sábado, 9 de julho de 2016

CLAUDIO COSTA: PHILOSOPHICAL TEXTS - TEXTOS FILOSÓFICOS


THIS "BLOG" HAS BEING THOUGHT OF AS A WAY TO MAKE MY WORK IN PHILOSOPHY ACCESSIBLE TO A WIDE PUBLICUM. THERE ARE NEARLY HUNDRED PAPERS, MOST OF THEM IN DRAFT FORM. SOME ARE INTRODUC-TORY. PAPERS WITH INTEREST FOR RESEARCH ARE MARKED WITH ONE OR MORE '#'. I WISH YOU HAVE SOME INTELECTUAL JOY IN READING THEM.



ESSE "BLOG" FOI PENSADO COMO UMA MANEIRA DE TORNAR MEU TRABALHO ACESSÍVEL A UM PÚBLICO AMPLO. SÃO CERCA DE CEM ARTIGOS. OS QUE FORAM ESCRITOS EM PORTUGUÊS SÃO EM SUA MAIORIA INTRODUTÓRIOS. ESPERO QUE SEJAM ÚTEIS.

INFORMATION ABOUT MYSELF:
I am full professor at the UFRN (Brazil) and researcher for the CNPq. My graduation was in medicine. I made my PhD in Germany and post-doctoral sabbaticals of one year in München (with Friedo Ricken), Konstanz (with Wolfgang Spohn), Oxford (with Richard Swinburne), Berkeley (with John Searle) and now in Göteborg (with Anna-Sofia Morin). I Published some papers in international journals and two books in English: The Philosophical Inquiry: Towards a Global Account  (UPA, 2002), developing a theory on the nature of philosophy, and Lines of Thought: Rethinking Philosophical Assumptions (CSP, 2014), which is a collection of papers on central issues of contemporary philosophy. Presently I am working in a book to be called  Philosophical Semantics: Towards a New Orthodoxy (CSP, 2016) aiming to defend some old ideas of the philosophy of language, which I believe are important an were falsely debunked by the philosophy of language of the last 60 years.




Photo of Magda's tree



SEM EXAGERO ALGUM
Minha poesia é tão hipermoderna que só daqui a mil anos, quando a espécie humana se alçar ao nível ômega, ela será capaz de compreendê-la!  Quanto à minha filosofia, essa então nem se fala. Essa vai até o osso. Sua força é como a da explosão de uma supernova. Ela é tão profunda, tão profunda mesmo, que espécimes do gênero humano, especialmente aqueles por muitos anos adestrados na miopia do academicismo cientificista da pigmeica filosofia contemporânea não conseguirão enxergar dela mais do que traços embaçados.
Por isso, ao temerário que pretender imergir em sua profundidade abissal eu previno: muito dificilmente conseguirá retornar outra vez à superfície sem explodir pelo simples efeito da despressurização cognitiva. Pois a distância entre a minha filosofia galáctica e quase toda a minúscula filosofia de nossa época é tão grande que não se mede em quilômetros, nem em léguas, nem em milhas... mas em anos-luz! Por isso resolvi denominá-la "filosofia clássica pós-analítica e anti-contemporânea."

## AN EXTRAVAGANT READING OF FREGEAN SEMANTICS (for the book "Philosophical Semantics"

This is a new revised draft. It still needs correction. It will be part of the book "Philosophical Semantics" to be published by Cambridge Scholars Publishing in 2016.



– 4 –
AN EXTRAVAGANT READING OF FREGEAN SEMANTICS


The sense of an expression – word, phrase or sentence – is part of what we understand when we understand the expression; it is part of the conventional significance of the expression within the language.
Michael Dummett

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, making 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, what leads to consequences regarding the explanation of the concept of truth. With the first idea I intend to show the true link between Wittgenstein’s semantics – as I have read him in the last chapter – and Frege’s semantics; a link that was already noted by Michael Dummett, though in a protocolar manner, without exploration of its pragmatic details. Moreover, it is worth to note that my aim here is not to produce a work of Fregean scholarship, but rather to reconsider his main categories with the aim of taking the best of then in order to go beyond him, providing the rather vague semantic insights gained in the last chapter with the more rigorous framework suggested by his semantic theory.
   As it is of general knowledge, Frege explains reference (Bedeutung) using a semantic intermediary link, which he called sense (Sinn). The schema below shows how Frege deals with these two main levels of sense (1) and of reference (2) 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 the 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 along the way how to purge Frege’s semantics of its greatest oddities.

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 the broadest sense, which includes abstract objects. 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 explicit what Frege really had in mind. There are also other terms like ‘semantic value’, ‘semantic role’ and ‘truth-value potential’ that 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 also 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 the truth (Tugendhat 1992, p. 231). The strongest reason, however, at least with regard to the reference of natural language terms, is that by introducing the term ‘Bedeutung’ Frege substantivated the verb ‘bedeuten’, no longer using it to express 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.[3]

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, which is the planet Venus. But in spite of this, a person can know the truth of (1) without knowing the truth of (2). 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). Sense is defined by Frege as the way in which an object gives itself to us (die Art des Gegebenseins des Gegenstandes), which is usually translated as the mode of presentation of an object. The senses of the singular terms ‘the Morning Star’ and ‘the Evening Star’ are different, because the first singular term presents Venus as the brightest celestial body, usually seen just before sunrise, while the second singular term presents the same planet Venus as the brightest celestial body, usually seen shortly after the sun goes down…
   Frege writes that words express their senses (drücken ihre Sinnen aus), while the senses determine (bestimmt) their reference, since the mode of presentation should show us how to find the object. This determination of reference through sense is given even in cases where the object of reference does not exist, for even in this case expressions preserve their senses. This fact points to a flaw in Frege’s idea that the 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 if considered as the intended mode of presentation instead of a mode of presentation given by the object (Textor 2010, p. 134); the sense is the way we intentionally present an object to ourselves, whether it exists or not. At any rate, an expression can have a sense without a reference, but cannot have a reference without its determination through a sense.
   The notion of sense in Frege is extended 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 expression is adequate. The Fregean sense has epistemological interest, for it constitutes the proper informative content of the linguistic expression. It is, in Dummett’s words, what we understand when we understand an expression (Dummet 1990, p. 92). The philosophical importance of Fregean semantics is largely due to the epistemological and ontological import of the concept of sense.
   Frege is a Platonist about sense. He conceives senses as abstract entities which he analyses in terms of constituents that are also senses. That is: his Platonism of senses prevents him to analyse senses in terms of other concepts. But it is just this task something that naturally imposes itself. For it seems very plausible to understand senses as semantic-cognitive criterial rules. Herein lies the fundamental difference that can be found between Fregean semantics and the reflections of the later Wittgenstein, who saw senses or meanings as episodic uses determined by rules.[4] The plausibility of the identification between senses and rules is particularly clear when we take numerical expressions as examples. Consider the following expressions:

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

   All these numeric expressions have the same reference, the number 2, but they have different Fregean senses or modes of presentation. At the same time, they are expressions of procedures, methods, semantic-cognitive rules or, more precisely, combinations of rules by means of which we identify the same number 2 as a result (see Runggaldier 1985, p. 91 ff.).
   By treating senses as semantic rules and these rules as conventions, we contrast them with what Frege called colourations and illuminations (Färbungen and Beleuchtungen), which are feelings often associated with representations (Vorstellungen) and sense-perceptions (Anschauungen), which as such belong all to an intrinsically subjective level (Frege 1892, p. 31). These ‘colouration’ and ‘illuminations’ are names for what we would today call expressive meanings, i.e., sensory and emotional states that we normally and customarily associate with many expressions. Thus, for example, the words ‘love’, ‘dog’ and ‘hell’, in the sentence ‘Love is a dog from hell’ (Bukowski) contrastively associate strong specific emotions.
   As Frege realised, the kind of appeal or lack of appeal that the emotional colourations associated with words have for different people depends correspondingly on the similarities and differences between their human natures, and does not require conventions to be communicated, as in the case of senses. This is why some people are impressed by certain poems, while others do not; and this is why it is so dificult to translate poetry, which greatly depends on the colourations acquired by expressions in a particular language and culture. Hence, colourations are not results of conventional rules; they are rather regularities due to the similarities in the human nature. If our understanding of Wittgenstein’s argument of private language is correct, then his attempt to explain phenomenalist language as a simple replacement of public behavioural criteria like ‘ouch!’ by words like ‘I am in pain’ is an attempt to assimilate the referential meaning of the phenomenalist language to its expressive meaning (I suppose that both can be legitimated).
   If against Frege we accept these understanding of meaning, we can easily solve the problem of communicability of senses. For the reason for the objectivity of senses – which Frege understand as the possibility of intersubjective acess – and for the consequent communicability of senses – in contrast to the relative lack of objectivity and communicability of colourations or illuminations – is that the senses depend on semantic-cognitive conventional rules usually agreed upon in a pre-reflexive manner. That is: senses or meanings result either from the direct application of conventions or from combinations of conventions, a point already considered as we discussed Wittgenstein’s views.
   Accepting that the sense of a singular term should be considered 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 accepting 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 made more explicit by the description ‘the brightest celestial body usually seen not far from the Sun just before the Sun rises.’ Frege has also seen this, since he has noted that a proper name like ‘Aristotle’ abbreviates a cluster of modes of presentation of the object that can be expressed by definite descriptions, which according to him may include (i) ‘the disciple of Plato’, (ii) ‘the teacher of Alexander the Great’, and (iii) ‘a person born in Stagira’ (Frege 1892, p. 28). If this is the case, then (i), (ii) and (iii) express different senses, different rules that in some way are determinating of the reference by helping us to identify Aristotle (see also Frege 1918-19, p. 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, in my judgement Kripke’s arguments can be successfully countered by the kind of descriptivist cluster theory resumed in the Appendix of the Chapter 1 of the present book.

Reference of a predicative expression
Frege has something to say about the reference of a predicative expression, which he calls a concept (Begriff) and 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 reference – while the reference itself could be called a property (e.g., a red patch) or some combination of properties (e.g., the colourful flethers of a bird).
   For a traditional philosopher 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 myriad of sensible patterns that are satisfied by the 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 a terminology inspired by Wittgenstein by saying that a concept is a semantic-cognitive rule or procedure for the formation of criteria or criterial configurations that can be satisfied by properties. And coming back to Frege’s terminology we could paraphrase the same idea by saying that the concept is simply the sense (Sinn) of the predicative term, which can be nothing but modes of presentation of its reference… What any one of these explanations suggest is that that the concept should be understood as the sense of the general term or predicative expression and not as its reference, as in Frege’s bizarre use of the term.
   To be fair with Frege, he also says that when an object falls under a concept the concept may be called a propriety (Eigenschaft) of the object (Frege 1892, p. 201),[6] and ‘property’ really seems to be the best name for the reference of a predicative expression. However, when no object falls under a concept, the unanswered question remains: what kind of being would be the Fregean concept, except another superfluous Platonic entity? To make things worst, for Frege the criterion of identity for his concepts is the sameness of extension, what means that predicative expressions with different senses but the same extension must refer to the same concept (Frege 2001, p. 31). So, for example, ‘…animal with kidney’ and ‘…animal with hearth’ must be predicative expressions referring to the same concept, since they have the same extension. But it is intuitively obvious that kidneys and hearths are designated by very different concepts!
   In addition to be an entity belonging to the realm of reference, Frege sees his concept as a function. 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 an argument replacing 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) 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 earth’ has the value true when the object Moon falls under it, and it has the value false when the object Jupiter falls under it.
   For Frege, concepts cannot be objects, neither collections of objects, nor extensions (Frege 2001, p. 26). The reason is that objects, collections of objects and extensions are complete entities (vollständig). That is, they do not require anything to complete them. A concept, by contrast, being a function, is seen by Frege as being necessarly open: he calls it an incomplete (unvollständig) or unsaturated (ungesättigt) entity, always needing to be completed by arguments, which in this case are 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 in 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 as the ultimate ground. This concept, for its part, is completed when some object, being in itself complet, falls under it, for instance, the object referred by the name ‘Bucephalus’. With metaphors like that of ‘unsaturation’ and ‘incompletness’ Frege hoped to open the way to the solution of the mistery of the logical distinction between subject and predicate, for the subject (the nominal expression) would refer to the saturated object, while the predicate (the predicative expression) would refer to the unsaturated concept.
   Unsaturated predicative expressions and saturated singular terms combine to form complete sentences like ‘Bucephalus is a horse’, which being complete must also be the name of an object that for Frege is the truth-value. According to some this would be confirmed by the possibility that we have of nominalising sentences in the form of definite descriptions, since the last are also singular terms. Thus, the sentence ‘Bucephalus is a horse’ can be transformed into the description ‘the horse named Bucephalus’, which appears in the sentence ‘The horse named Bucephalus belonged to Alexander’. However, the same can be done with general terms: ‘…is a horse’ can be nominalised as ‘the horse’ occurring in sentences like ‘The horse is an herbivorous animal.’ This shows that the former argument isn’t very telling.

Ontological level
The discussion regarding 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. Frege accepts the commonsensical idea that concepts can be empty. For him the predicate ‘... is a unicorn’ refers to an empty concept, since no material object falls under it. But this good choice when related to his other views entangles him in unnecessary difficulties. If the concept is an unsatured entity, what kind of entity is it? If it is an abstract entity, a kind of universalia ante res, we have, as Ernst Tugendhat once noted, not only concepts as referred abstract entities (incomplete platonic objects, like the reference of ‘…is a unicorn’), but also the abstract references of the substantivations of the corresponding conceptual expressions (like the complete reference of ‘the unicorn’) as references (Tugendhat & Wolf 1983, pp. 138-9). These admissions are ontologically abusive, to say the least.
   Finally, as it is by now clear that Frege uses the word ‘concept’ as a technical term that contrasts violently with our ordinary use of the word ‘concept’. For the ordinary language there is surely an empty concept expressed by the predicate ‘…is a unicorn’, but this predicate, though having a sense, has no reference at all. So, it is no wonder that Frege has nothing relevant to say about the sense of predicative expressions, since, as it seems, he has in advance emptied its role by absorving the semantic level in the ontological one.
   We conclude 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 intension, its connotation, its mode of presentation. It is obvious that ‘...is a unicorn’ does not have any reference; but it clearly has a sense intuitively expressing what we ordinarily understand by a concept. The best way to give a legitime role to the word ‘concept’ is to see it as the sense of a predicative expression understood as its ascription rule.

Referring to singularized properties
But if we drop Frege’s technical concept of concept, what is the reference of a predicative expression? I believe that in the present days the most feasible answer to this question consists in the appeal to the ontology of tropes. So, I propose to replace the reference of predicative expression by what we now call tropes, which I characterize as spatio-temporaly located properties or singularized proprieties, or (to simplify) s-properties, such as the white that we see when we look at the snow and which, in a way, is there (physically speaking as the property of the snow of reflecting all wavelengths of the visible spectrum). In its most coherent version – the so-called pure trope theory – the 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. In this sense, this smooth surface I am touching, the brown colour I am seeing, that sharp sound I just heard, and even (I suppose) the rectangular shape of the computer screem before me, are tropes. Not only physical things would be constituted by tropes, but also mental events like my headache. Even imperceptible things like atoms or physical forces could be derivatively constructed from tropes, and it is not even impossible that abstract entities like numbers could be explained as constructions derived from tropes. (For the exposition of my understanding of the trope theory see the Appendix of Chapter 3 of this book).[7]
   Moreover, it is easy to suggest that a nominalist kind of universal could be built up based on tropes or s-properties. In my view a universal could be defined as any chosen s-property model P* or any other property qualitatively identical[8] with the model P*, assuming that what we take as a model P* is at will and may vary according to the epistemic subject and even in the same epistemic subject on different occasions.[9] In this case the s-properties P1, P2… Pn are identified as instantiations of the universal only because they are qualitatively identical to the chosen s-property model P*. And material objects could in principle be understood as concurrent, that is, nearly co-located co-temporal bundles of varied s-properties.
   Although a pure ontology of tropes (s-properties) 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 parcimonious solution for ontological problems, which would free us from at least three traditional hindrances: (i) ostrich nominalist solutions, with their lack of explanations for wanting questions, (ii) abstract objects with contestable intelligibility like Platonic universals which their uneconomical multiplication of entities, and (iii) uncognoscible substances. Other options have occupied the philosophical minds for more than two millennia without progress able to make them more plausible and in my view the only reason why they seem to be still in the foreground is the longstanding weight of tradition. For such reasons, I accept the pure ontology of tropes as exposed in the Appendix of chapter 3 as the most plausible one. Finally, I will use the word ‘s-property’ instead of simply ‘property’ only because the philosophical tradition has too often hypostasised the word ‘property’ as referring to some scarcely intelligible non-empirical entity, viciating our philosophical language and stubbornly ignoring that in ordinary language the word ‘property’ was always properly usely to refer to tropes or compositions of tropes.
   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 or designates (or refer to) a singularised property, namely, the s-property (trope or tropes) of the whiteness of the Moon as given to observers. Secondarily, however, the predicate ‘…is white’ alludes to or conotates the fact that this s-property exemplifies the universal property of whiteness, understood in a particularist way as this same s-property that is being now considered or any other s-property that is like it. Summarizing, a predicative expression has a twofold function:

 (A) An ascriptive function: that of referring or denotating the s-property belonging to the object referred to by the subject term,
 (B) An allusive function: that of aluding or connotating any other s-properties that would be qualitatively identical with the s-property designated by the predicative expression, building what may be called the universal understood in an ontologically inoffensive way.
   Opposing much of the traditional logic, from this perspective the extension (Begriffsumfang) turns to be something that albeit necessary isn’t primarily associated with the predication. It can be derived from the application of the allusive function of the predicate plus additional knowledge, allowing us to find or infer the existence of a class of objects that has these s-properties or even a class of s-properties individually considered as elements of an extensional class. However, in both cases the extension of these classes is a secondary element usually only vaguelly inferred.

