Defining implication relation for classical logic
Li Fu
TL;DR
This work introduces IRL (Implication-Relation Logic), a two-valued system that treats implication as a non-truth-functional relation, aiming to fix the problematic material-implication reading in classical logic. By taking bi-implication as primitive and constructing a corresponding algebraic semantics via IRLA, the authors show that Disjunction-to-Implication is not derivable in IRL, while key classical laws such as LEM, LNC, ECQ, and transitivity of implication are preserved. They establish soundness, completeness, and an isomorphism between IRL and IRLA, and they analyze the relationship to CL, showing that CL is obtained by adding the disjunction-to-implication schema $\neg\phi\vee\psi\to(\phi\to\psi)$, with D2I generally independent in IRL unless no contingencies exist. The paper further extends IRL to natural deduction and first-order logic, discusses the decision problem (likely undecidable in the general setting), and offers interpretive guidance on ECQ and generalized ECQ within a contingency-aware framework, highlighting the practical impact on foundational logic where implication is not reducible to a truth-functional disjunction.
Abstract
In classical logic, "P implies Q" is equivalent to "not-P or Q". It is well known that the equivalence is problematic. Actually, from "P implies Q", "not-P or Q" can be inferred ("Implication-to-Disjunction" is valid), whereas from "not-P or Q", "P implies Q" cannot be inferred in general ("Disjunction-to-Implication" is not generally valid), so the equivalence between them is invalid in general. This work aims to remove the incorrect Disjunction-to-Implication from classical logic (CL). The logical system (the logic IRL) this paper proposes has the expected properties: (a) CL is obtained by adding Disjunction-to-Implication to IRL, and (b) Disjunction-to-Implication is not derivable in IRL; while (c) fundamental laws in classical logic, including law of excluded middle (LEM) and principle of double negation, law of non-contradiction (LNC) and ex contradictione quodlibet (ECQ), conjunction elimination and disjunction introduction, and hypothetical syllogism and disjunctive syllogism, are all retained in IRL.