortunately, the issue is not so simple. As we saw in the Appendix of Chapter I, the identification rule of a proper name requires for its application sufficient and predominant satisfaction of at least an inclusive disjunction of the two fundamental description-rules belonging to it, which are the localizing and the characterizing rules (Cf. Appendix to Chapter I). This identification rule, as we also saw, can be satisfied by an indeterminate range of external criterial configurations, in other words, tropes or configurations of tropes. This means that what Simons understood as a necessary nucleus of mutually founding tropes may change regarding only one individual in different counter-factual situations. Already considered examples are the Aristotle born 500 years later in Rome in one possible world and the Aristotle who in another possible world was born with cerebral paralysis in Stagira in 283 BC, son of Nicomachusand was unable because of his disorder to write any philosophy. Hence, the nucleus of mutually founding tropes may be different within limits established by the identification rule. Consequently, in the case of objects referred to by proper names there is no necessary condition in re – no unique real essence of the object – to be expected, but only a nominal essence given by its proper identification rule, even if based on verified regularities. Peripheral tropes, on their side, would be those referred to by our auxiliary descriptions like (i) ‘the teacher of Alexander’ and (ii) ‘the founder of the Lyceum.’ And it is clear that the tropes designated by relations like ‘…the teacher of…’ and ‘…the founder of…’ are dependent on the existence of individuals like ‘Aristotle,’ ‘Alexander’ and the ‘Lyceum’ in order to exist as components of the facts represented by statements (i) and (ii).
     Searching for a simpler example, I will now consider the singular term ‘this chair.’ I regard this phrase as an indexical name. This indexical name has an identification rule made up of two interconnected fundamental description-rules: a contextually dependent localizing description-rule establishing a spatio-temporal location (by means of the demonstrative ‘this’ and some indicative gesture) and a characterizing description-rule (by means of the sortal ‘chair’). This characterizing description-rule is simply the definition of a chair as a movable seat with a backrest made for only one person to sit on at a time. We can say that the complex criterion for the identification of chairs added to the spatio-temporal location is what in this case forms the indispensable nuclear structure of this designatum. Symptoms of this chair, such as its having four legs and two armrests, or its being made of wood, are peripheral combinations of tropes. Moreover, if I say ‘This chair is green,’ the trope of green (in the described fact) exists in dependence on the existence of a complex of compresent tropes that forms this chair and would not exist without their existence.
     These considerations allow us to better understand the independence-dependence relation regarding the compresent core of tropes of an object satisfying its identification rule and its contingent peripheral tropes. Consider, for example, the singular predicative sentence ‘Bucephalus is swift.’ The predicate ‘...is swift’ in this sentence applies to a contingent trope that constitutes swiftness, whose existence is here fully dependent on the existence of an object, Bucephalus, which is constituted by some core of compresent tropes constitutive of a living material object. On the other hand, the same distinction also applies to properties linked to individuals that are not properly material objects. A rainbow, for instance, is an individual (a cluster of compresent tropes), though not properly a material object. But consider the dynamic fact described by the statement ‘That rainbow is fading away.’ The fading away of a rainbow is a process-trope whose existence is dependent on the existence of the rainbow in itself.
     Consider now the true relational sentence ‘Bucephalus belongs to Alexander.’ Regarding this fact, the contingent relational complex trope of belonging to could not possibly be found if Bucephalus and Alexander didn’t exist as independent individuals formed by nuclei of compresent tropes. That is, the proper existence of the relation ‘…belongs to…’ is here indebted to the existence of two more stable essential nuclei of mutually founding tropes constituting the two objects Bucephalus and Alexander. These clusters of compresent tropes referred to by the names ‘Bucephalus’ and ‘Alexander’ are concrete psycho-physical individuals that certainly exist independently of the existence of the relatively contingent complex combinations of tropes constituting the trope of ‘…belongs to…,’ since to have ownership we need the previous existence of individuals having this particular relational property.
     As in the cases described above, things are easy when we apply the dichotomy independence/dependence to tropes that do not belong to the identifiable core of an object. So, consider once more our definition of a chair as a seat with a backrest made for only one person to sit on at a time, which gives much of what Simons would call the nucleus of mutually founding tropes for the object referred to by the nominal term ‘this chair.’
     Suppose now that I point to the chair and say ‘This chair has two armrests,’ since the tropical clusters constitutive of having two armrests do not belong to the definition that makes explicit the nucleus, its existence as something that the chair has is dependent on the chair’s existence. (Notice I am not claiming that the existence of the chair’s armrests in themselves is dependent on the chair, since armrests are material parts able to exist separately from chairs. I am claiming their existential dependence of the particular chair insofar as they belong to the whole fact described by the statement ‘This chair has two armrests.’ In this context, they exist not only as armrests, but also as parts of a whole, as ‘the two armrests belonging to this x,’ making their existence dependent.)
     A problem arises when predicates denote tropes belonging to definitional cores (with their ‘in’ and not ‘of’ properties). Suppose I say, ‘This chair has a backrest.’[16] Despite the tautological character of this statement, the trope ‘having a backrest’ can be considered in its existence to be dependent on the whole cluster of tropes that form the definitional cores of tropes distinctive of the kind of object spatio-temporally individuated as ‘this chair.’ Here one could object that the cluster of tropes constitutive of the core depends reciprocally on the backrest: after all, a chair without a backrest is no chair. However, the seat is still the main object referred to, because it still has the most relevant trope combination: it is still a seat that can be used by only one person at a time, while the backrest has lost its function.
     The example shows us that the dependence of components on what is defined turns out here to be a question of proportion. One piece of evidence for this is that if the proportion is the same – if the division of a cluster of tropes is equilibrated – the question of dependence vanishes and one cannot identify a main original object of predication anymore. To exemplify this, suppose someone saws the chair into two identical halves. Where is the subject? Now we cannot say that one chair-half belongs to the other chair-half in the same way as we say that this backrest belongs to that seat. And this is so because both have the same weight regarding dependence. All that one can do now is to make relational statements about two new objects like ‘These two chair-halves belong together’ or ‘The first chair-half can be joined with the second chair-half to form a whole chair.’
     Consider now the following statement: ‘Jupiter is orbited by many moons.’ In this case ‘…is orbited by many moons’ figures as a predicative expression. It is true that the many moons are solid clusters of tightly connected compresent tropes – they are no less material objects than Jupiter. But as material objects jointly designated by the predicative expression ‘…has many moons’ they are all existentially dependent on Jupiter in the sense that without the existence of Jupiter they would not be its moons. The many moons of Jupiter belong to what is predicated as a collection of material objects, each of which could exist independently as celestial bodies, though not as moons of Jupiter. Now, think about a similarly sized system of double stars that revolve around each other. Since they are equally independent, they are references of logical subjects related by their revolution around each other. My conclusion is that the considered criterion of dependence/independence, fragile as it may be, is the ultimate criterion for distinguishing logical subject from logical predicate in the unity of a statement.
     Finally, what about formal names and sentences? Consider the sentence ‘Three is an odd number.’ This sentence describes a mathematical fact. Considering here ideas about what confers existence, we can think the number three without thinking that it is also an odd number, or ‘the number two or any multiple of two added to the number one’, which is the definition of an odd number. But there is no ‘being odd’ independent of a number. Hence, the existence of oddness factually related to the existence of the number three is dependent on the number three that we are taking into consideration.
     Consider now the statement ‘Two is a natural number.’ One could argue that to be a natural number belongs to the definition of two as a kind of genus proximum, although not essentially to the (here seen as incomplete) definition of two as its differentia. Maybe this differentia could be given by our already suggested understanding of applied natural numbers as higher-order ideally countable tropical properties belonging to an effectively applicable conceptual rule (See sec. 4 of the Appendix of Chapter III). Repeating what I said there, consider the statement ‘This hat has three corners.’ Here the applied number 3 indicates that the possible conceptual rule identifying the corners of this hat not only has the tropical meta-property of being applicable (attributing existence), but also the tropical meta-property of being applicable three times in an additive way. Moreover, we can analytically express this conceptually dependent higher-order trope of 3 by means of the set of applications {a, {a}, {{a}}} understood as a spatio-temporally located higher-order set-trope.
     But how to represent the number 3 distinguishing it as the universal object that is common to all conceptual identifications of three singular entities, the three-in-itself? Here, if we wish to avoid speaking of a Russellian abstract set of all sets of the same kind, we can still construct the number 3 as a located model of tropical numerable trope-set {a, {a}, {{a}}} or any other strictly (equinumerous) located trope-set:

Number 3 in itself (Df.) = a chosen higher-order located set of tropical applicabilities {a, {a}, {{a}}}m used as a model or any other higher-order strictly similar (equinumerous) located set of tropical applicabilities of the form {a, {a}, {{a}}}m.

This definition still allows the predicate ‘…is a natural number’ to be ascribed to the whole definiens as an internal dependent addition (the genus) and the predicate ‘…is an odd number’ as an external dependent addition. In any case, even the name of a so-called abstract object, such as ‘the number three in itself’ cannot be moved to the predicate position here, insofar as it refers to something held as independent, being identifiable (existing) independently of its non-definitional predicates like ‘…is an odd number.’
     Understanding unsaturatedness as relative existential dependence suggests, therefore, that the tropes denoted by the predicate have an inevitable tie of dependence when considered in relation to the relevant individual within the fact referred to by the singular sentence. This gives us a better understanding of the asymmetrical tie between subject and predicate.
     Summarizing the argument, my point is that the independence/dependence distinction gives a sufficiently reasonable ontological ground (I guess the only one) to explain the logical distinction between the references of subject and predicate in singular predicative and relational sentences. The nominal term cannot be moved to the predicate position, because it refers to a core of compresent tropes that exists in relative independence of the less central tropes in and outside of the core, these less central tropes being able to be designated by predicative expressions.
     In my view, the proposed analysis also sheds light on the so-called problem of the unity of proposition. What really differentiates subject from predicate regarding the fact represented by the statement is the corresponding independence/dependence of their references. Moreover, what assures the unity of the thought-content expressed by the sentence is simply the existential dependence/independence in the factual unity (for instance, in the fact that Bucephalus is swift). And it is clear that these ties of dependence/independence will be more evident when the difference in relevance between the elements in question regarding the identity of the individuals is greater, and weaker when this difference is smaller, justifying occasional uncertainties.
     Finally, one could object that what really distinguishes the predicate from the subject in singular statements is simply that the subject is a singular term that identifies one particular object and distinguishes it from all others, while the predicate is a general term able to be applied to more than one object… It is this possible one-to-many relation that is at the base of the subject-predicate distinction! Nevertheless, I think this is true regarding the definition of singular and general terms, but not that it is at the ultimate ground of the distinction. The point is made clear when we consider that a nominal term can refer to many things, while a predicative expression can refer to only one, given the proper factual circumstances. To see this, imagine a possible world Ws where there are many identical Socrates’s with indistinguishable pasts debating in the streets of Athens. In this case, we can choose one Socrates and say, ‘That thing there is a Socrates,’ and ‘Socrates’ will belong to the predicative expression and be factually dependent on that thing there. And suppose we wish to predicate that a certain trope of blue is emerald blue. In this case we would say, ‘This blue is emerald.’ Now, what makes ‘… is a Socrates’ a predicate and ‘this blue’ a subject is that in the represented fact the Socrates exists in dependence on the indicated thing-place, and what makes the blue a subject is that it is spatio-temporally given independently of having taken the shade of emerald. And this could be so considered, even if by chance all the other Socrates’s disappeared and there were only one remaining Socrates, just like only one trope of blue. This is why I prefer to hold that a possible one-to-many relation is grounded on the independence/dependence existential relation, which is primary.
8. Sense of a predicative term
The independence/dependence relationship originating on the ontological level of reference is reflected on the semantic and linguistic levels. It is first reflected on the semantic-epistemic level of Fregean senses. We see this in the fact that the identification rule of the nominal term – its sense – is applied to its object independently of the ascription of tropes to the same object by the ascription rule – the sense – of the predicative expression, while the ascription rule of the predicative expression – its sense – depends on the prior application of the identification rule of the object referred to by the nominal term. Finally, on the level of linguistic signs, the same relation of independence/dependence is what makes the singular predicative sentence take its usual subject-predicate form.
     Our view of tropes as the designata of predicative expressions allows us to make some additions not present in Frege’s original semantic distinctions. The first is the suggestion that different predicative expressions with the same designata may be able to have different senses, paralleling the case of nominal terms like definite descriptions. Consider the following two sentences:

      1. Mont Blanc is white.
      2. Mont Blanc reflects all wavelengths of the visible spectrum.

The reference of the predicative expressions of sentences (1) and (2) – the trope or compositions of tropes that constitute the whiteness of Mont Blanc – remains the same, while the senses of the predicative expressions are different: a person may know that Mont Blanc is white without knowing that its surface reflects all wavelengths of the visible spectrum and vice versa. This means that there are differences in concepts as modes of presentation or ascription rules of the predicative expressions of sentences (1) and (2), although they have the same designatum.
     Another consequence of our understanding of predicative expressions as basically referring to tropes by means of their semantic-cognitive conceptual rules contradicts the Fregean expectation that the same sense cannot have more than one reference, which disfavors the potential for multi-referentiality inherent to predication. Consider the following sentences:

1.       The South Pole is white.
2.       Mont Blanc is white.

The predicate ‘...is white’ in sentences (1) and (2) obviously has the same sense in both, as in each case it expresses the same ascription rule. But the tropes of whiteness (of reflecting the combined wavelengths of the visible spectrum) of the South Pole are located at the South Pole itself, while the tropes of whiteness of Mont Blanc are located in its eternal snows. The same can be found in the application of relational predicates. This is explained by the fact that the different objects referred to by different singular terms have numerically different tropical configurations that satisfy the ascription rule of the same predicate by being qualitatively identical.
     Another easy problem is to know how predicative expressions are used in the case of general sentences: universals and existential statements. Regarding universal sentences, we see them as abbreviated expressions of a conjunction of singular sentences, each ascribing tropes to identified objects. For example: the universal sentence ‘All trees have roots’ would be analyzed as {Tree 1 has root & tree 2 has root &… & tree n has root}. Here the qualitatively identical tropical root-clusters Tr1 of 1, Tr2 of 2… Trn of n are considered in conjunction and jointly denoted by the universal sentence. This also means that qualitatively identical ascription rules able to denote the complex tropes of roots belonging to the objects trees also need to be conjoined in the sense that the universal sentence at least indicates the probable application of this conjunction of ascription rules to tropical root-clusters of trees, even if we are unable to really apply them to all trees in the world. Similar considerations can be made regarding existential sentences like ‘At least one tree has roots,’ which abbreviates a disjunction of ascription rules and refers to at least one real tropical root-cluster belonging to the multiplicity of trees as objects.
     We will come back to this issue in the last chapter, when we consider problems related to this attribution of truth-value to general statements. Anyway, to the objection that this will never give us the conditions for a truly universal quantification, the obvious answer is that the truth of the universal quantification is almost always enclosed within some domain and almost always is only probable. What ‘All living trees have roots’ really means is: ‘[Very probably] all living trees [on the planet Earth] have roots.’
9. Dependence of the predicative sense
As we have already noted, in the context of a singular predicative sentence, the identification rule of the singular term applies to the object as some core of compresent tropes, which is seen as existing independently in relation to its more or less dependent partial or peripheral tropes. Consequently, the identification rule is also able to be applied regardless of the application of contingent ascription rules, which means that this identification rule can be conceived as being applied in isolation. This explains its independence and why one could call it complete or saturated. The ascription rule, on its side, will be applied to a trope dependent on the core and consequently depending for its real application on the earlier application of the identification rule, lacking in this sense completeness. This is at most clear in the case of rules for contingent properties, like the conceptual rule for the predicate ‘swift’ when applied to Bucephalus.
     The same may also hold for the fundamental descriptions constitutive of the identification rule of the nominal term in the sentential context. Since the tropes belonging to the object to which the identification rule applies are ultimately dependent on the existence of this object as containing a kernel of tropes, even the ascription rules of predicative expressions already belonging to the identification rule of the object as part of this rule require prior application of the whole identification rule to identify the object in order to become themselves applicable as part of the identification (e.g. the statement ‘Aristotle was the author of the Metaphysics’). Because of this, the application of the predicate’s ascription rule is always dependent on the application of the identification rule of the singular term.[17]
     The general sense of a concept-word, which (diverging from Frege) we identify with the concept or ascriptive rule expressed by it, should then be a rule whose application to an object depends on the prior application of another rule. Hence, the ascription rule of the predicative expression is dependent, incomplete, unsaturated, in the sense that it demands the prior application of the identification rule of the singular term in order to be applied. It is necessary to identify, that is, in the empirical case to find some particular object in space and time, in order to be able to characterize it by ascribing the predicative rule to its appropriate trope. We must, for instance, first apply the rule that allows us to spatio-temporally locate the horse called Bucephalus in order to apply to it related tropes, and on that basis, the ascription rules of predicative terms. Thus, due to the independence of the object Bucephalus, we apply the ascription rules for the predicates ‘... is a horse,’ ‘... is black,’ ‘... is swift’… and also the ascription rules of more complex predicative expressions like ‘…a horse that belonged to the best Thessalonian breed’ to the tropical kernel constitutive of Bucephalus. And we also need to apply the identification rules for Bucephalus and Alexander in order to be able to apply the relational predicate ‘…belongs to…’ In a similar way, we need to apply the rule that allows us to mentally identify the number 3, in order to be able to apply to associated dependent tropes the ascription rules of predicative expressions like ‘…is odd,’ ‘…is a prime number,’ or ‘is the square root of 9,’ though it is not the case that the number 3 depends on these things in order to be identified as such. In the same way, the relational ascription rule for ‘3 < 7’ is only applicable in dependence on the independent application of the identification rules for the numbers 3 and 7.
     As I have very early noted (Ch. I, sec. 1), it would be a naive objection to think that after all it is possible to say things like ‘That is a horse’ or ‘There is a black thing,’ applying ascription rules of predicates without identifying Bucephalus. The reason is that a fully detailed identification of the reference as Bucephalus isn’t required at all. Indexicals such as ‘that’ and ‘there’ accompanied by some gesture of pointing are already able to identify some particular as anything spatio-temporally localizable which exists independently of further predication. As we already saw, this (non-absolute) independent way can be made explicit when the indexical is followed by a term designating countable things (sortals) such as ‘that object,’ ‘that animal,’ and this is enough. Therefore, not only does the trope designated by the predicate depend upon the previous existence of the object and its identification, but, as a consequence, also the effective applicability of the ascription rule of the predicate must be dependent upon the prior application of the identification rule to the relatively independent cluster of tropes. This is how the relation of semantic dependency – on the level of sense – mirrors the relation of ontological dependency – on the level of reference – solving the riddle of unsaturation.
10. The concept horse paradox
We can continue to make major revisions of Frege’s views in order to overcome difficulties arising from his semantic views, like the so-called concept horse paradox. Based on his view of a concept as the unsaturated reference of a predicate, Frege was led to the strange conclusion that one cannot name a concept. For him the sentence:

1.       The concept horse is not a concept,

is true. After all, ‘the concept horse’ appears here as a singular term – a definite description – and as such it must refer to something saturated, that is, an object and not a concept. The paradoxical point is that the denial of the true sentence (1), which is:

2.       The concept horse is a concept,

must for Frege be false! Nonetheless, (2) clearly sounds like an obviously true analytic sentence.
     From our perspective, the first thing to do is to treat nominalization as what it really is: an abbreviated way to speak about quantified concepts. What (1) really means is:

3.          For any x, if x is a concept horse, then x isn’t a concept,

which is obviously false. Regarding sentence (2) it really means:

4.          For any x, if x is a concept horse, then x is a concept,

which is obviously true. Using H to replace ‘… is a concept horse,’ which is the ascription rule able to designate the property-trope of horseness, and replacing ‘…is a concept’ with C, which is the ascription rule able to designate any property-trope in an undifferentiated way, we can formalize (3) as (5): (x) (Hx → ~Cx), which is false, and (4) as (6): (x) (Hx → Cx), which is true.
     What is the lesson of this analysis? If ‘the concept horse’ does not really work as a definite description – as a singular term – but rather as a hidden universal predication, Frege was wrong in maintaining that it cannot be a concept only because it now works as a definite description. Frege’s paradox results from an incomplete analysis of sentences like (1) and (2), and the true analyzed sentences are the corresponding harmless universal conditionals (3) and (4), the first being contradictorily false and the second tautologically true. If we agree that rightly analyzed ‘the concept horse’ expresses a universal predication and no real singular term, the whole paradox dissolves. It turns out to originate from the naïve mistake of thinking that if you put a predicate in the position of a subject, transforming it into a definite description, you necessarily transform it into a real singular term (See Appendix to this chapter).
11. Existence as a property of concepts
At this point, we can turn to Frege’s treatment of the concept of existence. Deepening an idea already present in Kant’s philosophy, he suggested that existence is a property (Eigenschaft) of a concept, namely, the property that at least one object would fall under it (Frege 1884, sec. 53). A similar idea was later advocated by Bertrand Russell in the suggestion that existence is the property of a propositional function of being true for at least one instance (1994: 232-3, 250-54.).
     Here I will not try to interpret the details of Frege’s often obscure remarks. Using more current terminology, I will follow an explanation taken from John Searle, who with his usual clarity brings us unmistakably to the point (2008: 176). Consider the sentence ‘Horses exist.’ This sentence can be analyzed as:

There is at least one ... such that (... is a horse).

