Core Logic: A Conspectus

This chapter presents an ‘absolutist’ view about logic—Core Logic. Core Logic is relevant, in a sense heretofore not satisfactorily explicated. The so-called loss of unrestricted transitivity of deduction in Core Logic brings with it epistemic gain. Core Logic suffices for Intuitionistic Mathematics, Classical Mathematics, the hypothetico-deductive testing of scientific theories against empirical evidence, and the reasoning involved in the logical and semantic paradoxes. Core Logic is the minimal inviolable core of logic that is needed, and suffices, for rational belief revision. Core Logic is the logic of ‘conceptual constitution’: it suffices for neo-logicist derivation of number-theoretic axioms from deeper logical principles governing the operator #xΦ(x) (the number of Φs). Core Logic has the nice property that its natural deductions and sequent proofs are structurally isomorphic, and it is obtained by smoothly generalizing an inferential reading of the truth-tables. There is an automated proof-finder for (propositional) Core Logic, whose decision problem is PSPACE-complete (like that of Intuitionistic Logic). Any classical core proof provides a means of ‘quasi-’effectively transforming any verifications of its premises, relative to a model M, into a verification of its conclusion, relative to M.