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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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X. An Abstract Approach to Bivalence


XAn Abstract Approach to Bivalence10

The principle of bivalence is usually expressed as the following statement (‘either or’ expresses the exclusive disjunction):

   (1)   every sentence is either true or false.

Any logic based on (1) is called two-valued or classical. Its traditional importance as the very foundation of formal logic consists in its relation to the so-called highest (the most fundamental) laws of thought. In classical metalogic, (1) is equivalent to the conjunction of two other basic rules of thought, namely the metalogical principle of excluded middle:

   (2)   every sentence is true or false;

and the metalogical principle of non-contradiction:

   (3)   no sentence is both true and false.

The adjectives ‘true’ and ‘false’ in (1), (2) and (3) are just metalogical (i.e. they belong to metalanguage and as such must be sharply distinguished from theorems A ∨ ¬A, ¬(A ∧ ¬A) and (A ∨ ¬A) ∧ ¬(A ∧ ¬A), that is, theorems of the propositional calculus formulated in the object-language of logic. Jan Łukasiewicz was perhaps the first logician who observed that (1) is metalogical (see Łukasiewicz 1930).11 It is important to see that the equivalence

   (4)   (1) ⇔ (2) ∧ (3)

holds for classical logic only.12

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