The problem with the concept of insaturation
The most serious problem with the idea of incompleteness or unsaturation is that it fails to serve its main purpose of distinguishing a predicative from a nominative expression. As it is well-known, between the object referred to by the subject and the property designated by the predicate, there is a well-known asymmetry: the nominative expression, the singular term, always refers to its object and cannot properly take the place of a predicate (e.g. ‘Socrates’ and ‘Socrates is wise’). On the other hand, we can easily subjectivate a predicate by means of nominalization (e.g. ‘… is wise’ and ‘Wisdom is a virtue’).  To make this clear, take for instance the sentences:

1.     There is Socrates.
2.     Wisdom is a property of Socrates.
3.     The husband of Xanthippe is Socrates.

In these sentences the term ‘Socrates’ occupies grammatically a predicative position, though continues to be logically used as a proper name, since these sentences can be respectively reformulated as:

1.     Socrates is that person pointed by me now.
2.     Socrates has the property of wisdom.
3.     The husband of Xanthippe = Socrates.[10]

One cannot really transform a singular term as such into a predicate, while predicates seem to be easily transformed in singular terms by means of substantivation. This asymmetry shows that subject and predicate play different roles in sentences, which demands an explanation. Cann really the Fregean distinction between saturation and unsaturation do something to explain this difference?
   At first view the answer is no. His distinction doesn’t 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, are ‘unsaturated’, ‘incomplete.’ After all, what is the difference between

[Bucephalus, Silver, Beauty, Gypsy… Pegasus] …is a horse

 And

 Bucephalus is... [black, restless, of best thessalian strain… swift]?

   ‘Bucephalus’ could be seen here as the name of a function under which properties like ‘black’, ‘restless’ and ‘swift’ fall as arguments, so that when this is the case we have the value ‘true’ as a result, and when this isn’t the case we have the value ‘false’ as a result. And the conclusion seems to be that both, the general term and the singular term can be seen as unsaturated, expressing functions that can be supplemented by a myriad of other arguments in order to get the True or the False as values.

Insaturation 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. Is there in this metaphor the hint of an answer that wasn’t sufficiently explored by Frege?
   In what follows I want to offer a reading of the reference of a predicative expression in terms of s-properties 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, namely, that which exists independently of other things (Aristotle 1994, sec. 5). Applied to material objects understood as (at least) compresent bundles of tropes, this definition suggests that these structures are in their existence comparatively more stable than their associated s-properties. In other words, it seems that the individuals typified by material things exist in a manner relatively independent of their s-properties in the composition of the kind of fact represented by true singular predicative or relational statements. In other words, the true dichotomy is that between independence and dependence, terms only rarely used by Frege. What distinguishes the reference of a general term, in the case of a predicative sentence or even a relational sentence, is that this reference is an s-property whose existence depends on a whole which is the compresent bundle of properties constituting the individual referred to by the singular term. The general thesis can be exposed as follows:

In the constitution of a fact represented by a true singular predicative statement the existence of the s-property ascribed by the predicative expression is relatively dependent of the existence of the object refered to by the nominal expression (what is also valid to the facts represented by objects related by singular relational statements).

   In order to explore this idea in more details I will assume the most plausible pure trope-theory of individuals, which is Peter Simons’ nuclear trope theory, according to which in the standard case individuals are formed by an essential nucleous or kernel of mutually founding tropes, which is necessary, circunded by a looser bundle of peripheral tropes, which are accidental, requiring the kernel of essential tropes for their existence. (Simons 1987, pp. 567-569) Using our example of the definition of a chair as a seat with a back made for only one person to sit on each time, we can say that the criterion for the identification of chairs is what forms the nuclear structure, while the symptoms of a chair, as to have four legs and arms or of being made of wood, forms the peripheral tropes.
   Simon’s nuclear theory allows us an easy explanation of the relation independence-dependence regarding the nuclear kern of an object versus its contingent s-properties. Consider, for example, the singular predicative sentence ‘Bucephalus is swift’. The predicate ‘...is swift’ in this sentence applies to a contingent s-property that constitute swiftness, whose existence depends on the existence of an object, Bucephalus, which is essentially formed by some kern of mutually founding tropes. Consider now the relational sentence ‘Bucephalus belongs to Alexander.’ The relational combination of tropes that constitutes the contingent s-property of belonging to could not possibly be found if Bucephalus and Alexander could not independently exist as particulars formed by kerns of mutually dependent tropes. That is, the possibility of existence of the relation ‘…belongs to…’ is here indebted to the existence of the more stable and non-dependent essential kern of mutually founding s-properties constituting the two objects Bucephalus and Alexander. These kerns of s-properties referred to by the names ‘Bucephalus’ and ‘Alexander’ are objects that certainly exist independently of the existence of the contingent combinations of tropes constituting the s-properties of ‘being swift’ or ‘belonging to someone’, since for being agile in the action and having an owner we need the previou existence of things that have these properties.
   Things are easy when we apply the dichotomy independence-dependence to the s-properties that do not belong to the own definitory kern. So, consider once more our definition of a chair as a seat with a backrest made for only one person to seat each time, which gives what Simons calls the kern of mutually founding tropes. Suppose now that I point to the chair and say ‘This chair has two arms,’ since the s-property of having two arms does not belong to the definition that makes explicit the kern of tropes, its existence is dependent of the existence of the chair as part of the fact. Notice that I am not saying that the proper existence of the arms of the chair is dependent of the chair in themselves, because arms of chairs are parts and can exist separated from chairs; but in the context given by the fact described by the statement ‘This chair has two arms’ they exist as dependent parts.
   The problem arises when we consider s-properties belonging to the definitory kern. Suppose that I say: ‘This chair has a backrest.’ I think that, despite of the triviality of the statement, the s-property of ‘having a backrest’ is dependent in its existence of the whole bundle of tropes that builds the nucleus of tropes distinctive of the object. Here one could object that the bundle of tropes constitutive of the kern reciprocally depends on the backrest: a chair without a backrest is no chair. But the point is that the referred object still exists, first as a seat and second as a material object. Even if you take out one foot of the seat you still have a kripled material object as reference of the singular term. This shows that the reference of the object exists in a way that is more independent than the reference of its s-property.
   Finally, what about formal sentences? Consider the sentence ‘9 is the result of the sum of two prime numbers’. This sentence describes a mathematical fact. Well, the property of being the result of the sum of two prime numbers cannot exist except in the dependence of the existence of numbers, which in the fact described is the number 9. Once more, the property ascribed by the predicate is dependent on the object referred to by the proper name. But what about a sentence as ‘9 is a number’? Regarding the mathematical fact described by this sentence the predicate ascribes to the subject a property that belongs to its definition, being therefore necessary to it. The case is similar to those of having a backrest and being a horse, which belong respectively to the definitions of the individuals chair and horse. But these properties exist all in dependent ways relatively to the objects referred to by the nominal terms, because there must be always something to be a natural number, to have a back and to be a horse (predications of analytic statements). And here too you can kriple the object: cut the backrest, shot the horse and give the number 9-2 instead of 9.
   Understanding unsaturation as a relative existential dependence suggests, therefore, that the s-properties referred to by predicates have an inevitable tie of dependence when considered relatively to the individual in the context of the fact referred to by the sentence, and that this gives us a better explanation (I guess, the only one) for the asymmetrical relation between the subject and its predicate. This explanation is the following: A singular term cannot become a predicative expression ascribing properties because its reference cannot become dependent, since a relative independence regarding the concerned properties belongs to the nature of its reference. Moreover, concerning a predicative expression, we can say that its nominalization does not turn it even into the name of an independent individual as it appears to be. For the nominalisation can be logically analysed by means of quantification. The sentence ‘Wisdom is a virtue’, for example, can be analysed as ‘For any human being x, if x has wisdom, x is virtuous’, where wisdom is shown work as a disguised predicate dependent on any human bein x on which it is predicated.
   It is also to be noted that the relationship of independence/dependence couldn’t be preserved if the references of predicative terms were extensions, understood as classes of objects to which the predicative terms apply, for these classes are independent, as Frege has noted. The relationship of independence/dependence is only preserved if we understand the reference of the predicate as s-properties.
   Summarising this section, my point is that the independence/dependence distinction gives a principled ontologically grounded explanation of the logical distinction between subject as singular term and predicate as general term. In the context of a sentence, the singular term cannot move to the predicate position, as far as what it refers to is something existing in relatively independent way, that is, it forms a kern of s-properties which exists in the independence of the the reated s-properties in and out the kern, while the opposite isn’t true in the context of the fact referred to by the sentence. Even the name of an abstract object such as the number ‘9’ cannot be moved to the predicate position, since it refers to something independent, being identifiable in the independence of its non-definitory predicates, say ‘…is greater than 5’. Finally, these ties of dependence/independence can be stronger or weaker, what justifies our frequent uncertainties about what is the singular and what is the general term of a sentence.

Sense of a predicative term
The relationship of independence/dependence originated in the ontological level of reference is reflected in the semantic and linguistic levels. First, the independence/dependence relation is reflected on the semantic-epistemic level of sense in the fact that the mode of presentation of the object referred to by its name, that is, the identification rule of the singular term, is applied in the independence of the ascription of its s-properties by the ascriptive rule builded by the predicative expression; on the other hand, the sense, the ascriptive rule of its predicative expression, depends upon the prior application of the identification rule of the object referred to by the nominal term. Second, concerning the linguistic signs, the same relation of independence/dependence would be what makes the singular predicative sentences to take the grammatical subject-predicate form.
   A consequence of this view is that, in the same way as two definite descriptions may have the same reference but different senses, two different predicative expressions may also have the same reference but different senses. Consider the 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 s-properties 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, in my understanding, that there are differences in concepts as modes of presentation or ascription rules for the predicates of the sentences (1) and (2).
   Another consequence of our understanding of predicative expressions as basically referring to s-properties contradicts the Fregean expectation that the same sense cannot have more than one reference. 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 s-property of whiteness (of reflecting all wavelengths of the spectrum or so) of the South Pole is located at the South Pole itself, while the s-property of the whiteness of Mont Blanc is located in its eternal snows. This is of course no isolated example: in the case of singular predicative or relational sentences (assuming that relations can be s-properties) the references differ in the dependence of the object denotated by means of the same ascription rule type (the same conceptual sense type) since each object is constituted by its own s-properties, which are only qualitatively identical.
   As we already noted, the predicative expression used in a singular sentence has a twofold function. The first and foremost function is what I called that of the ascriptive function (denotative or referential), since the conceptual ascription rule establishes the criteria that must be satisfied by the relevant s-properties in order to allow the application of the predicative expression. The second function is what I called the allusive (connotative) dimension, bringing with it the implicit allusion to a trope-theoretic universal. We get to this conclusion as we see that by applying the ascription rule and satisfying its criteria by means of the relevant s-properties, we simultaneously see that it can be extended to any other precisely similar property, what we do without establishing any closed set of s-properties.
   I can exemplify this dependency of senses with the predicate ‘…is white’ in the statement ‘The Taj Mahal is white.’ We must first apply the identification rule by pointing to the object Taj Mahal situated in a spatio-temporal region not far from the speaker and indicated by him when he speaks, while its complement, the word ‘white’, express as sense the rule demanding as a criterion for its application the s-property of being white. Hence, the whole identification rule of the subject term can be expressed as the condition that in the indicated spatio-temporal region there is an object that is a mausoleum with such and such relevant s-properties (see Appendix of chapter 1). Once this identification criterion is satisfied, the sense of the predicative expression ‘…is white’ comes in question as its ascription rule demanding for its application the satisfaction of the criterion that the given s-property is white.
   Once this has been done, we may consider that a precisely similar ascription could be done to any other material object that satisfies the ascription rule, building in this way the universal ‘white’ as some open set of criterial conditions belonging to the potentially applicable rules (building the nominalist equivalent of a ‘platonic’ universal), as much as the extension of the concept as the open set of s-properties that constitutes the materially given white entities in the world by the effective application of the ascription rules (building a nominalist equivalent of an ‘Aristotelian’ universal).
   But how are predicative expressions used in the case of general sentences, universals and existentials? Regarding the universal sentence, we consider them as the abbreviated expression of a conjunction of singular sentences, each of them ascribing s-properties to identified objects. For example: the universal sentence ‘All trees are made of wood’ would be analysed as ‘Tree 1 is made of wood, tree 2 is made of wood… tree n is made of wood’; thus, the qualitatively identical s-properties P1 of 1, P2 of 2… Pn of n, can be conjoined and jointly denoted by the universal sentence, which also means that the ascription rules able to give the s-properties P’s of being made of wood belonging to the objects trees, which are qualitatively identical rules, also need to be conjoined in the sense that the universal sentence at least points to the application of this conjunction of ascription rules, once we are unable to apply it to all trees in the world.[11] 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 a disjunction of s-properties of type P belonging to the multiplicity of trees as objects.

Dependence of the sense of predicative terms
The ontological distinction between independence/dependence (saturation/unsaturation) is also reflected on the epistemic level to which the senses belong. This is understandable if we consider the sense of the predicative term as an ascription rule. In the context of a sentence the identification rule of the singular term applies to the object, which is considered as existing independently in relation to its usually contingently possessed singularised properties. Consequently, the rule of identification is also liable to be applied regardless of the application of contingent ascription rules, what means that this rule of identification is the only conceivable in its application alone, in that sense being independent, complete, saturated.
   In the context of the sentence the same is valid even to s-properties belonging to the definition of the object. Since these s-properties as belonging to the objects to which the identification rules apply are ultimately dependent of the existence of these objects as bundles of s-properties, even the ascription rules of predicative expressions belonging to the definition of the object require prior application of the other definitory rules for identification of the object in order to become themselves applicable as part of the identification. This is why the ascription rule of the predicate is always dependent on the identification rule of the singular term.[12]
   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 on and, accordingly, 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 characterise it ascribing the predicative rule to its s-property. We need to apply the rule that allows us, for instance, to spatio-temporally locate the animal called Bucephalus in order to apply, on that basis, the ascription rules of predicative terms like ‘... 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 need to apply the rule that allows us to mentally identify the number 9, in order to be able to apply to it the ascription rules of predicative expressions like ‘…is uneven’ or ‘…is a prime number’, though it is not the case that the number 9 depends of being uneven or of being a prime number in order to be identified as something given to our minds.
   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’, without identifying Bucephalus, because a fully detailed identification isn’t demanded. Indexicals such as ‘that’ and ‘there’ already identify some particular in the form of something spatio-temporally localisable in a way independent of its predication. This independent way can be made explicit when the indexical is followed by a term designating countable things (a sortal) such as ‘that place’, ‘that object’, ‘that animal’, and this is sufficient. Therefore, not only is the reference of the predicate dependent of a previous existence of the object, but also its ascription rule, its conceptual sense. 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.

The concept horse paradox
If we accept this deep revision of Frege’s views, some problems of his philosophy can also be better understood and solved, like the so-called paradox of the concept horse. Based on his understanding of a concept as something unsaturated, Frege was led to the strange conclusion that one cannot name a concept. For him the sentence:

(1)    The concept of horse is not a concept,

is true. After all, ‘the concept horse’ is a singular term – a definite description – and as such, it must refer to something saturated, that is, an object. The paradoxical point is that the denial of (1), the sentence

(2)    The concept horse is a concept,

sounds like a true analytic sentence, while as the denial of (1) should sound false.
   In my view the first thing to do is to treat the nominalization as an abbreviated way to speak about quantified concepts. What (1) really means is:

(3)    For any x, if x is a concept of 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 of horse, then x is a concept,

which is true. Calling ‘the concept of horse’ (x) (Hx), where H symbolizes ‘… is a concept of horse’, and calling ‘…is a concept’ C, 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 of horse’ isn’t really a nominal term, but a hidden universal predication, Frege was wrong in saying 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 analysed sentences are the corresponding harmless universal conditionals (3) and (4), the first being contradictorily false and the second tautologically true. We conclude that rightly analysed ‘the concept of horse’ stands for a concept expressed in a predicative expression. The whole paradox originates from the illusion that putting the predicate in the position of the subject really transforms it into a singular term.

