segunda-feira, 14 de maio de 2018


DRAFT for a chapter of the book PHILOSOPHICAL SEMANTICS, to be published by Cambridge Scholars Publishing, 2018.

An Extravagant Reading 
of Fregean Semantics (1)

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

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

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

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

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

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

Sentences (1) and (2) concern to the same thing regarding the planet Venus. But in spite of this, a person can know the truth of (1) without knowing the truth of (2) and vice versa. Frege’s explanation for this is that although the two singular terms ‘the morning star’ and ‘the evening star’ refer to the same planet Venus, they convey different informative contents, that is, they have different senses (Sinne).[3] The word ‘sense’ is defined by Frege as an object’s way of being given (die Art des Gegebenseins des Gegenstandes), which is usually translated as a mode of presentation. The senses of the singular terms ‘the morning star’ and ‘the evening star’ are different, because ‘the morning star’ presents Venus as the brightest celestial body usually seen just before sunrise, while ‘the evening star’ presents the same planet Venus as the brightest celestial body usually seen shortly after sunset…
     Frege writes that words express their senses (drücken ihre Sinnen aus), while senses determine (bestimmen) their reference, since the mode of presentation should show us how to find the reference. Even in cases where the reference does not exist, this determination of reference through sense is given as a possibility, since even in this case the words preserve their senses. This fact points to a flaw in Frege’s idea that sense is the way an object presents itself to us, for in the case of empty terms there is no object to be presented to us. This is why sense can be better understood as the intended mode of presentation instead of as a mode of presentation given by the object (Textor 2010: 134); sense is the way we intentionally present an object or reference to ourselves, whether it exists or not. At any rate, for Frege an expression can have a sense without a reference, but cannot have a reference without its determination by means of a sense.
     Frege extended his notion of sense to other terms and to sentences. In the case of the senses of (declarative) sentences, he calls it cognitive or (more literally) epistemic value (Erkenntniswert). The last term is also appropriate. The Fregean concept of sense has epistemological interest, for it constitutes the proper informative content of the linguistic expression. It is what makes ‘the evening star’ and other expressions informative. Or, using Dummett’s words, ‘sense is what we understand when we understand an expression’ (1990: 92). The philosophical importance of Fregean semantics is largely due to the epistemological and ontological imports of the concept of sense (this is what distinguishes it from a more exclusively linguistic semantics like that of Ferdinand de Saussure.)
     Frege is a Platonist about sense. For this reason, he conceives senses as abstract entities which can only be analyzed in terms of constituents that are also senses. A consequence of his Platonism of senses is that it prevents him from analyzing senses in terms of other concepts. However, it is just this task that naturally imposes itself. For it seems very plausible to understand senses as semantic-cognitive criterial rules. We see here a fundamental difference between Fregean semantics and the semantic considerations of the later Wittgenstein, who regarded senses or meanings as depending on episodic uses of expressions determined by rules. Dummett was perhaps the first to defend the idea that senses are rules as the most natural reading of Frege’s use of the term senses. As he wrote in his book on Frege’s philosophy of language:
The sense of a word consists in a rule which, taken together with the rules constitutive of the senses of the other words, determines the condition for the truth of a sentence in which the word occurs. (1981b: 194; my italics)
And concerning the singular sentences in Frege, understanding with the term ‘criterion’ the condition of satisfaction of a semantic rule, he wrote:
To know the sense of a proper name is to have a criterion for recognizing, for any given object, whether or not it is the bearer (referent) of that name; to know the sense of a predicate is to have a criterion for deciding, for any given object, whether or not the predicate applies to that object; and to know the sense of a relational expression is to have a criterion for deciding, given any two objects taken in a particular order, whether or not the relation it stands for holds between the first object and the second. (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,
(7 + 3) – 8,
(874 – 870)/2
5 – 3

All these numerical expressions have the same reference: the number 2. But their senses or modes of presentation are in each case different. At the same time, they are expressions of procedures, methods, semantic-cognitive rules or, in most cases, combinations of such rules by means of which we reach the identification of the same number 2 as a result (See Runggaldier 1985: 91 f.).
     By treating senses as semantic-cognitive rules and these rules in the primary case as shared conventions, we contrast them with what Frege called colorations and illuminations (Färbungen and Beleuchtungen), which are feelings often associated with image representations (Vorstellungen) and sensory-perceptions (Anschauungen), as such all belonging to an intrinsically subjective level (Frege 1892: 31). These ‘colorations’ and ‘illuminations’ are names for what we would more often call expressive meanings, that is, sensory-emotional states that we normally and customarily associate with expressions. Thus, for example, the words ‘love,’ ‘dog’ and ‘hell’ in the sentence ‘Love is a dog from hell’ (Bukowski) contrastively associate words linked with strong specific emotions in order to create a weird epigrammatic effect.
     As Frege realized, the kind of appeal or lack of appeal that the colorations associated with words have for different persons depends correspondingly on similarities and differences in their human natures. Because of this, they do not require conventions to be communicated, as in the case of senses. This is why some people are emotionally moved by a certain poem, while others are not. Consequently, it is very difficult to translate poetry, which depends so much on colorations acquired by expressions in a particular language and form of life. Hence, colorations are not results of conventional rules; they are rather regularities originating from shared aspects of human nature within a historically developed cultural context. If my understanding of Wittgenstein’s argument against private language is correct, then his attempt to explain phenomenological language as a simple replacement of public behavioral criteria like uttering ‘ouch!’ under conditions that would cause pain with a sentence like ‘I feel pain’ is insufficient (1984d, sec. 244). It is an attempt to assimilate the referential meaning of the phenomenal language to its expressive meaning (I suppose that both can be legitimated).
     If in opposition to Frege we accept the view that sense is usually only something with the form of a rule (etwas Regelartiges), namely, a convention or a combination of conventions, we can easily solve the problem of the com­municability of senses that has long tormented philosophers like him. This is because the reason can easily be found for the objectivity (interpersonal accessibility) of senses, as well as for their consequent communicability. This reason is that Fregean senses are epistemic unities easily reducible to conventional semantic-cognitive rules or associations of them, and such conventions are interpersonally established and agreed upon in a pre-reflexive manner. Indeed, accepting the conclusions reached through our discussion of Wittgenstein’s views, senses typically result either from the direct application of interpersonally established semantic conventions or, more importantly, from associations or combinations of these conventions.
     Accepting that the sense of a singular term is the same thing as a rule understood as a conventional or conventionally grounded procedure that plays a decisive role in the identification of the object, it is easy to go further and accept that this rule can be typically expressed by means of definite descriptions. Hence, the sense or mode of presentation expressed by the singular term ‘the morning star’ is a conventional rule that can be understood as requiring as a criterial condition for the cognitive identification of the morning star that we see as the brightest celestial body not too far from the Sun just before or after the Sun rises. Concisely stated, this rule can be expressed by the definite description ‘the brightest celestial body that is seen close to where the Sun is about to rise.’ Without assuming that definite descriptions are expressions of rules, Frege also approached this in a note on the name ‘Aristotle’ (Frege 1892: 28). For him the proper name ‘Aristotle’ abbreviates a cluster of modes of presentation of the object that can be expressed by descriptions, which may include (i) ‘the disciple of Plato,’ (ii) ‘the teacher of Alexander the Great,’ and (iii) ‘a person born in Stagira.’ If this is the case, then (i), (ii) and (iii) express different senses, different rules that in one way or another help us to determine the reference of the proper name ‘Aristotle’ (Cf. also Frege 1918-19: 63).[5]
     Of course, there is a controversy about this issue, which arose from Kripke’s arguments against descriptivist views of proper names like Frege’s. However, it seems to me out of question that Kripke’s arguments are successfully countered by the kind of meta-descriptivist bundle theory suggested in the Appendix to Chapter I of the present book.[6]
3. Reference of a predicative expression
Frege has something to say about the reference of a predicative expression, which he calls a concept (Begriff) and which may include relations. This is odd because it seems natural to call a concept something like the sense of a conceptual expression – the mode of presentation of its designata – while the reference itself should be called a property (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 variety of sensory patterns that are satisfied by those things to which the concept applies (Cf. Kant 1988, B 180). Although Kant’s text on schematism is terminologically impenetrable, it is easy to paraphrase his intuition using the terminology we have developed based on Wittgenstein by saying that a concept is a semantic-cognitive rule or procedure that requires the satisfaction of criteria by particularized properties (p-properties) or tropes, which is also consonant with Dummett’s and Tugendhat’s analyses of singular statements. Consequently, we have good reasons to suspect that a concept should be the sense of a predicative expression, its mode of presentation, and not its reference, as in Frege’s strange use of the term.
     To be fair to Frege, he also says that when an object falls under a concept, the concept may be called a property (Eigenschaft) of the object (1892: 201), seemingly acknowledging that ‘property’ is the right term for the reference of a predicative expression. However, for him the criterion of identity for two concepts is the sameness of their value-range (Wertverlauf), what includes 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’ should be predicative expressions referring to the same concept since they have the same extension. But it is intuitively obvious that kidneys and hearts are very different concepts.
     In addition to belonging to the realm of reference, Frege also sees his concepts as functions. The mathematical concept of function can be defined as a rule that has as its input arguments and as its output values (for example: ‘3 + x = y’ is a function by means of which when we give as input the number 2 as the argument for x, we get as an output the number 5 as the value of y). For Frege, a concept is a function whose argument is the object that ‘falls under it’ (fällt unter etwas) or does not and whose value is a truth-value, which can be alternatively two abstract objects: ‘The True’ (das Wahre) when the object falls under the given concept and ‘The False’ (das Falsche) when it does not. For example, the concept designated by the conceptual term ‘ a satellite of the earth’ has the value true for the object Moon and the value false for the object Jupiter.
     Nevertheless, for Frege, concepts cannot be objects, either collections of objects, nor extensions (2001: 26). The reason is that objects, collections of objects and extensions are complete (vollständig) entities. That is, they do not require anything to complete them. A concept, by contrast, being a function, is seen by Frege as necessarily open: he calls it an incomplete (unvollständig) or unsaturated (ungesättigt) entity, needing to be completed by those arguments represented by the objects falling under the concept. In contrast, objects referred to by proper names are complete (vollständig), saturated (gesättigt) or independent (unabhängig).
     One could add that the saturated-unsaturated distinction can be found on three distinct levels: linguistic, semantic and referential. For instance: the predicate ‘…is a horse’ could be called an unsaturated linguistic expression (the unsaturatedness is shown by the gap ‘…’), expressing a supposedly unsaturated sense, which refers to an unsaturated concept (property) as the ultimate unsaturated ground. This unsaturated concept, for its part, becomes saturated when some object falls under it, for instance, the object named ‘Bucephalus’ referred to by the predicative sentence ‘Bucephalus is a horse.’
     With metaphors like those of ‘unsaturation’ and ‘incompleteness,’ Frege hoped to open the way to the solution of the problem of the logical distinction between the subject and predicate of a sentence. After all, the subject (the singular term) would refer to the saturated object, which would complete the unsaturated concept referred to by the predicate (general term).
      Unsaturated predicative expressions and saturated singular terms combine to form saturated singular sentences like ‘Bucephalus is a horse,’ which being complete must also be the name of an object, which for Frege is the truth-value of the sentence. This seem to be confirmed by the possibility we have of nominalizing sentences in the form of definite descriptions, since the latter are also singular terms (1879: § 3). Thus, the sentence ‘Bucephalus is a horse’ can be transformed in the description ‘the horse named Bucephalus,’ which appears in the sentence as ‘The horse named Bucephalus was black.’ The problem with this argument is that the same can also be done with general terms: ‘…is a horse’ can be nominalized as ‘the horse,’ as found in sentences like ‘The horse is an herbivorous animal.’ Hence, this argument isn’t persuasive. Anyway, we can accept that assertoric sentences are like proper names in the sense that they do not require completion as unities of meaning.
4. Ontological level
Discussing the unsaturated nature of the references of predicative expressions leads us to the question of the ontological nature of what Frege meant by a concept. If a concept is an unsaturated entity, what kind of entity is it? If it is an abstract entity, it seems that we should also have concepts as referred-to abstract entities of empty predicates, like ‘…is a yeti,’ which seems to be an ontologically abusive admission.
     Anyway, it is by now clear that Frege uses the word ‘concept’ as a technical term that contrasts too strongly with the word’s ordinary use. For our ordinary language intuition, there is surely an empty concept expressed by the predicate ‘…is a yeti,’ but this concept should be called empty because it is nothing but the sense of a predicate that has no reference at all! It is no wonder that Frege has nothing to say about the sense of predicative expressions, since he has beforehand emptied them by absorbing the semantic level into the ontological one.
     My final conclusion is that it is better to drop the Fregean technical notion of a ‘concept.’ This is a problematic remnant of ontological realism that does nothing to explain predication. Instead, I will understand the word ‘concept’ here in an intuitive way as the sense of the predicative expression: its mode of presentation of something. It is counter-intuitive to assume that ‘ a yeti’ must have any reference; but this predicate clearly has a sense intuitively expressing what we ordinarily understand by a concept, namely, that of the abominable snowman of the Himalayas. Thus, it seems that the best way to give a legitimate role to the word ‘concept’ is to see it as the sense of a predicative expression understood as its cognitive meaning, that is, its ascription rule.
5. Referring to particularized properties: trope theory
But if we drop Frege’s technical notion of concept, what is the reference of a predicative expression? I think that nowadays the most reasonable answer to this question consists in an appeal to the pure ontology of tropes proposed in the Appendix of Chapter III of this book, since it not only promises a parsimonial solution for ontological problems, but produces less difficulties than the traditional doctrines. Thus, I propose to replace Frege’s reference of predicative expressions with what we now call a trope, which I characterize simply as any spatiotemporally individualizable property, notwithstanding its degree of vagueness.
     There are many examples of tropes that are genetically primary and directly accessible to experience: the white color I see when I look at newly fallen snow on a sunny day, and which is there in my visual field, the smooth surface of this couch, the rectangular shape of my computer screen, its hardness or my headache. All these are tropes – spatiotemporally particularized properties or simply p-properties – that may range from simple objective or subjective qualities to complex ones, and from  homogeneous or heterogeneous complex tropes, like the music I listen to in the former case and the personality of a human being or a country’s political system or a social upheaval in that country in the latter, since all these things are in a less specific way also spatiotemporally localizable. Also very indirectly experienceable things like physical forces could be derivatively constructed from perceived tropes, since they are spatiotemporally localizable, and it is not inconceivable that even space and time, together with formal properties could eventually be reducible to tropes, as I tried to show in the Appendix of Chapter III.
     Moreover, it is easy to suggest a particularistic construction of universals built on the basis of particularized properties or tropes. In my view, a universal can be disjunctively defined as:

Any chosen trope model T* or any other trope strictly similar[7] to 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.[8] In this case, tropes T1, T2… Tn are identified as instantiations of the universal only because they are strictly similar (qualitatively identical) to an arbitrarily chosen trope model 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 cannot be the primary trope we intend to consider, since it must stand for the experienced one.
     A material object could be constructed as a cluster of tropes. It can in principle be understood as a cluster of tropes displaying at least compresence, that is, it must consist of a co-located and 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 in a broader way as inertial mass – that typically comprise material objects.
     I usually avoid using the word ‘property’ instead of ‘trope’, not because it isn’t the best one, but because the philosophical tradition has too often hypostasized this word as referring to some scarcely intelligible non-empirical entity, vitiating our philosophical language. This tradition has stubbornly ignored the fact that in ordinary language the word ‘property’ has always been used to refer to simple or complex, homogeneous or heterogeneous tropes. Anyway, I intend to use the word trope exactly as the word ‘property’ is ordinarily used. Thus, I explicitly include among the tropes complex tropes made up of different kinds of tropes, these complex tropes possibly being designated by composite predicates like ‘…a black horse of the best Thessalonian strain’ in the sentence ‘Bucephalus was a black horse of the best Thessalonian Strain.’ This does not make this complex trope (complex property) a singular material object, mainly because, as we will see later, a singular material object, taken as an individual, is seen as able to exist independently if compared with the trope to which it is tied (in a different possible world Alexander’s beloved horse, Bucephalus, could still exist even if he were just a tired old nag).
     According to the understanding of the reference of predicative terms that I am proposing, a predicative expression like ‘... is white’ in the sentence ‘The moon is white’ does not refer to any Fregean concept. It primarily ascribes, denotes, designates (or refers to) a particularized property, namely, a trope, which is the whiteness of the Moon as normally seen by observers on the Earth. Secondarily but distinctively, however, the predicate ‘…is white’ also alludes to (or connotes) the fact that this trope exemplifies the universal property of whiteness, here understood in the already explained particularist way as this same model trope that is being considered, or any other trope that is like it. Summarizing, a predicative expression has mainly a twofold function:

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

The allusive function is subsidiary to the ascriptive function: to identify a trope you do not necessarily need to grasp its role as an instance of a universal.[9] Better said, as particularized properties tropes have not only ontological, but also epistemic priority if compared with their role in the identification of universals.
     Furthermore – opposing the overwhelming influence of the logicist tradition – we have, as a still more subsidiary element: (C) the extension. Although relevant, differently from (A) and (B), extension isn’t primarily associated with predication. Extension doesn’t even need to be implicitly considered in the act of predication! However, it can be derived from the application of the allusive function of the predicate plus additional information, allowing us to infer or even find: (C1) an extension of tropes as the set of tropes strictly similar to the trope in question and (C2) an extension of objects as a set of objects having tropes strictly similar to the trope in question. However, in both cases the extension is a further element that is usually an only vaguely inferred set.[10] As a rule, you do not need to take it into consideration to use a predicate ascriptively.
6. Difficulty with the concept of unsaturation
The main objection to the idea of incompleteness or unsaturation is that it fails to serve its main purpose, which is that of distinguishing a predicative expression from a nominative or singular term. Between the object referred to by the subject and the property designated by the predicate, there seems to be an important functional asymmetry: the nominative term always refers to its object and cannot properly take the place of a predicate; on the other hand, it seems that we can easily turn a predicate into a subject by means of nominalization.[11] For instance, ‘Socrates’ in the statement ‘Socrates is wise’ always refers to its object and cannot properly take the place of a predicate, while ‘… is wise’ can be nominalized as ‘wisdom’ in a statement like ‘Wisdom is a virtue.’ To make the point more convincing, consider the following sentences:

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

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

1.        <Socrates> was a man who lived in Antiquity.
2.        <Socrates> has the property of being wise.
3.        <Socrates> is the husband of <Xantippe>.
4.        <Socrates> is in < that place>![12]

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

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


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

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

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

That is, some things can exist apart, and some cannot, and the former are substances.
     I am not here worried in questioning if there are substances, what they are and if they are ultimately able to exist apart. However, applied to individuals or material objects understood as (at least) clusters of tropes displaying compresence, the independence definition suggests that the objects typified by material things exist in a manner relatively independent of their tropes in the composition of facts understood as tropical arrangements in the world.[13] Moreover, I hold that the individual referred to as a subject is only independent relatively to its predicated trope-properties, because the relation of existential independence/dependence is here understood in a way restricted to the internal context of the fact represented by the statement.
     In other words, my suggestion 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 statement is that this reference is a trope (simple or complex, homogeneous or heterogeneous) whose existence as part of the fact depends on a cluster of selected compresent tropes constituting the individual referred to by the singular term, which is independent relatively to that trope. It seems that this fragile distinction is what that really distinguishes the references of logical subjects. Here are some clear examples supporting this view:[14]

