sábado, 20 de maio de 2017

## AN EXTRAVAGANT READING OF FREGEAN SEMANTICS




Last uncorrected draft of a chapter of the book Philosophical Semantics to be published in 2017 by CSP. 




– IV –
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 exposing the misperceptions that often almost unavoidably originate from the use of language on the relationships between concepts … then my ideography, further developed for these purposes, can become a useful tool for philosophers.]
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 clear that the basic concept of sense can be paraphrased in terms of semantic-cognitive rules and that the 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 – as I read him in the last chapter – and Frege’s semantics; a link already noted by Michael Dummett, though in a very general manner, devoid of 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, with the aim of providing a more rigorous framework for the rather vague semantic insights gained in the last chapter.
   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) of sense, and (2) of reference in the case of a predicative singular 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 eccentricities. My commonsensical reading of its main semantic elements in terms of conceptual rules will show how to purge Frege’s semantics of its greatest oddities.

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 about the reason why Frege would have chosen the word ‘Bedeutung.’ 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). It may be. But for me 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)           (points ...  designates...)        (what is said)
                                                                                ↓
                                                                      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 can be well 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 the 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); the sense is the way we intentionally present an object or reference to ourselves, whether it exists or not. At any rate, 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 meanings of 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 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. The result is that his Platonism of senses 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 the fundamental difference between Fregean semantics and the semantic considerations of the later Wittgenstein, who regarded senses or meanings as episodic uses determined by rules. Dummett was possibly the first to defend the idea that senses are rules as the most natural reading of Frege’s 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)

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, more precisely, 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 sense-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, i.e., 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 obtain 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 the similarities and differences between 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 moved by a certain poem, while others are not; and this is why it is so difficult to translate poetry, which depends so greatly on colorations acquired by expressions in a particular language and culture. Hence, colorations are not results of conventional rules; they are rather regularities originating from shared aspects of human nature. 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 phenomenological language to its expressive meaning (I suppose that both can be legitimated).
   If in opposition to Frege we accept the view that sense is only 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 and Husserl. This is because the reason could be easily found for the objectivity of sensible things (characterized for Frege by their possibility of interpersonal access), as well as the reason for the consequent communicability of senses (in contrast to the at least relative lack of objectivity and communicability of representations and colorations or illuminations). This reason would be that senses typically depend on conventional semantic-cognitive rules, usually interpersonally 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 conventions or from combinations of these conventions.
   Accepting that the sense of a singular term is the same as a rule seen as a conventional or conventionally grounded procedure which 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 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 usually seen not too far from the Sun just before the Sun rises.’ Without assuming that definite descriptions are expressions of rules, Frege also approached this in his 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 some way 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 that suggested by Frege. However, it seems indubitable to me that Kripke’s arguments can be successfully countered by the kind of meta-descriptivist bundle theory summarized 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 (e.g., a red patch) or some combination of properties (e.g., 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 of sensory patterns that are satisfied by 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 our terminology, inspired by Wittgenstein, by saying that a concept is a semantic-cognitive rule or procedure requiring criteria or criterial configurations that can be satisfied by particularized properties (p-properties) or tropes. Coming back to Frege’s semantics, we see that what all these comments suggest is that the concept should be the sense of the predicative expression, its mode of presentation, and not its reference, as in Frege’s bizarre 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),[7] seemingly acknowledging that ‘property’ is the right term for the reference of a predicative expression. However, for Frege 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 x, we get as 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) or not and whose value is a truth-value, which can be alternatively two abstract objects: ‘The True’ (das Wahre) when the object falls under and ‘The False’ (das Falsche) when 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 being 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 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 example: 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, is saturated when some object falls under it, for instance, the object named ‘Bucephalus’ referred to by the sentence ‘Bucephalus is a horse.’
   With metaphors like that of ‘unsaturation’ and ‘incompleteness,’ Frege hoped to open the way to the solution of the mystery of the logical distinction between subject and predicate of a sentence. After all, the subject (the singular term) would refer to the saturated object, while the predicate (the general term) would refer to the unsaturated concept.
   Unsaturated predicative expressions and saturated singular terms combine to form saturated sentences like ‘Bucephalus is a horse,’ which being complete must also be the name of an object, which for Frege is the truth-value of the sentence. It may seem that this would be confirmed by the possibility that 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’ occurring 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 could at least have concepts not only as referred-to abstract entities (incomplete Platonic entities as references to empty predicates like ‘…is a yeti’[8]), but also (maybe) as the abstract references of the corresponding nominalized conceptual expressions. However, such admissions seem to be ontologically abusive (Tugendhat & Wolf 1983: 138-9).
   Anyway, it is by now clear that Frege uses the word ‘concept’ as a technical term that contrasts starkly with our ordinary use of the word ‘concept.’ 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, for he has beforehand emptied them by absorbing the semantic level into the ontological one.
   My conclusion is that it is better to drop the Fregean technical notion of a ‘concept’. This is a problematic remnant of Platonism that does nothing to explain predication. Instead, we will understand the word ‘concept’ here in its intuitive way as the sense of the predicative expression: its mode of presentation. It is counterintuitive 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 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 presently the most reasonable answer to this question consists in an appeal to the ontology of tropes. 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.[9] There are many epistemically near and primary examples: the white color I see when I look at newly fallen snow on a sunny day, and which is there in some sense, this smooth surface, the rectangular shape of a computer screen, the hardness of a stone, my headache. All these are tropes – particularized properties or simply p-properties – that may range from simple qualities to complexes of tropes, like the process of condensation of water-vapor or the personality of a human being or a country’s political organization or a social upheaval, though these last things are also in a much more vague way spatio-temporally located. Even non-perceptible things like physical forces could be derivatively constructed from perceived tropes, and it is not even impossible that so-called abstract entities like numbers could be explained as constructions derived from spatio-temporally located properties called tropes. A pure ontology of tropes maintains that all reality must be constructed of tropes, which from a genetic-epistemological perspective are the building blocks of the world.[10]
   Moreover, it is easy to suggest a particularistic construction of universals build up based on particularized properties or tropes. In my view a universal can be defined as:

Any chosen trope model T* or any other trope strictly similar[11] with to model T*.

I suggest this assuming that the trope we take as the model T* is at our discretion and may vary according to the epistemic subject and even concerning the same epistemic subject on different occasions.[12] In this case, the 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 T*. 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 is not 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.
   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 universal doctrines of realism and nominalism. In return, it promises a really parsimonious solution for ontological problems, which would free us from at least three traditional hindrances: (i) ostrich nominalist solutions, with their lack of explanations for impositive questions, (ii) abstract objects of contestable intelligibility like Platonic universals which lead to uneconomical multiplication of entities, and (iii) non-cognoscible naked substances.
   Realism (Platonic or Aristotelian) – the most influential traditional doctrine – has occupied philosophical minds for more than two millennia without progress sufficient to considerably increase its plausibility. Thus, in my view the only reason why this doctrine still seems to hold a foreground of attention is the longstanding weight of tradition. For such reasons (under the assumption that the ontological enterprise makes sense) I accept a pure ontology of tropes as the most plausible solution, at least in the form exposed in the Appendix of Chapter III.
   Finally, I usually avoid the use of 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’ was 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 those complex tropes that are made of different kinds of tropes, being these complex tropes possibly 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 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 Bucephalus could still exist as Alexander’s beloved horse even if he were just a warn out 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 seen by observers on the Earth. Secondarily and independently, 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 belonging to the object referred to by the subject term,
 (B) An allusive function: that of alluding to or connoting any other tropes that would be strictly similar to the model-trope considered by the speaker as designated by the predicative expression, building what might be called the universal, here understood in an ontologically inoffensive 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.[13] As particularized properties, tropes have not only ontological, but also epistemic priority if compared with their role in building universals.
   Furthermore – opposing the overwhelming influence of the logic tradition – we have, as a still more subsidiary element: (C) the extension. Although extension is relevant, it isn’t primarily associated with predication. Extension doesn’t even need to be 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 class of tropes strictly similar to the trope in question and (C2) an extension of objects as a class of objects having tropes strictly similar to the trope in question. However, in both cases the extension is a further element usually only vaguely inferred.[14] 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 most serious problem with the idea of incompleteness or unsaturation is that it fails to serve its purpose 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.[15] 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.     Xanthippe’s husband is Socrates.
4.     There comes Socrates!

In these sentences, the name ‘Socrates’ seems to occupy a predicative position. However, this name clearly continues to be used logically as a proper name, since these sentences can be better reformulated as, respectively:

1.     Socrates was a man who lived in Antiquity.
2.     Socrates has the property of being wise.
3.     Socrates is the husband of Xanthippe.
4.     Socrates comes from that place![16]

One cannot effectively transform a singular term as such into a predicate, while predicates seem to be easily transformed by nominalization into singular terms. , at least at first view.[17] This asymmetry suggests that subjects and predicates play different logical roles in sentences, which requires explanation. 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 truth-value, which for the object Bucephalus is ‘The True’ and for the object Alexander is ‘The False.’ Now, with the same right we can apply similar reasoning to the second case. We can suggest that the name ‘Bucephalus is…’ refers to an object that is a function that may have as argument any property designated by any predicative expression, which if it is this property black has as a value The True and if it is the property white has The False as a value, since Bucephalus was in fact a black horse. The conclusion is that 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. 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 one of the Aristotelian definitions of substance, which is:

That which exists independently of other things (Aristotle 1984a, sec. 5).

Applied to individuals or objects understood as (at least) clusters of tropes displaying compresence, this definition suggests that in their existence these structures are comparatively much more stable than their associated tropes. That is, it seems that the objects typified by material things exist in a manner relatively independent of their tropes in the composition of the kinds of facts[18] represented by true singular predicative or relational statements. Moreover, I hold that the individual referred to as a subject is only relatively independent, because the relation of existential independence/dependence is here understood relatively to the internal context of the statement. I can say, ‘Jupiter is orbited by many moons’ and here ‘…is orbited by many moons’ figures as a predicative expression. Nevertheless, the many moons are solid clusters of tightly connected compresent tropes – material objects like Jupiter – which as moons are existentially dependent on Jupiter, figuring as designata of the predication! In other words: my take 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, unvaried and varied) whose existence in some way depends on a relatively independent cluster of selected compresent tropes… which constitute the individual referred to by the singular term. Consequently this fragile principium individuationis is the only thing that really distinguishes the references of logical subjects. Here are some examples[19]:

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.
Admundsen’s expedition to the South Pole depended on the existence of both Admundsen and the South Pole.

The general idea can be summarized as follows:

In the constitution of a fact represented by a true singular (predicative or relational) statement, the existence of thelex) trope ascribed by the predicative expression is dependent relatively to the existence of the compresent trope-cluster constitutive of the object(s) referred to by the nominal term(s).

