terça-feira, 17 de outubro de 2017




On my CV:
After a graduation in medicine I made my M.S. in philosophy at the UFRJ (Rio de Janeiro), Ph.D. at the University of Konstanz (Germany) and post-doctoral works at the Hochschule für Philosophie (Munich) and at the universities of Berkeley, Oxford, Konstanz, Göteborg, and at the École Normale Supérieure. 
My main articles published in international journals were collected and better developed in the book Lines of Thought: Rethinking Philosophical Assumptions (Cambridge Scholars Publishing, 2014). Also from interest may be a short theory on the nature of philosophy in the book The Philosophical Inquiry (UPA, 2002). Presently I am writting a book aiming to recuperate the credibility of the old orthodoxy in analytic philosophy of language. This book, to be called Philosophical Semantics, shall be also published by CSP in 2017/2. 

I am full professor at the Department of Philosophy of the UFRN, Natal, Brazil, though with ergonomic limitation.

Advertisement of some published books (see Amazon.usa):


This is an advanced draft of a chapter discussing Frege ti be published in the book Philosophical Semantics, by Cambridge Scholars Publishing in 2018/1

– IV –

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

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

The importance of Fregean semantics for the philosophy of language derives from its unique blend of theoretical simplicity, explanatory scope and philosophical relevance. In this chapter, I want to revise and reconstruct the essentials of Fregean semantics. I intend to make it clear that the basic concept of sense can be paraphrased in terms of semantic-cognitive rules and that the concept of existence can be reinterpreted in terms of the effective applicability of semantic-cognitive rules, leading to some unexpected consequences regarding the explanation of the concepts of verification, fact and truth. With the identification of senses with rules, I intend to show the real link between Wittgenstein’s semantics – as I understood him in the last chapter – and Frege’s semantics; a link already noted by Michael Dummett, though still devoid of pragmatic exploration. Anyway, my aim here is not to produce a work of Fregean scholarship. My aim is instead to reconstruct Frege’s semantic work with him, against him, and beyond him, in order to provide a more rigorous framework for the rather vague semantic insights gained in the last chapter.
  As is general knowledge, Frege explains reference (Bedeutung) using a semantic intermediary link, which he called sense (Sinn) (1891:14). The schema below shows how Frege deals with these two main levels (1) of sense, and (2) of reference in the case of a predicative singular assertive sentence (Satz) of the form Fa:

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

Although Fregean semantics was a development of unparalleled importance for contemporary philosophy of language, it is not free from well-known oddities. My intuitively natural reading of its main semantic elements in terms of conceptual rules will show how to purge Frege’s semantics of its most puzzling weirdnesses.

1. Reference of the singular term
Let’s start with singular terms. The reference of a singular term is, for Frege, the object itself, taken in an enlarged sense. The reference of the name ‘Moon’, according to him, is the Moon itself with its craters. To designate the reference, he uses the German word ‘Bedeutung,’ whose literal translation in English is ‘meaning.’ Most English translators have chosen words like ‘reference,’ ‘denotation,’ and ‘nominatum,’ in this way making clear what Frege really had in mind. There are also other terms, like ‘semantic value,’ ‘semantic role’ and ‘truth-value potential.’ These terms underline the contributions of the references of a sentence’s components to the truth-value of the sentence as a whole. Although the literal translation of ‘Bedeutung’ as ‘meaning’ remains the correct one, for the sake of clarity I will use the word ‘reference.’[1]
  There is also an interpreter’s discussion about the reason why Frege would have chosen the strange 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). It may be. But it seems clear to me that the strongest reason, at least with regard to the reference of natural language terms, is that by introducing the term ‘Bedeutung’ Frege substantivated the verb ‘bedeuten.’ In this way, the word no longer expresses the act of pointing at (deuten) or of designating (bezeichnen), but rather what is pointed at (die Bedeutung), what is designated (das Bezeichnete), that is, the reference itself.[2] In German these derivations could be diagrammed as follows:

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

This would have been the small semantic twist with which Frege turned the word ‘Bedeutung’ into a technical term – a twist that seems to betray some semantic referentialist influence.

2. Sense of the singular term
Now we come to what Frege understands as the sense of a singular term. To introduce it, compare the following two sentences:

