An algebraic type of structure is shown forth which is such that if it is a characteristic matrix for a logic, then that logic satisfies Meyer's weak variable sharing property. As a corollary, it is shown that RM and all its odd-valued extensions \(\mathbf{RM}_{2n\mathord{-}1}\) satisfy the weak variable sharing property. It is also shown that a proof to the effect that the "fuzzy" version of the relevant logic R satisfies the property is incorrect.

As is well known, the variable-sharing property (vsp) is, according to Anderson and Belnap, a necessary property of any relevant logic. In this paper, we shall consider two versions of the vsp, what we label the "weak vsp" (wvsp) and the "strong vsp" (svsp). In addition, the "no loose pieces property," a property related to the wvsp and the svsp, will be defined. Each one of these properties shall generally be characterized by means of a class of (...) logical matrices. In this way, any logic verified by an actual matrix in one of these classes has the property the class generally represents. Particular matrices (and so, logics) in each class are provided. (shrink)

In our paper “A general characterization of the variable-sharing property by means of logical matrices”, a general class of so-called “Relevant logical matrices”, RMLs, is defined. The aim of this paper is to define a class of simpler Relevant logical matrices RMLs′serving the same purpose that RMLs, to wit: any logic verified by an RML′has the variable-sharing property and related properties predicable of the logic of entailment E and of the logic of relevance R.

A restriction of R-Mingle with the variable-sharing property and the Ackermann properties is defined. From an intuitive semantical point of view, this restriction is an alternative to Anderson and Belnap’s logic of entailment E.

Many relevant logics are conservatively extended by Boolean negation. Not all, however. This paper shows an acute form of non-conservativeness, namely that the Boolean-free fragment of the Boolean extension of a relevant logic need not always satisfy the variable-sharing property. In fact, it is shown that such an extension can in fact yield classical logic. For a vast range of relevant logic, however, it is shown that the variable-sharing property, restricted to the Boolean-free fragment, still holds (...) for the Boolean extended logic. (shrink)

As it is well known, Relevance Logic R plus the axiom mingle (R-Mingle) does not have the variable-sharing property (vsp). The aim of this paper is to improve this result by defining a weak logic with the axiom mingle and not included in minimal logic BM lacking the vsp.

Let us refer by MK3 to Kleene’s strong 3-valued matrix. An implicative expansion of MK3 is natural if the conditional function defining it verifies modus ponens, assigns a designated value to a conditional whenever it assigns the same value to its antecedent and its consequent, and, finally, it coincides with the classical conditional function when restricted to the “classical” values ????\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {t}$$\end{document} and ????\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} (...) \begin{document}$$\mathfrak {f}$$\end{document}. Two are the main results of this paper. (1) It is proven that, from the viewpoint of functional strength, there is only one 3-valued natural implication expansion of MK3 with the variable-sharing property, the logic we dub L3VSP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ }^{\text{VSP}}$$\end{document}. (2) It is shown that L3VSP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ }^{\text{VSP}}$$\end{document} is a significant and strong logic that can be seen from different perspectives, one of them being to consider it an expansion of classical positive propositional logic. (shrink)

The logic TM is the result of adding the mingle axiom, M to Ticket Entailment logic, T. In the present study, it is proved that TM has the variable-sharing property . Ternary relational semantics for TM is provided. Finally, an interesting extension of TM with the vsp is briefly discussed.

The class of the logics free from paradoxes of relevance determined by all natural implicative expansions of Kleene’s strong 3-valued matrix with two designated values is defined. These logics are free from paradoxes of relevance in the sense that they have the "variable sharing property". They have "natural conditionals" in the sense that the function defining them coincideswith the classical function when restricted to the "classical values", satisfies Modus Ponens and, finally, assigns a designated value to a conditional (...) whenever its antecedent and its consequence are assigned the same value. These logics are defined by using a "two-valued" overdetermined Belnap-Dunn semantics. Thus, the interpretation of the three values is crystalline. The logics here introduced enjoy the properties customarily demanded of many-valued implicative logics except, of course, the satisfaction of the rule "Verum ex quodlibet". (shrink)

