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Logic and Its Philosophy

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Jan Woleński

This collection of essays examines logic and its philosophy. The author investigates the nature of logic not only by describing its properties but also by showing philosophical applications of logical concepts and structures. He evaluates what logic is and analyzes among other aspects the relations of logic and language, the status of identity, bivalence, proof, truth, constructivism, and metamathematics. With examples concerning the application of logic to philosophy, he also covers semantic loops, the epistemic discourse, the normative discourse, paradoxes, properties of truth, truth-making as well as theology, being and logical determinism. The author concludes with a philosophical reflection on nothingness and its modelling.

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V. Is Identity a Logical Constant and are there Accidental Identities?

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VIs Identity a Logical Constant and are there Accidental Identities?3

Propositional connectives and quantifiers are logical constants without any doubt. However, we speak about first-order logic with or without identity. Even this way of speaking suggests that identity has a special status to some extent. In fact, the status of identity is controversial. Wittgenstein says (Wittgenstein 1922, 5.5303):

   Roughly speaking, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all.

According to Wittgenstein, identity is not a relation. This view rises an important point: does identity hold between objects, which are numerically different, for example, two occurrences of ‘e’ in the word ‘different’? Tarski’s view of identity was radically opposite to that of Wittgenstein. The difference is well illustrated by the following quotation (Tarski 1941, p. 49):

   Among the logical concepts not belonging to sentential calculus, the concept of IDENTITY, or, of EQUALITY, is perhaps the one which has the greatest importance.

Wittgenstein’s and Tarski’s statements about identity can be rephrased without essential changes by replacing ‘identity as a relation’ by ‘identity as a predicate’ (I prefer the second way of speaking).

Formally speaking, identity is introduced into first-order logic by the axioms (I omit quantifiers in the front of formulas):

   (A1) x = x;

   (A2) (x = y) ⇔ (y = x);

   (A3) (x = y)...

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