Defining implication relation for classical logic
It addresses a foundational issue in logic for researchers and philosophers, but it is incremental as it modifies classical logic rather than introducing a new paradigm.
This paper tackles the problem that in classical logic, the equivalence between 'P implies Q' and 'not-P or Q' is invalid because Disjunction-to-Implication is not generally valid. It proposes a new logical system called IRL that removes this incorrect inference while retaining fundamental laws like the law of excluded middle and principle of double negation.
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.