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I introduce a novel and quite intuitive interpretation of the ternary relation that figures in the relational semantics of many relevance logics. Conceptually, my interpretation makes use only of incompatibility and parthood relations, defined over a set of states. In this way, the proposed interpretation—of the ternary relation and the conditional—extends Dunn's and Restall's works on negation and the Routley star operator. Therefore, the interpretation is unified, and hence not only intuitive but also parsimonious. Additionally, the interpretation provides us with a cogent argument to the effect that a ternary relation is indispensable, and we cannot merely do with a binary relation. [ABSTRACT FROM AUTHOR]

We study a version of public announcement logic with common knowledge based on the relevant logic |$\textsf {R}$|⁠. Public announcements, as represented in our framework, are not necessarily truthful and accepted by all agents, nor is it assumed that beliefs are preserved under announcements. We establish a completeness result with respect to a relational semantics, and we show that an alternative semantics based on information states is dual to the relational one. We add a question-forming inquisitive disjunction operator to the language and prove a completeness result with respect to the information semantics. [ABSTRACT FROM AUTHOR]

This paper offers general support for what Valerie Plumwood's paper is trying to achieve by supporting the rejection of each of her four "false laws of logic": exportation, illegitimate replacement, commutation (aka. permutation) and disjunctive syllogism. We start by considering her general characterizations of entailment, beginning with her stated definition of entailment as the converse of deducibility. However, this applies to a wide range of relevant logics and so is not able to be used as a criterion for deciding what laws to include in a logic. In this context, we examine the two key differences between deduction from premises to conclusion and entailment from antecedent to consequent. We also consider her use of sufficiency as a general characterizing feature. We then discuss Plumwood's syntactic criteria used to reject the first three of her false laws of logic and add the Relevance Condition in this context. We next consider semantic characterizing criteria for a logic. After making a case against using truth, we introduce Brady's logic MC of meaning containment. We then examine the content semantics for MC and use it to reject all of Plumwood's false laws of logic together with some others. We follow with the related Depth Relevance Condition, which is a syntactic criterion satisfied by MC. This clearly rejects the first three of these laws and many others, but not the fourth law. We conclude by giving our overall support for her general enterprise. [ABSTRACT FROM AUTHOR]

Pretabularity is the attribute of logics that are not characterised by finite matrices, but all of whose proper extensions are. Two of the first-known pretabular logics were Dummett's famous super-intuitionistic logic LC and the well-known semi-relevance logic RM (= R-Mingle). In this paper, we investigate Anderson and Belnap's relevance logic R with the extra axiom: (p → q) ∨ (q → p), which we name LR, and which is (much) weaker than RM, and so is not pretabular. This means that over LR, there may be some pretabular extensions other than RM, two of which this paper presents and with which it provides axiomatizations, characteristic algebras, and Routley-Meyer semantics. [ABSTRACT FROM AUTHOR]

This paper proposes a formalization of the class of sentences quantified by most, which is also interpreted as proportion of or majority of depending on the domain of discourse. We consider sentences of the form "Most A are B", where A and B are plural nouns and the interpretations of A and B are infinite subsets of N . There are two widely used semantics for Most A are B: (i) C (A ∩ B) > C (A \ B) and (ii) C (A ∩ B) > C (A) 2 , where C(X) denotes the cardinality of a given finite set X. Although (i) is more descriptive than (ii), it also produces a considerable amount of insensitivity for certain sets. Since the quantifier most has a solid cardinal behaviour under the interpretation majority and has a slightly more statistical behaviour under the interpretation proportional of, we consider an alternative approach in deciding quantity-related statements regarding infinite sets. For this we introduce a new semantics using natural density for sentences in which interpretations of their nouns are infinite subsets of N , along with a list of the axiomatization of the concept of natural density. In other words, we take the standard definition of the semantics of most but define it as applying to finite approximations of infinite sets computed to the limit. [ABSTRACT FROM AUTHOR]

The logic DAI of demodalised analytic implication has been introduced by J.M. Dunn (and independently investigated by R.D. Epstein) as a variation on a time-honoured logical system by C.I. Lewis' student W.T. Parry. The main tenet underlying this logic is that no implication can be valid unless its consequent is "analytically contained" in its antecedent. DAI has been investigated both proof-theoretically and model-theoretically, but no study so far has focussed on DAI from the viewpoint of abstract algebraic logic. We provide several different algebraic semantics for DAI, showing their equivalence with the known semantics by Dunn and Epstein. We also show that DAI is algebraisable and we identify its equivalent quasivariety semantics. This class turns out to be a linguistic and axiomatic expansion of involutive bisemilattices, a subquasivariety of which forms the algebraic counterpart of Paraconsistent Weak Kleene logic (PWK). This fact sheds further light on the relationship between containment logics and logics of nonsense. [ABSTRACT FROM AUTHOR]