In trying to explore this view in more detail, we can remember Peter Simons’ nuclear trope theory of material objects. According to this theory, individuals are in the standard case formed by an essential nucleus or kernel 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 case of material objects, that belonging to the nucleus isn’t only the trope of compresence (a dependent relational trope), but also those of tightness, form, volume and resistance to pressure or solidity and necessarily what the physics call inertial mass.
   Unfortunately, the issue is not so simple. According to our identification rule of proper names (see Appendix to chapter I), the object of reference of a singular term must be located by its identification rule. Regarding proper names, this identification rule requires for its application sufficient and predominant satisfaction of at least a disjunction of fundamental description-rules, which are the localizing and the characterizing rule. 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 counterfactual situations. Examples are the Aristotle born 300 years later in Rome in one possible world and the Aristotle who in another possible world died young and never wrote his opus, although he was born in Stagira in 283 BC. It may be different within limits established by the identification rule. Consequently, in the case of objects there is no necessary condition in re – no real essence of the object – to o be expected, but a nominal essence given by its identification rule. Peripheral tropes, on their side, would be those referred to by our auxiliary descriptions like ‘the teacher of Alexander,’ ‘the founder of the Lyceum,’ etc.
   Searching for a simple example, I will consider here the singular term ‘this chair.’ I consider this phrase as only an indexical name. It 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 moveable 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 plus a spatio-temporal location is what in this case forms an 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 exists in the dependence of the existence of the 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 formed by some kernel of mutually founding compresent tropes. This in fact also applies to properties of individuals that are not properly material objects. A rainbow, for instance, is an individual (a cluster of compresent tropes), though not a material object. The fading of a rainbow is a process-trope whose existence is dependent on the existence of the rainbow in itself.
   Consider now the relational sentence ‘Bucephalus belongs to Alexander.’ This 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 possibility of existence of the relation ‘…belongs to…’ is here indebted to the existence of the more stable essential nuclei of mutually founding tropes constituting the two objects Bucephalus and Alexander. These kerns of tropes referred to by the names ‘Bucephalus’ and ‘Alexander’ are concrete psycho-physical objects that certainly exist independently of the existence of the relatively contingent complex combinations of tropes constituting the trope of ‘…belongs to…,’ since for having ownership we need the previous existence of objects having the particularized relational property of ownership.
   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 Simons’ nucleus of mutually founding tropes for the object referred to by the singular term ‘this chair.’ Suppose now that I point to the chair and say ‘This chair has two armrests,’ since the tropes 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 that I am not claiming that the existence of the chair’s armrests in themselves is dependent on the chair, since armrests are parts able to exist separately from chairs. But in the context of the fact described by the statement ‘This chair has two armrests’ they only exist as dependent parts, since they are armrests belonging to this chair and not two armrests found separated from the chair in some other place.)
   A problem arises when predicates denote tropes belonging to definitional cores (with their ‘in’ and not ‘of’ properties). Suppose that I say, ‘This chair has a backrest.’[20] I think that, despite the tautological character of the statement, the trope ‘having a backrest’ can be considered in its existence to be dependent on the whole cluster of tropes that builds 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. But the seat is still the referred to object, because it has the most relevant tropes; it is still a seat that can be used by only one person at a time, while the backrest has no function. We see that dependence of components on what is defined turns out here to be a question of proportion. One 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 the original object of predication anymore. To exemplify this, suppose someone cuts the chair into two identical halves. One cannot say that one chair-half belongs to the other chair-half. 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 mutually’ or ‘The first chair-half can be joined with the second chair-half to form a whole chair.’
   Consider now the statement ‘The Sun has eight planets.’ Each of these planets is a compresent cluster of tropes with a kern, just like the Sun. However, this does not mean that we will analyze this statement as having nine subjects… No, although the predicate points to eight material objects, each one called a ‘…planet of the Sun’, as part of the fact they exist all in the dependence of the existence of the referred to logical subject Sun: they would not be planets of the Sun if the Sun didn’t existed. Consequently, even material objects may belong to what is predicated. The fact of something being a material object does not makes it something independent. 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. The given criterion of dependence/indepedence, fragile as it may be, is the only thing that allows us to distinguish the logical subject from the logical predicate in the unity of the sentence.
    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 ‘the number two or any multiple of two added to the number one’ (which is the definition of an odd number). Hence, the existence of oddness related to the existence of the number three is dependent on the number three that we are taking into consideration. Consider now the statement ‘Three is a natural number.’ One could argue that to be a natural number belongs to the definition of three as a kind of genus proximum, although not essentially to the (here seen as incomplete) definition of three as its differentia. Maybe this differentia could be given by our suggested attempt to define the number 3 (sec. 3 of the Appendix to chapter III), namely, as a certain triad understood as a second order conceptual property. Then the object number 3 (a universal) could be constructed as an experienced tropical triad defined in Newman’s way or any other strictly similar triad:

Number 3 (Df.) = a chosen triad {{}, {{}}, {}, {{}}}}* or any other triad strictly similar (equinumerous) to {{}, {{}}, {{}, {{}}}}*.

This definition still allows the predicate ‘…is a natural number’ to be ascribed to the whole subject as an internal dependent addition. In any case, even the name of a so-called abstract object, such as the number ‘3’ cannot be moved to the predicate position here, insofar as it refers to something held as independent, being identifiable in the independence of its non-definitional predicates like ‘…is an odd number.’ Another point (considered in Appendix to chapter III) is that numbers and their properties can in this way be considered as tropes.[21]
   Understanding unsaturatedness as relative existential dependence suggests, therefore, that the tropes denoted by the predicate have an inevitable tie of dependence when considered relative to the relevant individual within the fact referred to by the singular sentence. This gives us a better understanding of the asymmetrical relation between subject and predicate.
   Summarizing this section, 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 kern of compresent tropes which exists in the relative independence of the less central tropes in and outside of the nucleus that can be designated by predicates. On the other hand, the opposite isn’t true in the context of the fact referred to by the sentence: the predicate can be nominalized.
   However, nominalization is a tricky deal. The statement ‘Goodness is desirable’, nominalizing the predicate ‘…is good’ is better analyzed as the universal statement: All good (=G) things are desirable (=D)’ or (x) (Gx → Dx), which means that {Ga1 → Da1, Ga2 → Da2… Gan → Dan}. But if you take Ga1 → Da1, for instance, what you have is again the predicate G or ‘…is good’. In other words, nominalization is only a simplifying pragmatic device of ordinary language, and the predicate was in fact never transformed into a singular term. Nevertheless, even at this level the asymmetry remains, since the same procedure cannot be applied to the nominal term, since the predicate can apply to many different objects, while the nominal term is made to be applied to precisely one object, which must be distinguished from all others, what shows that it is the one-many distinction that is at botton of the asymmetry. Anyway, what really differentiates subject from predicate in the sentence is the corresponding independence/dependence of their references. Moreover, what assures the unity of the thought-content expressed by the sentence is simply the dependence/independence in the factual unity. 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.

8. Sense of a predicative term
The independence/dependence relationship originated on the ontological level of reference is reflected on the semantic and linguistic levels. It is first reflected on the semantic-epistemic level of sense. We see this in the fact that the identification rule of the nominal term is applied independently of the ascription of tropes to the object by the ascriptive rule of the predicative expression, while the sense of the ascriptive rule of the predicative expression depends upon 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 would be what makes the singular predicative sentence take the usual subject-predicate form.
   At this point, we can make some reasonable additions, originating from our view of tropes as the designata of predicative expressions. 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 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 the sentences (1) and (2) with the same designatum.
   Another consequence of our understanding of predicative expressions as basically referring to tropes by means of their conceptual rules contradicts the Fregean expectation that the same sense cannot have more than one reference, since the potential for multi-referentiality is inherent to predication. Consider the following sentences:

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

The predicates ‘...is white’ in sentences (1) and (2) obviously have the same sense, for they express the same ascription rule. But the tropes of whiteness (of reflecting all wavelengths of the 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 said of relational predicates.
   Another easy point 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 living trees have roots’ would be analyzed as {Tree 1 has roots & tree 2 has roots &… & tree n has roots}. Here the qualitatively identical tropes of rootness 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 rootness 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 tropes of rootness of many 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 is made of wood,’ which abbreviates a disjunction of ascription rules and refers to at least one trope of rootness belonging to the multiplicityy of trees as objects. We will come back to this issue in the last chapter, when we consider the attribution of truth-value to general statements. Anyway, to the objection that this will never give us the conditions for a truly universal quantification, we could answer that the truth of the universal quantification is usually only probable and enclosed under some domain. What ‘All living trees have roots’ means is in fact ‘[Very probably] all living trees [on the planet Earth] have roots.’

9. Dependence of the predicative sense
The ontological distinction between independence/dependence (saturation/un­sat­uration) is reflected on the semantic-epistemic level to which the senses belong as an outcome of the original ontological dependence. This is clear enough if we see the sense of the predicative expression as an ascription rule. In the context of a singular predicative sentence, the identification rule of the singular term applies to the object as some kern 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, saturated. The ascription rule, on its side, will be applied to a trope dependent on the kern and consequently depending on the earlier application of the identification rule, lacking in this sense completeness.
   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 core of tropes, even the ascription rules of predicative expressions already belonging to the identification rule of the object as part of this rule already require prior application of the whole identification rule for identification of the object in order to become themselves applicable as part of the identification. 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.[22]
   The general sense of a concept-word, which (diverging from Frege) we identify with the concept 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 need, for instance, first apply the rule that allows us to spatio-temporally locate the animal called Bucephalus in order to apply to its related tropes, on that basis, the ascription rules of predicative terms. Thus, due to the independence of the object Bucephalus, we apply ‘... is a horse,’ ‘... is black,’ ‘... is swift’… and also the complex rules of application of more complex predicates like ‘…a horse that belonged to the best Thessalonian breed.’ 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…’ And finally, we need to apply the rule that allows us to mentally identify the number 3, in order to be able to apply to it the ascription rules of predicative expressions like ‘…is odd’, ‘…is a prime number,’ or ‘is the quadrat 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.
   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 isn’t required at all. Indexicals such as ‘that’ and ‘there’ accompanied by some gesture of pointing already identify some particular as anything spatio-temporally localizable independently of any 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 depends upon the previous existence of the object and its identification, but, as a consequence, also the ascription rule of the predicate, its conceptual sense, 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 deep revisions of Frege’s views in order to solve problems in his philosophy, 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 my perspective, the first thing to do is to treat the 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 designates the trope of horseness, and replacing ‘…is a concept’ with C, designating the trope of a thought concept, 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 ‘the concept horse’ does not really work as a nominal 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. We conclude that rightly analyzed ‘the concept horse’ expresses an universal predication and no real singular term. The whole paradox originates from the illusion that always that you put a predicate in the position of a subject transforming it into a definite description you necessarily transform it into a 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 thinking, 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 to be 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, it is a sentence with the form Ǝx(Fx), usually expressing a second-order concept applied to a 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. Existence is so understood something objective, since this satisfaction is independent of our cognitively grasping it as the applicability (and not mere application) of a concept.
   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, that is, 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 ascription rule is effectively applicable in a domain of external objects. I add the adverb ‘effectively’ to make 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 applicable for sure. Moreover, the existence or effective applicability of a semantic-cognitive rule is always considered with regard to a certain domain of entities. 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 to the first case. The statement ‘Headaches exist’ applies to the second case. Thus, what is meant with 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. Furthermore, this effective applicability is usually within what we may call its 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, whose proper domain is a fictional one, 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 apply the predicate of 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 horses are effectively applicable in the fictional domain described in the libretto. Thus, there are fictional domains in the arts, domains of imaginable and plausible entities, domains of so-called abstract entities and their various sub-domains, like the domain of mathematical entities, of logical entities, etc., and the word ‘existence’ can be applied to concepts belonging to all of them.
   So, according to the view I support, 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. Finally, to say that an entity exists is, apart from certain exceptions, to say that its conceptual rule is effectively applicable tout court is to say that it is applicable in the already conventionally established as its most proper domain of application, although this rule can be implicitly or explicitly disclaimed. 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. It is normally assumed that the attribution of existence is made in its most proper domain.
   As we have 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 an ascription rule are apt to be satisfied by corresponding objective criterial configurations. These criterial configurations must be configurations of tropes usually belonging to more complex constructions from tropes called facts – another point against Frege that I intend to 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 our 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 conceptual rule that demands 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 proper domain.

In my judgment, the advantage of this last form of analysis is epistemological: we are able to better scrutinize the nature of our existence-assignments, as will be shown in our answers to objections.

12. Two naive objections
There are two naïve objections to my formulation of the higher-order view of existence, whose answer is revealing. The first is that the concept of effective applicability of a rule is 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 which are essentially mental, though often also inevitably physical, 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 verifying 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 being required is that if the rule existed, it would be effectively applicable. Thus, there is no doubt that the concept of effective applicability is not anthropomorphic.
   This answer makes it easier to refute a second naïve objection that proponents of the idea that existence is a property of things instead of concepts would be tempted to raise against our proposal. It would be the objection that 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 said to exist. However, this seems odd, since intuitively we think that existence in some way belongs to entities 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 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 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 world, and this 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, this couch’s trope of red 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 detailed 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. However, this also means that the couch’s trope of redness secondarily owns the meta-property of its ascription rule of being effectively applicable to it – it has this property-property. That is, since the property of existence is the ascription rule’s property of being effectively applicable to the trope of redness of the sofa, that property of the ascription rule is a meta-property of the trope of redness, since through the ascription rule it is indirectly and dependently applicable to the trope of redness belonging to the real empirical world. Finally, the higher-order property of existence or applicability of the rule must be where the rule is, that is, it must be spatio-temporally located, being therefore a trope. Existence isn’t an exception of our all-embracing trope ontology.
   In short: the meta-rule of existence is a dependent trope that also applies to the trope, even if in a subordinate way. Thus, one can argue that 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 the given entity belonging to the chosen domain of entities without being a proper constituent of this entity.

