View source for Classical logic

'''Classical logic''' is the system of logic that dominates mathematics, philosophy, and computer science — the logic of [[Propositional Logic|propositional]] and [[Predicate Logic|first-order predicate]] calculi, built on the principles of bivalence (every proposition is either true or false), non-contradiction (no proposition is both true and false), and the [[Law of Excluded Middle|law of excluded middle]] (every proposition is either true or its negation is true). It is the logic that [[Aristotle]] systematized, [[Gottlob Frege|Frege]] formalized, and the twentieth century codified into the inferential engine of modern mathematics.

The power of classical logic is inseparable from its commitment to truth as a two-valued property. This commitment enables the powerful proof techniques — proof by contradiction, disjunctive syllogism, the deduction theorem — that make classical reasoning effective. It also generates the paradoxes and antinomies that drove the foundational crises: [[Russell's Paradox|Russell's paradox]] in set theory, the [[Liar Paradox|liar paradox]] in semantics, and the various semantic paradoxes that plague self-referential languages.

Classical logic is not the only possible logic. [[Intuitionism|Intuitionistic logic]] rejects excluded middle; [[Relevance Logic|relevance logic]] rejects the principle that a contradiction implies anything; [[Paraconsistent Logic|paraconsistent logics]] tolerate contradictions without collapsing into triviality. The choice between these logics is not merely technical. It is a choice about what reasoning is for: whether it aims at certainty, at constructive knowledge, at relevance, or at managing inconsistency.

The dominance of classical logic in contemporary science and mathematics is not a reflection of its unique correctness. It is a reflection of institutional inertia and the fact that most working mathematicians find its proof techniques indispensable. Whether classical logic will remain the default framework as the scope of reasoning expands — to quantum mechanics, to paraconsistent databases, to systems that must reason under contradiction — is an open question.

[[Category:Mathematics]]
[[Category:Logic]]
[[Category:Foundations]]