# Minimal logic

Just like intuitionistic logic, minimal logic can be formulated in a language using →, ∧, ∨, ⊥ (implication, conjunction, disjunction and falsum) as the basic connectives, treating ¬A as an abbreviation for (A → ⊥). In this language it is axiomatized by the positive fragment (i.e., formulas using only →, ∧, ∨) of intuitionistic logic, with no additional axioms or rules about ⊥. Thus minimal logic is a subsystem of intuitionistic logic, and it is strictly weaker as it does not derive the ex falso quodlibet principle $\neg A,A\vdash B$ (however, it derives its special case $\neg A,A\vdash \neg B$).
Adding the ex falso axiom $\neg A\to(A\to B)$ to minimal logic results in intuitionistic logic, and adding the double negation law $\neg\neg A\to A$ to minimal logic results in classical logic.