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Carlo Penco Fregean Oscillation*

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Carlo Penco Fregean Oscillation* last draft 15.12.99** Carlo Penco ([email protected]) Fregean oscillation* In this paper I discuss an oscillation in Frege's conception of sense. In section A and B I will present two alternative ways to define sense and sense identity. The point given in A is a kind of thought experiment, applying Frege's criterion of intuitive difference of thoughts to two logically equivalent sentences; I claim that they should be considered different in sense. The point given in B aims to show how Frege used logical equivalence as a criterion of identity of thoughts. Then I conclude that, among different possible solutions, it is reasonable to think that Frege really oscillated between two alternative conceptions of sense. This oscillation can be better considered from the point of view of present day philosophy, after more than one hundred years of his first definition of the distinction between sense and reference. The distinctions towards which Frege was striving have become clearer and clearer to our sensibilities of scholars working in a world where the problem of limited rationality is imposing itself even in logical matter. A. Traditional setting of the definition of sense: 1892 In 1892 Frege defines the sense of a sentence as the thought expressed by it; in defining thought he elaborates what has been called the principle of intuitive difference of thoughts: (1) DEFINITION: thesense of a sentence is the thought expressed by the sentence (2) ARGUMENT: principle of intuitive difference of thought:: "If it is possible to understand two sentences and coherently believe what one expresses while not believing what the other express, * Paper given at the Unviersity of Genoa in a small conference with Eva Picardi and Mark Sainsbury. IU have to thank them and the audience for many good suggestions. then those sentences express different senses (different thoughts)" (Evans 1982). Sainsbury 1999 explains this idea, speaking of rational cotenability In belief contexts cosubstitutability requires identity of sense - not of reference. But we need to know exactly what a sense of a sentence is. And the definition that the sense of a sentence is the thought expressed by that sentence is only programmatic. We have only an intuitive negative criterion of difference of thoughts: sentences that are not substitutable salva veritate in belief contexts are supposed to have different senses. To have an positive criterion of identity we need to go the other way around: which kinds of sentences are substitutable in any contexts without loss of truth value? A criterion of sense-identity as general substitutability in indirect contexts has been widely discussed.1 Still it is easy to doubt that a clear definition of sense- identity can be attained in this way. However, even doubting the applicability of a positive criterion of sense-identity, the negative criterion of difference of thoughts is enough to provoke some deep problems in Frege's analysis. The negative criterion is the core of Frege's argument in "Uber Sinn un Bedeutung" ("....."). The argument, stated as it is, can be applied to different examples, even to logical ones. Take the following: (3) EXAMPLE John believs that A → B A → B ↔ ¬ (A & ¬ B) * John believes that ¬ (A & ¬ B) Now John, even if he has studies some logic and already knows the definition of negation and the definition of conditional 2, may easily hold that A → B and disbelieve that ¬ (A & ¬ B). Therefore we should conclude that A → B and ¬ (A & ¬ B) have different senses. This sounds awkward: the Fregean principle of identity of thoughts as rational cotenability would exclude as absurd the idea that the two sentences above express different thoughts. They must be rationally cotenable, and somebody who believes one and disbelieves the other is an irrational person who does not recognise the apparent identity of the thoughts he entertains. Probably we would say that 1 We may think of the traditional attempts by Carnap and model theoretic semantics. In the abundant literature on Frege this topic is also widely discussed. A good presentation is given in Beaney 1996.. 2 Without further specification, we might say that, knowing the definition of negation and conditional, we may concede John to know or to be immediately aware of the sense of the sentences containing it. John does not understand completely the sense of each side of the biconditional. This possibility hovewer would compel us to take sense as something which can never completely be grasped (a claim that Frege himself undertook in some passages). In this way we run the risk of undermining any viable conception of sense and any possibility of defining sense identity. Besides, we may rationally hold that "John believes that A → B" and "John does not believe that ¬ (A & ¬ B)". This could be even clearer if we think of much more complex logically equivalent