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

Sentences (1) and (2) concern to the same thing regarding the planet Venus. But in spite of this, a person can know the truth of (1) without knowing the truth of (2) and vice versa. Frege’s explanation for this is that although the two singular terms ‘the morning star’ and ‘the evening star’ refer to the same planet Venus, they convey different informative contents, that is, they have different senses (Sinne).[3] The word ‘sense’ is defined by Frege as an object’s way of being given (die Art des Gegebenseins des Gegenstandes), which is usually translated as a mode of presentation. The senses of the singular terms ‘the morning star’ and ‘the evening star’ are different, because ‘the morning star’ presents Venus as the brightest celestial body usually seen just before sunrise, while ‘the evening star’ presents the same planet Venus as the brightest celestial body usually seen shortly after sunset …
  Frege writes that words express their senses (drücken ihre Sinnen aus), while senses determine (bestimmen) their reference, since the mode of presentation should show us how to find the reference. Even in cases where the reference does not exist, this determination of reference through sense is given as a possibility, since even in this case the words preserve their senses. This fact points to a flaw in Frege’s idea that sense is the way an object presents itself to us, for in the case of empty terms there is no object to be presented to us. This is why sense can be better understood as the intended mode of presentation instead of as a mode of presentation given by the object (Textor 2010: 134); sense is the way we intentionally present an object or reference to ourselves, whether it exists or not. At any rate, for Frege an expression can have a sense without a reference, but cannot have a reference without its determination by means of a sense.
  Frege extended his notion of sense to other terms and to sentences. In the case of the senses of (declarative) sentences, he calls it cognitive or (more literally) epistemic value (Erkenntniswert). The last term is also appropriate. The Fregean concept of sense has epistemological interest, for it constitutes the proper informative content of the linguistic expression. It is what makes ‘the evening star’ and other expressions informative. Or, using Dummett’s words, ‘sense is what we understand when we understand an expression’ (1990: 92). The philosophical importance of Fregean semantics is largely due to the epistemological and ontological imports of the concept of sense (this is what distinguishes it from a properly 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 the fundamental difference between Fregean semantics and the semantic considerations of the later Wittgenstein, who regarded senses or meanings as depending on episodic uses of expressions determined by rules. Dummett was perhaps the first to defend the idea that senses are rules as the most natural reading of Frege’s use of the term senses. As he wrote in his book on Frege’s philosophy of language:

The sense of a word consists in a rule which, taken together with the rules constitutive of the senses of the other words, determines the condition for the truth of a sentence in which the word occurs. (Dummett 1981b: 194)

And concerning the singular sentences in Frege, understanding with the term ‘criterion’ the condition of satisfaction of a semantic rule, he wrote:

To know the sense of a proper name is to have a criterion for recognizing, for any given object, whether or not it is the bearer (referent) of that name; to know the sense of a predicate is to have a criterion for deciding, for any given object, whether or not the predicate applies to that object; and to know the sense of a relational expression is to have a criterion for deciding, given any two objects taken in a particular order, whether or not the relation it stands for holds between the first object and the second. (Dummett 1981b: 229)[4]

The identification between senses and rules proves particularly compelling when we take numerical expressions as examples. Consider the following expressions:

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

All these numerical expressions have the same reference: the number 2. But their senses or modes of presentation are in each case different. At the same time, they are expressions of procedures, methods, semantic-cognitive rules or, more precisely, combinations of such rules by means of which we reach the identification of the same number 2 as a result (see Runggaldier 1985: 91 f.).
  By treating senses as semantic-cognitive rules and these rules in the primary case as shared conventions, we contrast them with what Frege called colorations and illuminations (Färbungen and Beleuchtungen), which are feelings often associated with image representations (Vorstellungen) and 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, i.e., sensory-emotional states that we normally and customarily associate with expressions. Thus, for example, the words ‘love,’ ‘dog’ and ‘hell’ in the sentence ‘Love is a dog from hell’ (Bukowski) contrastively associate words linked with strong specific emotions in order to create an epigrammatic effect.
  As Frege realized, the kind of appeal or lack of appeal that the colorations associated with words have for different persons depends correspondingly on similarities and differences in their human natures. Because of this, they do not require conventions to be communicated, as in the case of senses. This is why some people are emotionally moved by a certain poem, while others are not. Consequently, it is very difficult to translate poetry, which depends so much on colorations acquired by expressions in a particular language and way 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 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 could easily be found for the objectivity (interpersonal accessibility) of senses, as well as for their consequent communicability. This reason would be that senses typically depend on conventional semantic-cognitive rules, usually 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 conventions or, more importantly, from 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 some way help us to determine the reference of the proper name ‘Aristotle’ (cf. also Frege 1918-19: 63).[5]
  Of course, there is a controversy about this issue, which arose from Kripke’s arguments against descriptivist views of proper names like Frege’s. However, it seems indubitable that Kripke’s arguments can be successfully countered by the kind of meta-descriptivist bundle theory summarized in the Appendix to Chapter I of the present book.[6]