As Searle notes, this sentence contains two components. One is expressed by the predicate ‘…is a horse,’ symbolically Hx (where we use x instead of ‘…’ and H replaces ‘is a horse’). The other component is the predication of existence expressed by the open sentence ‘there is at least one ... such that ...’ This predication can be symbolically expressed as Ǝx(...) (where Ǝx replaces ‘there is at least one … such that…,’ and the last ‘...’ is the gap to be filled by some concept applied to something, now in the usual sense of the word concept, which in this case is the concept horse symbolized as Hx. The result is that the whole sentence ‘Horses exist’ can be symbolized as Ǝx(Hx). This also means that the predication of existence Ǝx(...) is a meta-predication expressing a higher-order concept, a concept of a concept, a meta-concept under which other concepts can fall – in this case (Hx). Thus, Ǝx(Hx) instantiates the general form Ǝx(Fx), which usually expresses a second-order concept – the concept of existence – applied to some first-order concept. In a Fregean way of speaking, what this second-order concept does is to say of the first-order concept that at least one object falls under it, which also means that the first-order concept is satisfied or fulfilled by being applicable to at least one thing. So understood, existence is something objective, since this satisfaction is independent of our cognitively grasping it as the applicability (and not mere application) of a concept.
12. Existence as a property of conceptual rules
These last ways of speaking are more interesting to me because they could be paraphrased in accordance with my identification of concepts with senses of predicates, more precisely, with conceptual, semantic-cognitive ascription rules. This identification shows that existence is a property of these conceptual rules, namely, their property of being able to be satisfied, fulfilled, or simply applicable. For instance, when I say ‘Horses exist,’ I mean that the conceptual rule expressed by the predicate ‘…is a horse’ is applicable. More precisely, I mean that this conceptual or ascriptive rule is effectively applicable in a domain of external objects. I add the adverb ‘effectively’ or ‘definitely’ to make it clear that I do not use the word ‘applicable’ in a merely subjunctive sense, as referring to something that may be applied, but as referring to something that is effectively (definitely, warrantedly) applicable, which is continuously the case during some period of time (the period in which the object is said to exist). Moreover, the existence or effective applicability of a semantic-cognitive rule is always considered with regard to a certain domain of entities (a ‘universe of discourse’). The most fundamental domain is that of the real empirical world, be it the external (physical) world (Carnap’s thing-world) or the internal (psychological) world. The statement ‘Horses exist’ applies in the first domain. The statement ‘Headaches exist’ applies in the second domain. Indeed, what is normally meant by the predication of existence isn’t the applicability of the ascription rule of a general term as a mere possibility entertained only in our imagination, but also effective applicability of the rule within some empirically given domain of entities. Furthermore, this effective applicability is usually within what we might call its most proper domain of entities, which in the case of horses is a domain of external, physical entities, and in the case of headaches is a domain of internal, psychological entities. I consider this point here because there are dependent cases, like that of the Valkyries,[18] whose most proper domain is mythical – that of Norse mythology.
     As one could guess from the last example, there are other higher-order domains and sub-domains of entities within which we can predicate existence, even if only in a subsidiary sense. One can say, for instance, that the Valkyries’ horses exist in the fictional domain of Wagner’s opera The Valkyrie in the sense that the ascription rules for these fictional horses are effectively applicable in the fictional domain described in the libretto. There are also cases like the probable existence of life in other galaxies, which can be in principle verified. Thus, there are imaginary mythological domains, fictional domains in the arts, and domains of imaginable but also plausible entities. Moreover, there are domains of so-called abstract entities and their various sub-domains, like the domain of mathematical entities, of logical entities… It is simply a linguistic fact that we can apply the word ‘existence’ in any of these domains. What I intend to show in the following is that there is a unifying justification for this.
     According to the view I am supporting, to say that horses, rocks, trees and chairs exist is to confer effective applicability to the ascription rules of the respective concept-words ‘horse,’ ‘rock,’ ‘tree’ and ‘chair’ in the fundamental domain of material objects belonging to the objectively real external world. To say that thoughts, joys and pains exist is to ascribe effective applicability to the ascription rules expressed by the concept-words ‘thought,’ ‘joy’ and ‘pain’ in the subjectively real mental domain of entities. And to say that ‘totalitarianism,’ ‘corruption’ and ‘exploitation’ exist is to affirm the effective applicability of the ascription rules of these concept-words within the psycho-physical domain of social entities. The domain of entities to which such concept-words apply is usually assumed to be respectively physical, psychological and social. As a general rule, to say that an entity exists is to say that its conceptual rule is effectively applicable in the already conventionally established most proper domain of application. Thus, to give examples, the most proper domain of application of the conceptual word ‘horse’ is the real external world, while the most proper domain of application of ‘Valkyrie’ is a fictional one. That is, it is normally assumed that the attribution of existence is made in its most proper domain. But this assumption isn’t universal (when I say that there are horses in Wagner’s opera The Valkyrie, the concept horse isn’t being applied in its most proper domain).
     As already noted, a concept – understood as the semantic-cognitive ascription rule of a predicative expression – is able to generate subjective criterial configurations. Thus, to say that a concept-word is effectively applicable is to say that subjective criterial configurations generated by its ascription rule likely to be fulfilled by corresponding objective criterial configurations. These objective criterial configurations can be seen as configurations of tropes usually belonging to more complex tropical arrangements called facts – another point against Frege that I will explain and justify in some detail later.
     The parallel between the concept of existence in Frege and the more detailed concept of existence derived from my reconstruction of concepts as senses of predicates understood as ascription rules is straightforward:

 Concept of existence (Frege) =
A second-order concept that demands for its satisfaction that a first-order concept has at least one object that falls under it.

Concept of existence (reconstructed) =
A higher-order semantic-cognitive conceptual rule that has a criterion for its (effective) application that a lower-order conceptual (or ascriptive) semantic-cognitive rule is effectively applicable to at least one entity, this entity being a trope or a configuration of tropes, usually in what is conventionally viewed as its most proper domain.

In my judgment, the advantage of this last form of analysis is epistemological: we are better able to scrutinize the nature of our existence-assignments, as will be shown by the answers to objections.
13. Two naive objections
There are two naïve objections to the proposed formulation of the higher-order view of existence, which offer revealing answers. The first is that the concept of a rule’s effective applicability would be an anthropomorphic one, while things are said to exist in full independence of cognitive beings.
     However, this objection only arises if we confuse the concept of effective applicability (within a certain domain) with the concept of effective application. The application of a semantic-cognitive rule is an act or a series of acts that are essentially mental, though often also inevitably sensorimotor, resulting in judgments. The application of the conceptual rule for the identification of the planet Venus, for instance, really demands the existence of cognitive beings able to perform the application. Our judgment that the Moon circles the Earth depends on the experience of the application of a verifiability rule for the existence of this fact by ourselves or by someone who testifies to its application. On the other hand, the concept of effective applicability is not anthropomorphic. Even if there were no cognitive beings able to apply the identification rule for the concept Venus, this planet would continue to exist, since the ascription rule for the identification of Venus would still be effectively applicable to this object in its proper domain. The rule would still be applicable, even if no one had ever applied or even conceived it! The rule would be effectively applicable in a universe without any cognitive being able to conceive it, since all that is required is that if the rule existed, it would be effectively applicable. Thus, there is no doubt that the concept of effective applicability isn’t anthropomorphic.
     This answer makes it easier to refute a second naïve objection. This objection could easily be made by proponents of the idea that existence is a property of things instead of concepts. According to it, if existence is a property of conceptual rules, then it has nothing to do with the objects that fall under these concepts: existence seems to be something floating above things that are said to exist. However, this seems odd, since intuitively we think that existence must in some way belong to entities that we believe exist!
     The answer to this objection is that there is no contradiction between being a higher-order property of an entity and belonging to this entity. We make this clear by inverting the form of exposition. We can not only say that some ascription rules have the property of being effectively applicable to some entities belonging to a certain domain, but we can also say that some entities of a domain have the property of having their own ascription rules effectively applicable to them, meaning by this that these entities exist in their most proper domain. That is, when we say that entities such as horses exist, we also mean that at least one of these conceivable countable kinds of objects has the higher-order property or trope of having its ascription rule effectively applicable to it. In other words, we mean that it has the meta-property of existing in the actual external world, and that this meta-property is also a property of the object of the kind – even if of a second-order – since it is a property-property at the level of the object’s ascription rule and not intrinsic to it.
     In still other words, according to the higher-order view of existence, the red trope of a couch in front of me exists only insofar as this object (the couch) has the property of falling under the concept of being red in the Fregean way of speaking. But in a more natural way, we can say that the trope of redness of the indicated couch exists in the sense that the ascription rule of the concept-word ‘red’ has the meta-property of being effectively applicable to the couch’s trope of redness. Even in a world where this ascription rule does not instantiate, for instance, a world without cognitive beings to think and apply the rule, this rule would remain effectively applicable, because we know that if this rule were conceived, it would be effectively applicable. (Since the rule only instantiates in minds, the rule is in this case only a possibility; but even if the rule actually does not exist, the effective applicability of the rule actually exists as a higher-order dispositional trope). However, this also means that the couch’s trope of redness secondarily owns the meta-property of the effective applicability of its ascription rule to it – it owns this property-property dispositionally. That is, since the property of existence is the ascription rule’s property of being effectively applicable to the trope of redness located on the surface of the couch, that property of the ascription rule is a meta-property of this trope of redness. It is so because through the ascription rule, this property belongs indirectly and dependently to the trope of redness belonging to the real empirical world. Finally, the higher-order property of existence or effective applicability of the rule must be where the rule is, that is, it must be spatio-temporally located, being therefore a trope. Existence doesn’t need to be an exception to our all-embracing trope ontology.
     Considering that the meta-rule of existence is a dependent trope that also applies to the trope, even if in a subordinate way, one could still ask: how would it be possible in the case of a possible world where there is no cognitive being able to think this cognitive-semantic rule? The answer is: the property of having the rule effectively applicable to it is a dispositional and not an actual property. In the same way as an object is only dispositionally green at night, when colors cannot be seen, the greenness of this object will continue to exist as a disposition even in a world without cognitive beings.
      Summarizing: it is a peculiar feature of the concept of existence (and certainly of some other concepts) that, being owned by a first-order concept effectively applicable to some entity, it must also be owned by some given entity belonging to the chosen domain of entities without being a proper constituent of this entity.
14. Existence attributed to objects
The idea that existence is a property of concepts concerns not only what is meant by general terms, but also by singular terms, since both kinds of terms express conceptual senses, and their references can be said to exist. Since singular terms can be generally divided into proper names, definite descriptions and indexicals, I will briefly consider each of them, beginning with definite descriptions.
     Consider as an example the following definite description: ‘the inventor of the Maieutic’. Applying the logical device to treat some descriptions by replacing them with a predicate, we symbolize the predicate ‘the inventor of Maieutic’ with M, so that the statement ‘The inventor of the Maieutic existed’ can be analyzed as:

Ǝx [Mx & (y) (My → (y = x)].[19]

In this way, we are affirming the existence of at least one and not more than one inventor of the Maieutic. This means that the ascription rule that constitutes the concept (sense) expressed by the predicate ‘…is the inventor of the Maieutic’ has the property of being effectively applicable to only one human being, namely, Socrates, reducing the domain of application to only one member. This is the same as attributing existence to the inventor of the Maieutic.[20]
     Consider now the case of proper names. As we have seen, they should also have senses in the form of identification rules. Considering existence as the effective applicability of a semantic-conceptual rule in a chosen domain, the existence of the object referred to by a proper name should be established by the effective applicability of its identification rule, primarily in a proper contextualized domain of the external world.
     Although this issue cannot be properly addressed without a deeper investigation of the nature of proper names, we can start by applying the Fregean-Russellian formal device to the foregoing view. In order to do this we transform proper names into predicative expressions applied to only one particular, showing then that the senses of names themselves can be reduced to the conceptual senses of predicative terms. A first step in the attempt to arrive at this is to transform the proper name into a predicate. Thus, ‘Socrates’ in ‘Socrates exists’ can be transformed into a predicate in the sentence ‘There is something that socratizes,’ or ‘Ǝx(x socratizes).’[21] Taken literally, this suggestion is not only linguistically deplorable, but also formally deficient, since it leaves open the possibility that there is more than one Socrates.
     Nevertheless, I think that ‘Ǝx(x socratizes)’ points in the right direction by suggesting that the existence of a name’s bearer may be asserted by means of the conceptual senses of predicative terms. After all, the verb ‘to socratize’ can be seen as a kind of abbreviation of the predicative conceptual expressions included in the descriptions supposedly summarized by the proper name ‘Socrates.’ This is a reasonable strategy, insofar as we take seriously the bundle theory of proper names that was already fully present in one way or another in the writings of Frege, Russell and Wittgenstein, though it has been made more explicit by P. F. Strawson and particularly John Searle. According to this theory, the whole sense of a proper name is given by a cluster of definite descriptions. Having this in mind, we might suggest that the attribution of existence to Socrates in ‘Ǝx(x socratizes)’ could be seen as an abbreviation of a set of predicative expressions like:

Ǝx {x is inventor of Maieutic, x is mentor of Plato... x is Xantippe’s husband}.

Of course, this is still inadequate, since it not only demands that all predicates must be satisfied, but leaves open the possibility that these predicates could be applied to more than one object. However, this fault can easily be remedied by means of the formal device that allows us to establish a minimum of at least one effectively applicable definite description:

Ǝx {x and no other person invented the Maieutic, or x and no other person was the mentor of Plato or… or x and no other person was the husband of Xanthippe}.

Symbolizing the predicates ‘…is the inventor of the Maieutic’ as P1, ‘…is Plato’s mentor’ as P2, and ‘…is the husband of Xanthippe’ as Pn, the above sentence can still be symbolically formulated as follows:

Ǝx [(P1x & (y1) (P1y1 → (y1 = x)) ˅ (P2x & (y2) (P2y2 → (y2 = x)) ˅... ˅ (Pnx & (yn) (Pnyn → (yn = x))]

Here the supposed meaning of a proper name is disjunctively translated into the conceptual-senses of predicative expressions such as P1, P2… Pn, which according to our analysis are nothing but ascription rules expressed by predicates that we expect to be really applicable to one and the same thing. So analyzed, the attribution of existence to the object referred to by a proper name is made by saying that its sense, its identification rule, definitely applies in the assumed context. As this rule for the identification of a name was here analyzed in terms of a disjunctive set of rules for the application of predicates that must be applied to the same individual, we can easily explain existence as follows: The existence of the bearer of a proper name is the same as the effective applicability of at least one conceptual rule of a predicative expression to precisely one individual.
     Of course, here it could be objected that such a descriptivist attempt to explain the meaning of a proper name is doomed to failure. This must be so, not only because the applied formal device is limited, but also because it amounts to some version of the bundle theory of proper names with its well-known difficulties, already persuasively pointed out by Saul Kripke, Keith Donnellan and others...
     However, such a conclusion would be too hasty, and there are at least three reasons to oppose it. The first is that, contrary to a current bias, Kripke’s and Donnellan’s objections have not discredit the most comprehensively developed versions of descriptivist theories, and some criticism have already been answered with considerable success by John. R. Searle (1983, Ch. 9). A second reason is that Kripke’s alternative solution, the causal-historical view, could never be developed beyond a rough sketch.[22] These first two points lead us to the conclusion that bundle theory hasn’t yet been definitely refuted.[23] Indeed, perhaps it just needs a stronger defense.
15. The existence of objects and its identification rules
The third and really conclusive reason that I can oppose to the anti-descriptivist view is that the above presented formal analysis is a crude simplification when seen from the viewpoint of my own much more convincingly elaborated version of the bundle theory of proper names. This version has, as I believe I have demonstrated, greater explanatory power than any previous theory, answering in a more nuanced way the most diverse counter-examples. Although this theory is too complex to be discussed here at length, it was already summarized in some depth in the Appendix to Chapter I.
     Briefly repeating what I said there, my view is the following. The traditional bundle theory of proper names defended by Frege, Russell, Wittgenstein, P. F. Strawson, John Searle and others has a severe limitation that has been overlooked: the bundles have no internal order. The theory does not tell us which descriptions or combinations of descriptions are more or less important or even why some seem to be very important for the application of a name, while others are obviously irrelevant for it. Definite descriptions are nothing but expressions of rules that should help us to connect a proper name with its reference; we may call them description-rules. Regarding all this, my question was whether we cannot find the general form of a rule that we all implicitly know, which if applied to any bundle of descriptions associated with a proper name enables us to recognize the most relevant ones and decide in what ways the satisfaction of these descriptions makes this proper name applicable to some referent.
     When searching for the general form of a rule, the first thing to do is to classify the descriptions. There is a sensible, ordinary-language method to use in order to begin with: check how encyclopedias treat well-known proper names. We can thereby easily distinguish fundamental from merely auxiliary descriptions, which are accidental. In doing this we see that proper names are first and foremost attached to two fundamental forms of description, which I call localizing and characterizing description-rules. Here is how we can define them:

(A) Localizing description-rule: This is the description that gives the spatio-­temporal location and career of the object referred to by the proper name.
(B) Characterizing description-rule: This is the description that gives the characteristics of the object that we consider the most relevant to be referred to by the proper name – which gives us the reasons to use the name.

Consider, for instance, the name ‘Adolf Hitler.’ Here is what is said about its bearer in the first paragraph of a Wikipedia article:

Adolf Hitler (20 April 1889 – 30 April 1945) was born in Braunan an Inn, Austria. Later he was a German politician and leader of the Nazi Party. He was Chancellor of Germany from 1933 to 1945 and Führer of Nazi Germany from 1934 to 1945. As effective dictator of Nazi Germany, Hitler was at the center of World War II in Europe and the Holocaust.

It is usual in encyclopedias that the first thing we find is an abbreviation of the localizing description-rule, followed by an abbreviation of the characterizing description-rule, stating the reason why we remember the name. What follows in the Wikipedia article (as in many others) are more or less relevant details and explanations. We find a variety of definite and indefinite descriptions that are more or less irrelevant: accidental, auxiliary descriptions. Examples of them are that Hitler was ‘the lover of Eva Braun,’ ‘the son of Alois Hitler and Klara Pölzi’[24], ‘the person called “Adolf Hitler”,’[25] ‘the boy who was sent by his father Alois to the Realschule in Linz in September 1900.’ All this information given by encyclopedias will also be found in more extended form in biographies.
     You find a similar pattern if you search in encyclopedias for other proper names like ‘New York,’ ‘USA,’ ‘Eiffel Tower,’ ‘Niagara Falls,’ or ‘Milky Way.’ Of course, there are also the proper names of ordinary persons who are not famous enough to mention in encyclopedias. But the basic mechanism of reference remains the same. It is not difficult to see that the relevant information is given by their localizing descriptions and by the usually much more scattered characterizing descriptions. So, in most cases, if you wish to know who Sam is, you can probably get relevant information from his identity card, drivers license, employment record, police record (if any), school reports, club records… and most of all from details given by him, by his family and friends about his personality, character, education, interests, abilities, relationships, accomplishments, etc., which are linked together by just one spatio-temporal career.
     Now, my suggestion is that, although a conjunction of the localizing and the characterizing descriptions isn’t required in any possible world, as Kripke has clearly shown (1980: 62), an inclusive disjunction of the two fundamental description-rules must in some degree be satisfied to enable a proper name to refer to its object in any possible world. John Searle perceived this point many years ago, when he wrote:

…if none of the identifying descriptions believed to be true of some object proved to be true of some independently located object, then that object couldn’t be identical to the bearer of the name. (1969: 169)
Indeed, if we discovered records of a man named Adolf Hitler who was born in Gusental and lived in Austria from 1634 to 1689, worked as a shoemaker and had no political interests, we could safely conclude that he wasn’t our Adolf, since he does not satisfy any of the disjunction.
     Moreover, two other complementary conditions should be added. First, a condition of sufficiency that must be satisfied: the disjunction of these two fundamental descriptions must be at least sufficiently satisfied in order to enable a proper name to refer to its object in any possible world. So, you can imagine a possible world where there was no World War II but where Adolf Hitler was born on 20 April 1889 in Braunan an Inn as the son of Alois and Klara Hitler. However, he had the same career as Adolf Hitler up to the point where he was not rejected but rather accepted by the Vienna Academy of Fine Arts in 1907, becoming a rich landscape painter who lived a long uneventful life. In this case, we are inclined to say that this person is our Hitler in this counter-factual situation, although he satisfies only the localizing description-rule, and even this only partially. But he does satisfy the condition of sufficiency.
     The second important condition is that of predominance, demanding that a possible bearer of a proper name should satisfy fundamental descriptions in a more complete manner than any other competitor in a possible world, since by definition the bearer of a proper name cannot be more than just one specified object. Thus, suppose that in a very similar possible world there were twins Adolf and Rudolf Hitler, both born on… 20 April 1889… but only Rudolf went to Berlin, served in World War I and later headed the Nazi Party, starting World War II and the Holocaust, while Adolf became a farmer in his native Austria. We would choose Rudolf as the true Hitler, despite his different name, since the name Rudolf is associated with the disjunction of conditions belonging to the identification rule for our Adolf in a much stronger way than the name of his twin brother presented by the auxiliary description ‘the person called “Adolf Hitler”.’ This shows once more the low relevance of auxiliary descriptions.
     Finally, it is important to add that the object of reference belongs to the nearest relevant class ‘C’ that does not mix with the contents made explicit in the localizing condition (for instance, being a human being and not a politician for Adolf Hitler, and being an asteroid and not a planet for Venus).
     Bringing all this together, we are able to propose the following general form of any identification rule for proper names, a form that must be satisfied by any bundle of descriptions associated by the linguistic community with a given proper name:

General form of the identification rule for proper names:
A proper name called ‘N’ has a bearer
 iff
it is something that belongs to the nearest relevant class of referents C, so that more than any other entity of the kind C it sufficiently satisfies at least the conditions set by:
(A) its localizing description-rule,
 and/or
(B) its characterizing description-rule.
(Auxiliary descriptions can be added as helpful symptoms for the identification[26]).

Now we can apply this form to any well-known bundle of descriptions that we associate with a proper name in order to have its identification rule. When we link the general rule form with the bundle of descriptions associated with the proper name ‘Adolf Hitler’, we get the following identification rule for this person:

The proper name ‘Adolf Hitler’ has a bearer
 iff
the bearer is something that belongs to the class of human beings, so that sufficiently and more than any other human being he satisfies the following inclusive disjunction of conditions:
(A) being born on 20 April 1889 in Braunan an Inn… living the last part of his life in Germany… dying on 30 April 1945 in Berlin, and/or
(B) being the leader of the Nazi Party… dictator of Nazi Germany from 1934 to 1945… the person most responsible for World War II and the Holocaust.
(He would very probably also satisfy helpful auxiliary descriptions like being ‘the lover of Eva Braun,’ ‘the person called “Adolf Hitler”,’ ‘the Führer,’ etc.)

