Petr Cintula, Carles Noguera, Logic and Implication. An Introduction tothe General Algebraic Study of Non-classical Logics, vol. 57 of Trends in Logic,Springer, 2021, pp. 465+xxii; ISBN: 978-3-030-85674-8 (Hardcover) 117.69e, ISBN:978-3-030- 85675-5 (eBook) 93.08 e.

Abstract

A propositional logic, taken as a consequence relation ⊢, is weakly implicative if its language has a binary connective (primitive or defined) →, named weak implication, that satisfies for all formulas φ,ψ, δ the following four conditions: 1. ⊢ φ → φ, 2. φ,φ → ψ ⊢ ψ, 3. φ → ψ,ψ → δ ⊢ φ → δ, 4. φ → ψ,ψ → φ ⊢ ⋆(δ1 . . . , δi, φ, δi+2, . . . , δn) → ⋆(δ0 . . . , δi,ψ, δi+2, . . . , δn), for every connective ⋆ of the language, every 1 ≤ i ≤ n where n is the arity of ⋆ and all formulas δ0 . . . , δn. The concept was introduced by P. Cintula in [1] and since then it has been extensively studied by the authors of Logic and Implication. It is a weakening of Rasiowa’s concept [5] of implicative logic in that weakly implicative logics do not need to satisfy the condition φ ⊢ ψ → φ that in addition to 1–4 above characterize Rasiowa’s notion.