We consider a family of logical systems for representing entailment relations of various kinds. This family has its root in the logic of first-degree entailment formulated as a binary consequence system, i.e. a proof system dealing with the expressions of the form \, where both \ and \ are single formulas. We generalize this approach by constructing consequence systems that allow manipulating with sets of formulas, either to the right or left of the turnstile. In this way, it is possible (...) to capture proof-theoretically not only the entailment relation of the standard four-valued Belnap’s logic, but also its dual version, as well as some of their interesting extensions. The proof systems we propose are, in a sense, of a hybrid Hilbert–Gentzen nature. We examine some important properties of these systems and establish their completeness with respect to the corresponding entailment relations. (shrink)

In this paper, we look at applying the techniques from analyzing superintuitionistic logics to extensions of the cointuitionistic Priest-da Costa logic daC (introduced by Graham Priest as “da Costa logic”). The relationship between the superintuitionistic axioms- definable in daC- and extensions of Priest-da Costa logic (sdc-logics) is analyzed and applied to exploring the gap between the maximal si-logic SmL and classical logic in the class of sdc-logics. A sequence of strengthenings of Priest-da Costa logic is examined and employed to pinpoint (...) the maximal non-classical extension of both daC and Heyting-Brouwer logic HB . Finally, the relationship between daC and Logics of Formal Inconsistency is examined. (shrink)

In the field of logics of formal inconsistency, the notion of “consistency” is frequently too broad to draw decisive conclusions with respect to the validity of many theses involving the consistency connective. In this paper, we consider the matter of the axiom 0—i.e., the schema ◦ ◦ϕ—by considering its interpretation in contexts in which “consistency” is understood as a type of verifiability. This paper suggests that such an interpretation is implicit in two extracanonical LFIs—Sören Halldén’s nonsense-logic C and Graham Priest’s (...) cointuitionistic logic daC—drawing some interesting conclusions concerning the status of 0. Initially, we discuss Halldén’s skepticism of this axiom and provide a plausible counterexample to its validity. We then discuss the interpretation of the operator in Priest’s daC and show the equivalence of 0 to the intuitionistic principle of testability. These observations suggest that it may be fruitful for members of the LFI community to look outside the canon for evidence concerning the adoption of principles like 0. (shrink)

This paper is dedicated to developing a formalism that takes rejection seriously. Bilateral notation of signed formulas with force indicators is adopted to define signed consequences which can be viewed as the bilateral counterpart of Tarskian consequence relations. Its relation to some other bilateral approaches is discussed. It is shown how David Nelson’s logic N4 can be characterized bilaterally and the corresponding completeness result is proved. Further, bilateral variants of three familiar notions are considered and investigated: that of a fragment, (...) of definitional equivalence, and of a conservative extension. (shrink)

The work is concerned with the so called display property of display logic. The motivation behind it is discussed and challenged. It is shown using one display calculus for intuitionistic logic as an example that the display property can be abandoned without losing subformula, cut elimination and completeness properties in such a way that results in additional expressive power of the system. This is done by disentangling structural connectives so that they are no longer context-sensitive. A recipe for characterizing structural (...) extensions of display calculi is provided. (shrink)

This volume presents recent advances in philosophical logic with chapters focusing on non-classical logics, including paraconsistent logics, substructural logics, modal logics of agency and other modal logics. The authors cover themes such as the knowability paradox, tableaux and sequent calculi, natural deduction, definite descriptions, identity, truth, dialetheism and possible worlds semantics. The developments presented here focus on challenging problems in the specification of fundamental philosophical notions, as well as presenting new techniques and tools, thereby contributing to the development of the (...) field. Each chapter contains a bibliography, to assist the reader in making connections in the specific areas covered. Thus this work provides both a starting point for further investigations into philosophical logic and an update on advances, techniques and applications in a dynamic field. The chapters originate from papers presented during the Trends in Logic XI conference at the Ruhr University Bochum, June 2012. (shrink)