The paper is a continuation of A deductive-reductive form oflogic: general theory and intuitionistic case and considers the problem of definability of modal operators on the intuitionistic base. Contrary tothe classical case, it seems that the fact whether the connective is Heyting’sor Brouwerian is essential for the intuitionistic logic. The connective of possibility has the classical interpretation, i.e. w |= ✸α iff ∃t, if it is defined on the base of the logicwith Brouwerian connective of coimplication.

We consider a logic which is semantically dual (in some precise sense of the term) to intuitionistic. This logic can be labeled as “falsification logic”: it embodies the Popperian methodology of scientific discovery. Whereas intuitionistic logic deals with constructive truth and non-constructive falsity, and Nelson's logic takes both truth and falsity as constructive notions, in the falsification logic truth is essentially non-constructive as opposed to falsity that is conceived constructively. We also briefly clarify the relationships of our falsification logic to (...) some other logical systems. (shrink)

Two new first-order paraconsistent logics with De Morgan-type negations and co-implication, called symmetric paraconsistent logic (SPL) and dual paraconsistent logic (DPL), are introduced as Gentzen-type sequent calculi. The logic SPL is symmetric in the sense that the rule of contraposition is admissible in cut-free SPL. By using this symmetry property, a simpler cut-free sequent calculus for SPL is obtained. The logic DPL is not symmetric, but it has the duality principle. Simple semantics for SPL and DPL are introduced, and the (...) completeness theorems with respect to these semantics are proved. The cut-elimination theorems for SPL and DPL are proved in two ways: One is a syntactical way which is based on the embedding theorems of SPL and DPL into Gentzen’s LK, and the other is a semantical way which is based on the completeness theorems. (shrink)

This study introduces refutation-aware Gentzen-style sequent calculi and Kripke-style semantics for propositional until-free linear-time temporal logic. The sequent calculi and semantics are constructed on the basis of the refutation-aware setting for Nelson’s paraconsistent logic. The cut-elimination and completeness theorems for the proposed sequent calculi and semantics are proven.

In this study, falsification-aware semantics and sequent calculi for first-order classical logic are introduced and investigated. These semantics and sequent calculi are constructed based on a falsification-aware setting for first-order Nelson constructive three-valued logic. In fact, these semantics and sequent calculi are regarded as those for a classical variant of N3. The completeness and cut-elimination theorems for the proposed semantics and sequent calculi are proved using Schütte’s method. Similar results for the propositional case are also obtained.

Falsification-aware (hyper)sequent calculi and Kripke semantics for normal modal logics including S4 and S5 are introduced and investigated in this study. These calculi and semantics are constructed based on the idea of a falsification-aware framework for Nelson’s constructive three-valued logic. The cut-elimination and completeness theorems for the proposed calculi and semantics are proved.