segunda-feira, 14 de maio de 2018

Tradução da carta de Frege à Husserl sobre o conceito (Begriff)

CARTA À HUSSERL, 24.5.1891

(...) Tudo o que eu gostaria de dizer agora é que parece haver uma diferença de opinião entre nós sobre como uma palavra-conceitual (nome comum) é relacionada a objetos. O seguinte esquema deve tornar minha concepção clara:

Proposição           nome próprio          palavra conceitual
       ↓                         ↓                              ↓
Sentido                sentido                    sentido de uma
de uma                de um nome            palavra
Proposição          próprio                    conceitual
        ↓                         ↓                             ↓
BEDEUTUNG    BEDEUTUNG        BEDEUTUNG           →   Objeto que cai
Da proposição     do nome próprio      da palavra conceitual         sob o conceito
(valor-verdade)    (objeto)                   (conceito)

Com a palavra conceitual é preciso mais um degrau para se chegar ao objeto do que com o nome próprio e esse último degrau pode estar faltando – ou seja, o conceito (Begriff) pode ser vazio – sem a palavra conceitual cessar de ser cientificamente útil. Eu desenhei o passo do conceito ao objeto horizontalmente para indicar que estão no mesmo nível, que objetos e conceitos tem a mesma objetividade. No uso literário é suficiente que tudo tenha sentido; no uso científico é preciso ter Bedeutung. (...)
     Agora parece a mim que para você o esquema seria assim:

                                          Palavra conceitual
                                   Sentido da palavra conceitual
                                  Objeto caindo sob o conceito

Assim para você o número de degraus do nome próprio para o objeto seria o mesmo que o das palavras conceituais. A única diferença entre nomes próprios e palavras conceituais seria então que as primeiras poderiam se referir somente a um objeto e as segundas a mais de um. Uma palavra conceitual cujo conceito fosse vazio teria então de ser excluída da ciência assim como um nome próprio sem um objeto correspondente.

(Obs: Para Frege se o nome não tiver referência ele não servirá à ciência. Mas o predicado pode não ter referência e mesmo assim servir à ciência; logo, o predicado precisa ter como referência algo mesmo quando é vazio, ou seja, um “conceito”. Qual a razão de ser assim?)


 DRAFT for the book PHILOSOPHICAL SEMANTICS to be published by CSP in 2018

Part 2

21. The reference of a sentence as its truth-value
Now we leave our speculative excurse and come back to the more tangible Fregean semantics, considering what he has to say about the reference of a sentence. Here I have no compliments to make. Frege was the author of the insane idea that the references of sentences are their truth-values, so that the thoughts expressed by them should be modes of presentation of truth-values.
     How did he reach this strange conclusion? There are several reasons. First, he notes that sentences are independent, saturated, closed; they work in ways similar to those of names, and a truth-value is also closed, since it does not require complementation. Second, he says that the search for truth is what brings us from sense to reference. Third, he notes that sentences without reference lack truth-value: ‘Vulcan is a warm planet’ has no reference and for him no truth-value, since this hypothetical planet has been shown not to exist. Fourth, he also noted that conforming to the principle of compositionality – according to which the whole is a function of its parts – the reference must be what remains unchanged after we change the senses of a sentence’s components without changing their references. This is what happens, for instance, if we replace ‘Napoleon lost the Battle of Waterloo’ with ‘The man of destiny lost his last battle.’ Since the references of the sentence-components do not change, the reference of the whole sentence likewise does not change. Moreover, the truth-value of both sentences remains the same: The Truth. Hence, their reference must be their truth-value. The conclusion of all this is that in extensional languages the references of sentences must be their truth-value (1892: 34). For Frege, all true sentences have only one reference, which is the abstract object The True (das Wahre), while all false sentences also have only one reference, which is the abstract object The False (das Falsche).
     However, there are a number of well-known embarrassing objections to Frege’s identification of reference with truth-value that in my opinion completely disqualify his view. A first objection is that, contrary to any healthy intuition, Frege’s proposal frontally contradicts the meaning we normally give to the word ‘reference.’ It is intuitively obvious that the sentence ‘Napoleon was born on Corsica’ refers to something very different from the sentence ‘2 + 2 = 4,’ even if both are true. Moreover, if you replace ‘Venus is a planet & the Earth is a planet’ with ‘Mars is a planet & the Earth is a planet,’ both composite sentences remain true because of the truth of the partial sentences, but the reference of ‘Venus’ is totally different from the reference of ‘Mars,’ what runs against the principle of compositionality. Another objection is that we expect the references of components of our sentences to be on the same ontological level as the sentences’ references. But for a Fregean, this could not be the case: the reference of the name ‘Napoleon’ is the Napoleon of flesh and blood, while the reference of the sentence ‘Napoleon was born on Corsica’ must be the abstract object called The True. Moreover, Frege’s solution violates his own principle of compositionality. If the reference of a sentence is its truth-value, it cannot be established by its parts, since a truth-value has no parts. And even if it had parts, then all objects referred to by names in true sentences should be parts of The True, which would hardly make sense. There are also serious substitutability problems with Frege’s explanation of the references of sentences. The first is that if all true sentences refer to The True, and the name ‘The True’ also refers to The True, then in the conditional sentence ‘If it rains, then water falls from the sky,’ we can replace ‘it rains’ with ‘The True.’ But the result will be the sentence ‘If The True, then water falls from the sky,’ which should be true but is in fact unintelligible (Black 1954: 235-6). A second and fatal problem of the kind is that a multitude of obviously false identities between true sentences should be true. For example, ‘Paris is a city = snow is white’ should be a true assertoric composite sentence, since the two sentences refer to the same thing: The True. Under critical scrutiny, Frege’s view shows itself to be hopeless.
     The most charitable interpretation is that Frege uses the word ‘reference’ as truth-value because it is what counts, because the word Bedeutung (meaning) in German, more than in English, also means relevance, pointing to semantic relevance or meaningfulness (Cf. Tugendhat 1992b: 231).[1] Indeed, truth-value is of decisive relevance for logic, because it is what must be preserved in valid arguments. The logician does not need to know more than truth-value regarding the referring function of the sentences he is dealing with in order to evaluate inferential possibilities.
     A main problem with this interpretation is that it contradicts expected principles of Frege’s own theory. Since the reference (Bedeutung) of the parts of a singular sentence (general and singular terms) can be seen as their references in a literal sense (the concept and the object that can fall under it), truth-value as relevance satisfies the principle of compositionality in an odd, non-linear form, since relevance is normally only an adjective applied to truth-value. This is different from the principle of compositionality applied to senses in which the whole and its components are linearly arranged in the same epistemic domain. The attempt to tell us that a reference is mere qualification attributable to it is equivocal and confusive.
     Finally, when we take the truth-value for the reference of a sentence, this view can be – and in my judgment really has been – utterly misleading from an epistemological standpoint. Since truth considered as in some way belonging to thought has nothing to do with anything that can reasonably be understood as the reference of our statements, calling truth-value ‘the reference’ contributes to placing the relation between language and the world virtually beyond semantic reach.
22. Logical structure of facts
The Fregean account of the references of sentences as their truth-values turns out to be still less acceptable if we consider that a much more natural alternative is available, which, as Sir Anthony Kenny has noted, was not even mentioned by Frege (Kenny 2000: 133). This alternative, which the logical atomism of Wittgenstein and Russell tried to explore, consists in the appeal to facts. Since it is prima facie much more plausible that the references of sentences are facts, it is important for us to investigate the logical structure and ontological nature of facts.
     Considering first the logical structure of facts, a plausible view is that they correspond to the logical structure of the thoughts representing them, assuming that these thoughts are what declarative sentences express when logically analyzed, at least in accordance with the context of the linguistic practices where they occur. Nevertheless, even respecting linguistic practices we can go further, considering that they are placed within the factual language in general and accepting a form of atomism in which the bottom line of the analysis is the exposure of the logical components of what is stated in singular sentences where we can find identification rules of singular termini associatively used with ascription rules of predicative expressions. Singular empirical statements such as ‘Frege has a beard’ and ‘The cat is on the mat’ belong to this bottom line and respectively represent facts that should have the logical structure depicted by Fa and bRc.
     Elements a, b and c, as singular terms, refer to individuals constructed as clusters of appropriate compresent tropes, while F and R would also be seen as designating tropes, usually complex tropes forming complex criterial configurations dependent on the clusters to which they are tied. The ties between b, R and c, and between F and a, in turn, are only pseudo-relations, since admitting their existence as relational tropes would generate an inevitable infinite regress. As we already noted, individuals and their property-tropes are linked by ‘non-relational ties’ without any ontological addition (Cf. Appendix to Chapter III, sec. 1). Indeed, what could be the relational ties between the application of the ascription rule of ‘…was bearded’ to Aristotle with the already applied identification rule of Aristotle in the fact represented by the statement ‘Aristotle was bearded’?
     We should also pay attention to the somewhat trivial rule of analysis according to which we should not accept singular terms as components of complex predicative expressions. Thus, for instance, in a sentence like ‘Stockholm is the capital of Sweden’ we should not view ‘…is the capital of Sweden’ as a predicate, since Sweden is a proper name. Also inadequate would be to analyze ‘the capital of Sweden’ as a definite description contextually referring to Stockholm in our world, so that the analyzed sentence would have as its relational predicate ‘…is (the same as)…’ The most appropriate analysis would be to consider ‘…is the capital of…’ as a relational predicate completed by the proper names ‘Stockholm’ and ‘Sweden,’ separating the relational trope from the compresent bundles of tropes referred to by the proper names. Proper names are stronger identifiers than definite descriptions and should therefore be preferentially singled out in the logical analysis of thought.
     Furthermore, it also seems possible to analyze proper names and definite descriptions using Russell’s technique of transforming them into quantified predicative expressions, insofar as to a limited degree this device mirrors the neodescriptivist theory of proper names defended in this book, a similar procedure being possible regarding general terms. Anyway, such sub-sentential terms normally do not need to be analyzed when our task is to analyze sentences, since they are the proper elements of sentences, except when they are not what they seem to be, as in the case of nominalizations.
     Finally, we have composite facts represented by our extensional language, along with the general (universal, existential) facts to be analyzed as having the same structure of sets (conjunctions, disjunctions) of singular statements that make up general (universal, existential) statements, which, as we already noted, can be reduced to associations of singular predicative and relational statements. (I think that the philosophical problem of a hidden lingua mentis ends up in elements like those briefly pointed out in this section).
23. Ontological nature of facts
If we accept that the references of sentence-senses or thoughts are facts, then from an ontological perspective what empirical sentences represent must be empirical facts, most typically located in the external world, though possibly also located in the inner mental world. This assumption speaks for the correspondence or adequation theory of truth, according to which empirical facts are truth-makers normally seen as complex contingent arrangements of elements in the world, that is, usually contingent tropical arrangements associating tropical individuals and property-tropes.
     However, this assumption conflicts with Frege’s anti-correspondentialist view of truth. According to him, a fact would simply be a true thought (Frege 1918: 74). Following similar anti-correspondentialist lines, in a very influential article, P. F. Straw­son suggested that empirical facts are mere ‘pseudo-material correlates of the statement as a whole’ and not something in the world (1950: 6). According to him, empirical facts, unlike events or things, are not spatiotemporally localizable (‘the world is the totality of things, not of facts’). One reason for this is that the description of a fact usually begins with a that-clause. For instance, I can say ‘the fact that the book is on the table,’ but not ‘the fact of a book on the table.’ On the other hand, the description of an event typically lacks a that-clause: I can say ‘the event of a tsunami in Japan,’ but not properly ‘the event that there was a Tsunami in Japan.’ Facts are for Strawson what statements (when true) state, not what statements are about. They are
not, like things or happenings on the surface of the globe, witnessed or heard or seen, broken or overturned, interrupted or prolonged, kicked, destroyed, mended or noisy. (1950: 6)
The same is for him the case with states of affairs and situations. [2] Finally, to give a striking example, the event of Caesar’s crossing the Rubicon occurred in the year 47 BC, while the fact that he crossed the Rubicon did not occur in the year 47 BC, but it is still a fact today, since facts simply do not occur (Patzig 1980: 19-20).[3]
     An easy way to dispose of this argument could be the following. We need a word to describe the condition in the world that makes our thoughts true. The word ‘fact’ is available. So, why don’t we use it stipulatively in order to designate the truth-maker, whatever condition it is?[4]
     However, it seems clear to me that even this stipulative way to circumvent the problem is avoidable, since it is not difficult to show that the problem exists only in the imagination of philosophers. To begin with, of course not everything we may call a ‘fact’ is empirical in the usual sense of the word. It is hard to assign empirical status to the fact that 2 + 2 = 4, even if its supposed non-empirical character can be an object of controversy.[5] And we can say ‘It is a fact that the Sun is not green,’ although this seems to me only a linguistically modified way to say ‘There is no fact that the Sun is green’ or ‘The fact that the sun is green does not exist.’ What I want to defend here is that there is a privileged sense of the word ‘fact’ that involves references to more or less obvious empirical facts, particularly so-called observational facts, which should be considered objectively real: they exist in the external world and can be seen as the ultimate truth-makers of their statements.
     To begin with, it is good to remember that there is a well-known and very convincing reason to think that facts can be constituents of the empirical world. This is that many facts are said to act causally. Consider the following sentences:

(1)   The fact that the match was scratched caused the flame.
(2)   Thomas died because of the fact that he forgot to turn off the gas.
(3)   Because of the fact that today is a holiday, the class will be canceled.
(4)   The fact that Caesar crossed the Rubicon had important historical consequences.