This summarized identification rule gives us the core meaning of the proper name ‘Adolf Hitler.’ If we try to imagine an Adolf Hitler who does not minimally satisfy the fundamental localizing and/or characterizing conditions, we see that this is impossible. This was the case of the Adolf Hitler born in Gusental in 1634, who was a peaceful shoemaker and had nothing to do with politics. Surely, he could not be the person in a political socio-historical context whom we always mean by the name ‘Adolf Hitler,’ but someone else with the same name.
     This example also outlines the lack of relevance of auxiliary descriptions. Suppose that the Adolf Hitler born in Gusental in 1634 satisfies many of the best-known auxiliary descriptions: he was the lover of an Eva Braun, he was the son of an Alois Hitler and a Klara Pölzi, the person called ‘Adolf Hitler,’ the boy who was sent by his father Alois to the Realschule in Linz… The feeling elicited by these strange discoveries would be of deep puzzlement, not persuasion. For his Eva Braun could not be the well-known Eva Braun who also committed suicide in the Bunker… and even that his parents had the same name as those of the infamous Adolf Hitler would be merely a remarkable coincidence… (He could not, it is true, satisfy the description ‘the author of Mein Kampf ’; however, more than an auxiliary description, this is already part of the full characterizing description of our Adolf Hitler.) Anyway, at no point will this change our belief that he is not the person we are trying to identify.
     Since so understood the identification rule simply defines which object among all others owns the proper name by establishing the definitional criteria for identifying the proper name’s bearer in any possible world, it unavoidably also applies in any possible world where the name’s bearer exists, satisfying the fundamental requirement of the Kripkean definition of a rigid designator (1980: 48). The individually taken definite descriptions belonging to the bundle, particularly the auxiliary ones, on the other hand, being only loosely associated with the identification rule, can refer to other objects in different possible worlds, and are therefore only accidental or flaccid designators.[27]
     Moreover, one can insert a name correctly in a sufficiently vague discourse without knowing more than auxiliary and indefinite descriptions, even when they are wrong, as Kripke realized. This is the case at least insofar as these descriptions are convergent (rightly classified), making in this way what we should call a parasitical reference, which can be helpful in several ways. For instance, if someone already knows that Hitler was ‘some dictator’ or erroneously thinks that he was ‘a military general,’ this person already classifies him correctly as a man of power and can already apply the name correctly in sufficiently vague contexts and possibly learn more about him.
     Now, the existence of an object referred to by a proper name is the effective applicability of what can be called the identification rule of the proper name in its (in most cases) proper contextual domain. We know that Hitler existed because we know that his identification rule was effectively applied, hence applicable, in the political-historical context of Europe in the first half of the twentieth century. And as far as the identification rule for the proper name ‘Hitler’ is concerned, what allows us to say that the bearer of this name really existed is that it has the second-order property of being effectively applicable in its proper contextual domain.
16. Existence of spatio-temporal locations: indexicals
Finally, there is the problem of the application of the proposed analysis of existence to the reference of those singular terms that change their reference according to the context: the indexicals. I will consider them only very briefly. Take simple statements with indexicals as (pointing) There is a raven,’ Here is cold,’ ‘It rained yesterday,’ I am tired,’ I am here now’... What indexicals necessarily do is at least to indicate some spatio-temporal location relative to the speaker. Thus, ‘here’ points to the place where the speaker is, ‘now’ to the moment when he speaks, ‘yesterday’ to a period of time, the day before the day of the speaker’s utterance… Regarding indexicals like ‘I,’ ‘she,’ ‘he,’ ‘they,’ there is more to say than just this. Surely, these personal pronouns have more semantic content than just a plain spatio-temporal location, but this does not matter to us now.
     Consider now the indexical statement There is a raven,’ said if one found only one raven there. How should we analyze it? First, there is the spatio-temporal location usually pointed to by the speaker, which is relative to the spatio-temporal location of the speaker under adequate circumstances. All this must be assumed in applying the demonstrative ‘there.’ The identification rule for ‘there’ can here at least be summarized in the description (i): ‘the spatio-temporal location pointed to (or contextually shown) by the speaker when he utters the word.’ Now, what we call the existence of the spatio-temporal location indicated by the demonstrative ‘there’ must be the effective applicability of this identification rule of location in the restricted contextual domain established by the speaker when he speaks the sentence. Once the person has applied this identification rule, she is allowed to apply the ascriptive rule of the predicate ‘…is a raven’ to the existing location. Now, how would we symbolize this? Substituting the indicated spatio-temporal location by the predicate ‘…that is in time t and place p’ and replacing it by the symbol P, and replacing the predicate ‘…is a raven’ by the symbol R, the existential statement ‘There is a raven’ could be symbolized simply as Ǝx [(Lx & Rx) & (y) ((Ly & Ry) → (y = x))], which means: ‘There is precisely one Lx that is R.’ Although the location L figures here as a predicate, the condition of unity (any y = x) makes it a singularized spatio-temporal location supposedly analyzable in terms of tropes (See Appendix to Chapter III, sec. 3).
     There is another way to build indexicals, adding to them a sortal predicate, as in ‘that raven there’ in the sentence ‘That raven there is flying’ or ‘this chair’ in the sentence ‘This chair is comfortable.’ In these cases, we consider the phrases ‘that raven there’ and ‘this chair’ as referring to only one specific object, distinguishing it from all others. Hence, these phrases clearly work as singular terms and must be analyzed as expressing identification rules. Thus, replacing ‘…is a raven there’ with R and ‘…is flying’ with F, we can formalize the existential statement ‘That raven there is flying’ as Ǝx [Rx & (y) (Ry → (y = x)) & Fx].
     When we use indexical statements, the language ‘touches’ the world, which makes indexicals the primary, indispensable roots of reference. Because of this, although the sense still determines its reference, we can find here a double direction of fit. First, with the help of our sensory cognitions, we create the identification rule for the indexical that is for the first time used in a determinate context. Once formed, this identification rule (the Fregean sense) determines the spatio-temporal location, often together with the kind of object named by the sortal. Now, this new identification rule can be so established that it can be reapplied (not only later but immediately thereafter), soon forgotten, or maybe interpersonally conventionalized by association with new non-indexical terms of our language that can be a definite description, which can be later inserted in the bundle of definite descriptions that flexibilize the referential work to the most diverse circumstances and is typically abbreviated in the form of proper names.
17. Advantages of the higher-order view of existence
There are several advantages in conceiving existence as a higher-order property, that is, as a higher-order trope. One is that it gives a straightforward answer to what seems odd in the traditional forms of the ontological proof of God’s existence. So, according to Descartes, once we accept the definition of God (1) as the being with all perfections, and that (2) existence is a perfection, we must conclude (3) that God exists (1978, V: 65). But if existence is a (tropical) meta-property of objects and not a proper intrinsic first-order tropical-property constitutive of them, differing in this way from perfections like infinite goodness, omniscience and omnipresence, which should be intrinsic properties of God, the ontological proof is doomed to failure (Cf. Frege 1874, sec. 53).
     However, the greatest advantage of conceiving existence as a higher-order property is that we will not have problems with the denial of existence. Suppose that existence were a first-order property of an object. In a sentence like ‘Vulcan does not exist,’ the negation of existence should then be applied to the object itself, and we would first have to identify the object in order to deny that it has the property, the trope of existence. That is, if in order to identify an object, we first had to admit that it exists, we would be caught in a contradiction: we would have to admit the existence of Vulcan in order to deny its existence.
     According to our Fregean view, this contradictory conclusion isn’t necessary, because all we do by denying the existence of Vulcan is to assert that the ascription rule that forms the concept of Vulcan doesn’t have the meta-property of being effectively applicable in its proper contextualized domain of physical objects. Using Russellian devices, we could analyze the sentence ‘Vulcan does not exist’ as a shorthand way of saying (only to illustrate our point):

x [(x is a small planet orbiting the Sun between Mercury and the Sun) & (y) (if y is a small planet orbiting the Sun between Mercury and the Sun, then y = x)].[28]

What belong to the scope of ‘~Ǝx’ are concepts constitutive of the identification rule, which in my illustration consists of an ascription rule for a predicate that can be applied to only one and the same object. What ‘~Ǝx’ does is just to deny that this identification rule has the property of being effectively applicable to the corresponding physical object, which is to deny that an object existing only in our minds has the meta-property of also existing in reality.
18. Ubiquity of existence
The understanding of existence as the effective applicability of (semantic-cognitive) conceptual rules allows us to explain the almost unlimited extensions in the application of this concept. Why, given that existence is primarily attributed to properties and objects of the outside world or of psychological states, are we also allowed to say that supposed entities like hypothetical and fictional ones exist? Some believe that even contradictory objects exist. We can even say that everything exists, including all that can be conceived – at least as something that can be conceived. And even of existence itself it can be said that it exists. Indeed, it seems that in one way or another everything exists. How can this be possible?
     Concerning supposed entities, we need to distinguish at least two kinds: hypothetical entities that experience hasn’t yet shown to exist or has shown not to exist, and imaginary (including fictional) entities. Beginning with the first group, it is clear that we can find a sense in which they exist. Although the planet Vulcan has been shown not to exist in the real external world, its most proper domain, it surely has existed in the domain of the minds of many astronomers in the past who searched for it, as a hypothetical object… and it still exists in our minds, as a merely imaginary object. For Frege this would be a problem. But this is no problem for our proposed view, because our identification rules can also have the existence-endowing property of being applicable, at least partially, in imagination, that is, only in the dependent domain of conceivable things that we consider as possible or even plausible candidates for existence in the external world. If I imagine the hypothetical planet Vulcan orbiting the Sun, I apply the identification rule for that proper name (even if in a vague, sketchy, deficient way) to a merely conceivable state of affairs. Indeed, the French astronomer Le Verrier, who first named the planet, even had a precise identification rule according to which Vulcan should be a small planet orbiting close to the Sun at a distance of 21 million km, which he mathematically calculated in order to explain by means of Newtonian laws the perihelion precession of Mercury’s orbit. He applied this rule in the domain of what is conceivable, which means that Vulcan ‘existed’ in the restricted domain of the imagination of Le Verrier and other astronomers in his time, though not in its most proper domain – that of a concrete object belonging to the external world.
     Consider now the case of purely fictional entities. Ivan is a character in Dostoyevsky’s philosophical novel The Brothers Karamazov. He never existed in the real world; but he can be said to exist in the fictional world created in this novel, which is from the start fictional. In this domain, Ivan is the son of Fyodor Pavlovich and has two brothers, Dimitri and Alyosha. Ivan is a cerebral as much as a weak character, taking refuge from the inevitable confrontations of life in contemplation and inaction and creating resentful justifications for this; in the end, under the weight of his own conflicts, he descends into madness. These and other elements form parts of the rule for Ivan’s identification. We say that he exists in the story, insofar as this rule is effectively applicable only to him within this proper fictional domain. Differing from the case of hypotheses, existence in a fictional world excludes from the start existence in the real world. That Ivan said to Alyosha: ‘let the worms devour one another’ is true in its fictional domain, as this statement is made in the novel. But this utterance has no existence in the domain of the real external world, where it would be a displaced truth-bearer, since the novel was not written to fit into it.
     Saul Kripke gave examples of cases of fictional-fictional characters like Gonzago (2013: 250), who is a personage in Shakespeare’s Hamlet as a fictional character created by Hamlet in his play within a play ‘The Murder of Gonzago.’ There is a hierarchy here. We may say that Gonzago exists in a third-order domain of Hamlet’s play, requiring the effective applicability of a proper identification rule in this same domain. This third-order domain is supported by the existence of the plot of the fictional play Hamlet, forming a second-order domain. This play is in turn supported by the identification of some writer and writings in the first-order domain of our self-sustaining fundamental real empirical world.
     As with other merely imaginary entities like winged horses and unicorns, existence is here affirmed within a domain that is dependent, derivative or extended (Kripke 2013: 81), being supported by the fundamental form of existence, which concerns the effective applicability of cognitive rules in the domain of the real external (physical) or internal (psychological) world. Existence in these forms of usage is parasitic to the fundamental sense, though retaining its basic features (also Searle 1969: 78-9). In traditional philosophy, it was common to use the word ‘being’ instead of ‘existence’ for merely conceived existence. But I suspect that the real intention was often to underline the importance of conceived entities, underplaying or obstructing its derivative, parasitic character.
     What about the attribution of existence to contradictory imaginative conceptions like that of a round square? This case is really too hard to accept. We cannot combine the rule of identification of the square with the rule of identification of a circle, so that both can identify one and the same thing, since they are from the start incompatible. We cannot do this even in our imagination. Because of this impossibility, we must recognize that in a literal sense a round square cannot reasonably exist: we cannot have a contradictory combination of conceptual rules, because it cannot form a possibly applicable rule combination. Since conceptual ascription rules are what constitute their cognitive meanings, this conclusion agrees with our strongest intuition: contradictions do not exist because they lack cognitive meaning.[29]
     Finally, what about existence? Can we say that existence itself exists? Surely, we know that existence exists in the sense that we know that the concept-word ‘existence’ is effectively applicable to the property of effective applicability of conceptual rules in the most diverse domains, telling us that this property of effective applicability exists. Existence exists in the sense that we can build a meta-meta-rule of existence, whose criterion of application is the effective applicability of our meta-conceptual rules made for the attribution of existence as the property of effective applicability of lower-order conceptual rules. Since these meta-conceptual rules of existence are effectively applicable (since the entities belonging to their varied domains exist), the meta-meta-rule – which demands the effective applicability of meta-rules attributing effective applicability to first-order conceptual rules – is also effectively applicable. Consequently, it is safe to conclude that existence itself exists. Well, then, does the existence of existence also exist? Surely: since the meta-meta-rule of existence is effectively applicable to meta-rules of existence by saying that the latter are effectively applicable to the first order conceptual rules, insofar as the latter ones are effectively applicable, we can conclude that a meta-meta-meta-rule of existence (affirming the existence of existence in itself) is also effectively applicable to the meta-meta-rule of existence, making the latter consequently existent. Of course, one can continue acknowledging the existence of the existence of existence and so on, in an infinite regress, which is virtuous since it can always be stopped.
19. Answering some final objections
According to many present theorists, existence is a first-order predicate. A statement like ‘Horses exist’ should be analyzed in a form similar to ‘Horses are animals.’ Since they have developed objections against the traditional second-order view, I will answer at least some of them, as they were formulated by Collin McGinn (2000b: 21-30). The answers can be helpful in clarifying my own standpoint.
     The first one is against Russell’s proposal that to say something exists is to say that a propositional function – a property, a concept – is true for at least one instance. Roughly stated, the objection is that for one object to instantiate a property this object must already exist, an admission that would make Russell’s view circular, since it must already presuppose the existence of objects instantiating the property. For instance, if ‘Mars is a planet’ is true, it presupposes the existence of the planet Mars to instantiate the property expressed by ‘…is a planet’ in order to make the sentence true. Summarizing, there must already be existent objects in order to instantiate the properties ascribed to them by our conceptual words.
     This objection works insofar as one holds a Kripkean view of objects bearing proper names, since for him they cannot be defined by their own properties (1980: 52). Once we have analyzed an object as a variably accessible cluster of tropes displaying compresence, the objection appears to us in a different form. Since not only the ascriptive rules of predicative expressions, but also the identification rules of nominal terms are for us conceptual rules, our position should be generally stated as saying that existence is the effective applicability of any semantic-cognitive rules in some chosen domain or context. However, since these rules also apply to objects as compresent clusters of tropes, this means we cannot conceive any object as being given – that is, as existing – without simultaneously conceiving its identification rule as effectively applicable to it. Thus, for instance, the existence of a concrete object like the planet Mars is nothing but the effective applicability of its identification rule in its proper astronomical context. This means that we cannot separate the existence of the object in its proper context from the effective applicability of its identification rule in the same context, since this is what warrants the object’s existence. Now, if we assume that the attribution of truth to a singular predicative statement results from the applicability of the identification rule added to the applicability of the ascription rule, the attribution of properties and the admission of the object’s existence are correlative and cognitively simultaneous. Moreover, as the truth follows from the combined application of the first two rules, it is wrong to insist that the attribution of truth requires the attribution of any property prior to the attribution of existence to the trope-property and the object as a cluster of trope-properties. The conclusion is that the flaw in McGinn’s objection lies in the assumption that we can separate the instantiation of a property by an object from the attribution of existence to this same object.
     Now to the second of McGinn’s objections: uninstantiated properties are said to exist. But in order to exist, an uninstantiated property must fall under a higher-order property attributing its existence. This higher-order property must also exist, which means it must fall under a still higher-order property and so on infinitely. Consequently, the attribution of existence as a higher-order property is impossible, because it requires an infinite regress of properties to allow the attribution of existence.
     My answer is that I agree (partially) with the diagnosis, but not with the prognosis. The effective applicability of a semantic-cognitive (conceptual) rule in its most proper domain not only endows its reference with existence, but is in itself a second-order property or trope that can also be said to exist. And furthermore, a semantic-cognitive rule that is only imaginatively applicable not only endows its reference with existence in an imaginary domain, but can also be said to exist. The trope-property of existence exists, which means that we can say that the second-order property of effective applicability of a conceptual rule can be the object of a third-order rule predicating its effective applicability, and so on indefinitely. This of course leads to an infinite regress. However, it is a virtuous infinite regress, since the applicability of a conceptual rule such as existence is already warranted by the application to it of a higher-order rule, and we don’t need to bother with all the unlimited further applicabilities of applicabilities or existences of existences that the first existence-endowing rule can generate. The mark of a virtuous regress is that we may stop it without loss when we feel that we do not need further steps to what we intend to explain, and this is the case here (See Appendix of Chapter III, sec. 2).
     The third objection is that there are statements ascribing existence to particulars, such as ‘Venus exists,’ that resist the traditional paraphrase. We have already answered this objection in our treatment of proper names as conceptual identification rules.
     But there are other objections. Consider the statement ‘Something exists.’ Although this is a true statement, McGinn believes that it is not paraphraseable in terms of the higher-order view, since there is no property to be instantiated here, and if we try to translate into the standard form we get the gibberish ‘Ǝx(…x).’
     The answer to this objection is too easy. What ‘Something exists’ means is that there is at least one trope or tropical construction out of tropes that exists without a further determination on our side. That is, we can say that there is some semantic-conceptual rule that is applicable to some domain of entities, even if this rule remains unspecified. This possibility is even shown by our logical symbolism on an elementary level, since we can symbolize an undetermined property such as, say, F. In this way we can translate ‘Something exists’ symbolically as Ǝx(Fx). But there is nothing wrong with Ǝx(Fx). Paralleling existential universalization, we can reach this result by considering singular existential statements like ‘Venus exists.’ So, calling Venus V, if it is true that ‘Ǝx(Vx)’ this implies that some property exists or ‘Ǝx(Fx), namely, that some conceptual rule is effectively applicable. This assumption is of cognitively undetermined properties is harmless.
     McGuinn reminds us that there are also more complicated statements that seem to resist a higher-order understanding of existence, like:

1.       Some cities are purely imaginary.
2.       Some of the things you are talking about do not exist.
3.       There are things that do not exist…

Nonetheless, we can easily explain the predication of existence in such statements, insofar as we do not confuse the domains of application of the semantic-cognitive rules involved.
     Thus, statement (1) means that some cities that exist in the imaginary domain exist only in this domain. Hence, the effective applicability of rules allowing us to identify the imaginary cities of Chloe or Valdrada in the contextual domain of the book The Invisible Cities is sufficient for the attribution of existence in that purely fictional context. Statement (2) means that some things you are talking about exist only in imaginary domains, but not in the external world, that is, there are identification rules that are effectively applicable only in the unreal domain of one’s own discourse. For instance, the identification rule of the name ‘Vulcan’ in the statement ‘Vulcan is red’ is only applicable in the speaker’s (or hearer’s) imagination. Finally, statement (3) means that there is at least one thing that exists only in the mind but not in external reality. Indeed, it seems obvious that the identification rule for some objects and therefore for at least one of them, though effectively applicable in an imaginary, only conceivable domain, isn’t effectively applicable in the domain of external reality.
     The last of McGinn’s objections is that according to the higher-order view, nothing can exist without falling under some property or other, which rules out the existence of a thing that has no properties – a ‘bare existent.’ However, our empiricist commitment makes us see this not as a weakness, but rather as a further anti-metaphysical advantage of the higher-order view.
20. Reference of concepts again: a metaphysical excurse (Mill)
It is instructive to consider what happens when we compare the famous phenomenalist view of J. S. Mill, according to which ‘matter’ or ‘substance’ is nothing but ‘permanent possibilities of sensation’ with our view of existence in terms of the effective applicability of conceptual rules. The results will be no less speculative than Mill’s phenomenalism, but they may be telling.
     Mill’s great epistemological question was: If all that is experientially given to us are sensory phenomena, how can we justify our belief in the existence of an external world, an objective world constituted by substance or matter? – An external world that can exist even when there is no observer at all to perceive it?
     Mill’s answer to the question was a development of Berkeley’s unofficial view, according to which things that we know to exist when we are not perceiving them are nothing more than things that we are certain we would perceive under suitable circumstances.[30] As Berkeley wrote:
Existere is percipi or percipere… The horse is in the stable, the books are in the study as before. (1707-8, Notebok A, 429)
The table I write on, I say, exists, that is, I see and feel it; and if I were out of my study I should say it existed – meaning thereby that if I was in my study I might perceive it, or that some other spirit actually does perceive it. (1710, I, sec 3)
According to this view, esse is not only percipi, but also percipi possi. In a more explicit manner, what Mill suggests is that:
Matter or substance is not made up of actual sensations, but of groups of permanent (or guaranteed or certified) possibilities of sensation.
Mill justifies his identification of matter or substance with permanent possibilities of sensation in the following way. First, these possibilities of sensation are conditional certainties: they are not mere epistemic possibilities, but firm conditional expectations that are in direct or indirect ways based on experience. They are permanent in the sense that, once suitable circumstances are given, they would always be experienced insofar as they are said to exist. And they are guaranteed or certified in the sense that we have good reasons – observational or not – to have a firm expectation that under suitable circumstances they will be experienced again and again. This does not mean that the groups of permanent possibilities of sensations would depend for their existence on our past experience of them, because if that were so, they could not exist without us as subjects of knowledge, and we would fall like Berkeley into some radical form of idealism like his immaterialism (Berkeley 1710, 1713). This was not Mill’s intention. As he explains:
We mean [by permanent possibilities of sensation]… something which exists when we are not thinking of it; which existed before we have ever thought of it, and would exist if we were annihilated; and further that things exist that we never saw, touched or otherwise perceived, and things which never have been perceived by man. (Mill 1979, X: 178-177)
Thus, it is clear that Mill wished to avoid idealism: the permanent possibilities of sensations would exist even if cognitive beings able to perceive them never existed.
     These permanent possibilities are for Mill objective, differing from our actual constantly changing sensations, which are subjective. They are objective because they are grounded, he thinks, in our common public world, which makes us able to interpersonally agree on their existence. For him, even if different persons cannot have access to the same sensations, they can have access to the same possibilities of sensation. As he writes:
The permanent possibilities are common to us and to our fellow creatures, the actual sensations are not… The world of possible sensations succeeding one another according to laws is as much in other beings as it is in me; it has therefore an existence outside me; it is an external world. (Mill 1979, X: 181-2, my italics)
This is in summary Mill’s view on the nature of matter – a view that always seemed to me as much deeply suggestive as contentious.
     I think there is a serious confusion in Mill’s view, which can be made clear when we compare his insights with those of Berkeley. According to the non-official Berkeleyan view, the external world is constituted by sensations whose experience is continually (permanently) possible for us, even if we are not there to experience them. But if this is so, the material objects constituting the external world cannot be reduced to simple ‘groups of permanent possibilities of sensation,’ for possibilities as such, permanent or not, cannot be qualitatively distinguished one from the other in the same way as one material object can be distinguished from another. Material objects can be qualitatively very different from each other, they are multiple and varied, while possibilities are always the same, namely, mere possibilities. Consequently, possibilities (of sensations), permanent or not, cannot be the same as material things. Keeping this in mind, the only feasible way to express the Berkeleyan insight in Mill’s terminology seems to me to use it in the characterization of material objects, as follows:

Material objects (or substances) are nothing but multiple and varied groups of sensations whose effective experience is permanently (or guaranteed or certified to be) possible.

