Home Philosophy Advances in Proof-Theoretic Semantics
Abstract To determine what deductions are it does not seem sufficient to know that the premises and conclusions are propositions, or something in the field of propositions, like commands and questions. It seems equally, if not more, important to know that deductions make structures, which in mathematics we find in categories, multicategories and polycategories. It seems also important to know that deductions should be members of particular kinds of families, which is what is meant by their being in accordance with rules.
Keywords Deduction • Proposition • Command • Question • Category • Multicategory • Polycategory • Rule • Natural transformation • Proof-theoretic semantics • General proof theory • Categorial proof theory
In a terminology like that of the old logic, the notion of deduction will be for us primarily a hypothetical and not a categorical notion. (This use of categorical should not be confused with categorial, which is found later in this paper, and which, according to the Oxford English Dictionary , means “relating to, or involving, categories”; unfortunately, in mathematical category theory categorical dominates in the sense of categorial.) The distinction between categorical and hypothetical is found when we speak about categorical and hypothetical proofs. The latter is a proof under hypotheses, while the former depends on no hypothesis. Both may involve deduction, but we will be concerned here with deduction as found in hypothetical proofs.
Schroeder-Heister (together with P. Contu in , Sect. 4, in , Sect. 3, and in ; see also ) states that the reigning semantics—both classical semantics based on model theory and constructivist proof-theoretic semantics—is based on dogmas, the main one of which may be formulated succinctly by saying that categorical notions have primacy over hypothetical notions. We conform to this dogma when we take the notion of proposition, a categorical notion, to have primacy over the notion of deduction, a hypothetical notion. We conform to the same dogma when we take that, among functions of language, asserting, which is tied to propositions, is more basic than deducing.
The question for us here should not be what function of language is the most important in general, but what function of language is the most important for logic. Even if it were the case that asserting is the most important function of language in general, it could happen that, because of the specific goals it has, logic, though it takes into account the importance of asserting, gives precedence to the function of deducing. Even if asserting is the most important function of language in general, for a specific area another function may have precedence. In the nomenclature of a science wouldn't the most important function of language consist in naming rather than in asserting?
It is questionable however that there is a most important function of language in general. Following Frege's context principle from the introduction of the Grundlagen der Arithmetik “never to ask for the meaning of a word in isolation, but only in the context of a proposition” , and following the Wittgenstein of the Tractatus , as usually understood, the most important function of language should be asserting. The belief that there is such a function and that this function is asserting was however rejected by the Wittgenstein of the Philosophical Investigations . The later Wittgenstein said that, using his terminology, there may be language games, appropriate to particular forms of life, where various functions of language like commanding, or questioning, would have precedence over asserting, and it could not be said that these language games are less fundamental. They are not meaningful only because behind them lurks somehow the activity of asserting.
Philosophers, scientists, those living a theoretical life, were inclined since ancient times to give precedence to naming, and more recently, as it happened with Frege and Wittgenstein in the Tractatus, they gave precedence to asserting. (The late Frege wanted to fuse the two activities.) But does language acquire meaning primarily in theoretical life? Are not the quarters where that life is led (something like a university campus, or a leisurely residential upper-class quarter) rather lately built and not central quarters in the city language (see , Sect. 18)?
Even though it is not essential to agree with the later Wittgenstein on this point, it helps to do so if we want to claim without worry that in logic the most important function of language is deducing. It is strange that one has to defend nowadays this rather venerable opinion, but so many developments in the philosophy of modern logic and the philosophy of language spoke against it in the last two centuries.
Textbooks of logic in the second half of the twentieth century would often start with a definition of logic as a human endeavour concerned with deduction, and would practically not mention deduction in the remainder of the book. It is only as the century was moving to its close that natural deduction or related matters started getting ground in textbooks of logic.
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