Minimal logic

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Minimal logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson (1936). It is a variant of intuitionistic logic that rejects not only the classical law of excluded middle (as intuitionistic logic does), but also the principle of explosion (ex falso quodlibet).

Like intuitionistic logic, minimal logic can be formulated in a language using → (implication), ∧ (conjunction), ∨ (disjunction), and ⊥ (falsum) as the basic connectives, treating ¬A (negation) as an abbreviation for A → ⊥. In this language, it is axiomatized by the positive fragment (i.e., formulas using only →, ∧, and ∨) of intuitionistic logic, with no additional axioms or rules about ⊥. Thus minimal logic is a subsystem of intuitionistic logic[note 1], and it is strictly weaker as it does not derive the ex falso quodlibet principle (however, it derives its special case ).

Adding the ex falso axiom to minimal logic results in intuitionistic logic, and adding the double negation law to minimal logic results in classical logic (Troelstra and Schwichtenburg 2000:37)

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  1. ^ Recall that a statement in a subsystem S of a logic A is provable in S if and only if it is provable in A. Thus, any formula using only is provable in minimal logic if and only if it is provable in intuitionistic logic


  • Ingebrigt Johansson, 1936, "Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus." Compositio Mathematica 4, 119–136.
  • A.S. Troelstra and H. Schwichtenberg, 2000, Basic Proof Theory, Cambridge University Press, ISBN 0521779111