13. 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 term express senses, and their references can be said to exist. Since singular terms can be divided into definite descriptions, proper names and indexicals, I will briefly consider each of them, beginning with definite descriptions like ‘the inventor of the Maieutic.’
   Applying to the sentence the Russellian device to treat definite descriptions and replacing the predicate ‘…is the inventor of the Maieutic’ with M, the sentence ‘The inventor of the Maieutic existed’ can be analyzed as:

Ǝx [Mx & (y) (My → (y = x)].

In this way, we are affirming the existence of at least one and not more than one inventor of the Maieutic, which 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 (Socrates) who once existed.[23]
   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 Russell’s method (disregarding his failed metaphysics of meaning) 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).’[24] Taken literally, this suggestion is not only linguistically atrocious, but also defective, 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 that are 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 present in one way or another in the writings of Frege, Russell and Wittgenstein, and made more explicit by P. F. Strawson and John Searle. According to this theory, the whole sense of a proper name is given by a cluster of definite descriptions. To illustrate this, we can assume that the sentence ‘Ǝx(x socratizes)’ is an abbreviation of what could be more extensively and adequately expressed as:

Ǝx {x is inventor of Maieutic, x is mentor of Plato... x is Xanthippe's husband}.

Of course, this is still insufficient, 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 be easily remedied by means of the Russellian device of restricting the number of objects of predication to only one:

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 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 be still symbolically formulated as follows:

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

Here the meaning of a proper name is 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, effectively applies in the assumed context. As this rule for the identification of the 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 the proper name is the same as the effective applicability of a larger or smaller number of conceptual rules of predicative expressions to one and the same 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 Russellian device is limited, but also because it amounts to some version of the bundle theory of proper names with its well-known difficulties, which were already persuasively pointed out by Saul Kripke, Keith Donnellan and others.
   However, this 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 Donellan’s objections have few effects against the most explicitly developed versions of descriptivist theories, some of them having 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 in a satisfactory way. These two first points lead us to the conclusion that bundle theory hasn’t yet been conclusively refuted.[25] Indeed, perhaps it just needs a stronger defense.

14. Existence of objects and its identification rules
The third reason to be considered is that the above presented formal analysis is a crude simplification when seen from the viewpoint of my own much more elaborated version of the bundle theory of proper names. This version has (I hope to demonstrate) a greater explanatory power than any previous theory, making possible strong defenses against the most diverse counter­examples. Although the theory and its consequences are too complex to be discussed here at length, I have already summarized it 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 seems to have been overlooked: the bundles have no internal order. The theory does not tell us what description or combination of descriptions have more or less importance or even why some seem to be very important for applying the name, while others are obviously irrelevant for it. Definite descriptions are expressions of rules that should help us to connect the proper name with its reference; they are in this sense 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 allows us to recognize 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 do this: check how well-known proper names are treated in encyclopedias. If we do that, we will easily distinguish fundamental from merely auxiliary descriptions, which are accidental. If we do this, we will 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 reason to use the name.

Consider, for instance, the name ‘Adolph Hitler.’ Here is what is said about him 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 centre 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 – the accidental, auxiliary descriptions. Examples of them are that Hitler was ‘the lover of Eva Brown,’ ‘the son of Alois Hitler and Clara Pölzi’[26], ‘the person called “Adolf Hitler”,’[27] ‘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 extended form in biographies.
   You will find a similar pattern if you search in encyclopedias for other proper names like ‘New York,’ ‘USA,’ ‘Eiffel Tower,’ ‘Niagara Falls’ or ‘Venus’. 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 know who Sam is, you can probably get relevant information from his drivers license, employment record, police record (if any), school reports, club records and passport, perhaps adding information given by family and friends about his personality, character, education, interests, abilities, relationships, accomplishments, etc., which are linked together by only 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 noted this point 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 called 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 Hitler, since he does not satisfy any of the disjunction.
   Moreover, there are two other complementary conditions that need to be added. First, a condition of sufficiency that needs to 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 in 20 April 1889 in Braunan an In as the son of Alois and Clara. However, he had the same career as Adolf Hitler only up to the point where he was not rejected but rather accepted by the Vienna Academy of Fine Arts in 1907, becoming a rich painter who unfortunately died in his early thirties. In this case, we are inclined to say that this person is our Hitler in this counterfactual situation, although he satisfies only the localizing description-rule, and even this only partially. He satisfies 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 the considered 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 in… 20 April 1889… but only Rudolf went to Berlin, served in World War I and later headed the Nazi Party, presiding over World War II and the Holocaust, while Adolf chose to become a farmer in his native Austria. We would tend to think that Rudolf was 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 lack of relevance of the auxiliary descriptions.
   Finally, it is important to see that the object to be referred to belongs to the nearest relevant class that does not mix with the contents made explicit by the fundamental conditions (for instance, being a human being – and not a politician – for Adolf Hitler, and being a celestial body – and not a planet – for Venus).
   Bringing all this together, we are able to propose the following general form of the identification rule for proper names, a form that must be satisfied by any bundle of descriptions associated with a given proper name:

    General form of the identification rule for proper names:
In any possible world a proper name ‘N’ has a bearer iff it belongs to the nearest relevant class of referents, so that more than any other object it sufficiently satisfies at least the conditions set by (A), its localizing description-rule, and/or (B), its characterizing description-rule. (Auxiliary descriptions may be helpful in the identification).

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 we associate with the proper name ‘Adolf Hitler’, we get the following identification rule for this person:

In any possible world, the proper name ‘Adolf Hitler’ has a bearer
 iff
this bearer belongs to the class of human beings, so that sufficiently and more than any other person he satisfies the following disjunction of conditions:
(A)   being born in 20 April 1889 in Braunan an Inn… living the last part of his life in Germany… dying in 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 auxiliary descriptions like being ‘the lover of Eva Braun,’ ‘the person called  “Adolf Hitler”,’ etc.)

This 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 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 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 some homonymous.
   This example also outlines the lack of relevance of the 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 Clara 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 puzzlement, not persuasion. For his Eva Braun could not be the well-known Eva Braun who committed suicide in the Bunker… the identity of names of his parents would be only a curious coincidence… (He could not, it is true, satisfy the description ‘the author of Mein Kampf ’; however, this is no auxiliary description, but part of the full characterizing description of our Adolf.) Anyway, at no point will this changes our belief that he is not our Adolf.
   Since so understood the identification rule simply defines what object among all others owns the proper name by establishing the (definitional) criteria for identification of the bearer of the proper name in any possible world, it also applies in any possible world where the bearer of the name exists, satisfying the main point of the Kripkian 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, being therefore only accidental or flaccid designators.[28]
   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 as far 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. – If someone already knows that Hitler was ‘some dictator’ or thinks erroneously that he was ‘a militar,’ this person already classifies him correctly as a human being 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 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, being a conceptual rule, it has the second-order property of being effectively applicable in its proper contextual domain – a domain of human beings belonging to the external world – what allows us to say that the bearer of this name really existed.
   Finally, it is interesting to note that we can try to give a Russellian formulation for the form of the identification rule. Calling the predicate ‘…satisfies its localizing condition more than any other object’ L and the predicate ‘…satisfies its characterizing condition more than any other object’ C, we can say that a proper name N has a bearer if and only if, regarding the nearest relevant class of objects:

   Ǝx [(Lx ˅ Cx) & (y) ((Ly ˅ Cy) → (y = x))]

The difficulty (for the logic) is that this is an insufficiently adequate formal replacement of the general form of the identification rule for proper names, since it leaves unclear the satisfaction of the condition of sufficiency and takes for granted the existence of only one object satisfying the rule. But it is already enough to formally show that the existence of the object referred to by a proper name can also be seen as a property of concepts.

15. 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 point to 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 the period of time of the day before the day of the speaker’s utterance… There is more to say than just this regarding indexicals like ‘I,’ ‘she,’ ‘he,’ ‘they’. Plainly, these personal pronouns cover more than the simple spatio-temporal, but this does not matter to us now.
   Consider now the indexical statement There is a raven.’ How should we analyze it? First, there is the spatio-temporal location usually pointed by the speaker, which is relative to the spatio-temporal location of the speaker within some context. All this must be assumed in applying the demonstrative ‘there.’ The identification rule for ‘there’ can 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 we have applied this identification rule, we are allowed to apply the ascriptive rule of the predicate ‘…is a raven’ to the existing location. Now, how would we symbolize this? Calling the indicated spatio-temporal location L and the predicate ‘…is a raven’ R, it could be simply Ǝx [(Lx & Rx) & (y) ((Ly & Ry) → (y = x))], which means: ‘There is precisely one x spatio-temporally located in L, and this x is R.’ Although the location L figures here as a predicate, the condition of unity (any y = x) makes its reference an individual spatio-temporal location possibly 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 has two armrests.’ In these cases, considering that the phrases ‘that raven there’ and ‘this chair’ refer to only one specific object, distinguishing it from all others, these phrases work as singular terms and must be analyzed as expressing an identification rule. So, replacing ‘That raven there’ with R and ‘…is flying’ with F, we can formalize the sentence ‘That raven there is flying’ as Ǝx [Rx & (y) (Ry → (y = x)) & Fx].  
   When we use an indexical statement, the language ‘touches’ the world, which makes indexicals the ultimate, indispensable roots of reference. Because of this, although the sense still determines its reference, there is here a double direction of fit. First, with the help of our sensory cognitions, we create the identification rule for indexical when it is for the first time used in a determinate context. Once formed, this rule (the sense) determines the spatio-temporal location, often together with the sortal object. Now, a new identification rule can sooner or later be applied again, and may be interpersonally conventionalized by association with new non-indexical terms of our language that can be definite descriptions that can be flexibilized when abbreviated as proper names.

16. Advantages of the higher-order view of existence
There are several advantages in conceiving existence as a higher-order property, that is, 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, assuming 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 meta-property of objects and not a proper intrinsic first order property or trope belonging to 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 do not find problems with the denial of existence. Suppose that existence were a first-order property of an object. In this case, in a sentence like ‘Vulcan does not exist,’ the negation of existence should 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. But since in order to identify the object we first need 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.
   Anyway, according to our Fregean view, this conclusion isn’t necessary, because all we do by denying the existence of Vulcan is to admit that the ascription rule that forms the concept of Vulcan doesn’t have the 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 between Mercury and the Sun) & (y) (if y is a small planet orbiting between Mercury and the Sun, then y = x)].

What falls under the scope of ‘~Ǝx’ are the concepts constitutive of the identification rule, which in my illustration consists of an ascription rule for a predicate that must be applied to only one and the same object. What ‘~Ǝx’ does is just to deny the property of this rule of being effectively applicable to the corresponding physical object.[29]

17. The 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 merely imaginary ones exist? Some believe that 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 fictional entities. Beginning with the first ones, it is clear that we can find a sense in which they exist. Although the planet Vulcan has been show not to exist in the real external world, its proper domain, it surely has existed in other domains, such as in the minds of many astronomers in the past who searched for it… 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 the proposed view, because our identification rules can also be applicable, at least partially, 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 in my imagination (even if in a vague and fictive 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 of his time, though not in its 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. In opposition to 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. It has no sense or existence in the domain of the real external world, 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 play 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 application of a proper identification rule in this domain, which is supported by the existence of the plot of the fictional play Hamlet in a second-order domain. This play is in turn supported by the identification of some writer and written papers in the first-order domain of our self-sustaining 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 ways of use 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 conceived existence. But I think that the intention was to underline the importance of conceived entities, underplaying or obstructing its derivative 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 round thing, so that both can identify one and the same thing, since they are from 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 rules because it cannot build 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.[30]
   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 to 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 applicability of meta-rules to first order conceptual rules – also applies. Consequently, it is safe to conclude that existence itself exists. Well, then, does the existence of existence also exist? Surely: since we can conceive a meta-meta-meta-rule of existence demanding effective applicability of the meta-meta-rule of existence to meta-rules of existence, we can conclude that the meta-meta-meta-rule of existence is also applicable to the foregoing meta-meta-rule, making it consequently existent, and so on in an infinite regress, which is virtuous since always stoppable.