3. Reference of a predicative expression
Frege has something to say about the reference of a predicative expression, which he calls a concept (Begriff) and which may include relations. This is odd, because it seems natural to call a concept something like the sense of a conceptual expression – the mode of presentation of its designata – while the reference itself should be called a property (e.g., a red patch) or some combination of properties (e.g., a bird’s colorful feathers).
  A traditional philosopher like Kant understood the concept as immediately related to a schema, which, as I understand him, is a rule able to lead to the formation of a manifold 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 borrowed from 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. Coming back to Frege’s semantics, we see that what all these comments suggest is that the concept should be the sense of the predicative expression, its mode of presentation, and not its reference, as in Frege’s bizarre use of the term.
  To be fair to Frege, he also says that when an object falls under a concept, the concept may be called a property (Eigenschaft) of the object (1892: 201),[7] seemingly acknowledging that ‘property’ is the right term for the reference of a predicative expression. However, for Frege the criterion of identity for two concepts is the sameness of their value-range (Wertverlauf), or of their extension, which means that predicative expressions with different senses but the same extension must refer to the same concept (2001: 31). So, for instance, ‘…animal with a kidney’ and ‘…animal with a heart’ must be predicative expressions referring to the same concept, since they have the same extension. But it is intuitively obvious that kidneys and hearts are very different concepts.
  In addition to belonging to the realm of reference, Frege also sees his concepts as functions. The mathematical concept of function can be defined as a rule that has as its input arguments and as its output values (for example: ‘3 + x = y’ is a function by means of which when we give as input the number 2 as the argument 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) 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 not. For example, the concept designated by the conceptual term ‘...is a satellite of the earth’ has the value true for the object Moon and the value false for the object Jupiter.
  For Frege, concepts cannot be objects, neither collections of objects, nor extensions (2001: 26). The reason is that objects, collections of objects and extensions are complete (vollständig) entities. That is, they do not require anything to complete them. A concept, by contrast, being a function, is seen by Frege as necessarily open: he calls it an incomplete (unvollständig) or unsaturated (ungesättigt) entity, needing to be completed by those arguments represented by the objects falling under the concept. In contrast, objects referred to by proper names are complete (vollständig), saturated (gesättigt) or independent (unabhängig).
  One could say that the saturated-unsaturated distinction can be found on three distinct levels: linguistic, semantic and referential. For instance: the predicate ‘…is a horse’ could be called an unsaturated linguistic expression (the unsaturatedness is shown by the gap ‘…’), expressing a supposedly unsaturated sense, which refers to an unsaturated concept (property) as the ultimate unsaturated ground. This unsaturated concept, for its part, becomes saturated when some object falls under it, for instance, the object named ‘Bucephalus’ referred to by the predicative sentence ‘Bucephalus is a horse.’
  With metaphors like those of ‘unsaturation’ and ‘incompleteness,’ Frege hoped to open the way to the solution of the mystery of the logical distinction between the subject and predicate of a sentence. After all, the subject (the singular term) would refer to the saturated object, which would complete the unsaturated concept referred to by the predicate (general term).
   Unsaturated predicative expressions and saturated singular terms combine to form saturated singular sentences like ‘Bucephalus is a horse,’ which being complete must also be the name of an object. For Frege this object is the truth-value of the sentence. It was already noted that this would be confirmed by the possibility we have of nominalizing sentences in the form of definite descriptions, since the latter are also singular terms. Thus, the sentence ‘Bucephalus is a horse’ can be transformed in the description ‘the horse named Bucephalus,’ which appears in the sentence as ‘The horse named Bucephalus was black.’ The problem with this argument is that the same can also be done with general terms: ‘…is a horse’ can be nominalized as ‘the horse’, as found in sentences like ‘The horse is an herbivorous animal.’ Hence, the argument isn’t persuasive.

4. Ontological level
 Discussing the unsaturated nature of the references of predicative expressions leads us to the question of the ontological nature of what Frege meant by a concept. If a concept is an unsaturated entity, what kind of entity is it? If it is an abstract entity, it seems that we could at least have concepts not only as referred-to abstract entities (incomplete Platonic entities as references to empty predicates like ‘…is a yeti’[8]), but also (maybe) as the abstract references of corresponding nominalized conceptual expressions.[9] However, such admissions seem to be ontologically abusive (Tugendhat & Wolf 1983: 138-9).
  Anyway, it is by now clear that Frege uses the word ‘concept’ as a technical term that contrasts strongly with our ordinary use of the word ‘concept.’ For our ordinary language intuition there is surely an empty concept expressed by the predicate ‘…is a yeti,’ but this concept should be called empty because it is nothing but the sense of a predicate that has no reference at all! It is no wonder that Frege has nothing to say about the sense of predicative expressions, for he has beforehand emptied them by absorbing the semantic level into the ontological one.
  My conclusion is that it is better to drop the Fregean technical notion of a ‘concept.’ This is a problematic remnant of Platonism that does nothing to explain predication. Instead, we will understand the word ‘concept’ here in an intuitive way as the sense of the predicative expression: its mode of presentation of something. It is counter-intuitive to assume that ‘...is a yeti’ must have any reference; but this predicate clearly has a sense intuitively expressing what we ordinarily understand by a concept, namely, that of the abominable snowman of the Himalayas. Thus, it seems that the best way to give a legitimate role to the word ‘concept’ is to see it as the sense of a predicative expression understood as its cognitive meaning, that is, its ascription rule.

5. Referring to particularized properties: trope theory
But if we drop Frege’s technical notion of concept, what is the reference of a predicative expression? I think that nowadays the most reasonable answer to this question consists in an appeal to the ontology of tropes. Thus, I propose to replace Frege’s reference of predicative expressions with what we now call a trope, which I characterize simply as any spatio-temporally individualizable property. (For a less summarized exposition of my understanding of trope theory, see the Appendix to chapter III of this book.)
  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 and my headache. All these are tropes – particularized properties or simply p-properties – that may range from simple qualities to complex homogeneous or heterogeneous tropes, like the music I listen to and the condensation of water vapor for the former case and the personality of a human being or a country’s political system or a social upheaval for the latter, though such things are in a less specific way also spatio-temporally located. Even very indirectly experienceable things like physical forces could be derivatively constructed from perceived tropes, and it is not even impossible that so-called abstract entities like numbers could be explained as constructions derived from spatio-temporally located properties called tropes. A pure ontology of tropes maintains that all reality must be constructed of tropes, which from a genetic-epistemological perspective are the world’s true building blocks.
  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[10] to model T*.

