quarta-feira, 1 de fevereiro de 2017

# A STRAVAGANT READING OF FREGEAN SEMANTICS (1) (sense and reference, predication)

This is a corrected draft for the chapter on Frege of the book PHILOSOPHICAL SEMANTICS to be published in 2017 by Cambridge Scholars Publishing.

– 4 –

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 domination of the word over the human spirit by laying bare the misconceptions that through the use of language often almost unavoidably arise concerning the relations 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 domination 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 reconstruct and revise the essentials of Fregean semantics. I intend to make clear that the basic concept of sense can be paraphrased in terms of semantic-cognitive rules and that the concept of existence can be reinterpreted in terms of the effective applicability of semantic-cognitive rules, leading to unexpected consequences regarding the explanation of the concepts of verification, fact and truth. With the identification of senses with rules, I intend to show the real link between Wittgenstein’s semantics – as I read him in the last chapter – and Frege’s semantics; a link already noted by Michael Dummett, though in a very general manner, devoid of pragmatic exploration. Anyway, my aim here is not to produce a work of Fregean scholarship. My aim is instead to reconstruct Frege’s work with him, against him, and beyond him, with the aim of providing a more rigorous framework for the rather vague semantic insights gained in the last chapter.
   As is general knowledge, Frege explains reference (Bedeutung) using a semantic intermediary link, which he called sense (Sinn) (1891:14). The schema below shows how Frege deals with these two main levels (1) of sense, and (2) of reference in the case of a predicative singular sentence of the form Fa:

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

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

1. The 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 here use the word ‘reference’.[1]
   There is also an interpreter’s discussion about the reason why Frege would have chosen the word ‘Bedeutung’. A widespread interpretation is that one of the meanings of ‘Bedeutung’ (as well as of ‘meaning’ or ‘signification’) is relevance or importance, since reference is what matters most for truth (Tugendhat 1992: 231). For me it is clear the strongest reason, at least with regard to the reference of natural language terms, is that by introducing the term ‘Bedeutung’ Frege substantivated the verb ‘bedeuten’. In this way, the word no longer expresses the act of pointing at (deuten) or of designating (bezeichnen), but rather what is pointed at (die Bedeutung), what is designated (das Bezeichnete), that is, the reference itself.[2] In German these derivations could be diagrammed as follows:

Bedeutet ... → deutet ...   bezeichnet.      was gedeutet, bezeichnet wird/
(means)           (points ...  means ...)             (what is said)
                                                                      die Bedeutung
                                                                      (meaning = reference)

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

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

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

Sentences (1) and (2) refer 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 the way in which an object gives itself to us (die Art des Gegebenseins des Gegenstandes), which is well translated as a mode of presentation. The senses of the singular terms ‘the morning star’ and ‘the evening star’ are different, because ‘the morning star’ presents Venus as the brightest celestial body usually seen just before sunrise, while ‘the evening star’ presents the same planet Venus as the brightest celestial body usually seen shortly after sunset …
   Frege writes that words express their senses (drücken ihre Sinnen aus), while the senses determine (bestimmen) their reference, since the mode of presentation should show us a way to find the reference. Even in cases where the reference does not exist, this determination of reference through sense is given, since even in this case the words preserve their senses. This fact points to a flaw in Frege’s idea that sense is the way an object presents itself to us, for in the case of empty terms there is no object to be presented to us. This is why sense can be better understood as the intended mode of presentation instead of as a mode of presentation given by the object (Textor 2010: 134); the sense is the way we intentionally present an object or reference to ourselves, whether it exists or not. At any rate, an expression can have a sense without a reference, but cannot have a reference without a determination by means of sense.
   Frege extended his notion of sense to other terms and to sentences. In the case of the meaning of sentences, he calls it cognitive or (more literally) epistemic value (Erkenntniswert). The last term is also appropriate. The Fregean concept of sense has epistemological interest, for it constitutes the proper informative content of the linguistic expression. It is what makes ‘the evening star’ and other expressions informative – in Dummett’s words, sense is what we understand when we understand an expression (Dummett 1990: 92). The philosophical importance of Fregean semantics is largely due to the epistemological and ontological imports of the concept of sense. (This is what distinguishes it from a linguistic semantics like that of Ferdinand de Saussure.)
   Frege is a Platonist about sense. For this reason he conceives senses as abstract entities which can only be analyzed in terms of constituents that are also senses. The result is that his Platonism of senses prevents him from analyzing senses in terms of other concepts. However, it is just this task that naturally imposes itself. For it seems very plausible to understand senses as semantic-cognitive criterial rules. We see here the fundamental difference between Fregean semantics and the semantic considerations of the later Wittgenstein, who regarded senses or meanings as episodic uses determined by rules. Dummett was probably the first to defend the idea that senses are rules as the most natural reading of Frege’s senses. As he wrote in his book on Frege’s philosophy of language:

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

And concerning the singular sentences in Frege, understanding by 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 ff.).
   By treating senses as semantic-cognitive rules and these rules in the primary case as shared conventions, we contrast them with what Frege called colorations and illuminations (Färbungen and Beleuchtungen), which are feelings often associated with image representations (Vorstellungen) and sense-perceptions (Anschauungen), which for him belong as such all to an intrinsically subjective level (Frege 1892: 31). These ‘colorations’ and ‘illuminations’ are names for what we would more often call expressive meanings, i.e., sensory-emotional states that we normally and customarily associate with expressions. Thus, for example, the words ‘love’, ‘dog’ and ‘hell’, in the sentence ‘Love is a dog from hell’ (Bukowski) contrastively associate words linked with strong specific emotions in order to obtain an epigrammatic effect.
   As Frege realized, the kind of appeal or lack of appeal that the colorations associated with words have for different persons depends correspondingly on the similarities and differences between their human natures, and because of this they do not require conventions to be communicated, as in the case of senses. This is why some people are impressed by a certain poem, while others are not; and this is why it is so difficult to translate poetry, which depends greatly on colorations acquired by expressions in a particular language and culture. Hence, colorations are not results of conventional rules; they are rather regularities resulting from similarities in human nature. If my understanding of Wittgenstein’s argument against private language is correct, then his attempt to explain phenomenological language as a simple replacement of public behavioral criteria like uttering ‘ouch!’ under conditions that would cause pain with a sentence like ‘I feel pain’ is insufficient (1984d, sec. 244). It is an attempt to assimilate the referential meaning of the phenomenological language to its expressive meaning (I suppose that both can be legitimated).
   If against Frege we accept that sense is primarily a convention or a combination of conventions, we can easily solve the problem of com­municability of senses that has long tormented philosophers like Frege and Husserl. For the reason for the objectivity of senses (characterized for Frege by their possibility of interpersonal access) and the reason for the consequent communicability of senses (in contrast to the at least relative lack of objectivity and communicability of representations and colorations or illuminations) would be found. This reason would be that senses typically depend on semantic-cognitive conventional rules usually interpersonally agreed upon in a pre-reflexive manner. Indeed, accepting the conclusions reached through our discussion of Wittgenstein’s views, senses or meanings originally result either from the direct application of interpersonally established conventions or from combinations of these conventions.
   Accepting that the sense of a singular term is the same as a rule seen as a conventional or conventionally grounded procedure which plays a decisive role in the identification of the object, it is easy to go further and accept that this rule can be typically expressed by means of definite descriptions. Hence, the sense or mode of presentation expressed by the singular term ‘the morning star’ is a conventional rule that can be understood as requiring as a criterial condition for the cognitive identification of the morning star that we see the brightest celestial body not too far from the Sun just before or after the Sun rises. In a shortened way, this rule can be expressed by the definite description ‘the brightest celestial body usually seen not too far from the Sun just before the Sun rises’. Without admitting that definite descriptions are expressions of rules, Frege has also seen this point in his note on the name ‘Aristotle’ (Frege 1892: 28). For him the proper name ‘Aristotle’ abbreviates a cluster of modes of presentation of the object that can be expressed by descriptions, which may include (i) ‘the disciple of Plato’, (ii) ‘the teacher of Alexander the Great’, and (iii) ‘a person born in Stagira’. If this is the case, then (i), (ii) and (iii) express different senses, different rules that in some way help us to determine the reference of the proper name ‘Aristotle’ (see also Frege 1918-19: 63).[5]
   Of course, there is a controversy about this issue, which arose from Kripke’s arguments against descriptivist views of proper names like that suggested by Frege. However, it seems indubitable to me that Kripke’s arguments can be successfully countered by the kind of meta-descriptivist bundle theory summarized in the appendix of Chapter 1 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 designatum – 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).