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

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

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

Hence, it is important to see that the considered existential tropical dependence is relative to the fact it is a constituent (Cf. section 23).
     In trying to explore this view in more detail, we can begin by remembering Peter Simons’ nuclear trope theory of material objects. According to this theory, individuals are in the standard case formed by an essential nucleus or core of mutually founding tropes, which is necessarily surrounded by a looser cluster of accidental peripheral tropes, so that these peripheral tropes require the nucleus of essential tropes for their existence (See Appendix to Chapter III, sec. 3). To this we should add, as already noted for the relevant case of material objects, that belonging to the nucleus are typically tropes like those of hardness, form, volume and resistance to pressure or solidity, a trope that in physics was better elaborated under the label of inertial mass, all of them related by the dependent trope of compresence.
     Unfortunately, the issue is not so simple. As we saw in the Appendix of Chapter I, the identification rule of a proper name requires for its application sufficient and predominant satisfaction of at least one inclusive disjunction of the two fundamental description-rules belonging to it, which are the localizing and the characterizing rules (Cf. Appendix to Chapter I). This identification rule, as we also saw, can be satisfied by an indeterminate range of independent 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 one only individual in different counter-factual situations. Already considered examples are the Aristotle born 500 years later in Rome in one possible world and the Aristotle who in another possible world was born with cerebral paralysis in Stagira in 283 BC, son of Nicomachusand was unable because of his disorder to write any philosophy. Hence, the nucleus of mutually founding tropes may be different within limits established by the identification rule. Consequently, in the case of objects referred to by proper names there is no necessary condition in re – no unique real essence of the object – to be expected, but only a nominal essence given by its proper identification rule, even if grounded on verified regularities. Peripheral tropes, on their side, would be those referred to by our auxiliary descriptions like (i) ‘the teacher of Alexander’ and (ii) ‘the founder of the Lyceum.’ And it is clear that the tropes designated by relations like ‘…the teacher of…’ and ‘…the founder of…’ are dependent on the existence of individuals like ‘Aristotle,’ ‘Alexander’ and the ‘Lyceum’ in order to exist as components of the facts represented by statements (i) and (ii).
     Searching for a simpler example, I will now consider the singular term ‘this chair.’ I regard this phrase as an indexical name. This indexical name has an identification rule made up of two interconnected fundamental description-rules: a contextually dependent localizing description-rule establishing a spatiotemporal 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 non-vehicular seat with a backrest made for only one person to sit on at a time. We can say that the complex criterion for the identification of chairs added to the spatiotemporal location is what in this case forms the indispensable nuclear structure of this designatum. Symptoms of this chair, such as its having four legs and two armrests, or its being made of wood, are peripheral combinations of tropes. Moreover, if I say ‘This chair is green,’ the trope of green (in the described fact) exists in dependence on the existence of a complex of compresent tropes that forms this chair and would not exist without their existence.
     These considerations allow us to better understand the corresponding 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 ‘ swift’ in this sentence applies to a contingent trope that constitutes swiftness, whose existence is here fully dependent on the existence of an object, Bucephalus, which is constituted by some core of compresent tropes constitutive of a living material object. On the other hand, the same distinction also applies to properties linked to individuals that are not properly material objects. A rainbow, for instance, is an individual (a cluster of compresent tropes), though not properly a material object. But consider the dynamic fact described by the statement ‘That rainbow is fading away.’ The fading away of a rainbow is a process-trope whose existence is dependent on the existence of the rainbow in itself.
     Consider now the true relational sentence ‘Bucephalus belongs to Alexander.’ Regarding this fact, the contingent relational complex trope of belonging to could not possibly be found if Bucephalus and Alexander didn’t exist as independent individuals formed by nuclei of compresent tropes. That is, the proper existence of the relation ‘…belongs to…’ is here indebted to the existence of two more stable essential nuclei of mutually founding tropes constituting the two objects Bucephalus and Alexander. These clusters of compresent tropes referred to by the names ‘Bucephalus’ and ‘Alexander’ are concrete psycho-physical individuals that certainly exist independently of the existence of the relatively contingent complex combinations of tropes constituting the trope of ‘…belongs to…’ since to have ownership we need the previous existence of individuals having this particular relational property.
     A problem arises when we have independent countable things or sortals designated by predicative expressions. 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. Suppose now that I point to the chair and say, (i) ‘This chair has two armrests.’ Since the tropical clusters constitutive of having two armrests do not belong to the definition that makes explicit the nucleus, its existence as something that the chair should be dependent on the chair’s existence. However, the predicate ‘…has two armrests’ exists in the independence of the object referred to by the subject ‘this chair,’ since they can be separated from the chair, differently from its color or size. The solution for this problem is simply to see the above logical analysis as incomplete. The right analysis must take roughly the form: (ii) ‘<This chair> has <its first armrest here> and <its second armrest there>’ pointing to the armrests, where ‘x having y and z’ is the main property-trope, which is dependent on this chair and its armrests.
     A related problem arises when predicates denote sortals belonging to definitional cores. Suppose I say, (i) ‘This chair has a backrest,’ where ‘…has a backrest’ is the predicative expression. The problem is not only that having a backrest belongs intrinsically to the object referred to by the singular term, but that the backrest exists independently of the chair. One can saw the backrest and say things like ‘This backrest is green,’ using ‘this backrest’ to refer to an individual. To this case, I suggest a similar solution. A more complete analysis of the sentence (i) will be (ii) ‘<This backrest> belongs (intrinsically) to <this chair>’, where ‘x belongs intrinsically to y’ means that it belongs definitionally to the sortal ‘chair’ used to characterize the located individual y.
   Very complex tropes (homogeneous or heterogeneous, mixed or not) are also existentially dependent on the individuals to which they are bounded. Consider some examples:

(1)     <Céline> had a strange personality.
(2)     <India> has a democratic system.
(3)     <The ancient Spartan State> was extremely militarized.
(4)     <The Vienna Philharmonic Orchestra> played the 5th Symphony.
(5)     <The Irish potato famine> was caused by <the late blight>.

None of these tropes could survive alone. They need to be attached to some localizeable and characterizeable individual to which they belong.
     Finally, what about formal names and sentences? Consider the sentence ‘Three is an odd number.’ This sentence describes a mathematical fact. Considering here ideas about what confers existence, we can think the number three without thinking that it is also an odd number, or ‘the number two or any multiple of two added to the number one’, which is the definition of an odd number. But there is no ‘being odd’ independent of a number. Hence, the existence of oddness factually related to the existence of the number three is dependent on the number three that we are taking into consideration.
     Consider now the statement ‘Two is a natural number.’ One could argue that to be a natural number belongs to the definition of two as a kind of genus proximum, although not essentially to the (here seen as incomplete) definition of two as its differentia. Maybe this differentia could be given by our already suggested understanding of applied natural numbers as higher-order tropical properties of actual or idealized counting belonging to an effectively applicable conceptual rule (See sec. 4 of the Appendix of Chapter III). Repeating what I said there, consider the statement ‘This hat has three corners.’ Here the applied number 3 indicates that the possible conceptual rule identifying the corners of this hat not only has the tropical meta-property of being applicable (attributing existence), but also the tropical meta-property of being applicable three times in an additive way (a counting process). Moreover, we can analytically express this conceptually dependent higher-order trope of 3 by means of the set of applications {a, {a}, {{a}}} understood as a spatiotemporally located higher-order numerical set-trope.
     But how to represent the number 3 distinguishing it as the universal object that is common to all conceptual identifications of three singular entities, the three-in-itself? Here, if we wish to avoid speaking of a Russellian abstract set of all sets of the same kind, we can still construct the number 3 as a located model of tropical numerable trope-set {a, {a}, {{a}}} or any other strictly (equinumerous) located trope-set:

Number 3 in itself (Df.) = a chosen higher-order located numerical set-trope of counting {a, {a}, {{a}}}* used as a model or any other higher-order strictly similar located numerical set-trope.

This definition still allows the predicate ‘…is a natural number’ to be ascribed to the whole definiens as an internal dependent addition (a genus) and the predicate ‘…is an odd number’ as an external dependent addition. In any case, even the name of a so-called abstract object, such as ‘the number three in itself’ cannot be moved to the predicate position here, insofar as it refers to something held as independent, being identifiable (existing) independently of its non-definitional predicates like ‘…is an odd number.’
     Understanding unsaturatedness as relative existential dependence suggests, therefore, that the tropes denoted by the predicate have an inevitable tie of dependence when considered in relation to the relevant individual within the fact referred to by the singular sentence. This gives us a better understanding of the asymmetrical tie between subject and predicate.
     Summarizing the argument, my point is that the independence/dependence distinction gives a sufficiently reasonable ontological ground (I guess the only one) to explain the logical distinction between the references of subject and predicate in singular predicative and relational sentences. The nominal term cannot be moved to the predicate position because it refers to a core of compresent tropes that exists in relative independence of the less central tropes in and outside of the core, these less central tropes being able to be designated by predicative expressions.
     In my view, the proposed analysis also sheds light on the so-called problem of the unity of proposition. What really differentiates subject from predicate regarding the fact represented by the statement is the corresponding independence/dependence of their references. Moreover, what assures the unity of the thought-content expressed by the sentence is simply the existential dependence/independence in the factual unity (for instance, in the fact that Bucephalus is swift). And it is clear that these ties of dependence/independence will be more evident when the difference in relevance between the elements in question regarding the identity of the individuals is greater, and weaker when this difference is smaller, justifying occasional uncertainties.
     Finally, one could object that what really distinguishes the predicate from the subject in singular statements is simply that the subject is a singular term that identifies one particular object and distinguishes it from all others, while the predicate is a general term able to be applied to more than one object… It is this possible one-to-many relation that is at the base of the subject-predicate distinction!
     Nonetheless, although this is true regarding a formal definition of singular and general terms, I believe that what gives a reason for this distinction is the relation of independence/dependence between subject and predicate. What defines an individual is that because of the uniqueness of its existence it can be referred to by a nominal term by satisfying its condition of sufficiency. And what defines a property-trope is its existential dependence on some individual (object). The individual is by definition non-repeatable. On the other hand, the property-trope is repeatable, insofar as qualitatively the same property-trope can be, by its lack of existential dependence, tied to many individuals. But this is so as a consequence of the fact that the existence of the property-trope must always be dependent on the existence of individuals, disregarding what individuals. In the end, it is the difference in nature between individuals (objects) and property-tropes (attributes) that is responsible for the one-to-many relation.