Existence as a property of concepts
At this point we can get to Frege’s treatment of the concept of existence. Deepening an idea already present in Kant, he suggested that existence is a property (Eigenschaft) of a concept, namely, the property that at least one object would fall under it (Frege 1988, sec. 53). A similar idea was later defended by Bertrand Russell in the suggestion that existence is the property of a propositional function to be true for at least one instance (Russell 1994, pp. 232-3, pp. 250-54.).
   Here I will not try to interpret the details of Frege’s often obscure remarks. Using a more current terminology, I will follow an explanation taken from John Searle, which with his usual clearity brings us unmistakenly to the point. Consider the sentence ‘Horses exist’. This sentence can be analysed as:

There is at least one ... such that ... is a horse. (Searle 2008, p. 176)

As Searle notes, this sentence contains two components. One is expressed by the predicate ‘…is a horse’, symbolically Hx (where x replaces ‘…’ and H replaces ‘is a horse’). The other component is the predication of existence, expressed by the open sentence ‘there is at least one ... such that ...’ This predicate can be symbolically expressed as Ǝx(...) (where Ǝ replaces ‘there is at least one x’, and ‘...’ is the gap to be filled by some concept applied to something in the usual sense of the word concept. The result is that the whole sentence ‘Horses exist’ can be symbolised as Ǝx(Hx). This also means that the predication of existence Ǝx(...) is a meta-predication expressing a higher order concept, a concept of a concept, a metaconcept under which other concept can fall, in case (Hx). Hence, a sentence with the form Ǝx(Fx) typically expresses a second-order concept applied to a first order concept. For Frege what this second-order concept does is to say that at least one object falls under the first order concept to which it applies, what also means that the first order concept is satisfied or fulfilled or that it applies to at least one thing. Existence is in this sense something objective, since this satisfaction is independent of our subjective grasping of this concept or of its applicability.
   These last ways of speaking are more interesting to us, because they could be paraphrased in accordance with our identification of concepts with senses of predicates or their conceptual or ascription rules, showing that existence is a property of these rules, namely, the property of being satisfied, fulfilled, or simply applicable. Thus, when I say ‘Horses exist’, I mean that the concept expressed by the predicate ‘…is a horse’ is effectively applicable in the domain of external objects. I add the adverb ‘effectively’ to ‘applicable’ in order to make clear that I do not mean this last word merely subjunctively, as referring to something that may be applied, but as referring to something that is applicable for sure.
   Moreover, as it was noted above, the existence or effective applicability of a semantic-cognitive rule is always considered with regard to a certain domain of entities, the fundamental one being the domain concerning the real empirical world, be it the external (physical) world (Carnap called it the thing-world) or the internal (psychological) world. This is the case of the statement ‘Horses exist’. Typically, what is in this case meant with the predication of existence isn’t only the applicability of the ascription rule of a general term as a mere possibility entertained only in our imagination, but also an effective applicability of the rule in its most proper domain of entities, namely, the domain of external, physical objects.
   However, there are others higher order domains and sub-domains of entities in which we apply the predicate of existence, even if only in a derivative sense. One can say, for instance, that the horses from the Vakyries exist in the fictional domain of Wagner’s opera Die Valküre in the sense that the ascription rules for these horses are effectively applicable in the fictional domain described in the libreto. So, there are fictional domains in the arts, domains of imaginable and plausible entities, domains of the so-called abstract entities and its various sub-domains, like the domain of mathematical entities, of logical entities, etc. and the word ‘existence’ can be applied to entities belonging to all of them.
   So, according to the view that I am considering, to say that horses, rocks, trees and chairs exist is to affirm the effective applicability of the ascription rules of the concept-words ‘horse’, ‘rock’, ‘tree’ and ‘chair’ in the fundamental domain of objects of the objectively real external world; to say that thoughts, joys and pains exist is to affirm the effective applicability of the ascription rules constitutive of the concept-words ‘thought’, ‘joy’ and ‘pain’ in the subjectively real mental domain of entities; and to say that ‘totalitarianism’, ‘corruption’ and ‘exploitation’ exist is to affirm the effective applicability of the ascription rules of these concept-words in the psycho-physical domain of social entities. The domain of entities to which these concept-words apply is usually assumed as respectively physical, psychological and social. Finally, to say that an entity exist is – ceteris paribus – to say that its conceptual rule is effectively applicable in what was in many cases already conventionally established as its most proper domain of application, e.g. the most proper domain of application of the conceptual word ‘horse’ is the real external world and the most proper domain of application of ‘a valkyre’ is a fictional one.  
   As we have already noted, a concept, that is, the semantic-cognitive ascription rule of a predicative expression, is able to generate subjective criterial configurations. Thus, to say that a concept-word is effectively applicable is to say that subjective criterial configurations generated by the ascription rule are satisfied or fulfilled by its criterial configurations that we are assuming as being objectively given, namely, configurations of s-properties usually belonging to more complex entities – a point that I intend to explain and justify more satisfactorily in the last chapter, when we will need to tackle the problem of perception.
   The parallel between the concept of existence in Frege, and the concept of existence that I am proposing in my reconstruction of sense as a conceptual or ascription rule is straightforward:

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

     Concept of existence (reconstructed) =
A higher order conceptual rule that demands for its application that a first order conceptual (or ascription) rule is effectively applicable to at least one entity, being this entity an s-property or some construction out of s-properties in a certain domain of entities, usualy what is conventioned as the proper domain for the s-property or the construction out of s-properties.

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

Prima facie objetions
There could be some telling objections to this lighly revisionist understanding of the orthodox view of existence. A first objection could be that the concept of the effective applicability of a rule is an anthropomorphic one, while things are said to exist in full independence of cognitive beings. However, this objection would only arise if we confused the concept of (effective) applicability (within a certain domain) with just the concept of application. The application of a semantic-cognitive rule is an act or a series of acts, which are essentially mental and often also physical, resulting in judgements. The application of the conceptual rule for the identification of the planet Neptune, for instance, really demands the existence of cognitive beings able to apply it. And our judgment that the moon exists depends on the experience of the application of a rule by ourselves or by someone who testifies to its applicability. On the other hand, the concept of (effective) applicability is not anthropomorphic. Even if there were no cognitive beings able to apply the identification rule for the concept Uranus, this planet would continue to exist, since the ascription rule for the identification of Uranus would still be effectively applicable to this object in its proper domain. The rule would still be applicable, even if it were never conceived or effectively applied, even if no cognitive being would have existed in our universe to think that Uranus exist. These remarks make clear that in opposition to the concept of application of a rule, the concept of effective applicability isn’t anthropomorphic.
   Based on such considerations, it is possible to rebut an objection that the proponents of the idea that existence is a property of things instead of concepts could easily raise, namely, that if existence is a property of concepts, then it hasn’t anything more to do with the objects that fall under these concepts. But this seems odd, leading us to think that existence isn’t something in some way belonging to the objects we claim to exist!
   The answer to this objection lies in a consideration of the peculiarity of the conceptual property that we call existence. We can not only say that some ascription rules have the property of being effectively applicable to some entities belonging to a certain domain, but, mutatis mutandis, we can also say that some entities have the property of having their ascription rules effectively applicable to them, meaning by this that these entities exist in their proper domain. That is, when we say that entities, for example, objects in the external world like horses, exist, we also mean that at least one under these conceivable objects has the property of having its ascription rule effectively applicable to it, that is, it has the property of existing in the actual world, and this property is also a property of the object – even if of a second order – since it is dependent of the property of the ascription rule of being aplicable to it.
   Thus, according to the higher order view of existence, the s-property of red of this couch exists only insofar as this object (the couch) has the property of falling under the concept of being red in a Fregean way of speaking. Or, speaking in a more detailed way, we can say that the s-property the redness of the pointed out couch exists only in the sense that the ascription rule of the concept-word ‘red’ has the metaproperty of being effectively applicable to its s-property of being red; but this also means that the s-property of the couch of being red owns secondarily the meta-property of its ascription rule of being effectively applicable to it. But since this meta-property of being effectively applicable is the property of existence, and this metaproperty is applicable to the ascription rule (since it is effectively applicable), this metaproperty is indirectly and dependently applicable to an s-property of redness that belomgs to the real empirical world.
   In short: the meta-rule of existence also applies to the s-property, even if in a secondary way. Thus, one can argue that it is a peculiar feature of the concept of existence (and certainly of some others) that, being owned by a first order concept, it must also be owned by the given entities belonging to the chosen domain of entities without being a constitutive element of these entities.

Existence attributed to objects
The idea that existence is a property of concepts concerns not only what is meant by general terms, but also by singular terms, since both kinds of term express senses and their references can be said to exist. Since singular terms can be divided in definite descriptions, proper names and indexicals, I will shortly consider them all, beginning with definite descriptions.
   Following Russell’s treatment of definite descriptions and replacing the predicate ‘…is inventor of Maieutics’ by M, the sentence ‘The inventor of the Maieutic existed’ can be analysed as:

Ǝx [Mx & (y) (My → (y = x)].

   In this way we are affirming the existence of at least one and not more than one inventor of the Maieutic, what means that the ascription rule that constitutes the concept expressed by the predicate ‘…is the inventor of the Maieutic’ has the property of being effectively applicable to exactly one human being, a being (Socrates) that exists or has existed.[13]
   Consider now the case of proper names. As we have seen, they should also have senses in the form of identification rules. Considering existence as the effective applicability of a semantic-conceptual rule in a chosen domain, the existence of the object referred to by a proper name should be established by the effective applicability of its identification rule.
   Although this issue cannot be properly addressed without a deeper investigation of the nature of proper names, we can begin by giving a suggestive defence of the foregoing view à lá Russell, by transforming proper names into predicative expressions applied to one only particular, showing then that the senses of names themselves can be reduced to the conceptual senses of predicative terms. A first step in the attempt to arrive at this is to transform the proper name into a predicate. Thus, ‘Socrates’ in ‘Socrates exists’ can be predicated in the sentence ‘There is something that socratises’, or ‘Ǝx(x socratises)’, as W. V-O. Quine suggested. Even if suggestive, this is not only linguistically abominable, but also inadequate, since it leaves open the possibility that there is more than one Socrates.[14]
   Despite this, ‘Ǝx(x socratises)’ points in the right direction by suggesting that the existence of the bearer of a name may be asserted by means of the conceptual senses of predicative terms. For the verb ‘to socratise’ can be seen as an abbreviation of the predicative expressions that are included in those descriptions summarised by the proper name ‘Socrates’. This is a good strategy, if we believe in the cluster theory of proper names in one or other way already present in Frege, Russel and Wittgenstein, and made more explicit by P. F. Strawson and John Searle, according to which the whole sense of a proper name is given by a cluster of definite descriptions. To illustrate this, we can assume that the sentence ‘Ǝx(x socratises)’ would be an abbreviation of what could be more extensively and adequately expressed as:

Ǝx (x is inventor of maieutics, x is mentor of Plato... x is Xanthippe's husband).

   Of course, this is still insufficient, since it leaves open the possibility that these predicates could be applied to more than one object. However, this can easily be remedied by means of the Russellian device of restricting the number of objects of predication to exactly one:

Ǝx (x and no other invented Maieutic, or x and no other was the mentor of Plato… or x and no other was the husband of Xanthippe).

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

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

   It is important to note that here the meaning of a proper name is translated into the conceptual-senses of predicative expressions such as P1, P2… Pn, which according to our analysis are nothing but semantic ascription rules expressed by predicates that we expect to be really applicable to one and the same thing. So analysed, the attribution of existence to the object referred to by a proper name is made by saying that its sense, its identification rule, effectively applies in that assumed domain. As this rule for the identification of the name was here analysed in terms of a disjunctive set of rules for the application of predicates that must be applied to the same individual, the existence of the bearer of the proper name becomes the same as the effective applicability of conceptual rules of predicative expressions to one and the same individual.
   Of course, here could be objected that all these descriptivist attempts to explain the meaning of a proper name are doomed to failure, because they amount to some version of the cluster theory of proper names with its well-known difficulties, which were already convincingly pointed out by Saul Kripke, Keith Donnellan, Michael Devitt and others.
   However, there are two points that need to be considered. The first is that, contrary to a current bias, these objections have few effects against the most expliciply developed versions of descriptivist theories, some of them having already been answered by John. R. Searle with relative success (Searle 1993, ch. 9). In other words, the cluster theory may not have found a decisive defence, but it has not yet been satisfactorily refuted.[15]

Existence of objects and its identification rules
The second point to be considered is that the analysis presented above is a simplification when seen from the point of view of our own much more elaborated version of the cluster theory of proper names. This version has (in my humble opinion) greater explanatory power than any previous theory, allowing decisive answers to the counter­examples. Although the theory is too complex to be exposed now in any convincing detail, I have already summarized in some deep in the Appendix to chapter 1.
   Briefly repeating what I said there, my take is the following. The traditional cluster theory of proper names, defended by Frege, Russell, Wittgenstein, P. F. Strawson, John Searle and others has a severe limitation that seems to have passed unseen: the clusters have no internal order. They do not tells us what description or combination of descriptions are fundamental and what descriptions are secondary or even irrelevant for the application of the name. Definite descriptions are expressions of rules aiming to connect the proper name with its reference; they are in this sense description-rules. But what we still need to ask is if there isn’t a meta-descriptive rule that applied to any cluster of descriptions that we associate with a proper name allows us to know in what ways the satisfaction of these descriptions make the proper name applicable. In order to find this meta-descriptive rule we need first to distinguish the fundamental from the merely auxiliary descriptions, which are accidental. There is a natural ordinary language method to do it: looking how proper names are treated in encyclopaedias. If we do it, we will see that proper names are first and foremost attached with two fundamental descriptions, which I call the localizing and the characterizing description-rules. Here is how we can define these two kinds of description:

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

Consider, for instance, the name ‘Adolph Hitler.’ Here is what is told from him in the first paragraph of the Wikipedia article:

Adolf Hitler (20 April 1889 – 30 April 1945) was an Austrian-Born German politician who was the leader of the Nazi Party. He was Chancellor of Germany from 1933 to 1945 and Führer of Nazi Germany from 1934 to 1945. As effective dictator of Nazi Germany, Hitler was at the centre of World War II in Europe and the Holocaust.

As it is usual in the encyclopaedias, the first things we find are an abbreviation of the localizing description-rule followed by an abbreviation of the characterizing description-rule, stating the reason why we remember the name. What follows in the article are more or less relevant details and explanations. There we find a variety of definite and indefinite descriptions that are more or less irrelevant – the accidental auxiliary descriptions. For example: Hitler was ‘the lover of Eva Brown’. And all that can be largely extended in biographies.
   The same you will find if you look at other proper names like ‘New York’, ‘Eiffel Tour’, ‘Niagara Falls’ or ‘Venus’ in encyclopaedias. Of course, there are also the proper names of most of us, which do not figure in encyclopaedias. But the basic mechanism of reference remains the same. It isn’t difficult to see that the relevant information is given by their localizing descriptions and in the usually much more scattered characterizing descriptions. So, in usual cases if you will know who Sam is, you will probably get the relevant information looking at his identity card, his employment record card, his passport, maybe adding to this the information given by his family and acquaintances about his personality, character, formation, interests, relationships, deeds, etc.
   Now, my suggestion was that, although a conjunction of the localizing and the characterizing descriptions isn’t required in any possible world, as Kripke has clearly shown, a disjunction of the two fundamental descriptions must be in some degree satisfied in order to allow a proper name to refer to its object in any possible world.  Moreover, there are two other complementary conditions that need to be added to it. First, a condition of sufficiency that needs to be satisfied: the disjunction of these two fundamental descriptions must be at least sufficiently satisfied in order to allow a proper name to refer to its object in any possible world (suppose that in a possible world there was no Nazism and only one Adolph Hitler who died some days after birth). Second, a condition of predominance, demanding that that satisfaction is made in a more complete manner than any other competitor in the considered possible world, since by definition the bearer of a proper name must be always one unique specified object (an Adolf Hitler who lived in the XVII Century and was a saddler would not satisfy this condition).
   Here is how the meta-descriptive rule applicable to the cluster of descriptions associated to any proper name can be formulated (including conditions os sufficiency and predominance):

MD-rule for the application of proper names:
In any possible world in which a proper name ‘N’ has a bearer, this bearer must belong to the nearest relevant class of referents, so that it more than any other object sufficiently satisfies the conditions set by its localysing description-rule and/or its characterising descrition-rule. (To this auxiliary descriptions may helpfully be added).