It does not seem possible that pseudo-material correlates (which I suppose to be abstract contents) can be causally active in the empirical world, producing these effects. But conceding the empirical nature of facts (1) to (4) solves the problem in obvious ways. Scratching a match is a fact-event causing a flame. The situational fact created by Thomas’ forgetting to turn off the gas caused his death. The fact-circumstance that today is a holiday causes the cancellation of a class. The fact-event of crossing the Rubicon established a state of affairs that causally determined decisive political changes in the Roman Empire.
     Furthermore, I have a key-argument to regenerate the idea that empirical facts are correlates of true thoughts, as the classical correspondence theory of truth has held. According to the view I propose, empirical facts are contingent tropical arrangements in the external and/or internal world in general. Similar would be the case with facts apparently as simple as those referred to by sentences like ‘Frege had a beard,’ ‘The Eiffel Tower is in Paris,’ and also facts constituted by combinations of such facts.
     My argument against Strawson’s opposition between non-spatiotemporal facts and spatiotemporal events begins by showing that there is a serious confusion in his argument. He treats facts (as much as states of affairs and situations) as opposed to events. His schema is:

FACTS                              x                            EVENTS
Pseudo-material                             Spatiotemporal
correlates                                       phenomena

But this can easily be contested. We begin to be suspicious when we perceive that every event can be called a fact, but not every fact can be called an event. For instance: I can replace ‘the event of the sinking of the Titanic’ with ‘the fact of the sinking of the Titanic,’ but I cannot replace ‘the fact that Mt. Everest is more than 8,000 m. high’ with ‘the event of Mt. Everest being more than 8,000 m. high.’ Strawson’s opposition isn’t symmetrical. Now, since events can be called facts, it is much more reasonable to consider events as particular kinds of facts than to oppose the two, as Strawson did. Indeed, my proposal is that the word ‘fact’ is an umbrella term that encompasses events, occurrences, processes, as much as situations, circum­stances, states of affairs, etc. And the reason for this proposal is that we can call all these things facts, but we cannot call all these things states of affairs or events. We see that events are sub-types of facts and that linguists could classify the word ‘event’ as a hyponym of the word ‘fact.’ Considering things in this way, we can distinguish two great sub-classes of facts:

1.     STATIC FACTS: Can be formal or empirical, the latter when clearly located in space and time. As a whole, static facts do not change while they last. Typical of static facts is that the relationships between their tropical components do not decisively change during the period of their existence. They are truth-makers of a static kind. And ordinary language has names for them: they are called (with different semantic nuances) ‘states,’ ‘situations,’ ‘conditions,’ ‘circumstances,’ ‘states of affairs,’ ‘ways things are,’ etc.
2.     DYNAMIC FACTS: These are always empirical. They change while they last. The relationships between the elements constitutive of them change decisively during the period of their existence, so that they have a beginning, followed by some kind of development that comes to an end after a certain amount of time. We will see that they work as truth-makers of a dynamic kind. And ordinarily they can be called (with different semantic nuances) ‘events,’ ‘episodes,’ ‘occurrences,’ ‘occasions,’ ‘pro­cesses,’ ‘transformations,’ etc.

Facts said to be formal, like the fact that 7 × 8 = 56, are static in the harmless sense that they do not need to be considered as spatiotemporally located. They are not of concern to us here. Many facts are empirical and static, insofar as the relationships between the elements constitutive of them do not change during their existence. Static facts are usually called ‘states,’ ‘situations,’ ‘conditions,’ ‘circumstances,’ ‘states of affairs’… with different nuances of meaning. Examples of static facts are my state of poor health, the situation that I am lying in bed, the circumstance that the airport is closed, the state of affairs that the Mona Lisa is in the Louvre or that the Earth orbits the Sun. The Earth’s movement of revolving around the Sun does not count because it is an internal cyclical relationship that remains the same during the fact’s existence: as a whole, this state of affairs does not change while it lasts (although each orbital period counts as an event).
     Dynamic facts, on the other hand, can be called ‘events,’ ‘episodes,’ ‘occurrences,’ ‘occasions,’ ‘processes,’… They are defined by changes in their overall composition and in relations among their elements during the period of their existence. World War II, viewed a process, for instance, began with a rapid expansion of the territories dominated by Nazi Germany and was marked by events like the Battle of Britain, the Battle of Stalingrad and the Normandy invasion – it had an unforeseeable history. Dynamic facts are usually called events when their duration is comparatively short, occurrences when their duration isn’t as short, processes when their duration is longer. Examples of events are an explosion or a lightning flash in a storm. An example of an occurrence is a volcanic eruption. The process of global warming is a very slow natural process, slower than the process of economic globalization. We can predict the stages of many events and processes, although many are also unpredictable in such a way that (unlike static facts) we cannot grasp them in their integrity before they end. Important is to see that all these things can be individually called events, occurrences, occasions, happenings, processes… and also facts, since they are all nothing but empirical facts – 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. Thus, we can speak about the event of Caesar’s crossing the Rubicon, but not about the event that he crossed the Rubicon. We can speak about the process of climate change, but not about the process that the climate changes… But this isn’t the case regarding static facts, which are typically (though not necessarily) described as beginning with that-clauses. So, I can speak about the state of affairs that my book is on the table or that I am lying on the bed, although I can also speak about the state of affairs of my book being on the table and of my lying on the bed. The conclusion is that if that-clauses have some function it is that of excluding dynamic facts and emphasizing static facts. Moreover, since the hyperonymic term ‘fact’ can be applied to both – static facts as much as dynamic facts – it is reasonable to suppose that this term inherits the property of being used indifferently, with or without a that-clause. Indeed, 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 we can also say: ‘Caesar crossing the Rubicon was an event’, as much as ‘It is a fact that Caesar crossed the Rubicon,’ referring less precisely to the event. We can summarize these relationships in a schema:

(a)   Static facts (states of affairs…): can be well stated with or without a that-clause.
(b)  Dynamic facts (events…): cannot be well stated with a that-clause.
(c)   Facts in general: admit both cases, because being all-embracing they do not differentiate between (a) and (b).

Now, what about the fact that Caesar crossed the Rubicon? Isn’t this fact timeless? The answer is that this is a good case of a misleading statement. In most cases, it is not understood as the description of an event, but as an illustrative way of referring to a static social fact: the state of affairs established by the movement of Caesar’s army onto Roman territory, violating the law that prohibited this and forcing the Roman state to declare war against him. Only occasionally is the phrase ‘crossing the Rubicon’ understood in its literal sense, as the physical event of crossing the river, which comprises Caesar’s sequential locations in relation to the river from t1 to tn.
     Due to the nature of dynamic facts like events and processes, we say that they not only are, but also occur in time, while of static facts we only say that they are located in time while they last. It seems, therefore, that because philosophers such as Strawson did not realize that events are sub-types of facts, seeing only that we may say of events that they occur in time, they hastily concluded that only events (and things) are located in time, opposing them to timeless facts. But that this isn’t true can be shown even by inter-substitutivity salva veritate: it is correct to say that the event, the occurrence of Caesar’s crossing the Rubicon, was a fact and that this fact occurred in 47 BC, as a concrete dynamic fact. On the other hand, the static social fact, the political state of affairs established by Caesar’s crossing the river was far more enduring. Being a static fact, it was the political situation that led, as is well-known, to the fall of the Republic. However, it seems clear that the state of affairs brought about by the crossing of the Rubicon was spatially limited to the Roman Empire and temporally limited to the time from Caesar’s crossing the Rubicon to his coronation as Caesar and up until his assassination. It was not something that existed in Greenland or that endured until the present, even if in a misleading way our ordinary language can be confusive by allowing us to use the present tense to speak about historical facts.
     The relevant conclusion is that by having the broadest scope, the so often vilipended word ‘fact’ remains the ideal candidate for the role of ultimate truth-maker in a correspondence theory of truth. Facts are universal truth-makers.
24. Church’s slingshot argument
As already noted, for Frege a sentence’s reference is its truth-value. To refute the charge that this view is implausible, the Fregean logician Alonzo Church devised a slingshot argument. He wanted to show that by means of inter-substitutability of co-referentials we can prove that the most diverse sentences can only have a truth-value as their reference.
     Church’s argument is in my view equivocal, but telling. Its basic assumption is that when one constituent expression is replaced by another, so that their partial references (the references of their singular terms) are interchangeable, the reference of the whole sentence does not change. I will begin by explaining his slingshot argument, underlining its supposedly co-referential definite descriptions (Church 1956: 25):

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

According to him, if it is plausible that sentences (2) and (3) are, if not synonymous, at least co-referential sentences, then (1) has the same reference as (4). Since (4) seems to concern a fact completely different from (1), it seems that the only thing left as the same reference is the truth of both sentences. Hence, The True is the only referent of all these sentences.
      However, the argument proves to be unsustainable when we pay attention to what should be the real reference of each singular term of these sentences. In sentence (1) the proper name ‘Sir Walter Scott’ and the definite description ‘the author of Waverley’ are two singular terms expressing different modes of presentation of the same human being. These modes of presentation make what we could call two partial references to Walter Scott, namely, references that must be partial relatively to the whole reference of the sentence. In sentence (2) again, the nominal expression ‘Sir Walter Scott’ and the definite description ‘the man who wrote the twenty-nine Waverley novels altogether’ both refer in different ways to the same Walter Scott. The third sentence is the tricky one. Its reference is unclear: Walter Scott? The number 29? Both in one? The combination Scott-29? The answer appears when we paraphrase sentence (3) so that it gives back in a transparent way its complete informative content. Now, carefully considering the confusing sentence (3), we see that the only way to reveal its content in a transparent way without any addition or loss of sense is to split the sentence into the following conjunction of two sentences: (5) ‘29 is the number of Waverley novels and Sir Walter Scott is the man who wrote that many Waverley novels altogether.’ Sentence (5) makes explicit all the content wrapped up in sentence (3). For the sake of clarity, replacing in (5) ‘=’ for ‘is (the same as)’ and ‘&’ for ‘and,’ we can still unpack (3) as:

6.     (29 = the number of Waverley novels) & (Sir Walter Scott = the man who wrote the many Waverley novels altogether).[6]

That is: Sentence (3) confusingly compresses nothing less than a conjunction of two identity sentences, each with its own proper partial references given by the singular terms flanking their identity signs. They are the number 29 in the first sentence and Walter Scott in the second. Finally, we come to the analysis of sentence (4): ‘29 is the number of counties in Utah,’ which means the same as the identity sentence (7) ‘29 = the number of counties in Utah.’ Here, each singular term that flanks the identity sign has the number 29 as its partial reference. So analyzed, the derivation appears as:

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

Now, although all these sentences are true, Church’s argument has by now lost its initial plausibility. Sentences (1) and (2) have as the partial references made by their singular terms Walter Scott under different guises. However, sentence (3) is a conjunction of two identity sentences, each with its own very distinct partial reference. The object referred to by the flanking terms of the first identity sentence of (3) is the number 29 (as the number of Waverley novels), while the object referred to by the flanking terms of the second identity sentence of (3) is Sir Walter Scott (as the man who wrote the Waverley novels). Finally, sentence (4) has as a partial reference made by its singular terms only the number 29 (as the number of counties in Utah), without referring to Walter Scott, as it should. That is:

In the composed sentence (3), the second sentence of the conjunction is the only one that preserves as the partial reference made by its singular terms the references of (1) and (2), while (4) is an identity sentence that has as the partial reference made by its singular terms only the partial reference of the first sentence of (3). However, this is precisely what should not occur because the preserved partial reference has nothing to do with the partial references made by the singular terms of the sentences (1) and (2) and supposedly with the whole references of these sentences.