   For a traditional philosopher such as Kant, a concept is seen as immediately related to a schema, which, as I understand him, is a rule able to lead to the formation of a manifold of sensory patterns that are satisfied by things to which the concept applies (see Kant 1988, B 180). Although Kant’s text on schematism is terminologically impenetrable, it is easy to paraphrase his intuition using our Wittgensteinian terminology by saying that a concept is a semantic-cognitive rule or procedure for the demanding of criteria or criterial configurations that can be satisfied by singularized properties or tropes. Coming back to Frege’s semantics, we could paraphrase the same idea by saying that the concept should be simply the sense (Sinn) of the predicative expression, which should be reducible to modes of presentation of its reference… What all these remarks suggest is that the concept should be the sense of the predicative expression, of the general term, 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 (Frege 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 extension, understood as a value-range (Wertverlauf), which means that predicative expressions with different senses but the same extension must refer to the same concept (Frege 2001: 31). So, for instance, ‘…animal with a kidney’ and ‘…animal with a heart’ must be predicative expressions referring to the same concept, since they have the same extension. But it is intuitively obvious that kidneys and hearts are very different concepts.
   In addition to belonging to the realm of reference, Frege also sees his concepts as functions. The mathematical concept of function can be defined as a rule that has as its input arguments and as its output values (for example: ‘3 + x = y’ is a function by means of which when we give as input the number 2 as the argument x, we get as output the number 5 as the value of y). For Frege a concept is a function whose argument is the object that ‘falls under it’ (fällt unter) or not and whose value is a truth-value, which can be alternatively two abstract objects: ‘The True’ (das Wahre) or ‘the False’ (das Falsche). 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 (Frege 2001: 26). The reason is that objects, collections of objects and extensions are complete (vollständig) entities. That is, they do not require anything to complete them. A concept, by contrast, being a function, is seen by Frege as being necessarily open: he calls it an incomplete (unvollständig) or unsaturated (ungesättigt) entity, needing to be completed by those arguments represented by the objects falling under the concept. In contrast, objects referred by proper names are complete (vollständig), saturated (gesättigt) or independent (unabhängig).
   One could say that the saturated-unsaturated distinction can be found on three distinct levels: linguistic, semantic and referential. For example: the predicate ‘…is a horse’ could be called an unsaturated linguistic expression (the unsaturatedness is shown by the gap ‘…’), expressing a supposedly unsaturated sense, which refers to an unsaturated concept (property) as the ultimate unsaturated ground. This concept, for its part, is completed when some object, being in itself complete, falls under it, for instance, the object referred by the name ‘Bucephalus’ referred to by the sentence ‘Bucephalus is a horse’.
   With metaphors like that of ‘unsaturation’ and ‘incompleteness’, Frege hoped to open the way to the solution of the mystery of the logical distinction between subject and predicate. After all, the subject (the singular term) would refer to the saturated object, while the predicate (the general term) would refer to the unsaturated concept.
   Unsaturated predicative expressions and saturated singular terms combine to form complete sentences like ‘Bucephalus is a horse’, which as it is complete must also be the name of an object, which for Frege is the truth-value of the sentence. It may seem that this would be confirmed by the possibility that we have of nominalizing sentences in the form of definite descriptions, since the last are also singular terms. Thus, the sentence ‘Bucephalus is a horse’ can be transformed in the description ‘the horse named Bucephalus’, which appears in the sentence as ‘The horse named Bucephalus was black’. The problem with this argument is that the same can also be done with general terms: ‘…is a horse’ can be nominalized as ‘the horse’ occurring in sentences like ‘The horse is an herbivorous animal.’ Hence, the argument isn’t persuasive.