   What about the other definite descriptions like ‘the lover of Eva Braun’ or ‘the boy who was sent by his father Alois to the Realschule in Linz in September 1900’? The answer is that they are merely auxiliary descriptions. One can insert a name correctly in a sufficiently vague discourse without knowing more than auxiliary descriptions, even when they are wrong, as far as they are convergent (rightly classified) making in this way a parasitical reference, and this can be helpful in several ways. Nonetheless, they can only be meaningfully when applied in the dependence of the existence of applicable fundamental descriptions. It matters very little, for example, if Hitler was not the lover of Eva Braun, or if his father didn’t send him to the Realschule. But the localising and characterising descriptions are fundamental in the sense that they cannot be both completely inapplicable. If someone tells us about a person called Adolph Hitler, who was born in Mittelweg and who lived in Germany from 1634 to 1689, working as a shuhemaker and having nothing to do with politics, we would say that he isn’t the person we in the usual political-historical contexts meant with the name ‘Adolph Hitler’; it must be an homonymous, since this shuhemaker does not satisfy anything from the fundamental descriptions.
   Finally, we can try to give a Russelian formulation for an application of MD-rule. Calling the predicate ‘…satisfies its localizing condition more than any other object’ = L and the predicate ‘…satisfies its characterizing condition more than any other object’ = C, we can say that a proper name N has a bearer if and only if regarding the nearest relevant class of objects:

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

   This is not a very satisfactory formal replacement the MD-rule, since it assumes the condition of sufficiency and already takes for granted the existence of only one object that satisfies the rule, which is derived from the condition of predominance. But it is enough to show that from our perspective the existence of the object referred to by a proper name can be also seen as a property of concepts.
   The existence of an object referred to by a proper name is the effective applicability of what can be called the identification rule of the proper name in its (proper) domain, which is derived by the replacement of the conceptual expressions L and/or C by the fundamental description-rules that we associate with the proper name that we are considering. So, Adolph Hitler is the object belonging to the class of human beings that in a possible world Wx satisfies sufficiently and more than any other human being the conditions expressed by the localizing description-rule of being the Austrian-Born person who lived from 20 April 1889 to 30 April 1945… and the characterizing description-rule of being the leader of the Nazi Party and German dictator from 1933 to 1945, leading Germany to the World War II and being responsible for the Holocaust…
   Since so understood the rule of identification establishes the criteria of identification for the bearer of the proper name in any possible world, it also applies in any possible world where the bearer of the name exists, satisfying the main Kripkian definition of what is a rigid designator; the definite descriptions belonging to the cluster, on the other hand, being only loosely associated with the identification rule, can refer to other objects in different possible worlds being therefore accidental designators.
   An important feature of the identification rule of a proper name is that it picks out the same object by means of many possibly different s-properties as far as they are able to satisfy sufficiently the criteria imposed by the rule. So, the identification rule for Aristotle applies in a possible world where he lived five hundert years later in Rom, but wrote the Aristotelian opus; it applies to Aristotle in a possible world where he died as a child and never wrote the Aristotelian opus; it applies to the first twin in a world where there are two twin Aristotle, but where only one of them wrote the Aristotelian opus, and so on. The criterial configurations vary, as much as the s-properties that satisfy them. In this sense there is no ‘essential property’, but an ‘essential rule’ roughly corresponding to Locke’s nominal essence – a rule that allows the same object to be given to us under an indetermined variety of modes of presentation.
   As I have explained in the Appendix of Chapter 1 on proper names, when taken in isolation descriptions associated with a proper name, particularly the auxiliary ones, are not criteria making the application of the name necessary. They are mere symptoms, making its application more or less probable. So, ‘the lover of Eva Braun’ makes the application of the proper name ‘Hitler’ more probable, since it is only a symptom – and this explains the accidentality of such associated descriptions. There are possible worlds where Hitler existed without being the lover of Eva Braun. And without the previous application of at least part of the fundamental descriptions these auxiliary descriptions are of no worth.
   Finally, as far as the identification rule for the proper name ‘Hitler’, being a conceptual rule, has the second order property of has been effectively applicable to its proper domain, the domain of the objects belonging to the external world, we will can say that the bearer of this name really existed. Because the identification rule derived from the MD-rule applies in any world in which the referent exists, it makes the proper name a rigid designator.

Existence of spatio-temporal locations by indexicals
Finally, there is the problem of the application of the proposed analyse of existence to indexicals, a problem that I will consider here only very briefly. Consider simple statements with indexicals as ‘There is a raven’, ‘Here is cold’, ‘Tomorrow will rain’, ‘I am tired’, ‘I am here now’... What these indexicals do is at least to establish the existence of some spatio-temporal location in relation to the speaker. There is more than this regarding a pure indexical like ‘I’, which covers more than ‘the shire spatio-temporal location of the emissor of the word ‘I’ when this word is spoken’, but this does not mind to us now.
   Consider now the statement ‘There is a raven’. First there is the spatio-temporal location gestually indicated by the speaker, which is relative to the spatio-temporal location of the speaker within a context. The existence of all this must be assumed in the application of the demonstrative ‘there’. The identification rule for ‘there’ can be at least summarized in the descrition ‘the spatio-temporal location pointed out by the speaker when he utters the word.’ Now, what we call the existence of the spatio-temporal location indicated by the demonstrative ‘there’ must be the effective applicability of the identification rule of location in the domain expressed by the utterance. Once we have applied this identification rule, we are allowed to apply the ascriptive rule of the predicate ‘…is a raven’ to the existing location.  Now, how would we symbolise this? Calling the pointed out spatio-temporal location L and the predicate ‘…is a raven’ R, a way to state it symbolically could be simply ‘Ǝx (Lx & Rx) & (y) [(Ly & Ry) → (y = x)]’, what means: ‘There is precisely one x spatio-temporaly located in L that is R.’
   When we use an indexical statement the language, so to speak, touches the world. Although the sense of the concept still determines (bestimmt) its reference, there is here a double direction of fit: For here the rabe is what determines what conceptual ascription rule will be applied to it. The same is true regarding the sense of the indexical term ‘there’: the location is what determines what fortuitous identification rule we attach to the demonstrative in order to determine the spatio-temporal location of the object.

Advantages of the higher order view of existence
There are several advantages of conceiving existence as a higher order property. One is that it gives a straightforward answer to what seems odd in the traditional forms of the ontological proof of God’s existence. So, according to Descartes, if (1) God is the being with all perfections and (2) existence is a perfection, we can conclude (3) that God exists. But since existence isn’t a property of objects, differing in this way from perfections like infinite goodness and omniscience, which should be property of objects, the ontological proof is doomed to failure (Frege 1988, sec. 53).
   Nonetheless, the great advantage of conceiving the existence as a higher order property is that we do not find problems with the denial of existence. Suppose that existence were a first order property of an object. In this case, in a sentence like ‘Vulcan does not exist’, the negation of existence should be applied to the object itself, and we would first have to identify the object in order to deny that it has the property of existence. But since in order to identify the object we need to admit that it exists, we would be caught in a contradiction. That is, we would have to admit the existence of Vulcan in order to deny its existence. However, according to a Fregean perspective, this conclusion isn’t necessary, because all that we do by denying the existence of Vulcan is to admit that the ascription rule that builds the concept Vulcan doesn’t has the metaproperty of being effectively applicable in its proper domain of physical objects. Using the devices of the theory of descriptions, we could analyse the sentence ‘Vulcan does not exist’ as a shorthand way of saying (only to illustrate our point):

x (x is the planet that has an orbit between Mercury and the Sun) & (y) (if y is a planet with an orbit between Mercury and the Sun, then y = x).

   What falls under the scope ‘~Ǝx’ are the concepts constitutive of the identification rule, which in our illustration consists of an ascription rule for a predicate that must be applied to only one and the same object. What ‘~Ǝx’ does is just to deny the applicability of this rule to a physical object.[16]
   The understanding of existence as the applicability of conceptual rules allows us to explain the nearly unlimited extensions in the application of this concept. Why, given that existence is primarily attributed to properties and objects of the outside world or psychological states, are we also allowed to say that imaginary objects exist? Some believe that contradictory objects exist. We can even say that everything exists, meaning by this that all that can be conceived does exist, at least as something that can be conceived. And even of existence itself can it be said that it exists. Indeed, it seems that in one way or another everything exists. How can this be possible?
   Concerning imaginary entities there are two mains kinds: hypothetical entities that are improbable or that experience has shown not to exist and purelly fictional entities. Beginning with the first ones, it is clear that we can find a sense in which they exist. Although the planet Vulcan has show not to exist in the real external world, its proper domain, he surely existed in other domains as in the mind of many astronomers of the past who have searched for it; and it still exists in our minds as an imaginary object. This is no problem for the proposed view because our identification rules can also be applicable, partially or completely, only in the dependent domain of conceivable things that we consider as possible or even plausible candidates for the existence in the external world. If I imagine the hypothetical planet Vulcan orbiting the Sun, I apply the identification rule for that proper name in my imagination (even if in a vague and precarious way). Indeed, the French scientist Le Verrier, who named the planet, had even a localizing rule according to which Vulcan should be a small planet orbiting close to the Sun at a distance of 21 million km, which he mathematically calculed in order to explain by means of Newtonian mechanics the perihelion precession of Mercury’s orbit. This rule he applied in the domain of what is conceivable, what means that Vulcan existed in the domain of the imagination of Le Verrier and other astronomes of his time, though not in its proper domain, that of a concrete object belonging to the external world.
   Consider now the case of purelly fictional entities. Dorothy, the personage of Frank Baum’s tale The Wizard of Oz never existed in the real world; but she still exists in the small fictional world created in this story; this is its proper domain, which is from start fictional. In this domain Dorothy exists because we have her identification rule as being an eight-years-old girl whose house was blown away by a tornado and who found three friends, the cowardly lion, the scarecrow and the tin man… This rule demonstrates itself effectively applicable within its proper fictional domain of entities, which is described in the story. Moreover, the existence in a fictional world excludes the existence in the real world, so that ‘Dorothy was saved by the Wizard of Oz’ is true in the fictional domain of the story, but is from the start false in the domain of the real external world, since in our real world the Dorothy of Frank Baum does not exist.
   Kripke gave as example the case of fictional fictional characters like Gonzago (2013, p. 250), who is a personage that appears in Shakespeares’ piece Hamlet as a fictional character created by the personage Hamlet in his piece ‘The Murder of Gonzago’, which is a piece within the piece. We may say that Gonzago exists in a third order domain, requiring the application of its identification rule added with the identification rule of the existence of the plot of the fictional piece Hamlet. Moreover, we apply the higher order predicate of existence in all these cases only in dependent secondary (or terciary way in the case of Gonzago), assuming that we first apply terms referring to the real external world, which includes, for instance, the author of the piece, who are said to exist. As with other fictional entities like Pegasus and Unicorns, the existence is here affirmed within a domain that is dependent, derivative or extended, presupposing the basic form of existence, which concerns the effective applicability of cognitive-rules in the domain of the real external (physical) or internal (psychological) world, a point also in my view correctly made by Kripke (Kripke 2013, p. 81).
   What about the attribution of existence to contradictory imaginative conceptions like ‘the round square’? This case is really too hard to cauen. We cannot combine the rule of identification of the square with the rule of identification of a round thing, so that both can identify only one and the same thing. This we cannot do even in the imagination. Because of this impossibility, we must recognise that a round square in no reasonable way exist: we cannot make any application of a contradictory combination of rules. Since the conceptual ascription rules are what constitute their epistemic meanings, this conclusion agrees with our strongest intuition: contradictions do not exist, since they lack epistemic meaning.[17]
   Finally, what about existence? Can we say that existence itself exists? Surely, we know that existence exists in the sense that we know that the concept-word ‘existence’ is effectively applicable to the effective applicability of conceptual rules to the most diverse domains, telling us that the property of effective applicability exists. Existence exists in the sense that we can build a meta-metarule of existence, whose criterion of application is the effective applicability of meta-conceptual rules for the attribution of existence; since this meta-conceptual rules of existence are effectively applicable (since the entities belonging to their varied domains exist), the meta-metarule – which demands the applicability of metarules to their conceptual rules – also applies. Consequently, it is safe to conclude that existence itself exists. Moreover, exists also the existence of the existence? Surelly, since we can conceive a meta-meta-metarule of existence demanding effective applicability of the meta-metarule of existence to metarules of existences, we can conclude that the meta-meta-metarule of existence is also applicable to the foregoing meta-metarule, being consequently existent, and so in an infinite regress, which is virtuous since stoppable.

Answering objections
According to many present authors, existence is a first order predicate. A statement like ‘Horses exist’ should be analysed in a form similar to ‘Horses are animals.’ Since they have developed objections against the traditional view of philosophers like Frege and Russell, I will answer here at least some of them as they were put forward by Collin McGinn (McGinn 2000, pp. 21-30). The answers are helpful in clarifying my own view.  
   The first one is againt Russell’s suggestion that to say that something exist is to say that a propositional function (property, concept) is true for at least one instance. Put roughly the objection is that for one object to instantiate a property this object already needs to exist, an admission that would make Russell’s view circular, since it must already presuppose the existence of objects instantiating the property. For example, if ‘Mars is a planet’ is true, it presuposes the existence of the planet Mars to instantiate the property expressed by ‘…is a planet’ in order to make the sentence true. Sumarizing: there must already be existent objects to instantiate the properties ascribed to them by our conceptual words.
   This objection works as far one sustains the Kripkian view of the objects bearing proper names, since they are not defined by their properties. Since we have already analysed objects as bundles of s-properties, the objection must appears to us in a different form. Since not only ascribing rules of predicative terms, but also identification rules of singular terms, are for us conceptual rules, our position should be generally stated as saying that existence is the effective applicability of semantic-cognitive rules in some choosen domain. But this means that we cannot conceive any object being given – that is, existing – without its identification rule being effectively applicable to it. So, the existence of a concrete object like the planet Mars is the effective applicability of its identification rule in its proper domain of physical objects. But if it is so we cannot separate the existence of the object in its proper domain from the effective applicability of its identification rule in its domain, since it is this rule that establish what the object must be in its domain in any possible world. Consequently, the flaw in the objection is to suppose that we can separate the giveness or existence of the object from what McGinn calls the instantiation of its properties, since these are nothing but internal or external s-properties of the object.
   The second objection is that uninstantiated properties are said to exist. But in order to exist the uninstantiated property must fall under a higher order property attributing its existence. This higher order property must also exist, what means that it must fall into a still higher property and so infinitely. If there were no infinite regress of properties, the first property could not be said to exist, from this resulting the impossibility of attributing existence to anything.
   My answer is that I agree with the diagnostic but not with the prognostic. The effective applicability of a semantic-cognitive (conceptual) rule in its proper domain not only makes its reference existent, but is in itself a second order property that can be also said to exist. And even some semantic-cognitive rule that is only imaginatively applicable not only makes its reference existent in an imaginary domain, but can be said to exist by having the second order property of being applicable. Moreover, in any case the property of existence exists, what means that we can say that the applicability of the rule is applicable and so indefinitely. This leads us to an infinite regress, of course. But this isn’t a vicious infinite regress, but a virtuous one, since we don’t need to consider all the ilimited applicabilities of applicatilities in order to admit that some higher order rule is applicable, once its applicability is experienced. The mark of a virtuous regress is that we may stop it without loss when we feel that we do not need further cases, and this is just the what we have here (it is like Plato’s idea of his ideas: it doesn’t require the idea of the idea of his ideas and so successively in order to exist).
   The third objection is that there are statements that resist the traditional paraphrase. First there are statements ascribing existence to particulars, as ‘Venus exist.’ This objection we have already answered in our treatment of proper names and it makes sense only to non-descriptivists like McGinn. But there are others. Consider the statement ‘Something exists’. Although this is a true statement, McGinn thinks that it is not paraphraseable in terms of the higher order view, since there is no property to be instantiated here and if we try to translate in the standard form we get the gibberish ‘Ǝx(…x).’
   The answer is the following. What ‘Something exists’ means is that there is at least one s-property or construction of s-properties that exists, without that we determine what property is there. In other words, there is some semantic-conceptual rule that is applicable to some domain of entities, though this rule isn’t specified. It is usual in logic to symbolize an undetermined property as F. Hence, if we translate this symbolically we get Ǝx(Fx) and not Ǝx(…x). But there is nothing wrong with Ǝx(Fx). Often we come to a similar result aplying existential quantification to singular sentences like ‘Venus exists’: Calling Venus V, if it is true that ‘Ǝx(Vx)’, this imply that some property exists or ‘Ǝx(Fx), namely, that some conceptual rule is effectively applicable. There is no mistery in accepting the existence of undetermined properties.
   There are also complicated statements that seems to resist the higher order understanding of existence like:

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

But they can be solved as far as we do not confuse the domains of application of the semantic-cognitive rules involved. So, (1) can mean that there are some cities that exist only in the domain of imaginary, which means that certain identification rules for cities like Chloe or Valdrada are effectively applicable only in the fictional domain of the book The Invisible Cities of Italo Calvino; (2) can mean that there are things that exist only in your imagination, but not in the external world,  i.e., there are identification rules that are effectively applicable only in the domain of your talk, and (3) means that there are things that may exist only in the mind but not in the external reality, that is, the identification rule of some objects is effectively applicable in a imaginary fictional domain but not in the the domain of external reality.
   The last of McGinn’s objection is that according to the higher order view nothing can exist without falling under some property or other, what rules out the existence of a thing that has no properties – a ‘bare existent’. However, our empiricist commitments makes us see this not as a weakness, but as a further anti-metaphysical advantadge of the higher order view.

