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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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I. Semantic Loops

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ISemantic Loops

Michael Heller introduced (see Heller 1999, pp. 90, 99–100) the idea of semantic loops. They are essentially involved in interactions between languages and their metalanguages (I will use the letter L as referring to a language and the symbol ML as denoting a metalanguage of L). As it is well-known, such interactions lead to semantic antinomies related to the self-referential use of expressions. The famous Liar antinomy (LA for brevity) is perhaps the most paradigmatic example. Consider the sentence:

   (λ) the sentence denoted by (λ) is false.

A simple inspection shows that (λ) is true if and only if it is false (it will be demonstrated below). The situation illustrated by (λ) can be metaphorically characterized as a closed semantic loop, because we pass from truth to falsehood and back without any possibility of leaving the loop (or a circle, if you prefer this, more popular, figure of speech; see also a historico-terminological digression below). On the other hand, the language/metalanguage distinction cannot be liquidated, because we need to speak about languages and their various properties.

Although ML can be effectively reduced to L in the case of syntax (for example, via the method of Gödel numbering), this is impossible in the case of semantics; Tarski showed that doing semantics of L in ML requires that the latter is essentially richer than the former. Thus, closed semantic loops operate somehow between L and ML. As Heller indirectly (as he speaks about loops...

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