4. Ontological level
Discussion regarding the unsaturated nature of 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 at least could have not only concepts as referred to abstract entities (incomplete Platonic entities as references to empty predicates like ‘…is a yeti’[8]), but also may be the abstract references of the substantiated corresponding conceptual expressions (take the reference of ‘yeti’). However, the admission seems to be ontologically abusive. (Tugendhat & Wolf 1983: 138-9).
   Anyway, it is by now clear that Frege uses the word ‘concept’ as a technical term that contrasts starkly with our ordinary use of the word ‘concept’. For our ordinary language intuitions 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 isn’t satisfied by any real object, having 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 we shall drop the Fregean technical concept of concept for the best. Instead, we will understand the word ‘concept’ here in its intuitive way as the sense of the predicative expression: its mode of presentation. It is nearly certain that ‘...is a yeti’ does not have any reference; but it clearly has a sense intuitively expressing what we ordinarily understand by a concept, namely, that of the abominable snowman of the Himalayas. Thus, it seems that the best way to give a legitimate role to the word ‘concept’ is to see it as the sense of a predicative expression understood as its ascription rule.

5. Referring to singularized properties (tropes)
But if we drop Frege’s technical concept of concept, what is the reference of a predicative expression? I think that today 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 expression by what we now call a trope, which I characterize simply as a spatio-temporally individualizable property. There are many possible examples: the white color I see when I look at newly fallen snow on a sunny day, and which is there in some sense, this smooth surface, the rectangular shape of a computer screen, the hardness of a stone, my headache. All these are tropes or particularized properties that may range from simple qualities to complexes of tropes, like the process of condensation of water-vapor or the state of affairs of the democratic organization of a country, once these things are also in a more vague way spatio-temporally located. Even non-perceptible things like physical forces could be derivatively constructed from perceived tropes, and it is not even impossible that so-called abstract entities like numbers could be explained as constructions derived from spatio-temporally located properties called tropes. A pure ontology of tropes maintains that all reality must be built up of tropes, which from a genetic-epistemological perspective are the building blocks of the world.[9] (For the exposition of my personal understanding of trope theory, see the appendix of Chapter 3 of this book).
   Moreover, it is easy to suggest that a particularistic construction of universals could be built up based on singularized properties or tropes. In my view a universal could be defined as:

any chosen trope model T* or any other trope strictly similar[10] with the model T*,

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, the tropes T1, T2… Tn are identified as instantiations of the universal only because they are strictly similar (qualitatively identical) to the chosen trope model T*.
   A material object could be constructed as a bundle of tropes. It can in principle be understood as a cluster of tropes displaying at least compresence, that is, it must be made 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 mass… that typically comprise material objects.
   Although a pure ontology of tropes is a very new ontological achievement and brings a wide range of unsolved problems with it, it does not produce more difficulties than the traditional universal doctrines of realism and nominalism. In return, it promises a really parsimonious[12] solution for ontological problems, which would free us from at least three traditional hindrances: (i) ostrich nominalist solutions, with their lack of explanations for apparently well justified questions, (ii) abstract objects of contestable intelligibility like Platonic universals which lead to uneconomical multiplication of entities, and (iii) non-cognoscible naked substances.
   Realism (Platonic or Aristotelian) – the most influential traditional doctrine – has occupied philosophical minds for more than two millennia without progress sufficient to considerably increase its plausibility. Thus, in my view the only reason why this doctrine still seems to hold the foreground of attention is the longstanding weight of tradition. For such reasons (under the assumption that ontological investigation makes sense) I accept the pure ontology of tropes as the most plausible solution, in the form exposed in the appendix of Chapter 3.
   Finally, I will usually avoid the use of the word ‘property’, not because it isn’t the best one, but because the philosophical tradition has too often hypostasized this word as referring to some scarcely intelligible non-empirical entity, vitiating our philosophical language. This tradition has stubbornly ignored the fact that in ordinary language the word ‘property’ was always been used to refer to simple or composed tropes. Anyway, I use the word trope exactly as the word property is ordinarily used. Thus, I explicitly include among the tropes those tropes that are complexes of different kinds of tropes that are designated by complex 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 (this complex property) a singular material object because, as we will see later, the singular material object can exist independently if compared with the complex trope to which it is tied (in a different possible world Bucephalus could still exist as the beloved horse of Alexander even if he were just a lame 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, denotesdesignates (or refer to) a singularized property, namely, a trope, which is the whiteness of the Moon as seen by observers on the Earth. Secondarily and independently, however, the predicate ‘…is white’ also alludes to (or connotes) the fact that this trope exemplifies the universal property of whiteness, understood in a particularist way as this same model trope that is being considered, or any other trope that is like it. Summarizing, a predicative expression has mainly a twofold function:

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

The allusive function is subsidiary to the ascriptive function: to identify a trope you do not necessarily need to grasp its role as an instance of a universal.[13] As singularized properties, tropes have not only ontological, but also epistemic priority.
   Furthermore – opposing much of logic tradition – from this perspective we have still more subsidiarily: (C) the extension. Although extension is something whose consideration is unavoidable, it isn’t primarily or even secondarily associated with predication. Extension doesn’t even need to be considered in the act of predication. However, it can be derived from the application of the allusive function of the predicate plus additional knowledge, allowing us to infer or even find: (C1) the extension of the predicate as a class of strictly similar tropes and (C2) an extension of objects having tropes strictly similar to the trope in question. However, in both cases the extension is a further element usually only vaguely inferred.[14] As a rule, you do not need to take it into consideration to use a predicate ascriptively.

6. Difficulty with the concept of unsaturation
The most serious problem with the idea of incompleteness or unsaturation is that it fails to serve its purpose of distinguishing a predicative expression from a nominative or singular term. Between the object referred to by the subject and the property designated by the predicate, there is a well-known asymmetry: the nominative term always refers to its object and cannot properly take the place of a predicate; on the other hand, we can easily turn a predicate into a subject by means of nominalization. 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’.[15]  To make the point more convincing, consider the following sentences:

1.     A man who lived in the Antiquity was called Socrates.
2.     Wisdom is a property of Socrates.
3.     Xanthippe’s husband is Socrates.

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

1.     Socrates was a man who lived in the Antiquity.
2.     Socrates has the property of being wise.
3.     Socrates is the husband of Xanthippe.