I suggest this assuming that the trope we take as the model T* is at our discretion and may vary according to the epistemic subject and even concerning the same epistemic subject on different occasions.[11] 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 & co-temporal cluster of tightly connected varied tropes. Moreover, there are some general characterizing property-tropes like unity, displaceability, volume, solidity, resistance to pressure – scientifically explained as inertial mass – that typically comprise material objects.[12]
  Concerning tropes, I usually avoid using the word ‘property,’ not because it isn’t the best one, but because the philosophical tradition has too often hypostasized this word as referring to some scarcely intelligible non-empirical entity, vitiating our philosophical language. This tradition has stubbornly ignored the fact that in ordinary language the word ‘property’ has always been used to refer to simple or complex tropes. Anyway, I intend to use the word trope exactly as /the word ‘property’ is ordinarily used. Thus, I explicitly include among the tropes complex tropes made up of different kinds of tropes, these complex tropes possibly being designated by composite predicates like ‘…a black horse of the best Thessalonian strain’ in the sentence ‘Bucephalus was a black horse of the best Thessalonian Strain.’ This does not make this complex trope (complex property) a singular material object, mainly because, as we will see later, a singular material object can exist independently, compared with the complex trope to which it is tied (in a different possible world Alexander’s beloved horse, Bucephalus, could still exist even if he were just a tired old nag).
  According to the understanding of the reference of predicative terms that I am proposing, a predicative expression like ‘... is white’ in the sentence ‘The moon is white’ does not refer to any Fregean concept. It primarily ascribes, denotes, designates (or refers to) a particularized property, namely, a trope, which is the whiteness of the Moon as normally seen by observers on the Earth. Secondarily but distinctively, however, the predicate ‘…is white’ also alludes to (or connotes) the fact that this trope exemplifies the universal property of whiteness, here understood in the already explained particularist way as this same model trope that is being considered, or any other trope that is like it. Summarizing, a predicative expression has mainly a twofold function:

 (A) An ascriptive function: that of ascribing or denoting the trope 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 nominalist way.

The allusive function is subsidiary to the ascriptive function: to identify a trope you do not necessarily need to grasp its role as an instance of a universal.[13] Better said, as particularized properties tropes have not only ontological, but also epistemic priority if compared with their role in building universals.
   Furthermore – opposing the overwhelming influence of the logic tradition – we have, as a still more subsidiary element: (C) the extension. Although relevant, differently from (A) and (B), extension it isn’t primarily associated with predication. Extension doesn’t even need to be implicitly considered in the act of predication! However, it can be derived from the application of the allusive function of the predicate plus additional knowledge, allowing us to infer or even find: (C1) an extension of tropes as the set of tropes strictly similar to the set 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 seen as an open set only vaguely inferred.[14] As a rule, you do not need to take it into consideration to use a predicate ascriptively.

6. Difficulty with the concept of unsaturation
The great objection against the idea of incompleteness or unsaturation is that it fails to serve its main purpose, which is that of distinguishing a predicative expression from a nominative or singular term. Between the object referred to by the subject and the property designated by the predicate, there is a well-known asymmetry: the nominative term always refers to its object and cannot properly take the place of a predicate; on the other hand, we can easily turn a predicate into a subject by means of nominalization.[15] For instance: ‘Socrates’ in ‘Socrates is wise’ always refers to its object and cannot properly take the place of a predicate, while ‘… is wise’ can be nominalized as ‘wisdom’ in a statement like ‘Wisdom is a virtue.’ To make the point more convincing, consider the following sentences:

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

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

1.     Socrates was the name of 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![16]

One cannot effectively transform a singular term as such into a predicate, while predicates seem to be easily transformed by nominalization into singular terms.[17] This asymmetry suggests that subjects and predicates play different logical roles in sentences, which requires explanation. The question is: can the Fregean distinction between saturated and unsaturated 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 similar reasoning to the second case. Hence, one can suggest that the name ‘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 Bucephalus was in fact a black horse. The unavoidable conclusion seems to be that in a singular predicative sentence both the general and the singular terms can be viewed as unsaturated in the sense of denoting functions that can be supplemented by a myriad of arguments able to bring in ‘The True’ or ‘The False’ as the resulting values!

7. Unsaturation as ontological dependence
Notwithstanding, I think that the metaphor of unsaturation is not exhausted so easily. In chemistry, a carbon compound is said to be unsaturated when it contains carbon-carbon bonds that can be broken by the addition of hydrogen atoms, which make it a saturated compound. The hydrogen atoms aren’t said to be unsaturated. Isn’t there a hint in the metaphor of an answer that was not sufficiently explored by Frege?
  In what follows, I offer a reading of the reference of a predicative expression in terms of tropes that enables us to make a useful paraphrase of the Fregean distinction between saturation and unsaturation. This paraphrase is inspired by one of the Aristotelian definitions of substance, which can be expressed as:

That which exists independently of other things (See Aristotle 1984, vol. 1, Categories, sec. 5).

