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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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II. Logic as Calculus vs. Logic as Universal Medium, and Syntax vs. Semantics

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IILogic as Calculus vs. Logic as Universal Medium, and Syntax vs. Semantics

The distinction between logic as calculus (LoC) and logic as language (LoL) was introduced by van Heijenoort (see Van Heijenoort 1967). He regarded this distinction as a very useful device for the history of logic, closely resembling Leibniz’s famous pair of calculus ratiocinator and lingua characteristica. According to van Heijenoort, logic under Frege’s understanding was rather a kind of lingua characteristic, contrary to the view of Boole or Schröder seeing logic as a calculus. Russell and Whitehead followed Frege’s view, making some changes forced by the theory of logical types. A new tradition, more related to Schröder than to Frege, was initiated by Löwenheim in 1915 (see also Badesa 2004), who proved the first model-theoretic theorem (if a formula of first-order logic is valid in a denumerable domain, it is valid in every domain). Generally speaking, the opposition between LoC and LoL was replaced by the axiomatic approach to logic and its set-theoretic account. Skolem, Gödel and Herbrand made essential contributions to this new world of logic. In particular, van Heijenoort finds Herbrand’s work as a possible third way. Van Heijenoort seems to consider LoC and LoL as ideal types. For instance, he stresses that Frege characterized logic not only as a lingua characteristica but also as a calculus ratiocinator. Although van Heijenoort does not give any general characteristics of both types of logic, he points out some of their important features,...

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