One cannot effectively transform a singular term as such into a predicate, while predicates seem to be easily transformed into singular terms by nominalization. This asymmetry suggests that subject and predicate play different logical roles in sentences, which requires explanation. Can the Fregean distinction between saturation and unsaturation really do anything to explain this 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, Beauty, Gypsy… Pegasus] …is a horse


 Bucephalus is... [black, strong, restless, swift… of 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 the truth-value, which for the object Bucephalus is The True and for the object Alexandre is The False. Now, with the same right we can apply a similar reasoning to the second case. We can say that the name ‘Bucephalus is…’ refers to an object that is a function that may have as an argument any concept-property designated by any predicative expression, which being this concept black has as a value The True and being the concept white has The False as a value. The conclusion seems to be that both the general and the singular terms can be viewed as unsaturated, denoting functions that can be supplemented by a myriad of arguments able to bring in The True or The False as the resulting values.

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

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

Applied to 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[16] represented by true singular predicative or relational statements.[17] In other words: the true dichotomy 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) 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. Here are some examples[18]:

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

The general idea can be summarized as follows:

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

In trying to explore this view in more detail, we can consider Peter Simon’s nuclear trope theory of material objects. According to this theory, individuals are in the standard case formed by an essential nucleus or kernel of mutually founding tropes, which is necessarily surrounded by a looser bundle of accidental peripheral tropes, so that these peripheral tropes require the nucleus of essential tropes for their existence (see appendix to chapter 3). To this we should add, as already noted for the case of material objects, that tropes like those of compresence (a dependent relational trope), tightness, form, volume, resistance to pressure (mass), solidity, possibility of movement, are typically part of the nucleus.
   Unfortunately, the issue is not so simple. According to the identifying rule of proper names already examined in the appendix of chapter 1, the object of reference of a singular term must be located by its identifying rule. Regarding proper names, this identifying rule requires for its application sufficient and predominant satisfaction of at least a disjunction of fundamental description rules, which are the localizing and the characterizing rule. This identifying rule, as we also saw, can be satisfied by an undetermined variety of external criterial configurations, that is, of tropes or configurations of tropes. This means that what Simons understood as a necessary nucleus of mutually founding tropes may change regarding only one individual in different counterfactual situations. Examples are the Aristotle born 300 years later in Rome in one possible world and the Aristotle who in another possible world died young and never wrote his opus, although he was born in Stagira in 283 BC. It may be different within limits established by the identifying rule. Peripheral tropes, on their side, would be those referred to by auxiliary descriptions like ‘the teacher of Alexander’, ‘the founder of the Lyceum’, etc.
   Searching for a simple example, I will consider here the singular term ‘this chair’. This indexical name has an identifying rule with 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 seat with a backrest made for only one person to sit on at a time. We can say that the criterion for the identification for chairs plus a spatio-temporal location is what in this case forms an indispensable nuclear structure. Symptoms of a chair, such as its having four legs and two armrests, or its being made of wood, are peripheral combinations of tropes.
   These considerations allow us to better understand the independence-dependence relation regarding the nuclear core of an object versus its contingent peripheral tropes. Consider, for example, the singular predicative sentence ‘Bucephalus is swift’. The predicate ‘...is swift’ in this sentence applies to a contingent trope that constitutes swiftness, whose existence is here fully dependent on the existence of an object, Bucephalus, which is formed by some kernel of mutually founding compresent tropes. This in fact also applies to properties of individuals that are not properly material objects. A rainbow, for instance, is an individual (a cluster of compresent tropes), though not a material object. The fading of a rainbow is a process-trope whose existence is dependent upon the existence of the rainbow in itself.
   Consider now the relational sentence ‘Bucephalus belongs to Alexander.’ The contingent relational complex trope of belonging to could not possibly be found if Bucephalus and Alexander didn’t exist as independent particulars formed by nuclei of compresent and mutually dependent tropes. That is, the possibility of existence of the relation ‘…belongs to…’ is here indebted to the existence of the more stable essential nuclei of mutually founding tropes constituting the two objects Bucephalus and Alexander. These nuclei of tropes referred to by the names ‘Bucephalus’ and ‘Alexander’ are concrete psycho-physical objects that certainly exist independently of the existence of the relatively contingent complex combinations of tropes constituting the trope of ‘…belongs to…’, since for having ownership we need the previous existence of objects having the singularized relational property of ownership.