8. Sense of a predicative term

The independence/dependence relationship originating on the ontological level of reference is reflected on the semantic and linguistic levels. It is first reflected on the semantic-epistemic level of Fregean senses. We see this in the fact that the identification rule of the nominal term – its sense – is applied to its object independently of the ascription of tropes to the same object by the ascription rule – the sense – of the predicative expression, while the ascription rule of the predicative expression – its sense – depends on the prior application of the identification rule of the object referred to by the nominal term. Finally, on the level of linguistic signs, the same relation of independence/dependence is what makes the singular predicative sentence take its usual subject-predicate form.
     Our view of tropes as the designata of predicative expressions allows us to make some additions not present in Frege’s original semantic distinctions. The first is the suggestion that different predicative expressions with the same designata may be able to have different senses, paralleling the case of nominal terms like definite descriptions. Consider the following two sentences:

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

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

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

The predicate ‘ white’ in sentences (1) and (2) obviously has the same sense in both, as in each case it expresses qualitatively identical ascription rules. But the tropes of whiteness (of reflecting the combined wavelengths of the visible spectrum) of the South Pole are located at the South Pole itself, while the tropes of whiteness of Mont Blanc are located in its eternal snows. The same can be found in the application of relational predicates. This is explained by the fact that the different objects referred to by different singular terms have numerically different tropical configurations that satisfy qualitatively identical ascription rules of the same predicative expression.
9. Dependence of the predicative sense
As we have already noted, in the context of a singular predicative sentence, the identification rule of the singular term applies to the object as some core of compresent tropes, which seen as a whole exists independently in relation to its more or less dependent partial or peripheral tropes. Consequently, the identification rule is also able to be applied regardless of the application of contingent ascription rules, which means that this identification rule can be conceived as being applied in isolation. This explains its independence and why one could call it complete or saturated. The ascription rule, on its side, will be applied to a trope dependent on the core and consequently depending for its real application on the earlier application of the identification rule, lacking in this sense completeness. This is at most clear in the case of rules for contingent properties, like the conceptual rule for the predicate ‘swift’ when applied to Bucephalus.
     The same may also hold for the fundamental descriptions constitutive of the identification rule of the nominal term in the sentential context. Since the tropes belonging to the object to which the identification rule applies are ultimately dependent on the existence of this object as containing a kernel of tropes, even the ascription rules of predicative expressions already belonging to the identification rule of the object as part of this rule require prior application of the whole identification rule to identify the object in order to become themselves applicable as part of the identification (e.g. the statement ‘Aristotle was the author of the Metaphysics’). Because of this, the application of the predicate’s ascription rule is always dependent on the application of the identification rule of the singular term.[15]
     The general sense of a concept-word, which (diverging from Frege) we identify with the concept or ascriptive rule expressed by it, should then be a rule whose application to an object depends on the prior application of another rule. Hence, the ascription rule of the predicative expression is dependent, incomplete, unsaturated, in the sense that it demands the prior application of the identification rule of the singular term in order to be applied. It is necessary to identify, that is, in the empirical case to find some particular object in space and time, in order to be able to characterize it by ascribing the predicative rule to its appropriate trope. We must, for instance, first apply the rule that allows us to spatiotemporally locate the horse called Bucephalus in order to apply to it related tropes, and on that basis, the ascription rules of predicative terms. Thus, due to the independence of the object Bucephalus, we apply the ascription rules for the predicates ‘... is a horse,’ ‘... is black,’ ‘... is swift’… and also the ascription rules of more complex predicative expressions like ‘…a horse that belonged to the best Thessalonian breed’ to the tropical kernel constitutive of Bucephalus. And we also need first to apply the identification rules for Bucephalus and Alexander in order to be able to apply the relational predicate ‘…belongs to…’ In a similar way, we need to apply the rule that allows us to mentally identify the number 3, in order to be able to apply to associated dependent tropes the ascription rules of predicative expressions like ‘…is odd,’ ‘…is a prime number,’ though it is not the case that the number 3 depends on these things in order to be identified as such. In the same way, the relational ascription rule for ‘3 < 7’ is only applicable in dependence on the independent application of the identification rules for the numbers 3 and 7.
     As I have very early noted (Ch. I, sec. 1), it would be a naive objection to think that after all it is possible to say things like ‘That is a horse’ or ‘There is a black thing,’ applying ascription rules of predicates without identifying Bucephalus. The reason is that a fully detailed identification of the reference as Bucephalus isn’t required at all. Indexicals such as ‘that’ and ‘there’ accompanied by some gesture of pointing are already able to identify some spatiotemporally localizable spot which exists independently of further predication, being therefore in a technical sense an object or individual. As we already saw, this relative independency of the indexical identification rule can be made explicit when the indexical is followed by a term designating countable things, that is, a sortal, such as ‘that object,’ ‘that animal,’ since we localize with the demonstrative and characterize with the sortal. Therefore, not only does the trope designated by the predicate depend upon the previous existence of the object and its identification, but, as a consequence, also the effective applicability of the ascription rule of the predicate must be dependent upon the prior application of the identification rule to the relatively independent cluster of tropes. This is how the relation of semantic dependency – on the level of sense – mirrors the relation of ontological dependency – on the level of reference – solving the riddle of unsaturation.
10. The concept horse paradox
We can continue to make major revisions of Frege’s views in order to overcome difficulties arising from his semantic views, like the so-called concept horse paradox. Based on his view of a concept as the unsaturated reference of a predicate, Frege was led to the strange conclusion that one cannot name a concept. For him the sentence:

1.        The concept horse is not a concept,

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

2.        The concept horse is a concept,

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

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

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

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

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

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

As Searle notes, this sentence contains two components. One is expressed by the predicate ‘…is a horse,’ symbolically Hx (where we use x instead of ‘…’ and H replaces ‘is a horse’). The other component is the predication of existence expressed by the open sentence ‘there is at least one ... such that ...’ This predication can be symbolically expressed as Ǝx(...) (where Ǝx replaces ‘there is at least one … such that…,’ and the last ‘...’ is the gap to be filled by some concept applied to something, now in the most proper ususal 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 metapredication expressing a higher-order concept, a concept of a concept, a metaconcept under which other concepts can fall – in this case (Hx). Thus, Ǝx(Hx) instantiates the general form Ǝx(Fx), which usually expresses a second-order concept – the concept of existence – applied to some first-order concept. In a Fregean way of speaking, what this second-order concept does is to say of the first-order concept that at least one object falls under it, which also means that the first-order concept is satisfied or fulfilled by being applicable to at least one thing. So understood, existence is something objective, since this satisfaction is independent of our cognitively grasping it as the applicability (and not mere occasional application) of a concept.
12. Existence as a property of conceptual rules
These last ways of speaking are more interesting to me because they could be paraphrased in accordance with my identification of concepts with senses of predicates, more precisely, with conceptual, semantic-cognitive ascription rules. This identification shows that existence can be 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 definitely applicable. More precisely, I mean that this conceptual or ascriptive rule is, if it is given, effectively applicable in a domain of external objects. I add the adverb ‘effectively’ or ‘definitely’ to make it clear that I do not use the word ‘applicable’ in a merely subjunctive sense, as referring to something that may be applied, but as referring to something that is effectively (definitely, warrantedly) applicable, which is continuously the case during some period of time (the period in which the object is said to exist). Moreover, the own ascription rule must be seen as a possibility, not as an actuality, since things exist in the independence of their semantic-conceptual characterizing’s rule existence. Furthermore, the existence or effective applicability of a semantic-cognitive rule is always considered with regard to a certain domain of entities (a ‘universe of discourse’). The most fundamental domain is that of the real empirical world, be it the external (physical) world (Carnap’s thing-world) or the internal (psychological) world. The statement ‘Horses exist’ applies in the first domain. The statement ‘Headaches exist’ applies in the second domain. Indeed, what is normally meant by the predication of existence isn’t the applicability of a possibly given ascription rule of a general term as a mere possibility entertained only in our imagination, but also an effective applicability of the rule within some empirically given domain of entities. Furthermore, this effective applicability is usually within what we might call its most proper domain of entities, which in the case of horses is a domain of external, physical objects, and in the case of headaches is a domain of internal, psychological states. I consider this point here because there are subsidiary cases, like that of the Valkyries,[16] whose most proper domain is mythical – that of Norse mythology.
     As one could guess from the last example, there are other higher-order domains and sub-domains of entities within which we can predicate existence, even if only in a subsidiary sense. One can say, for instance, that Valkyries’ horses exist in the fictional domain of Wagner’s opera The Valkyrie in the sense that the ascription rules for these fictional horses are effectively applicable in the fictional domain described in the libretto. There are also cases like the probable existence of life in other galaxies, which can be in principle verified. Thus, there are imaginary mythological domains, fictional domains in the arts, and domains of imaginable but also plausible entities. Moreover, there are domains of so-called abstract entities and their various sub-domains, like the domain of mathematical entities, of logical entities… It is simply a linguistic fact that we can apply the word ‘existence’ in any of these domains. What I intend to show in the following is that there is a unifying justification for this.
     According to the view I am supporting, to say that horses, rocks, trees and chairs exist is to confer effective applicability to the ascription rules of the respective concept-words ‘horse,’ ‘rock,’ ‘tree’ and ‘chair’ in the fundamental domain of material objects belonging to the objectively real external world. To say that thoughts, joys and pains exist is to ascribe effective applicability to the ascription rules expressed by the concept-words ‘thought,’ ‘joy’ and ‘pain’ in the subjectively real mental domain of entities. And to say that ‘totalitarianism,’ ‘corruption’ and ‘exploitation’ exist is to affirm the effective applicability of the ascription rules of these concept-words within the psycho-physical domain of social entities. The domain of entities to which such concept-words apply is usually assumed to be respectively physical, psychological and social. As a general rule, to say that an entity exists is to say that its conceptual rule is effectively applicable in the already conventionally established most proper domain of application. Thus, to give examples, the most proper domain of application of the conceptual word ‘horse’ is the real external world, while the most proper domain of application of ‘Valkyrie’ is a fictional one. That is, it is normally assumed that the attribution of existence is made in its most proper domain. But this assumption isn’t necessary (when I say that there are horses in Wagner’s opera The Valkyrie, the concept horse isn’t being applied in its most proper domain).
     As already noted, a concept – understood as the semantic-cognitive ascription rule of a predicative expression – is able to generate dependent, subjective criterial configurations. Thus, to say that a concept-word is effectively applicable is to say that dependent criterial configurations generated by its ascription rule are able to be fulfilled by corresponding independent, objective criterial configurations. These objective criterial configurations (external or not) can be seen as configurations of tropes usually belonging to more complex tropical arrangements called facts – another point against Frege that I will explain and justify in some detail later.
     The parallel between the concept of existence in Frege and the more detailed concept of existence derived from my reconstruction of concepts as senses of predicates understood as ascription rules is straightforward:

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

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

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

Ǝx [Mx & (y) (My → y = x)].[17]

In this way, we are affirming the existence of at least one and not more than one inventor of the Maieutic. This means that the ascription rule that constitutes the concept (the sense) expressed by the predicate ‘…is the inventor of the Maieutic’ has the property of being effectively applicable to only one human being, namely, Socrates, reducing the domain of application to only one member. This is the same as attributing existence to the inventor of the Maieutic.[18]
     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 possible 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 possible identification rule, primarily in a proper contextualized domain of the external world.
     Although this issue cannot be properly addressed without a deeper investigation of the nature of proper names, we can start by applying the Fregean-Russellian formal device to the foregoing view. In order to do this, we transform proper names into predicative expressions applied to only one particular, showing then that the senses of names themselves can be reduced to the conceptual senses of predicative terms. A first step in the attempt to arrive at this is to transform the proper name into a predicate. Thus, ‘Socrates’ in ‘Socrates exists’ can be transformed into a predicate in the sentence ‘There is something that socratizes,’ or ‘Ǝx(x socratizes).’[19] Taken literally, this suggestion is not only linguistically deplorable, but also formally deficient, since it leaves open the possibility that there is more than one Socrates.
     Nevertheless, I think that ‘Ǝx(x socratizes)’ points in the right direction by suggesting that the existence of a name’s bearer may be asserted by means of the conceptual senses of predicative terms. After all, the verb ‘to socratize’ can be seen as a kind of abbreviation of the predicative conceptual expressions included in the descriptions supposedly summarized by the proper name ‘Socrates.’ This is a reasonable strategy, insofar as we take seriously the bundle theory of proper names that was already fully present in one way or another in the writings of Frege, Russell, and Wittgenstein, though it has been made more explicit by P. F. Strawson and particularly John Searle. According to this theory, the whole sense of a proper name is given by a cluster of definite descriptions. Having this in mind, we might suggest that the attribution of existence to Socrates in ‘Ǝx(x socratizes)’ could be seen as an abbreviation of a set of predicative expressions like:

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

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

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

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

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

Here the supposed meaning of a proper name is disjunctively translated into the conceptual-senses of predicative expressions such as P1, P2… Pn, which according to our analysis are nothing but ascription rules expressed by predicates that we expect to be really applicable to one and the same thing. So analyzed, the attribution of existence to the object referred to by a proper name is made by saying that its sense, its identification rule, definitely applies in the assumed context. As this rule for the identification of a name was here analyzed in terms of a disjunctive set of rules for the application of predicates that must be applied to the same individual, we can easily explain existence as follows: The existence of the bearer of a proper name is the same as the effective applicability of at least one conceptual rule of a predicative expression to precisely one individual.
     Of course, here it could be objected that such a descriptivist attempt to explain the meaning of a proper name is doomed to failure. This must be so, not only because the applied formal device is limited, but also because it amounts to some version of the bundle theory of proper names with its well-known difficulties, already persuasively pointed out by Saul Kripke, Keith Donnellan, and others...
     However, such a conclusion would be too hasty, and there are at least three reasons to oppose it. The first is that, contrary to a current bias, Kripke’s and Donnellan’s objections have not discredit the most comprehensively developed versions of descriptivist theories, and some criticism has already been answered with considerable success by John. R. Searle (1983, Ch. 9). A second reason is that Kripke’s alternative solution, the causal-historical view, could never be developed beyond a rough sketch.[20] These first two points lead us to the conclusion that bundle theory hasn’t yet been definitely refuted.[21] Indeed, perhaps it just needs a stronger defense.
15. The existence of objects and its identification rules
The third and really conclusive reason that I can oppose to the anti-descriptivist view is that the above presented formal analysis is still a crude simplification when seen from the viewpoint of the new version of the bundle theory of proper names I have exposed in the Appendix to Chapter I. This version has, as I believe I have demonstrated, a much greater explanatory power than any previous theory, answering in a more nuanced way the most diverse counter-examples.
     Briefly repeating what I said there, my view is the following. The traditional bundle theory of proper names defended by Frege, Russell, Wittgenstein, P. F. Strawson, John Searle and others has a severe limitation that has been overlooked: the bundles have no internal order. The theory does not tell us which descriptions or combinations of descriptions are more or less important or even why some seem to be very important for the application of a name, while others are obviously irrelevant for it. Definite descriptions are nothing but expressions of rules that should help us to connect a proper name with its reference. I called them description-rules. Regarding all this, my question was whether we cannot find the general form of a rule that we all implicitly know, which if applied to any bundle of descriptions associated with a proper name enables us to recognize the most relevant ones and decide in what ways the satisfaction of these descriptions makes this proper name applicable to some referent.
     When searching for the general form of a rule, the first thing to do is to classify the descriptions. There is a sensible, ordinary-language method to use in order to begin with: check how encyclopedias treat well-known proper names. We can thereby easily distinguish fundamental from merely auxiliary descriptions, which are accidental. In doing this we see that proper names are first and foremost attached to two fundamental forms of description, which I call localizing and characterizing description-rules. Here is how we can define them:

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

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

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

It is usual in encyclopedias that the first thing we find is an abbreviation of the localizing description-rule, followed by an abbreviation of the characterizing description-rule, stating the reason why we remember the name. What follows in the Wikipedia article (as in many others) are more or less relevant details and explanations. We find a variety of definite and indefinite descriptions that are more or less irrelevant: accidental, auxiliary descriptions. Examples of them are that Hitler was ‘the lover of Eva Braun,’ ‘the son of Alois Hitler and Klara Pölzi’[22], ‘the person called “Adolf Hitler”,’[23] ‘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 a more extended form in biographies.
     You find a similar pattern if you search in encyclopedias for other proper names like ‘New York,’ ‘USA,’ ‘Eiffel Tower,’ ‘Niagara Falls,’ or ‘Milky Way.’ Of course, there are also the proper names of ordinary persons who are not famous enough to mention in encyclopedias. But the basic mechanism of reference remains the same. It is not difficult to see that the relevant information is given by their localizing descriptions and by the usually much more scattered characterizing descriptions. So, in most cases, if you wish to know who Sam is, you can probably get relevant information from his identity card, drivers license, employment record, police record (if any), school reports, club records… and most of all from details given by him, by his family and friends about his personality, character, education, interests, abilities, relationships, accomplishments, faults, etc., which are linked together by just one spatiotemporal career.
     Now, my suggestion is that, although a conjunction of the localizing and the characterizing descriptions isn’t required in any possible world, as Kripke has clearly shown (1980: 62), an inclusive disjunction of the two fundamental description-rules must in some degree be satisfied to enable a proper name to refer to its object in any possible world. John Searle perceived this point many years ago when he wrote:

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

General form of the identification rule for proper names:
A proper name called ‘N’ has a bearer
it is something that belongs to the nearest relevant class of referents C, so that more than any other entity of the kind C it sufficiently satisfies at least the conditions set by:
(A) its localizing description-rule,
(B) its characterizing description-rule.
(Auxiliary descriptions can be added as helpful symptoms for the identification[24]).