18. 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 view of philosophers like Frege and Russell, I will answer at least some of them, as they were formulated by Collin McGinn (2000b: 21-30). The answers are helpful in clarifying my own view.
   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. Furthermore, 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 Kripkian 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 variable cluster of tropes displaying compresence, the objection appears to us in a different way. Since not only the ascribing 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 semantic-cognitive rules in some chosen domain or context. However, since they also apply to objects as compresent clusters of tropes, this means that we cannot conceive any object as being given – that is, as existing – without simultaneously conceiving its identification rule as 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 physical context. 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 endows the object with 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 attribution of truth follows from the application of the first two rules, it is wrong to demand that the attribution of truth requires the attribution of any property prior to the attribution of existence of this property and the object as a cluster of properties (tropes). 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 objection: uninstantiated properties are said to exist. But in order to exist, a uninstantiated property must fall under a higher-order property attributing its existence. This higher-order property must also exist, which means that 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 of a first-order property.
   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 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 also a semantic-cognitive rule that is only imaginatively applicable not only endows its reference with existence in an imaginary domain, but can be said to exist by having the higher-order property (or trope) effectively applicable in this domain. In any case, the property (or trope) of existence exists, which means that we can say that the applicability of the rule has the property of being applicable, and so on indefinitely. This leads us to an infinite regress, of course. But this is a virtuous infinite regress, since once the applicability of a conceptual rule is admitted by the application of a higher-order rule to it – a rule that is satisfied by the instantiation of this applicability, an existence-endowing rule – we don’t need to bother with all the unlimited further applicabilities of applicabilities or existences of existences that the existence-endowing rule generates. 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).
   The third objection is that there are statements that resist the traditional paraphrase. First, there are statements ascribing existence to particulars, such as ‘Venus exists.’ 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).’
   To this objection the answer is too easy. What ‘Something exists’ means is that there is at least one trope or 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. But this possibility is shown even by our logical symbolism on an elementary level. McGinn seems to have forgotten that in logic we can symbolize an undetermined property as F. That is, if we translate ‘Something exists’ symbolically, we get Ǝx(Fx). Moreover, there is nothing wrong with Ǝx(Fx). Often we reach this result by applying existential generalizations to singular statements like ‘Venus exists.’ Calling Venus V, if it is true that ‘Ǝx(Vx)’ this implies by existential generalization that some property exists or ‘Ǝx(Fx), namely, that some conceptual rule is effectively applicable. Hence, there is no mystery in accepting the existence of undetermined properties.
   He reminds us that there are also more complicated statements which 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 them, 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 are sufficient for the attribution of existence in that purely fictional context. Statement (2) can mean that there are things that exist only in the imagination, but not in the external world, that is, there are identifying rules that are effectively applicable only in the ineffective 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 our imagination. Finally, statement (3) means that there is at least one thing that exists only in the mind but not in external reality. It attributes existence to at least one x as existing in the mind, though not in external reality. Indeed, it seems obvious that the identification rule for some objects, though effectively applicable in an imaginary, fictional domain, are not 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.

19. 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 know with certainty we would perceive under suitable circumstances.[31] As Berkeley wrote:

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. (Berkeley 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 or could be 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 reasons – observational or not – to have a firm expectation that under suitable circumstances they will be experienced 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. 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. 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 italic)

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.
   Nonetheless, 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 one from the 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 as) possible.

This would meet the requirement of multiplicity and diversity of material objects and their presentations, because each material object would be constituted by innumerable different groups of sensations that could always be possibly distinctly experienced under suitable circumstances. 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. Consider the expressions:

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

Expressions (1) and (2) seem to 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 criterial configurations.

Although (4) is only a case of (3) and (5), it seems clear that when we interpret existence as (4) we are saying something similar 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 approached to existence. That is:

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

This point 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-reflexive) 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 tropes. Now, when we interpret these variable (supposedly objective) 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 warranted 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 moveable 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 tropes. But the criterial configurations (the subjective, 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 not only (i) when its conceptual rule is effectively applicable, but this effective applicability is only the case when (ii) tropes 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 contents of sensations whose experienceability is warranted or permanently possible. Moreover, as Mill suggested, (iii) this experienceability must be (at least in principle) 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. At least 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 contribute to warranting the attribution of external existence.
   There is, however, an important and seemingly fatal objection to Mill’s view of matter, which is made more acute by the Berkleyan correction that I have made[32]: It is that the group of sensations or configurations of sensory criteria that satisfy a conceptual rule are by their nature inevitably psychological. Even sensations that are warranted as permanently possible (sensibilia) must be psychological in a dispositional way. This means that if we follow this path, we may fall into some kind 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. 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 experienceability… But all this seems to be insufficient to perform the magic of making sensations (or even ‘contents of sensations’) match what they aren’t, as a supposed elements of a non-mental objective external world of material tropes. This is a pressing objection, whose answer will be given only in the final chapter of this book, as a consequence of our discussion of the correspondence theory of truth in its relation to direct realism.
   Notwithstanding, we can now anticipate something of the way we can deal with the problem. Having in mind our improved views of existence, we can ask: What is, in more conventional language, the 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 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 for sure not what Plato called the 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 of this rule specularly projected 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 (though inverted) would be able to give unity to the multiple and variable criterial configurations through 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 (and not necessarily as it is in itself) must have the structure of an inverted specular tree with branches whose ramifications end up in objective criterial configurations that (supposedly) should match correspondingly subjective criterial configurations. Furthermore, these objective criterial configurations should be nothing but tropes and constructions out of them (objects, facts). Of course, this objective structure should be putative, so that the rule could always be corrected or improved as an effect of new information regarding such specular objective counterpart. This sounds as wild metaphysics, at least for the moment.

20. The reference of a sentence as its truth-value
Now we leave this speculative excurse and come back to the more tangible Fregean semantics, considering what he has to say about the reference of the sentence. Here there is no compliment to be made. Frege was the author of the insane idea that the references of sentences are their truth-values, and the thoughts expressed by them are 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 a way similar to those of names, and their truth-value is also closed. Loke the object refered to by a name 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 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 ‘Napoleon lost his last battle’; both sentences remain true. Since the references of the sentence-components do not change, the reference of the whole sentence likewise does not change. The truth-value remains the same: The Truth. Hence, the reference of these two sentences 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 view 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’ by ‘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 sentence’s 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 his solution sounds false, because it in fact violates Frege’s own principle of compositionality. If the reference of a sentence is its truth-value, it cannot be established by its parts, since truth-value has no parts. And even if it had, 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 both partial sentences refer to the same thing: The True. So interpreted Frege’s view is hopeless.
   The most charitable interpretation is that Frege uses the word ‘reference’ as truth-value because it is what gives them value for us, because the word Bedeutung (meaning) in Germany, more than in English, also means relevance, what points to semantic relevance or meaningfulness (e.g., Tugendhat 1992b: 231).[33] 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.
   The main problem with this view is that it is in dissonance with expected principles of Frege’s own theory. Since the reference (Bedeutung) of the parts of a singular sentence are really their references – the concept and the object that can fall under it – the truth-value as relevance satisfies the principle of compositionality in a non-linear form, since relevance is only an adjective applied to the truth-value. This is unconvincing if compared with the principle of compositionality applied to senses, in which the whole and its components are linearly arranged in the same ontological realm. Furthermore, why not to say that the applicability/non applicability of the proper name is what contributes respectively to the truth-values true and false of the thought-content more properly than the existence/non-existence of its referred object?
   Finally, when we take the truth-value for the reference of sentence, this view can be – and in my judgment really has been – seriously misleading from an epistemological standpoint. Since truth as in some way belonging to thought has nothing to do with anything that can reasonably be understood as the reference of our statements, viewing truth as reference leaves the relation between language and the world virtually outside of philosophical reach.

21. 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). Since it is more plausible that the references of sentences are facts, it is important for us to investigate the logical structure and ontological nature of facts.
   Concerning the logical structure of facts, the most plausible view is that they correspond to the logical structure of the thoughts representing them, assuming these thoughts are what declarative sentences express when logically analyzed in accordance with the context in which they are presented. But the botton line is are the logic structures constitutive of thought. Singular empirical statements such as ‘Plato has a beard’ and ‘The cat is on the mat’ respectively represent facts that should have the logical structure Fa and bRc. The elements a, b and c, as particulars, refer to clusters of appropriate compresent tropes, while F and R would be also seen as tropes, usually complex tropes forming complex configurations dependent on the related clusters. The links b-R-c and F-a, in turn, are only pseudo-relations, since the admission of their existence as relational tropes would generate an inevitable infinite regress. As we already noticed, individuals and their tropes are linked by ‘non-relational ties’ without any ontological addition (cf. Appendix to Chapter III).
   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 analysed sentence would be ‘Stockholm = the capital of Sweden.’ The most appropriate analyse would be to consider ‘…is the capital of…’ as a relational predicate completed by the proper names ‘Stockholm’ and ‘Sweden.’
   It seems also possible to analyze proper names using Russell’s technique of transforming them into quantified predicative expressions, insofar as to a limited degree this device mirrors my own defense of a neo-descriptivist theory of proper names. The structure of facts must correspond with the structure of the so-analyzed sentences that express the structure of thought.
   Finally, we have 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, whose structures will be better clarified later (in my view the philosophical problem of a hidden lingua mentis ends in the logical analysis).

22. 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 theory of truth, according to which empirical facts are truth-makers normally seen as complex contingent arrangements of elements in the world, which are nothing but arrangements of tropes and their combinations.
   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).[34] 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 being the case with states of affairs and situations.[35] 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).
   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 truthmaker, whatever condition it is?[36]
   However, it seems clear 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 properly empirical. It is hard to assign empirical status to the fact that 2 + 2 = 4. And we can say it is a fact that the Sun is not green, although this seems only a linguistically different way to say that there is no fact that the Sun is green. What I want to defend here is that there is a privileged sense of the word ‘fact’ that involves reference to empirical facts, particularly so-called observational facts, which should be considered objectively real: they exist in the external world and they are possible truthmakers.
   There is a well-known and very convincing reason to think that facts can be constituents of the empirical world. It 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 (supposedly abstract contents) can be causally active in the empirical world producing these effects. 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’ forgetfulness of the gas being turned on caused his death; the fact-circumstance that today is a holiday causes the canceling of a class; the fact-event of crossing the Rubicon concretized a state of affairs that causally determined decisive political changes in the Roman Empire.
   Furthermore, I believe 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 this view, empirical facts are contingent arrangements of elements 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 clusters of compresent tropes referred to by proper names). This would be the case with facts as simple as those referred to by the sentences ‘Frege had a beard,’ ‘Lydia suffers from agoraphobia’ or ‘the book is on the table,’ but also with their combinations.
   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
              Non spatio-temporal                            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 example: I can replace ‘the event of the sinking of the Titanic’ by ‘the fact of the sinking of the Titanic,’ but I cannot replace ‘the fact that Mt. Everest is more than 8,000 m. high’ by ‘the event of Mt. Everest being more than 8,000 m. high.’ 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 or whatever. So considered, events are sub-types of facts: Linguists would say that the word ‘event’ is a hyponym of the word ‘fact.’ Considering things in this way, it is easy to distinguish two great sub-classes of facts:

1.     STATIC FACTS: Can be formal or empirical, the latter when clearly located in space and time. On the 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 truthmakers of a static kind. And ordinary language has names for them: they are called (with different semantic nuances) ‘states’, ‘situations,’ ‘circumstances,’ ‘conditions’, ‘states of affairs,’ etc.
2.     DYNAMIC FACTS: 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 truthmakers of a dynamic kind. And ordinarily they can be called (with different semantic nuances) ‘events,’ ‘episodes’, ‘occurrences,’ ‘occasions,’ ‘pro­cesses,’ etc.