This would meet the requirement of multiplicity and diversity proper to material objects and their presentations, because each material object would be constituted by innumerable different groups of sensations that under suitable circumstances could always be possibly distinctly experienced. But if the permanent possibility of sensations is not the material object, what is?
     I believe it is a way to point to the external existence of the material object. This answer emerges when we consider Mill’s view in the light of our reconstruction of Frege’s concept of existence, according to which existence is the effective applicability of a conceptual or semantic-cognitive rule. If this is so, it seems that the permanent (guaranteed, certified) possibility of groups of sensations could be approximated to the existence of such groups of sensations and the last ones to material objects; these warranted groups of sensations would be the same as the criterial configurations warranting the applicability of the rule. Consider the expressions:

1.   Permanent (guaranteed, certified) possibilities of groups of sensations.
2.   Effective experienceability of groups of sensations.

Expressions (1) and (2) say the same thing in different words. Now, compare them to the following expressions of existence in our reconstruction of Frege’s view:

3. Effective applicability of a conceptual rule.
4. Effective applicability of a conceptual rule to groups of given sensory contents.
5. Effective applicability of a conceptual rule to given (external) criterial configurations or tropes.

Although (4) is only a variation of (3), it seems clear that when we interpret existence as (4) we are saying something at least equivalent to (2): the effective experienceability of groups of sensations. Since (2) is only a different way to say (1), the permanent (guaranteed, certified) possibility can be approximated to existence. One could even suggest:

Existence is the effective (permanent, guaranteed, certified) possibility of groups of sensations.

The point in question is made clearer when we consider the general structure of our conceptual rules of ascription and identification. We already know that these rules have the form of semantic-criterial rules that bring us to some (usually pre-reflexively achieved) cognition, given by the satisfaction of variable subjective criterial configurations (supposedly) by means of their match with objective criterial configurations, which should be nothing but configurations of external tropes. Now, when we interpret these variable criterial configurations as being the same as Mill’s groups of sensations, as we have reconstructed them, we can speak of existence as the effective, guaranteed, certified, permanent possibilities of groups of sensations as consistent with the effective applicability of a conceptual rule. For instance: In order to be applied to a real located object, the conceptual rule for the concept chair demands the satisfaction of criterial configurations. These criterial configurations are established by the definition of a chair as a movable seat with a backrest made for only one person to sit on at a time, which we could decompose in terms of subjective sensory criterial configura­tions that must be satisfied by matching objective criterial configura­tions or configurations of given external tropes. But the criterial configurations (the subjective ones, at least) could be reduced to groups of sensations whose experience is permanently (guaranteed, certified as) possible.
     Now, Mill’s insights can help us deepen our reconstruction of the Fregean concept of existence. A material object exists only:

(i)    when its conceptual rule is effectively applicable, but this effective applicability is only the case when
(ii)   criteria for the application of its identification rule can be objectively given to us at least in the form of groups of what we may call external contents of sensation whose experience is warranted or permanently possible. Moreover, as Mill suggested,
(iii) this possibility of experience must be (at least in principle and indirectly) interpersonally accessible by allowing agreement in the description of the experience;
(iv) this experience can be more or less direct;
(v)  it is (usually) independent of our will; and
(vi) it is also experienced as following causal laws regarded as typical of things belonging to the external world.

It seems that all these things together contribute to building the condition of an effective application of a semantic-cognitive rule in the domain of the external world – they are contributing to warrant the attribution of external existence.
     There is, however, an important and seemingly fatal objection to Mill’s view of matter, which is made more serious by the Berkeleyan correction I made above.[31] It is that the group of sensations or configurations of sensory criteria that satisfy a conceptual rule are by their nature inevitably psychological. It seems clear that even sensations or contents of sensations that are warranted as permanently possible must be psychological in a dispositional way. This means that if we follow this path, we end up falling into some form of Berkeleyan idealism in which there is no objective, external material world to be contrasted with our subjective world of sensations or sensory criteria. No really non-mental external trope needs to be there to match the apparently satisfied internal criterial conditions, as suggested in statement (5). It is true that, as Mill noted, his possible sensations are independent of our will, that they follow the regularities of nature, even that they appear to be interpersonally accessible under circumstances that warrant their experience (under suitable circumstances they are described as being experienced simultaneously by different subjects, etc.) However, all this seems insufficient to perform the magic of turning sensations qua sensations into something they aren’t, namely, supposed elements of a non-mental objective external world of material objects with their own tropical-properties. This is an important objection, whose answer will be given only in the final chapter of this book, as a consequence of our discussion of the adequation theory of truth in its relation to direct realism.
     Notwithstanding, I can now anticipate something of the way I intend to deal with the problem. Having in mind our improved views of existence, we can ask: What is, in more conventional language, an existing external material object? One too daring answer would be: the external object (as it is thought) must be the identification rule in itself, insofar as it is effectively applicable; in this way the multiplicity and diversity of objects would be explained by the multiplicity and diversity of identification rules... However, this cannot be, since a semantic-cognitive rule is also something essentially mental, and we are definitely not what Plato called friends of ideas.
     Looking for a less daring answer, we can suggest that what we understand as the material object is not the semantic-cognitive rule, but is supposed to have the same structure as this rule projected in a specular way onto the external world. There is a reason for this suggestion: It seems that only something with a structure similar to its semantic-cognitive rule would be able to give unity to the multiple and variable criterial configurations by means of which external entities are able to give themselves to us in our experience of them. Figuratively speaking, if the semantic-cognitive rule has the form of a tree with branches whose ramifications end in criterial conditions internal to the rule, then the object of its application, as we believe it to be, must have the structure of an inverted specular tree with branches whose ramifications end in objective criterial configurations that (supposedly) should match the corresponding subjective criterial configurations. Furthermore, these objective criterial configurations should be nothing but external tropes and constructions out of them (objects, facts). Of course, this objective structure should be putative, so that the rule could always be improved or corrected as a response to new information regarding such specular objective counterparts. We will come back to this suggestion toward the end of this book.
21. The reference of a sentence as its truth-value
Now we leave our speculative excurse and come back to the more tangible Fregean semantics, considering what he has to say about the reference of a sentence. Here I have no compliments to make. Frege was the author of the at first view insane idea that the references of sentences are their truth-values, so that the thoughts expressed by them should be modes of presentation of truth-values.
     How did he reach this strange conclusion? There are several reasons. First, he notes that sentences are independent, saturated, closed; they work in ways similar to those of names, and a truth-value is also closed, since it does not require complementation. Second, he says that the search for truth is what brings us from sense to reference. Third, he notes that sentences without reference lack truth-value: ‘Vulcan is a warm planet’ has no reference and for him no truth-value, since this hypothetical planet has been shown not to exist. Fourth, he also noted that conforming to the principle of compositionality – according to which the whole is a function of its parts – the reference must be what remains unchanged after we change the senses of a sentence’s components without changing their references. This is what happens, for instance, if we replace ‘Napoleon lost the Battle of Waterloo’ with ‘The man of destiny lost his last battle.’ Since the references of the sentence-components do not change, the reference of the whole sentence likewise does not change. Moreover, the truth-value of both sentences remains the same: The Truth. Hence, their reference must be their truth-value. The conclusion of all this is that in extensional languages the references of sentences must be their truth-value (1892: 34). For Frege, all true sentences have only one reference, which is the abstract object The True (das Wahre), while all false sentences also have only one reference, which is the abstract object The False (das Falsche).
     However, there are a number of well-known embarrassing objections to Frege’s identification of reference with truth-value that in my opinion completely disqualify his view. A first objection is that, contrary to any healthy intuition, Frege’s proposal frontally contradicts the meaning we normally give to the word ‘reference.’ It is intuitively obvious that the sentence ‘Napoleon was born on Corsica’ refers to something very different from the sentence ‘2 + 2 = 4,’ even if both are true. Moreover, if you replace ‘Venus is a planet & the Earth is a planet’ with ‘Mars is a planet & the Earth is a planet,’ both composite sentences remain true because of the truth of the partial sentences, but the reference of ‘Venus’ is obviously different from the reference of ‘Mars.’ Another objection is that we expect the references of components of our sentences to be on the same ontological level as the sentences’ references. But for a Fregean, this could not be the case: the reference of the name ‘Napoleon’ is the Napoleon of flesh and blood, while the reference of the sentence ‘Napoleon was born on Corsica’ must be the abstract object called The True. Moreover, one could also argue that Frege’s solution sounds false, because it in fact violates his own principle of compositionality. If the reference of a sentence is its truth-value, it cannot be established by its parts, since a truth-value has no parts. And even if it had parts, then all objects referred to by names in true sentences should be parts of The True, which would hardly make sense. A further objection is that there are serious substitutability problems with Frege’s explanation of the references of sentences. If all true sentences refer to The True, and the name ‘The True’ also refers to The True, then in the conditional sentence ‘If it rains, then water falls from the sky,’ we can replace ‘it rains’ with ‘The True.’ But the result will be the sentence ‘If The True, then water falls from the sky,’ which should be true but is in fact unintelligible (Black 1954: 235-6). Finally, to make things still worse, a multitude of obviously false identities between true sentences should be true. For example, ‘Paris is a city = snow is white’ should be a true assertive composite sentence, since the two sentences refer to the same thing: The True. Under critical scrutiny, Frege’s view shows itself to be hopeless.
     The most charitable interpretation is that Frege uses the word ‘reference’ as truth-value because it is what counts, because the word Bedeutung (meaning) in German, more than in English, also means relevance, pointing to semantic relevance or meaningfulness (e.g., Tugendhat 1992b: 231).[32] Indeed, truth-value is of decisive relevance for logic, because it is what must be preserved in valid arguments. The logician does not need to know more than truth-value regarding the referring function of the sentences he is dealing with in order to evaluate inferential possibilities.
     A main problem with this interpretation is that it contradicts expected principles of Frege’s own theory. Since the reference (Bedeutung) of the parts of a singular sentence (general and singular terms) can be seen as their references in a literal sense (the concept and the object that can fall under it), truth-value as relevance satisfies the principle of compositionality in an odd, non-linear form, since relevance is normally only an adjective applied to truth-value. This is different from the principle of compositionality applied to senses in which the whole and its components are linearly arranged in the same epistemic domain.
     Finally, when we take the truth-value for the reference of a sentence, this view can be – and in my judgment really has been – utterly misleading from an epistemological standpoint. Since truth considered as in some way belonging to thought has nothing to do with anything that can reasonably be understood as the reference of our statements, calling truth-value ‘the reference’ contributes to placing the relation between language and the world virtually beyond semantic reach.
22. Logical structure of facts
The Fregean account of the references of sentences as their truth-values turns out to be still less acceptable if we consider that a much more natural alternative is available, which, as Sir Anthony Kenny has noted, was not even mentioned by Frege (Kenny 2000: 133). This alternative, which the logical atomism of Wittgenstein and Russell tried to explore, consists in the appeal to facts. Since it is prima facie much more plausible that the references of sentences are facts, it is important for us to investigate the logical structure and ontological nature of facts.
     Considering first the logical structure of facts, the most plausible view is that they correspond to the logical structure of the thoughts representing them, assuming that these thoughts are what declarative sentences express when logically analyzed, at least in accordance with the context of the linguistic practices where they occur. Nevertheless, even respecting linguistic practices we can go further, considering that they are placed within the factual language in general and accepting a form of atomism in which the bottom line of the analysis is the exposure of the logical components of what is stated in singular sentences where we can find identification rules of singular termini associatively used with ascription rules of predicative expressions. Singular empirical statements such as ‘Frege has a beard’ and ‘The cat is on the mat’ belong to this bottom line and respectively represent facts that should have the logical structure depicted by Fa and bRc.
     Elements a, b and c, as singular terms, refer to individuals constructed as clusters of appropriate compresent tropes, while F and R would also be seen as designating tropes, usually complex tropes forming complex configurations dependent on the clusters to which they are tied. The ties between b, R and c, and between F and a, in turn, are only pseudo-relations, since admitting their existence as relational tropes would generate an inevitable infinite regress. As we already noted, individuals and their property-tropes are linked by ‘non-relational ties’ without any ontological addition (Cf. Appendix to Chapter III). Indeed, what are the relational ties between the application of the ascription rule of ‘…was bearded’ to Aristotle with the already applied identification rule of Aristotle in the fact represented by the statement ‘Aristotle was bearded’?
     We should also pay attention to the somewhat trivial rule of analysis according to which we should not accept singular terms as components of complex predicative expressions. Thus, for instance, in a sentence like ‘Stockholm is the capital of Sweden’ we should not view ‘…is the capital of Sweden’ as a predicate, since Sweden is a proper name. Also inadequate would be to analyze ‘the capital of Sweden’ as a definite description contextually referring to Stockholm in our world, so that the analyzed sentence would have as its relational predicate ‘…is (the same as)…’ The most appropriate analysis would be to consider ‘…is the capital of…’ as a relational predicate completed by the proper names ‘Stockholm’ and ‘Sweden,’ separating the relational trope from the compresent bundles of tropes referred to by the proper names.
     Furthermore, it also seems possible to analyze proper names and definite descriptions using Russell’s technique of transforming them into quantified predicative expressions, insofar as to a limited degree this device mirrors the neodescriptivist theory of proper names defended in this book, a similar procedure being possible regarding general terms. Anyway, such sub-sentential terms normally do not need to be analyzed when our task is to analyze sentences, since they are the proper elements of sentences, except when they are not what they seem to be, as in the case of nominalizations.
     Finally, we have composite facts represented by our extensional language, along with the general (universal, existential) facts to be analyzed as having the same structure of sets (conjunctions, disjunctions) of singular statements that make up general (universal, existential) statements, which, as we already noted, can be reduced to associations of singular predicative and relational statements. (I think that the philosophical problem of a hidden lingua mentis ends up in elements like those shortly pointed out in this section).
23. Ontological nature of facts
If we accept that the references of sentence-senses or thoughts are facts, then from an ontological perspective the references of empirical sentences – what they represent – must be empirical facts, most typically located in the external world, though possibly also located in someone’s inner mental world. This assumption speaks for the correspondence or adequation theory of truth, according to which empirical facts are truth-makers normally seen as complex contingent arrangements of elements in the world, that is, usually contingent tropical arrangements.
     However, this assumption conflicts with Frege’s anti-correspondentialist view of truth. According to him, a fact would simply be a true thought (Frege 1918: 74). Following similar anti-correspondentialist lines, in a very influential article P. F. Straw­son suggested that empirical facts are mere ‘pseudo-material correlates of the statement as a whole’ and not something in the world (1950: 6). According to him, empirical facts, unlike events or things, are not spatio-temporally localizable (‘the world is the totality of things, not of facts’). One reason for this is that the description of a fact usually begins with a that-clause. For instance, I can say ‘the fact that the book is on the table,’ but not ‘the fact of a book on the table.’ On the other hand, the description of an event typically lacks a that-clause: I can say ‘the event of a tsunami in Japan,’ but not properly ‘the event that there was a Tsunami in Japan.’ Facts are for Strawson what statements (when true) state, not what statements are about. They are
not, like things or happenings on the surface of the globe, witnessed or heard or seen, broken or overturned, interrupted or prolonged, kicked, destroyed, mended or noisy. (1950: 6)
The same is for him the case with states of affairs and situations. [33] Finally, to give a striking example, the event of Caesar’s crossing the Rubicon occurred in the year 47 BC, while the fact that he crossed the Rubicon did not occur in the year 47 BC, but it is still a fact today, since facts simply do not occur (Patzig 1980: 19-20).[34]
     An easy way to dispose of this argument could be the following. We need a word to describe the condition in the world that makes our thoughts true. The word ‘fact’ is available. So, why don’t we use it stipulatively in order to designate the truth-maker, whatever condition it is?[35]
     However, it seems clear to me that even this stipulative way to circumvent the problem is avoidable, since it is not difficult to show that the problem exists only in the imagination of philosophers. To begin with, of course not everything we may call a ‘fact’ is empirical in the usual sense of the word. It is hard to assign empirical status to the fact that 2 + 2 = 4, even if its non-empirical character can be an object of controversy.[36] And we can say ‘It is a fact that the Sun is not green,’ although this seems to me only a linguistically modified way to say ‘There is no fact that the Sun is green’ or ‘The fact that the sun is green does not exist.’ What I want to defend here is that there is a privileged sense of the word ‘fact’ that involves references to more or less obvious empirical facts, particularly so-called observational facts, which should be considered objectively real: they exist in the external world and can be seen as the ultimate truth-makers of their statements.
     To begin with, it is good to remember that there is a well-known and very convincing reason to think that facts can be constituents of the empirical world. This is that many facts are said to act causally. Consider the following sentences:

(1)  The fact that the match was scratched caused the flame.
(2)  Thomas died because of the fact that he forgot to turn off the gas.
(3)  Because of the fact that today is a holiday, the class will be canceled.
(4)  The fact that Caesar crossed the Rubicon had important historical consequences.

It does not seem possible that pseudo-material correlates (which I suppose to be abstract contents) can be causally active in the empirical world, producing these effects. But conceding the empirical nature of facts (1) to (4) solves the problem in obvious ways. Scratching a match is a fact-event causing a flame. The situational fact created by Thomas’ forgetting to turn off the gas caused his death. The fact-circumstance that today is a holiday causes the cancellation of a class. The fact-event of crossing the Rubicon established a state of affairs that causally determined decisive political changes in the Roman Empire.
     Furthermore, I have a key-argument to regenerate the idea that empirical facts are correlates of true thoughts, as the classical correspondence theory of truth has held. According to the view I propose, empirical facts are contingent tropical arrangements in the external and/or internal world in general (in the simplest case arrangements of more or less complex predicative or relational tropes contingently tied with material clusters of compresent tropes referred to by nominal terms). Similar would be the case with facts apparently as simple as those referred to by sentences like ‘Frege had a beard,’ ‘The Eiffel Tower is in Paris,’ and also facts constituted by combinations of such facts, which are elementary in the context of their linguistic practices.
     My argument against Strawson’s opposition between non-spatio-temporal facts and spatio-temporal events begins by showing that there is a serious confusion in his argument. He treats facts (as much as states of affairs and situations) as opposed to events. His schema is:

FACTS                              x                            EVENTS
Pseudo-material                            Spatio-temporal
correlates                                       phenomena

But this can easily be contested. We begin to be suspicious when we perceive that every event can be called a fact, but not every fact can be called an event. For instance: I can replace ‘the event of the sinking of the Titanic’ with ‘the fact of the sinking of the Titanic,’ but I cannot replace ‘the fact that Mt. Everest is more than 8,000 m. high’ with ‘the event of Mt. Everest being more than 8,000 m. high.’ Strawson’s opposition isn’t symmetrical. Hence, it is much more reasonable to consider events as particular kinds of facts than to oppose the two, as Strawson did. Indeed, my proposal is that the word ‘fact’ is an umbrella term that encompasses events, occurrences, processes, as much as situations, circum­stances, states of affairs, etc. And the reason for this proposal is that we can call all these things facts, but we cannot call all these things states of affairs or events. We see that events are sub-types of facts, and that linguists could classify the word ‘event’ as a hyponym of the word ‘fact.’ Considering things in this way, we can distinguish two great sub-classes of facts:

1.     STATIC FACTS: Can be formal or empirical, the latter when clearly located in space and time. As a whole, static facts do not change while they last. Typical of static facts is that the relationships between their tropical components do not decisively change during the period of their existence. They are truth-makers of a static kind. And ordinary language has names for them: they are called (with different semantic nuances) ‘states,’ ‘situations,’ ‘conditions,’ ‘circumstances,’ ‘states of affairs,’ ‘ways things are,’ etc.
2.     DYNAMIC FACTS: These are always empirical. They change while they last. The relationships between the elements constitutive of them change decisively during the period of their existence, so that they have a beginning, followed by some kind of development that comes to an end after a certain amount of time. We will see that they work as truth-makers of a dynamic kind. And ordinarily they can be called (with different semantic nuances) ‘events,’ ‘episodes,’ ‘occurrences,’ ‘occasions,’ ‘pro­cesses,’ ‘transformations,’ etc.

Facts said to be formal, like the fact that 7 × 8 = 56, are static in the harmless sense that they do not need to be considered as spatio-temporally located. They are not of concern to us here. Many facts are empirical and static, insofar as the relationships between the elements constitutive of them do not change during their existence. Static facts are usually called ‘states,’ ‘situations,’ ‘conditions,’ ‘circumstances,’ ‘states of affairs’… with different nuances of meaning. Examples of static facts are my state of poor health, the situation that I am lying in bed, the circumstance that the airport is closed, the state of affairs that the Mona Lisa is in the Louvre or that the Earth orbits the Sun. The Earth’s movement of revolving around the Sun counts as a static fact, because it is an internal cyclical relationship that remains the same during the fact’s existence: as a whole, this state of affairs does not change while it lasts (although each orbital period counts as an event).
     Dynamic facts, on the other hand, can be called ‘events,’ ‘episodes,’ ‘occurrences,’ ‘occasions,’ ‘processes,’… They are defined by changes in their overall composition and in relations among their elements during the period of their existence. World War II, viewed a process, for instance, began with a rapid expansion of the territories dominated by Nazi Germany and was marked by events like the Battle of Britain, the Battle of Stalingrad and the Normandy invasion – it had an unforeseeable history. Dynamic facts are usually called events when their duration is comparatively short, occurrences when their duration isn’t as short, processes when their duration is longer. Examples of events are an explosion or a lightning flash in a storm. An example of an occurrence is a volcanic eruption. The process of global warming is a very slow natural process, slower than the process of economic globalization. We can predict the stages of many events and processes, although many are also unpredictable in such a way that (unlike static facts) we cannot grasp them in their integrity before they end. Important is to see that all these things can be individually called events, occurrences, occasions, happenings, processes… and also facts, since they are all nothing but empirical facts – truth-makers of a dynamic kind.
     We are now able to find what seems to be the real reason why we use a that-clause in the description of facts, but not in the description of events. When we speak of dynamic facts, we do not use a that-clause. Thus, we can speak about the event of Caesar’s crossing the Rubicon, but not about the event that he crossed the Rubicon. We can speak about the process of climate change, but not about the process that the climate changes… But this isn’t the case regarding static facts, which are typically (though not necessarily) described as beginning with that-clauses. So, I can speak about the state of affairs that my book is on the table or that I am lying on the bed, although I can also speak about the state of affairs of my book being on the table and of my lying on the bed. The conclusion is that if that-clauses have some function it is that of excluding dynamic facts and emphasizing static facts. Moreover, since the hyperonymic term ‘fact’ can be applied to both – static facts as much as dynamic facts – it is reasonable to suppose that this term inherits the property of being used indifferently, with or without a that-clause. Indeed, you can say, ‘It is a fact that Mount Vesuvius is located near Naples’ (referring to a state of affairs), as much as ‘It is a fact that Mount Vesuvius has erupted’ (referring to an event). And we can also say: ‘Caesar crossing the Rubicon was an event’, as much as ‘It is a fact that Caesar crossed the Rubicon,’ referring less precisely to the event. We can summarize these relationships in a schema:

(a)  Static facts (states of affairs…): can be well stated with or without a that-clause.
(b)  Dynamic facts (events…): cannot be well stated with a that-clause.
(c)   Facts in general: admit both cases, because being all-embracing they do not differentiate between (a) and (b).

