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# Logic and Its Philosophy

## 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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# VI. Naturalism and Genesis of Logic

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VINaturalism and Genesis of Logic

It is traditionally accepted that we differentiate between logica docens and logica utens, that is, between theoretical logic (logic as theory) and applied or practical logic. Both can be defined with the use of the concept of logical consequence. The first is a set of consequences of an empty set, symbolically LOGT = Cn∅, provided that the operation Cn satisfies the well-known general Tarski’s axioms, i.e. denumerability of the language (a set of sentences) L, X ⊆ CnX (the inclusion axiom; X, Y are sets of sentences of L), if X ⊆ Y, then Cn X ⊆ CnY (monotonicity of Cn), CnCnX = Cn (idempotence of Cn) and, if A ∈ CnX, then there is a finite set Y ⊆ X such that A ∈ CnY (Cn is finitary). Cn is a mapping of the type 2L → 2L, that is, transforming subsets of L into its subsets. In order to make things simpler, I assume that Cn is based on classical logic. LOGT can be also defined as the only common part of the consequences of all sets of sentences. Otherwise speaking, logic is the only non-empty intersection of the family of all subsets of L. What follows from it is that logic is included in the consequences of each set of sentences, which underlines its universal character. If CnX ⊆ X, then X = CnX due to the inclusion axiom. Moreover, if CnX ⊆ X, we say that that X is closed by the consequence operation of logical consequence. This is the definition...

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