   Like in the cases above, things are easy when we apply the dichotomy independence/dependence to the tropes that do not belong to the own definitional 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 Simon’s nucleus of mutually founding tropes for the object referred to by the singular term ‘this chair’. Suppose now that I point to the chair and say ‘This chair has two armrests,’ since the tropes of having two armrests do not belong to the definition that makes explicit the nucleus, its existence as something that the chair has depends on the chair’s existence. (Notice that I am not claiming that the existence of the chair’s armrests in themselves are dependent on the chair, since armrests are parts able to exist separately from chairs. But in the context of the fact described by the statement ‘This chair has two armrests’ they exist as dependent parts, since they are the armrests belonging to this chair and not two unknown armrests.)
   The problem arises when predicates denote tropes belonging to definitional cores (with their ‘in’ and not ‘of’ properties). Suppose that I say, ‘This chair has a backrest.’ I think that, despite the tautological character of the statement, the trope ‘having a backrest’ can be considered in its existence dependent on the whole of the mutually founding bundle of tropes that builds the definitional cores of tropes distinct from the object. Here one could object that the cluster of tropes constitutive of the core depends reciprocally on the backrest: after all, a chair without a backrest is no chair. But the seat is still the referred to object, because it has the most important tropes; it is still a seat that can be used by only one person at a time. We see that dependence of components on what is defined turns out here to be a question of proportion. One evidence for this is that if the proportion is the same, if the division of mutual founding of tropes is equilibrated, the question of dependence vanishes and one cannot identify the original object of predication anymore. To exemplify this, suppose someone cuts a chair into two identical halves. One cannot say that one chair-half belongs to the other chair-half, because both have the same weight regarding dependence. All that one can do now is to make relational statements about two new objects like ‘These two chair-halves belong together mutually’ or ‘The first chair-half can be joined with the second chair-half to form a whole chair’.
   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 as what confers existence, we can think the number three without thinking that it is ‘the number two or any multiple of two added to the number one’ (which is the definition of an odd number). Hence, the existence of oddness related to the existence of the number three is dependent on the number three that we are taking into consideration. Consider now the statement ‘Three is a natural number’. One could argue that to be a natural number belongs to the definition of three as a kind of genus proximum, although not essentially to the (here seen as incomplete) definition of three as its differentia. This differentia could be given (supposedly) by Zermelo-Fraenkel’s set theoretical definition as ‘3 (Df) = {{},{{}},{{},{{}}}}’, which still allows the predicate ‘…is a natural number’ to be ascribed to the whole subject as an internal dependent addition, because it has less weight.????? Otherwise we can achieve only things like the identity: ‘3 = {{},{{}},{{},{{}}}}.’ In any case, even the name of a so-called abstract object, such as the number ‘3’ cannot be moved to the predicate position here, insofar as it refers to something held as independent, being identifiable in the independence of its non-definitional predicates like ‘…is an odd number’.[19]
   Understanding unsaturation as a relative existential dependence suggests, therefore, that the tropes denoted by the predicate have an inevitable tie of dependence when considered relative to the individual within the fact referred to by the singular sentence. This gives us a better understanding of the asymmetrical relation between the subject and its predicate.
   Summarizing this section, my point is that the independence/dependence distinction gives a sufficiently reasonable ontological ground (I guess, the only one) for the explanation of the logical distinction between the references of subject and predicate in the singular predicative and relational sentences, along with a ground for the functional asymmetry between predicate and subject. The nominal term cannot be moved to the predicate position, because it refers to a nucleus of compresent tropes which exists in the relative independence of the less central tropes in and outside of the nucleus that can be designated by predicates. On the other hand, the opposite isn’t true in the context of the fact referred to by the sentence: the predicate can play the role of subject if we find a predicative expression dependent upon it, what is semantically and ontologically grounded. This fact explains the asymmetry. And what assures the unity of the thought-content is simply the represented factual unity. Finally, it is clear that these ties of dependence/independence will be more evident when the difference in relevance (weight, centrality) between the elements in question regarding the identity of the individuals is greater, and weaker when this difference is smaller, what justifies occasional uncertainties.

8. Sense of a predicative term
The relationship of independence/dependence originated in the ontological level of reference is reflected on the semantic and linguistic levels. The inde­pendence/dependence relation is reflected on the semantic-epistemic level of sense. We see this in the fact that the identifying rule of the nominal term is applied independently of the ascription of tropes to the object by the ascriptive rule of the predicative expression, while the sense the ascriptive rule of the predicative expression depends upon the prior application of the identifying rule of the object referred to by the nominal term. Finally, on the level of linguistic signs, the same relation of independence/dependence would be what makes the singular predicative sentence to usual take the grammatical subject-predicate form.
   At this point, we can make some reasonable additions, originated from our view of tropes as references of predicative expressions. The first is the suggestion that different predicative expressions with the same reference may be able to have different senses, paralleling the case of nominal terms like definite descriptions. Consider the following two sentences:

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

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

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

The predicates ‘...is white’ in sentences (1) and (2) obviously have the same sense, for they express the same ascription rule. But the tropes of whiteness (of reflecting all wavelengths of the spectrum) of the South Pole are located at the South Pole itself, while the tropes of whiteness of Mont Blanc are located in its eternal snows. Moreover, in the case of singular predicative or relational sentences, the references differ in the dependence of the pair of objects denoted by means of the same ascription rule type (the same conceptual sense type), since each object is constituted by its own tropes.
   Another point is to know how predicative expressions are used in the case of general sentences: universals and existential statements. Regarding universal sentences, we consider them as the abbreviated expression of a conjunction of singular sentences, each of them ascribing tropes to identified objects. For example: the universal sentence ‘All trees are made of wood’[20] would be analyzed as {Tree 1 is made of wood & tree 2 is made of wood &… & tree n is made of wood}; thus, the qualitatively identical tropes of woodness Tw1 of 1, Tw2 of 2… Twn 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 tropes of woodness 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, once we are unable to really apply it to all trees in the world. Similar considerations can be made regarding existential sentences like ‘At least one tree is made of wood’, which abbreviate a disjunction of ascription rules and refers to at least one trope of woodness belonging to the multiplicity of trees as objects.
   To the objection that this will never give us the conditions for a truly universal quantification, we could answer that the truth of the universal quantification is usually only probable and enclosed under some domain. What ‘All trees are made of wood’ means is in fact ‘Very probably all trees on the planet Earth are made of wood.’ Moreover, the so-called all fact (the requirement that the considered set of trees is the set of all trees, suggesting the recognition of a higher order general fact) has no mystery, because the all fact is nothing but a singular fact of a given particular domain. In the above case it is the fact that the trees in question must belong to the planet Earth. To give an easier example, if I say that all the coins in my pocket are two euro coins, the all fact is given by the domain established by the fact that my shirt has a pocket where I put coins (the vague boundaries of most domains does not change my point). (For contrasting views see Russell 1918, V; Armstrong 1997, Ch. 13)

9. The dependence of the predicative sense
The ontological distinction between independence/dependence (saturation/un­sat­uration) is reflected on the semantic-epistemic level to which the senses belong as an outcome of the original ontological dependence. This is clear enough if we see the sense of the predicative term as an ascription rule. In the context of a singular predicative sentence, the identifying rule of the singular term applies to the object as some core of mutually founding 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 liable 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 nucleus and consequently in dependence on the earlier application of the identification rule, lacking in this sense completeness.
   The same may be also valid for the fundamental descriptions constitutive of the identifying rule of the nominal term in the sentence context. Since the tropes belonging to the object to which the identifying 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 identifying rule of the object require prior application of the fundamental rules for identification of the object in order to become themselves applicable as part of the identification. Consequently, the ascription rule of the predicate is always dependent on the identifying rule of the singular term.[21]
   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. The ascription rule of the predicative expression is dependent, incomplete, unsaturated, in the sense that it demands the prior application of the identifying 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 appropriated trope. We need, for instance, first apply the rule that allows us to spatio-temporally locate the animal called Bucephalus in order to apply to its tropes, on that basis, the ascription rules of predicative terms. Thus, due to the independence of the object Bucephalus, we apply ‘... is a horse’, ‘... is black’, ‘... is swift’… and also the complex rules of application of more complex predicates like ‘…a horse that belonged to the best Thessalonian breed’. And we also need to apply the identifying rules for Bucephalus and Alexander in order to be able to apply the relational predicate ‘…belongs to…’ And finally, we need to apply the rule that allows us to mentally identify the number 3, in order to be able to apply to it the ascription rules of predicative expressions. Expressions that may be ‘…is odd’ or ‘…is a prime number’, though it is not the case that the number 3 depends on being odd or on being a prime number in order to be identified as such. In the same way, the relational ascription rule for ‘3 < 7’ is only applicable in the dependence of the independent application of the identification rules for the numbers 3 and 7.
   It would be a naive objection to think that after all it is possible to say things like ‘That is a horse’ or ‘There is a black thing’, applying ascription rules of predicates without identifying Bucephalus. The reason is that a fully detailed identification isn’t required. Indexicals such as ‘that’ and ‘there’ accompanied by some gesture of pointing already identify some particular as something 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 (a sortal) such as ‘that object’, ‘that animal’, and this is enough. Therefore, not only does the trope designated by the predicate depend upon a previous existence of the object and its identification, but, as a consequence, also the ascription rule of the predicate, its conceptual sense, must be dependent upon the prior application of the identifying rule to the independent bundle of tropes. This is how the relation of semantic dependency – on the level of sense – mirrors the relation of ontological dependency – on the level of reference, solving the riddle of unsaturation.

