Elementary Syntax

The completeness of the Propositional Calculus, from Stephen Cole Kleene’s “Mathematical Logic” (Section 12, Theorem 14 for the curious). If a formula is valid, or tautological, then it is derivable in the classical propositional calculus using no assumptions other than the axioms and modus ponens.

Duality rules for the propositional calculus from Stephen Cole Kleene’s “Mathematical Logic” (section 6 for the curious). E and F are formulas consisting only of atomic sentences and their negations, using only conjunction and disjunction for operations. E’ and F’ come from E and F simply by replacing all conjunctions with disjunctions, and all disjunctions with conjunctions. The
‘⊨’ can be read as ‘entails’ means the following formula is valid; it is true for all possible values of the atomic sentences composing it.