Now, what about the fact that Caesar crossed the Rubicon? Isn’t this fact timeless? The answer is that this is a good case of a misleading statement. In most cases, it is not understood as the description of an event, but as an illustrative way of referring to a static social fact: the state of affairs established by the movement of Caesar’s army onto Roman territory, violating the law that prohibited this and forcing the Roman state to declare war against him. Only occasionally is the phrase ‘crossing the Rubicon’ understood in its literal sense, as the physical event of crossing the river, which comprises Caesar’s sequential locations in relation to the river from t1 to tn.
     Due to the nature of dynamic facts like events and processes, we say that they not only are, but also occur in time, while of static facts we only say that they are located in time while they last. It seems, therefore, that because philosophers such as Strawson did not realize that events are sub-types of facts, seeing only that we may say of events that they occur in time, they hastily concluded that only events (and things) are located in time, opposing them to timeless facts. But that this isn’t true can be shown even by inter-substitutivity salva veritate: it is correct to say that the event, the occurrence of Caesar’s crossing the Rubicon, was a fact and that this fact occurred in 47 BC, as a concrete dynamic fact. On the other hand, the static social fact, the political state of affairs established by Caesar’s crossing the river was far more enduring. Being a static fact, it was the political situation that led, as is well-known, to the fall of the Republic. However, it seems clear that the state of affairs brought about by the crossing of the Rubicon was spatially limited to the Roman Empire and temporally limited to the time from Caesar’s crossing the Rubicon to his coronation as Caesar and up until his assassination. It was not something that existed in Greenland or that endured until the present, even if in a misleading way our ordinary language can be confusive by allowing us to use the present tense to speak about historical facts.
     The relevant conclusion is that by having the broadest scope, the so underrated word ‘fact’ remains the ideal candidate for the role of ultimate truth-maker in a correspondence theory of truth. Facts are universal truth-makers.
24. Church’s slingshot argument
As already noted, for Frege a sentence’s reference is its truth-value. To refute the charge that this view is implausible, the Fregean logician Alonzo Church devised a slingshot argument. He wanted to show that by means of inter-substitutability of co-referentials we can prove that the most diverse sentences can only have a truth-value as their reference.
     Church’s argument is in my view equivocal, but telling. Its basic assumption is that when one constituent expression is replaced by another, so that their partial references (the references of their singular terms) are interchangeable, the reference of the whole sentence does not change. I will begin by explaining his slingshot argument, underlining its supposedly co-referential definite descriptions (Church 1956: 25):

1.     Sir Walter Scott is the author of Waverley.
2.     Sir Walter Scott is the man who wrote the twenty-nine Waverley novels altogether.
3.     Twenty-nine is the number such that Sir Walter Scott is the man who wrote that many Waverley novels altogether.
4.     Twenty-nine is the number of counties in Utah.

According to him, if it is plausible that sentences (2) and (3) are, if not synonymous, at least co-referential sentences, then (1) has the same reference as (4). Since (4) seems to concern a fact completely different from (1), it seems that the only thing left as the same reference is the truth of both sentences. Hence, The True is the only referent of all these sentences.
      However, the argument proves to be unsustainable when we pay attention to what should be the real reference of each singular term of these sentences. In sentence (1) the proper name ‘Sir Walter Scott’ and the definite description ‘the author of Waverley’ are two singular terms expressing different modes of presentation of the same human being. These modes of presentation make what we could call two partial references to Walter Scott, namely, references that must be partial relatively to the whole reference of the sentence. In sentence (2) again, the nominal expression ‘Sir Walter Scott’ and the definite description ‘the man who wrote the twenty-nine Waverley novels altogether’ both refer in different ways to the same Walter Scott. The third sentence is the tricky one. Its reference is unclear: Walter Scott? The number 29? Both in one? The combination Scott-29? The answer appears when we paraphrase sentence (3) so that it gives back in a transparent way its complete informative content. Now, carefully considering the confusing sentence (3), we see that the only way to reveal its content in a transparent way without any addition or loss of sense is to split the sentence into the following conjunction of two sentences: (5) ‘29 is the number of Waverley novels and Sir Walter Scott is the man who wrote that many Waverley novels altogether.’ Sentence (5) makes explicit all the content wrapped up in sentence (3). For the sake of clarity, replacing in (5) ‘=’ for ‘is (the same as)’ and ‘&’ for ‘and,’ we can still unpack (3) as:

6.     (29 = the number of Waverley novels) & (Sir Walter Scott = the man who wrote the many Waverley novels altogether).[37]

That is: Sentence (3) confusingly compresses nothing less than a conjunction of two identity sentences, each with its own proper partial references given by the singular terms flanking their identity signs. They are the number 29 in the first sentence and Walter Scott in the second. Finally, we come to the analysis of sentence (4): ‘29 is the number of counties in Utah,’ which means the same as the identity sentence (7) ‘29 = the number of counties in Utah.’ Here, each singular term that flanks the identity sign has the number 29 as its partial reference. So analyzed, the derivation appears as:

1.     Sir Walter Scott = the author of Waverley.
2.     Sir Walter Scott = the man who wrote the 29 Waverley novels altogether.
3.     (5) (29 = the number of Waverley novels) & (Sir Walter Scott = the man who wrote the many Waverley novels altogether).
4.     (6) 29 = the number of counties in Utah.

Now, although all these sentences are true, Church’s argument has by now lost its initial plausibility. Sentences (1) and (2) have as the partial references made by their singular terms Walter Scott under different guises. However, sentence (3) is a conjunction of two identity sentences, each with its own very distinct partial reference. The object referred to by the flanking terms of the first identity sentence of (3) is the number 29 (as the number of Waverley novels), while the object referred to by the flanking terms of the second identity sentence of (3) is Sir Walter Scott (as the man who wrote the Waverley novels). Finally, sentence (4) has as a partial reference made by its singular terms only the number 29 (as the number of counties in Utah), without referring to Walter Scott, as it should. That is: In the composed sentence (3), the second sentence of the conjunction is the only one that preserves as the partial reference made by its singular terms the references of (1) and (2), while (4) is an identity sentence that has as the partial reference made by its singular terms only the partial reference of the first sentence of (3). However, this is precisely what should not occur because the preserved partial reference has has nothing to do with the partial references made by the singular terms of the sentences (1) and (2) and supposedly with the whole references of these sentences.
     In other words we can say that in a surreptitious way the replacements slide equivocally from having as partial reference Walter Scott in (1) and (2), to a Walter Scott, together with the number 29 in (3), and to the number 29 in (4). This means, according to the principle of compositionality applied to complete sentences, that the references of sentences (1) and (4) should indeed be very different. Initially the flaw is not easy to spot, because sentence (3) contains both these partial references conjoined in a grammatically confusing way. We have the impression that the partial references of (3) seem to be something like an amalgam of Walter Scott and 29, say, a ‘Scott-29,’ while they are and must in fact be totally distinct. The replacements would only respect the compositionality principle, warranting the sameness of the sentences’ references, if the argument could prove that the partial references of all the sentences could be replaced without furtively inviting the reader to conjoin in sentence (3) partial references to completely different objects.
25. Sub-facts and grounding facts
If we take the whole reference of the sentence as not a truth-value but a fact, we get much more intuitive results. In what follows, I will consider Church’s intended derivation, not only to introduce facts as referents of sentences, but also to introduce a very useful distinction between sub-facts and grounding facts. As will be seen, this distinction fills a gap in Frege’s explanation.
     We need to distinguish at least two facts referred to by identity sentences. The first is the sub-fact: it is the perspectival fact as the appearance immediately revealed through a particular mode of presentation expressed by the statement. I will call it a sub-fact and make the diversified sub-facts responsible for differences in the modes of presentation constitutive of the different sentences senses (thoughts, rules) concerning one and the same object, e.g., Walter Scott and the author of Waverley. This is why Church’s sentences (1) and (2) can be seen as expressing different senses or thoughts. They evoke different perspectival sub-facts. They indirectly represent different sub-facts, since (i) being Sir Walter Scott is not the same thing as (ii) being the author of Waverley and (iii) being the man who wrote the 29 Waverley novels altogether… In this way, sentences (1) and (2) respectively show two different sub-facts that contain perspectival objects of reference that as such differ from one another. Using the term ‘being’ to indicate that we are speaking about a matching correlat, the sub-facts represented by:

(1)     Sir Walter Scott is the author of Waverley.
(2)     Sir Walter Scott is the man who wrote the 29 Waverley novels altogether.

Can be respectively represented as follows:

(1a) Being Sir Walter Scott ≠ being the author of the Waverley novels.
(2a) Being Sir Walter Scott ≠ being the man who wrote the 29 Waverley novels altogether.

These sub-facts are of differences since Sir Walter Scott could have not written the Waverley novels or any novel in the first case, and he could have written a different number of Waverley novels in the second. (If you accept that there is a relational trope of identity, you should accept that there are here relational tropes of difference.)
     Nonetheless, it is also clear that (1) and (2) are also identity sentences. Each of these sentences can be understood as referring under different guises to only one object, the person called Walter Scott, justifying the employment of the ‘is’ of identity. In this sense, sentences (1) and (2) represent an identity, which can be directly expressed by ‘Walter Scott = Walter Scott.’ They can represent the self-identity of Walter Scott considered in full, as the ultimate bearer of all descriptions (under all possible perspectives) that we might intend to use to refer to it. Among the descriptions we associate with the name ‘Walter Scott’ we can select ‘the person with the title of Sir named “Walter Scott”’ (that is, ‘Sir Walter Scott’), ‘the author of Waverley’ and, certainly, ‘the man who wrote the 29 Waverley novels altogether,’ that is, the constituent expressions of (1) and (2). Now, this primary fact expressing the self-identity of Walter Scott considered in full is what I call the grounding fact. Characteristic of the grounding fact is that it must be able to unify all the sub-facts, all the facets revealed by its multiple modes of presentation. This is what remiting us to sentences of the form a = a make sentences with the form a = b identity sentences.
     Now, consider one of these definite descriptions more carefully, for instance, ‘the author of Waverley.’ As we saw, the mode of presentation is intentional and internal, considering that the reference can be absent. But when the mode of presentation isn’t empty, as in this case, it also exposes something external, evoking what I could spell out as ‘being the author of the Waverley novels.’ This should be seen as an objective entity, a sub-object mediating our reference to the object Walter Scott that belongs to the grounding fact of identity, namely, that Walter Scott is (the same as) Walter Scott.
     As well, ‘the author of Ivanhoe’ (who was also Walter Scott) is a mode of presentation of the sub-object ‘being the author of Ivanhoe,’ though it ultimately refers to Walter Scott. Now, take the sentence:

(a)     The author of the Waverley novels is the author of Ivanhoe.

This sentence evokes two different sub-objects that together form the contrastive sub-fact that being the author of Waverley is not the same as being the author of Ivanhoe. But this sub-fact also consists of two modes by which the same object is given, whose identity is the grounding fact that can be directly represented by the sentence ‘Being Walter Scott [in full] = Being Walter Scott [in full]’, where ‘in full’ here means that we are considering all the conceivable modes of presentation, beyond the limited knowledge of this or that particular speaker.
     However, this is not all. It seems clear that the sentence (a) must also be able to express the two thoughts representing the two kinds of facts considered. First we have a derived thought expressible by the sentence (a1) ‘Being the author of Waverley novels isn’t the same as being the author of Ivanhoe,’ representing directly the sub-fact and indirectly the grouning fact.Second we have the basal thought directly expressible by the sentence (a2) ‘Being Walter Scott [in full] = being Walter Scott [in full]’ representing the grounding fact directly.[38]
     According to the foregoing analysis, when I say ‘The author of Waverley novels is the author of Ivanhoe,’ I am saying two things. First, by means of intentional modes of presentation I am expressing the derived thought evoking an objective difference. This thought can be expressed by the sentence ‘Being the author of Waverley ≠ (isn’t) being the author of Ivanhoe’, representing a derived fact. Indeed, it is an objective factual difference that someone writing Waverley is not the same as the same person writing Ivanhoe, even if they are both the same person (he has written different stories at different places and times…). However, since when I say ‘The author of Waverley is the author of Ivanhoe I use an ‘is’ of identity, I also mean the basal thought expressible by the sentence ‘The author of Waverley = the author of Ivanhoe,’ indicating that under different guises I am presenting the grounding fact that ‘Being Walter Scott = Being Walter Scott.’ It is because of the two – the grounding fact along with the sub-fact – that identities of the kind a = b are able to express identities in their differences.
     Now, assuming the kind of neo-descriptivism proposed in Appendix I of this book, we can make explicit the above-mentioned doubling of the presented facts by stating each of the four sentences of Church’s reasoning as follows:

(1a) Sentence expressing the derived thought representing the sub-fact: Being Sir Walter Scott ≠ being the author of Waverley.
(1b) Sentence expressing the basal thought representing the grounding fact: Being Walter Scott [in full] = being Walter Scott [in full].

(2a) Sentence expressing the derived thought representing the sub-fact: Being Sir Walter Scott ≠ being the man who wrote the 29 Waverley novels altogether.
(2b) Sentence expressing the basal thought representing the grounding fact: Being Walter Scott [in full] = being Walter Scott [in full].

(3a) Sentence expressing the derived thought representing the sub-fact: (Being 29 ≠ being the number of Waverley novels) & (Being Sir Walter Scott ≠ being the man who wrote the 29 Waverley novels altogether).
3b) Sentence expressing the basal thought representing the grounding fact: (Being 29 = being 29) & (Being Walter Scott = being Walter Scott).

(4a) Sentence expressing the derived thought representing the sub-fact: Being 29 ≠ being the number of counties in Utah.
(4b) Sentence expressing the basal thought representing the grounding fact: Being 29 = being 29.

The sub-facts show why the semantic contribution of each referential component in identities with the form a = b, due to the semantic-cognitive rules constitutive of the derived thought, can be different. The sub-fact that being Sir Walter Scott isn’t the same as being someone who wrote 29 Waverley novels discriminates more than the sub-fact that being Scott isn’t the same as someone writing the Waverley novels. And regarding true sentences, this discrimination isn’t just a mentally considered mode of presentation, a cognitive rule, but also the representation of something objectively or factually given in the external world (corresponding to different ‘ways the object gives itself to us,’ using Frege’s words). The evocations of these sub-facts all lead us to a grounding fact of identity showing that in the end different senses referring to primarily qualitatively different phenomena refer secondarily to something numerically identical. On the other hand, in sentences with the form a = a, such as ‘the morning star = the morning star,’ the sub-fact is already an identity ‘Being the morning star = being the morning star.’ The corresponding grounding-fact, additionally, may also be the same identity, if not the identity ‘Being Venus = being Venus,’ depending on the speaker’s intention.
26. Taking seriously the sentence’s reference as a fact
I think I have shown that the most plausible option concerning the nature of reference is to side with philosophers like Russell and the earlier Wittgenstein. These philosophers assumed that the reference of a statement is a fact – a fact that in the usual case is understood as a contingent arrangement of cognitively-independent tropical components commonly given (completely or partially) in the external world, although they can also belong to an internal (psychologically accessible) reality. Facts would satisfy the Fregean condition that the reference of a sentence is an object: they are in some sense independent, complete, closed. They would satisfy his condition that thoughts expressed by sentences are modes of presentation of their references, the latter – particularly as sub-facts – being as numerous and diverse as their thoughts. Finally, unlike truth-values, facts would clearly satisfy the principle of compositionality: they would always vary in accordance with variations in the references of the senses of component parts of the sentences as we understand them.
     If we assume the answer given above, we are able to solve a vexing problem concerning which fact the thought expressed by a sentence refers to. Consider the following sentences:

1.     The morning star is the morning star.
2.     The morning star is the evening star.
3.     Venus is the morning star.
4.     Venus is the second planet orbiting the Sun.
5.     Venus is the brightest planet visible in the sky.
6.     Venus is the only planet in our solar system shrouded by an opaque layer of highly reflective sulfuric acid clouds.
7.     The morning star is the only planet in our solar system shrouded by an opaque layer of highly reflective sulfuric acid clouds…

On the one hand, it is intuitively correct to say that each of these sentences refers to a different fact. Sentence (1) is tautological, proclaiming the self-identity of the morning star, while sentences (2) to (7) provide information on different factual contents regarding the planet Venus. On the other hand, since all singular terms composing these identity sentences have the same ultimate reference, the planet Venus, it also seems clear that in the end all these identity sentences must have the same reference, representing the same fact in the world. How can we reconcile these two seemingly correct views?
     The answer departs from the distinction already made in the last section: first, there must be a privileged grounding fact able to be described that can be identified as the ultimate truth-maker of all these identity sentences about the planet Venus. Second, this grounding fact must in some way contain the facts immediately indicated by the different cognitive values of sentences (1) to (7) above as its perspectival sub-facts. My suggestion is that this last task can be accomplished by the references of identity sentences, insofar as the identification rules of their singular terms are considered in full, including all their fundamental and complementary descriptions.
     Now, assuming our proposed view of proper names’ meanings as abbreviations of bundles of descriptions centered in those constituting their fundamental identification rules, then the proper name ‘Venus’ in full includes in its most complete content all the already known modes of presentation. This means that definite descriptions such as ‘the morning star,’ ‘the second planet orbiting the Sun,’ ‘the brightest planet visible in the sky,’ etc. can be made at least probable by applying the concept of Venus in full. (I say ‘made at least probable’ because in the case of the application of most identification rules, any particular description-rule of the bundle might remain unsatisfied.) If this view is correct, then there is only one sentence that could describe the grounding fact as the ultimate truth-maker or verifier of any identity sentence concerning the planet Venus, including the sentences from (1) to (7) above. We can present it as the grounding fact (8) that being Venus with all its known sub-factual identificational implications is being Venus with all its known sub-factual identificational implications, represented by the basal thought expressed by the sentence:

(9) Venus [in full] = Venus [in full]

My contention is that rightly understood this sentence summarizes the most complete basal thought able to represent the single grounding fact, which considered in its entirety can be regarded as the truth-maker for any identity sentence about the planet Venus. (To represent sub-facts we have the already called derived thoughts.)
     It is not hard to explain why things are so. If the full meaning of the proper name ‘Venus’ is understood as an abbreviation of the whole bundle of descriptions regarded as uniquely identifying its object (Cf. Appendix of Chapter I, sec. 4), then this proper name should include descriptions like ‘the morning star,’ ‘the evening star,’ ‘the second planet orbiting the Sun,’ ‘the most brilliant planet visible in the sky,’ ‘the only planet in our solar system shrouded by an opaque layer of highly reflective sulfuric acid clouds,’ and many others. Consequently, from the sentence ‘Venus [in full] = Venus [in full]’ we can inferentially derive sentence (2) ‘The morning star = the evening star.’ We do this simply by replacing the first occurrence of the name ‘Venus’ with the definite description ‘the morning star,’ which the name ‘Venus’ (in full) abbreviates, and the second occurrence of the name ‘Venus’ (in full) with the description ‘the evening star,’ which the name Venus also abbreviates. In a similar way, we can obviously (inductively, at least) infer all the other above presented co-referential identities from (1) to (7). Thus, rightly understood the sentence ‘Venus [in full] = Venus [in full]’ should express the basal thought able to represent a fact complex enough to comprehend all the sub-facts represented by each of the thoughts expressed by the above sentences, which may be seen here as contingent a posteriori. (To convince yourself of this, look at the meaning of ‘Venus’ as presented in any encyclopedia, since it aims to offer an abbreviation of Venus in full.)
     In order to better support what I am suggesting, I can also use numerical identities like the following:

1.       2 + 2 = 2 + 2
2.       2 + 2 = 1 + 1 + 1 + 1
3.       2 + 2 = 4
4.       4     = √16
5.       2 + 2 = (14 – 6) / 2

Of course, here the identity sentence expressing the basal thought representing the grounding fact would be:

6.     The number 4 [in full] = the number 4 [in full]

But could the sub-facts expressed by sentences (1) to (5) be derived from (6)? Obviously, the answer must be in the affirmative, since we are dealing with a deductive system. After all, I wrote the five sentences above simply based on deductive inferences from my knowledge of the grounding fact that being the number 4 = being the number 4!
      However, one could still object that a sentence like ‘Venus [in full] = Venus [in full]’ is a tautology: a necessary truth. How could a necessary truth ground contingent truths like, ‘Venus is the brightest planet visible in the sky’?
     My answer to this objection is that for a privileged user of the word (a Venus specialist) who is supposed to know all the relevant information about Venus, this proper name expresses an identification rule that can be approximatively summarized as follows:

IR-Venus: Our proper name ‘Venus’ has a bearer, iff this bearer belong to the class of celestial bodies that satisfy sufficiently and more than any other the condition of being the second planet orbiting the sun between Mercury and the Earth at the time of its identification. (To this it is helpful to add very probably applicable auxiliary descriptions like ‘the brightest planet visible in the sky,’ ‘a planet somewhat smaller than the earth,’ ‘the morning star,’ ‘the evening star,’ etc.)