Applied to individuals or material objects understood as (at least) clusters of tropes displaying compresence, this definition suggests that in their existence these structures are comparatively much more stable than their associated tropes. That is, it seems that the objects typified by material things exist in a manner relatively independent of their tropes in the composition of the kinds of facts[18] represented by true singular predicative or relational statements. Moreover, I hold that the individual referred to as a subject is only relatively independent, because the relation of existential independence/dependence is here understood in a way restricted to the internal context of the fact represented by the statement.
  Summarizing: my thesis is that the true dichotomy distinguishing subject from predicate is between independence and dependence, terms only rarely used by Frege. Thus, what distinguishes the designatum of a predicative expression in the fundamental case of a predicative or relational sentence is that this reference is a trope (simple or complex, homogeneous or heterogeneous) whose existence in some way depends on a relatively independent cluster of selected compresent tropes… which constitute the individual referred to by the singular term. In my view, this fragile principium individuationis is the only thing that really distinguishes the references of logical subjects. Here are some examples supporting this view:[19]

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

Concerning singular statements, my thesis can be summarized as follows:

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

In trying to explore this view in more detail, we can 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 case of material objects, that belonging to the nucleus are typically tropes like those of hardness, form, volume and resistance to pressure or solidity and (mainly) what physics calls 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 a disjunction of the two fundamental description-rules belonging to it, which are the localizing and the characterizing rules (cf. Appendix to Chapter I). This identification rule, as we also saw, can be satisfied by an indeterminate range of external criterial configurations, in other words, tropes or configurations of tropes. This means that what Simons understood as a necessary nucleus of mutually founding tropes may change regarding only one individual in different counter-factual situations. Already examined examples are the Aristotle born 300 years later in Rome in one possible world and the Aristotle who in another possible world was born with cerebral palsy in Stagira in 283 BC, son of Nicomachus, and 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, which we hope mirrors some real essence. 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 tropes  designated by relations like ‘…the teacher of…’ and ‘…the founder of…’ are respectively dependent on the individuals ‘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. It has an identification rule made up of two interconnected fundamental description-rules: a contextually dependent localizing description-rule establishing a spatio-temporal location (by means of the demonstrative ‘this’ and some indicative gesture) and a characterizing description-rule (by means of the sortal ‘chair’). This characterizing description-rule is simply the definition of a chair as a moveable seat with a backrest made for only one person to sit on at a time. We can say that the complex criterion for the identification of chairs added to the spatio-temporal location is what in this case forms the indispensable nuclear structure of this designatum. Symptoms of this chair, such as its having four legs and two armrests, or its being made of wood, are peripheral combinations of tropes. Moreover, if I say ‘This chair is green,’ the trope of green (in fact) exists in dependence on the existence of a complex of compresent tropes that forms this chair and would not exist without their existence.
  These considerations allow us to better understand the independence-dependence relation regarding the compresent core of tropes of an object satisfying its identification rule and its contingent peripheral tropes. Consider, for example, the singular predicative sentence ‘Bucephalus is swift.’ The predicate ‘...is swift’ in this sentence applies to a contingent trope that constitutes swiftness, whose existence is here fully dependent on the existence of an object, Bucephalus, which is formed 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 a material object. 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 relational sentence ‘Bucephalus belongs to Alexander.’ This contingent relational complex trope of belonging to could not possibly be found if Bucephalus and Alexander didn’t exist as independent individuals formed by nuclei of compresent tropes. That is, the proper existence of the relation ‘…belongs to…’ is here indebted to the existence of the more stable essential nuclei of mutually founding tropes constituting the two objects Bucephalus and Alexander. These 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 particularized relational property.
  As in the cases described above, things are easy when we apply the dichotomy independence/dependence to tropes that do not belong to the identifiable core of an object. So, consider once more our definition of a chair as a seat with a backrest made for only one person to sit on at a time, which gives much of Simons’ nucleus of mutually founding tropes for the object referred to by the nominal term ‘this chair.’ Suppose now that I point to the chair and say ‘This chair has two armrests,’ since the tropes of having two armrests do not belong to the definition that makes explicit the nucleus, its existence as something that the chair has is dependent on the chair’s existence. (Notice I am not claiming that the existence of the chair’s armrests in themselves is dependent on the chair, since armrests are parts also able to exist separately from chairs, but in the context of the fact described by the statement ‘This chair has two armrests.’ In this context they do not exist only as armrests, as parts of a whole, as ‘the two armrests belonging to this x,’ which makes their existence dependent.)
  A problem arises when predicates denote tropes belonging to definitional cores (with their ‘in’ and not ‘of’ properties). Suppose I say, ‘This chair has a backrest.’[20] Despite the tautological character of this statement, the trope ‘having a backrest’ can be considered in its existence to be dependent on the whole cluster of tropes that builds the definitional cores of tropes distinctive of the kind of object spatio-temporally individuated as ‘this chair.’ Here one could object that the cluster of tropes constitutive of the core depends reciprocally on the backrest: after all, a chair without a backrest is no chair. However, the seat is still the main object referred to, because it has the most relevant trope combination: it is still a seat that can be used by only one person at a time, while the backrest has lost its function.
  The example shows us that the dependence of components on what is defined turns out here to be a question of proportion. One piece of evidence for this is that if the proportion is the same – if the division of a cluster of tropes is equilibrated – the question of dependence vanishes and one cannot identify a main original object of predication anymore. To exemplify this, suppose someone saws the chair into two identical halves. Where is the subject? Now we cannot say that one chair-half belongs to the other chair-half. And this is so because both have the same weight regarding dependence. All that one can do now is to make relational statements about two new objects like ‘These two chair-halves belong together’ or ‘The first chair-half can be joined with the second chair-half to form a whole chair.’
  Consider now the following statement: ‘Jupiter is orbited by many moons.’ In this case ‘…is orbited by many moons’ figures as a predicative expression. It is true that the many moons are solid clusters of tightly connected compresent tropes – they are no less material objects than Jupiter. But as material objects jointly designated by the predicative expression ‘…has many moons’ they are all existentially dependent on Jupiter in the sense that without the existence of Jupiter they would not be its own moons. The many moons of Jupiter belong to what is predicated as a collection of material objects, each of which could exist independently as celestial bodies, but not as moons of Jupiter. Now, think about a similarly sized system of double stars that revolve around each other. Since they are equally independent, they are references of logical subjects related by their revolution around each other. My conclusion is that the given criterion of dependence/independence, fragile as it may be, is the basic criterion for distinguishing logical subject from logical predicate in the unity of a statement.
  Finally, what about formal names and sentences? Consider the sentence ‘Three is an odd number.’ This sentence describes a mathematical fact. Considering here ideas about what confers existence, we can think the number three without thinking that it is also ‘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 attempt to define the numeral 3 (sec. 3 of the Appendix to Chapter III) as a higher-order tropical property belonging to an effectively applicable conceptual rule. We could express this dependent trope of 3 in Zermelo’s fashion as {{{{}}}}. In a sentence such as ‘This hat has three corners’, the numeral three indicates that the conceptual rule identifying the corners of this hat has the meta-property (trope) of being applicable three times, which cognitively requires some temporal process of counting.[21]
  But how to represent the number 3 in abstract arithmetic, that is, to distinguish it as the universal object that is common to all conceptual identifications of three singular entities? Here, if we wish to avoid speaking of a class of classes of the same kind… we can construct the number 3 as an experienced tropical counting expressible by means of a Zermelo’s triad or any other strictly similar result of a tropical counting:

Number 3 (Df.) = a chosen tropical enumeration of the form {{{{}}}}* or any other tropical enumeration strictly similar (equinumerous) to {{{{}}}}*.

This definition still allows the predicate ‘…is a natural number’ to be ascribed to the whole definiens as an internal dependent addition (the genus) and the predicate ‘…is an odd number’ as an external dependent addition. In any case, even the name of a so-called abstract object, such as ‘the number three in itself’ cannot be moved to the predicate position here, insofar as it refers to something held as independent, being identifiable (existing) independently of its non-definitional predicates like ‘…is an odd number.’
  Understanding unsaturatedness as relative existential dependence suggests, therefore, that the tropes denoted by the predicate have an inevitable tie of dependence when considered relative 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 the predicative expressions. On the other hand, in the context of the fact referred to by the sentence the opposite is true, since the predicate can be nominalized.
  However, it is worth noting that nominalization is a tricky deal. The statement ‘Goodness is desirable,’ nominalizing the predicate ‘…is good’ is better analyzed as the universal statement: All good (=G) things are desirable (=D)’ or (x) (Gx → Dx), which means that {(Ga1 → Da1) & (Ga2 → Da2) &… & (Gan → Dan)}. But if you take Ga1 → Da1, for instance, what you have is again the predicate G or ‘…is good’ and the singular term a1. In other words, nominalization is only a simplifying pragmatic device of natural language, and the predicate was in fact never transformed into a singular term. Nevertheless, even at this deeper level the asymmetry remains, since a similar procedure cannot be applied to a true nominal term like Socrates.
  In my view this analysis also solves 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 dependence/independence in the factual unity (for instance, in the fact that Bucephalus is swift). And it is clear that these ties of dependence/independence will be more evident when the difference in relevance between the elements in question regarding the identity of the individuals is greater, and weaker when this difference is smaller, justifying occasional uncertainties.
  Finally, one could object that what really distinguishes the predicate from the subject in singular statements is simply that the subject is a singular term that identifies one particular object and distinguishes it from all others, while the predicate is a general term able to be applied to more than one object. It is this possible one-to-many relation that is at the base of the subject-predicate distinction! Nevertheless, I think this is true regarding the definition of singular and general terms, but not that it is at the base of the distinction. The point is made clear when we consider that a nominal term can refer to many things, while a predicative expression can refer to only one, given the proper factual circumstances. To see this, imagine a possible world Ws where there are many identical Socrates’s with almost the same pasts debating in the streets of Athens. In this case we can find one Socrates and say, ‘That human being is a Socrates,’ and ‘Socrates’ will belong to the predicative expression. And suppose we wish to predicate that a certain trope of blue is emerald blue. In this case we would say, ‘This blue is emerald.’ What makes ‘a Socrates’ a predicate and ‘this blue’ a subject is the fact that in the represented fact the Socrates exists in dependence on the indicated place, and what makes the blue a subject is that it is spatio-temporally given independently of having taken the shade of emerald. And this could be so considered, even if by chance all the other Socrates’s disappeared and there were only one remaining Socrates, just like only one trope of blue. This is why I prefer to hold that a possible one-to-many relation is grounded on the independence/dependence existential relation, which is primary.