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

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

This summarized identification rule gives us the core meaning of the proper name ‘Adolf Hitler.’ If we try to imagine an Adolf Hitler who does not minimally satisfy the fundamental localizing and/or characterizing conditions, we see that this is impossible. This was the case of the Adolf Hitler born in Gusental in 1634, who was a peaceful shoemaker and had nothing to do with politics. Surely, he could not be the person in a political socio-historical context whom we always mean by the name ‘Adolf Hitler,’ but someone else with the same name.
     This example also outlines the lack of relevance of auxiliary descriptions. Suppose that the Adolf Hitler born in Gusental in 1634 satisfies many of the best-known auxiliary descriptions: he was the lover of an Eva Braun, he was the son of an Alois Hitler and a Klara Pölzi, the person called ‘Adolf Hitler,’ the boy who was sent by his father Alois to the Realschule in Linz… The feeling elicited by these strange discoveries would be of deep puzzlement, not persuasion. For his Eva Braun could not be the well-known Eva Braun who also committed suicide in the Bunker… and even that his parents had the same name as those of the infamous Adolf Hitler would be merely a remarkable coincidence… (He could not, it is true, satisfy the description ‘the author of Mein Kampf ’; however, more than an auxiliary description, this is already part of the full characterizing description of our Adolf Hitler.) Anyway, at no point will this change our belief that he is not the person we are trying to identify.
     Since so understood the identification rule simply defines which object among all others owns the proper name by establishing the definitional criteria for identifying the proper name’s bearer in any possible world, it unavoidably also applies in any possible world where the name’s bearer exists, satisfying the fundamental requirement of the Kripkean definition of a rigid designator (1980: 48). The individually taken definite descriptions belonging to the bundle, particularly the auxiliary ones, on the other hand, being only loosely associated with the identification rule, can refer to other objects in different possible worlds and are therefore only accidental or flaccid designators.[25]
     Moreover, one can insert a name correctly in a sufficiently vague discourse without knowing more than auxiliary and indefinite descriptions, even when they are wrong, as Kripke realized. This is the case at least insofar as these descriptions are convergent (rightly classified), making in this way what we should call a parasitical reference, which can be helpful in several ways. For instance, if someone already knows that Hitler was ‘some dictator’ or erroneously thinks that he was ‘a military general,’ this person already classifies him correctly as a man of power can already apply the name correctly in sufficiently vague contexts and possibly be corrected and learn more about him.
     Now, the existence of an object referred to by a proper name is the effective applicability of what can be called the identification rule of the proper name in its (in most cases) proper contextual domain. We know that Hitler existed because we know that his identification rule was effectively applied, hence applicable, in the political-historical context of Europe in the first half of the twentieth century. Moreover, what allows us to say that the bearer of the proper name ‘Hitler’ exists is that the property-tropes that belong to this object satisfy an identification rule that by this reason has the property of being effectively applicable to it, a property that is actual if the rule is instantiated in some mind, but that would be only potential if the rule were never instantiated in any mind (what is almost impossible to imagine in the present case, but would be easily imaginable concerning an object like a primitive animal living in a distant planet.) This property of the potentially existing identification rule is a higher-order property of the object, endowing it with existence in the real world and not as something only conceivable.
16. Existence of spatiotemporal 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’... The indexicals minimal task is to indicate some spatiotemporal location relative to the speaker. Thus, ‘here’ points to the place where the speaker is, ‘now’ to the moment when the speaker speaks, ‘yesterday’ to a period of time, the day before the day of the speaker’s utterance… And regarding indexicals like ‘I,’ ‘she,’ ‘he,’ ‘they,’ there is more to say than just this. Surely, these personal pronouns have more semantic content than just a plain spatiotemporal location, but this does not matter to us now.
     Consider now the indexical statement There is a raven,’ said when one found only one raven there. How should we analyze it? Of course, we can transform ‘There’ in the definite description ‘the spatiotemporal location pointed to (or contextually shown) by the speaker when he utters the word,’ which expresses a one-foot localizing identification rule followed by the countable predicate, the sortal ‘…is a raven’ with its ascription rule. But in order to show our existential commitment, we need more. We need to analyze the definite description replacing the indicated spatiotemporal location by the predicate ‘…is in time t and place p’ symbolized by L, replacing then the predicate ‘…is a raven’ with the symbol R. With help of this we can symbolize ‘There is a raven’ as Ǝx [(Lx & Rx) & (y) ((Ly & Ry) → y = x)], which means: ‘There is precisely one x that is in L and is a R.’ Although the location L figures here as a predicate, the condition of unity (any y = x) makes it a singularized spatiotemporal location supposedly also analyzable in terms of tropes (See Appendix to Chapter III, sec. 3).
     There is another common way to expose our existential commitment in indexical statements. It is when we add to them a sortal predicate, as in ‘that raven there’ in the sentence ‘That raven there is flying’ or ‘this chair’ in the sentence ‘This chair is comfortable.’ In these cases, we consider the phrases ‘that raven there’ and ‘this chair’ as referring to only one specific object, distinguishing it from all others. Hence, these phrases work as singular terms and must be analyzed as expressing identification rules. Replacing ‘…is a raven there’ with R and ‘…is flying’ with F, we can also formalize it as the existential statement Ǝx [Rx & (y) (Ryy = x) & Fx].
     Indexical statements are important because when we use them the language, so to speak, ‘touches’ the world, which makes indexicals the indispensable roots of reference. Because of this, although the sense still determines its reference, we can find here a double direction of fit. First, with the help of our sensory cognitions, we create the identification rule for the indexical that is for the first time used in a determinate context. Once formed, this identification rule (a Fregean sense) determines the spatiotemporal location, often together with the kind of object characterized by the sortal. Now, this new identification rule can be so established that it can be reapplied (not only later, but immediately thereafter), soon forgotten, or maybe interpersonally conventionalized by association with a non-indexical singular term of our language, normally a definite description. To this description, others can be later joined, building that bundle of descriptions able to flexibilize the referential work to many diverse circumstances which is typically abbreviated as a proper name.
17. Advantages of the higher-order view of existence
There are several advantages in conceiving existence as a higher-order property, that is, as a higher-order trope. One is that it gives a straightforward answer to what seems odd in the traditional forms of the ontological proof of God’s existence. So, according to Descartes, once we accept the definition of God (1) as the being with all perfections, and that (2) existence is a perfection, we must conclude (3) that God exists (1978, V: 65). But if existence is a (tropical) meta-property of objects and not a proper intrinsic first-order tropical-property constitutive of them, differing in this way from perfections like infinite goodness, omniscience, and omnipresence, which should be intrinsic properties of God, the ontological proof is doomed to failure (Cf. Frege 1874, sec. 53).
     However, the greatest advantage of conceiving existence as a higher-order property is that we will not have problems with the denial of existence. Suppose that existence were a first-order property of an object. In a sentence like ‘Vulcan does not exist,’ the negation of existence should then be applied to the object itself, and we would first have to identify the object in order to deny that it has the property, the trope of existence. That is, if in order to identify an object, we first had to admit that it exists, we would be caught in a contradiction: we would have to admit the existence of Vulcan in order to deny its existence.
     According to our Fregean view, this contradictory conclusion isn’t necessary, because all we do by denying the existence of Vulcan is to assert that the ascription rule that forms the concept of Vulcan doesn’t have the meta-property of being effectively applicable in its proper contextualized domain of physical objects. Only to illustrate the point, we could analyze the sentence ‘Vulcan does not exist’ as a shorthand way of saying:

x [(x is a small planet orbiting the Sun between Mercury and the Sun) & (y) (if y is a small planet orbiting the Sun between Mercury and the Sun, then y = x)].[26]

