Equivalent overdetermined and underdetermined bivalent Belnap–Dunn type semantics for the logics determined by all natural implicative expansions of Kleene’s strong 3-valued matrix with only one designated value are provided.

This paper is a sequel to ‘Belnap-Dunn semantics for natural implicative expansions of Kleene's strong three-valued matrix with two designated values’, where a ‘bivalent’ Belnap-Dunn semantics is provided for all the expansions referred to in its title. The aim of the present paper is to carry out a parallel investigation for all natural implicative expansions of Kleene's strong 3-valued matrix now with only one designated value.

In this introduction to the special issue “40 years of FDE”, we offer an overview of the field and put the papers included in the special issue into perspective. More specifically, we first present various semantics and proof systems for FDE, and then survey some expansions of FDE by adding various operators starting with constants. We then turn to unary and binary connectives, which are classified in a systematic manner. First-order FDE is also briefly revisited, and we conclude by listing (...) some open problems for future research. (shrink)

One of the problems we face in many-valued logic is the difficulty of capturing the intuitive meaning of the connectives introduced through truth tables. At the same time, however, some logics have nice ways to capture the intended meaning of connectives easily, such as four-valued logic studied by Belnap and Dunn. Inspired by Dunn’s discovery, we first describe a mechanical procedure, in expansions of Belnap-Dunn logic, to obtain truth conditions in terms of the behavior of the Truth and the False, (...) which gives us intuitive readings of connectives, out of truth tables. Then, we revisit the notion of functional completeness, which is one of the key notions in many-valued logic, in view of Dunn’s idea. More concretely, we introduce a generalized notion of functional completeness which naturally arises in the spirit of Dunn’s idea, and prove some fundamental results corresponding to the classical results proved by Post and Słupecki. (shrink)

The relationships between various modal logics based on Belnap and Dunn’s paraconsistent four-valued logic FDE are investigated. It is shown that the paraconsistent modal logic \, which lacks a primitive possibility operator \, is definitionally equivalent with the logic \, which has both \ and \ as primitive modalities. Next, a tableau calculus for the paraconsistent modal logic KN4 introduced by L. Goble is defined and used to show that KN4 is definitionally equivalent with \ without the absurdity constant. Moreover, (...) a tableau calculus is defined for the modal bilattice logic MBL introduced and investigated by A. Jung, U. Rivieccio, and R. Jansana. MBL is a generalization of BK that in its Kripke semantics makes use of a four-valued accessibility relation. It is shown that MBL can be faithfully embedded into the bimodal logic \ over the non-modal vocabulary of MBL. On the way from \ to MBL, the Fischer Servi-style modal logic \ is defined as the set of all modal formulas valid under a modified standard translation into first-order FDE, and \ is shown to be characterized by the class of all models for \. Moreover, \ is axiomatized and this axiom system is proved to be strongly sound and complete with respect to the class of models for \. Moreover, the notion of definitional equivalence is suitably weakened, so as to show that \ and \ are weakly definitionally equivalent. (shrink)

We present a novel treatment of set theory in a four-valued paraconsistent and paracomplete logic, i.e., a logic in which propositions can be both true and false, and neither true nor false. Our approach is a significant departure from previous research in paraconsistent set theory, which has almost exclusively been motivated by a desire to avoid Russell’s paradox and fulfil naive comprehension. Instead, we prioritise setting up a system with a clear ontology of non-classical sets, which can be used to (...) reason informally about incomplete and inconsistent phenomena, and is sufficiently similar to ${\mathrm {ZFC}}$ to enable the development of interesting mathematics. We propose an axiomatic system ${\mathrm {BZFC}}$, obtained by analysing the ${\mathrm {ZFC}}$ -axioms and translating them to a four-valued setting in a careful manner, avoiding many of the obstacles encountered by other attempted formalizations. We introduce the anti-classicality axiom postulating the existence of non-classical sets, and prove a surprising results stating that the existence of a single non-classical set is sufficient to produce any other type of non-classical set. Our theory is naturally bi-interpretable with ${\mathrm {ZFC}}$, and provides a philosophically satisfying view in which non-classical sets can be seen as a natural extension of classical ones, in a similar way to the non-well-founded sets of Peter Aczel [1]. Finally, we provide an interesting application concerning Tarski semantics, showing that the classical definition of the satisfaction relation yields a logic precisely reflecting the non-classicality in the meta-theory. (shrink)