This article introduces, studies, and applies a new system of logic which is called 'HYPE'. In HYPE, formulas are evaluated at states that may exhibit truth value gaps (partiality) and truth value gluts (overdeterminedness). Simple and natural semantic rules for negation and the conditional operator are formulated based on an incompatibility relation and a partial fusion operation on states. The semantics is worked out in formal and philosophical detail, and a sound and complete axiomatization is provided both for the propositional and the predicate logic of the system. The propositional logic of HYPE is shown to contain first-degree entailment, to have the Finite Model Property, to be decidable, to have the Disjunction Property, and to extend intuitionistic propositional logic conservatively when intuitionistic negation is defined appropriately by HYPE's logical connectives. Furthermore, HYPE's first-order logic is a conservative extension of intuitionistic logic with the Constant Domain Axiom, when intuitionistic negation is again defined appropriately. The system allows for simple model constructions and intuitive Euler-Venn-like diagrams, and its logical structure matches structures well-known from ordinary mathematics, such as from optimization theory, combinatorics, and graph theory. HYPE may also be used as a general logical framework in which different systems of logic can be studied, compared, and combined. In particular, HYPE is found to relate in interesting ways to classical logic and various systems of relevance and paraconsistent logic, many-valued logic, and truthmaker semantics. On the philosophical side, if used as a logic for theories of type-free truth, HYPE is shown to address semantic paradoxes such as the Liar Paradox by extending non-classical fixed-point interpretations of truth by a conditional as well-behaved as that of intuitionistic logic. Finally, HYPE may be used as a background system for modal operators that create hyperintensional contexts, though the details of this application need to be left to follow-up work. [ABSTRACT FROM AUTHOR]

We show that most of the paraconsistent logics which have been investigated in the literature have no finite characteristic matrices, and in the most important cases not even finite characteristic non-deterministic matrices (Nmatrices). [ABSTRACT FROM AUTHOR]

There is a relatively recent trend in treating negation as a modal operator. One such reason is that doing so provides a uniform semantics for the negations of a wide variety of logics and arguably speaks to a longstanding challenge of Quine put to non-classical logics. One might be tempted to draw the conclusion that negation is a modal operator, a claim Francesco Berto (Mind, 124(495), 761-793, 2015) defends at length in a recent paper. According to one such modal account, the negation of a sentence is true at a world x just in case all the worlds at which the sentence is true are incompatible with x. Incompatibility is taken to be the key notion in the account, and what minimal properties a negation has comes down to which minimal conditions incompatibility satisfies. Our aims in this paper are twofold. First, we wish to point out problems for the modal account that make us question its tenability on a fundamental level. Second, in its place we propose an alternative, non-modal, account of negation as a contradictory-forming operator that we argue is superior to, and more natural than, the modal account. [ABSTRACT FROM AUTHOR]

This article presents some theorems for syntactic and semantic embeddings of a Gentzen-type sequent calculus MLn for multilattice logic into a Gentzen-type sequent calculus LK for classical logic and vice versa. These embedding theorems are used to prove cut-elimination, decidability and completeness theorems for MLn, as well as a modified Craig interpolation theorem. Some of these results are then extended to the first-order system FMLn with implications and co-implications. [ABSTRACT FROM AUTHOR]

This article addresses and resolves some issues of relational, Kripke-style, semantics for the logics of bounded lattice expansions with operators of well-defined distribution types, focusing on the case where the underlying lattice is not assumed to be distributive. It therefore falls within the scope of the theory of Generalized Galois Logics (GGLs), introduced by Dunn, and it contributes to its extension. We introduce order-dual relational semantics and present a semantic analysis and completeness theorems for non-distributive lattice logic with n-ary additive or multiplicative operators (n-ary boxes and diamonds), with negation operators modally interpreted as impossibility and unnecessity (falsifiability), as well as with implication connectives. Order-dual relational semantics shares with the generalized Kripke frames, or the bi-approximation semantics approach, the use of both a satisfaction and a co-satisfaction (refutation) relation, but it also responds to the recently voiced concerns of Craig, Haviar and Conradie about the relative non-intuitiveness of the 2-sorted semantics of the aforementioned approaches. In this article, we provide a standard (classical) interpretation (or dual interpretation) of modalities and natural interpretations of both negation and implication, despite the absence of distribution. Thereby, our results contribute in creating the necessary background for research in non-distributive logics with modalities variously interpreted as dynamic, temporal etc, by analogy to the classical case. [ABSTRACT FROM AUTHOR]