What belongs to the scope of ‘~Ǝx’ are concepts constitutive of the identification rule, which in this illustration consists of an ascription rule for a predicate that can be applied to only one and the same object. What ‘~Ǝx’ does is just to deny that this identification rule has the property of being effectively applicable to the corresponding physical object, which is to deny that an object existing only in our minds has the (meta-)property of also existing in reality.
18. Ubiquity of existence
The understanding of existence as the effective applicability of (semantic-cognitive) conceptual rules allows us to explain the almost unlimited extensions in the application of this concept. Why given that existence is primarily attributed to properties and objects of the outside world or of psychological states, are we also allowed to say that supposed entities like hypothetical and fictional ones exist? Some believe that even contradictory objects exist. We can even say that everything exists, including all that can be conceived – at least as something that can be conceived. And even of existence itself, it can be said that it exists. Indeed, it seems that in one way or another everything exists. How can this be possible?
     Concerning supposed entities, we need to distinguish at least two kinds: hypothetical entities that experience hasn’t yet shown to exist or has shown not to exist, and imaginary (including fictional) entities. Beginning with the first group, it is clear that we can find a sense in which they exist. Although the planet Vulcan has been shown not to exist in the real external world, its most proper domain, it surely has existed in the domain of the minds of many astronomers in the past who searched for it, as a hypothetical object… and it still exists in our minds, as a merely imaginary object.
     For Frege, this would be a problem. But this is no problem for our proposed view because our identification rules can also have the existence-endowing property of being applicable, at least partially, in imagination, that is, only in the dependent domain of conceivable things that we consider as possible or even plausible candidates for existence in the external world. If I imagine the hypothetical planet Vulcan orbiting the Sun, I apply the identification rule for that proper name (even if in a vague, sketchy, deficient way) to a merely conceivable state of affairs. Indeed, the French astronomer Le Verrier, who first named the planet, even had a precise identification rule according to which Vulcan should be a small planet orbiting close to the Sun at a distance of 21 million km, which he mathematically calculated in order to explain by means of Newtonian laws the perihelion precession of Mercury’s orbit. He applied this rule in the domain of what is conceivable, which means that Vulcan ‘existed’ in the restricted domain of the imagination of Le Verrier and other astronomers in his time, though not in its most proper domain – that of a concrete object, a planet belonging to the external world.
     Consider now the case of purely fictional entities. Ivan is a character in Dostoyevsky’s philosophical novel The Brothers Karamazov. He never existed in the real world; but he can be said to exist in the fictional world created in this novel, which is from the start fictional. In this domain, Ivan is the son of Fyodor Pavlovich and has two brothers, Dimitri and Alyosha. Ivan is a cerebral as much as a weak character, taking refuge from the inevitable confrontations of life in contemplation and inaction and creating resentful justifications for this; in the end, under the weight of his own conflicts, he descends into madness. These and other elements form parts of the rule for Ivan’s identification. We say that he exists in the story, insofar as this rule is effectively applicable only to him within this proper fictional domain. Differing from the case of hypotheses, existence in a fictional world excludes from the start existence in the real world. That Ivan said to Alyosha: ‘let the worms devour one another’ is true in its fictional domain, as this statement is really made in the novel. But this utterance has no existence in the domain of the real external world, where it would be a displaced truth-bearer since the novel was not written to fit into it.
     Saul Kripke gave examples of cases of fictional-fictional characters like Gonzago (2013: 250), who is a personage in Shakespeare’s Hamlet as a fictional character created by Hamlet in his play within a play ‘The Murder of Gonzago.’ There is a hierarchy here. We may say that Gonzago exists in a third-order domain of Hamlet’s play, requiring the effective applicability of a proper identification rule in this same domain. This third-order domain is supported by the existence of the plot of the fictional play Hamlet, forming a second-order domain. This play is in turn supported by the identification of some writer and writings in the first-order domain of our self-sustaining fundamental real empirical world.
     As with other merely imaginary entities like winged horses and unicorns, existence is here affirmed within a domain that is dependent, derivative or extended (Kripke 2013: 81), being supported by the fundamental form of existence, which concerns the effective applicability of cognitive rules in the domain of the real external (physical) or internal (psychological) world. Existence in these forms of usage is parasitic to the fundamental sense, though retaining its basic features (also Searle 1969: 78-9). In traditional philosophy, it was common to use the word ‘being’ instead of ‘existence’ for merely conceived existence. But I suspect that the real intention was often to underline the importance of conceived entities, underplaying or obstructing its derivative, parasitic character.
     What about the attribution of existence to contradictory imaginative conceptions like that of a round square? This case seems really too hard to accept. We cannot combine the rule of identification of the square with the rule of identification of a circle so that both can identify one and the same thing, since they are from the start incompatible. We cannot do this even in our imagination. Because of this impossibility, we must recognize that in a literal sense a round square cannot reasonably exist: we cannot have a contradictory combination of conceptual rules, because it cannot form a possibly applicable rule combination. Since conceptual ascription rules are what constitute their cognitive meanings, this conclusion agrees with our strongest intuition: contradictions do not exist because they lack cognitive meaning.[27]
     Finally, what about existence? Can we say that existence itself exists? Surely, we know that existence exists in the sense that we know that the concept-word ‘existence’ is effectively applicable to the property of effective applicability of conceptual rules in the most diverse domains, telling us that this property of effective applicability exists. This means that 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 metaconceptual rules made for the attribution of existence as the property of effective applicability of lower-order conceptual rules. Since there are meta-conceptual rules of existence which are effectively applicable (since entities belonging to their varied domains exist), the meta-meta-rule – which demands the effective applicability of meta-rules attributing effective applicability to first-order conceptual rules – is also effectively applicable. Consequently, it is safe to conclude that existence itself exists. Well, then, does the existence of existence also exist? Surely: since the meta-meta-rule of existence is effectively applicable to meta-rules of existence by saying that the latter are effectively applicable to the first order conceptual rules, insofar as the latter ones are effectively applicable, we can conclude that a meta-meta-meta-rule of existence (affirming the existence of existence in itself) is also effectively applicable to the meta-meta-rule of existence, making the latter consequently existent. Of course, one can continue acknowledging the existence of the existence of existence and so on, in an infinite regress, which is virtuous since it can always be stopped.
19. Answering some final objections
According to many present theorists, existence is a first-order predicate. A statement like ‘Horses exist’ should be analyzed in a form similar to ‘Horses are animals.’ Since they have developed objections against the traditional second-order view, I will answer at least some of them, as they were formulated by Collin McGinn (2000b: 21-30). The answers can be helpful in clarifying my own standpoint.
     The first one is against Russell’s proposal that to say something exists is to say that a propositional function – a property, a concept – is true for at least one instance. Roughly stated, the objection is that for one object to instantiate a property this object must already exist, an admission that would make Russell’s view circular, since it must already presuppose the existence of objects instantiating the property. For instance, if ‘Mars is a planet’ is true, it presupposes the existence of the planet Mars to instantiate the property expressed by ‘…is a planet’ in order to make the sentence true. Summarizing, there must already be existent objects in order to instantiate the properties ascribed to them by our conceptual words.
     This objection works insofar as one holds a Kripkean view of objects bearing proper names, since for him they cannot be defined by their own properties (1980: 52). Once we have analyzed an object as a widely accessible cluster of tropes displaying compresence, the objection appears to us in a different form. Since not only the ascriptive rules of predicative expressions, but also the identification rules of nominal terms are for us conceptual rules, our position should be generally stated as saying that existence is the effective applicability of any semantic-cognitive rules in some chosen domain or context. However, since these rules also apply to objects as compresent clusters of tropes, this means we cannot conceive any object as being given – that is, as existing – without simultaneously conceiving its identification rule as effectively applicable to it. Thus, for instance, the existence of a concrete object like the planet Mars is nothing but the effective applicability of its identification rule in its proper astronomical context. This means that we cannot separate the existence of the object in its proper context from the effective applicability of its identification rule in the same context, since this is what warrants the object’s existence. Now, if we assume that the attribution of truth to a singular predicative statement results from the applicability of the identification rule added to the applicability of the ascription rule, the attribution of properties and the admission of the object’s existence are conceptually correlative and cognitively simultaneous. Moreover, as the truth follows from the combined application of the first two rules, it is wrong to insist that the attribution of truth requires the attribution of any property prior to the attribution of existence to the trope-property and the object as a cluster of trope-properties. The conclusion is that the flaw in McGinn’s objection lies in the assumption that we can separate the instantiation of a property by an object from the attribution of existence to this same object.
     Now to the second of McGinn’s objections: uninstantiated properties are said to exist. But in order to exist, an uninstantiated property must fall under a higher-order property attributing its existence. This higher-order property must also exist, which means it must fall under a still higher-order property and so on infinitely. Consequently, the attribution of existence as a higher-order property is impossible, because it requires an infinite regress of properties to allow the attribution of existence.
     My answer is that I agree (partially) with the diagnosis, but not with the prognosis. The effective applicability of a semantic-cognitive (conceptual) rule in its most proper domain not only endows its reference with existence, but is in itself a second-order property or trope that can also be said to exist. And furthermore, a semantic-cognitive rule that is only imaginatively applicable not only endows its reference with existence in an imaginary domain, but can also be said to exist. The trope-property of existence exists, which means that we can say that the second-order property of effective applicability of a conceptual rule can be the object of a third-order rule predicating its effective applicability, and so on indefinitely. This, of course, leads to an infinite regress. However, it is a virtuous infinite regress, since the applicability of a conceptual rule such as existence is already warranted by the application to it of a higher-order rule, and we don’t need to bother with all the unlimited further applicabilities of applicabilities or existences of existences that the first existence-endowing rule can generate. The mark of a virtuous regress is that we may stop it without loss when we feel that we do not need further steps to what we intend to explain, and this is the case here (See Appendix of Chapter III, sec. 2).
     The third objection is that there are statements ascribing existence to particulars, such as ‘Venus exists,’ that resist the traditional paraphrase. We have already answered this objection in our treatment of proper names as conceptual identification rules.
     But there are other objections. Consider the statement ‘Something exists.’ Although this is a true statement, McGinn believes that it is not paraphrasable in terms of the higher-order view, since there is no property to be instantiated here, and if we try to translate into the standard form we get the gibberish ‘Ǝx(…x).’
     The answer to this objection is too easy. What ‘Something exists’ means is that there is at least one trope or tropical construction out of tropes that exists without a further determination on our side. That is, we can say that there is some semantic-conceptual rule that is applicable to some domain of entities, even if this rule remains unspecified. This possibility is even shown by our logical symbolism on an elementary level, since we can symbolize an undetermined property such as, say, F. In this way we can translate ‘Something exists’ symbolically as Ǝx(Fx). But there is nothing wrong with Ǝx(Fx). Paralleling existential universalization, we can reach this result by considering singular existential statements like ‘Venus exists.’ So, calling Venus V, if it is true that ‘Ǝx(Vx)’ this implies that some property exists or ‘Ǝx(Fx), namely, that some conceptual rule is effectively applicable. This assumption of cognitively undetermined properties is harmless.
     McGuinn reminds us that there are also more complicated statements that seem to resist a higher-order understanding of existence, like:

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

Nonetheless, we can easily explain the predication of existence in such statements, insofar as we do not confuse the domains of application of the semantic-cognitive rules involved.
     Thus, statement (1) means that some cities that exist in the imaginary domain exist only in this domain. Hence, the effective applicability of rules allowing us to identify the imaginary cities of Chloe and Valdrada in the contextual domain of the book The Invisible Cities is sufficient for the attribution of existence to them in that purely fictional context. Statement (2) means that some things you are talking about exist only in imaginary domains, but not in the external world, that is, there are identification rules that are effectively applicable only in the unreal domain of one’s own discourse. For instance, the identification rule of the name ‘Vulcan’ in the statement ‘Vulcan is red’ is only applicable in the speaker’s (or hearer’s) imagination. Finally, statement (3) means that there is at least one thing that exists only in the mind but not in external reality. Indeed, it seems obvious that the identification rule for some objects and therefore for at least one of them, though effectively applicable in an imaginary, only conceivable domain, isn’t effectively applicable in the domain of external reality.
     The last of McGinn’s objections is that according to the higher-order view, nothing can exist without falling under some property or other, which rules out the existence of a thing that has no properties – a ‘bare existent.’ However, our empiricist commitment makes us see this not as a weakness, but rather as a further anti-metaphysical advantage of our understanding of the higher-order view.
20. Reference 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 must be nothing more than things that we are certain we would perceive under suitable circumstances.[28] As Berkeley wrote:
Existere is percipi or percipere… The horse is in the stable, the books are in the study as before. (1707-8, Notebok A, 429)
The table I write on, I say, exists, that is, I see and feel it; and if I were out of my study I should say it existed – meaning thereby that if I was in my study I might perceive it, or that some other spirit actually does perceive it. (1710, I, sec 3)
According to this view, esse is not only percipi, but also percipi possi. In a more explicit manner, what Mill suggests is that:
Matter or substance is not made up of actual sensations, but of groups of permanent (or guaranteed or certified) possibilities of sensation.
Mill justifies his identification of matter or substance with permanent possibilities of sensation in the following way. First, these possibilities of sensation are conditional certainties: they are not mere epistemic possibilities, but firm conditional expectations that are in direct or indirect ways based on experience. They are permanent in the sense that, once suitable circumstances are given, they would always be experienced insofar as they are said to exist. And they are guaranteed or certified in the sense that we have good reasons – observational or not – to have a firm expectation that under suitable circumstances they will be experienced again and again. This does not mean that the groups of permanent possibilities of sensations would depend for their existence on our past experience of them, because if that were so, they could not exist without us as subjects of knowledge, and we would fall like Berkeley into some radical form of idealism (Berkeley 1710, 1713). This was not Mill’s intention. As he explains:
We mean [by permanent possibilities of sensation]… something which exists when we are not thinking of it; which existed before we have ever thought of it, and would exist if we were annihilated; and further that things exist that we never saw, touched or otherwise perceived, and things which never have been perceived by man. (1979, X: 178-177)
Thus, it is clear that Mill wished to avoid idealism: the permanent possibilities of sensations would exist even if cognitive beings able to perceive them never existed.
     These permanent possibilities are for Mill objective, differing from our actual constantly changing sensations, which are subjective. They are objective because they are grounded, he thinks, in our common public world, which makes us able to interpersonally agree on their existence. For him, even if different persons cannot have access to the same sensations, they can have access to the same possibilities of sensation. As he writes:
The permanent possibilities are common to us and to our fellow creatures, the actual sensations are not… The world of possible sensations succeeding one another according to laws is as much in other beings as it is in me; it has therefore an existence outside me; it is an external world. (1979, X: 181-2, my italics)
This is in summary Mill’s view on the nature of matter – a view that always seemed to me as much deeply suggestive as contentious.
     I think there is a serious confusion in Mill’s view, which can be made clear when we compare his insights with those of Berkeley. According to the non-official Berkeleyan view, the external world is constituted by sensations whose experience is continually (permanently) possible for us, even if we are not there to experience them. But if this is so, the material objects constituting the external world cannot be reduced to simple ‘groups of permanent possibilities of sensation,’ for possibilities as such, permanent or not, cannot be qualitatively distinguished one from the other in the same way as one material object can be distinguished from another. Material objects can be qualitatively very different from each other, they are multiple and varied, while possibilities are always the same, namely, mere possibilities. Consequently, possibilities (of sensations), permanent or not, cannot be the same as material things. Keeping this in mind, the only feasible way to express the Berkeleyan insight in Mill’s terminology seems to me to use it in the characterization of material objects, as follows:

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

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

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

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

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

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

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

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

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

It seems that all these things together contribute to building the condition of an effective application of a semantic-cognitive rule in the domain of the external world – they are contributing to warrant the attribution of external existence.
     There is, however, an important and seemingly fatal objection to Mill’s view of matter, which is made more serious by the Berkeleyan correction I made above.[29] It is that the group of sensations or configurations of sensory criteria that satisfy a conceptual rule are by their nature inevitably psychological. It seems clear that even sensations or contents of sensations that are warranted as permanently possible must be psychological in a dispositional way. This means that if we follow this path, we end up falling into some form of Berkeleyan idealism in which there is no objective, external material world to be contrasted with our subjective world of sensations or sensory criteria. No really independent non-mental external trope needs to be there to match the apparently satisfied dependent criterial conditions, as suggested in statement (5). It is true that, as Mill noted, his possible sensations are independent of our will, that they follow the regularities of nature, even that they appear to be interpersonally accessible under circumstances that warrant their experience (under suitable circumstances they are described as being experienced simultaneously by different subjects, etc.). However, all these things do not seem to help because of the possibility of skeptical scenarios: they can all be unwittingly imagined, as in the dreams. They seem, therefore, insufficient to perform the magic of turning sensations qua sensations into something they aren’t, namely, supposed elements of a non-mental objective external world of material objects with their own tropical-properties. This is an important objection, whose answer will be given only in the final chapter of this book, as a consequence of our discussion of the adequation theory of truth in its relation to direct realism.
     Notwithstanding, I can now anticipate something of the way I intend to deal with the problem. Having in mind the suggested view of existence, we can ask: What warrants an object’s external existence or reality? One answer could be: the joint satisfaction of conditions (i) to (vi) by (5) and nothing more. This would be all that we need to identify the external reality with the contents of our experiences, for there is no way to verify whether or not there is some radical skeptical truth concerning our whole external world, which under normal circumstances makes radical skeptical doubt senseless. (Ch. VI, sec. 30)
     An associated question is: What is in this context an external material object? A too daring answer would be: the external object (as it is thought) must be the identification rule in itself, insofar as it is effectively applicable; in this way, the multiplicity and diversity of objects would be explained by the multiplicity and diversity of identification rules... However, this cannot be, since a semantic-cognitive rule is also something essentially mental, and we are definitely not what Plato called ‘friends of ideas.’
     Looking for a less daring answer, we can suggest that what we understand as the material object is not the semantic-cognitive rule, but is supposed to have the same structure as this rule projected in a specular way onto the external world. There is a reason for this suggestion: It seems that only something with a structure similar to its semantic-cognitive rule would be able to give unity to the multiple and variable criterial configurations by means of which external entities are able to give themselves to us in our experience of them. Figuratively speaking, if the semantic-cognitive rule has the form of a tree with branches whose ramifications end in criterial conditions dependent to the rule, then the object of its application, as we believe it to be, must have the structure of an inverted specular tree with branches whose ramifications end in independent criterial configurations that (supposedly) should match the corresponding subjective criterial configurations. Furthermore, these objective criterial configurations should be nothing but external tropes and constructions out of them (objects, properties, facts). Of course, this objective structure should be putative, so that the rule could always be improved or corrected as a response to new information regarding such specular objective counterparts. (Ch. VI, sec. 34)

[1] On the thorny issue of how to translate ‘Bedeutung’, see Beaney 1997: 36 f.
[2] Searching in the literature, the only place where I have found a similar view on this point is Kneale & Kneale 1985: 495.
[3] One can read singular terms like ‘the morning star’ as definite descriptions or as proper names (like ‘The Morning Star’). I prefer to read them here as definite descriptions, since for proper names we can use the words ‘Phosphorus.’
[4] As shown in the introduction, Ernst Tugendhat later defended a similar understanding of the meanings of singular statements in a more systematic and detailed way, though refraining from doing it as a reconstruction of Frege’s semantics.
[5] If we compare these two passages, it becomes clear that in opposition to Kripke’s interpretation (1980, Lecture I), Frege already had in mind the essentials of the later bundle theory of proper names. The same can be said of Russell (Cf. Russell 1911, Ch. 5).
[6] Assuming Kripke’s views, François Recanati replaces senses with mental files as supposedly non-descriptive modes of presentation (2012: 34). However, since these files are clusters of information and not subjective Vorstellungen, they should be able to be linguistically expressed by means of descriptions, bringing us back to the descriptivist standpoint. For this reason, it seems that semantic-cognitive rules are able to do the same job with higher explanatory potential and (as we will see) with important epistemological consequences. Moreover, these rules or combinations of rules do not need to contain less information than files. They can be as informational, durable, transitory, changeable and flexible as required by the context.
[7] 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 spatiotemporally located between tropes, even if, as an internal relation, it is a subordinate trope.
[8] I suggested this disjunctive construction of universal by means of tropes as the best way to circumvent the usual but problematic definition of a universal as a set or sum of tropes that are strictly similar, one with the other (See Appendix of Chapter III).
[9] Here I agree with Keith Campbell, who has suggested an epistemic primacy of identification over the generalizing function (1990: 24-25).
[10] Even D. C. Williams portrayed things misleadingly here. For him ‘Socrates is wise’ (or any Fa) means ‘The concurrence [togetherness] sum (Socrates) includes a trope that is a member of the similarity set.’ (my italics, 1953: 11)
[11] There are several asymmetries. The most discussed is probably the asymmetry of subjects and predicates regarding negation: you can negate the predicate, but not the subject (nominal term) (Strawson 1971, Ch. 5).  The answer seems to me clear. The negation of the predicate means the admission of the inapplicability of the ascription rule to the object identified by the identification rule. However, since the application of the ascription rule is dependent on the application of the identification rule, whenever you negate the application of the identification rule of the subject you also negate the applicability of the ascription rule and in this way the whole statement. Hence, it is impossible to negate the subject as the nominal term alone.
[12] Notice that the demonstrative ‘that’ does not have here the function of a constituent of the identification rule of Socrates, but expresses the identification rule of a certain place. In indexical statements like ‘This is a daisy,’ the demonstrative ‘this’ expresses a one-foot identification rule, localizing a place in time, while the sortal ‘daisy’ is placed as part of the predicate ‘…is a daisy,’ expressing the ascription rule. It is different from ‘This daisy is yellow’, in which the sortal ‘daisy’ is the characterizing part of the identification rule, whose localizing part is given by the demonstrative ‘this.’ The logical form of the statement ‘This is Socrates’ is already revealed by the relational statement ‘<This spatiotemporal place> is where <Socrates> is located.’ (For the role of localization and characterization in identification rules, see Appendix to Chapter I, sec. 1.)
[13] Ignoring Frege’s theses that the reference of a sentence is a truth-value and that a fact is a true thought, I will in the present context call the sentence’s reference a fact. This choice will be justified in the sections 21 to 23 of this chapter.
[14] I take these examples from Mulligan et al. (1984: 300, 301 and 306), though their point wasn’t the same.
[15] 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)
[16] The Valkyries were maidens who served the god Odin, choosing the soldiers on battlefields worthy of admission to Valhalla.
[17] It is easy to see that singular statements implicitly attribute existence to their objects, since a predicative statement with the form Fa could be written as Ǝx [Fx & (y) (Fy → (y = x)) & (x = a)] in order to make this attribution more explicit.
[18] Socrates lived in Greece from 470 to 399 BC. But usually the time and place of existence are abstracted when we talk about existence, since existence is essentially only the effective applicability of the conceptual rule, not the time of its applicability.
[19] It was W. V-O. Quine who suggested using the name Pegasus as a way to change a name into a predicate such as ‘the thing that pegasizes’ (1948/9: 27).
[20] There are less successful attempts, like Michael Devitt’s interesting book Designation (1981).
[21] 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.
[22] In some cases, like ‘Queen Elizabeth II,’ the family and even genetic origin is part of the localizing description, although this isn’t necessarily so (See Appendix to Chapter II).
[23] What symbolic form a proper name receives is contingent. What makes this form necessary is the identification rule that we attach to it. In a possible world where the name attached to the identification rule for the name Hitler was attached to the name Hartman, this different name would mean what we mean by the name Hitler.
[24] Remembering that there is no sharp boundary between fundamental and auxiliary descriptions.
[25] One could object that rules are changeable and that if we change the identification rule, it ceases to be a rigid designator, unaware that auxiliary descriptions can be changed as much as one will. Nonetheless, if we change the fundamental rules so that the set of possible worlds to which the proper name applies can be distinguished as a different one, we are not applying the same proper name anymore. However, you may introduce changes like additions to the fundamental description-rules insofar as this only specifies the identification better, and thus affecting nothing essential, only adding the application or non-application to possible worlds where the applicability of the rule was in an earlier stage indeterminate. (Cf. Appendix to Chapter I, sec. 7)
[26] This is again a didactic simplification (See Appendix of Chapter I).
[27] However, if the assertion that there are round squares were merely an equivocal manner of saying that we can syntactically combine the adjectives ‘square’ and ‘round,’ that is, a misleading way of saying that there is a syntactical rule allowing the combination of these incompatible words, then it makes some sense to attribute existence. But in this case, what we are trying to say will be more correctly expressed by the meta-linguistic sentence: ‘The rule for constructing the phrase “round square” is applicable, and therefore, the phrase “round square” exists as a grammatical construction.’ The Meinongian Sosein is reduced here to the recognition of a syntactical triviality.
[28] 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)
[29] I believe that Mill’s confusion in the definition of matter was in fact an attempt to evade the objection of idealism open to Berkeley.

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