Facts said to be formal, like the fact that 7 × 8 = 56, are static in the innocuous sense that they don’t need to be seen as spatio-temporally located. They are not our major concern 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 unhealthy state, the situation that I am lying in bed, the circumstance that the airport is closed, the state of affairs that Venice has many canals or that the Earth orbits the sun. The Earth’s movement of revolving around the sun counts as belonging to a static fact because it is an internal cyclical relationship that remains the same during the fact’s existence (but each orbital period is an event).
   Dynamic facts, on the other hand, are defined by continuing changes in the relations among their elements during the period of their existence. The process of World War II, for instance, 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 shorter, occurrences when their duration isn’t as short, processes when their duration is longer. Examples of events are an explosion or the lightning flash under dark clouds. An example of an occurrence is a volcanic eruption. The process of global warming is a very slow natural process, slower them the economic process of 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 entirely grasp them before they end. Important is to see that all these things can not only be called events, occurrences, occasions, happenings, processes… but also facts, since they are nothing but empirical facts and truthmakers of a dynamic kind.
   We are now able to find what may 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, but I can also speak about the state of affairs of my book being on the table and of my lying on the bed. The seemingly conclusion is that if that-clauses have some function it is that of emphasize static facts and exclude dynamic 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 the term ‘fact’ inherits the property of being used indifferently, with or without the that-clause. 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 you 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 stated with or without that-clause.
(b) Dynamic facts (events…): cannot be stated with a that-clause.
(c)  Facts in general: admit both cases because 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 durable. 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 dictator and up until his assassination. It was not something that existed in Greenland or that endures until the present, even if we often use the present tense to speak about historical facts.
   The relevant conclusion is that by having the broadest scope, the underrated word ‘fact’ remains the ideal candidate for the role of truthmaker in a correspondence theory of truth. It is so because facts are the universal truthmakers.

23. Church’s slingshot argument
As we 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 invented a slingshot argument. He wanted to show that by means of intersubstitutability of co-referentials we can prove that the most diverse sentences can have only one reference, namely, their truth-value.
   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 initially expose his example of a 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 altogether twenty-nine Waverley novels.
3.     Twenty-nine is the number such that Sir Walter Scott is the man who wrote altogether that many Waverley novels.
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). As (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 the 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 – they are partial references to Walter Scott. In sentence (2) again, the nominal expression ‘Sir Walter Scott’ and the definite description ‘the man who wrote the altogether twenty-nine Waverley novels’ both refer to the same Walter Scott. The third sentence is the tricky one. Its reference is unclear: Walter Scott? The number 29? Both in one? An amalgam like Scott-29? The answer appears when we paraphrase sentence (3) so that it gives back in a transparent way its complete informative content. Now, considering the confuse sentence (3) carefully, we see that the only way to reveal its content in a transparently clear 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 altogether that many Waverley novels.’ Or for the sake of clarity, replacing ‘=’ for ‘is (the same as)’ and ‘&’ for ‘and,’ we have:

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 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 the 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 term that flanks the identity sign has the number 29 as a 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 altogether 29 Waverley novels.
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 these sentences are all true, Church’s argument has by now lost all its initial plausibility. Sentences (1) and (2) have as partial references Walter Scott under different guises. However, sentence (3) is a conjunction of two identity sentences, each one 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 of its terms only the number 29 (as the number of counties in Utah), without referring to Walter Scott, as it should be. This means that in the composed sentence (3), the second sentence of the conjunction is the only one that preserves as its partial reference the partial references of (1) and (2), while (4) is an identity sentence that has as its partial reference only the partial reference of the first sentence of (3), which clearly has nothing to do with the partial references of sentences (1) and (2) and supposedly with their references. That is, in a subrreptitious way the replacements slip equivocally from 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 the complete sentences, that the references of sentences (1) and (4) should be different. Initially the flaw isn’t 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) are something like an amalgam of Walter Scott and 29, say, a ‘Scott-29,’ while they are in fact totally distinct. The replacements would only respect the compositionality principle, warranting the sameness of the sentences’ references, only if the argument could prove that the partial references of all the sentences could be replaced without surreptitiously inviting the reader to conjoin in sentence (3) partial references that are to completely distinct objects.

24. Facts: 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 helpful 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 perspectival fact: the fact as what is immediately revealed through a particular mode of presentation expressed by the sentence. I will call it a sub-fact and make the different sub-facts responsible for differences in the modes of presentation constitutive of the different sentence-senses or Fregean thoughts about one and the same thing, e.g., the planet Venus, Walter Scott, or the number 29. This is why Church’s sentences (1) and (2) can be seen as expressing different senses or thoughts, namely, by evoking different perspectival sub-facts. They expose 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 altogether 29 Waverley novels … In this way, sentences (1) and (2) respectively show two sub-facts that contain perspectivals objects of reference that differ from one another. Here they are, using the term ‘being’ to indicate that we are speaking about something objective:

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

Nonetheless, it is obvious that (1) and (2) are also identity sentences. Each of these sentences can be understood as referring under different guises to only one object, justifying their ‘is’ of identity. In this sense, sentences (1) and (2) also represent identities, which can be directly expressed by ‘Walter Scott = Walter Scott.’ They express the self-identity of Scott considered in full, as the ultimate bearer of all descriptions (and all possible perspectives) to be attached to it. Among the descriptions attached to the name ‘Walter Scott’ we can find ‘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 altogether 29 Waverley novels,’ expressions that constitute (1’) and (2’). This primary fact that unifies all the sub-facts revealed by its multiple modes of presentation is what I call the grounding fact. It is what makes sentences with the form a = b identity sentences.
   Now, consider these issues in more detail: 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 it also exposes something external, for instance, ‘the author of Waverley’ evokes ‘being the author of the Waverley novels,’ which is, we could say, an objective sub-object mediating our reference to the object 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:

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 ≠ (isn’t 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 ‘Scott [in full] = Scott [in full].’
   In other words: when I say ‘The author of Waverley is the author of Ivanhoe,’ I am saying two things. First, with the intentional modes of presentation I am evoking an objective difference that can be represented by the sentence ‘Being the author of Waverley ≠ (isn’t) being the author of Ivanhoe.’ Indeed, it is an objective factual difference that a person who is writing Waverley is not the same as a person writing Ivanhoe, even if they are both the same person (he was writing different stories at different 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 author of Waverley = the author of Ivanhoe,’ indicating that under different guises I am presenting the grounding fact that ‘Walter Scott = 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 what I call the identities in their differences.[38]
   Now, assuming the kind of neo-descriptivism proposed in appendix 1 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:

(1) Sub-fact: Being Sir Walter Scott ≠ being the author of Waverley. (1) Grounding fact: Walter Scott = Walter Scott.

(2) Sub-fact: Being Sir Walter Scott ≠ being the man who wrote the altogether 29 Waverley novels.
(2) Grounding fact: Walter Scott = Walter Scott.

(3) Sub-fact: (Being 29 ≠ being the number of Waverley novels) & (Being Sir Walter Scott ≠ being the man who wrote the altogether 29 Waverley novels).
(3) Grounding fact: (29 = 29) & (Walter Scott = Walter Scott).

(4) Sub-fact: Being 29 ≠ being the number of counties in Utah.
(4) Grounding fact: 29 = 29.

The sub-facts show why the semantic-cognitive contribution of each referential component in identities with the form a = b is different. For instance, the sub-fact that Sir Walter Scott wrote 29 Waverley novels discriminates more than the sub-fact that he wrote the Waverley novels, and in true sentences this discrimination isn’t just a mentally considered mode of presentation, but also something objectively, factually given (in Frege’s words, ‘the way the object gives itself to us’). The evocations of these sub-facts all lead us to the grounding fact that in the end all the different senses refer to something numerically identical. On the other hand, in sentences with the form a = a, as ‘the morning star = the morning star,’ the sub-fact is an identity ‘the morning star = the morning star,’ while the grounding-fact may be the same identity or the identity ‘Venus = Venus’, depending from the context involved.

25. Taking seriously the sentence’s reference as a fact
I hope I have shown that the most plausible option concerning the nature of reference is to follow philosophers like Russell and Wittgenstein, who assumed that the reference of a statement is a fact – a fact that in the empirical case is understood as a contingent arrangement of cognitive-independent tropical components more commonly given (completely or partially) in the external world. 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 last ones being (as sub-facts) 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.
   If we assume this answer, questions arise. The first is the following: how do we establish 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.     The morning star is Venus.
4.     Venus is the second planet orbiting the Sun.
5.     Venus is the most brilliant 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 linguistically 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 partial references, the planet Venus, it seems that in the end all these sentences must also have the same reference, representing the same fact…
   The point already touched in the last section comes back: in the last cases there must be a privileged grounding fact able to be described that can be identified as the ultimate truthmaker of all these identity sentences about the planet Venus. 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 task can be accomplished by the references of identity sentences between proper names, insofar as they are their identification rules considered in full, that is, including all their fundamental and complementary constituents.
   Now, assume our proposed view is correct of proper names’ meanings as abbreviations of bundles of descriptions applicable according to fundamental identification rules. Then the proper name ‘Venus’ in full (i.e., considering all its conventionally assumed descriptions beyond the limited knowledge of this or that particular speaker) 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 most brilliant 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 according to most identification rules, any description of the bundle may be empty). If my view is correct, then there is a sentence that could describe the grounding fact. This is the ultimate truthmaker or verifier of any identity sentence concerning the planet Venus, including the sentences from (1) to (7) above. Here is how we can present it:

8.     Being Venus = Being Venus.

My contention is that rightly understood this sentence can express an ideal grounding thought able to refer to the single grounding fact, which – once regarded in its entirety – can serve as the truthmaker for any identity sentence about the planet Venus.
   It is not hard to explain why. If the full meaning of the proper name ‘Venus’ (‘Venus’ in full) is understood as an abbreviation of a bundle of descriptions that uniquely identifies its object, then this proper name must include descriptions like ‘the morning star,’ ‘the second planet orbiting the Sun,’ ‘the most brilliant planet visible in the sky,’ etc. Consequently, from the sentence ‘Venus [in full] = Venus [in full]’ we can inferentially derive the 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’ abbreviates, and the second occurrence with the description ‘the evening star,’ which the name ‘Venus’ also abbreviates. We can also inductively infer all the other above exposed co-referential identities. In this way, the sentence ‘Venus [in full] = Venus [in full]’ would ideally be able to represent a fact complex enough to involve the sub-facts represented by each of the thoughts expressed by the above sentences, which may be seen here as contingent a posteriori. (cf. the meaning of ‘Venus’ as presented in an encyclopedia.)
   In order to further 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 grounding fact would be:

6.     Being the number 4 = being the number 4

But could the sub-fact expressed by sentences (1) to (5) be derived from (6)? The answer must be in the affirmative, since we are dealing with a deductive system. After all, I have written the five sentences above simply by conceiving deductive inferences from ‘4 [in full] = 4 [in full]’!
   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 tentative 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 as its proper meaning the following identification rule:

IR-Venus: In any possible world our proper name ‘Venus’ has a bearer, iff this bearer belongs to the class of celestial bodies that satisfy sufficiently and more than any other the condition of being the second planet orbiting the sun, located between Mercury and the Earth… (To this it is helpful to add, e.g. probably applicable auxiliary descriptions like ‘a planet somewhat smaller than the earth,’ ‘the morning star,’ ‘the evening star’, ‘the most brilliant planet visible in the sky,’ etc.)