8. Sense of a predicative term
The independence/dependence relationship originating on the ontological level of reference is reflected on the semantic and linguistic levels. It is first reflected on the semantic-epistemic level of sense. We see this in the fact that the identification rule of the nominal term is applied independently of the ascription of tropes to the object by the ascriptive rule of the predicative expression, while the constitutive sense of the ascriptive rule of the predicative expression 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) with 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, since the potential for multi-referentiality is inherent to predication. Consider the following sentences:

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

The predicate ‘...is white’ in sentences (1) and (2) obviously has the same sense in both, as in each case it expresses the same ascription rule. But the tropes of whiteness (of reflecting the combined wavelengths of the visible spectrum) of the South Pole are located at the South Pole itself, while the tropes of whiteness of Mont Blanc are located in its eternal snows. The same can be found in relational predicates.
  Another easy problem is to know how predicative expressions are used in the case of general sentences: universals and existential statements. Regarding universal sentences, we see them as abbreviated expressions of a conjunction of singular sentences, each ascribing tropes to identified objects. For example: the universal sentence ‘All trees have roots’ would be analyzed as {Tree 1 has root & tree 2 has root &… & tree n has root}. Here the qualitatively identical tropical root-clusters Tr1 of 1, Tr2 of 2… Trn of n are considered in conjunction and jointly denoted by the universal sentence. This also means that qualitatively identical ascription rules able to denote the complex tropes of roots belonging to the objects trees also need to be conjoined in the sense that the universal sentence at least indicates the probable application of this conjunction of ascription rules to tropical root-clusters of trees, even if we are unable to really apply them to all trees in the world. Similar considerations can be made regarding existential sentences like ‘At least one tree has roots,’ which abbreviates a disjunction of ascription rules and refers to at least one trope of roots belonging to the multiplicity of trees as objects.
  We will come back to this issue in the last chapter, when we consider problems related to this attribution of truth-value to general statements. Anyway, to the objection that this will never give us the conditions for a truly universal quantification, the obvious answer is that the truth of the universal quantification is almost always enclosed under some domain and very often is only probable. What ‘All living trees have roots’ really means is ‘[Very probably] all living trees [on the planet Earth] have roots.’

9. Dependence of the predicative sense
The ontological distinction between independence/dependence (saturation/un­sat­uration) is reflected on the semantic-epistemic level to which the senses belong as a consequence of the original ontological dependence.
  This is clear enough if we see the sense of the predicative expression as an ascription rule. In the context of a singular predicative sentence, the identification rule of the singular term applies to the object as some core of compresent tropes, which is seen as existing independently in relation to its more or less dependent partial or peripheral tropes. Consequently, the identification rule is also able to be applied regardless of the application of contingent ascription rules, which means that this identification rule can be conceived as being applied in isolation. This explains its independence and why one could call it complete, 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 core 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.[22]
  The general sense of a concept-word, which (diverging from Frege) we identify with the concept expressed by it, should then be a rule whose application to an object depends on the prior application of another rule. Hence, the ascription rule of the predicative expression is dependent, incomplete, unsaturated, in the sense that it demands the prior application of the identification rule of the singular term in order to be applied. It is necessary to identify, that is, in the empirical case to find some particular object in space and time, in order to be able to characterize it by ascribing the predicative rule to its appropriate trope. We must, for instance, first apply the rule that allows us to spatio-temporally locate the horse called Bucephalus in order to apply to it related tropes, and on that basis, the ascription rules of predicative terms. Thus, due to the independence of the object Bucephalus, we apply ‘... is a horse,’ ‘... is black,’ ‘... is swift’… and also the complex rules of application of more complex predicates like ‘…a horse that belonged to the best Thessalonian breed’ to the tropes (simple or complex, homogeneous or heterogeneous) linked with the object. And we also need to apply the identification rules for Bucephalus and Alexander in order to be able to apply the relational predicate ‘…belongs to…’ And finally, we need to apply the rule that allows us to mentally identify the number 3, in order to be able to apply to associated dependent tropes the ascription rules of predicative expressions like ‘…is odd,’ ‘…is a prime number,’ or ‘is the square root of 9,’ though it is not the case that the number 3 depends on these things in order to be identified as such. In the same way, the relational ascription rule for ‘3 < 7’ is only applicable in dependence on the independent application of the identification rules for the numbers 3 and 7.
  As I have already 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 isn’t required at all. Indexicals such as ‘that’ and ‘there’ accompanied by some gesture of pointing are already able to identify some particular as anything spatio-temporally localizable independently of any predication. As we already saw, this (non-absolute) independent way can be made explicit when the indexical is followed by a term designating countable things (sortals) such as ‘that object,’ ‘that animal,’ and this is enough. Therefore, not only does the trope designated by the predicate depend upon the previous existence of the object and its identification, but, as a consequence, also the ascription rule of the predicate, its most proper conceptual sense, must be dependent upon the prior application of the identification rule to the relatively independent cluster of tropes. This is how the relation of semantic dependency – on the level of sense – mirrors the relation of ontological dependency – on the level of reference – solving the riddle of unsaturation.