We introduce, an automated reasoning tool that analyses matrices generated by Slaney’s. The tool searches these matrices for a target logic to find one that witnesses the logic’s satisfaction of the Variable Sharing Property. We describe the theoretical background behind and highlight optimisations that enable our tool to determine whether a given matrix witnesses the property in polynomial time. We then give some example data sets for particular relevant logics, a theoretical analysis on the sizes of matrices generated (...) by which do not witness variable sharing, and close with an appendix explaining how to install and use. (shrink)

A challenge for relevant logicians is to delimit their area of study. I propose and explore the definition of a relevant logic as a logic satisfying a variable-sharing property and closed under detachment and adjunction. This definition is, I argue, a good definition that captures many familiar logics and raises interesting new questions concerning relevant logics.

We characterize the non-trivial substructural logics having the variable sharing property as well as its strong version. To this end, we find the algebraic counterparts over varieties of these logical properties.

In this paper, we examine a number of relevant logics’ variable sharing properties from the perspective of theories of topic or subject-matter. We take cues from Franz Berto’s recent work on topic to show an alignment between families of variable sharing properties and responses to the topic transparency of relevant implication and negation. We then introduce and defend novel variable sharing properties stronger than strong depth relevance—which we call cn-relevance and lossless (...) cn-relevance—showing that the properties are satisfied by the weak relevant logics B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{B}}$$\end{document} and BM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf{B}}}{{\textbf{M}}}$$\end{document}, respectively. We argue that such properties address a sort of semantic lossiness of strong depth relevance. (shrink)

However broad or vague the notion of connexivity may be, it seems to be similar to the notion of relevance even when relevance and connexive logics have been shown to be incompatible to one another. Relevance logics can be examined by suggesting syntactic relevance principles and inspecting if the theorems of a logic abide to them. In this paper we want to suggest that a similar strategy can be employed with connexive logics. To do so, we will suggest some (...) class='Hi'>properties that seem to be hinted at in Nelson’s work. Following this strategy will ideally shed some light over the notion of content and will also help make a clear comparison between relevance and connexive logics. (shrink)

The family of relevant logics can be faceted by a hierarchy of increasingly fine-grained variable sharing properties—requiring that in valid entailments $A\to B$, some atom must appear in both A and B with some additional condition (e.g., with the same sign or nested within the same number of conditionals). In this paper, we consider an incredibly strong variable sharing property of lericone relevance that takes into account the path of negations and conditionals in which an (...) atom appears in the parse trees of the antecedent and consequent. We show that this property of lericone relevance holds of the relevant logic $\mathbf {BM}$ (and that a related property of faithful lericone relevance holds of $\mathbf {B}$ ) and characterize the largest fragments of classical logic with these properties. Along the way, we consider the consequences for lericone relevance for the theory of subject-matter, for Logan’s notion of hyperformalism, and for the very definition of a relevant logic itself. (shrink)

A propositional logic has the variable sharing property if φ → ψ is a theorem only if φ and ψ share some propositional variable(s). In this note, I prove that positive semilattice relevance logic and its extension with an involution negation have the variable sharing property (as these systems are not subsystems of R, these results are not automatically entailed by the fact that R satisfies the variable sharing property). Typical proofs of the (...) variable sharing property rely on ad hoc, if clever, matrices. However, in this note, I exploit the properties of rather more intuitive arithmetical structures to establish the variable sharing property for the systems discussed. (shrink)

The present paper is a sequel to Robles et al. :349–374, 2020. https://doi.org/10.1007/s10849-019-09306-2). A class of implicative expansions of Kleene’s 3-valued logic functionally including Łukasiewicz’s logic Ł3 is defined. Several properties of this class and/or some of its subclasses are investigated. Properties contemplated include functional completeness for the 3-element set of truth-values, presence of natural conditionals, variable-sharing property and vsp-related properties.