Kripke's theory of truth (Kripke, The Journal of Philosophy 72(19), 690-716; 1975) has been very successful but shows well-known expressive difficulties; recently, Field has proposed to overcome them by adding a new conditional connective to it. In Field's theories, desirable conditional and truth-theoretic principles are validated that Kripke's theory does not yield. Some authors, however, are dissatisfied with certain aspects of Field's theories, in particular the high complexity. I analyze Field's models and pin down some reasons for discontent with them, focusing on the meaning of the new conditional and on the status of the principles so successfully recovered. Subsequently, I develop a semantics that improves on Kripke's theory following Field's program of adding a conditional to it, using some inductive constructions that include Kripke's one and feature a strong evaluation for conditionals. The new theory overcomes several problems of Kripke's one and, although weaker than Field's proposals, it avoids the difficulties that affect them; at the same time, the new theory turns out to be quite simple. Moreover, the new construction can be used to model various conceptions of what a conditional connective is, in ways that are precluded to both Kripke's and Field's theories. [ABSTRACT FROM AUTHOR]

A notorious difficulty with modal extensions over a non-distributive propositional basis is to construct canonical Kripke models (time flow structures, when a temporal interpretation is intended) that respect the well-established intuitions about the meaning of modal (temporal) operators. Indeed, advances in modal logics over a non-distributive propositional basis over the last decade or so address a number of issues of significance, such as Sahlqvist (algorithmic) correspondence and completeness, yet they do this while resting on a notion of frame and model, canonical or otherwise, that compromises the intuitive semantics of the modal operators in important ways. This becomes particularly apparent when a dynamic, or a temporal reading of boxes and diamonds is intended. This article is restricted to the simplest temporal logic, a Priorean Tense Logic over a negation and implication free non-distributive propositional basis. We build up to this system by considering separately systems with modal operators in isolation, or in related groups, and we prove, for each system, completeness via a traditional canonicity argument. We establish that the absence of a distributivity assumption of conjunctions over disjunctions and conversely has no effect on the interpretation of boxes and diamonds, which are interpreted exactly as in classical normal modal logics. The motivation of this work resides in the realization that at least part of the reason for the manifest dissatisfaction with studying and applying modal logics over a non-distributive propositional basis is due precisely to the fact that a large number of semantic intuitions, results and techniques familiar from distributive logics seem to have to be inescapably abandoned. We argue in this article that this is not necessarily the case. The results we present can be applied, e.g. in studying dynamic, or temporal extensions of Orthomodular Quantum Logic, as well as in the relational semantics for Substructural Logics. [ABSTRACT FROM AUTHOR]

This paper briefly overviews some of the results and research directions. In the area of substructural logics from the last couple of decades. Substructural logics are understood here to include relevance logics, linear logic, variants of Lambek calculi and some other logics that are motivated by the idea of omitting some structural rules or making other structural changes in LK, the original sequent calculus for classical logic. [ABSTRACT FROM AUTHOR]

The hallmark of the deductive systems known as ‘conceptivist’ or ‘containment’ logics is that for all theorems of the form, all atomic formulae appearing inalso appear in. Significantly, as a consequence, the principle of Addition (the inference tofrom) fails. While often billed as a formalisation of Kantian analytic judgements, once semantics were discovered for these systems, the approach was largely discounted as merely the imposition of a syntactic filter on unrelated systems. In this paper, we examine a number ofprima facieunrelated deductive contexts in which Addition fails and attempt to harmonise them by developing a computational interpretation of conceptivist logics. [ABSTRACT FROM PUBLISHER]

A modification of Kripke’s theory of truth is proposed and it is shown how this modification solves some of the problems of expressive weakness in Kripke’s theory. This is accomplished by letting truth values be grounded in facts about other sentences’ ungroundedness. [ABSTRACT FROM PUBLISHER]