I think this is a kind of ‘one-foot’ identification rule, since the characterizing rule is the only fundamental one and includes what would count in the localizing rule (being a planet). For suppose we have as localizing rule ‘a brilliant planet somewhat smaller than the earth.’ In this case one can imagine that if there is an only brilliant planet somewhat smaller than the Earth, this planet would be Venus, since one term of the conjunction of fundamental rules is satisfied. But this is absurd, since we can divise a possible world where there is only one lost planet somewhat smaller than the earth and this planet should then be Venus. On the other hand, the characterizing rule contains the essential of the localizing rule: Venus is a planet. If Venus loses its atmosphere or a major share of its mass (or even never had them), insofar as it remains the second planet from the sun, it will still be our Venus. Indeed, so understood the identification rule for Venus must be applicable in any possible world where the planet Venus exists.
   The case of Venus is somewhat like the case of the lines ‘a⁀b-a⁀c’ drawn to localize the center of a triangle without any further property. The characterizing description is of minor relevance. However, without the localizing condition established by the identification rule of Venus, it would be impossible to identify Venus. The application of many other descriptions does not create 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 loses its reflective atmosphere, it may cease to be the brightest planet, but will not cease to be Venus. If Venus loses half of its mass but remains in the same orbit, it still does not cease to be Venus. But if Venus loses nearly all its mass and becomes a small asteroid and not a true planet, we could say that it is what once was Venus. If Mercury never existed, Venus would be the first planet of the solar system and even if it were called ‘Venus,’ it seems that it would not really be our Venus, since at least for some time it needs to be the second planet from the sun. If in another possible world the second planet had been hurled out of the solar system thousands of years ago (Kripke 1980: 57-58), it could still with right 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. Anyway, even in a terrain of vagueness limits are set.
   What I said about identity sentences also applies to other singular predicative and relational sentences. Consider the following sentences:

1. Bucephalus was a living being.
2. Bucephalus was an animal.
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 five sentences refers to 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 is more completely referred to by sentence (6), since the truth of all the others can be implied by the truth of this sentence. Indeed, (6) is nothing but a shortened expression of the identification rule for Bucephalus, with a localizing and a characterizing description. The sub-facts presented by sentences (1) to (5) are all included in the grounding fact presented 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 because the ‘was’ only adds relevant predications. Statement (7) ‘Bucephalus once swam across the river Granicus’ is a different case: the auxiliary description ‘a horse who once swam over the river Granicus’ isn’t a relevant part of the fundamental description-rule. Nevertheless, it still can be derived from (7) considered in full, since it is (by ideal speakers) believed to be the case.

26. The problem of identity in difference
There is a final point concerning the relationship between the sub-fact and the grounding fact. It concerns the less than satisfactory way that Frege solved the puzzle of identity. As he writes, 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, once the modes of presentation are different and once 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 more clearly, consider again Frege’s sentence (i) ‘The morning star = (is the same as) 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) = the brightest star in the evening sky, understood as referring to the second planet orbiting the Sun between Earth and Mercury (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 a mediated sense that is identical; this last sense is that of (in short) being the second planet orbiting the Sun, namely, Venus. It is important to note that this last sense is not yet the reference, but still a cognitive identification rule constituting mainly the core sense of the name ‘Venus.’ It is only because both expressions flanking the identity sign indirectly evoke the same identification rule for the planet Venus that we are allowed to place an identity symbol between them! In order to make the point clearer we can use the following schema:

Sentence:    The morning star            =             the evening star.

immediate   IR: the brightest            ≠             IR: the brightest
sense:          star in the morning                        star in the evening.
                                    ↓               sub-fact:                  ↓
                   Being the morning star isn’t being the evening star

mediated     IR: The second planet…  =           IR: the second planet…
sense:          (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 immediate senses of the descriptions (the Fregean senses), evoking a relational sub-fact showing the differences between them. This sub-fact can be described 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.’ Furthermore, the word ‘is’ (the same as) indicates the identity of the implicitly intended mediated senses that can be expressed by the statement ‘Venus [in full] = Venus [in full].’ This statement points to the grounding fact that is constituted by Venus’ self-identity, which can be here better described as: ‘Being Venus is the same as being Venus.’
   A somewhat different example is the sentence ‘The morning star is Venus.’ Here the schema is:

Sentence:        The morning star            =         Venus.

Immediate:      IR: the brightest            ≠          IR: the second
sense                star at dawn                              planet from the Sun.
                                        ↓                 sub-fact:           ↓
                         Being the morning star isn’t being Venus.

Mediated         IR: the second planet      =          IR: the second planet
Sense:              (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. The first exposing the phenomenal sub-fact 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 same as) represents the ultimate grounding fact that being Venus is the same as being Venus.
   Now, how should we deal with cases in which the mediated sense responsible for the identity, like the planet Venus, lacks a proper name? Consider the identities (i) ‘Everest = Chomolungma,’ (ii) ‘a⁀b = a⁀c’ (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 a same imaginary mountain). I would answer this by creating a conjoint sense, a conjoint identification rule, respectively the ‘Everest-Chomolungma,’ the ‘a⁀b-a⁀c,’ and the ‘Afla-Ateb,’ which in fact produces respectively three new nominative expressions. By the law of identity it is now obvious that ‘Everest-Chomolungma = Everest-Chomolungma,’ ‘a⁀b-a⁀c = ‘a⁀b-a⁀c,’ and ‘Afla-Ateb = Afla-Ateb’ respectively represent the three different grounding facts. This is what sustains the identity of sentences (i), (ii) and (iii).
   We can apply a similar analysis to identities between concept-words in identities of the form (x) (Fx = Gx). Consider the identity ‘Water is H2O.’ The schema will be:

Sentence:       Water                        = (is)     amount of H2O.
                          ↓                                                    ↓
Immediate     Aqueous                    ≠            amount of
sense:            liquid…                                   hydroxide of oxygen.
                          ↓                        sub-fact:              ↓
                      Being water isn’t being an amount of H2O.

Mediated       Amount of                =              amount of
Sense:            hydroxide of oxygen                hydroxide of oxygen.
                              ↓               grounding fact:        ↓
                        Being H2O is the same as being H2O.

As already noted (Appendix to Chapter II), the concept-word ‘water’ has two nuclei of meaning: a superficial one, of an aqueous liquid (transparent, tasteless, odorless, etc.), and a deep one, a substance called by chemists hydroxide of oxygen or H2O. Regarding the immediate sense, what it presents is the sub-fact of the difference: the obvious sub-fact that being (phenomenally) an aqueous liquid isn’t the same thing as being (microphysically) a quantity of H2O molecules. Regarding the mediated sense the two nuclei of meaning, aqueous liquid and hydroxide of oxygen, and the whole bunch of descriptions that form them, is what is at stake. The sentence representing the grounding fact can be expressed as ‘aqueous-liquid-hydroxide-of-oxygen = aqueous-liquid-hydroxide-of-oxygen’, and the grounding fact it represents can be said to be the fact that being the aqueous liquid-hydroxide of oxygen is the same as being itself. It is because of this deep identity that we can say that the two different modes of presentation of an aqueous liquid and of H2O are those of the same substance. Finally, the whole identification rule for aqueous-liquid-hydroxide-of-oxygen requires the knowledge of the microstructure of water and the chemical theory supporting it.
   A last example is the identity ‘Heat in gases is molecular kinetic energy.’ Note that since heat is not only ambiguous – it can mean a subjective feeling, but also external temperature as it is sensed (heat1) – the best choice would be to use the expression ‘measured temperature’ (heat2, that we, as very unprecise termometers, can also measure in our surroundings), since surely the subjective feeling of heat cannot in any way be identified with molecular kinetic energy. Moreover, since molecules can have different mass and speeds, a more precise sentence would be ‘Temperature in gases (heat) is the average kinetic energy of its molecules.’ This sentence expresses as immediate sense the following difference: ‘Temperature in a gas (heat2) ≠ average kinetic energy of its molecules.’ This sense-thought-rule refers to the sub-fact that the (macrophysic) temperature that we can feel and measure by termometers is something phenomenally different from the (mycrophysic) average kinetic energy of the molecules of a gas like the air around us. In a next step, we come to the mediated sense, namely, the sense establishing the identity, which can be rescued by the sentence ‘Average kinetic energy – temperature (heat2) – of an amount of gas = average kinetic energy of – temperature (heat2) of an amount of gas’. Here both sense-thoughts-rules refer to the same grounding fact, what requires as assumption the acceptance of the kinetic theory of gases.
   This doubling of sense levels explains much of Saul Kripke’s in my view illusory discovery of the necessary a posteriori. But in order to fully understand the confusion involved, we need to add the context of our speech acts. A first thing to notice is that in different contexts we can enhance or magnify or emphasize the immediate Fregean perspectival sense or thought of a sentence (representing a sub-fact), or we can enhance or magnify or emphasize the mediated sense or thought of the sentence (representing the grounding fact).[39] Here I need to speak again of the contexts of interest of the linguistic agents, meaning by them contextualized practical aims.
   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 immediate senses (modes of presentation, identification rules) for the external, phenomenally given objects, the morning star and the evening star and considering the difference between being the brightest star in the morning and being the brightest star in the evening. If we do this, we leave Venus’ identity in the background. This can be the case, for instance, if we contemplate the beauty of the starry sky at night and, after localizing the evening star, we say to a child that it is also the same thing as the morning star. In this case, we react like Frege regarding this thought as a contingent a posteriori discovery, since we are well aware that we are emphasizing the different modes of presentation of the same object, a difference that as such represents an empirical sub-fact made by two different aspectual presentations of the same thing. We are emphasizing our representation of the phenomenally given sub-fact that being the morning star isn’t the same as being the evening star.
   Nonetheless, in a scientific context of interest, say, one in which we use a telescope to study the surface of Venus, when we consider the sentence ‘The morning star is also the evening star,’ what we may have in mind and emphasize is the identity of both stars. This is the mediated sense representing the grounding fact of the self-identity of Venus particularly emphazised in Kripke’s writings. In this case, we read the statement as preferentially meaning that ‘[in full] Venus = Venus [in full],’ which is a necessary a priori statement, since it affirms the grounding fact that being Venus is the same as being Venus. It leaves the different guises of sense in the background, as secondary effects of an astronomical theory that is assumed to be true.
   Now, consider the statement ‘Water is H2O’.[40] In a popular context of interest formed by fishermen interested in digging a well to find water for drinking and washing, this sentence is read as emphasizing the sub-fact that the word ‘water’ is a precious aqueous liquid (transparent, tasteless, odorless, drinkable…) and that this is not exactly the same as something being constituted by H2O. Since what is emphasized here is the difference between the senses of ‘aqueous liquid’ and ‘H2O’, the statement is seen as contingent a posteriori. On the other hand, when the context of interest is scientific, e.g., formed by chemists measuring the acidity of a sample of water, the word ‘water’ in the sentence ‘Water is H2O’ can be read as meaning the same thing as hydroxide of oxygen. In this case, the whole sentence is seen as preferentially representing the grounding fact expressed by the identity ‘Hydroxide of oxygen = H2O,’ which is the same as ‘H2O = H2O,’ that is, a necessary a priori tautology.
   A similar emphasization can be found in the statement ‘Heat is molecular movement’. If we emphasize the ordinary meaning of heat, the difference between heat2 (temperature) that we are able to feel and the average kinetic energy of a gas, the sense emphasized is contingent a posteriori and the fact referred to is a sub-fact of a difference: it is not necessarily so and it was learned by experience. On the other hand, if we assume the kinetic theory of gases as true in a scientific context in which we are measuring temperatures, the statement can be understood as emphasizing the mediated sense of the identity ‘Temperature of a gas (its average kinetic energy) = average kinetic energy in a gas (its temperature)’, which refers to the grounding fact of an assumed identity. The conceptual rule for temperature and for the average kinetic energy are blended in one only rule.
    Now, I think Kripke is misleadingly conjoining the a posteriority of emphasizing the immediate sense-thought-proposition with the necessity of the mediated sense-thought-proposition, 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, and not a = b. A Wittgensteinian therapist would probably conclude that Kripke was the victim of a deep grammatical ambivalence. 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 also justifies the self-identity of the grounding fact.

27. Sense of a sentence: the thought
Now it is time to go on to the sense of a sentence. Here Frege hit the bull’s eye! He was lucky in suggesting that the meaning of the whole sentence is the thought (Gedanke) it expresses. He reached this result by applying his principle of compositionality of senses, whereby combined in a certain way the senses of its 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, by the same token we can also say that the thought expressed by the resulting sentence is different. 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 – as in the sentence ‘12 x 12 = 144’ in the utterance: ‘The sentence “12 x 12 = 144” expresses a true thought.’ Frege had the latter meaning in mind. In this usage, the word ‘thought’ means simply what the sentence (statement) says, which Frege has conceived of as some sort of eternal 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.’[41]
   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’ expresses 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 counted as expressing 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 combinations of semantic-cognitive rules able to constitute correspondingly different verifiability procedures.