Being relevant to some topic can be informally understood as making a difference or having something to contribute. Given a sequent Δ⇒Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \Rightarrow \Gamma $$\end{document}, we can say, somewhat schematically, that a component of the sequent is relevant to the sequent when it contributes to the validity of the sequent. Different ways of making precise the idea of contributing to the validity and different understandings of the components of a sequent lead (...) to a hierarchy of explications of relevance. I identify four key explications, called gaunt validity, perfect validity, relevant validity, and perfectibility. Each is shown to enjoy an interesting variable sharing property. Furthermore, if we begin with a standard sequent calculus for classical logic and introduce some simple constraints on the rules, the result is a fragment of classical logic that proves exactly the gauntly valid sequents. (shrink)

Dunn–McCall logic $$\mathbf {RM}$$ RM is by far the best understood and the most well-behaved logic in the family of logics developed by the school of Anderson and Belnap. However, it is not considered to be a relevant logic by the relevant logicians, since it fails to have the variable-sharing property. Instead, $$\mathbf {RM}$$ RM is usually characterized as being “semi-relevant,” without explaining what this notion means. In this paper we suggest a plausible definition of semi-relevance, and show (...) that according to it, $$\mathbf {RM}$$ RM is a strongly maximal semi-relevant logic having a conjunction, a disjunction, and an implication. We also review and prove the most important nice properties of $$\mathbf {RM}$$ RM, especially strong completeness results about it (the full proofs of which are difficult to find in the literature). (shrink)

Relevant logics infamously have the property that they only validate a conditional when some propositional variable is shared between its antecedent and consequent. This property has been strengthened in a variety of ways over the last half-century. Two of the more famous of these strengthenings are the strong variable sharing property and the depth relevance property. In this paper I demonstrate that an appropriate class of relevant logics has a property that might naturally be characterized as the (...) supremum of these two properties. I also show how to use this fact to demonstrate that these logics seem to be constructive in previously unknown ways. (shrink)

The concept of shared mental models refers to the shared understanding among team members about how they should behave in different situations. This article aimed to develop a new shared mental model measure, specifically designed for the refereeing context. A cross-sectional study was conducted with three samples: national and regional football referees (n = 133), national football referees and assistant referees and national futsal referees (n = 277), and national futsal referees (n = 60). The proposed version of the Referee (...) Shared Mental Models Measure (RSMMM) has 13 items that are reflected on a single factor structure. The RSMMM presented good validity evidence both based on the internal structure and based on relations to other variables (presenting positive associations with team work engagement, team adaptive performance, and team effectiveness). Such promising psychometric properties point to an optimistic outlook regarding its use to measure shared mental models in futsal and football referee teams. -/- . (shrink)

In this article, I provide Urquhart-style semilattice semantics for three connexive logics in an implication-negation language (I call these “pure theories of connexive implication”). The systems semantically characterized include the implication-negation fragment of a connexive logic of Wansing, a relevant connexive logic recently developed proof-theoretically by Francez, and an intermediate system that is novel to this article. Simple proofs of soundness and completeness are given and the semantics is used to establish various facts about the systems (e.g., that two of (...) the systems have the variable sharing property). I emphasize the intuitive content of the semantics and discuss how natural informational considerations underly each of the examined systems. (shrink)

Val Plumwood and Richard Sylvan argued from their joint paper The Semantics of First Degree Entailment and onward that the variable sharing property is but a mere consequence of a good entailment relation, indeed they viewed it as a mere negative test of adequacy of such a relation, the property itself being a rather philosophically barren concept. Such a relation is rather to be analyzed as a sufficiency relation free of any form of premise suppression. Suppression of premises, (...) therefore, gained center stage. Despite this, however, no serious attempt was ever made at analyzing the concept. This paper shows that their suggestions for how to understand it, either as the Anti-Suppression Principle or as the Joint Force Principle, turn out to yield properties strictly weaker than that of variable sharing. A suggestion for how to understand some of their use of the notion of suppression which clearly is not in line with these two mentioned principles is given, and their arguments to the effect that the Anderson and Belnap logics T, E and R are suppressive are shown to be both technically and philosophically wanting. Suppression-freedom, it is argued, cannot do the job Plumwood and Sylvan intended it to do. (shrink)

Relevant logics have traditionally been viewed as paraconsistent. This paper shows that this view of relevant logics is wrong. It does so by showing forth a logic which extends classical logic, yet satisfies the Entailment Theorem as well as the variable sharing property. In addition it has the same S4-type modal feature as the original relevant logic E as well as the same enthymematical deduction theorem. The variable sharing property was only ever regarded as a necessary (...) property for a logic to have in order for it to not validate the so-called paradoxes of implication. The Entailment Theorem on the other hand was regarded as both necessary and sufficient. This paper shows that the latter theorem also holds for classical logic, and so cannot be regarded as a sufficient property for blocking the paradoxes. The concept of suppression is taken up, but shown to be properly weaker than that of variable sharing. (shrink)