     In other words, we can say that in a surreptitious way the replacements slide equivocally from having as partial reference Walter Scott in (1) and (2), to a Walter Scott, together with the number 29 in (3), and to the number 29 in (4). This means, according to the principle of compositionality applied to complete sentences, that the references of sentences (1) and (4) should indeed be very different. Initially, the flaw is not easy to spot, because sentence (3) contains both these partial references conjoined in a grammatically confusing way. We have the impression that the partial references of (3) seem to be something like an amalgam of Walter Scott and 29, say, a ‘Scott-29,’ while they are and must, in fact, be totally distinct. The replacements would only respect the compositionality principle, warranting the sameness of the sentences’ references, if the argument could prove that the partial references of all the sentences could be replaced without furtively inviting the reader to conjoin in sentence (3) partial references to completely different objects.
25. Sub-facts and grounding facts
If we take the whole reference of the sentence as not a truth-value but a fact, we get much more intuitive results. In what follows, I will consider Church’s intended derivation, not only to introduce facts as referents of sentences, but also to introduce a very useful distinction between sub-facts and grounding facts. As will be seen, this distinction fills a gap in Frege’s explanation.
     We need to distinguish at least two facts referred to by identity sentences. The first is the sub-fact: it is the perspectival fact as the appearance immediately revealed through a particular mode of presentation expressed by the statement. I will call it a sub-fact and make the diversified sub-facts the objective correlates responsible for differences in the modes of presentation constitutive of the different sentences senses (thoughts, rules) concerning one and the same object, e.g., Walter Scott and the author of Waverley. This is why Church’s sentences (1) and (2) can be seen as expressing different senses or thoughts. They evoke different perspectival sub-facts. They indirectly represent different sub-facts, since (i) being Sir Walter Scott is not the same thing as (ii) being the author of Waverley and (iii) being the man who wrote the 29 Waverley novels altogether… In this way, sentences (1) and (2) respectively show two different sub-facts that contain perspectival objects of reference that as such differ from one another. Using the term ‘being’ to indicate that we are speaking about a matching correlate, the sub-facts represented by:

(1)     Sir Walter Scott is the author of Waverley.
(2)     Sir Walter Scott is the man who wrote the 29 Waverley novels altogether.

Can be respectively represented as follows:

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

These sub-facts are of contingent differences since Sir Walter Scott could have not written the Waverley novels or any novel in the first case, and he could have written a different number of Waverley novels in the second. (If you accept that there is a relational trope of identity, you should accept that there are here relational tropes of difference.)
     Nonetheless, it is also clear that (1) and (2) are identity sentences. This is so because these sentences can be understood as referring under different guises to only one object, the person called Walter Scott, justifying the employment of the ‘is’ of identity. In this sense, sentences (1) and (2) represent an identity, which can be expressed simply by ‘Walter Scott = Walter Scott.’ That is, they can represent the self-identity of Walter Scott considered in full, as the ultimate bearer of all descriptions (under all possible perspectives) that we might intend to use to refer to it. Among the descriptions we associate with the name ‘Walter Scott’ we can select ‘the person with the title of Sir named “Walter Scott”’ (that is, ‘Sir Walter Scott’), ‘the author of Waverley’ and, certainly, ‘the man who wrote the 29 Waverley novels altogether,’ that is, the constituent expressions of (1) and (2). Now, this primary fact expressing the self-identity of Walter Scott considered in full is what I call a grounding fact. Characteristic of the grounding fact is that it must be able to unify all the sub-facts, all the facets revealed by its multiple modes of presentation. This is what remitting us to sentences of the form a = a make sentences with the form a = b identity sentences.
     Now, consider one of these definite descriptions more carefully, for instance, ‘the author of Waverley.’ As we saw, the mode of presentation is intentional and internal, considering that the reference can be absent. But when the mode of presentation isn’t empty, as in this case, it also exposes something external, evoking what I could spell out as ‘being the author of the Waverley novels.’ This should be seen as an objective entity, a sub-object mediating our reference to the object Walter Scott that belongs to the grounding fact of identity, namely, that Walter Scott is (the same as) Walter Scott.
     As well, ‘the author of Ivanhoe’ (who was also Walter Scott) is a mode of presentation of the sub-object ‘being the author of Ivanhoe,’ though it ultimately refers to Walter Scott. Now, take the sentence:

(a)     The author of the Waverley novels is the author of Ivanhoe.

This sentence evokes two different sub-objects that together form the contrastive sub-fact that being the author of Waverley is not the same as being the author of Ivanhoe. But this sub-fact also consists of two modes by which the same object is given, whose identity is the grounding fact that can be directly represented by the sentence ‘Being Walter Scott [in full] = Being Walter Scott [in full],’ where ‘in full’ here means that we are intending to consider all the conceivable modes of presentation of the object Walter Scott, far beyond the limited knowledge of this or that particular speaker.
     Moreover, it seems clear that the sentence (a) must also be able to express the two thoughts representing the two kinds of facts considered. First, we have a derived thought expressible by the sentence (a1) ‘Being the author of Waverley novels isn’t the same as being the author of Ivanhoe,’ representing directly the sub-fact and indirectly the grounding fact. Second, we have the basal thought directly expressible by the sentence (a2) ‘Being Walter Scott [in full] = being Walter Scott [in full]’ representing the grounding fact directly.[7]
     According to the foregoing analysis, when I say ‘The author of Waverley novels is the author of Ivanhoe,’ I am saying two things. First, by means of intentional modes of presentation, I am expressing the derived thought evoking an objective difference. This thought can be expressed by the sentence ‘Being the author of Waverley ≠ (isn’t) being the author of Ivanhoe,’ representing a derived fact. Indeed, it is an objective factual difference that a person writing Waverley is not the same as a person writing Ivanhoe, even if they are both the same person (he was writing different stories at different places and times…). However, since when I say ‘The author of Waverley is the author of Ivanhoe I use an ‘is’ of identity, I also mean the basal thought expressible by the sentence ‘The author of Waverley = the author of Ivanhoe,’ indicating that under different guises I am presenting the grounding fact that ‘Being Walter Scott = Being Walter Scott.’ It is because of the two – the grounding fact along with the sub-fact – that identities of the kind a = b are able to express identities in their differences.
     Now, assuming the kind of neo-descriptivism proposed in Appendix I of this book, we can make explicit the above-mentioned doubling of the presented facts by stating each of the four sentences of Church’s reasoning as follows:

(1a) Sentence expressing the derived thought representing the sub-fact: Being Sir Walter Scott ≠ being the author of Waverley.
(1b) Sentence expressing the basal thought representing the grounding fact: Being Walter Scott [in full] = being Walter Scott [in full].

(2a) Sentence expressing the derived thought representing the sub-fact: Being Sir Walter Scott ≠ being the man who wrote the 29 Waverley novels altogether.
(2b) Sentence expressing the basal thought representing the grounding fact: Being Walter Scott [in full] = being Walter Scott [in full].

(3a) Sentence expressing the derived thought representing the sub-fact: (Being 29 ≠ being the number of Waverley novels) & (Being Sir Walter Scott ≠ being the man who wrote the 29 Waverley novels altogether).
3b) Sentence expressing the basal thought representing the grounding fact: (Being 29 = being 29) & (Being Walter Scott = being Walter Scott).

(4a) Sentence expressing the derived thought representing the sub-fact: Being 29 ≠ being the number of counties in Utah.
(4b) Sentence expressing the basal thought representing the grounding fact: Being 29 = being 29.

The sub-facts show why the semantic contribution of each referential component in identities with the form a = b, due to the semantic-cognitive rules constitutive of the derived thought, can be different. The sub-fact that being Sir Walter Scott isn’t the same as being someone who wrote 29 Waverley novels discriminates more than the sub-fact that being Scott isn’t the same as someone writing the Waverley novels. And regarding true sentences, this discrimination isn’t just a mentally considered mode of presentation, a cognitive rule, but also the representation of something objectively or factually given in the external world (corresponding to different ‘ways the object gives itself to us,’ using Frege’s words). The above presented evocations of sub-facts all lead us to two grounding facts of identity showing how many different senses referring immediately to qualitatively different sub-facts refer mediately to something numerically identical. On the other hand, in sentences with the form a = a, such as ‘the morning star = the morning star,’ the sub-fact is already the identity ‘Being the morning star = being the morning star.’ The corresponding grounding-fact, additionally, may also be the same identity, if not the identity ‘Being Venus = being Venus,’ depending on the speaker’s intention.
26. Taking seriously the sentence’s reference as a fact
I think I have shown that the most plausible option concerning the nature of reference is to side with philosophers like Russell and the earlier Wittgenstein. These philosophers assumed that the reference of a statement is a fact – a fact that in the usual case is understood as a contingent arrangement of cognitively-independent tropical components commonly given (completely or partially) in the external world, although they can also belong to an internal (psychologically accessible) reality. Facts would satisfy the Fregean condition that the reference of a sentence is an object: they are in some sense independent, complete, closed. They would satisfy the condition that thoughts expressed by sentences should also be modes of presentation of their references, the latter – particularly as sub-facts – being as numerous and diverse as their thoughts. Finally, unlike truth-values, facts would smoothly satisfy the principle of compositionality: they would always vary in accordance with variations in the references of the senses of component parts of the sentences as we understand them.
     If we assume the answer given above, we are able to solve a vexing problem concerning which fact the thought expressed by a sentence refers to. Consider the following sentences:

1.     The morning star is the morning star.
2.     The morning star is the evening star.
3.     Venus is the morning star.
4.     Venus is the second planet orbiting the Sun.
5.     Venus is the brightest planet visible in the sky.
6.     Venus is the only planet in our solar system shrouded by an opaque layer of highly reflective sulphuric acid clouds.
7.     The morning star is the only planet in our solar system shrouded by an opaque layer of highly reflective sulphuric acid clouds…

On the one hand, it is intuitively correct to say that each of these sentences refers to a different fact. Sentence (1) is tautological, proclaiming the self-identity of the morning star, while sentences (2) to (7) provide information on different factual contents regarding the planet Venus. On the other hand, since all singular terms composing these identity sentences have the same ultimate reference, the planet Venus, it also seems clear that in the end all these identity sentences must have the same reference, representing the same fact in the world. How can we reconcile these two seemingly correct views?
     The answer departs from the distinction already made in the last section: first, there must be a privileged grounding fact able to be described that can be identified as the ultimate truth-maker of all these identity sentences about the planet Venus. Second, this grounding fact must in some way contain the facts immediately indicated by the different cognitive values of sentences (1) to (7) above as its perspectival sub-facts. My suggestion is that this last task can be accomplished by the references of identity sentences, insofar as the identification rules of their singular terms are considered in full, including all their fundamental and auxiliary descriptions.
     Now, assuming our proposed view of proper names’ meanings as abbreviations of bundles of descriptions centered in those constituting their fundamental identification rules, then the proper name ‘Venus’ in full includes in its most complete content all the already known modes of presentation. This means that definite descriptions such as ‘the morning star,’ ‘the second planet orbiting the Sun,’ ‘the brightest planet visible in the sky,’ etc. can be made at least probable by applying the concept of Venus in full. (I say ‘made at least probable’ because, in the case of the application of most identification rules, any particular description-rule of the bundle might be wrong and remain unsatisfied.) If this view is correct, then there is only one sentence that could describe the grounding fact as the ultimate truth-maker or verifier of any identity sentence concerning the planet Venus, including the sentences from (1) to (7) above. We can present it as the grounding fact (8) that being Venus with all its known sub-factual identificational implications is being Venus with all its known sub-factual identificational implications, represented by the basal thought expressed by the sentence:

(9) Venus [in full] = Venus [in full]

My contention is that rightly understood this sentence summarizes the most complete basal thought able to represent the single grounding fact, which considered in its entirety can be regarded as the truth-maker for any identity sentence about the planet Venus. (To represent sub-facts we have the already called derived thoughts.)
     It is not hard to explain why things are so. If the full meaning of the proper name ‘Venus’ is understood as an abbreviation of the whole bundle of descriptions regarded as uniquely identifying its object (Cf. Appendix of Chapter I, sec. 4), then this proper name should include descriptions like ‘the morning star,’ ‘the evening star,’ ‘the second planet orbiting the Sun,’ ‘the most brilliant planet visible in the sky,’ ‘the only planet in our solar system shrouded by an opaque layer of highly reflective sulphuric acid clouds,’ and many others. Consequently, from the sentence ‘Venus [in full] = Venus [in full]’ we can inferentially derive sentence (2) ‘The morning star = the evening star.’ We do this simply by replacing the first occurrence of the name ‘Venus’ with the definite description ‘the morning star,’ which the name ‘Venus’ (in full) abbreviates, and the second occurrence of the name ‘Venus’ (in full) with the description ‘the evening star,’ which the name Venus also abbreviates. In a similar way, we can obviously (inductively, at least) infer all the other above presented co-referential identities from (1) to (7). Thus, rightly understood the sentence ‘Venus [in full] = Venus [in full]’ should express the basal thought able to represent a fact complex enough to comprehend all the sub-facts represented by each of the thoughts expressed by the above sentences, which may be seen here as contingent a posteriori. (To convince yourself of this, look at the meaning of ‘Venus’ as presented in any encyclopedia, since it aims to offer an abbreviation of Venus in full.)
     In order to better support what I am suggesting, I can also use numerical identities like the following:

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

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

6.     The number 4 [in full] = the number 4 [in full]

But could the sub-facts expressed by sentences (1) to (5) be derived from (6)? Obviously, the answer must be in the affirmative, since we are dealing with a deductive system. After all, I wrote the five sentences above simply based on deductive inferences from my knowledge of the grounding fact that being the number 4 = being the number 4!
      However, one could still object that a sentence like ‘Venus [in full] = Venus [in full]’ is a tautology: a necessary truth. How could a necessary truth ground contingent truths like, ‘Venus is the brightest planet visible in the sky’?
     My answer to this objection is that for a privileged user of the word (a Venus specialist) who is supposed to know all the relevant information about Venus, this proper name expresses an identification rule that can be approximatively summarized as follows:

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

As in the case of Venus-Hesperus (Appendix of Chapter I, sec. 10 (iii)), this is a kind of ‘one-foot’ identification rule, since the localizing rule is the only fundamental one and includes what would count in the characterizing rule (being a planet). For suppose we have as a characterizing rule ‘a bright planet somewhat smaller than the earth.’ In this case, one can imagine that if there were only one bright planet somewhat smaller than the Earth, this planet would be Venus, since one term of the inclusive disjunction of a fundamental identifying rule is already satisfied. But if this were true, since we can imagine a possible world where there is just one bright planet somewhat smaller than the Earth with an orbit outside the Earth’s and no second planet, this planet should then be Venus, what is absurd. On the other hand, the localizing rule contains the essential content of the characterizing rule: Venus is a planet. If Venus were to lose its atmosphere or a major share of its mass (or in a different possible world never had them), insofar as it had been discovered to be the second planet from the Sun and the Earth the third, it would still be our Venus! Indeed, so understood it seems that the identification rule for Venus is applicable in any possible world where the planet Venus can be said to exist or to have existed.
     The case of Venus is somewhat like the case of the lines ‘aᴖb-aᴖc’ drawn to localize the center of a triangle without any call for a characterizing property; the characterizing description can be irrelevant or non-existent. By the same token, without the localizing condition established by the identification rule of Venus as the second planet, it would be impossible to identify Venus. The application of many other descriptions does not produce criteria, but only symptoms of the planet’s existence, since they make the applicability of the descriptions only more or less probable. Auxiliary descriptions like ‘the brightest planet in the sky’ are symptoms, like ‘the highly reflective clouds of sulfuric acid’ that cause this brightness. If Venus lost its reflective atmosphere, it might cease to be the brightest planet, but would still not cease to be Venus. If Venus lost half of its mass but remained in the same orbit, it still would not cease to be Venus. But if for some reason Venus lost nearly all its mass and became a small orbiting object only a few miles in diameter, no longer large enough to be called a planet, we could only say that it once was Venus. If in a possible world Mercury never existed, Venus would be the first planet of the solar system and even if it were called ‘Venus,’ it seems clear that it would not really be our Venus, unless it had once been the second planet from the Sun (Venus) for at least some period of time. Indeed, if in another possible world the second planet were hurled out of the solar system thousands of years ago (Kripke 1980: 57-58), it could still rightly be recognized as our Venus, since it once satisfied its identification rule. We see that the condition of sufficiency applied to the one-foot identification rule of Venus is more demanding than in the usual two-foot case. We can see that limits can be set even in a swampy terrain where vagueness prevails.
     What I said about identity sentences also applies to other singular predicative and relational sentences. Consider the following ones:

1. Bucephalus was a material thing.
2. Bucephalus was a living being.
3. Bucephalus was a horse.
4. Bucephalus was a black horse of the best Thessalonian strain.
5. Bucephalus was a massive black horse of the best Thessalonian strain, owned by Alexander the Great.
6. Bucephalus: (355 BC – 326 BC) was the most famous horse of Antiquity; it was a massive black horse of the best Thessalonian strain, owned by Alexander the Great.
7. Bucephalus once swam across the river Granicus.

One could say that each of the first six sentences expresses different derived thoughts representing different sub-facts by means of increasingly detailed modes of presentation expressed by their respective predicative expressions. However, relative to them there is a grounding fact that in a summarized form is represented by the basal thought expressed by sentence (6), since the truth of all the others can be implied by the truth of this thought. Indeed, (6) is nothing but an abbreviated expression of the identification rule for Bucephalus, with a localizing and a characterizing description and by these means furnishing a summarized definitional criterion. The sub-facts represented by sentences (1) to (5) are all included in the grounding fact represented by sentence (6). These facts are the immediate satisfiers of the diverse modes of presentation of Bucephalus given by each sentence. And the progression from (1) to (6) increases the complexity, insofar as new relevant predications are added. Statement (7) ‘Bucephalus once swam across the river Granicus’ is a different case: the very contingent auxiliary description ‘the horse Bucephalus who once swam across the river Granicus’ isn’t a relevant part of the fundamental description-rule (even if he didn’t cross, he would still be our Bucephalus). Nevertheless, it can still be derived from (6) considered in full, since this is believed (by privileged speakers) to be historically the case.
27. The riddle of identity in difference
There is a final point concerning the relationship between the sub-fact and the grounding fact. It concerns the unsatisfactory way that Frege solved the puzzle of identity. As he wrote, unlike sentences with the form a = a, a sentence with the form a = b is informative because it refers to the same object by means of different modes of presentation, by means of the different senses of a and b (1892: 26). However, we can still ask how this identity is possible, since the modes of presentation are different and since we are not intending to speak about the mere self-identity of the reference, as Frege also acknowledged. I call this ‘the riddle of identity in difference.
      To see the problem clearly, consider again Frege’s sentence (i) ‘The morning star = (is) the evening star.’ A more fully unpacked cognitive sense of (i) can be presented as:

The brightest star in the morning sky, understood as referring to the second planet orbiting the Sun between Earth and Mercury (Venus) = (is) the brightest star in the evening sky, understood as referring to the second planet orbiting the Sun between Earth and Mercury (Venus).

Here I have not [j1] [c2] underlined non-definitional expressions of what I call immediate senses presenting perceptual sub-objects like the morning and the evening star, though I have underlined expressions of what I call mediated senses, which here are definitional. The immediate senses build the derived thought representing a sub-fact (that being the morning star isn’t being the evening star), while the mediated senses essentially build the basal thought representing the grounding fact (that Venus is Venus). Here we have the hidden reason for the riddle of identity in difference: the immediate senses of the expressions flanking the identity sign in (i) are obviously different, but they both evoke the underlined mediated, in fact primary or leading sense (essentially building the basal thought that Venus is Venus) with the form a = a.
     Obviously, this last sense, the basic thought that the second planet orbiting the Sun… is the second planet orbiting the Sun… is not yet the reference, since it is constituted by the expression of the self-identity of the cognitive identification rule constituting the core definitional sense of the name ‘Venus’ and its conventionalized surroundings (Venus in full). It is only because both expressions flanking the identity sign in (i) implicitly evoke the same proper identification rule for the planet Venus that we are allowed to place an identity sign between them! In order to make the point still clearer we can appeal to the following schema:

Sentence:               The morning star                  = (is)       the evening star.
Derived:                IR: the brightest                   ≠              IR: the brightest
thought:                 star in the morning                               star in the evening.
sub-fact:                Being the morning star isn’t being the evening star.

Basal                      IR: The second planet…     =              IR: the second planet…
thought                    (Venus)                                                  (Venus).
grounding fact:    Being Venus is the same as being Venus.

In sum: the singular terms ‘Morning Star’ and ‘Evening Star’ are responsible for the difference present in what I call the immediate senses of the descriptions (the Fregean senses) constituting a derived thought evoking a relational sub-fact showing the differences between two sub-references. Expressing the derived thought we describe the sub-fact as: ‘being the brightest star seen in the morning sky differs in place and time from being the brightest star seen in the evening sky’ (one can even point to the two opposite sides of the sky in which alternately one or the other appears every twelve hours). Furthermore, the ‘is’ understood as ‘is the same as’ is the only indication of the identity of the implicitly intended mediated senses building the basal thought expressed by the sentence ‘The second planet orbiting the Sun between Earth and Mercury (Venus) = the second planet orbiting the Sun between Earth and Mercury (Venus).’ These mediated senses have multiple guises that are implicit in the names flanking the identity sign in the statement ‘Venus [in full] = Venus [in full]’ expressing the basal thought that could be known in full only by specialists. The statement expressing the derived thought is contingent a posteriori, while the statement expressing the basal thought can be seen as a necessary priori.
     A somewhat different example is the sentence ‘The morning star is Venus.’ Here the schema is:

Sentence:               The morning star                  = (is)       Venus.
Derived: IR: the brightest                   ≠              IR: the second
Thought:               star at dawn                                         planet.
sub-fact: Being the morning star isn’t being Venus.
Basal                      IR: the second planet          =              IR: the second planet
Thought:                (Venus)                                                  (Venus).
grounding fact:    Being Venus is the same as being Venus.

It is by now clear that the identity expressed by sentences of the kind a = b is an identity in difference. This means that in fact we have two levels of sense or thought. The first is the derived thought. It represents the perspectival sub-fact with its sub-objects expressing a difference (Being the morning star isn’t the same as being the second planet from the Sun). The second, intermediated by the first one and indicated by the ‘is’ of identity, is the basal thought representing the ultimate grounding fact that being Venus is the same as being Venus, which has the sub-facts as facets, as manifestations. The derived thought is contingent a posteriori, while the basic thought is necessary a priori since it is tautological.
     Now, how should we deal with cases in which the elements of the basal thought responsible for the identity, like the planet called ‘Venus’ in the statement above, lack a proper name? Consider the identities (i) ‘Everest = Chomolungma,’ (ii) ‘aᴖb = aᴖc’ (concerning Frege’s example of two different ways to name the center of a triangle), (iii) ‘Afla = Ateb’ (the two names that Frege gave for the same imaginary mountain). In order to get an answer, we need at first consider that the derived Fregean senses are thoughts of a difference, evoking different contingent sub-objects. But these sentences also implicitly evoke a basal conjoining sense, a conjoining identification rule, which refers to what we might call respectively the ‘Everest-Chomolungma,’ the ‘aᴖb-aᴖc,’ and the ‘Afla-Ateb,’ which in fact are three new nominative expressions. The law of identity makes obvious that:

(1)  ‘Mt. Everest is Chomolungma’ so understood can be replaced by ‘Everest[-Chomolungma] = [Everest-]Chomolungma,’
(2)  ‘aᴖb = aᴖc’ can be replaced by ‘aᴖb[-aᴖc] = ‘[aᴖb-]aᴖc,’ and
(3)  ‘Afla = Ateb’ can be replaced by ‘Afla[-Ateb] = [Afla-]Ateb.’

These three replacing basal thoughts respectively represent the three different grounding facts as the full self-identities that they are. This is respectively what sustains the identities expressed by the ‘is’ in the sentences (i), (ii) and (iii).
     We can apply a similar analysis to identities between concept-words of the form (x) (Fx = Gx). Consider the identity ‘Heat in gases is molecular kinetic energy.’ Note that the word ‘heat’ is ambiguous. It can mean a mere subjective feeling (heat1), like the feeling of increased bodily heat after exercise, which cannot be identified with molecular kinetic energy. But in the present case ‘heat’ means external temperature as it is normally felt by people (heat2). A third sense is independent of our sensations: it is heat as ‘measured temperature’ determined by thermometers (heat3) (in the sense of heat2, our bodies work as course, imprecise thermometers). Moreover, since molecules can have different masses and speeds, the most precise identity sentence would be ‘Temperature in a gas (heat3) is the average kinetic energy of its molecules.’ This sentence expresses two different modes of presentation of the same thing, that is, a derived thought that can be expressed by means of the following difference:

(i)    Temperature in a gas (heat3) ≠ average kinetic energy of its molecules.