As already explained (Appendix of Chapter I, sec. 10 (iii)), this is a kind of ‘one-foot’ identification rule, since the localizing rule is the only fundamental one and includes what would count in the characterizing rule (being a planet). For suppose we have as a characterizing rule ‘a bright planet somewhat smaller than the earth.’ In this case, one can imagine that if there were only one bright planet somewhat smaller than the Earth, this planet would be Venus, since one term of the inclusive disjunction of fundamental identifying rules is already satisfied. But if this were true, since we can imagine a possible world where there is just one bright planet somewhat smaller than the earth with an orbit outside the Earth’s and no second planet, this planet should then be Venus, what is absurd. On the other hand, the localizing rule contains the essential content of the characterizing rule: Venus is a planet. If Venus were to lose its atmosphere or a major share of its mass (or in a different possible world never had them), insofar as it had been discovered to be the second planet from the Sun and the Earth the third, it would still be our Venus! Indeed, so understood it seems that the identification rule for Venus is applicable in any possible world where the planet Venus can be said to exist or to have existed.
     The case of Venus is somewhat like the case of the lines ‘aᴖb-aᴖc’ drawn to localize the center of a triangle without any call for a characterizing property; the characterizing description can be irrelevant or non-existent. By the same token, without the localizing condition established by the identification rule of Venus as the second planet, it would be impossible to identify Venus. The application of many other descriptions does not produce criteria, but only symptoms of the planet’s existence, since they make the applicability of the descriptions only more or less probable. Auxiliary descriptions like ‘the brightest planet in the sky’ are symptoms, like ‘the highly reflective clouds of sulfuric acid’ that cause this brightness. If Venus lost its reflective atmosphere, it might cease to be the brightest planet, but would still not cease to be Venus. If Venus lost half of its mass but remained in the same orbit, it still would not cease to be Venus. But if for some reason Venus lost nearly all its mass and became a small orbiting object only a few miles in diameter, no longer large enough to be called a planet, we could only say that it once was Venus. If in a possible world Mercury never existed, Venus would be the first planet of the solar system and even if it were called ‘Venus,’ it seems clear that it would not really be our Venus, unless it had once been the second planet from the Sun (Venus) for at least some period of time. Indeed, if in another possible world the second planet were hurled out of the solar system thousands of years ago (Kripke 1980: 57-58), it could still rightly be recognized as our Venus, since it once satisfied its identification rule. We see that the condition of sufficiency applied to the one-foot identification rule of Venus is more demanding than in the usual two-foot case. We can see that limits can be set even in a swampy terrain where vagueness prevails.
     What I said about identity sentences also applies to other singular predicative and relational sentences. Consider the following sentences:

1. Bucephalus was a material thing.
2. Bucephalus was a living being.
3. Bucephalus was a horse.
4. Bucephalus was a black horse of the best Thessalonian strain.
5. Bucephalus was a massive black horse of the best Thessalonian strain, owned by Alexander the Great.
6. Bucephalus: (355 BC – 326 BC) was the most famous horse of Antiquity; it was a massive black horse of the best Thessalonian strain, owned by Alexander the Great.
7. Bucephalus once swam across the river Granicus.

One could say that each of the first six sentences express different derived thoughts representing different sub-facts by means of increasingly detailed modes of presentation expressed by their respective predicative expressions. However, relative to them there is a grounding fact that in a summarized form is represented by the basal thought expressed by sentence (6), since the truth of all the others can be implied by the truth of this thought. Indeed, (6) is nothing but an abbreviated expression of the identification rule for Bucephalus, with a localizing and a characterizing description and by these means furnishing a summarized definitional criterion. The sub-facts represented by sentences (1) to (5) are all included in the grounding fact represented by sentence (6). These facts are the immediate satisfiers of the diverse modes of presentation of Bucephalus given by each sentence. And the progression from (1) to (6) increases the complexity, insofar as new relevant predications are added. Statement (7) ‘Bucephalus once swam across the river Granicus’ is a different case: the very contingent auxiliary description ‘the horse Bucephalus who once swam across the river Granicus’ isn’t a relevant part of the fundamental description-rule (even if he didn’t crossed, he would still be our Bucephalus). Nevertheless, it can still be derived from (6) considered in full, since this is believed (by privileged speakers) to be historically the case.
27. The riddle of identity in difference
There is a final point concerning the relationship between the sub-fact and the grounding fact. It concerns the unsatisfactory way that Frege solved the puzzle of identity. As he wrote, unlike sentences with the form a = a, a sentence with the form a = b is informative because it refers to the same object by means of different modes of presentation, by means of the different senses of a and b (1892: 26). However, we can still ask how this identity is possible, since the modes of presentation are different and since we are not intending to speak about the mere self-identity of the reference, as Frege also acknowledged. I call this ‘the riddle of identity in difference.
      To see the problem clearly, consider again Frege’s sentence (i) ‘The morning star = (is) the evening star.’ A more fully unpacked cognitive sense of (i) can be presented as:

The brightest star in the morning sky, understood as referring to the second planet orbiting the Sun between Earth and Mercury (Venus) = (is) the brightest star in the evening sky, understood as referring to the second planet orbiting the Sun between Earth and Mercury (Venus).

Here I have not [j1] underlined expressions of what I call non-definitional immediate senses containing perceptual sub-objects like the morning and the evening star, though I have underlined expressions of what I call mediated senses, which here are definitional. The immediate senses build the derived thought representing a sub-fact (that being the morning star isn’t being the evening star), while the mediated senses essentially build the basal thought representing the grounding fact (that Venus is Venus). Here we have the hidden reason for the riddle of identity in difference: the immediate senses of the expressions flanking the identity sign in (i) are obviously different, but they both evoke the underlined mediated, in fact primary or leading sense (essentially building the basal thought that Venus is Venus) that is identical.
     Obviously, this last sense, the basic thought that the second planet orbiting the Sun… is the second planet orbiting the Sun… is not yet the reference, since it is constituted by the expression of the self-identification of the cognitive identification rule constituting the core definitional sense of the name ‘Venus’ and its conventionalized surroundings (Venus in full). It is only because both expressions flanking the identity sign in (i) implicitly evoke the same proper identification rule for the planet Venus that we are allowed to place an identity sign between them! In order to make the point still clearer we can appeal to the following schema:

Sentence:              The morning star                 = (is)       the evening star.
Derived:                IR: the brightest                   ≠              IR: the brightest
thought:                star in the morning                              star in the evening.
sub-fact:                Being the morning star isn’t being the evening star

Basal                      IR: The second planet…    =              IR: the second planet…
thought                    (Venus)                                                  (Venus).
grounding fact:   Being Venus is the same as being Venus.

In sum: the singular terms ‘Morning Star’ and ‘Evening Star’ are responsible for the difference present in what I call the immediate senses of the descriptions (the Fregean senses) constituting a derived thought evoking a relational sub-fact showing the differences between two sub-references. Expressing the derived thought we describe the sub-fact as: ‘being the brightest star seen in the morning sky differs in place and time from being the brightest star seen in the evening sky’ (one can even point to the two opposite sides of the sky in which alternately one or the other appears every twelve hours). Furthermore, the word ‘is’ (the same as) is the only indication of the identity of the implicitly intended mediated senses building the basal thought expressed by the sentence ‘The second planet orbiting the Sun between Earth and Mercury (Venus) = the second planet orbiting the Sun between Earth and Mercury (Venus).’ These mediated senses have multiple guises that are implicit in the names flanking the identity sign in the statement ‘Venus [in full] = Venus [in full]’ expressing the basal thought that could be known in full only by specialists. The statement expressing the derived thought is contingent a posteriori, while the statement expressing the basal thought can be seen as a necessary priori.
     A somewhat different example is the sentence ‘The morning star is Venus.’ Here the schema is:




Sentence:              The morning star                 = (is)       Venus.
Derived: IR: the brightest                   ≠              IR: the second
Thought:               star at dawn                          planet from the Sun.
sub-fact:                Being the morning star isn’t being Venus.
Basal                      IR: the second planet        =              IR: the second planet
Thought:               (Venus)                                                  (Venus).
grounding fact:   Being Venus is the same as being Venus.

It is by now clear that the identity expressed by sentences of the kind a = b is an identity in difference. This means that in fact we have two levels of sense or thought. The first is the derived thought. It represents the perspectival sub-fact with its sub-objects expressing a difference (Being the morning star isn’t the same as being the second planet from the Sun). The second, intermediated by the first one and indicated by the ‘is’ of identity, is the basal thought representing the ultimate grounding fact that being Venus is the same as being Venus, which has the sub-facts as facets, as manifestations. The derived thought is contingent a posteriori, while the basic thought is necessary a priori, since it is tautological.
     Now, how should we deal with cases in which the elements of the basal thought responsible for the identity, like the planet called ‘Venus’ in the statement above, lack a proper name? Consider the identities (i) ‘Everest = Chomolungma,’ (ii) ‘aᴖb = aᴖc’ (concerning Frege’s example of two different ways to name the center of a triangle), (iii) ‘Afla = Ateb’ (the two names that Frege gave for the same imaginary mountain). How should we deal with them?
     I would answer first by considering that the derived Fregean senses are thoughts of a difference, evoking different contingent sub-objects. But these sentences also implicitly evoke a basal conjoining sense, a conjoining identification rule, which refers to what we might call respectively the ‘Everest-Chomolungma,’ the ‘aᴖb-aᴖc,’ and the ‘Afla-Ateb,’ which in fact are three new nominative expressions. The law of identity makes obvious that:

(1) ‘Mt. Everest is Chomolungma’ can be replaced by ‘Everest[-Chomolungma] = [Everest-]Chomolungma,’
(2) ‘aᴖb = aᴖc’ can be replaced by ‘aᴖb[-aᴖc] = ‘[aᴖb-]aᴖc,’ and
(3) ‘Afla = Ateb’ can be replaced by ‘Afla[-Ateb] = [Afla-]Ateb.’

These three replacing basal thoughts respectively represent the three different grounding facts as the full self-identities that they are. This is respectively what sustains the identities expressed by the ‘is’ in the sentences (i), (ii) and (iii).
     We can apply a similar analysis to identities between concept-words of the form (x) (Fx = Gx). Consider the identity ‘Heat in gases is molecular kinetic energy.’ Note that the word ‘heat’ is ambiguous. It can mean a mere subjective feeling (heat1), like the feeling of increased bodily heat after exercise, which cannot be identified with molecular kinetic energy. But in the present case, ‘heat’ means external temperature as it is normally felt by people (heat2). A third sense is independent of our sensations: it is heat as ‘measured temperature’ determined by thermometers (heat3) (in the sense of heat2, our bodies work as course, imprecise thermometers). Moreover, since molecules can have different masses and speeds, the most precise identity sentence would be ‘Temperature in a gas (heat3) is the average kinetic energy of its molecules.’ This sentence expresses two different modes of presentation of the same thing, that is, a derived thought that can be expressed by means of the following difference:

(i)    Temperature in a gas (heat3) ≠ average kinetic energy of its molecules.

This secondary thought refers only to the sub-fact that the (macro-physical) temperature that we can measure with a thermometer (and feel as heat2) is something phenomenally different from the (micro-physical) average kinetic energy of the molecules of a gas such as the air around us.
     In a next step, we are able to consider the basal thought establishing a tautological identity based on conventions. This thought can be expressed by the whole more complete assertive sentence:

(ii)   [Average kinetic energy-temperature-] heat3 of a quantity of gas = average kinetic energy-temperature [-heat3 of a quantity of gas].

     We can exceptionally read (ii) in two ways, (a), considering only what is outside the brackets as explicitly emphasized, which expresses the derived thought of a difference and represents the sub-fact of the difference above. But we normally read (ii) as (b) emphasizing the whole, including what is in brackets. So understood, this is a basal thought referring to a grounding fact of definitional self-identity. This identity requires as an assumption the acceptance of the kinetic theory of gases, which makes (b) a tautology. This means that if we read (ii) in the sense (a), disconsidering what is in the brackets, we can see it as a contingent a posteriori thought, since it can be denied without contradiction, while if we read (ii) in the sense (b) it can be considered necessary a priori, since it cannot be denied without contradiction.
     Consider now the sentence ‘Water is H2O.’ I think Avrum Stroll was right when he noted that here the ‘is’ expresses constitution; the sentence means, ‘Water is made of H2O’ rather than ‘Water is the same as H2O’ (1996, 46 f.). However, this does not make a relevant difference for what I am trying to say and context can lead us easily to read again the ‘is’ as expressing identity.
     As already noted (Appendix to Chapter II), the concept-word ‘water’ has two nuclei of meaning: a superficial one, that of an aqueous liquid (transparent, tasteless, odorless, etc.), and a deep one, a substance called by chemists dihydrogen oxide or H2O (which includes much more than the simple chemical structure). This means that the complete sense of water must include the two nuclei. However, as in fact the presence of only one nucleus already allows us, in a proper context, to call the substance water, the most embracing criterial rule for the application of the general term ‘water’ demands sufficient satisfaction of the (summarized) inclusive disjunction:

(Water is an) aqueous liquid and/or (water is) dihydrogen oxide (H2O).

Philosophers have created a pseudo-problem by insisting that the criterion of application of the conceptual word ‘water’ must be either aqueous liquid or dihydrogen oxide, as if it were a dilemma.[39]
     Now, assuming that the ‘is’ is one of constitution and not of identity, the statement (i) ‘Water is H2O’ in fact means: (ii) ‘Aqueous liquid and/or dihydrogen oxide… is made of dihydrogen oxide.’ Since it could be that water isn’t made of dihydrogen oxide and only the first statement of the disjunction is true, it is possible for the whole statement to be false, which makes it a contingent a posteriori truth and not a necessary a posteriori truth, as Kripke would like it to be. However, as we will see in the next section, in some contexts statement (i) is rather seen as a necessary a priori truth.
28. Contexts of interest: no need for a necessary
 a posteriori
This double core sense of the general term ‘water’ helps to explain Saul Kripke’s in my view as much insightful as illusory discovery of the necessary a posteriori. But in order to better understand the confusions involved, we need to add to the sentences the contexts in which they are spoken.
     A first point to notice is that in the case of a sentences of the kind a = b uttered in different contexts we can enhance or magnify or emphasize its immediate (Fregean) perspectival sense that builds a derived thought (representing a sub-fact), or we can enhance or magnify or emphasize its mediated sense that builds the basal thought (representing a grounding fact).[40] Thus, in cases like ‘Water is H2O’ we can emphasize the immediate core sense of the concept-word ‘water’ as an aqueous liquid or its mediated core sense as dihydrogen oxide. Here I need to speak again of the contexts of interest of the linguistic agents, meaning thereby contextualized practical aims from which we can infer what is meant.
     Two contexts of interest are important regarding the main examples above: the popular context of interest and the scientific one. Thus, considering the sentence ‘The morning star is the evening star,’ we can contextually emphasize the derived thought composed by immediate senses (modes of presentation, identification rules) representing the external, phenomenally given objects, considering the difference between being the brightest star in the morning and the brightest star in the evening. If we do this, we leave the identity ‘Venus = Venus’ in the background. This can be the case, for instance, when contemplating the beauty of the starry sky at night and, after localizing the evening star, we tell a child that it is also the morning star. In this case, we think like Frege. We emphasize the different modes of presentation of the same object, a difference that as such represents nothing but an empirical sub-fact made by two different aspectual presentations of the same thing. We regard the thought that the morning star is the evening star as contingent a posteriori, since it mainly represents the sub-fact of the difference, although we are also aware that we are emphasizing the different ways by means of which the same thing presents itself to us.
     Nonetheless, in a scientific context of interest, such as one in which astronomers use a telescope to study the surface of Venus, when they consider the sentence ‘The morning star is also the evening star,’ what they usually have in mind and emphasize is the numerical identity of the object of both modes of presentation. These are the mediated senses constituting the basal thought representing the grounding fact of the self-identity of Venus, which Kripke particularly emphasized in his writings. In this case, we read the statement as preferentially meaning the basal thought that ‘Venus [in full] = Venus [in full],’ which is a necessary a priori statement, since it above all affirms the tautological grounding fact that being Venus is the same as being Venus. It leaves the different guises of sense in the background, as secondary effects, insofar as we assume the truth of our scientific astronomical views.
     Now, consider again the statement ‘Water is H2O’.[41] In a popular context of interest which arises when fishermen decide to dig a well to obtain fresh water for drinking and washing, this statement is read as emphasizing the sense of the word ‘water’ as a precious aqueous liquid (transparent, tasteless, odorless, drinkable… the popular nucleus of meaning), and it is for them a contingent matter that it is made of H2O insofar as it satisfies their practical aims. Because of this, the statement is seen as contingent a posteriori, since it means ‘This aqueous liquid is made of H2O,’ expressing a derived thought representing preferentially a sub-fact that does not demand that water is necessarily H2O, being deniable without contradiction.
     On the other hand, when the context of interest is scientific, for instance, formed by chemists measuring the acidity of a sample of water, the word ‘water’ in the sentence ‘Water is H2O’ can be read as emphasizing the sense of water as dihydrogen oxide (the scientific nucleus of meaning). In this case, the whole sentence is seen as preferentially expressing a thought representing a grounding fact expressed by the identity ‘Dihydrogen oxide = H2O,’ which is the same as ‘H2O = H2O,’ that is, a necessary a priori tautology based on our scientific assumptions.
     I think that philosophers like Kripke, by considering ‘Water is H2O’ a necessary a posteriori statement, simply confuse (i) the a posteriority of the statement which emphasizes that water is an aqueous liquid made of H2O with (ii) the a priori necessity of the statement that emphasizes that water as dihydrogen oxide is the same as H2O, mixing the a posteriority of (i) with the necessity of (ii).
     A somewhat different emphasis can be found in the statement ‘Heat is molecular movement,’ here understood as ‘Heat = molecular movement.’ If we emphasize the ordinary immediate senses, the derived thought, the difference between heat2 (heat as it is normally felt) and the average kinetic energy of a gas, the sense emphasized is contingent a posteriori, and the fact referred to is something learned by experience. This could be the case even using heat3 (temperature) as a fallible measure of average kinetic energy.
     On the other hand, if we assume the truth of the kinetic theory of gases in a scientific context in which we are measuring temperatures, the statement can be understood as emphasizing the mediated sense of the identity expressible by: ‘Temperature of a gas [-average kinetic energy] = average kinetic energy [-temperature of a gas],’ which is read here as expressing the basal thought representing the grounding fact of an assumed identity, being therefore a necessary a priori thought. In this reading, our conceptual rules for temperature and for average kinetic energy are blended into a single identification rule which assumes the kinetic theory of gases.
     It seems clear to me that in identities of the kind a = b, Kripke misleadingly conjoined the a posteriority of the emphasized derived identity thought with the necessity of the emphasized basal identity thought, concluding that the identities between nominal and conceptual terms have a necessary a posteriori nature that is only metaphysically explicable. However, if these names or concept-words serve as rigid designators applying to the same entities in all possible worlds, this is explained by their assumed mediated senses, which are of the kind a = a (or a[b] = [a]b) and not a = b. A Wittgensteinian therapist would conclude that in the considered cases Kripke was the victim of deep grammatical ambiguities. Finally, insofar as the terms a and b used in identity sentences are viewed as rigid designators applying to the same ultimate object in all possible worlds where it exists, this is also only justified by the self-identity of a grounding fact.
29. Sense of a sentence: the thought
Now it is time consider the sense of a sentence. Here is Frege at his best! He made the right decision in suggesting that the meaning of the whole sentence is the thought (Gedanke) it expresses. To reach this conclusion, he applied his compositionality principle: combined in the right way, the senses of the component terms constitute the sense of the whole sentence. If, for instance, in the sentence ‘The morning star is a planet’ we replace the description ‘the morning star’ with the description ‘the evening star,’ which is co-referential though having a different sense, the reference of the sentence does not change; but the sense of the sentence must change. Indeed, the sense of the sentence ‘The evening star is a planet’ is different. However, the only other thing that has changed is what we call the thought expressed by the resulting sentence. Consequently, the sense of a sentence must be the thought it expresses. (Frege 1892: 32)
     The word ‘thought’ is ambiguous. One can use it to describe a psychological process of thinking, as in the utterance ‘I was just thinking of you!’ But it also seems to designate something independent of specific mental occurrences – a content of thought – such as the thought expressed by the sentence ‘12 x 12 = 144’ in the utterance: ‘The sentence “12 x 12 = 144” expresses a true thought.’ Frege had the latter sense in mind. In this usage, the word ‘thought’ means simply what the sentence (statement) says, which Frege conceived of as some sort of eternal (timeless) Platonic entity. The terminology here counts, because the word ‘thought’ is the only term in ordinary language that has a sense corresponding to more technical terms like ‘proposition’ or ‘propositional content.’[42]    
     Frege has a criterion for deciding what belongs to a thought. For him everything that contributes to determining the truth-value of a sentence should belong to its thought. Thus, using his own example, the sentences ‘Alfred hasn’t arrived’ and ‘Alfred hasn’t arrived yet’ express the same thought, since the word ‘yet’ means only an expectation regarding Alfred’s arrival without contributing to the sentence’s truth-value (Frege 1918: 64). The sentences ‘The morning star is Venus’ and ‘The evening star is Venus’ can be considered to express different thoughts, because although the singular terms that make up these two identity sentences all refer to the same planet, they do this by means of different modes of presentation. That is, they make us follow different paths in determining their truth-value, or, as I prefer to think, they make us follow different associations of semantic-cognitive rules able to constitute correspondingly different verifiability procedures.
30. The thought as the truth-bearer
Another quite plausible Fregean thesis was that the primary bearer of truth is not the sentence, but rather the thought (proposition) expressed by it. I agree with this view. Although we can say that sentences, beliefs and even things and persons are true, they all seem to be true in a derivative sense.
     Consider the cases of things and persons. A useful test to identify secondary uses is that when a word is derivatively used we can often replace it with a more appropriate word. If we say that a diamond is false, what we mean is that it is only an imitation diamond: a fake or counterfeit of a real diamond that deceives us so much that we can think false thoughts about it. When we say that Socrates was ‘true’ as a person, what we mean is that he was a truthful, trustworthy or reliable person, someone with integrity. But this is not always so. When we say that Sam’s belief is true, we mean firstly a subjective psychological attitude of the believer concerning a (dispositional) thought that happens to be true, which leads us again to the truth of a thought in a Fregean sense.
     One reason for preferring to say that the thought is the truth-bearer concerns the logical behavior of this concept. We deal with our concept of truth as an ‘as if’ directive idea, so that the real or actual truth-value of a thought is naturally conceived of as something invariant: if something is (really) true, it is always true; if something is (really) false, it is always false. Obviously, we can always err in claiming something to be true (das Fürwahrhalten) and can later discover it is false, and we can err in believing something to be false (das Fürfalschhalten) when it is actually true – this is often the case, and this possibility is inevitable, due to our inherent epistemic fallibility. But when we discover our error, we correct ourselves, in the first case not by claiming that the thought was previously true and now has become false, but by saying that it was always false, and in the second case we correct ourselves not by claiming that the thought was previously false and now has become true, but by saying that it was always true. What changed was our truth-claim, not the truth-value. Moreover, it is fundamental to perceive that our inherent fallibility in holding thoughts to be true does not affect the invariability or immutability of the truth-value of the thought or proposition taken as a directive idea. It must be so, because it is beyond our fallible capacities to know with absolute certainty whether we have achieved this ideal, if we have indeed achieved it. This is how the logical grammar of our concept of truth works (and beyond this, of our concept of knowledge). If one wants to change something so fundamental, then to prevent confusion I propose to invent new terms instead, like ‘hturt’ and ‘eslaf’.
     Now, if the actual truth-value is immutable, its truth-bearer must also be unchanging, able to remain the same in order to retain this same truth-value independently of the time or place where we discovered it. Indeed, for Frege a really true thought remains true forever, just as a really false thought remains false forever. These entities are even abbreviated as ‘truths’ and ‘falsities’ respectively. Thus, it is deeply ingrained in our conceptual grammar that the entity that can be primarily called true or false must remain the same and possess the same truth-value, so that what may change is only our cognitive grasp of it, our believing in its truth-value (das Fürwahrhalten). If this is so, then only the thought has the necessary stability to be a proper truth-bearer; for a thought is, according to Frege, unchangeable and eternal (a-temporal), being eternally (a-temporally) true or false independently of our grasping (fassen) it.
     Consider now the case of sentences as candidates for truth-bearers. Ambiguous sentences can express different Fregean thoughts, such as ‘John saw the man on the mountain with a telescope.’ In this case, the truth-value of the thought will be able to change according to the different thoughts or interpretations that we assign to the sentence. But if the truth-bearer were the sentence, the truth-value should remain the same, which cannot be correct. This is obvious in the case of indexical utterances like ‘I am in pain,’ which has different truth-values depending on the speaker.[43] The same sentence can change its sense-thought when uttered by different persons, and even when uttered by the same person at different times; correspondingly, what may change with the change in thought is the truth-value. Hence, thoughts and their truth-values are co-variant, while sentences and their truth-values are not, which leads us to the conclusion that the primary bearer of truth-value must be the thought or proposition.
     One could suppose that perhaps the sentence-token would be the truth-bearer, since it would be a different one depending on the time and place of the utterance, changing with the truth-value. However, we still have cases in which different sentences (token or not) say the same thing – express the same thought – in this way preserving the same truth-value. Consider, for example, the following statements, ‘It is raining,’ ‘Il pleut,’ ‘Es regnet,’ ‘Chove’… uttered in the same context. They all say the same thing, express the same thought, and all have the same truth-value, while their sentence-tokens are quite different. Indeed, the only justification for insisting on the immutability of the truth-value of these four different sentence-tokens (and types) is that their primary truth-bearer is the thought expressed by them, since what they say – their senses, their thoughts – is what remains the same. Finally, this is the case not only for indexical sentences, but also for synonymous eternal sentences expressed in the most diverse languages.
     Likewise, beliefs, understood in a psychological sense, can only be derivative truth-bearers: if someone who believes something dies, his personal belief also ceases to exist. Consequently, the truth-bearer must be the content of his belief. It must be his belief-content and not his belief in a dispositional psychological sense, since a belief-content remains, even after death... But this is so only because we understand the belief-content as the same as a Fregean thought, a propositional content.
     The core of the foregoing arguments can be summarized as follows: thoughts and their truth-values are not just invariantly related; when thoughts vary, they maintain a relationship of co-variance with their truth-values. This relationship is missing in the relationships between sentences or psychological beliefs and their truth-values. Because of this, the proper bearer of truth must be the thought (proposition, propositional content, belief-content), not the sentence or a psychological disposition to agree on a truth-value.
31. Facts as true thoughts?
As already noted, Frege also proposed that what we call a fact is the same thing as a true thought, because when a scientist discovers a true thought, he claims to have discovered a fact. As he writes:
‘Facts! Facts! Facts!’ exclaims the researcher of nature, when he wants to proclaim the need for a secure basis of science. What is a fact? A fact is a thought that is true. (Frege 1918: 74)
Indeed, when we say ‘John stated several relevant facts in his speech,’ we are speaking about facts as true thoughts. However, there is no warrant that this is not a derivative use of the word ‘fact.’ A researcher of nature can well exclaim ‘Facts! Facts! Facts!’ understanding by a fact simply what corresponds to the true thought, namely, some objectively given tropical arrangement. After all, it seems natural to think that if someone discovers a true thought, it is because he has a fortiori discovered the fact corresponding to it.
     A more decisive argument against thoughts as true facts came from J. L. Austin, who made it clear that Frege’s identification does not resist all linguistic replacements (1990: 170-171). If the sentence ‘What he affirms is true’ had the same sense as ‘What he affirms is a fact,’ then the replacement of ‘what he affirms’ with ‘his affirmation’ should be allowed without any change of sense. But, ‘His affirmation is true’ preserves the meaning, while ‘His affirmation is a fact’ makes sense only as a meta-linguistic sentence referring to the occurrence of his affirmation, and not to the content of the affirmation itself. The reason for this can only be that the true content of an affirmation – the Fregean thought – cannot be properly identified with a fact.
     The main reason why Frege believed that a fact is a true thought is that he advocated a conception of truth as redundancy, rejecting the correspondence theory. However, on the one hand, his arguments against correspondence or theory (Frege 1918: 59-60) are unconvincing.[44] On the other hand, correspondence theory remains the most influential view. It is the most natural and historically influential conception of truth, suggesting that propositions or thoughts are true when they correspond to facts as arrangements of elements in the world (Rasmussen 2014; Vision 2004). Moreover, the view of truth as correspondence is commonsensical, agreeing with our methodological principle of the primacy of common knowledge. Because of this, I will defend this theory in the last chapter of this book.
     Finally, I think I have found a plausible way to explain why some are tempted to say that facts are true thoughts. It seems that the source of confusion resides in a persistent ambiguity of our own natural language. Dictionaries in very different languages present us a variety of trivial meanings for the word ‘truth.’ However, two general meanings are always emphasized. I call them: thought-truth and fact-truth. Here are their definitions, according to the best dictionaries:

(a)  Thought-truth: Truth as consisting of things being as we believe they are, as conformity or accordance or correspondence of the thought with the fact it represents.
(b)  Fact-truth: Truth as the actual, real, existing fact in the world.[45]

It is regarding the philosophically most proper sense (a) that we have singled out the thought as the primary bearer of truth. This is shown clearly in sentences like ‘His words are true,’ ‘Tell me the truth.’ In the factual sense (b), we single out facts in the world as secondary truth-bearers in the sense of being real, and we use sentences like ‘The mentioned occurrence was true (was real),’ ‘We are searching for the true facts (the real facts),’ ‘The truth (the fact) is out there.’
     As we have already seen, there are good reasons to think that sense (a) is primary while sense (b) is derivative, since in this last case we can replace the word ‘truth’ with more adequate ones like ‘reality,’ ‘existence,’ ‘actuality’… Anyway, ‘truth’ is very often used not only as ‘correspondence with facts’ but also replacing ‘an existing fact in the world.Thus, we can easily be misled by some extraneous motivation and confuse the two usages, mistakenly concluding that facts are true thoughts. This is what seems to have originated Frege’s confusion, giving us another example of equivocity as a common way of transgressing the internal limits of language (Ch. III, sec. 11).
32. The thought as a verifiability rule
As the application of the ascription rule (sense of the predicate) is subsidiary to the application of the identification rule (sense of the nominative term), the rule for applying the singular sentence (its sense or thought) can be seen as an association of semantic-cognitive rules. I identify this association with what Ernst Tugendhat called the verifiability rule of the statement (1976: 259, 484, 487-8). However, if the thought is an association of rules, then what results from such an association – the verifiability rule – must also have the character of a rule, even if it isn’t something previously conventionalized. Combining this with our acceptance of the correspondence view of truth, this means that the thought should be a kind of associated or combined semantic-cognitive rule – a verifiability rule – whose function is to make us aware of a corresponding fact to which it is applied.[46]
     This reasoning unavoidably leads us to the controversial idea of ‘verificationism,’ more precisely (and still worse) to semantic verificationism: the doctrine first proposed by Wittgenstein, according to which the (cognitive, informative) sense of a sentence is the rule or method or procedure used in its verification (1980: 29). Wittgenstein’s idea was soon appropriated by the philosophers of logical positivism. However, after varied attempts to give it a precise formulation, it was in the end abandoned due to strong criticism, internal and external to the logical-positivist circle, which led to it being considered by many as unsustainable. This is presently the received view, even if sophisticated philosophers have never really abandoned the idea that some form or other of verificationism is indispensable (Cf. Misak 1995). Indeed, in the next chapter of this book I intend to offer a reply to the main objections that philosophers have made against semantic verificationism, showing that these objections were not directed against the right form of verificationism, but rather against a straw-man called the ‘principle of verifiability,’ as it was wrongly construed by logical positivists.
     I am introducing semantic verificationism in this chapter speculatively, as an alternative and in fact as the most natural way to analyze Frege’s discovery of the thought as the cognitive sense (epistemic value, informative content) of a sentence. Now, suppose that the combined semantic-cognitive rule that constitutes the thought as expressed in an assertive sentence is its verifiability rule, as complex as it may be. Then the verifiability rule is also the most proper truth-bearer. Then, if we show that this verifiability thought-sense-rule is effectively applicable to the expected fact, this makes it true, which allows us to say derivatively that the sentence expressing it is also true. If, on the other hand, we show that this thought-sense-rule, though conceivable, isn’t effectively applicable to the expected fact, this makes it false and likewise the sentence expressing it. Moreover, if we cannot formulate a verifiability rule able to be at least in principle applicable to the fact, if we cannot even conceive its application, we must conclude that the declarative sentence is devoid of meaning, devoid of sense or thought, even if it may in some cases seem to have meaning.
     I think that this way to understand the truth of a thought is in line with Frege’s remark that although he regarded truth as the property of a thought, it does not seem to be a property in the usual sense of the word (Frege 1918: 61). Indeed, truth does not add anything to the combined cognitive rule called ‘the thought,’ except something dispositional, namely, its effective applicability as a verifiability rule in the appropriate context for its application. Moreover, the proposed identity between the Fregean concept of thought and the concept of a verifiability rule is also supported by the Fregean proposal that the identification criterion for what belongs to a thought is that it must have at least some role in the establishment of the thought’s truth-value.
     Nonetheless, there is another way to understand the property of effective applicability of the verifiability rule, which is to identify it with the existence of the fact. To reach this conclusion, we need only consider that the existence of an object (an independent cluster of compresent tropes) is the higher-order property of effective applicability of an identification rule expressed by a nominal term, and that the existence of a property – a dependent property-trope – is the higher-order property of effective applicability of the ascription rule of a predicative expression.     If we accept this, then by symmetry the existence of a singular fact should be the higher-order property of effective applicability of the verifiability rule of the singular declarative sentence to which it applies. It seems that we could say, in an almost Hegelian fashion, that existence is the truth of the concept, while the truth is the existence of the thought…
33. Frege’s Platonism
It is important to remember that for Frege thoughts and the senses that compose them are Platonic entities belonging to a third ontological realm, which is neither psychological nor physical (Frege 1918). For him, taking (a) the criterion of objectivity as being inter-subjectivity and independence of will, and taking (b) the criterion of reality as existence in space and time, we combine them in order to get three ontological realms:

1. Realm of the objective and real
2. Realm of the subjective and real
3. Realm of the objective but non-real

The first realm is that of physical entities such as concrete objects, which are objective and real. These entities satisfy criteria (a) and (b): they are objective, since they are interpersonally accessible and independent of our will, and they are real, since they are located in space and time. The second realm is that of psychological entities, mental states that he calls representations (Frege uses the word ‘Vorstellungen’ in a way that could be easily translated as qualia). These entities satisfy criterion (b) but not (a): they are subjective and real. By not being interpersonally accessible, they are subjective and often dependent on the will. However, they are still real, because they are in the mind and, consequently, in time and (we can add) space. There is, finally, a third realm, that of thoughts (propositions) and their constitutive senses. This realm satisfies criterion (a) but not (b). For Frege thoughts are objective but not real. Thoughts are objective, because, true or false, they are always interpersonally accessible: we can all agree, for example, that the Pythagorean Theorem expresses a true thought in Euclidean geometry. However, this third realm of thoughts is not real, because according to him thoughts are abstract things that cannot be found in space or time. Thus, the thought (the sense) of Pythagoras’ theorem is objective but non-real.
     There are, however, problems. One of them, noted by Frege, is that although for him thoughts are eternal (timeless), immutable, forever true or false, and not created but only grasped (gefasst) by us, they must have some kind of causal effect: they must be able to cause our grasping them in order to make judgments and act in the external world (Frege 1918: 77). How this interaction with something non-spatio-temporal is possible remains an unexplained mystery.
     Frege was aware of the difficulties, but the main reason why he felt he had to introduce this third realm of thoughts is that thoughts are interpersonally accessible, that is, they are objective, which makes them able to be communicable. Representations (Vorstellungen), on the other hand, are rather subjective psychological states that can vary depending on personal psychology and according to him are not interpersonally accessible and there­fore not communicable. Thus, for him the right way to explain how it is possible that we are able to share the same thoughts in conversation is to strictly distinguish thoughts from mere psychological representations, placing them in a supposedly shareable Platonic realm. In addition, if thoughts were on the level of representations, they would be dependent on changeable personal psychology and would lack their required stability as truth-bearers.
34. Avoiding Frege’s Platonism
Despite the above-suggested arguments, few today would accept Frege’s appeal to Platonism. After all, the Fregean form of Platonism not only commits us to an infinite multiplication of objective entities (all the infinite variety of true and false thoughts and their constitutive senses), but also seems to lack intelligibility. The price that Frege was willing to pay in order to avoid psychological subjectivism seems too high for us today.
     In my judgment, if we understand senses as rules, which usually are implicitly established conventions, there is a clear way to bring the empiricist view of thoughts as having a psychological-empirical nature in line with the view that as truth-bearers they must have stability and the possibility of being communicated. In order to establish this conclusion, I want to apply again the same strategy inspired by the ontological particularism of English empiricists, which I used in the construction of universals by means of tropes.[47] This is understandable, since according to trope ontology, a thought must be made up of internal tropes: the mental tropes constitutive of some conventionally grounded verifiability rule whose application is at least conceivable. In order to accomplish this, I need only show that something like Fregean Platonic thoughts (objective non-real truth-bearers…), which I call f-thoughts (‘f’ from Fregean) can be defined in terms of psychological (real and subjective) p-thoughts (‘p’ from psycho­logical), though typically based on intersubjective linguistic conventions. In other words, I suggest that we can warrant the existence and stability of f-thoughts without hypostasizing them as Platonic entities and even without resorting to classes of p-thoughts if we replace them with what I call spreadable thoughts or s-thoughts. We can do this by means of the following disjunctive definition, which is as simple as it is efficacious:

An s-thought (Df) = a given tropical p-thought X* (used as the model) embodied in some mind or any other tropical p-thought Y qualitatively identical to X*, embodied in the same mind or in any other mind.

The s-thought is my empiricist version of what Frege should have meant with his f-thought (objective non-real thought). The p-thought X* can be any X thought that someone decides to use as a model. The aim of this definition of an s-thought is that any supposed f-thought is reduced to mental p-thoughts without depriving it of its epistemic objectivity (inter-subjectivity) grounded on conventional rules, along with its expected stability or immutability. This procedure works at least insofar as my criticism of the private language argument is acceptable, though I have no doubts about this (See Ch. III, sec. 13).
     The so defined s-thought – which is the same as a verifiability rule, a tropical thought-content or simply a proposition – though usually distributed across space and time, doesn’t need to have any particular spatio-temporal location and can be seen as the most proper truth-bearer. For example: the s-thought or thought-content expressed by the sentence ‘The Eiffel Tour is made of iron’ can be instantiated as the p-thought that I have in mind when writing this sentence. However, it can also be instantiated by, say, the p-thought that you have in mind when you read it, such as by any qualitatively identical p-thought that I, we, or any other person can have at any place or time, insofar as it is considered an f-thought, namely, a model for any qualitatively identical p-thoughts. Characterized by the disjunction between qualitatively identical thoughts embodied in individual minds, the s-thought is apt to be regarded in abstraction from any particular human mind that causally instantiates it. This is what really occurs when we think an f-thought, and it is this abstraction from singular human minds caused by its spreading character that gave Frege the impression that he had found a Platonic entity outside of space and time.
     As with model-tropes in the construction of universals, it is not necessary to have only one particular model as the object of interpersonal consideration. To the contrary, what we need to do is simply to single out the first thought given to us by memory and use it arbitrarily as a model: first the one, and then any other that we recognize as being precisely (qualitatively) identical to the first, and we can choose any of them as a new model. In some way language is only the vehicle of communication that allows the reproduction of qualitatively identical psychological p-thoughts in the minds of hearers, insofar as they are rooted in the usually implicit interpersonal conventions we have attached to their semantic components.
     With the help of the above definition, we avoid not only appealing to psychologically specific occurrences of thoughts, but also the most expected alternative, which would be to explain one s-thought in terms of a sum or set of p-thoughts qualitatively identical to each other. This could lead us not only to the problem of defining sets, but also to the problem that sets and sums have or could have size, while thoughts cannot. If an s-thought were a set of p-thoughts, even if considered an open set, it would grow ever larger, the greater the number of people there were who grasped it.
     Under the proposed definition, in order to exist, an s-thought must always have at least one psychological occurrence. The s-thought is not less psychological than any p-thought, since it cannot be considered independently of its instantiation in at least one mind. This means that when we say that we both had the same idea, or the same thought, this is merely a manner of speaking. What we really mean is only that there is a qualitative identity between the (tropical) psychological verifiability p-thought-contents rules that we have respectively instantiated in our minds. We share the s-thought in the sense that we instantiate qualitatively identical p-thoughts. This has the advantage of bringing Fregean thoughts out of the ethereal Platonic heaven back to the concrete psychological realm without making any serious commitment to the transient psychology of individual minds.
     This understanding of the true nature of thought-contents explains something that Frege was unable to explain satisfactorily, namely, why and how they may have causal powers. Since as an open disjunction of p-thoughts, s-thoughts only exist as psychological instantiations of p-thoughts, this enables them to play a causal role: they can cause other psychological states and, combined with desires, human actions and their effects in the external world.
     At this point one could raise an objection of multiple realizability: the same p-thought could be differently realized in different human brains, making the qualitative identity of p-thoughts impossible. I agree with the very probable multiple realizability of p-thoughts but disagree that this makes their qualitative identity impossible. There is no reason why we cannot present things that can be considered qualitatively identical on a linguistic or even psychological level and different on a neuro-physiological level, in the same way as different devices can have different internal mechanisms and perform exactly the same tasks.[48] Moreover, my suggestion is that s-thoughts are constituted of p-thoughts that are internal tropical verifiability rules, which although complex, ramified and variable, are also able to be satisfied by foreseeable external tropical configurations.
     In my judgment, one of the most unyielding and deceitful philosophical errors in ontology has always been seeing numerical identity where there is only qualitative identity. It is true that we can ask for the meaning of the general term ‘chair’ using the definite article ‘the’ in the phrase ‘the chair.’ But this is only a linguistic device that changes nothing! In a similar way we can speak of the geometrical form of circularity, and of the number 2 in the singular… But this is just for the sake of simplicity of expression. What we are ultimately able to have in mind in all these cases are occurrences of qualitatively identical meanings, that is, of qualitatively identical concepts of chairs, circles, and cognitive arithmetical concepts of duality, and not something more, since we don’t need something more to get something more, as Murray Gell-Mann would say.[49] In the same way, we can talk about the thought expressed by ‘12 x 12 = 144,’ but if we do not intend a specific occurrence of this thought, we are only referring to some occurrence, but without taking into account or having to specify which occurrence and in what mind. We speak in the singular of the thought that 12 x 12 = 144 for reasons of simplicity.
     The adoption of the definition of s-thoughts proposed above, which is easily generalizable to all kinds of Fregean senses, seems to me the only plausible abstraction we can arrive at without committing any of various forms of reification that have infested ontology throughout its long history.
     At this point a stubborn Fregean defender can still ask: how is it possible that the psychologically dependent definition of s-thoughts suggested above could be able to ensure the objectivity of s-thoughts, their interpersonal accessibility or communicability? As we saw, Frege concluded that if we regard thoughts as psychological representations, as is the case with p-thoughts, they would unavoidably be subjective, and we could not compare them with each other. However, it still seems clear that Frege was too hasty when he admitted that his f-thoughts belong to a third realm of Platonic entities. One could note that there is no doubt that what Frege calls representations (phenomenal mental contents) have in fact possibilities of interpersonal communication, even if limited.[50] But much more important is something that Frege has completely excluded from consideration, namely, that senses and s-thoughts, without being Platonic entities, could be understood as rule-complexes built upon adequate associations of interpersonally accepted conventions established with the help of public signs that are communicable precisely because of their grounding interpersonal character. That is, because s-thoughts are verifiability rules rooted in linguistically shareable interpersonal conventions, they can well be able to satisfy Frege’s demand for objectivity as interpersonal accessibility followed by the possibility of communication and truth-evaluation.
     It may at first sight seem implausible that language is capable of repeatedly being reproduced in other minds and even in the same mind with the same subjective pattern, the same thought-content, the same recognizable instantiation of an adequate association of conventionally established semantic-cognitive rules attached to our words. However, compare by analogy this case with that of genetic information able to endlessly reproduce the same characteristics in successive biological individuals.[51] Why cannot the conventions and ways they can be combined in the constitution of p-thoughts do a similar job, even if only inferentially? More than this (and probably also in the case of genetic information), it is easy to suppose that there are corrective mechanisms able to interpersonally and intra-personally impose a limit on divergence from conventionalized standards. There is no reason, except an anti-empiricist bias, to think that things could not be that way.
     Finally, let us apply to s-thoughts John Searle’s important distinction between what is ontologically objective/subjective and what is epistemologically objective/sub­jective (Searle 1999: 43-45). Searle noted that we have a strong tendency to take what is epistemologically subjective for what is only ontologically subjective. However, something can be ontologically objective – for instance, ‘How justifiable was the First World War?’ – without ceasing to be epistemologically subjective, because it is not easy to reach common agreement about this issue. In contrast, a phenomenon can be ontologically subjective without ceasing to be epistemologically objective – for instance, the stabbing pain caused by a seizure of acute pancreatitis – because everyone (doctors and patients alike) will agree on the form and existence of this pain, even if the patient alone knows exactly how it feels.
     Something of the kind can also be said not only of Fregean subjective mental representations, but also of s-thoughts. In themselves they are ontologically subjective, since we admit that they are psychological events instantiated in one mind or another. But even so, they do not cease to be epistemologically objective, since we are capable of interpersonally agreeing about them and their truth-values. We can agree that an objectively assertive sentence like ‘The Eiffel Tower is made of metal’ expresses a true s-thought that is epistemologically objective, even though as an s-thought ontologically subjective, since it is distributed among the minds of those who think it. Like any s-thought, it remains epistemologically objective, given that it is grounded on conventions associating words with things in the world, which makes it fully measurable and communicable. An arithmetical sentence like ‘2 + 3 = 5’ is epistemologically objective (since we are able to agree inter-subjectively on its truth-value), but it also expresses an ontologically subjective s-thought, and as I tried to show in speaking of numbers, it seems to be a thin kind of trope sustained by lower-order tropes. On the other hand, a sentence like ‘Love is the Amen of the universe’ (Novalis), unlike an s-thought, has no truth-value. It is only suggestive and expressive. Like poetry, it is based on non-conventional subjective coloration, being susceptible only to emotive-aesthetic appreciation with differing degrees of subjective interpersonal agreement.
     Regarding ontology, Frege was no exception. Like Husserl, Bolzano and several other continental philosophers of his time with mathematical training, he believed that the ontologically subjective character of psychologically conceived thought-contents would inevitably be condemned to epistemological subjectivity. But this was a mistake.
35. Further ontological consequences
Our ultimately psychological reformulation of Fregean thoughts has some interesting ontological consequences. If the thought of the Pythagorean Theorem isn’t an eternal (timeless) entity belonging to a Platonic realm, always true or false, where and when does it exist? The answer is that if there is at least one occurrence of its thought or any other qualitatively identical occurrence, regardless of the bearer, something like the Pythagorean theorem acquires an existence dependent on minds. It is not an existence dependent on any of the many particular minds that will eventually think it, since it would continue to exist without having been thought by this or that particular mind. In fact, since this thought has been thought by both you and me and certainly by many others in the past, its existence must be spread over space and time. It must be distributed over the space and time occupied by the heads of mathematicians starting with Pythagoras himself, and perhaps ending in the head of some cognitive being at some unknown future time. This is what gives the impression that the thought is something abstract, beyond the psychological realm.
     Another consequence of the proposed view is that unlike the Platonic entity that Frege called a ‘thought,’ our s-thought of the Pythagorean theorem did not in fact exist before Pythagoras thought it for the first time (supposing he was the first), and will cease to exist if it ceases to be thought by anyone. The Pythagorean theorem certainly exists, has existed and will continue to exist in the sense that it is thought, has been thought and will probably be thought in the future, referring to occurrences of this thought, but without having to take into account who thinks it.
     One could object that this result sounds strange: it seems that the Pythagorean Theorem applies independently of minds. However, this strangeness can be softened by the fact that nobody can truly deny it. One cannot have the true thought, ‘The theorem according to which the sum of the squares of the shorter sides of a right triangle equals the square of the hypotenuse has been thought in the past and now is no longer thinkable.’ And the reason is that this judgment will already be an occurrence of the thought of the Pythagorean Theorem and insofar will falsify what it states. Anyway, the conclusion remains that the s-thought of this theorem would not have come into existence if nobody had ever thought it. Putting this more incisively: it would not exist in a world without cognitive beings.
     The last remark suggests the following objection. Imagine a possible world Ww similar to ours, with planets, stars and galaxies, but without any cognitive being. In Ww the s-thoughts that there are planets, stars and galaxies could not have been thought and, s-thoughts, being the primary bearers of truth, could not be true. Nevertheless, it seems very reasonable to think that in this world the fact that there are planets, stars and galaxies would still be true, even though there would be no cognitive beings to think this.
     It seems to me that the right answer to the strangeness is that here we are again victims of a confusion between thought-truth and fact-truth. As we saw, the first is the truth applied to the primary bearer of the truth, which is the s-thought, while the second is a derivative but very common application of truth to the real existing thing or fact in the world, as a secondary bearer of truth, meaning a real thing or fact. Indeed, that there would be planets, stars and galaxies in a mindless world would still be true as a fact in Ww. Hence, the applicability of the Pythagorean Theorem would still be a fact-truth in Ww, even though neither their s-thoughts nor their truth in the form of correspondence would exist. The flexibility of natural language has once again misled us.
     Still another objection that could be made against the idea that the bearers of truth are non-Platonic s-thoughts is the following. Many truths have been discovered. Pythagoras is credited with discovering the theorem that bears his name; Archimedes was one of the discoverers of the law of the lever, according to which magnitudes are in equilibrium at distances inversely proportional to their weights. However, if something is discovered, then logically it must have existed before being discovered. Consequently, the above-described thoughts must already have existed before their discovery.
      Again, the answer is that this naïve objection results from a confusion between the thought as the primary bearer of truth on the one hand, and the fact as a derived bearer of truth on the other. This is clear in the case of typical empirical truths. That the law of the lever was always applicable in principle is surely true. However, this is only a fact-truth! Its thought-truth was only part of the empirical (mental) world after scientists like Archimedes conceived it. Similarly, common sense tells us that the fact expressed by the Pythagorean Theorem must always have existed. However, our s-thought of it only came into existence after the theorem was thought by Pythagoras and since then has been thought by many others. Real facts, in their turn, as long lasting as they may be, are not the primary bearers of truth, but rather their truth-makers or verifiers. They exist independently and are said to be true only in the derived sense (b) of fact-truths, not in the sense (a) of thought-truths. They are what occurrences of their thoughts represent. Hence, in the most proper and demanding sense, no truths or falsehoods would exist in a world where there were no minds to think them. The most we could think of in this direction is to say that if the law of the lever were thought in Ww, it would be recognized as true.
      An s-thought that has never been thought does not exist and thus cannot be true. The same holds for falsehoods. Consider the thought ‘The Colossus of Rhodes is floating in the Sargasso Sea.’ In all probability this thought has never been thought before the present moment. But the moment we think that it has never been thought before, we are already thinking it, and we can even attribute falsehood to it. Even the s-thought ‘The world could exist, even if there were no minds to think about it’ is only a true thought insofar as there are minds to think it.
36. A short digression on contingent futures
Before we finish, it is interesting to examine the Aristotelian problem of contingent futures in the light of our conclusions (1984, vol. 1, De Interpretatione, sec. 9). According to a plausible interpretation of Aristotle the following argument is valid:

Argument A
1.     Necessarily, it is true or false that there will be a sea-battle tomorrow.
2.     If (1) is true, then the future is predetermined and there are no chance events.
3.     Therefore, the future is fixed and there are no chance events.

It seems that for Aristotle this conclusion would be unacceptable, because if the future were predetermined, then there would be no chance events, and if there were no chance events, there would be no free will. Hence, according to a traditional interpretation, he thought that although this argument is sound, premise (1) is false because it exemplifies the principle of bivalence, and the principle of bivalence – according to which any significant proposition is either true or false – is not applicable to future events (only to present and past ones).[52]
     I cannot agree with this conclusion, since I believe that we should preserve a strongly understood principle of bivalence for s-thoughts.[53] But premise (1) can be questioned from a different perspective. Suppose, first, that outside any context we consider the s-thought expressed by the sentence ‘There will be a sea battle tomorrow,’ which we can abbreviate as ├p. Is this statement true or false? The answer is the following: if taken literally, ├p is unable to express any s-thought because a verifiability s-thought rule is something to which we must possibly attribute a truth-value. Normally ‘There will be a sea battle tomorrow’ is an incomplete indexical statement, so that without any further contextual information we are totally at a loss for the task of associating p with any appropriate truth-maker in order to assign it a truth-value.
     Moreover, one could argue that the sentence ├p (as much as ├~p) is misleading and causes confusion, like argument A, because ├p only seems to express cognitive thought-content. The reason for this is that ├p is very easily confused with the meaningful sentence ├p*: ‘[It is likely that] a sea-battle will take place tomorrow,’ stated when there are reasons to think so. For example: having broken Japanese naval codes and having lured the Japanese fleet into an ambush at Midway, the Americans already knew on the night of June 3, 1942, that on June 4 there would almost certainly be a major naval battle. The sentence ├p* is easily confused with ├p, because ├p* is almost always abbreviated as ├p: ‘A sea-battle will take place tomorrow.’
     For example: suppose that American Admiral Nimitz had said on June 3:

Tomorrow there will be a sea-battle.

Everyone would understand that he was saying that all the factual evidence was leading to the conclusion that the expected battle would begin on June 4. This probability – made explicit or not – is in this case objectively measurable in terms of verification by actual empirical evidence, so that the assertion ├p* expresses an s-thought that is held to be true, for it is true that, with the information already available, it was very probable that a sea-battle would occur the next day. Indeed, the utterance ‘It is likely that a naval battle will take place tomorrow’ could be regarded as definitely true on the night of June 3, 1942, without violating any principle of bivalence!
     Suppose now, by contrast, that I am standing on the calm beach of Praia Bonita in Northeastern Brazil, looking out across the Atlantic Ocean, and without any reason I say ├q*: ‘A naval battle will take place in this region tomorrow,’ meaning by it ‘It is likely that a naval battle will take place in this region tomorrow.’ This statement can be regarded as definitely false, since there are many different reasons to believe that this kind of event is extremely improbable in this region and at this time.
     The conclusion is that in the absence of a context (and not in the above senses of ├p* or ├q*), the sentence ├p would be a linguistic bluff devoid of any meaning or justification. Aristotle would be right in rejecting the application of the principle of bivalence to it, not because this principle has exceptions, but simply because it expresses no s-thought, no proposition, no verifiability rule. All that this sentence does is to induce us to imagine a naval battle that takes place tomorrow, as if there were hidden verifiability criteria. However, insofar as no context is furnished, no real criteria can be given. Statements like ├p*,├~p* and ├q*, on the other hand, aim to say something probabilistic about the future that can be confirmed and made true by criterial reasons already found in the present. But from such statements premise (2) and the conclusion (3) of the argument A do not follow, because all that such statements can warrant, if true, is the inductive probability of a sea-battle.
     The upshot is that the metaphysical riddle about contingent futures can be eliminated if we consider with enough care what we are really able to mean by affirming s-thoughts regarding the future.
37. Conclusion
My first aim in this chapter was to insert in the framework of Fregean semantics the results of my reconstruction of how should we understand Wittgenstein’s view of (cognitive) meaning as use determined by rules, in order to better distinguish the most relevant forms of semantic-cognitive rules and their functions. This insertion requires strong corrections in Frege’s own framework. Even if most results could only be sketched here, they nonetheless seem to me clearly more plausible than Frege’s own original views.










[1] On the thorny issue of how to translate ‘Bedeutung’, see Beaney 1997: 36 f.
[2] Searching in the literature, the only place where I have found a similar view on this point is Kneale & Kneale 1985: 495.
[3] One can read singular terms like ‘the morning star’ as definite descriptions or as proper names (like ‘The Morning Star’). I prefer to read them here as definite descriptions, since for proper names we can use the words ‘Phosphorus.’
[4] As shown in the introduction, Ernst Tugendhat later defended a similar understanding of the meanings of singular statements in a more systematic and detailed way, though refraining from doing it as a reconstruction of Frege’s semantics.
[5] If we compare these two passages, it becomes clear that in opposition to Kripke’s interpretation (1980, Lecture I), Frege already had in mind the essentials of the later bundle theory of proper names. The same can be said of Russell (Cf. Russell 1911, Ch. 5).
[6] Assuming Kripke’s views, François Recanati replaces senses with mental files as supposedly non-descriptive modes of presentation (2012: 34). However, since these files are clusters of information and not subjective Vorstellungen, they should be able to be linguistically expressed by means of descriptions, bringing us back to the descriptivist standpoint. For this reason, it seems that our complex semantic-cognitive rules are apt to do the same job in an explanatorily more powerful way and (as we will see) with important epistemological consequences. Moreover, our rules or combinations of rules do not need to contain less information than files. They can be as informational, durable, changeable and flexible as required by the context.
[7] Although a pure ontology of tropes is a very new ontological achievement and brings with it a wide range of unsolved problems, it does not produce more difficulties than the traditional doctrines of realism and nominalism. In return, it promises an as reasonable as parsimonious solution for ontological problems, which would free us from traditional hindrances (see Appendix to Chapter III).
[8] Mere similarity would not do, since this concept is intransitive. Strict similarity means here the same as qualitative identity, which is transitive. Strict similarity must also be a trope, since it is spatio-temporally located between tropes, even if, as an intrinsic relation, it is a subordinate trope.
[9] I suggested this disjunctive construction of universal by means of tropes as the best way to circumvent the usual but problematic definition of a universal as a set or sum of tropes that are strictly similar, one with the other (See Appendix of Chapter III).
[10] Keith Campbell has suggested an epistemic primacy of identification over the generalizing function (1990: 24-25).
[11] Even D. C. Williams portrayed things misleadingly here. For him ‘Socrates is wise’ (or any Fa) means ‘The concurrence [togetherness] sum (Socrates) includes a trope that is a member of the similarity set.’ (my italics, 1953: 11)
[12] There are several asymmetries. The most discussed is probably the asymmetry of subjects and predicates regarding negation: you can negate the predicate, but not the subject (nominal term) (Strawson 1971, Ch. 5).  The answer seems to me obvious. The negation of the predicate means the admission of the inapplicability of the ascription rule to the object identified by the identification rule. However, since the application of the ascription rule is dependent on the application of the identification rule, whenever you negate the application of the identification rule of the subject you also negate the applicability of the ascription rule and in this way the whole statement. Hence, it is impossible to negate the subject as the nominal term alone.
[13] Notice that the demonstrative ‘that’ does not have here the function of a constituent of the identification rule of Socrates, but is itself part of the identification rule of a certain place. In indexical statements like ‘This is a daisy’ and ‘There is the Golden Gate Bridge,’ demonstratives have as their main role the location of the criteria satisfying the characterizing dimension of the identification rule of the daisy and the Golden Gate Bridge. The logical form of the statement ‘This is Socrates’ is already revealed by the relational statement ‘<This spatio-temporal place> is where <Socrates> is located.’
[14] Ignoring Frege’s theses that the reference of a sentence is a truth-value and that a fact is a true thought, I will in the present context call the sentence’s reference a fact. This choice will be justified in the sections 21 to 23 of this chapter.
[15] I take these examples from Mulligan et al. (1984: 300, 301 and 306), though their point wasn’t the same.
[16] Note that if I said, ‘This chair has this backrest’, pointing to each of them, ‘this chair’ and ‘this backrest’ could work as nominal terms and ‘…has…’ as a relational predicate.
[17] As Ernst Tugendhat wrote: ‘‘Fa’ is just the case to the extent that the rule of identification for ‘a’ is followed, and based on this result ‘F,’ is applicable in accordance with its rule of application’. (Tugendhat & Wolf 1983: 235)
[18] The Valkyries were maidens who served the god Odin, choosing the soldiers on battlefields worthy of admission to Valhalla.
[19] It is easy to see that singular statements implicitly attribute existence to their objects, since a predicative statement with the form Fa could be written as Ǝx [Fx & (y) (Fy → (y = x)) & (x = a)] in order to make this attribution explicit.
[20] Socrates lived in Greece from 470 to 399 BC. But the time and place of existence are abstracted when we talk about existence, since existence is essentially only the effective applicability of the conceptual rule, not the time of its applicability.
[21] It was W. V-O. Quine who suggested using the name Pegasus as a way to change a name into a predicate such as ‘the thing that pegasizes’ (1948/9: 27).
[22] There are less successful attempts, like Michael Devitt’s interesting book Designation.
[23] David Braun and Marga Reimer, two renowned specialists, made a balanced comparison of descriptivist and causal-historical views in their respective articles for the Stanford Encyclopedia of Philosophy. The results were inconclusive.
[24] In some cases, like ‘Queen Elizabeth II,’ the family and even genetic origin is part of the localizing description, although this isn’t necessarily so (See Appendix to Chapter II).
[25] What symbolic form a proper name receives is contingent. What makes this form necessary is the identification rule that we attach to it. In a possible world where the name attached to the identification rule for the name Hitler was attached to the name Hartman, this different name would mean what we mean with the name Hitler. Meta-linguistic theories of proper names have equivocally tried to reduce the proper name’s meaning to such auxiliary descriptions (e.g., Katz 1990).
[26] In fact there is no sharp boundary between fundamental and auxiliary descriptions.
[27] One could object that rules are changeable and that if we change the identification rule, it ceases to be a rigid designator. Nonetheless, if we change the rule so that the set of possible worlds to which the proper name applies can be distinguished as a considerably different one, we are not applying the same proper name anymore. However, you may introduce changes like additions to the fundamental description-rules insofar as this only specifies the identification better, and thus affecting nothing essential, only adding the application or non-application to possible worlds where the applicability of the rule was in an earlier stage indeterminate. (Cf. Appendix to Chapter I, sec. 7)
[28] This is again a didactic simplification (See Appendix of Chapter I).
[29] However, if the assertion that there are round squares were merely an equivocal manner of saying that we can syntactically combine the adjectives ‘square’ and ‘round,’ that is, a misleading way of saying that there is a syntactical rule allowing the combination of these incompatible words, then it makes some sense to attribute existence. But in this case, what we are trying to say will be more correctly expressed by the meta-linguistic sentence: ‘The rule for constructing the phrase “round square” is applicable, and therefore, the phrase “round square” exists as a grammatical construction.’ The Meinongian Sosein is reduced here to the recognition of a syntactical triviality.
[30] In accord with Berkeley’s official view, things that are not actually perceived by us exist because they are continuously being perceived by God. (Urmson 1983)
[31] I believe that Mill’s confusion in the definition of matter was in fact an attempt to evade the objection of idealism open to Berkeley.
[32] See Frege, Letter to Russell of 28.12.1912.
[33] Without offering a justification, Strawson writes: ‘a situation or state of affairs is, roughly, a set of facts, not a set of things.’ (1950: 8)
[34] For an important reply, see J. L. Austin, ‘Unfair to Facts’ (1961, Ch. 5). It seems to me at least curious that the posthumously published arguments of Austin against Strawson’s view have had so little impact.
[35] John Searle once proposed something approaching this answer: ‘…we neither have nor need a thick metaphysical notion of “fact.” Anything sufficient to make a statement true is a fact. Thus the fact that there are no three-headed cats is as much a fact as the fact that the cat is on the mat.’ (1998: 392)
[36] See Appendix of Chapter III, sec. 4.
[37] This also gives back the whole sense of Church’s still more convoluted original sentence: ‘The number such that Sir Walter Scott is the man who wrote that many Waverley Novels altogether is twenty-nine.’
[38] I think that the mode of presentation of the sub-fact can be approximated with what defenders of two-dimensionalism call a primary intention (here called derived thought) while the mode of presentation of the grounding fact can be approximated with what they call a secondary intention (here called basal thought) (Cf. Chalmers 2002). But the here presented explanation seems to me more perspicuous and natural.
[39] For instance, A. J. Ayer in the first case and Hilary Putnam in the second. (I atempted a more detailed analysis of the concept of water in Costa 2014, Ch. 3.)
[40] The concept of emphasization was fruitfully applied in Jürgen Habermas’s excellent work on universal pragmatics (Habermas 1976).
[41] The example was already considered in the Addendum of the Appendix to Chapter II in this book.
[42] As Tyler Burge wrote: ‘the word “thought” is the best substitute for ‘proposition’ for the naturalness of its semantics within the scope appropriate to the linguistic philosophy.’ (Burge, 2005: 227-8)
[43] For Frege, in the case of indexical sentences, the context of the utterance belongs to the expression of thought. See also addendum of the Appendix to Chapter II, sec. 8.
[44] According to his main argument, if you say that the truth of p is its correspondence with reality, you need to admit that p must have the property N in order to be true by correspondence with reality, and that to have the property N in order to be true by corresponding with reality will demand the property N’ and so successively. The answer (already given by Aquinas) is that to say that p is true by corresponding to reality, and to say that p has the property N due to being true by corresponding to reality are one and the same thing, consequently N is redundant. (Cf. Künne 2003: 129-133).
[45] For instance: ‘truth (principle): that which is true in accordance with the fact or reality’; ‘truth (fact): the actual fact about the matter’… (Oxford-Cambridge Dictionary).
[46] See Tugendhat’s verificationist correspondentialism in 1983: 235-6.
[47] See Appendix to Chapter III, sec. 2.
[48] As T. W. Polger has shown, in order to illustrate the flaw of the multiple realizability argument, we can explain how a carburetor has the function of mixing fuel and air for a combustion engine; but it is a multiply realizable device: it can be made of various different materials with various designs, provided it functions properly. (2004: 19-20).
[49]  Against this, one could ask: haven’t we learned that geometry deals with perfect circles and that arithmetic deals with entirely abstract numbers? Take the case of circles. The answer is, of course, in the negative, because we can make a new circle more perfect than the last one, and another even more perfect, and this process can continue indefinitely. The perfect circle is like the actual infinite: it does not exist. It is nothing more than a projection of our awareness of the possibility of making increasingly perfect empirical circles without any end in sight. Geometry does not work with actual perfect circles, but with potentially perfect circles.
[50] Against Frege, we could hold that to some extent even imagetic representations can be expressed through language and by its means could be subjectively identified and re-identified as being the same. It is true that a mental state that only one person is capable of having, for instance, a sort of epileptic aura, is not communicable, except indirectly, metaphorically. But it seems very plausible that typical mental states, such as feelings, images, sensations, are things that all of us are able to communicate and learn to identify in ourselves through induction by exclusion, added to induction by analogy and reinforced by a great variety of interpersonally accessible physical states strongly intermingled with them (Cf. Ch. III, sec. 8; See also Costa 2011, Ch. 3).
[51] Biological mutations are accidents whose occurrence should be evolutionarily calibrated. Species are only likely to survive if they can mutate to the right degree in the right period of time in order to adapt to environmental changes. Too many mutations, as well as too few, would be dangerous for species survival. It seems possible that an unchanging species with no mutation is conceivable, but it would be unable to adapt to changing external conditions.
[52] For a different interpretation, see Christopher Shields, 2007: 186-190.
[53] I mean a principle of bivalence understood as a different formulation of the principle of non-contradiction.






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