formulas, which require a certain amount of calculation to detect their truth- functional equivalence. In fact it is apparent that it may happen that John, even if he knows what "¬ ", "→" and "&" mean, may not know that A → B and ¬ (A & ¬ B) have the same truth condition and that are translatable into each other. The difference from the original example by Frege is that the limited knowledge of the (ignorant) astronomers regards an empirical matter, while the limited knowledge of John regards a logical matter; but, once we get involved in the problem of belief, we have to take into account the attitudes of speakers and their limited knowledge even in logical matters. I assume therefore that the requirement of rational cotenability should be kept in a weak way, dealing with a notion of human rationality, trying to keep in our picture also the failures a person may have regarding logical matter. May John rationally hold that the two sentences above express different thoughts? He may, if we admit the possibility of mistakes in our limited ability of computation. Certainly it is not rational to believe p and ¬ p; however, when the computation required to understand that a certain sentence is logically equivalent to another is too complex for our limited ability, we may rationally - and provisionally - hold the two sentences express different thoughts, unless a better computation shows us the contrary. In other words, when we see two sentences that appear to be very different in logical form, and whose immediate composition we recognise as holding correctly, it is reasonable to suspend judgement untile we can check whether or not they are equivalent. A rational agent should keep a reasonable attitude: if he already believes one sentence, he may disbelieve the other, with the reservation of further inquiry (we can consider this step as a typical case of default reasoning). If we interpret the principle (2) in a very weak sense, accepting to deal with the limited knowledge and limited rationality of John, who may not be able to perform completely the relevant operations we have therefore to conclude that: A → B has a different sense than ¬ (A & ¬ B) B. Setting of the definition of thought in 1906 In a letter to Husserl in 1906 Frege explicitly says that A → B and ¬ (A & ¬ B) express the same thought. In this letter he gives a definition of thought as "the content shared by equipollent sentence"; equipollent sentences may differ psychologically (difference in tone); a set of equipollent sentences may be given in normal form. However different forms may have different uses (for pragmatic reason to make deduction more perspicuous). I give here the relevant quotations for the following three points: (i) DEFINITION OF SENSE "equivalent sentences have something in common in their content, and this is what I call the thought they express... The rest I call the colouring and the illumination of the thought [...] (ii) DIFFERENCES AMONG EQUIPOLLENT SENTENCES Judged psychologically, the analysing proposition is of course always different from the analysed one, and all logical analysis can be brought to a halt by the objection that the two sentences are merely equipollent (...) For it will not be possible to draw a clear recognisable limit between merely equipollent and congruent sentences. (iii) NORMAL FORM AND USE OF DIFFERENCES "[Given logical analysis] all that would be needed would be a single standard sentence for each system of equipollent sentences, and any thought could be communicated by such a standard sentence. For given a standard sentence everyone would have the whole system of equipollent sentences, and he could make the transition to any one of them whose illumination was particularly to his taste." On this background Frege asks whether A → B and ¬ (A & ¬ B) are equipollent, that is, given the definitions above, whether they express the same thought: "With regard to the question whether the sentence "if A then B" is equipollent with the sentence "it is not the case the A without B" [...] we have four combinations: True True / True False / False True / False False Of these the first, third and fourth are compatible with the sentence "If A then B", but not the second. We therefore obtain by negation: A is true and B is false, or: A holds, without B holding, just as on the right hand side ["¬ (A→B) ↔ A & ¬ B" and "¬ (¬ (A & ¬ B) ↔ A & ¬ B)" ] [...] If we consult my Begriffsschrift, which is now 28 years old, we find the answer to such a question without further ado" The conclusion is that the two sentences are equipollent and therefore express the same thought. Frege recalls here his original Begriffsschrift; however, Frege writes this passage just a few years after his definition of thought as the sense of a sentence. We may therefore conclude that the sense of a sentence is considered by Frege as the truth condition given by the peculiar conventions in his logical system (see also the classical quotation by Frege 1893, §32).
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