This secondary thought refers only to the sub-fact that the (macro-physical) temperature that we can measure with a thermometer (and feel as heat2) is something phenomenally different from the (microphysical) average kinetic energy of the molecules of a gas such as the air around us.
     In a next step, we are able to consider the basal thought establishing a tautological identity based on conventions. This thought can be expressed by the whole more complete assertoric sentence:

(ii)   [Average kinetic energy-temperature-] heat3 of a quantity of gas = average kinetic energy-temperature [-heat3 of a quantity of gas].

     Now, we can read the sentence ‘Heat in gases is molecular kinetic energy’ as something made explicit by sentence (ii), which can be read in two ways: (a) considering only what is outside the brackets as explicitly emphasized, which expresses the derived thought of a difference and represents the sub-fact of the difference above; (b) emphasizing the whole, including what is in brackets. Understanding (ii) as (b), what we have is a basal thought referring to a grounding fact of definitional self-identity. This identity requires as an assumption the acceptance of the kinetic theory of gases, which makes (b) a tautology. This means that if we read (ii) in the sense (a), disregarding what is in the brackets, we can see it as a contingent a posteriori thought, since it can be denied without contradiction, while if we read (ii) in the sense (b) it can be considered necessary a priori, since it cannot be denied without contradiction.
     Consider now the sentence ‘Water is H2O.’ I think Avrum Stroll was right when he noted that here the ‘is’ expresses constitution; the sentence usually means ‘Water is made of H2O’ rather than ‘Water is the same as H2O’ (1996, 46 f.). However, this does not make a relevant difference for what I will try to say and context can lead us easily to read again the ‘is’ as expressing identity.
     As already noted (Appendix to Chapter II), the concept-word ‘water’ has two nuclei of meaning: a superficial one, that of an aqueous liquid (transparent, tasteless, odorless, etc.), and a deep one, a substance called by chemists dihydrogen oxide or H2O (which includes much more than the simple chemical structure). This means that the complete sense of water must include the two nuclei. However, as in fact the presence of only one nucleus already allows us, in a proper context, to call the substance water, the most embracing criteria for the application of the general term ‘water’ demands sufficient satisfaction of the (summarized) inclusive disjunctive rule:

DR: (Water is an) aqueous liquid and/or (water is) dihydrogen oxide (H2O).

Philosophers have created a pseudo-problem by insisting that the criterion of application of the conceptual word ‘water’ must be either aqueous liquid or dihydrogen oxide, as if it were a dilemma.[8]
     Now, assuming that the ‘is’ is one of constitution and not of identity, the statement (i) ‘Water is H2O’ in fact means: (ii) ‘Aqueous liquid and/or dihydrogen oxide… is made of dihydrogen oxide.’ Since it could be that water isn’t made of dihydrogen oxide and only the first statement of the DR is true, it is possible for the whole statement to be false, which makes it a contingent a posteriori truth and not a necessary a posteriori truth, as Kripke would like it to be. However, as we will see in the next section, in some contexts statement (i) is rather seen as a necessary a priori truth.
28. Contexts of interest: no need for a necessary
 a posteriori
This double core sense of the general term ‘water’ helps to explain Saul Kripke’s in my view as much insightful as illusory discovery of the necessary a posteriori. But in order to better understand the confusions involved, we need to add to the sentences the contexts in which they are spoken.
     A first point to notice is that in the case of a sentences of the kind a = b uttered in different contexts we can enhance or magnify or emphasize its immediate (Fregean) perspectival sense that builds a derived thought (representing a sub-fact), or we can enhance or magnify or emphasize its mediated sense that builds the basal thought (representing a grounding fact).[9] Thus, in cases like ‘Water is H2O’ we can emphasize the immediate core sense of the concept-word ‘water’ as an aqueous liquid or its mediated core sense as dihydrogen oxide. Here I need to speak again of the contexts of interest of the linguistic agents, meaning thereby contextualized practical aims from which we can infer what is meant.
     Two contexts of interest are important regarding the main examples above: the popular and the scientific one. Thus, considering the sentence ‘The morning star is the evening star,’ we can contextually emphasize the derived thought composed by immediate senses (modes of presentation, identification rules) representing the external, phenomenally given objects, considering the difference between being the brightest star in the morning and the brightest star in the evening. If we do this, we leave the identity ‘Venus = Venus’ in the background. This can be the case, for instance, when contemplating the beauty of the starry sky at night and, after localizing the evening star, we tell a child that it is also the morning star. In this case, we think like Frege. We emphasize the different modes of presentation of the same object, a difference that as such represents nothing but an empirical sub-fact made by two different aspectual presentations of what we believe to be the same thing. We regard the thought that the morning star is the evening star as contingent a posteriori, since it mainly represents the sub-fact of the difference, although we are also aware that we are emphasizing the different ways by means of which the same thing presents itself to us.
     Nonetheless, in a scientific context of interest, such as one in which astronomers use a telescope to study the surface of Venus, when they consider the sentence ‘The morning star is also the evening star,’ what they usually have in mind and emphasize is the numerical identity of the object of both modes of presentation. These are the mediated senses constituting the basal thought representing the grounding fact of the self-identity of Venus, which Kripke particularly emphasized in his writings. In this case, we read the statement as preferentially meaning the basal thought that ‘Venus [in full] = Venus [in full],’ which is a necessary a priori statement, since what we above all affirm is the tautological grounding fact that being Venus is the same as being Venus. It leaves the different guises of sense in the background, as secondary effects, insofar as we assume the truth of our scientific astronomical views.
     Now, consider again the statement ‘Water is H2O’.[10] In a popular context of interest which arises when fishermen decide to dig a well to obtain fresh water for drinking and washing, this statement is read as emphasizing the sense of the word ‘water’ as a precious aqueous liquid (transparent, tasteless, odorless, drinkable… the popular nucleus of meaning), and it is for them a contingent matter that it is made of H2O insofar as it satisfies their practical aims. Because of this, the statement is seen as contingent a posteriori, since it means ‘This aqueous liquid is made of H2O,’ this expressing a derived thought representing a sub-fact that does not demand that water is necessarily H2O, being deniable without contradiction.
     On the other hand, when the context of interest is scientific, for instance, formed by chemists measuring the acidity of a sample of water, the word ‘water’ in the sentence ‘Water is H2O’ can be read as emphasizing the sense of water as dihydrogen oxide (the scientific nucleus of meaning). In this case, the whole sentence is seen as preferentially expressing a thought representing a grounding fact expressed by the identity ‘Water [H2O] = H2O [water],’ which has the form a = a, that is, of a necessary a priori tautology based on our scientific assumptions.
     I think that philosophers like Kripke, by considering ‘Water is H2O’ a necessary a posteriori statement, simply confuse (i) the aposteriority of the statement which emphasizes that water is an aqueous liquid made of H2O with (ii) the a priori necessity of the statement that emphasizes that water as dihydrogen oxide is the same as H2O, mixing the aposteriority of (i) with the necessity of (ii).
     A somewhat different emphasis can be found in the statement ‘Heat is molecular movement,’ here understood as ‘Heat = molecular movement.’ If we emphasize the ordinary immediate senses, the derived thought, the difference between heat2 (heat as it is normally felt) and the average kinetic energy of a gas, the emphasized sense or thought is contingent a posteriori, and the fact referred to is something learned by experience. This could be the case even using heat3 (temperature) as a fallible measure of average kinetic energy.
     On the other hand, if we assume the truth of the kinetic theory of gases in a scientific context in which we are measuring temperatures, the statement can be understood as emphasizing the mediated sense of the identity expressible by: ‘Temperature of a gas [-average kinetic energy] = average kinetic energy [-temperature of a gas],’ insofar as it is read as expressing the basal thought representing the grounding fact of an assumed identity, being therefore a (conditional) necessary a priori thought. In this reading, our conceptual rules for temperature and for average kinetic energy are blended into a single identification rule which assumes the kinetic theory of gases.
     It seems to me that by considering identities of the kind a = b, Kripke misleadingly conjoined the aposteriority of the emphasized derived identity thought with the necessity of the emphasized basal identity thought, concluding that the identities between nominal and conceptual terms have a necessary a posteriori nature that is only metaphysically explicable. However, if these names or concept-words serve as rigid designators applying to the same entities in all possible worlds, this is explained by their assumed mediated senses, which are of the kind a = a (or a[b] = [a]b) and not only a = b as representing a difference. A Wittgensteinian therapist would conclude that in the considered cases Kripke was the victim of deep grammatical ambiguities. Finally, insofar as the terms a and b used in identity sentences are viewed as rigid designators unavoidably applying the same ultimate object in all possible worlds where it exists, this is also only justified by the self-identity of a grounding fact.
29. Sense of a sentence: the thought
Now it is time consider the sense of a sentence. Here is Frege at his best! He made the right decision in suggesting that the meaning of the whole sentence is the thought (Gedanke) it expresses. To reach this conclusion, he applied his compositionality principle: combined in the right way, the senses of the component terms constitute the sense of the whole sentence. If, for instance, in the sentence ‘The morning star is a planet’ we replace the description ‘the morning star’ with the description ‘the evening star,’ which is co-referential though having a different sense, the reference of the sentence does not change; but the sense of the sentence must change. Indeed, the sense of the sentence ‘The evening star is a planet’ is different. However, the only other thing that has changed is what we use to call the thought expressed by the resulting sentence. Consequently, the sense of a sentence must be the thought it expresses. (Frege 1892: 32)
     The word ‘thought’ is ambiguous. One can use it to describe a psychological process of thinking, as in the utterance ‘I was just thinking of you!’ But it also seems to designate something independent of specific mental occurrences – a content of thought – such as the thought expressed by the sentence ‘12 x 12 = 144’ in the utterance: ‘The sentence “12 x 12 = 144” expresses a true thought.’ Frege had the latter sense in mind. In this usage, the word ‘thought’ means simply what the sentence (statement) says, which Frege conceived of as some sort of eternal (timeless) Platonic entity. A way to make the difference explicit would be to call the Fregean thought a thought-content. The terminology here counts because the word ‘thought’ is the only term in ordinary language that has a sense corresponding to more technical terms like ‘proposition’ or ‘propositional content.’[11]    
     Frege has a criterion for deciding what belongs to a thought. For him, everything that contributes to determining the truth-value of a sentence should belong to its thought. Thus, using his own example, the sentences ‘Alfred hasn’t arrived’ and ‘Alfred hasn’t arrived yet’ express the same thought, since the word ‘yet’ means only an expectation regarding Alfred’s arrival without contributing to the sentence’s truth-value (Frege 1918: 64). The sentences ‘The morning star is Venus’ and ‘The evening star is Venus’ can be considered to express different thoughts because although the singular terms that make up these two identity sentences all refer to the same planet, they do this by means of different modes of presentation. That is, they make us follow different paths in determining their truth-value, or, as I prefer to think, they make us follow different associations of semantic-cognitive rules able to constitute correspondingly different verifiability procedures.
30. The thought as the truth-bearer
Another quite plausible Fregean thesis was that the primary bearer of truth is not the sentence, but rather the thought (proposition) expressed by it. I agree with this view. Although we can say that sentences, beliefs and even things and persons are true, they all seem to be true in a derivative sense.
     Consider the cases of things and persons. A useful test to identify secondary uses is that when a word is derivatively used we can replace it with a more appropriate word. If we say that a diamond is false, what we mean is that it is only an imitation diamond: a fake or counterfeit of a real diamond that deceives us so much that we can think false thoughts about it. When we say that Socrates was ‘true’ as a person, what we mean is that he was a truthful, trustworthy or reliable person, someone with integrity. But this is not always so. When we say that Sam’s belief is true, although we secondarily mean that he has a subjective psychological attitude concerning his (dispositional) thought – of finding it true – we primarily mean that his thought is true in a Fregean sense.
     One reason for preferring to say that the thought is the truth-bearer concerns the logical behavior of this concept. We deal with our concept of truth as an ‘as if’ directive idea, so that the real or actual truth-value of a thought is naturally conceived of as something invariant: if something is (really) true, it is always true; if something is (really) false, it is always false. Obviously, we can always err in judging and claiming something to be true (as das Fürwahrhalten) and can later discover it is false, and we can err in believing something to be false (das Fürfalschhalten) when it is actually true – this is often the case, and this possibility is inevitable, due to our inherent epistemic fallibility. But when we discover our error, we correct ourselves, in the first case not by claiming that the thought was previously true and now has become false, but by saying that it was always false, and in the second case we correct ourselves not by claiming that the thought was previously false and now has become true, but by saying that it was always true. What changed was our truth-claim expressing our judgment, not the truth-value. Moreover, it is fundamental to perceive that our inherent fallibility in holding thoughts to be true does not affect the invariability or immutability of the truth-value of the thought or proposition in itself. It must be so because it is beyond our fallible capacities to know with absolute certainty whether we have achieved this ideal, if we have indeed achieved it. This is how the logical grammar of our concept of truth works (and, beyond this, the grammar of our own concept of knowledge). If one wants to change something so fundamental, then to prevent confusion one should invent new terms instead, like ‘hturt’ and ‘eslaf’.
     Now, if the actual truth-value is immutable, its truth-bearer must also be unchanging, able to remain the same in order to retain this same truth-value independently of the time or place where we discovered it. Indeed, for Frege a really true thought remains true forever, just as a really false thought remains false forever. These entities are even abbreviated as ‘truths’ and ‘falsities’ respectively. Thus, it is deeply ingrained in our conceptual grammar that the entity that can be primarily called true or false must remain the same and possess the same truth-value so that what may change is only our cognitive grasp of it, our believing in its truth-value (das Fürwahrhalten). If this is so, then only the thought has the necessary stability to be the archetypical truth-bearer; for a thought is, according to Frege, unchangeable and eternal (a-temporal), being eternally (a-temporally) true or false independently of our grasping (fassen) it.
     Consider now the case of sentences as candidates for truth-bearers. Ambiguous sentences can express different Fregean thoughts, such as ‘John saw the man on the mountain with a telescope.’ In this case, the truth-value of the thought will be able to change according to the different thoughts or interpretations that we assign to the sentence. But if the truth-bearer were the sentence, the truth-value should remain the same, which cannot be correct. This is obvious in the case of indexical utterances like ‘I am in pain,’ which has different truth-values depending on the speaker.[12] The same sentence can change its sense-thought when uttered by different persons, and even when uttered by the same person at different times; correspondingly, what may change with the change in thought is the truth-value. Hence, thoughts and their truth-values are co-variant, while sentences and their truth-values are not, which leads us to the conclusion that the primary bearer of truth-value must be the thought or proposition.
     One could suppose that perhaps the sentence-token would be the truth-bearer, since it would be a different one depending on the time and place of the utterance, changing with the truth-value. However, we still have cases in which different sentences (token or not) say the same thing – express the same thought – in this way preserving the same truth-value. Consider, for example, the following statements, ‘It is raining,’ ‘Il pleut,’ ‘Es regnet,’ ‘Chove’… uttered in the same context. They all say the same thing, express the same thought, and all have the same truth-value, while their sentence-tokens are quite different. Indeed, the only justification for insisting on the immutability of the truth-value of these four different sentence-tokens (and types) is that their primary truth-bearer is the thought expressed by them, since what they say – their senses, their thoughts – is what remains the same. Finally, this is the case not only for indexical sentences but also for eternal sentences with the same content, though expressed in different languages.
     Likewise, beliefs, understood in a psychological sense, can only be derivative truth-bearers: if someone who believes something dies, his psychological belief also ceases to exist. Consequently, the truth-bearer must be the content of his belief. It must be his belief-content and not his belief in a dispositional psychological sense, since only the belief-content isn’t transitory. But this is so only because we understand the belief-content as the same as a Fregean thought, a propositional content.
     The core of the foregoing arguments can be summarized as follows: thoughts and their truth-values are not just invariantly related; when thoughts vary, they maintain a relationship of co-variance with their truth-values. This relationship is missing in the relationships between sentences or psychological beliefs and their truth-values. Because of this, the proper bearer of truth must be the thought (proposition, propositional content, belief-content), not the sentence or some personal psychological disposition to agree on a truth-value.
31. Facts as true thoughts?
As already noted, Frege also proposed that what we call a fact is the same thing as a true thought, because when a scientist discovers a true thought, he claims to have discovered a fact. As he wrote:
‘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. (1918: 74)
Indeed, when we say ‘John stated several relevant facts in his speech,’ we are speaking about facts as true thoughts. However, there is no warrant that this is not a derivative use of the word ‘fact.’ A researcher of nature can well exclaim ‘Facts! Facts! Facts!’ understanding by a fact simply what corresponds to the true thought, namely, some objectively given tropical arrangement. After all, it seems natural to think that if someone discovers a true thought, it is because he has a fortiori discovered the fact corresponding to it.
     A more decisive argument against thoughts as true facts came from J. L. Austin, who made it clear that Frege’s identification does not resist all linguistic replacements (1990: 170-171). If the sentence ‘What he affirms is true’ had the same sense as ‘What he affirms is a fact,’ then the replacement of ‘what he affirms’ with ‘his affirmation’ should be allowed without any change of sense. But, ‘His affirmation is true’ preserves the meaning, while ‘His affirmation is a fact’ makes sense only as a meta-linguistic sentence referring to the occurrence of his affirmation, and not to the content of the affirmation itself. The reason for this can only be that the true content of an affirmation – the Fregean thought – cannot be properly identified with a fact.
     The main reason why Frege believed that a fact is a true thought is that he advocated a conception of truth as redundancy, rejecting the correspondence theory. However, on the one hand, his arguments against correspondence theory (1918: 59-60) are unconvincing.[13] On the other hand, correspondence theory remains the prima facie most plausible view. It is the most natural and historically influential conception of truth, suggesting that propositions or thoughts are true when they correspond to facts as arrangements of elements in the world (Rasmussen 2014; Vision 2004). Moreover, the view of truth as correspondence is commonsensical, agreeing with our methodological principle of the primacy of common knowledge. Because of this, I will defend this theory in the last chapter of this book.
     Finally, I think I have found a plausible way to explain why some are tempted to say that facts are true thoughts. It seems that the source of confusion resides in a persistent ambiguity of our own natural language. Dictionaries in very different languages present us a variety of trivial meanings for the word ‘truth.’ However, two general meanings are almost invariably emphasized. I call them: thought-truth and fact-truth. Here are their definitions, according to the best dictionaries:

(a)   Thought-truth: Truth as consisting of things being as we believe they are, as conformity or accordance or correspondence of the thought with the fact it represents.
(b)  Fact-truth: Truth as the actual, real, existing fact in the world.[14]

It is regarding the philosophically most proper sense (a) that we have singled out the thought as the primary bearer of truth. This usage is shown clearly in sentences like ‘His words are true,’ ‘Tell me the truth.’ In the factual sense (b), we single out facts in the world as secondary truth-bearers in the sense of being real, and we use sentences like ‘The mentioned occurrence was true (was real),’ ‘We are searching for the true facts (the real facts),’ ‘The truth (the fact) is out there.’ The possibility of more adequate semantic replacements indicates the derivative character of fact-truths.
     As we have already seen, there are good reasons to think that sense (a) is primary while sense (b) is derivative, since in this last case we can replace the word ‘truth’ with more adequate ones like ‘reality,’ ‘existence,’ ‘actuality’… Anyway, ‘truth’ is very often used not only as ‘correspondence with facts’ but also replacing ‘an existing fact in the world.Thus, we can easily be misled by some extraneous motivation and confuse the two usages, mistakenly concluding that facts are true thoughts. This is what seems to have originated Frege’s confusion, giving us another example of equivocity as a common way of transgressing the internal limits of language (Ch. III, sec. 11).
32. The thought as a verifiability rule
As the application of the ascription rule (sense of the predicate) is subsidiary to the application of the identification rule (sense of the nominative term), the rule for applying the singular sentence (its sense or thought) can be seen as an association of semantic-cognitive rules. Ernst Tugendhat has identified this association with the verifiability rule in the case of the singular predicative statement (1976: 259, 484, 487-8), which implies the suggestion that this view can be generalized to all meaningful statements (See 1983: 235-6). Indeed, if the thought is an association of rules, then what results from such an association – the verifiability rule – must also have the character of a rule, even if it isn’t something previously conventionalized. Combining this with our acceptance of the correspondence view of truth and our salvaging of the fact as the universal truth-maker, this means that the thought should be a kind of associated or combined semantic-cognitive rule – a verifiability rule – whose function is to make us aware of a corresponding fact to which it is applied.[15]
     This reasoning unavoidably leads us back to the controversial idea of ‘verificationism,’ more precisely (and still worse) to semantic verificationism: the doctrine first proposed by Wittgenstein, according to which the (cognitive, informative) sense of a sentence is the rule or method or procedure used in its verification (1980: 29). As it is well-known, Wittgenstein’s idea was soon appropriated by the philosophers of logical positivism. However, after varied attempts to give it a precise formulation, it was in the end abandoned due to strong criticism, internal and external to the logical-positivist circle, which led to it being considered by many as unsustainable. This is presently the received view, even if sophisticated philosophers have never really abandoned the idea that some form or other of verificationism is indispensable (Cf. Misak 1995). Indeed, in the next chapter of this book I intend to offer replies to the main objections that philosophers have made against semantic verificationism, showing that these objections were not directed against its correct form, but rather against a straw-man called the ‘principle of verifiability,’ as it was wrongly construed by logical positivists.
     I am introducing semantic verificationism in this chapter speculatively, as an alternative and in fact as the most natural way to analyze Frege’s discovery of the thought as the cognitive sense (epistemic value, informative content) of a sentence. Now, suppose that the combined semantic-cognitive rule that constitutes the thought as expressed in an assertoric sentence is its verifiability rule, as complex as it may be. Then the verifiability rule in itself is the most proper truth-bearer. Then, if we show that this verifiability sense-thought rule is effectively applicable to the expected fact, this makes the rule true, which allows us to say derivatively that the sentence expressing it is also true. If, on the other hand, we show that this thought-sense-rule, though conceivable, isn’t effectively applicable to the expected fact, this makes it false and likewise the sentence expressing it. Moreover, if we cannot formulate a verifiability rule able to be at least in principle applicable to the fact, if we cannot even conceive its application, we must conclude that the declarative sentence is devoid of meaning, devoid of sense or thought, even if it may in some cases seem to have meaning.
     I think that this way to understand the truth of a thought is in line with Frege’s remark that although he regarded truth as the property of a thought, it does not seem to be a property in the usual sense of the word (Frege 1918: 61). Indeed, truth does not add anything to the combined cognitive rule called ‘the thought,’ except something dispositional, namely, its effective applicability as a verifiability rule in the appropriate context for its application. Moreover, the proposed identity between the Fregean concept of thought and the concept of a verifiability rule is also supported by the Fregean proposal that the identification criterion for what belongs to a thought is that it must have at least some role in the establishment of the thought’s truth-value.[16]
33. Frege’s Platonism
It is important to remember that for Frege thoughts and the senses that compose them are Platonic entities belonging to a third ontological realm, which is neither psychological nor physical (Frege 1918). For him, taking (a) the criterion of objectivity as being inter-subjectivity and independence of will, and taking (b) the criterion of reality as existence in space and time, we combine them in order to get three ontological realms:

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

The first realm is that of physical entities such as concrete objects, which are objective and real. These entities satisfy criteria (a) and (b): they are objective, since they are interpersonally accessible and independent of our will, and they are real since they are located in space and time. The second realm is that of psychological entities, mental states that he calls representations (Frege uses the word ‘Vorstellungen’ in a way that could be easily translated as qualia). These entities satisfy criterion (b) but not (a): they are subjective and real. By not being interpersonally accessible, they are subjective and often dependent on the will. However, they are still real, because they are in the mind and, consequently, in time and (we can add) space. There is, finally, a third realm, that of thoughts (propositions) and their constitutive senses. This realm satisfies criterion (a) but not (b). For Frege thoughts are objective but not real. Thoughts are objective, because, true or false, they are always interpersonally accessible: we can all agree, for example, that the Pythagorean Theorem expresses a true thought in Euclidean geometry. However, this third realm of thoughts is not real, because according to him thoughts are abstract things that cannot be found in space or time. Thus, the thought (the sense) of Pythagoras’ theorem is objective but non-real.
     There are, however, problems. One of them, noted by Frege, is that although for him thoughts are eternal (timeless), immutable, forever true or false, and not created but only grasped (gefasst) by us, they must have some kind of causal effect: they must be able to cause our grasping them in order to make judgments and act in the external world (Frege 1918: 77). How this interaction with something non-spatiotemporal is possible remains an unexplained mystery.
     Frege was aware of the difficulties, but the main reason why he felt he had to introduce this third realm of thoughts is that thoughts are interpersonally accessible, that is, they are objective, which makes them able to be communicable. Representations (Vorstellungen), on the other hand, are rather subjective psychological states that can vary depending on personal psychology and according to him could never become interpersonally accessible and there­fore are not communicable. Thus, for him the right way to explain how it is possible that we are able to share the same thoughts in conversation is to strictly distinguish thoughts from mere psychological representations, placing them in a supposedly shareable Platonic realm. In addition, if thoughts were on the level of representations, they would be dependent on changeable personal psychology and would lack their required stability as truth-bearers.
34. Avoiding Frege’s Platonism
Despite the above-suggested arguments, few today would accept Frege’s appeal to Platonism. After all, the Fregean form of Platonism not only commits us to an infinite multiplication of objective entities (all the infinite variety of true and false thoughts and their constitutive senses) but also seems to lack intelligibility. The price that Frege was willing to pay in order to avoid psychological subjectivism seems too high for us today.
     In my judgment, if we understand senses as rules, which usually are implicitly established conventions or something derived from them, there is a clear way to bring the empiricist view of thoughts as having a psychological-empirical nature in line with the view that as truth-bearers they must have stability and the possibility of being communicated. In order to establish this conclusion, I want to apply again the same strategy inspired by the ontological particularism of English empiricists, which I used in the construction of universals by means of tropes.[17] This is understandable since according to trope ontology, a thought should be made up of, at least dispositional, internal tropes: the mental tropes constitutive of some conventionally grounded verifiability rule whose application is at least conceivable. In order to accomplish this, I need only show that something like Fregean Platonic thoughts (objective non-real truth-bearers…), which I call f-thoughts (‘f’ from Fregean) can be defined in terms of psychological (real and subjective) p-thoughts (‘p’ from psycho­logical), though typically based on intersubjective linguistic conventions. In other words, I suggest that we can warrant the existence and stability of f-thoughts without hypostasizing them as Platonic entities and even without resorting to classes of p-thoughts if we replace them with what I call extensible thoughts or e-thoughts. We can do this by means of the following disjunctive definition, which is as simple as it is efficacious:

An e-thought (Df) = a given tropical p-thought X* (used as the model) embodied in some mind or any other tropical p-thought Y qualitatively identical to X*, embodied in the same mind or in any other mind.

The e-thought is our empiricist version of what Frege should have meant with his f-thought (objective non-real thought). The p-thought X* can be any X thought that someone decides to use as a model. The aim of this definition of an e-thought is that any supposed f-thought is reduced to mental p-thoughts without depriving it of its epistemic objectivity (mainly inter-subjectivity) grounded on conventional rules, along with its expected stability or immutability. This procedure works at least insofar as my criticism of the private language argument is acceptable, though I have no doubts about this (See Ch. III, sec. 13).
     The so defined e-thought – which is the same as a verifiability rule, a tropical thought-content or simply a proposition – though usually distributed across space and time, doesn’t need to have any particular spatiotemporal location and can be seen as the most proper truth-bearer. For example: the e-thought or e-thought-content or e-thought-content-rule expressed by the sentence ‘The Eiffel Tour is made of iron’ can be instantiated as the p-thought that I have in mind when writing this sentence. However, it can also be instantiated by, say, the p-thought that you have in mind when you read it, such as by any qualitatively identical p-thought that I, we, or any other person can have at any place or time, insofar as it is considered an f-thought, namely, a model for any qualitatively identical p-thoughts. Characterized by the disjunction between qualitatively identical thoughts embodied in individual minds, the e-thought is apt to be regarded in abstraction from any particular human mind that causally instantiates it. This is what really occurs when we think an f-thought, and it is this abstraction from singular human minds resulting from the spreading character of the real thought-contents that gave Frege the impression that he had found a Platonic entity outside of space and time.
     As with model-tropes in the construction of universals, it is not necessary to have only one particular model as the object of interpersonal consideration. To the contrary, what we need to do is simply to single out the first thought given to us by memory and use it arbitrarily as a model: first the one, and then any other that we recognize as being precisely (qualitatively) identical to the first, and we can choose any of them as a new model. In some way language is only the vehicle of communication that allows the reproduction of qualitatively identical psychological p-thoughts in the minds of hearers, insofar as they are rooted in the usually implicit interpersonal conventions we have attached to their semantic components. Since p-thoughts are tropes, we have simply applied to p-thoughts the same strategy we applied to singular tropes, as we needed to construct universals based on them. The e-thought verifiability rules are p-thoughts read as universals.
     With the help of the above definition, we avoid not only appealing to psychologically specific occurrences of thoughts, but also the most expected alternative, which would be to explain one e-thought in terms of a sum or set of p-thoughts qualitatively identical to each other. This could lead us not only to the problem of defining sets, but also to the problem that sets and sums have or could have size, while thoughts cannot. If an e-thought were a set of p-thoughts, even if considered an open set, it would at the ontological level grow ever larger, the greater the number of people there were who grasped it.
     Under the proposed definition, in order to exist, an e-thought must always have at least one psychological occurrence. The e-thought is not less psychological than any p-thought, since it cannot be considered independently of its instantiation in at least one mind. This means that when we say that we both had the same idea, or the same thought, this is merely a manner of speaking. What we really mean is only that there is a qualitative identity between the (tropical) psychological verifiability p-thought-contents rules that we have respectively instantiated in our minds. We share the e-thought in the sense that we instantiate qualitatively identical p-thoughts. This has the advantage of bringing Fregean thoughts out of the ethereal Platonic heaven back to the concrete psychological realm without making any serious commitment to the transient psychology of individual minds.
     This understanding of the true nature of thought-contents explains something that Frege was unable to explain satisfactorily, namely, why and how they may have causal powers. Since as an open disjunction of p-thoughts, e-thoughts only exist as psychological instantiations of p-thoughts, this enables them to play a causal role: they can cause other psychological states and, combined with desires, human actions and their effects in the external world.
     At this point one could raise an objection of multiple realizability: the same p-thought could be differently realized in different human brains, making the qualitative identity of p-thoughts impossible. I agree with the very probable multiple realizability of p-thoughts but disagree that this makes their qualitative identity impossible. There is no reason why we cannot present things that can be considered qualitatively identical on a linguistic or even psychological level and different on a neurophysiological level, in the same way as different devices can have different internal mechanisms and perform exactly the same tasks.[18] Moreover, my suggestion is that e-thoughts are constituted of p-thoughts that are internal tropical verifiability rules, which although complex, ramified and variable, are also able to be satisfied by foreseeable independent tropical configurations.
     In my judgment, one of the most unyielding and deceitful philosophical errors in ontology has always been seeing numerical identity where there is only qualitative identity. It is true that we can ask for the meaning of the general term ‘chair’ using the definite article ‘the’ in the phrase ‘the chair.’ But this is only a linguistic device that changes nothing! In a similar way, we can speak of the geometrical form of circularity, and of the number 2 in the singular… But this is just for the sake of simplicity of expression. What we are ultimately able to have in mind in all these cases are occurrences of qualitatively identical meanings, that is, of qualitatively identical concepts of chairs, circles, and cognitive arithmetical concepts of duality, and not something more, since we don’t need something more to get something more.[19] In the same way, we can talk about the thought expressed by ‘12 x 12 = 144,’ but if we do not intend a specific occurrence of this thought, we are only referring to some occurrence, but without taking into account or having to specify which occurrence and in what mind. We speak in the singular of the thought that 12 x 12 = 144 for reasons of simplicity.
     The adoption of the definition of e-thoughts proposed above, which is easily generalizable to all kinds of Fregean senses, seems to me the only plausible abstraction we can arrive at without committing any of various forms of reification that have infested ontology throughout its long history.
     At this point, a stubborn Fregean defender can still ask: how is it possible that the psychologically dependent definition of e-thoughts suggested above could be able to ensure the objectivity of e-thoughts, their interpersonal accessibility or communicability? As we saw, Frege concluded that if we regard thoughts as psychological representations, as is the case with p-thoughts, they would unavoidably be subjective, and we could not compare them with each other. However, it still seems clear that Frege was too hasty when he admitted that his f-thoughts belong to a third realm of Platonic entities. One could note that there is no doubt that what Frege calls representations (phenomenal mental contents) have in fact possibilities of interpersonal communication, even if limited.[20] But much more important is something that Frege hasnt considered at all, namely, that senses and e-thoughts, without being Platonic entities, could be understood as rule-complexes built upon adequate associations of interpersonally accepted conventions established with the help of public signs that are communicable precisely because of their grounding interpersonal character. That is, because e-thoughts are verifiability rules rooted in linguistically shareable interpersonal conventions, they can well be able to satisfy Frege’s demand for objectivity as interpersonal accessibility followed by the possibility of communication and truth-evaluation.
     It may, at first sight, seem implausible that language is capable of repeatedly being reproduced in other minds and even in the same mind with the same subjective pattern, the same thought-content, the same recognizable instantiation of an adequate association of conventionally established semantic-cognitive rules attached to our words. However, compare by analogy this case with that of genetic information able to endlessly reproduce the same characteristics in successive biological individuals.[21] Why cannot the conventions and ways they can be combined in the constitution of p-thoughts do a similar job, even if only inferentially? More than this (and probably also in the case of genetic information), it is easy to suppose that there are corrective mechanisms able to interpersonally and intra-personally impose a limit on divergence from conventionalized standards (See Ch. V, sec. 11). There is no reason, except an anti-empiricist bias, to think that things could not be that way.
     Finally, let us apply to e-thoughts John Searle’s important distinction between what is ontologically objective/subjective and what is epistemologically objective/sub­jective (Searle 1999: 43-45). Searle noted that we have a strong tendency to take what is epistemologically subjective for what is only ontologically subjective. However, something can be ontologically objective – for instance, ‘How justifiable was the First World War?’ – without ceasing to be epistemologically subjective, because it is not easy to reach a common agreement about this issue. In contrast, a phenomenon can be ontologically subjective without ceasing to be epistemologically objective – for instance, the stabbing pain caused by a seizure of acute pancreatitis – because everyone (doctors and patients alike) will agree on the form and existence of this pain, even if the patient alone knows exactly how it feels.
     Something of the kind can also be said not only of Fregean subjective mental representations, but also of e-thoughts. In themselves they are ontologically subjective, since we admit that they are psychological events instantiated in one mind or another. But even so, they do not cease to be epistemologically objective, since we are capable of interpersonally agreeing about them and their truth-values. We can agree that an objectively assertoric sentence like ‘The Eiffel Tower is made of metal’ expresses a true e-thought that is epistemologically objective, even though as an e-thought ontologically subjective, since it is distributed among the minds of those who think it. Like any e-thought, it remains epistemologically objective, given that it is grounded on conventions associating words with things in the world, which makes it fully measurable and communicable. An arithmetical sentence like ‘2 + 3 = 5’ is epistemologically objective (since we are all able to inter-subjectively agree on its truth-value), but it also expresses an ontologically subjective e-thought, and as I tried to show in speaking of numbers, it seems to be a thin kind of tropical arrangement sustained by lower-order tropes. On the other hand, a sentence like ‘Love is the Amen of the universe’ (Novalis), unlike an e-thought, has no truth-value. It is only suggestive and expressive. Like poetry, it is based on non-conventional subjective coloration, being susceptible only to emotive-aesthetic appreciation with differing degrees of subjective interpersonal agreement.
     Regarding ontology, Frege was no exception. Like Husserl, Bolzano and several other continental philosophers of his time with mathematical training, he believed that the ontologically subjective character of psychologically conceived thought-contents would inevitably be condemned to epistemological subjectivity. But this was a mistake.
35. Further ontological consequences
Our ultimately psychological reformulation of Fregean thoughts has some interesting ontological consequences. If the thought of the Pythagorean Theorem isn’t an eternal (timeless) entity belonging to a Platonic realm, always true or false, where and when does it exist? The answer is that if there is at least one occurrence of its thought or any other qualitatively identical occurrence, regardless of the bearer, something like the Pythagorean theorem acquires an existence dependent on minds. It is not an existence dependent on any of the many particular minds that will eventually think it since it would continue to exist without having been thought by this or that particular mind. In fact, since this thought has been thought by both you and me and certainly by many others in the past, its existence must be spread over space and time. It must be distributed over the space and time occupied by the heads of mathematicians starting with Pythagoras himself and perhaps ending in the head of some cognitive being at some unknown future time. This is what gives the impression that the thought is something abstract, beyond the psychological realm.
     Another consequence of the proposed view is that unlike the Platonic entity that Frege called a ‘thought,’ our e-thought of the Pythagorean theorem did not in fact exist before Pythagoras thought it for the first time (supposing he was the first), and will cease to exist if it ceases to be thought by anyone. The Pythagorean theorem certainly exists, has existed and will continue to 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 could object that this result sounds strange: it seems that the Pythagorean Theorem applies independently of minds. However, this strangeness can be softened by the fact that nobody can truly deny it. One cannot have the true thought, ‘The theorem according to which the sum of the squares of the shorter sides of a right triangle equals the square of the hypotenuse has been thought in the past and now is no longer thinkable.’ And the reason is that this judgment will already be an occurrence of the thought of the Pythagorean Theorem and insofar will falsify what it states. Anyway, the conclusion remains that the e-thought of this theorem would not have come into existence if nobody had ever thought it. Putting this more incisively: it would not exist in a world without cognitive beings.
     The last remark suggests the following objection. Imagine a possible world Ww similar to ours, with planets, stars, and galaxies, but without any cognitive being. In Ww the e-thoughts that there are planets, stars and galaxies could not have been thought and, e-thoughts, being the primary bearers of truth, could not be true. Nevertheless, it seems very reasonable to think that in this world the fact that there are planets, stars and galaxies would still be true, even though there would be no cognitive Beings to think this.
     It seems to me that the right answer to the strangeness is that here we are again victims of a confusion between thought-truth and fact-truth. As we saw, the first is the truth applied to the primary bearer of the truth, which is the e-thought, while the second is a derivative but very common application of truth to the real existing thing or fact in the world, as a secondary bearer of truth, meaning a real thing or fact. Indeed, that there would be planets, stars and galaxies in a mindless world would still be true as a fact in Ww. Hence, the applicability of the Pythagorean Theorem would still be a fact-truth in Ww, even though neither their e-thoughts nor their truth in the form of correspondence would exist. The flexibility of natural language has once again misled us.
     Still another objection that could be made against the idea that the bearers of truth are non-Platonic e-thoughts is the following. Many truths have been discovered. Pythagoras is credited with discovering the theorem that bears his name; Archimedes was one of the discoverers of the law of the lever, according to which magnitudes are in equilibrium at distances inversely proportional to their weights. However, if something is discovered, then logically it must have existed before being discovered. Consequently, the above-described thoughts must already have existed before their discovery.
      Again, the answer is that this naïve objection results from a confusion between the thought as the primary bearer of truth on the one hand, and the fact as a derived bearer of truth on the other. This is clear in the case of typical empirical truths. That the law of the lever was always applicable in principle is surely true. However, this is only a general fact-truth! Its thought-truth was only part of the empirical (mental) world after scientists like Archimedes conceived it. Similarly, common sense tells us that the fact expressed by the Pythagorean Theorem must always have existed. However, our e-thought of it only came into existence after the theorem was thought by Pythagoras and since then has been thought by many others. Real facts, on their turn, as long lasting as they may be, are not the primary bearers of truth, but rather their truth-makers or verifiers. They exist independently and are said to be true only in the derived sense (b) of fact-truths, not in the sense (a) of thought-truths. They are what occurrences of their thoughts represent. Hence, in the most proper and demanding sense, no truths or falsehoods would exist in a world where there were no minds to think them. The most we could think of in this direction is to say that if the law of the lever were thought in Ww, it would be recognized as true.
      An e-thought that has never been thought does not exist and thus cannot be true. The same holds for falsehoods. Consider the thought ‘The Colossus of Rhodes is floating in the Sargasso Sea.’ In all probability this thought has never been thought before the present moment. But the moment we think that it has never been thought before, we are already thinking it, and we can even attribute falsehood to it. Even the e-thought ‘The world could exist, even if there were no minds to think about it’ is only a true thought insofar as there are minds to think it.
36. A short digression on contingent futures
Before we finish, it is interesting to examine the Aristotelian problem of contingent futures in the light of our conclusions (1984, vol. 1, De Interpretatione, sec. 9). According to a plausible interpretation of Aristotle the following argument is valid:

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

It seems that for Aristotle this conclusion would be unacceptable, because if the future were predetermined, then there would be no chance events, and if there were no chance events, there would be no free will. Hence, according to a traditional interpretation, he thought that although this argument is sound, premise (1) is false because it exemplifies the principle of bivalence, and the principle of bivalence – according to which any significant proposition is either true or false – is not applicable to future events (only to present and past ones).[22]
     I cannot agree with this conclusion, since I believe that we should preserve a strongly understood principle of bivalence for e-thoughts.[23] But premise (1) can be questioned from a different perspective. Suppose, first, that outside any context we consider the e-thought expressed by the sentence ‘There will be a sea battle tomorrow,’ which we can abbreviate as ├p. Is this statement true or false? The answer is the following: if taken literally, ├p is unable to express any e-thought because a verifiability e-thought rule is something to which we must possibly attribute a truth-value. Normally ‘There will be a sea battle tomorrow’ is an incomplete indexical statement, so that without any further contextual information we are totally at a loss for the task of associating p with any appropriate truth-maker in order to assign it a truth-value.
     Moreover, one could argue that the sentence ├p (as much as ├~p) is misleading and causes confusion, like argument A, because ├p only seems to express cognitive thought-content. The reason for this is that ├p is very easily confused with the meaningful sentence ├p*: ‘[It is likely that] a sea-battle will take place tomorrow,’ stated when there are reasons to think so. For example: having broken Japanese naval codes and having lured the Japanese fleet into an ambush at Midway, the Americans already knew on the night of June 3, 1942, that on June 4 there would almost certainly be a major naval battle. The sentence ├p* is easily confused with ├p, because ├p* is 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, so that the assertion ├p* expresses an e-thought that is held to be true, for it is true that, with the information already available, it was very probable that a sea-battle would occur the next day. Indeed, the utterance ‘It is likely that a naval battle will take place tomorrow’ could be regarded as definitely true on the night of June 3, 1942, without violating any principle of bivalence!
     Suppose now, by contrast, that I am standing on the calm beach of Praia Bonita in Northeastern Brazil, looking out across the Atlantic Ocean, and without any reason I say ├q*: ‘A naval battle will take place in this region tomorrow,’ meaning by it ‘It is likely that a naval battle will take place in this region tomorrow.’ This statement can be regarded as definitely false, since there are many different reasons to believe that this kind of event is extremely improbable in this region and at this time.
     The conclusion is that in the absence of a context (and not in the above senses of ├p* or ├q*), the sentence ├p would be a linguistic bluff devoid of any meaning or justification. Aristotle would be right in rejecting the application of the principle of bivalence to it, not because this principle has exceptions, but simply because it expresses no e-thought, no proposition, no verifiability rule. All that this sentence does is to induce us to imagine a naval battle that takes place tomorrow, as if there were hidden verifiability criteria. However, insofar as no context is furnished, no real criteria can be given. Statements like ├p*,├~p* and ├q*, on the other hand, aim to say something probabilistic about the future that can be confirmed and made true by criterial reasons already found in the present. But from such statements premise (2) and the conclusion (3) of the argument A do not follow, because all that such statements can warrant, if true, is the inductive probability of a sea-battle.
     The upshot is that the metaphysical riddle about contingent futures can be eliminated if we consider with enough care what we are really able to mean by affirming e-thoughts regarding the future.
37. Conclusion
My first aim in this chapter was to insert in the framework of Fregean semantics the results of my reconstruction of Wittgenstein’s view of (cognitive) meaning as given by the application of semantic-cognitive rules in order to better distinguish the most relevant forms of meaning-rules and their functions. This insertion requires strong corrections in Frege’s own framework. Even if the results are complex and could only be sketched here, they nonetheless seem to me clearly more auspicious than Frege’s own original views.

[1] See Frege, Letter to Russell of 28.12.1912.
[2] Without offering a justification, Strawson writes: ‘a situation or state of affairs is, roughly, a set of facts, not a set of things.’ (1950: 8)
[3] For an important reply, see J. L. Austin, ‘Unfair to Facts’ (1961, Ch. 5). It seems to me at least curious that the posthumously published arguments of Austin against Strawson’s view have had so little impact.
[4] John Searle once proposed something approaching this answer: ‘…we neither have nor need a thick metaphysical notion of “fact.” Anything sufficient to make a statement true is a fact. Thus the fact that there are no three-headed cats is as much a fact as the fact that the cat is on the mat.’ (1998: 392)
[5] See Appendix of Chapter III, sec. 4.
[6] This also gives back the whole sense of Church’s still more convoluted original sentence: ‘The number such that Sir Walter Scott is the man who wrote that many Waverley Novels altogether is twenty-nine.’
[7] I think that the mode of presentation of the sub-fact can be approximated with what defenders of two-dimensionalism call a primary intention (here called derived thought) while the mode of presentation of the grounding fact can be approximated with what they call a secondary intention (here called basal thought) (Cf. Chalmers 2002). Anyway, the present suggestion is clearly more perspicuous and natural.
[8] For instance, A. J. Ayer in the first case and Hilary Putnam in the second. (See also Costa 2014, Ch. 3.)
[9] The concept of emphasization was fruitfully applied in Jürgen Habermas’s excellent work on universal pragmatics (Habermas 1976).
[10] The example was already considered in the Addendum of the Appendix to Chapter II in this book.
[11] As Tyler Burge 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: 227-8)
[12] For Frege, in the case of indexical sentences, the context of the utterance belongs to the expression of thought. See also addendum of the Appendix to Chapter II, sec. 8.
[13] According to his main argument, if you say that the truth of p is its correspondence with reality, you need to admit that p must have the property j in order to be true by correspondence with reality, and that to have the property j in order to be true by corresponding with reality will demand the property j’ and so successively. The answer (already given by Aquinas) is that to say that p is true by corresponding to reality, and to say that p has the property j due to being true by corresponding to reality are one and the same thing; consequently, N is redundant. (Cf. Künne 2003: 129-133).
[14] For instance: ‘truth (principle): that which is true in accordance with the fact or reality’; ‘truth (fact): the actual fact about the matter’… (Oxford-Cambridge Dictionary).
[15] See Tugendhat’s verificationist correspondentialism in 1983: 235-6.
[16] Nonetheless, there is an at least seemingly alternative way to understand the property of effective applicability of the verifiability rule, which is to identify it with the existence of the fact. To reach this conclusion, we need only consider that the existence of an object (an independent cluster of compresent tropes) is the higher-order property of effective applicability of an identification rule expressed by a nominal term, and that the existence of a property – a dependent property-trope – is the higher-order property of effective applicability of the ascription rule of a predicative expression. If we accept this, then by symmetry the existence of a singular fact should be the higher-order property of effective applicability of the verifiability rule of the singular declarative sentence to which it applies. It seems that we could say, in an almost Hegelian fashion, that existence is the truth of the concept, while the truth is the existence of the thought… We have here two alternative understandings of the property of effective applicability of a verifiability rule, what generates a dilemma that will only be solved in the beginning of chapter VI.

[17] See Appendix to Chapter III, sec. 2.
[18] As T. W. Polger has shown, in order to illustrate the flaw of the multiple realizability argument, we can explain how a carburetor has the function of mixing fuel and air for a combustion engine; but it is a multiply realizable device: it can be made of various different materials with various designs, provided it functions properly. (2004: 19-20).
[19] The phrase is from Murray Gell-Mann. Against this, however, one could ask: haven’t we learned that geometry deals with perfect circles and that arithmetic deals with entirely abstract numbers? Take the case of circles. The answer is, of course, in the negative, because we can make a new circle more perfect than the last one, and another even more perfect, and this process can continue indefinitely. The perfect circle is like the actual infinite: it does not exist. It is nothing more than a projection of our awareness of the possibility of making increasingly perfect empirical circles without any conceivable end. Geometry does not work with actual perfect circles, but with potentially perfect circles.
[20] Against Frege, we could hold that to some extent even imagetic representations can be expressed through language and by its means could be subjectively identified and re-identified as being the same (e.g., a police sketch or a Photofit). It is true that a mental state that only one person is capable of having, for instance, a sort of epileptic aura, is not communicable, except indirectly, metaphorically. But it seems very plausible that typical mental states, such as feelings, images, sensations, are things that all of us are able to communicate and learn to identify in ourselves through induction by exclusion, added to induction by analogy and reinforced by a great variety of interpersonally accessible physical states strongly intermingled with them (Cf. Ch. III, sec. 8; See also Costa 2011, Ch. 3).
[21] Biological mutations are accidents whose occurrence should be evolutionarily calibrated. Species are only likely to survive if they can mutate to the right degree in the right period of time in order to adapt to environmental changes. Too many mutations, as well as too few, would be dangerous for species survival. It seems possible that an unchanging species with no mutation is conceivable, but it would be unable to adapt to changing external conditions.
[22] For an alternative interpretation, see Christopher Shields, 2007: 186-190.
[23] I mean a principle of bivalence understood as a different formulation of the principle of non-contradiction.