Reference of concepts again (a metaphysical excurse)
It is instructive to consider what happens when we compare the famous phenomenalist view of J. S. Mill, according to which ‘matter’ or ‘substance’ is nothing more than ‘permanent possibilities of sensation’ with our view of existence in terms of the effective applicability of conceptual rules. The results will be no less speculative than Mill’s phenomenalism itself, but they may be telling.
   Mill’s main epistemological question was: if all that is experientially given to us are sensory phenomena, how can we justify our belief in the existence of an external world, an objective world constituted by substance or matter? – An external world that can exist even when there is no observer at all to perceive it?
   Mill’s answer to this question was a development of Berkeley’s unofficial view, according to which things that exist when they are not perceived by us are nothing more than things that we are aware that would be perceived by us under suitable circumstances.[18] As Berkeley writes:

The table I write on, I say, exists, that is, I see and feel it; and if I were out of my study I should say it existed – meaning thereby that if I was in my study I might perceive it, or that some other spirit actually does perceive it. (Berkeley 1975, I, sec 3)

   According to this view, esse is not only percipi, but also percipi possi. In a more explicit manner, what Mill suggests is that:

Matter or substance is not made up of actual sensations, but of groups of permanent (or guaranteed or certified) possibilities of sensation.

   Mill justifies his identification of matter or substance with permanent possibilities of sensation in the following way. First, these possibilities of sensation are conditional certainties: they are not mere epistemic possibilities, but firm conditional expectations that are or could be based on experience. They are permanent in the sense that given suitable circumstances they would always be experienced, insofar as they exist. And they are guaranteed or certified in the sense that we have reasons – observational or not – to have a firm expectation that under the right circumstances their will be experienced again. This does not mean that the groups of permanent possibilities of sensations would depend, for their existence, on our past experience of them because if it were so, they could not exist without us as subjects of knowledge, and we would fall like Berkeley into a radical form of idealism (immaterialism). This is not Mill’s intention. As he explains:

We mean [by permanent possibilities of sensation]… something which exists when we are not thinking of it; which existed before we have ever thought of it, and would exist if we were annihilated; and further that things exist that we never saw, touched or otherwise perceived, and things which never have been perceived by man. (Mill 1979, X. pp. 178-177)

Thus, it is clear that Mill wish to avoid idealism: the permanent possibilities of sensations would exist even if cognitive beings able to perceive them have never existed.
   These permanent possibilities are for Mill objective, differing from our actual passing sensations, which are subjective. These possibilities are objective because they are grounded, he thinks, in our common public world, which means that we are able to intersubjectively agree on their existence. Even if different persons cannot have access to the same sensations they can have access to the same possibilities of sensation… As he writes:

The permanent possibilities are common to us and to our fellow creatures, the actual sensations are not… The world of possible sensations succeeding one another according to laws is as much in other beings as it is in me; it has therefore an existence outside me; it is an external world. (Mill 1979, X, pp. 181-2)

This is in summary Mill’s view on the nature of matter – a view that always seemed to me both: deeply suggestive and controversial.
   Nonetheles, I think that there is a serious confusion in Mill’s view, which can be made clear when we compare his remarks with those of Berkeley. According to the non-official Berkeleyian view, the external world is constituted by sensations whose experience is continually (permanently) possible for us, even if we are not there to experience them. But if this is so, the material objects constituting the external world cannot be reduced to simple ‘groups of permanent possibilities of sensation’, for the possibility as such, permanent or not, cannot be qualitatively distinguished one from another possibility, in the same form as a material object from another. Possibilities are always the same thing, namely, mere possibilities, while the things constituting the external world, whether they share some permanent possibility of being experienced or not, are multiple and varied. Holding this in mind, the correct way to express the Berkeleyian insight in Mill’s terms must be the following:

Material objects (or substances) are nothing but groups of sensations whose effective experience is permanently (or guaranteed or certified as) possible.

   This would satisfy the required multiplicity and diversity, because each material object would be constituted by a different group of sensations that could be always possibly experienced. But if the permanent possibility is not the material object, what is it?
   The answer emerges when we consider Mill’s view in the light of our reconstruction of Frege’s concept of existence, according to which existence is the effective applicability of a conceptual rule. If this is so, why would the effective possibility of sensations not be approximated to what we have called existence? Why could we not say that the expression ‘groups of permanent possibilities of sensations’ means the same thing as ‘groups of effectively experienceable sensations’, which means something like ‘groups of existing sensations’?
   My point is made clearer when we consider the general structure of our conceptual rules of ascription and identification. Assuming that existence is the effective applicability of a semantic-cognitive rule, what form they have? The answer is: It is a conceptual rule that brings us to some (usually pre-reflexive) cognition given by the satisfaction of variable criterial configurations by s-properties. Moreover, these variable criterial configurations can be groups of sensations. Consequently, when we speak of existence as the effective applicability of groups of sensations, we are indirectly speaking of the effective applicability of conceptual rules as a result of the satisfaction of their criterial configurations.
   For example, to be applied to a real object, the conceptual rule for the concept chair demands as a criterial configuration the satisfaction of the s-proprieties that what is observed is a seat with a backrest made for only one person to sit on. Surely, these criterial configurations can be satisfied by the sensory-perception in variable ways, but they always end up in the requirement of groups of sensations, like those of colours, forms, hardness and movements. Now, the requirement that the conceptual rule should be effectively applicable implies the effective possibility that its criterial configurations are satisfied, namely, that the configurations of s-properties that satisfy them are always possibly given to experience, configurations that can appear to us as groups of sensations. And this means that the effective or permanent possibility of having groups of sensations implies nothing but the existence of the object, since it implies the effective applicability of the semantic-cognitive rule – that groups of the sensory configurations that we have when we experience the object are guaranteed as effectively, permanently actualisable whenever the appropriate conditions for its experience are given.
   Now, Mill’s insights can help us to deepen our reconstruction of the Fregean concept of existence. An empirical object exists not only when its conceptual rule is effectively applicable, but also when s-properties for the application of this rule can be objectively given to us at least in the form of groups of contents of sensations whose experienceability is permanently possible. Moreover, this experienceability must be (at least in principle) intersubjectively accessible. It can be more or less directly given, it is (usually) independent of our will, and is also experienced as following causal laws regarded as typical of things belonging to the external world. We can say that all these things together build the condition of an effective application of a semantic-cognitive rule in the domain of the external world.
   Finally, if existence can be approached to ‘permanent possibility of sensations’, so that existence is the effective (and permanent) applicability of the conceptual rule that includes such possibilities of sensations, then what is matter? What is, in the modern language, the existing external material object? One too daring answer would be: the object, as it is thought by us, must be the identification rule in itself, as far as it is applicable. But this cannot be, since any semantic-cognitive rule is something essentialy mental. However, we have at hand a less daring answer: The material object isn’t the semantic-cognitive rule, but is supposed to have the same structure of the rule projected into the external, intersubjective world. The reason for this supposition is clear enough: only something with a structure similar to its semantic-cognitive rule would be able to give unity to the inumerous aspects, facets, that is, to the multiplicity of variable criterial configurations throught which material objects are able to give themselves to us in our experience of them. Figuratively speaking, if the semantic-cognitive rule has the form of a tree, whose branches end up in criterial conditions internal to the rule, the object, as it is understood by us – and not necessarily as it is in itself – must have the structure of an inverted specular tree, able to form the objective criterial configurations that satisfy the criterial conditions, namely, the corresponding s-properties or constructions out of them. Of course, this objective structure is putative, the rule can be corrected and improved after new information and so also its specular objectual counterpart.
   There is, finally, an important and seemingly fatal objection to Mill’s view of matter, which is made more acute by the corrections we have made. It is that the group of sensations or configurations of sensory criteria that satisfy a conceptual rule are by definition psychological. Even sensations that are warranted as permanently possible (sensibilia) must be psychological in a dispositional way. This means that if we follow this path we fall into some kind of idealism in which we have no external objectivity wold to be contrasted with our subjective world of sensations or sensory criteria, no real s-property to match the criterial conditions. It is true that these possible sensations are given independently of our will and follow the regularities of nature, that they seem to be intersubjectively accessible under circunstances that warrant their experienciability, but it seems that they remain nonetheless belonging to the subjective side of experience. This is a pressing objection, whose answer will be given only in the final chapter of this book, as a consequence of our discussion of the correspondence theory of truth in its relation with the direct realism.

The reference of a sentence as its truth-value
Now we let the realm of speculation and come back to Frege’s semantics thematizing the reference of the sentence (Satz). For Frege the reference of sentences is their truth-values. How comes he to this strange result? First, he noted that sentences without reference lack truth-value. Moreover, according to the principle of compositionality the reference must be what remains unchanged after we change the senses of the components of the sentence without changing their references. This happens when we replace ‘The Evening Star is illuminated by the Sun’ with ‘The Morning Star is illuminated by the Sun’. Here the references of the sentence-components do not change. Hence, the reference of the whole sentence should also not change. But what hasn’t also changed? Frege’s reply is: their truth-value. Both sentences remain true. Indeed, as he emphazises, it is the search for truth what brings us from sense to reference. Hence, he concludes that in extensional languages the references of sentences must be their truth-value.
  A first consequence of this conclusion is that for Frege all true sentences have only one reference, which is the abstract object The True (das Wahre), while all false sentences have only one reference, which is the abstract object The False (das Falsche). Independently of this argument interpreters have noted that Frege has choosen the truth-value as the reference of sentences because it is what gives them value to us, because of its Bedeutubg in the sense of semantic relevance or meaningfulness.[19] Indeed, truth-value is of decisive importance for logic because it is what must be preserved in valid arguments; the logician does not need to know more than the truth-value about the referring function of the sentences he is dealing with (e.g., Tugendhat 1992b).
   Nonetheless, in spite of any theoretical advantage that this suggestion can provide to the logician, it remains from the point of view of what we understand by ‘reference’ utterly implausible. Trying to refute the accusation of implausibility, the Fregean philosopher Alonzo Church mounted a slingshot argument, attempting to show that by means of intersubstitutability we can prove that the reference of the most diverse sentences must be only one, namely, their truth-value. The assumption of the argument is that if a constituent expression is replaced by another with the same partial reference, the reference of the whole sentence does not change. I will expose his example of a slingshot argument underlining its supposedly co-referential definite descriptions (Church 1956, p. 25):

1.      Sir Walter Scott was the author of Waverley.
2.      Sir Walter Scott is the man who wrote the 29 Waverley novels altogether.
3.      29 is the number such that Sir Walter Scott is the man who wrote that many Waverley novels altogether.
4.      29 is the number of counties in Utah.

   Assuming it to be plausible that sentences (2) and (3) are, if not synonymous, at least co-referential sentences, then (1) has the same reference as (4). As (1) concerns a fact completely different from (4), it seems that the only thing left as a reference is the truth of both sentences.
   However, in my view the argument proves to be unsustainable when we consider what should be the true logic subjects of these sentences. All sentences are identity sentences. The first one identifies the singular terms ‘Sir Walter Scott’ and ‘the author of Waverley’. The second sentence identifies ‘Sir Walter Scott’ and ‘the man who wrote the 29 Waverleys novels altogether’, both referring to the same object. The third sentence is the tricky one:  is there a clearer way to paraphrase the confusive sentence (3) ‘29 is the number such that Sir Walter Scott is the man who wrote that many Waverley novels altogether’? The only way to expose its informative content clearly without any loss of sense is to split it into the following conjunction of two sentences: (5) ’29 is the number of Waverleys novels and Sir Walter Scott is the man who wrote that many Waverleys novels altogether.’ Or, for the sake o clarity, replacing ‘is (the same as)’ for ‘=’:

(5)     ‘(29 = the number of Waverleys novels) & (Walter Scott = the man who wrote the many Waverley novels altogether).’

What we now have is a conjunction of two identity sentences, each with ist own partial reference. Finally, we come to the analysis of the sentence (4): ’29 is that number of counties in Utah’, what means the same as the identity sentence (6) ‘29 = The number of counties in Utah’. So analysed the argument appears as:

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

Although all these sentences are true, the claim that the reference remains the same now loses its intuitive appeal. Sentences (1) and (2) have as objects of reference Sir Walter Scott, under different guises. But (3) is a conjunction of two sentences with two different references: the object refered by the first is the number 29 (as the number of Waterley’s novels), while the object refered by second is Sir Walter Scott (as the man who wrote the Waverley’s novels). And (4) has as object of reference only the number 29 (as the number of counties in Utah) without referring to Scott. This means that sentence (3) has as its second object of reference the same object of (1) and (2), while (4) has as its object of reference only the the first object of reference of (3). Consequently, the replacements slips into equivocity by changing the object of reference from Sir Walter Scott to the number 29. Initially this flaw isn’t easy to note because the sentence (3) contains both objects of reference conjoined in a grammatically confusive way. The replacements would only be allowed if the objects of reference could remain the same in all the sentences, because only in this case we could say that the whole reference of each of these sentences remains the same.
   Not only can the slingshot argument be debunked, but there are a number of well-known embarrassing objections to Frege’s identification between reference and truth-value. A first one is that there are substitutibility problems with Frege’s view on the reference of sentences. If all true sentences refer to an object called ‘the truth’ and the name ‘the truth’ also refers to the truth, then in the conditional sentence ‘If it rains, then water falls from the sky’, we can replace ‘it rains’ with ‘the true’, getting the sentence ‘If the truth, then water falls from the sky’, which should be true, even though it is in fact unintelligible (Black 1954, pp. 235-6). Another difficulty is that many obviously false identities should be true. For example, ‘Paris is a city = snow is white’ should be a true assertive sentence, since both partial sentences refer to the same thing: The Truth. Moreover, contrary to any healthy intuition, Frege’s proposal frontally contradicts the meaning we normaly give to the word ‘reference’. It seems clear that the sentence ‘Napoleon was born on Corsica’ refers to domething very different from the sentence ‘2 + 2 = 4’, even if both are true. Furthermore, it should be expected that the reference of the components of our sentences should belong to the same ontological level as their references. But this isn’t the case: the reference of the name ‘Napoleon’ is the Napoleon of flesh and blood, while the reference of the sentence ‘Napoleon was born on Corsica’ must be the abstract thing called The Truth. Finally, one could argue that from the perspective of Fregean semantics, his solution sounds false because it in fact violates a principle of compositionality, whereby the whole depends on its parts, so that a change in a part produces a change in the whole. If the reference of a sentence is its truth-value, it cannot be established by its parts, since the truth-value is simple and has no part. The components of the sentence, however, have their own indefinitely variable references.
   I conclude that the most charitable interpretation consists in saying that Frege is here using the word ‘reference’ in a new, technical sense, though recognising that this way to use the word is dangerously misleading, since it has virtually nothing to do with what can be reasonably understood with the word.

The reference of the sentence as a fact
The Fregean account of the references of sentences as their truth-values turns out to be still less acceptable when we consider that a much more natural alternative is available, which, as Anthony Kenny has noted, was not even mentioned by Frege (Kenny 2000, p. 133). We can, as Wittgenstein, Russell and others did, suppose that the reference of a statement is always a fact, generally understood as a contingent arrangements of elements given in the world. Facts would satisfy the Fregean condition that the reference of a sentence is an object: they are independent, complete, closed. Moreover, facts would satisfy the principle of compositionality: they could always vary in accordance with the variations in the references of the component parts of the sentences. Finally, as we will see, in a way they could vary with the unlimited possible variations in the epistemic values of different whole sentences.
   If we assume this answer, questions arise. The first is the following: how do we establish what fact the thought expressed by a sentence refers to? Consider the following sentences:

1.     The Morning Star is the Morning Star.
2.     The Morning Star is the Evening Star.
3.     The Morning Star is Venus.
4.     Venus is the most brilliant planet visible in the sky.
5.     Venus is the second planet orbiting the sun.
6.     Venus is the only planet in our solar system shrouded by an opaque layer of highly reflective sulphuric acid clouds.