In this article, we present relational semantics for the Lambek–Grishin calculus and various extensions. Following the approach of generalized Kripke semantics described in Gehrke (2006, Studia Logica, 84, 241–275), we consider semantics based on the generalized Kripke frames naturally associated with the algebraic semantics of the logics in question via their representation theory. This approach is based on canonicity and correspondence as in the classical modal logic setting. Traditional Kripke semantics for the Lambek–Grishin calculus have the drawback that, as soon as additional axioms or additional connectives are present, one may have to start over to obtain semantics for such richer logics. The advantage of our approach via canonical extensions of LG-algebras is that each additional axiom that lifts to the canonical extension can be handled in a modular way, whereas additional connectives modularly slot in as additional relational components. All groups of axioms presented by Grishin in Grishin (1983, Symmetric categorial Grammar) are canonical, and we obtain Sahlqvist-style correspondence results for each of these. The modular set-up allows us to augment these results by the correspondence results for associativity, commutativity, weakening and contraction given in Dunn, Gehrke, and Palmigiano (2005, J. Symbol. Logic., 70, 713–740), as well by results for additional connectives such as lattice operations and linear logic-type negation. This allows a clear comparison of the various logics and a fully modular family of completeness results. [ABSTRACT FROM PUBLISHER]

One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley-Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing a general conception of conditionality that may unify the three given conceptions. [ABSTRACT FROM AUTHOR]

A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra–Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended to other modal logics. [ABSTRACT FROM AUTHOR]

In this work, an intuitionistic propositional logic with a Galois connection (IntGC) is introduced. In addition to the intuitionistic logic axioms and inference rule of modus ponens, the logic contains only two rules of inference mimicking the performance of Galois connections. Both Kripke-style and algebraic semantics are presented for IntGC, and IntGC is proved to be complete with respect to both of these semantics. We show that IntGC has the finite model property and is decidable, but Glivenko's Theorem does not hold. Duality between algebraic and Kripke semantics is presented, and a representation theorem for Heyting algebras with Galois connections is proved. In addition, an application to rough L-valued sets is presented. [ABSTRACT FROM PUBLISHER]

Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics and relation algebras. Some details of certain incompleteness results, however, pinpoint where relevance logics and relation algebras diverge. To carry out these semantic investigations, we define a new tableaux formalization and new sequent calculi (with the single cut rule admissible) for various relevance logics. [ABSTRACT FROM AUTHOR]

The article discusses the implication of semantic position, which is an evaluative stance with respect to how someone's reasoning ought to be appraised and what criterion of argument-goodness ought to be applied. It cites a case of a boy who was confused of the two ants and mistakenly recognizes that the two ants are the same. Thus, the error of which a confused person is guilty is not the error of believing something false de re. However, philosopher Hartry Field's theory change and the indeterminancy of reference recommends a supervaluational approach to the problem of confusion. However, the author believes that the confused sentences of a confused person can be usefully evaluated as True of False because such sentences do not fit the world tightly enough to be evaluated.

We study logic programs under Gelfond's translation in the context of modal logic S5. We show that for arbitrary logic programs (propositional theories where logic negation is associated with default negation) ground nonmonotonic modal logics between T and S5 are equivalent. Furthermore, we also show that these logics are equivalent to a nonmonotonic logic that we construct using the well known F O U R bilattice. We will call this semantic GNM-S5 as a reminder of its origin in the logic S5. Finally we show that, for normal programs, our approach is closely related to theWell-Founded-by-Cases Semantics introduced by Schlipf and the WFS+ proposed by Dix. We prove that GNM-S5 has the properties of classicality and extended cut. While WFS+ also supports classicality it fails to satisfy the extended cut principle, an important property available in other semantics such as stable models. Hence, we claim that GNM-S5 is a good candidate for defining a nonmonotonic semantics closer to the direction of classical logic. [ABSTRACT FROM PUBLISHER]

Further to the connections between meaning and use, it seems useful to consider the (explanation of the immediate) consequences one is allowed to draw from a proposition as something directly related to its meaning/use. And indeed, Wittgenstein's references to the connections between meaning and the consequences, as well as between use and consequences are sometimes as explicit as his celebrated ‘definition’ of meaning as use given in the Investigations. Here we attempt to collect some of these references, discussing how an intuitive basis for the construction of a more convincing proof-theoretic semantics (than, say, assertability conditions semantics) for the mathematical language can arise out of this connection meaning/use/(explanation of the immediate) consequences.1 [ABSTRACT FROM PUBLISHER]

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