Epistemic Gain

Core Logic avoids the Lewis First Paradox, even though it contains ∨-Introduction, and a form of ∨-Elimination that permits core proof of Disjunctive Syllogism. The reason for this is that the method of cut-elimination will unearth the fact that the newly combined premises form an inconsistent set. A new formal-semantical relation of logical consequence, according to which B is not a consequence of A,¬A, is available as an alternative to the conventionally defined relation of logical consequence. Nevertheless we can make do with the conventional definition, and still show that (Classical) Core Logic is adequate unto it. Although Core Logic eschews unrestricted Cut, nevertheless (i) Core Logic is adequate for all intuitionistic mathematical deduction; (ii) Classical Core Logic is adequate for all classical mathematical deduction; and (iii) Core Logic is adequate for all the deduction involved in the empirical testing of scientific theories.