From the one hand, it seems clear that each of these sentences refers to a different fact. Sentence (1) is an empty tautology, while sentence (2) is informative; and the information conveyed by the sentences that follows are all different. However, since all singular terms composing these identity sentences have the same reference, it seems that in the end all these sentences must also have the same reference, pointing to the same fact.
   The question arises: is there a privileged grounding fact able to be described that can be identified as the same truth-maker of all the identity sentences about the planet Venus, including in some way the facts stated by the different epistemic values of the sentences (1) to (6) above as its sub-facts? My suggestion is that this task can be accomplished by the references of identity sentences between proper names. Assuming our proposed view of proper names as abbreviations of clusters of descriptions as essentially correct, then the proper name ‘Venus’ ideally (i.e., abstracting the limited knowledge of concrete speakers) includes in its content all the known modes of presentation of this word. This means that definite descriptions such as ‘the Morning Star’, ‘the Evening Star’, ‘the second planet orbiting the Sun’ and even ‘the only planet in our solar system shrouded by an opaque layer of highly reflective clouds of sulphuric acid’ can be at least made probable by the concept of Venus (I say made probable because most of them are symptoms and are not seen as necessarily applicable). Well, in that case there is indeed a sentence that could describe the grounding fact, which is the ultimate truth-maker of any identity sentence concerning the planet Venus, including the sentences from (1) to (6) above. Here it is:

7.     Venus = Venus.

My claim is that this sentence is able to express the grounding thought (in principle included in the others) able to refer to the single grounding fact, which if regarded in its entirety, is able to work as the truth-maker for any identity sentence about the planet. Indeed, if the proper name ‘Venus’ is understood as an abbreviation of a cluster of descriptions that uniquely identify our object, then this proper name replaces the descriptions ‘the Morning Star’, ‘the Evening Star’, ‘the second planet orbiting the Sun’, etc. Consequently, we can derive as probable from the sentence ‘Venus is Venus’ the sentence (2) ‘The Morning Star is the Evening Star’, simply by replacing the first occurrence of the name ‘Venus’ with the definite description ‘the Morning Star’, which the name ‘Venus’ abbreviates, and the second with the description ‘the Evening Star’, which the name ‘Venus’ also abbreviates. In a similar way we can easily get all the co-referential identities presented above. In this way the sentence ‘Venus is (the same as) Venus’ would be ideally able to represent a fact complex enough to contain the sub-facts that form the truth-makers of each one of the thoughts expressed by the above sentences. 
   In order to reinforce what I am suggesting, we can use instead numerical identities like the following:

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

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

6.     4 = 4

   But could the sub-fact expressed by sentence (5) be derived from (6)? The answer is obviously ‘yes’, since we are handling with a deductive system. I have even wrote the five sentences above conceiving deductive derivations from 4 = 4.
   In the case of empirical facts, being the derivation based on symptoms, the probability is variable. It is sure the fact that Venus is the second planet of the solar system. But it is somewhat less sure the fact that Venus is the only planet in our solar system shrouded by an opaque layer of highly reflective clouds of sulphuric acid. And it is still less sure the supposed fact that the core of Venus is partially liquid. Moreover, one could object that a sentence like ‘Venus is Venus’ is a tautology dispensing verification: a necessary truth. How could a necessary truth ground contingent truths like ‘Venus is the most brilliant planet visible in the sky’?
   My tentative answer to this objection is that for a privileged user of the word (the astronomer) who is supposed to master all pertinent information about Venus, this proper name expresses its identification rule, according to which it has been identified by astronomers during a definite period of time as the second planet to orbit the sun (localizing and main fundamental rule) and a planet somewhat smaller than the earth (characterizing rule). Without the sufficient and predominant application of this identification rule it would be difficult if not impossible to identify Venus. The application of many other descritiptions, however, do not make criteria, but symptoms of the existence of the planet, since they make the application of the descriptions only more or less probable. Contingent truths like ‘Venus is the most brilliant planet’ hang on these symptoms, in the case, the highly reflexive sulfuric acid clouds. If Venus loses its dense atmosphere it may cease to be the most brilliant planet but does not cease to be Venus.
   What I said about identity sentences applies also to others singular predicative and relational sentences. Consider the following predicative sentences:

1. Bucephalus is a material object.
2. Bucephalus is an animal.
3. Bucephalus is a horse.
4. Bucephalus is a black horse
5. Bucephalus is a black horse of the best Thessalian strain.

Each of these sentences refers to different sub-facts by means of more and more detailed modes of presentation expressed by their respective predicative expression. But the grounding fact is the same and it can be said to be more completely referred by the sentence 4, since the reference of all the others can be implied from this sentence. These sub-facts are all included in the fact represented by this sentence.
   A final point concerns the logical structure of facts (and sub-facts). The most plausible answer is that they have a structure that corresponds to the logical groud-structure of the thoughts representing them. Basic empirical statements as ‘Plato has a beard’ and ‘The cat is on the mat’ represent respectively facts that have the logical structure Fa and bRc. Their elements a, b and c, as particulars, are bundles of s-properties to be analysed in accordance with Peter Simons’ nuclear trope theory, while F and R would be analysed as s-properties themselves, possibly constituted by combinations of s-properties or tropes. The links b-R-c and F-a, on their side, are only pseudo-relational, since the admission of their existence as relational s-properties would generate a Bradleyan regress. The individuals and their s-properties should be seen as directly linked one with the other without ontological addition.

The ontological status of facts
If we accept that the references of sentence-senses or thoughts are facts, then from an ontological perspective the reference of empirical sentences must be empirical facts, often located in the external world. This assumption speaks for the correspondence theory of truth, according to which empirical facts are truth-makers seen as contingent arrangements of elements in the world. However, this assumption runs against Frege’s anti-correspondentialist view of truth. This is a reason why according to him a fact would be nothing more than a true thought (Frege 1918, p. 74) Following similar anti-correspondentialist lines, P. F. Strawson in an influential article suggested that empirical facts are mere ‘pseudo-material correlates of the statement as a whole’ and not something in the world (Strawson 2004, p. 151; Strawson 1998, p. 402). His most incisive argument was that empirical facts, in opposition to events or things, are not spatio-temporally localisable. One evidence of this is that the description of a fact usually begins with a that-clause, for instance (I can say ‘the fact that the book is on the table’, but not ‘the fact of a book on the table’), while the description of an event typically lacks a that-clause (I can say ‘the event of a tsunami in Japan’, but not very properly ‘the event that there was a Tsunami in Japan’). Moreover, in opposition to events or things, I cannot create, destruct, testimony, avoid, kick, repair, point at, see or hear a fact, the same being the case with states of affairs and situations.[20] Finally, to give a striking example, the event of Caesar’s crossing the Rubicon occurred in the year 47 BC, but the fact that he crossed the Rubicon did not occur in the year 47 BC; it is still a fact today, since facts simply do not occur (Patzig 1980, pp. 19-20).
   An efficacious way to dispose of this argument was proposed by the correspondentialist John Searle. For him we need a word to describe the thing in the world that makes our thoughts true. The word ‘fact’ is available. So, why do not use it estipulatively in order to designate the truth-maker, whatever is it? (Searle 1998).
   However, it seems clear to me that even this way to circumvent the problem is avoidable, since we can show that the problem exists only in the imagination of philosophers. To begin with, of course I’m not saying that everything we may call a ‘fact’ is objectively real. The fact that 2 + 2 = 4 is not said to be empirical. And we can say that it is a fact that the Sun is not green, though this seems only a queer way to say that the fact that it is not the case that rhe sun is green, namely, that the fact that the Sun is green does not exist. What I want to defend is that empirical facts, particularly regarding so-called observational facts, should be considered objectively real: they exist in the external world as the basic truth-makers.
   There is a first and relatively obvious argument to show that facts can be constituents of the empirical world. It is the property of many of them of acting causally. Consider the following sentences:

(1)   The fact that the match was scratched caused the flame.
(2)   Thomas died because of the fact that he forgot to turn off the gas.
(3)   Because of the fact that today is holiday, today there will be no class.
(4)   The fact that Cesar crossed the Rubicon had important historical consequences.

   It does not seem possible that a pseudo-material correlate (supposedly a kind of abstract content) can in itself act causally in the empirical world producing an effect. Obviously, the admission of the empirical nature of the facts (1) to (4) solves the problem: scratching the match is a fact-event causing the flame; the situational fact created by Thomas’ forgetfulness of the gas turned on caused his death; the fact-circumstance that today is holyday causes the absence of class today; the fact-event of crossing the Rubicon concretized a state of affairs that determined causally important political changes in the Roman empire.
   Furthermore, I have a key-argument to regenerate the idea that empirical facts are correlates of true thoughts, so that the empirical facts that we represent by means of affirmative sentences may be contingent arrangements of elements in the external world or in the world (external and/or internal) in general (supposedly, a combination of more or less complex s-properties and s-relations designated by predicates contingently linked with the bundles of s-properties referred to by proper names, using the word ‘link’ in Wittgenstein’s sense as the glides of a chain, what makes them devoid of ontological import. This would be the case with facts as simple as those referred to by the sentences ‘Frege wore a beard’ and ‘The book is on the table’.
   My argument against Strawson’s opposition between non-spatio-temporal facts and spatio-temporal events shows that there is confusion in his argument. He treats facts (as much as states of affairs and situations) as being opposed to events. But this can be contested. Any event can be called a fact, though not any fact can be called an event. For example: I can replace ‘the event of the sinking of the Titanic’ by ‘the fact of the sinking of the Titanic’, but I cannot replace ‘the fact that the Everest is more than 8.000 m. high’ by ‘the event that the Everest is more than 8,000 m. high’. Hence, it is much more reasonable to consider the event as a kind of fact than to oppose both as Strawson did. Indeed, my proposal is that the word ‘fact’ is an umbrella term that encompasses events, occurrences, processes, as much as situations, circumstances, states of affairs… And the reason for my proposal is that we can call all these things facts, but we cannot call all these things states of affairs or events or whatsoever. So considered, events are sub-types of facts: the word ‘event’ should be seen as a hyponym of the word ‘fact’. The reason for my proposal is that when we treat facts as arrangements of elements, we see that there are two great sub-classes of facts:

1.     STATIC FACTS: Can be formal or empirical, the latter when clearly located in space and time. Static facts do not change while they last. Typical of static facts is that the relationships between the elements constitutive of them do not decisively change during the period of their existence. In the ordinary language they may be called ‘states’, ‘situations’, ‘circumstances’, ‘states of affairs’, etc.
2.     DYNAMIC FACTS: Are always empirical. They change while they last. The relationships between the elements constitutive of them change decisively during the period of their existence, so that they have a beginning, followed by some kind of development that after a while comes to an end. Ordinarily they are called ‘events’, ‘occurrences’, ‘occasions’, ‘processes’, etc.

Formal facts, like the fact that 7 × 7 = 49, are static in the innocuous sense that they aren’t spatio-temporaly located. They are not our major concern here. Many facts are empirical and static, since the relationships between the elements constitutive of them do not change during their existence. Static facts are usually called ‘states’, ‘situations’, ‘conditions’, ‘circumstances’, ‘states of affairs’, with different nuances of meaning. For example, my unhealthy state, the situation that I am lying in bed, the circumstance that the airport is closed, the state of affairs that Venice is full of canals or that the earth orbits the sun (which counts as a static fact, since the element of revolving around the sun is a cyclic relationship that remains the same during the fact’s existence...).
   Dynamic facts, on the other hand, are defined by irrevocable changes in the relations among their elements during the period of their existence; the process of the World War II, for instance, was marked by events like the war of Britain, the defeat of Stalingrad and the invasion of Normandy – it had an imprevisible history. Dynamic facts are usually called events when their duration is comparatively shorter, occurrences when their duration isn’t as short, and processes when their duration is longer. Examples of events are a lightning flash under dark clouds, the explosion of a bomb. Example of an occurrence is the eruption of a volcano. The process of global warming is a very slow natural process, slower then the economic process of globalization. We can predict the stages of many events and processes, although many are also unpredictable in a way that (differently from static facts) we cannot have an entire grasp of them before their end. Important is to see that all these things can not only be called events, occurrences, occasions, happenings, processes… but also facts, since they are nothing but empirical facts and truth-makers of a dynamic kind.
   We are now able to find what seems to be the real reason why we use a that-clause in the description of facts but not in the description of events. When we speak of dynamic facts, we do not use a that-clause. So, we can speak about the event of Caesar’s crossing the Rubicon, but not about the event that he crossed the Rubicon. We can speak about the process of climate change, but not about the process that the climate changes… But this isn’t the case regarding static facts, which are typically (though not necessarily) described beginning with that-clauses. So, I can speak about the state of affairs that my book is on the table or that I am lying on the bed, but I can also speak about the state of affairs of my book being on the table and of my being lying on the bed. Conclusion: that-clauses are used to emphasise static facts and to eschew dynamic facts. And since the hyperonymic term ‘fact’ can be applied to both – the static facts as much as the dynamic facts – it seems plausible to think that the term ‘fact’ inherits the property of being used indifferently, with or without the that-clause: you can say ‘It is a fact that Mount Vesuvius is located near Naples’ (referring to a state of affairs), as much as ‘It is a fact that Mount Vesuvius has erupted’ (referring to an event). And you can say: ‘Cesar crossing the Rubicon was an event’ as much as ‘It is a fact that Cesar crossed the Rubicon’, referring less precisely to the event.
   Caesar’s crossing the Rubicon is in turn a very misleading statement: it is usually not understood as the description of an event, but as an illustrative way of referring to a static social fact: the state of affairs established by the movement of Caesar’s army onto Italian territory, violating the law that prohibited this and forcing the Roman state to declare war against him. Rarely is ‘crossing the Rubicon’ understood in the literal sense, as the physical event of crossing the river, which comprises Caesar’s locations in relation to the river from t1 to tn.
   Due to the nature of dynamic facts like events and processes, we say that they not only are, but also occur in time, while of static facts we say only that they are located in time while they last. It seems, therefore, that because philosophers such as Strawson did not realise that events are sub-types of facts, seeing only that we may say of events that they occur in time, they hastily concluded that only events are located in time, opposing them to timeless facts. But that this isn’t true can be shown even by inter-substitutivity salva veritate: it is not incorrect to say that the event, the occurrence of Caesar’s crossing the Rubicon was a fact and that this fact occurred in 47 BC, as a concrete dynamic fact. On the other hand, the political state of affairs established by the crossing of the river was far more durable, because it was a static fact, the political situation that produced, as is well-known, the fall of the Republic. However, it seems clear that the state of affairs brought about by the crossing of the Rubicon was spatially limited to the Roman Empire and temporally limited to the time from the crossing of the Rubicon to the empowerment of Cesar as a dictator until his assassination. It was not something that existed in Australia or that endures until the present, even if we have the habit to use the present tense to speak about historical facts. I conclude that, by having the broadest scope, the concept-word ‘fact’ remains the ideal candidate for the role of truth-maker in a correspondence theory of truth. In this sense facts are the universal truth-makers.

Sense of sentences: the thought
Now it is time to go on to the sense of the sentence. Here Frege scored well! He was lucky in suggesting that the meaning of the whole sentence is the thought (Gedanke) expressed by it. He came to this result by applying his principle of compositionality of senses, whereby the sense of a complex expression is formed by the senses of its component expressions combined in a certain way. If, for instance, in the sentence ‘The Morning Star is a planet’ we replace for the phrase ‘the Morning Star’ ‘the Evening Star’, which is co-referential, although having a different sense, the reference of the sentence does not change; but the sense of the sentence must change. And indeed, the sense of the sentence ‘The Evening Star is a planet’ is a different one. But what we have changed with the change of the sense of the sentence is the thought that the sentence expresses. Consequently the sense of a sentence must be the thought expressed by it.
   The word ‘thought’ is ambiguous. It can be used to describe a psychological process of thinking, as in the utterance ‘I was just thinking of you!’ But it also seems to designate something independent of specific mental occurrences – a content of thought – as the sentence ‘12 . 12 = 144’ in the utterance: ‘The sentence “12 . 12 = 144” expresses a true thought.’ Frege had the latter meaning of the word ‘thought’ in mind. In this usage, the word means simply what the sentence (or statement) says, which Frege has conceived of as an unchangeable platonic entity. The terminology here counts, because the word ‘thought’ is the only term in natural language that has one sense corresponding to more technical terms like ‘proposition’ and ‘propositional content’.[21]
   Frege has a criterion for deciding what belongs to a thought. For him everything that contributes to determining the truth-value of a sentence should belong to its thought. Thus, using his own example, the sentences ‘Alfred hasn’t arrived’ and ‘Alfred hasn’t arrived yet’ express the same thought, once the word ‘yet’ expresses only an expectation regarding the arrival of Alfred without contributing to its truth-value (Frege 1918, p. 64). On the other hand, the sentences ‘The Morning Star is Venus’ and ‘The Evening Star is Venus’ can be counted as expressing different thoughts: the singular terms that make up these two identity sentences all refer to the same planet, but by means of different modes of presentation, that is, by following different paths in the determination of their truth-value, or, as we could finally suggest, by following different combinations of semantic-cognitive rules producing verification procedures that are correspondingly different.