28. The thought as the truth-bearer
Another quite plausible Fregean thesis was that the bearer of truth is not the sentence, but rather the thought expressed by it. I agree with this. Although we can say that sentences, beliefs and even things and persons are true, they all seem to be true in a derived 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 forgery deceiving us into having false thoughts about it. When we say that Socrates was ‘true,’ what we mean is that he was a truthful, trustworthy or reliable person, someone with integrity. But it 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. Our concept of truth works as a normative ideal so that the real or actual truth-value of a thought is conceived of as something invariant: if something is true, it is always true; if something is false, it is always false. Obviously, we can always err in holding something to be true (das Fürwahrhalten), believing in a falsity instead, and vice versa – this possibility is inevitable, due to our inherent epistemic fallibility. But when we discover the error, we correct ourselves, not by saying the thought was previously true and now turned false, but by saying that it was always false! 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 normative ideal, even because it is beyond our fallible capacities even to know whether we have reached this ideal or not. So is the grammar of our concept of truth.
   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 grasped it. Indeed, for Frege a true thought (if really true) remains true forever, just as a false thought (if really false) 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 with the same truth-value, so that what may change is only our cognitive grasp of it, our believing in its truth-value (unser 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 (atemporal), being eternally (atemporally) true or false independently of our grasping (fassen) it.
   Consider now the case of sentences as candidates for truth-bearers. Identical sentences can express different Fregean thoughts, 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 thought that we give to the sentence. But if the truth-bearer were the sentence, the truth-value should remain the same. This is obvious in the case of indexical sentences like ‘I am in pain,’ which express different thoughts depending on the speaker.[42] 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, what leads us to the conclusion that the primary bearer of the truth-value must be the thought.
   One can suppose that perhaps the sentence-token would be the truth-bearer, since it would be a different one in accordance with 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,’ ‘Llueve’… uttered in the same context. They say the same thing, express the same thought and all have the same truth-value, while their sentence-tokens seem quite different. Indeed, the only justification for the forceful sameness of truth-value of these four different statements is that their 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.
   Also beliefs, understood in a psychological sense, can only be derivative truth-bearers: if someone who believes something dies, his belief also disappears. Consequently, what is the truth-bearer must be the content of his belief, his belief-content and not his belief in a dispositional psychological sense, since the belief-content remains after his death. But a belief-content is the same as a propositional content or a thought.
   The kern 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; a relationship that is missing in the relationships between sentences or 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 believe.

29. 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. But this can be a derivative use of the word. A researcher of nature can well exclaim ‘Facts! Facts! Facts!’ understanding by a fact simply what corresponds to the true thought, namely, some given arrangement of tropes and constructions out of them. 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.
   Another argument against thoughts as true facts came from J. L. Austin, who made it clear that the Fregean 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’ is a metalinguistic sentence referring to the occurrence of his utterance, 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 hidden reason why Frege believed that the fact is a true thought is that he advocated a conception of truth as redundancy, rejecting the correspondence theory of truth. However, on the one hand, his arguments against correspondence theory (Frege 1918: 59-60) are unconvincing.[43] On the other hand, correspondence theory still remains highly influential as the most natural and plausible conception of truth, suggesting that propositions or thoughts are true when they correspond to facts as arrangements of elements in the world (Rasmussen 2014). Moreover, the view of truth as correspondence is commonsensical, staying in conformity with our methodological principle of primacy of the common knowledge. These are reasons that justify my endeavor to defend this theory later in this book.
   Finally, I think I can explain the reason 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 almost always emphasized:

(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.


In the most proper sense (a), we say that a thought is true in sentences like ‘His words are true,’ ‘Tell me the truth.’ In the factual sense (b), we say that the fact in the world is true 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).’[44]
   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,’ ‘actuality,’ authenticity.’ However, since ‘truth’ is very often used not only as ‘correspondence with facts’ but also as ‘an existing fact in the world,’ it is easy, if one is motivated, to confuse both and believe – considering that both, facts and thoughts, can be said to be true – that facts are true thoughts. This seems to have originated Frege’s confusion, giving us another example of hypostasis as a way of transgressing the internal limits of language (Ch. III, sec. 11).

30. The thought as a verifying 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 a combination of semantic-cognitive rules, which I identify with what Ernst Tugendhat called the verifiability rule of the statement (1976: 259, 484, 487-8). However, if the thought is a combination of rules, then what results from such a combination, the verifiability rule, must also have the character of a rule, even if not of a previously conventionalized rule. Combining this with our acceptance of the correspondence view of truth, this means that the thought should be a kind of combined semantic-cognitive rule whose function is to make us aware of a corresponding fact to which it is applied.[45]
   This reasoning leads us to the cursed word ‘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 for its verification (1980: 29). This doctrine was soon appropriated by logical positivism and after attempts to give it a precise formulation abandoned due to strong criticism, internal and external to logical positivism, which led to it being considered by many unsustainable. This is now the received view, even if sophisticated philosophers have never completely abandoned the idea that some form of verificationism is indispensable (cf. Misak 1995). Indeed, in the next chapter of this book I intend to offer a reply to the objections that philosophers have made against semantic verificationism, showing that these objections were not directed against the right form of verificationism but against the straw-man named ‘principle of verifiability’ such as it was constructed by the logical positivists.
    I will introduce 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 sense (epistemic value, informative content) of a sentence. Suppose the combined semantic-cognitive rule that constitutes the thought expressed in an assertive sentence is its verifying rule, as complex as it may be. Then if we show that this rule is effectively applicable to a fact, this makes this thought-sense-rule 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 isn’t effectively applicable to the expected fact, this makes the thought-sense-rule 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 sentence is devoid of meaning, lacking sense or thought, even if it may in some cases seem to have a sense.
   I think that this way to understand the truth of a thought is in line with Frege’s remark that although treating 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 its effective applicability as a verifying rule in an appropriate context. Moreover, the proposed identity between the Fregean concept of sense-thought and the concept of a verifying 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 if the higher-order property of effective applicability of a conceptual rule is the existence of an object (cluster of tropes) or a property (trope). If we accept this, then by symmetry the higher-order property of effective applicability of the verifiability rule should be the existence of the fact to which it applies. We could almost say, in a Hegelian fashion, that existence is the truth of the concept, while the truth is the existence of the thought…
   Anyway, there is a dilemma here, because we have two readings of truth:

1)     Truth is the property of a verifiability rule of being effectively applicable to a fact, which seems to be a way to understand correspondence theory.
2)     Truth is the property of the verifiability rule of being effectively applicable to a fact, which amounts to the attribution of existence to a fact.

These two interpretations of truth may be equivalent, but they are not the same. Which is the correct one? The seemingly paradoxical provisional answer that I am able to give is that (1) and (2) take into account different senses of the word truth. Sense (1) is of thought-truth: truth as a property of the thought or the verifying rule of being effectively applicable in order to correspond to a fact. Sense (2) is that of fact-truth, truth as the higher-order property of the thought or verifying rule of being effectively applicable to the fact, which means the same thing as to attribute existence or reality to a fact. Truth of a thought and the existence of a fact are twin concepts.
   In this way I think we have deciphered the ambiguity correctly captured by dictionaries. Thought-truth concerns truth alone and points to the correspondence of the thought with the fact. In the case of fact-truth the word ‘truth’ can be replaced by words like ‘existence’ or ‘reality,’ which are even more suitable, pointing to the existence to the corresponding fact, which can be only known by the application of the verification rule constitutive of its thought.

31. Frege’s Platonism
It is important to remember that, for Frege, thoughts (including 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 as (a) the criterion of objectivity inter-subjectivity and independence of will, and taking as (b) the criterion of reality existence in space and time, we 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 (Vorstellungen, a word that we could here translate as qualia). These entities satisfy criterion (b) but not (a): they are subjective and real. They are subjective by not being interpersonally accessible and are often dependent on the will. However, they are still real, because they are in the mind and, consequently, in time and (although he does not say) in space. There is, finally, a third realm, that of thoughts (usually called 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 interpersonally accessible: we can all agree, for example, that the Pythagorean Theorem expresses a true thought. 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 here. One of them, noted by Frege, is that though for him thoughts are eternal (timeless), immutable, forever true or false, and not created but 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). However, how this interaction is possible remains an unexplained mystery.
   Frege was aware of the difficulties, but the main reason 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 effectively communicable. Representations (Vorstellungen), on the other hand, are rather subjective psychological states, which 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 is to strictly distinguish thoughts from mere psychological representations, placing thoughts 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.

32. Avoiding Frege’s Platonism
Despite the above-suggested arguments, today few 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 true and false thoughts and their constitutive senses), but also seems to lack intelligibility. The price that Frege was willing to pay in order not to fall into psychologist subjectivism seems too high for us today.
   In my judgment, there is a 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 show this, I want to apply again the same strategy inspired by the ontological particularism of English empiricists from Locke to Hume, which I used in the construction of universals by means of tropes.[46] This is understandable, since according to trope ontology, a thought must be made up of internal tropes. In order to accomplish this, I need only show that Fregean Platonic thoughts (objective non-real truth-bearers…), which I call f-thoughts (‘f’ from Fregean), can be defined in terms of psychological p-thoughts (‘p’ from psycho­logical). Hence, I suggest that we can warrant the existence and stability of f-thoughts by means of what I call s-thoughts (‘s’ from spreadable) without hypostasizing them as Platonic entities and even without resorting to classes of p-thoughts. We can do this by means of the following definition, which is as simple as efficacious:

An s-thought X (Df) = a given p-thought X embodied in some mind or any other 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). This definition reduces supposed f-thoughts to p-thoughts without depriving them of their objectivity (inter-subjectivity) and expected stability or immutability by interpreting them as s-thoughts.
   The so defined s-thought, which is the same as what is called a thought-content or simply as a proposition, though distributed across space and time, has no particular spatio-temporal location and can be seen as the most proper truth-bearer. For example: the s-thought (or thought-content) expressed in the sentence ‘The Eiffel Tower is made of metal’ 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 other time. Characterized by the disjunction between qualitatively identical thoughts embodied in any individual mind, the s-thought is regarded in abstraction from the particular human minds that causally instantiate it.
   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 do is simply to pick out the first thought given to us by memory and use it arbitrarily as a model: first the one and then some other, which we recognize as being identical to the first, so that 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 a qualitatively identical psychological p-thought in the minds of hearers, insofar as they are rooted in conventions we have attached to their semantic components.
   With the help of the above definition, we avoid not only appealing to specific occurrences of thoughts, but also the most expected alternative, which would be to explain one s-thought in terms of a sum or class of p-thoughts qualitatively identical to each other. This could lead us not only to the problem of defining classes, but to the problem that classes have size, while thoughts do not. If an s-thought were a class of p-thoughts, 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 psychological p-thought-contents that we have respectively instantiated in our minds. This has the advantage of bringing Fregean thoughts out of the ethereal Platonic heaven back to the psychological realm without making a 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. As an open disjunction of p-thoughts, s-thoughts only exist as psychological instantiations of p-thoughts, which enables them to play a causal role: this 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 psychological level and different on a neuro-physiological level, just in the same way as different devices can have different internal mechanisms and perform exactly the same tasks.
   In my view, 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 common name ‘the chair’ using the definite article, that we can speak of the geometrical form of circularity, and that we can speak of the number 2 in the singular – but this is just for the sake of simplicity of expression. What we actually have in mind are occurrences of qualitatively identical meanings, of qualitatively identical concepts of chairs, of circles, and probably of cognitive arithmetical concepts of dualities, and nothing more.[47] In the same way, we can talk about the thought expressed by ‘7 + 5 = 12,’ 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 7 + 5 = 12 because there is no reason to consider the individual persons who think it.
   The adoption of the definition of s-thoughts proposed above, which is easily generalizable to all kinds of Fregean senses, is in my view 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, the Fregean question turns back: 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 that Frege was too hasty when he admitted that 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 limited possibilities of interpersonal communication.[48] But more important is to note that senses and s-thoughts, without being Platonic entities, are something more than subjective mental states: they are rule-complexes built upon combinations of interpersonally agreed upon conventions made with the help of public signs that precisely because of their interpersonal character are communicable. That is, because s-thoughts are verifiability rules rooted in interpersonal conventions, they can well be able to satisfy Frege’s demand for objectivity as interpersonal accessibility followed by the possibility of communication and evaluation.
   It may at first sight seem implausible that language is capable of repeatedly reproducing in other minds and even in the same mind the same subjective pattern, the same thought-content, the same recognizable instantiation of a combination of conventionally established semantic rules attached to our words. However, compare by analogy this case with the case of genetic information able to endlessly reproduce the same characteristics in successive biological individuals.[49] Why cannot the conventions and ways they can be combined in the constitution of p-thoughts do the same job? 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 conventional standards. There is no reason, except an anti-empiricist bias, to think that things could not be that way.
   Finally, let us apply the distinction made by John Searle between what is ontologically objective/subjective and what is epistemologically objective/sub­jective (Searle 1999: 43-45) to the objectivity of s-thoughts. Searle noted that we have a strong tendency to take what is epistemologically subjective for what is only ontologically subjective. However, one thing 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 the issue. In contrast, a phenomenon can be ontologically subjective without ceasing to be epistemologically objective – for example, the knife-like pain caused by a seizure of acute pancreatitis – because everyone (doctors and patients) 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 regarding the nature of s-thoughts. They are in themselves 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. After all, 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 scattered 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. 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 expressive. It has only non-conventional subjective coloration, being susceptible only to emotive-aesthetic appreciation with differing degrees of subjective interpersonal agreement.
   On this point Frege was no exception: like Husserl, Bolzano and several other continental philosophers 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.