10. The concept horse paradox
We can continue to make deep revisions of Frege’s views in order to solve some mistakes of 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 our perspective, the first thing to do is to treat the nominalization as what it really is: an abbreviated way to speak about quantified concepts. What (1) really means is:

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

which is intuitively 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. Formalizing ‘The concept horse’ as (x) (Hx), where H replaces ‘… is a concept horse’, designating the trope of horseness, and replacing ‘…is a concept’ by C designating the trope of a 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 moral of this analysis? If the ‘the concept horse’ isn’t really a nominal term, but a hidden universal predication, Frege was wrong in maintaining that it cannot be a concept only because it works as a definite description. Frege’s paradox results from an incomplete analysis of sentences like (1) and (2), and the true analyzed sentences are the corresponding harmless universal conditionals (3) and (4), the first being contradictorily false and the second tautologically true. We conclude that rightly analyzed ‘the concept horse’ expresses universal predications and no real singular term. The whole paradox originates from the illusion that putting the predicate in the position of the subject really transforms it into a 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. I will read them here as definite descriptions, since for proper names we can use words like ‘Phosphorus’.
[4] As I showed in the introduction, Ernst Tugendhat later defended a similar understanding of the meaning of singular statements more systematically, without citing Frege.
[5] If we compare these two passages, it turns 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 (see Russell 1911, ch. 5).
[6] Taking Kripke seriously, François Recanati replaces senses by mental files as supposed non-descriptive modes of presentation (2012: 34). Independently of the progress made by this move, it is worth noting that these files, being clusters of information and not subjective Vorstellungen, should also be conventionally grounded. Consequently, they should be able to be linguistically expressed by means of descriptions, which brings us back to the descriptivist controversy. For this reason it seems clear to me that complex semantic-cognitive rules are apt to do a similar job in an explanatorily more natural and convincing way.
[7] François Recanati prefers to use the word ‘property’ (propriété) instead of concept in his summary of Frege’s semantics (Recanati 2008: 34). 
[8] This can be rejected by considering that for Frege a concept must have an extension. But even in this case we would have problems: for in the case of a false sentence like ‘Göteborg is the capital of Sweden’, nothing would fall under the concept of ‘…is the capital of Sweden’, what means that this concept-property should be seen again as an abstract entity. Frege seems to me in this point a kind of naïve Aristotelian realist.
[9] The pure theory of tropes was first introduced into philosophy by D. C. Williams (1952), and has since then sparked growing interest.
[10] Mere similarity would not do, since this concept is intransitive. Strict similarity means here qualitative identity, which is transitive. The strict similarity must also be a trope, since it is spatio-temporally located between tropes, even if it is a dependent trope.
[11] This trope-model way of constructing the universal is suggested in order to circumvent the usual but problematic definition of a universal as a set of tropes that are strictly similar, one with the other. A definition using the concept of set is problematic mainly because sets are quantitative and changeable, being bigger or smaller, growing or diminishing, which isn’t the case with the universal. An additional point is that usually the trope-model needs to be intermediated by memory: we don’t bring with us physical patterns to compare things with, but have a memory of them. The memory-trope is not the primary trope we intend to consider, since it must stand for the experienced one.
[12] See appendix to Chapter 3.
[13] This primacy of identification over the predicative function was emphasized by Keith Campbell (1990: 24-25).
[14] D. C. Williams also set things misleadingly here. For him ‘Socrates is wise’ (or any Fa) means ‘The concurrence [togetherness] sum (Socrates) includes a trope that is a member of the similarity set’. (my italics, 1953: 11)
[15]   The asymmetry remains even if 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’.
[16] Ignoring Frege`s thesis that the reference of a sentence is a truth-value, I will in the present context call it a fact. This choice will be justified later in this chapter.
[17] I hold that the individual referred to as a subject is relatively independent because the relation of independence/dependence is here relative to the internal context of the statement. I can say, ‘Jupiter is orbited by many moons’ and ‘…is orbited by many moons’ figures as a predicative expression. Nevertheless, these many moons are solid clusters of tightly connected compresent tropes – material objects – which are as moons dependent on Jupiter, figuring as designatum of the predication.
[18] I take these examples from Mulligan et al. (1984: 300, 301 and 306), though their point isn’t the same.
[19] On the other hand, in a sentence such as ‘My hat has three corners’, the numeral three designates a higher-order property of the hat.
[20] Here I treat this statement as a non-analytic realization.
[21] As already noted in the introductory chapter, this dependency that an ascription rule of a predicative expression has on a prior application of the identifying rule of a singular term was noted by Ernst Tugendhat in his analysis of the conditions needed for a true predicative singular statement. As he 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)

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