The thought as the truth-bearer
Another quite plausible Fregean thesis was that the bearer of truth is not the sentence, but the thought expressed by it. Although we can say that sentences, beliefs and even things and persons are true, they all seem to be true in a derived sense. According to this reasoning, when we say that a diamond is false, what we mean is only that it is inauthentic, deceiving us and making us having false thoughts about it: When we say that Socrates was ‘true’, what we mean is only that he was a person used to tell the truth, that he was reliable; when we say that Sam’s belief is true, we mean a subjective psychological attitude of the believer, even if well-grounded, and when we say that a sentence is true, this is only a derived way to say that the thought it expresses is true.
   One reason for preferring to say that the thought is the truth-bearer concerns the logic grammar of the concept. Our concept of truth works as a normative ideal so that the the truth-value should be of something invariant: if something is true, it is forever true; if something is false, it is forever false. We can fail to grasp the truth and grasp a falsity instead, and vice versa; so our holding something as true is fallible. But this does not affect the invariability or imutability of the truth as an ideal. Now, if it is so, the truth bearer must also be something invariable, able to remain the same in order to retain the same truth-value independently of the time of our grasping of it. And indeed, a true thought (if true) remains for ever true, as much as a false thought (if false) remains for ever false. They are even shorted out as ‘truths’ and ‘falsities’ respectively. It is something deeply ingrained in our conceptual grammar that the entity that can be primarily called true or false remains the same and with the same truth value, so that what may change is only our cognitive grasp of it, our believing in its truth-value (unseres für wahr halten). If it is so, then only the thought has the necessary stability to be the truth-bearer; for the thought is, according to Frege, unchangeable and eternal (atemporal), being eternally (atemporally) true or false in the independence of our grasping (fassen) of it.
   Consider now the sentences. Identical sentences-tokens can express different Fregean thoughts, but in this case the truth-values of the thought will accompany the thought and not the sentence… This is obvious in the case of indexical sentences like ‘I am in pain’, which express different thoughts depending of the speaker. These sentences can change their truth-value when uttered by different persons, and even by the same person in different times. If the sentence were the truth-bearer, the truth-value of the same sentence wouldn’t be able to change in this form. Moreover, if the sentence were the truth-bearer then the truth bearer could change without changing its truth-value. For example, the different sentences ‘Il pleut’, ‘Es regnet’, ‘Chove’, when uttered in the same context, remain with the same truth-value. Hence, the sentence does not have the kind of stability expected of a truth-bearer. Moreover, the only justification for the truth-value of these sentences remaining the same is that their bearer is the thought expressed by them, since the thought remains the same. And this is the case not only for indexical sentences, but also for identical eternal sentences expressed in different languages.[22] Concluding: the thought and the truth-values not only are invariants but have a relationship of co-variance that is missing in the relationship between sentences and truth-values. Because of this the proper bearer of truth must be the thought (or the proposition) and not the sentence.[23]
   As we already noted, Frege also suggested that what we call a fact is the the same as a true thought, because when a scientist discovers a true thought, he claims to have discovered a fact. As he writes:

‘Facts! Facts! Facts!’ exclaims the researcher of nature, when he wants to proclaim the need for a secure basis of science. What is a fact? A fact is a thought that is true. (Frege 1918, p. 74)

   However, this conclusion isn’t forceful, for a scientist can also say the same thing – and it seems with more property – understanding by a fact simply what corresponds to the true thought, namely, some real arrangements of elements in the world. After all, it is natural to think that if someone discovers a true thought, it is because a fortiori he has discovered the fact corresponding to it. Moreover, J. L. Austin has shown that the Fregean identification does not resist linguistic replacements, showing changes in meaning. If the sentence ‘What he affirms is true’ had the same sense as ‘What he affirms is a fact,’ then the replacement of ‘what he affirms’ to ‘his affirmation’ should be allowed without change in the sense (Austin 1990, pp. 170-171). But while ‘His affirmation is true’ preserves the meaning, ‘His affirmation is a fact’ is a metalinguistic sentence referring to the fact of an utterance of him, and not to the content of the affirmation anymore. The reason for this would be that even if the content of its affirmation (the Fregean thought) is true, it cannot be in itself a fact.
   The ultimate reason why Frege believed that the fact is a true thought seems to be that he advocated a conception of truth as redundancy, rejecting the correspondential theory. But not only his arguments against the correspondence theory (Frege 1918, pp. 59-60) are unconvincing (see Künne 2003, pp. 129-133), but the correspondence view remains the most natural and plausible conception of truth, suggesting that facts are combinations of elements in the world, able to be at least aspectually represented by their thoughts, which, when this happens, are called true. Moreover, the view of truth as correspondence is in conformity with our common sense methodological point of departure, and in the final chapter of this book I will endeavour to defend it.
   Finally, it is worth to note that a source of confusion between true thoughts and facts seems to reside in the ambiguity of our natural language. Although dictionaries present us a variety of meanings for the word ‘truth’, there are two pervasive meanings that are the most emphazised:

(a)  Truth as consisting in things being as we believe that they are, as conformity with reality: the correspondence to fact.
(b) Truth as the actual state of affairs: the real existing thing or fact.[24]

   In sense (a) we say that a thought is true, uttering sentences like ‘His words are true’, while in sense (b) we say that the fact in the world is true in the sense of being real, uttering sentences like ‘the mentioned occurrence was true (it was real)’, ‘we are searching for the truth (for the real facts).’
   For philosophers there are good reasons to think that the sense (b) of the world ‘truth’ is derivative, since in this case we can replace the word ‘truth’ by more adequate ones. So, instead to say that facts in the world are true we can say that they are real, they exist; instead of saying that the diamond is true we can say that it is authentic; and instead of saying that the person is true we can say that he or she is reliable. Hence, it seems that we use the word ‘true’ in such cases derivatively because the existing fact, the authentic diamond and the reliable person lead us to have true thoughts and to say true things. Now, since ‘truth’ can mean not only ‘correspondence with facts’ but also ‘an existing fact in the world’, it is easy to confuse the first and most proper sense of the word with the second derivative sense and believe – considering that facts and thoughts can be said to be true – that facts are true thoughts, as Frege seems to have done.

The thought as a verifying rule
If we analysed Frege’s senses of singular and general terms as semantic-cognitive rules, thoughts, as the senses of sentences, must be for us combinations of semantic rules. But if the thought is a combination of rules, then what results from such a combination must also have the character of a rule, even if it is not a previously conventionalised rule. Combining this with our acceptance of the correspondence view of truth this means that the thought is also a kind of rule, whose function is to make us aware of the fact referred to by it.[25]
   This reasoning leads us directly to the cursed word of our scientist world called ‘verificationism’, more precisely (and still worst) to semantic verificationism: the doctrine first proposed by Wittgenstein, according to which the (cognitive, informative) sense of a sentence is its rule or method or procedure of verification. This doctrine is the present days considered by many unsustainable, even though the received wisdom against it has never been very critically scrutinized. Indeed, I intend to provide the first serious criticism of this received wisdom as something corrupted by positivist-scientist formalist prejudices in the next chapter of this book, making plausible that there is nothing wrong with this doctrine except for its intrinsic philosophical difficulty.
   Anyway, I will introduce this view here speculatively, as an alternative and in fact as the most natural way to analyse what can be meant by Frege’s discovery of the thought as the sense or epistemic value of the sentence. Suppose that the semantic-cognitive rule that constitutes the thought expressed by an assertive sentence is its verifying rule. If we show that this rule is effectively applicable to a fact, it makes the thought-sense-rule true, and we may also derivatively say that the sentence expressing it is true. If, on the other hand, we show that this rule isn’t effectively applicable, it makes the thought-sense-rule false and we may derivatively say that the sentence expressing it is false. Finally, if we cannot build up a verifying rule able to be at least in principle aplicable, we must conclude that the sentence is devoid of meaning, that is, devoid of thought-sense-rule, even if it may seem to have a sense. I think that this way to understand the truth of a thought is in line with Frege’s remark that although treating truth as the property of a thought it does not seem to be a property in the usual sense of the word (Frege 1918, p. 61); indeed, truth does not add anything to the thought-sense-rule, except its effective applicability as a verifying rule in a chosen domain.
   The proposed identity between the Fregean concept of thought and the concept of a verifying rule is supported by the Fregean suggestion that the criterion for the identification of what belongs to a thought is for it to have at least some role in the establishment of its truth. That is, the thought expressed be a sentence is the same as its epistemic sense or the informative content, which should be its rule of verification. And the identifying criterion for this thought-sense-rule (procedure, method) must be what allows the recognition of its truth-value, what should be whatever is able to warrant it its effective applicability to the corresponding fact.
   But what is the property of this verifying rule – the thought – of being true? Well, there are two ways of answering the question. First, we can get an answer by reflecting on what we have said about the property of existence of what is referred by a conceptual (ascription or identification) rule. Since this existence should be the property of the effective (and not only supposed) applicability of the concept expressed by the conceptual word, by symmetry the same should be said about the property of thoughts of being true. Truth should be the property of a verifying rule of being effectively applicable to its object, which should be the fact (or sub-fact) that satisfies the rule. What the sentence ‘p is true’ expresses should be thought of p, which is a verifiability rule followed by its ascription of truth, which is nothing but the higher order ascription of the property of effective applicability to the sense-thought-rule. We could even say, in a Hegelian fashion, that existence is the truth of the concept while the truth is the existence of the thought. According to this reasoning existence and truth are twin concepts.

Frege’s Platonism
It is important to remember that for Frege thoughts (including the senses that compose them) are Platonic entities belonging to a third ontological realm, which is neither psychological nor physical. For him, taking as the criterion of objectivity the intersubjectivity and independence of will, and taking as the criterion of reality existence in space and time, we get three ontological realms:

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

   The first realm is that of physical entities, such as concrete objects, which are objective and real. These entities are objective, since they are intersubjectively accessible and independent of our will; and they are real, since they are located in space and time. The second realm is that of psychological entities, mental states that he calls representations (Vorstellungen, a word that we could today translate as qualia). These entities are subjective and real. They are subjective by not being interpersonally accessible and are often dependent on the will. However, they are still real, because they are in the heads of those who have them and, consequently, in time and space. There is, finally, a third realm, that of thoughts (understood as propositions) and their constitutive senses. This realm is for Frege objective but not real. It is objective, because thoughts, true or false, are inter-subjectively accessible; we can all agree, for example, that the Pitagorean theorem expresses a true thought. But this third realm of thoughts isn’t real, because according to him as abstract entities thoughts cannot be found in space or time. Thus, the thought (the sense) of Pythagoras’ theorem is objective but non-real.
   Although for Frege thoughts are eternal (timeless), immutable, forever true or false, and not created but grasped (gefasst) by us, they must have some kind of indirect causal effect: by grasping them we must be able to make judgements, and by making judgements we are able to act in the external world.  How this is possible remains unexplained.
   The reason Frege has to introduce this third realm of thoughts is that thoughts are communicable and, to be effectively communicable, they need to be objective, that is, in some way interpersonally accessible. Representations (Vorstellungen), on the other hand, are rather subjective psychological states, which can vary in dependence of the personal psychology and are not interpersonally accessible. Thus, for Frege the only way to explain how it is possible that we are able to share the same thought is to distinguish it strictly from mere psychological representations. As well, if thoughts were on the level of representations, they would be dependent on the changeable personal psychology and would lack the required stability of truth-bearers.

Avoiding Frege’s Platonism
Despite the above suggested arguments, only few would today accept Frege’s appeal to Platonism. After all, the Fregean form of Platonism not only commits us to an infinite multiplication of objective entities (all the infinite true and false thoughts), but also seems to lack intelligibility. The price that Frege was willing to pay in order not to fall into psychologist subjectivism seems today too high.
   In my view, Frege’s Platonist solution is unnecessary because the whole problem was wrongly formulated. For there is a way to conform the view that thoughts have a psychological nature to the view that as a truth-bearer they must have stability and the possibility of being communicated. In order to show this, I want to apply a strategy inspired by the ontological particularism of the English empiricists from Locke to Hume, for whom the universal does not exist beyond the similarities among mental ideas.[26] In order to do this, I wish to show that Fregean thoughts (objective non-real truth-bearers…), which I call f-thoughts (‘f’ from Fregean), can be defined in terms of psychological p-thoughts (‘p’ from psychological). I suggest that we can warrant the existence and stability of what I call s-thoughts (‘s’ from spreadable) without hypostasising them as Platonic entities and even without resorting to classes of p-thoughts by means of the following definition:

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

   The s-thought is my version of what Frege should have meant with his f-thought (objective non-real thought). This definition reduces the supposed f-thoughts to p-thoughts without forcing them to lose their objectivity (intersubjectivity) and expected stability or imutability by interpreating them as s-thoughts.
   The so defined s-thought has no particular spatio-temporal location and can be seen as the truth-bearer. For example: the s-thought expressed in the sentence ‘The Eiffel Tower is made of metal’ can be instantiated as the p-thought that I have in mind when writing this sentence, but also by, say, the p-thought that you have in your mind when you read it, or yet again, by any qualitatively identical p-thought that I, we or any other person can have at any other time. Characterised by the disjunction between the qualitatively identical thoughts embodied in any individual mind, the s-thought comes to be regarded in abstraction from the particular human minds that casually instantiate it. With this we avoid not only the appeal to specific occurrences of thoughts, but also the most expected alternative, which would be to explain one s-thought in terms of a sum or class of p-thoughts qualitatively identical to each other, which could lead us not only to the problem that classes are candidates for abstract entities, but to the problem that classes have size while thoughts have no size. If a thought were seen as a class it would be made bigger and bigger the greater the number of people that would have been grasped it.
   Under the proposed definition, in order to exist an s-thought must always have at least one psychological instantiation. The s-thought is no less psychological than any p-thought, since it cannot be considered independently of its instantiation in at least one mind. Thus, when we say that we both had the same idea or have reached the same thought, what we mean is only that there is a qualitative identity between the psychological p-thought-contents that we have respectively instantiated in our minds. This has the advantage of bringing the Fregean thoughts from the Platonic heaven back to the psychological realm, without commitment to a transient psychology of a particular cognitive being. This understanding of the true nature of the s-thoughts explains something that Frege was unable to explain satisfactorily, namely, why they may have causal powers. As an open disjunction of p-thoughts, s-thoughts only exist as psychological instantiations of p-thoughts, and as such they can cause others psychological states and finally human actions and their effects in the external world.
   At this point one could rise the objection of multiple realizability: the same p-thought could be differently realized in different human brains impeding the qualitative identity of p-thoughts. I agree with the probable multiple realizability of p-thoughts, but disagree that this makes their qualitative identity impossible. First as a matter of fact: we frequently agree that we have the same psychological process. Second because we can present things that are qualitatively identical in a psychological level and qualitatively different in neurophysiological level, in a similar way as the number 5 can be represented as ‘2 + 3’, ‘2 + (1 + 1 + 1)’, etc.
   In my view, one of the most ingrained and deceitful philosophical errors in ontology has been seeing numerical identity where there is only qualitative identity. It is true that we can speak of the number 2 in the singular, that we can speak of the geometrical form of circularity, and that we can ask for the meaning of the common name ‘chair’ using the definite article – but this is just for the sake of simplicity of expression. What we actually can have in mind are instances of qualitatively identical cognitive arithmetical concepts of the number 2, of qualitatively identical geometrical concepts of circles[27], of qualitatively identical meaning occurrences of the word 'chair’ and nothing more. In the same way, we can talk about the thought ‘7 + 5 = 12’, but if we do not intend a specific occurrence of this thought, we are only referring to some occurrence, but without taking into account or specifying which occurrence it is. The true reason why we speak in the singular of the thought ‘7 + 5 = 12’ is that there is no reason to consider the individual persons who think it.
   The adoption of the definition of s-thoughts proposed above, which is easily generalisable to all kinds of Fregean senses, is in my judgment the only plausible abstraction that we can arrive at without falling into any of the various forms of reification that have infested ontology throughout its long history.
   Here arises, however, the Fregean question: how is it possible that the above suggested psychologically dependent definition of s-thoughts could be able to ensure the objectivity of s-thoughts, their intersubjective acessibility or communicability? As we saw, for Frege if thoughts were regarded as psychological representations, as is the case for p-thoughts, they would unavoidably be subjective, unable to be compared with each other. However, the need that Frege feels to admit that thoughts belong to a third realm of Platonic entities is too hasty. There is no doubt that what Frege calls representations, psychological mental contents, can largely be expressed through language and by its means be able to be subjectively identified and re-identified as being the same. It is true that a mental state that only one person is capable of having, for instance, some epileptic aura, isn’t communicable, except indirectly, metaphorically. But it seems an unquestionable fact that typical mental states, such as feelings, images, sensations, are things that all of us are able to communicate and learn to identify in ourselves, through induction by exclusion in some cases, and, in others, through induction by analogy reforced by interpersonally accessible physical states strongly intermingled with them (Costa 2011, ch. 3). Finally, we have almost forgotten the main point: senses are for us rules or conventions that are in the end intersubjectively grounded and are able to be intersubjectively corrected. Though s-thoughts can possibly involve mental imagery, s-thoughts are verificational rules rooted in intersubjective conventions, what satisfies Frege’s criterion of objectivity as intersubjective accessibility and evaluation.
   Concerning s-thoughts it is also important to remember that it is not necessary to have one only particular model as the object of interpersonal consideration. On the contrary. What we do is simply to alternate a variety of qualitatively identical models that are usually given to us by memory: first the one and then some other, which we recognise as being identical to the first, and then we can use some of them as the model and so on. And language is only the vehicle of communication that allows the reproduction of a qualitatively identical psychological p-thought in the minds of hearers.
   It may at first sight seem implausible that language is capable of repeatedly reproducing in other minds and even in the same mind the same subjective pattern, the same thought content, the same recognisable instantiation of a combination of conventionally established semantic rules. However, compare this case with the case of genetic information able to indefinitely reproduce the same characteristics in successive biological individuals:[28] why the conventions and the ways they can be combined in the constitution of p-thoughts could not render the same? In addition, it is easy to suppose that there are correcting mechanisms able to interpersonally and intra-personally correcting deviances of the standard. There is no reason, except an anti-empiricist bias, to think that things could not be that way.
   Finally, let us apply the distinction made by John Searle between what is ontologically objective/subjective and what is epistemically objective/subjective to the objectivity of s-thoughts (Searle 1999, pp. 43-45). Searle noted that we have a strong tendency to take what is epistemically subjective for what is only ontologically subjective. However, one thing can be ontologically objective – for example, the enduring social effects of the Napoleonic wars – without ceasing to be epistemically subjective, because we are not able to reach common agreement about it. In contrast, a phenomenon can be ontologically subjective without ceasing to be epistemically objective – for example, the knife-like pain caused by a seizure of acute pancreatitis – because everyone (doctors and patients) will agree on the form and existence of this pain. Something similar can be said about the nature of s-thoughts. They are ontologically subjective, since we admit that they are psychological events instantiated in one or other mind. But even so, they do not cease to be epistemically objective. After all, we are capable of both intersubjectively agree about them and their truth values. We can agree that the utterance ‘I am having a knife-like pain in my abdomen’ is true in the case of a patient with a diagnostic of acute pancreatitis. And an objective assertive sentence like ‘The Eiffel Tower is made of metal’ expresses an s-thought that we all also recognise as being true. This thought, like any other s-thought, is ontologically subjective, since it can only be psychologically instantiated. However, like any s-thought, it remains epistemologically objective, given that both the proposition and its truth-value are fully measurable and reportable, since they are structured by our conventions and based on our knowledge of the facts. On the other hand, a sentence like ‘Love is the Amen of the universe’ (Novalis), unlike an s-thought, has no truth value. It has only colouration, being susceptible only to an aesthetic appreciation with a variable degree of interpersonal agreement.
   Frege was no exception: like Husserl, Bolzano and many other continental philosophers with mathematical training, he believed that the ontologically subjective character of the psychologically conceived contents of thought would inevitably be condened to epistemic subjectivity. But this was a mistake.

