Abstract

The article presents a full proof‐theoretic semantics for natural deduction based on an extended inversion principle: the elimination rule for an operator q may invert the introduction rule for q, but also vice versa, the introduction rule for a connective q may invert the elimination rule for q. Such an inversion—extending Prawitz' concept of inversion—gives the following theorem: Inversion for two rules of operator q (intro rule, elim rule) exists iff a reduction of a maximum formula for q exists. The inversion theorem is specified to two logics, Lambek calculus (LC) and intuitionistic linear logic (ILL), with four propositional connectives: two multiplicatives (implication and conjunction) and two additives (conjunction and disjunction), with two quantifiers and two modals. LC is defined by using elimination rules by composition. ILL is defined by using usual general elimination rules. Elimination rules by composition are an exciting alternative to general elimination rules; in some cases, they do not need permutations.