In this study, we introduce Gentzen-type sequent calculi BDm and BDi for a modal extension and an intuitionistic modification, respectively, of De and Omori’s extended Belnap–Dunn logic BD+ with classical negation. We prove theorems for syntactically and semantically embedding BDm and BDi into Gentzen-type sequent calculi S4 and LJ for normal modal logic and intuitionistic logic, respectively. The cut-elimination, decidability, and completeness theorems for BDm and BDi are obtained using these embedding theorems. Moreover, we prove the Glivenko theorem for embedding (...) BD+ into BDi and the McKinsey–Tarski theorem for embedding BDi into BDm. (shrink)

Gentzen-type sequent calculi GBD+, GBDe, GBD1, and GBD2 are respectively introduced for De and Omori’s axiomatic extensions BD+, BDe, BD1, and BD2 of Belnap–Dunn logic by adding classical negation. These calculi are constructed based on a small modification of the original characteristic axiom scheme for negated implication. Theorems for syntactically and semantically embedding these calculi into a Gentzen-type sequent calculus LK for classical logic are proved. The cut-elimination, decidability, and completeness theorems for these calculi are obtained using these embedding theorems. (...) Similar results excluding cut-elimination results are also obtained for alternative Gentzen-type sequent calculi gBD+, gBDe, gBD1, and gBD2 for BD+, BDe, BD1, and BD2, respectively. These alternative calculi are constructed based on the original characteristic axiom scheme for negated implication. (shrink)

We investigate the notion of classical negation from a non-classical perspective. In particular, one aim is to determine what classical negation amounts to in a paracomplete and paraconsistent four-valued setting. We first give a general semantic characterization of classical negation and then consider an axiomatic expansion BD+ of four-valued Belnap–Dunn logic by classical negation. We show the expansion complete and maximal. Finally, we compare BD+ to some related systems found in the literature, specifically a four-valued modal logic of Béziau and (...) the logic of classical implication and a paraconsistent de Morgan negation of Zaitsev. (shrink)

This paper introduces the logic _Q__L__E__T_ _F_, a quantified extension of the logic of evidence and truth _L__E__T_ _F_, together with a corresponding sound and complete first-order non-deterministic valuation semantics. _L__E__T_ _F_ is a paraconsistent and paracomplete sentential logic that extends the logic of first-degree entailment (_FDE_) with a classicality operator ∘ and a non-classicality operator ∙, dual to each other: while ∘_A_ entails that _A_ behaves classically, ∙_A_ follows from _A_’s violating some classically valid inferences. The semantics of _Q__L__E__T_ (...) _F_ combines structures that interpret negated predicates in terms of anti-extensions with first-order non-deterministic valuations, and completeness is obtained through a generalization of Henkin’s method. By providing sound and complete semantics for first-order extensions of _FDE_, _K3_, and _LP_, we show how these tools, which we call here the method of _anti-extensions + valuations_, can be naturally applied to a number of non-classical logics. (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)

In one of their papers, Michael De and Hitoshi Omori observed that the notion of classical negation is not uniquely determined in the context of so-called Belnap-Dunn logic, and in fact there are 16 unary operations that qualify to be called classical negation. These varieties are due to different falsity conditions one may assume for classical negation. The aim of this paper is to observe that there is an interesting way to make sense of classical negation independent of falsity conditions. (...) We discuss two equivalent semantics, and offer a Hilbert-style system that is sound and complete with respect to the semantics. (shrink)