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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.
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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.
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JANSANA, Ramon. 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. Studia Logica. 2023. Vol. 111, num. 4, pags. 709-715. ISSN 0039-3215. [consulted: 29 of October of 2026]. Available at: https://hdl.handle.net/2445/217688