33. 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, which does not mean that it is dependent on any of the many minds that will eventually think it. 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 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 reason why this may sound strange is that nobody can truly deny it. One cannot think: ‘The theorem according to which the sum of the squares of the shorter sides of a right triangle equals the square of the hypotenuse is something which existed in the past and now no longer exists,’ for this judgment will already be an occurrence of the thought of the Pythagorean theorem and insofar falsify what it states. Nevertheless, the s-thought of this theorem would not have come into existence if nobody had ever thought it. Thus, it would not exist in a world without cognitive beings.
   This 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, as the primary bearers of truth, could not be true. Nevertheless, it seems obvious that in this world the fact that there are planets, stars and galaxies would still be true, even though there would be no sentient beings to think this.
   Our answer is that here we are again victims of the confusion between thought-truth and fact-truth. The first is the truth applied to the primary bearer of the truth, which is the s-thought. The second is a derived but as we already saw very common application of truth to the real existent thing or fact in the world, as a secondary bearer of truth. Indeed, that there would be planets, stars and galaxies in a mindless world would still be true as a fact, and the applicability of the Pythagorean theorem would still be true as a fact 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 are discovered. Pythagoras discovered 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 they were discovered.
   Again, the answer is that this 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 seems clear in the case of empirical truths. That the law of the lever was always applicable in principle is surely true. However, this is a fact-truth; the thought-truth of it first came into the world when 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. Facts, in their turn, as long-lasting as they may be, are not the primary bearers of truth, but rather their truthmakers or verifiers. They are said to be true only in the sense (b) of fact-truths, not in the sense (a) of thought-truths. They are what occurrences of their thoughts represent. Hence, in a most demanding sense, no truth or falsehood 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 Wt, it would be recognized as truth.
   An s-thought that has never been thought does not exist and thus cannot be true. The same with falsities: suppose the thought ‘The Colossus of Rhodes is floating in the Sargasso Sea’ had never been thought before the present moment. The moment we think that it has never been thought before, we are already thinking it, and we can attribute falsity 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 because there are minds to think it.

34. 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. According to a plausible interpretation of Aristotle (1984, vol. I, Ch. IX), 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 with 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 isn’t applicable to future events (only to present and past ones).[50]
   I cannot agree with this, since I believe that we should preserve the principle of bivalence for s-thoughts. But (1) can be questioned from a different perspective. Suppose 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 this: if taken literally,├p is unable to express an s-thought because an s-thought, a thought-content, a proposition, is something to which we must possibly attribute a truth-value, and without any further contextual information we are totally at a loss for the task of associating p with any appropriate truthmaker in order to assign it a truth-value.
   However, 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 the 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* almost always appears 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 available to us, it was very probable that a sea-battle would occur the next day. Indeed, the utterance ‘It is likely that a naval battle will occur tomorrow’ could be regarded as definitely true on the night of June 3, 1942, without breaking any principle of bivalence.
   Suppose now, by contrast, that I am on the calm beach of Praia Bonita, 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 I have all kinds of reasons to believe that this kind of event is extremely improbable in this region and at this time.
   The conclusion is that taken in the absence of a context (and not in the sense of ├p* or ├q*) the sentence ├p is 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. 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, in as much as no context is furnished, no real criteria can be given. Statements like ├p*,├~p* and ├q*, on the other hand, say something probabilistic about the future that can be confirmed and made true by criterial reasons already found today.
   It seems that in principle the metaphysical problem about contingent futures can be eliminated when we consider with enough care what we are really able to mean by affirming thought-contents regarding the future.

35. Conclusion 
My first aim in this chapter was to insert in the framework of Fregean semantics the results of my reading of Wittgenstein’s view of cognitive meaning as use in accordance with rules, in order to better distinguish the most relevant forms of semantic-cognitive rules. This requires strong corrections in Frege’s own framework. Even if most results could only be sketched here, they nonetheless seem to me much 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. I will read them here as definite descriptions, since for proper names we can use words like ‘Phosphorus’.
[4] As it was 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 of doing it as a reconstruction of Frege’s thought.
[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 supposed non-descriptive modes of presentation (2012: 34). Independently of the intrinsic interest of Recanati’s prolific work, it is worth noting that these files, being clusters of information and not subjective Vorstellungen, should also be conventionally grounded and akin to our cognitive-semantic rules. Consequently, they should be able to be linguistically expressed by means of descriptions, which brings us back to the descriptivist standpoint. For this reason it seems clear to me that complex semantic-cognitive rules are apt to do a similar job in an explanatorily more natural and convincing way.
[7] François Recanati prefers to use the word ‘property’ (propriété) instead of ‘concept’ in his summary of Frege’s semantics (2008: 34). 
[8] This can be rejected by considering that for Frege a concept must have an extension. But even in this case we would have problems: for in the case of a false sentence like ‘Göteborg is the capital of Sweden’, nothing would fall under the concept of ‘…is the capital of Sweden’, which means that again this concept-property should be seen as an abstract entity.
[9] For the exposition of my understanding of trope theory, see the Appendix to chapter III of this book.
[10] The pure theory of tropes was first introduced into philosophy by D. C. Williams (1952), and has since then sparked growing interest.
[11] 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 it is a subordinate trope.
[12] This trope-model way of constructing the universal is suggested in order 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. Sets and sums are quantitatively changeable, and the universal will grow ever larger, the more members its set has. Moreover, the similarities between the tropes of the set must also be similar, producing an infinite regression of higher-order similarities.
[13] This epistemic primacy of identification over the generalizing function was already pointed out by Keith Campbell (1990: 24-25). It will justify the search for an independent criterion to distinguish predicate from subject in the next sections.
[14] Even D. C. Williams set 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)
[15] 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 be simple. The negation of the predicate is the inapplicability of the ascription rule to the object identified by the identification rule. 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 whole statement. Hence, it is impossible to negate the subject or name alone.
[16] Notice that the demonstrative ‘there’ does not have here the function of a constituent of the identification rule of Socrates, but is itself an identification rule of an individual place. In statements like ‘This is a daisy’ and ‘There is the Golden Gate Bridge’, demonstratives have as their main role of localizing constituents of the identification rules of the daisy and the Golden Gate Bridge.
[17] We can show that the nominalized predicate is in fact a disguised universal predication: the sentence ‘Wisdom is a virtue’, for instance, could be analysed as, ‘For any human being x, if x has wisdom, then x is virtuous.’ However, the asymmetry remains at this deeper level, since we cannot analyse a nominal term in a similar general way.
[18] Ignoring Frege’s thesis that the reference of a sentence is a truth-value, I will in the present context call it a fact. This unexpected choice will be justified later in this chapter.
[19] I take these examples from Mulligan et al. (1984: 300, 301 and 306), though their point isn’t the same.
[20] Note that if I said ‘This chair has this backrest’, this chair and this backrest would work as names and …has… as a relational predicate.
[21] In a sentence such as ‘My hat has three corners’, the number three indicates that a higher-order property of the rule for identification of corners is applicable to the three different corners of the hat.
[22] 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)
[23] Although the existence of something is always given in terms of some time and place, we normally abstract existence from these external elements.
[24] It was W. V-O. Quine who suggested, using the name Pegasus, to form a predicate such as ‘the thing that pegagizes’ (1948/9: 27)).
[25] 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.
[26] In some cases, like ‘Queen Elizabeth II,’ family origin is part of the localizing description, but this is not necessarily so (see Appendix to Chapter II).
[27] What symbolic form the 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).
[28] To the objection that if we change the rule, it ceases to be a rigid designator. But if we change the rule so that the set of possible worlds to which the proper name applies also changes, we are not applying the same proper name anymore. However, you may introduce changes like additions in the fundamental description-rules insofar as this only determines the identification better, or corrections and additions in the auxiliary rules that are innocuous. These changes are innocuous insofar as they do not affect the set of possible worlds to which the proper name applies, for in this case they do not affect the rule’s rigidity.
[29] This is again a merely pedagogical simplification (see Appendix of Chapter I).
[30] 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 adjectives, 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, therefore, the phrase “round square” exists as a grammatical construction.’ The Meinongian Sosein is reduced here to the recognition of a syntactical triviality.
[31] 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)
[32] I believe that Mill’s uncomfortable definition of matter was in fact an attempt to evade the objection of idealism open to Berkeley.
[33] See Frege, Letter to Russell of 28.12.1912.
[34] For him ‘a situation or state of affairs is, roughly, a set of facts, not a set of things’ (1950: 8).
[35] For a 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.
[36] 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)
[37] This also gives back the whole sense of the original still more convoluted sentence from Church: ‘The number such that Sir Walter Scott is the man who wrote that many Waverley Novels altogether is twenty-nine.’
[38] It seems that the mode of presentation of the sub-fact can be approximated with what defenders of two-dimensionalism call primary intention, while the mode of presentation of the grounding fact can be approximated with what they call a secondary intention. (cf. Chalmers 2002).
[39] The concept of emphasization was fruitfully applied in Jürgen Habermas’ excellent work on universal pragmatics (Habermas 1976).
[40] The example was already considered in the Addendum of the Appendix to Chapter II in this book.
[41] As Tyler Burge wrote in ‘Sinning Against Frege’: ‘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)
[42] For Frege, in the case of indexical sentences, the context of the utterance belongs to the expression of the thought. For discussion, see addendum of the Appendix to chapter II, sec. 8.
[43]  According to the 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 F in order to be true by corresponding with the reality, and that to have the property F in order to be true by corresponding with reality will demand the property F’ and so successively. The answer is that to say that p is true by corresponding to reality, and to say that p has the property F due to being true by corresponding to reality are one and the same thing, so that F is redundant. (For a discussion, see Künne 2003: 129-133)
[44] 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)
[45] See Tugendhat’s verificationist correspondentialism in 1983: 235-6.
[46] See Appendix to chapter III.
[47]  One could object: 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 without end. 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.
[48] Against Frege, we could hold that to a great extent even 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 in some cases, and, in others, through induction by analogy reinforced by a great variety of interpersonally accessible physical states strongly intermingled with them (Costa 2011, Ch. 3).
[49] Biological mutations are accidents whose incidence should be evolutionarily calibrated. Only species that mutate to the right degree in the right period of time and corresponding to environmental changes are likely to survive. Too many mutations, as well as too few, would be dangerous. So it seems that an unchanging species with no relevant mutations is conceivable, but they would be unable to adapt to changing external conditions.
[50] For a sophisticated alternative interpretation in which Aristotle does not reject the principle of bivalence, see Christopher Shields, 2007: 186-190.

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