Further ontological consequences
This ultimately psychological reconstruction of the Fregean thoughts has interesting ontological consequences. If the thought of the Pythagorean teorem isn’t an eternal (timeless) entity of a Platonic realm, always true or false, where and when is it? The answer is that being at least one occurrence of thought, or any other qualitatively identical occurrence, regardless of the bearer, the Pythagorean teorem acquires an existence that is dependent on minds, although remaining independent of any of the many minds that eventually think it. Since this thought was thought by both you and me, and certainly by many others in the past, its existence must be extended over space and time. This existence can be seen as distributed over the space and time occupied by the heads of mathematicians starting with Pythagoras himself, and perhaps ending in the head of some cognitive being at some unknown future time. This is what gives the impression that the thought is something abstract, beyond the psychological realm.
   A consequence from this view is that unlike the Platonic entity that Frege has called ‘thought’, our s-thought of the pytagoric theorem in fact did not exist before Pythagoras had it for the first time (supposing he was the first), and will cease to exist when it ceases to be thought by anyone. The Pythagorean theorem certainly exists, has existed and will exist in the sense that it is thought, has been thought and will probably be thought in the future, referring to occurrences of this thought, but without having to take into account who thinks it. One reason why this sounds strange is because nobody can truly think so. One cannot think: ‘The theorem according to which the sum of the squares of the shorter sides of a right triangle equals the square of the hypotenuse is something which existed in the past and now no longer exists’, for this judgment will already date the occurrence of the thought of the Pythagorean theorem and insofar falsify what is said. Nevertheless, it remains the outcome that the s-thought of this theorem would not have come into existence if nobody had ever thought it in a world of no cognitive beings.
   This brings us to the following objection. Imagine a possible world Wn similar to our, though with stars and planets, but without thinking beings. In Wn the s-thoughts of the Pythagorean teorem and and that there are stars and planets could not have been thought and, as the primary bearers of the truth, could not be true. But it seems obvious that also in this world not only the Pythagoren theorem but also the fact that the sun is a star would be true...
   A reasoble answer is that there is here confusion between the primary bearer of the truth – which is the s-thought – and a secondary but as we already saw very usual bearer of the truth – the real thing or fact in the world, which is reported in any good dictionary. Indeed, that there are planets and stars would be still true as a fact in Wn, and the applicability of the Pythagorean teorem would still be true as a fact in Wn, though their s-thought would not exist. It is the flexibility of natural language that once again misleads us.
   Still one objection that could be made against the idea that the bearers of truth are non-Platonic s-thoughts is the following. Many truths are discovered. Pythagoras discovered the theorem that bears his name; Archimedes was one of the discoverers of the law of the lever, according to which magnitudes are in equilibrium at distances inversely proportional to their weights. But if something is discovered, it logically must have existed before being discovered. Consequently, the above described thoughts already existed before they were discovered.
   Again, the answer is that this objection results from a confusion between the thought as the primary bearer of truth on the one hand, and the fact as the secondary bearer of truth on the other. It seems clear in the case of empirical truths: that the law of the lever was always applicable is true... but the thought of it first came into the world when scientists like Archimedes conceived of it. Similarly, the fact expressed by the Pythagorean teorem has always existed, but our s-thought of it only came into existence after the theorem was thought by Pythagoras and since then by many others. Such facts, however, as long-lasting as they may be, are in the most proper sense not the bearers of truth, but the supposed makers of truth or verifiers. They are what occurrences of their thoughts represent, which means that the truth of their thoughts cannot have existed before them. In the most proper sense, no truth or falsehood would exist if there were no minds to think of them.
   However, would this mean that before conscious beings appeared on Earth the thought that the Moon is white wasn’t true? The answer is yes and no. The thought couldn’t be true, since it didn’t exist. But we can say that this was a true fact. And we could even say, I believe, that it was true that the Moon was white in the sense that if the thought of the Moon being white were thought it would be true because it would correspond to a fact that would make it true. And this seems to be all that we can really mean when we say that the Moon would still be white even if there were no cognitive being to think it.
   An s-thought that has never been thought does not exist and thus cannot be true. The same with falsities: suppose that the thought ‘The Colossus of Rhodes is floating in the Sargasso Sea’ had never been thought before. As soon as we think that it has never been thought before, we are already thinking it, and we see that it is surely false. Even the s-thought ‘The world could exist, even though there were no minds to think about it’ is only true because there are minds to think it.

A short digression on contingent futures
Before we finish, it is curious to examine the Aristotelian problem of contingent futures in the light of these conclusions. According to Aristotle (1984, vol. 1, ch. 9), the following argument is valid:

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

For Aristotle this conclusion would be unacceptable because if the future is fixed then there is no chance, and if there is no chance there is no free will. Hence, he thought that although this argument is sound, the premise (1) is false because it exemplifies the principle of bivalence and the principle of bivalence isn’t applicable to future events (only to present and past ones).
   I cannot agree with this, since I think that a s-thought must obey the principle of bivalence. Suppose that outside any context we consider the s-thought expressed by the sentence ‘There will be a sea battle tomorrow’, which we can abbreviate as ├p. Is this true or false? The answer is: taken literally├p is unable to express an s-thought because an s-thought is something to which we must possibly attribute a truth-value and without any further contextual information we are totally unable to correlate p with any truth-maker in order to assign it a truth-value.
   However, one could argue that the sentence ├p is misleading and brings us to confusions like the argument A because ├p only seems to express cognitive content. The reason for this is in my view that ├p is very easily confused with the sentence ├q: ‘It is likely that a sea-battle will take place tomorrow’, when there are reasons to think so. For example: having broken the Japanese naval codes and having lured the Japanese fleet into an ambush at Midway, the Americans already knew on the night of June 3, 1942 that on June 4 there would almost certainly be a major naval battle. The sentence ├q is easily confused with ├p, because ├q appears almost always abbreviated as ├p: ‘A sea-battle will take place tomorrow’.
   For example: suppose that American Admiral Nimitz had said on June 3:

Tomorrow there will be a sea-battle.

   Everyone would understand that he was saying that all the factual evidence was leading to the conclusion that the expected battle would begin on June 4. This probability – made explicit or not – is in this case objectively measurable in terms of verification by actual empirical evidence that the assertion ├q is expresses a probabilistic true s-thought. Indeed, the utterance ‘It is likely that a naval battle will occur tomorrow’ was true in the night of June 3, 1942, without breaking the principle of bivalence. In a similar way, if I am on a beach looking at the Atlantic Ocean and say without any reason ‘A naval battle will take place tomorrow’, meaning by it ‘It is likely that a naval battle will occur tomorrow’, this statement will be seen as false, since I have all kinds of reasons to believe that this kind of event is very improbable in this place and time.
   The conclusion is that taken literally (and not in the sense of ├q) the sentence ├p is a bluff devoid of meaning and justification and Aristotle was right in rejecting the application of the principle of bivalence to it. All that this sentence does is to induce us to imagine a naval battle taking place tomorrow, as if there was hidden verification criteria. However, as much as no context is furnished, no such criteria can be given. The statement ├q, on one hand, says something probabilistic about tomorrow that can be confirmed or rejected by criterial reasons already given today.
   It seems that the whole metaphysical trouble about contingent futures can be eliminated when we consider with enough care what we are really able to mean by using a sentence on the future.

Conclusion 
My first aim in this chapter was to insert in the framework of the Fregean semantics the results of my reading of Wittgenstein’s view of meaning as use in accordance with rules, in order to distinguish and giving some explanation of the most relevant forms of semantic-cognitive rules. This required strong corrections in Frege’s own framework. Even if some results are admittedly vague and speculative, they nonetheless seem to me no less plausible than Frege’s own original views.






[1] On the thorny issue of how to translate ‘Bedeutung’, see Michael Beaney’s introduction of his The Frege Reader, p. 36 ff.
[2] Searching in the literature, the only place where I have found a similar interpretation of this point was in W. Kneale & M. Kneale’s book, The Development of Logic, p. 495.
[3] See the introduction of the distinction in Gottlob Frege, ‘Funktion und Begriff’, p. 14 (original pagination).


[4] We owe this revised reading of Frege’s sense mainly to Michael Dummett (see Dummett 1981, p. 229). But a similar view was later more explicitly defended by Ernst Tugendhat.
[5] If we compare these two passages, it turns clear that contrary to Kripke’s interpretation (1980, Lecture I) Frege had already in mind the essentials of the later cluster theory of proper names. The same can be said of Russell (see Russell 1980, ch. 5).
[6] François Recanati prefers to use the word ‘property’ (propriété) instead of concept in his summary of Frege’s semantics (Recanati 2008, p. 34). 
[7] The pure theory of tropes was first introduced in the philosophy by A. D. Williams, having since then sparked growing interest.
[8] I prefer to use the phrase ‘qualitative identity’ in order to avoid the confusion with the ordinary concept of similarity, which is intransitive. See Appendix to chapter 3.
[9] I suggest this trope-model way of constructing the universal in order to circumvent the usual but problematic definition of a universal as a set of tropes that are exactly similar, one with the other. This 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 concept. 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 the memory of them. The memory-trope is not the ultimate trope we intend to consider, since it stays for the experienced one.
[10] I do not agree with Frege’s acceptance of typical singular terms as parts of predicates. A sentente like ‘Stockholm is the capital of Sweden’ cannot be fully analysed as having as predicate ‘…is the capital of Sweden’, but as having the relational predicate ‘…is the capital of….’
[11] To the objection that this will never gives us the necessary conditions to the universal quantification we could answer that the truth of the universal quantification is nearly always only probable. What ‘All trees are made of wood’ means is ‘Very probably all trees (of this planet) are made of wood.’ Moreover, the so-called general fact has no mistery because it is a singular fact about a particular concrete domain, like that of my pocket: when I say that all five coins in my pocket are of one euro what I mean is that I have four coins of one euro in my pocket and nothing more (though most domains are highly blurred this does not changes my point). (For contrasting views see Russell 1918, V; Armstrong 1997, ch. 13)
[12] This dependency that a ascription rule of a predicative expression has of a prior application of the identification rule of a singular term was sharply noted by Ernst Tugendhat in his analysis of the real conditions of a predicative singular statement: ‘‘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 1983, p. 235).
[13] I take the tense as an addition exterior to the existence, that is, the time and place in which something exists (existed, will exist) is something complementary that we add to the affirmation of its existence.
[14] The alternative formulation Ǝx (x = Socrates) is still less workable, since it leads us to contradictory results. For a sentence as ‘Vulcan does not exist’ we get ~Ǝx (x = Vulcan); but applying the Universal Instantiation to ‘Vulcan’ we get to the contradictory sentence Ǝyx (x = y), while ~Ǝx (x vulcanizes) is only bizarre.
[15] David Braun and Marga Reimer made a balanced careful comparison between the two views in their respective articles for the Stanford Encyclopedia of Philosophy. The result was indecibility.
[16] This is again a simplification. See Appendix of Chapter 1.
[17] However, if the assertion that there are round squares could be merely an equivocal manner to say that we can syntactically combine the adjectives ‘square’ and ‘round’, that is, a misleading way of saying that there is a syntactic rule allowing the mere combination of these adjectives, then it makes some sense to attribute existence. But in this case what we are trying to say is more correctly expressed by the metalinguistic sentence: ‘The rule for constructing the phrase “round square” is applicable; therefore, the phrase “round square” exists as a grammatical construction’. The Meinongian Sosein seems to be reduced here to the recognition of a syntactic triviality.
[18] According with Berkeley’s official view, things that are not actually perceived by us exist because they are continuously being perceived by God (Urmson 1983).
[19] Also Frege, Letter to Russell from 28.12.1912.
[20] For counterexamples, see J. L. Austin, ‘Unfair to Facts’. It seems to me at least curious that the posthumously published arguments of J. L. Austin against Strawson’s view have had so little impact.
[21] As Tyler Burge in ‘Sinning against Frege’ wrote, ‘the word “thought” is the best substitute for ‘proposition’ for the naturalness of its semantics within the scope appropriate to the linguistic philosophy’ (Burge, 2005, pp. 227-8).
[22] If instead of sentences-token we appealed to sentences-type as truth-bearers, the problem would be worst, since the same sentence-type could be true and false, infringing the law of non-contradiction.
[23] For Frege in the case of indexical sentences the context of the utterance belongs to the expression of the thought. I published a defence of Frege’s view in my 2014c (See also Perry 1977, 1979)
[24] If you search the word ‘truth’ in any good dictionary in very different languages these two senses – truth as correspondence and truth as the real thing or fact – are always enphazized.
[25] See my exposition of Tugendhat’s verificationist correspondentialism in the introduction of this book.
[26] In its plain form the insight is clearly expressed by Berkeley in the following passage: ‘...an idea, that if considered in itself is private, becomes general by being made to represent or be in the place of all other particular ideas of the same type. ... a private line becomes general by being made a sign, so that the name line, which considered absolutely is private, to be a sign is made general.’ George Berkeley, Principles of Human Knowledge, introduction, section 12. See the more sophisticated but also less clear view of David Hume in A Treatise of Human Nature, book I, part 1, section VII.
[27]  One could object: how can we deal with the geometrical circles of geometry if we are always dealing with imperfect empirical circles? Don’t we need a Platonic ideal pattern? The answer is ‘no’, because in principle we can make a circle more perfect than the last one, and another still more perfect, and this process can continue without end. The perfect circle is like the actual infinite, it does not exist. It is nothing more than the projection of our awareness of the possibility of making more and more perfect empirical circles without any end in sight. Geometry only works as if a perfect circle existed and uses it as a limiting concept. For we clearly see that ‘qualitatively identical’ cannot mean ‘absolutely qualitatively identical’. The identity here depends on our ability to discern identity and our practical aims. My car has for me (qualitatively) the same colour as yours, but this identity cannot withstand a sufficiently close inspection.
[28] Biological mutations are accidents whose incidence should be evolutionarily calibrated. Only species that are able to mutate in the right amount at the right period of time can survive. An unchanging species with no relevant mutation is probably possible, but would not have the adaptability necessary for survival under changing external conditions.