10. The concept horse paradox
We can continue to make major revisions of Frege’s views in order to solve problems in his philosophy, like the so-called concept horse paradox. Based on his view of a concept as the unsaturated reference of a predicate, Frege was led to the strange conclusion that one cannot name a concept. For him the sentence:

1.     The concept horse is not a concept,

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

2.     The concept horse is a concept,

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

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

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

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

which is obviously true. Using H to replace ‘… is a concept horse,’ which designates the trope of horseness, and replacing ‘…is a concept’ with C, designating the trope of a thought concept, we can formalize (3) as (5): (x) (Hx → ~Cx), which is false, and (4) as (6): (x) (Hx → Cx), which is true.
  What is the lesson of this analysis? If ‘the 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).

[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 words like ‘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). Independently of the intrinsic interest of Recanati’s prolific work, it is worth noting that these files, being clusters of information and not subjective Vorstellungen, should also be conventionally grounded, which makes them akin to our cognitive-semantic rules. Consequently, they should be able to be linguistically expressed by means of descriptions, which brings us back to the descriptivist standpoint. For this reason it seems to me that our complex semantic-cognitive rules are apt to do a similar job in an explanatorily more powerful way and (as we will see) with important epistemological consequences.
[7] François Recanati prefers to use the word ‘property’ (propriété) instead of ‘concept’ in his summary of Frege’s semantics (2008: 34). 
[8] This can be rejected by considering that for Frege a concept must have an extension. But even in this case we would have problems: for in the case of a false statement like ‘Göteborg is the capital of Sweden’, nothing would fall under the concept of ‘…is the capital of Sweden’, which means that again this concept-property should be seen as an abstract entity.
[9] Although not for Frege, since a nominalized expression is for him saturated and cannot refer to a concept (cf. section 10 of this chapter).
[10] Mere similarity would not do, since this concept is intransitive. Strict similarity means here the same as qualitative identity, which is transitive. Strict similarity must also be a trope, since it is spatio-temporally located between tropes, even if it is a subordinate trope.
[11] I suggested this disjunctive way of constructing the universal by means of tropes in order to circumvent the usual but problematic definition of a universal as a set or sum of tropes that are strictly similar, one with the other (see Appendix of Chapter III).
[12] Although a pure ontology of tropes is a very new ontological achievement and brings with it a wide range of unsolved problems, it does not produce more difficulties than the traditional universal doctrines of realism and nominalism. In return, it promises a really parsimonious solution for ontological problems, which would free us from at least three traditional hindrances: (i) ostrich nominalist solutions, with their lack of explanations for self-imposing questions, (ii) abstract objects of contestable intelligibility like Platonic universals which lead to non-parsimonious multiplication of entities, and (iii) non-cognoscible naked substances. Realism (Platonic or Aristotelian) – the most influential traditional doctrine – has occupied philosophical minds for more than two millennia without progress sufficient to considerably increase its plausibility. Thus, in my view the only reason why some kind of realism still seems to hold the foreground of attention is the longstanding weight of tradition. For such reasons (under the assumption that the ontological enterprise makes sense), I accept a pure ontology of tropes as the most plausible solution, at least in the form explained in the Appendix of Chapter III.

[13] An epistemic primacy of identification over the generalizing function was already suggested by Keith Campbell (1990: 24-25).
[14] 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)
[15] There are several asymmetries. The most discussed is probably the asymmetry of subjects and predicates regarding negation: you can negate the predicate, but not the subject (nominal term) (Strawson 1971, Ch. 5).  The answer seems obvious. The negation of the predicate means the admission of the inapplicability of the ascription rule to the object identified by the identification rule. However, since the application of the ascription rule is dependent on the application of the identification rule, whenever you negate the application of the identification rule of the subject you also negate the applicability of the ascription rule and in this way the whole statement. Hence, it is impossible to negate the subject or name alone.
[16] Notice that the demonstrative ‘that’ does not have here the function of a constituent of the identification rule of Socrates, but is itself an identification rule of an individual place. In indexical statements like ‘This is a daisy’ and ‘There is the Golden Gate Bridge,’ demonstratives have as their main role the location of the criteria satisfying the characterizing rule of the daisy and the Golden Gate Bridge. The logical form of the statement ‘This is Socrates’ is already revealed by the relational statement ‘This spatio-temporal place is where Socrates is located.
[17] We can show that the nominalized predicate is in fact a disguised universal predication: the sentence ‘Wisdom is a virtue’, for instance, could be analyzed as, ‘For any human being 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 a similar way.
[18] 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 later in this chapter.
[19] I take these examples from Mulligan et al. (1984: 300, 301 and 306), though their point isn’t the same.
[20] Note that if I said, ‘This chair has this backrest’, pointing to them, this chair and this backrest could work as names and …has… as a relational predicate.
[21] One could object that if the identification of numbers requires a cognitive temporal process of counting, what about very large numbers? I would answer that they result from extensions of this primitive process of counting, even if they are in fact not countable by our finite minds.
[22] As Ernst Tugendhat wrote: ‘‘Fa’ is just the case to the extent that the rule of identification for ‘a’ is followed and, based on this result, ‘F’ is applicable in accordance with its rule of application’. (Tugendhat & Wolf 1983: 235)