The aim of this paper is technically to study Belnap's four-valued sentential logic . First, we obtain a Gentzen-style axiomatization of this logic that contains no structural rules while all they are still admissible in the Gentzen system what is proved with using some algebraic tools. Further, the mentioned logic is proved to be the least closure operator on the set of {Λ, V, ⌝}-formulas satisfying Tarski's conditions for classical conjunction and disjunction together with De Morgan's laws for negation. It (...) is also proved that Belnap's logic is the only sentential logic satisfying the above-mentioned conditions together with Anderson-Belnap's Variable-Sharing Property. Finally, we obtain a finite Hilbert-style axiomatization of this logic. As a consequence, we obtain a finite Hilbert-style axiomatization of Priest's logic of paradox. (shrink)

It is shown that the logic $$\textbf{subDL}$$ satisfies the signed variable sharing property, as well as the depth relevance property. The proofs given are by way of translations into logics which are known to satisfy the requisite properties.

Relevant propositional dynamic logics have been sporadically discussed in the broader context of modal relevant logics, but have not come up for sustained investigation until recently. In this paper, we develop a philosophical motivation for these systems, and present some new results suggested by the proposed motivation. Among these, we’ll show how to adapt some recent work to show that the extensions of relevant logics by the extensional truth constants \ are complete with respect to a natural class of ternary (...) relation models to show a similar result for the constant-free versions of the logic. In addition, we prove that the logics in question satisfy the variable sharing property, vindicating the claim that they really are relevant logics. (shrink)

Ackermann’s motivational spin on his theory of rigorous implication is analyzed and it is shown to contain en equivalent idea to Plumwood’s notion of suppression freedom. The formal properties these ideas back turn out to be properly weaker than Belnap’s variable sharing property, but it is shown that they can be strengthen in various ways. Some such strengthenings, it is shown, yield properties which are equivalent to Belnap’s, and thus provide for new ways of motivating Belnap’s (...) fundamental relevance principle. (shrink)

We introduce structural rules mingle, and investigatetheorem-equivalence, cut- eliminability, decidability, interpolabilityand variable sharing property for sequent calculi having the mingle.These results include new cut-elimination results for the extendedlogics: FLm (full Lambek logic with the mingle), GLm(Girard's linear logic with the mingle) and Lm (Lambek calculuswith restricted mingle).

In this article, I consider the positive logic of Boolean groups (i.e. Abelian groups where every non-identity element has order 2), where these are taken as frames for an operational semantics à la Urquhart. I call this logic BG. It is shown that the logic over the smallest nontrivial Boolean group, taken as a frame, is identical to the positive fragment of a quasi-relevance logic that was developed by Robles and Méndez (an extension of this result where negation is included (...) is also discussed). It is proved that BG satisfies the variable sharing property and is Halldén complete, while failing to satisfy the disjunction property. Sound and complete subscripted tableaux are presented for BG. An axiomatization (in the positive language extended with fusion) is presented, which is sound with respect to BG and, with respect to a related ternary relational semantics, complete; the problem of identifying a complete axiomatization for BG itself is left open. Connections between BG and the quasi-relevance logic KR are also discussed. (shrink)

A logic is relevant if it is a set of formulas closed under adjunction, modus ponens and enjoys the variable sharing property. It is connexive if it has some or all of the versions of Aristotle’s thesis and Boethius’ thesis. It is connexively acceptable if it does not sanction such formulas as the negation of the self-identity axiom. Finally, it is strictly connexive if it does not validate some formula implying (or being implied by) its own negation. Now, (...) it is known that addition of Aristotle’s thesis to a relevant logic does not necessarily result in an acceptable connexive logic. Then, the aim of this paper is to examine the prospects of defining relevant acceptable strictly connexive logics based upon weak relevant logics. (shrink)

The computer is one of the most complex artifacts ever built. Given its complexity, it can be described from many different points of view. The aim of this paper is to investigate the representational structure and multifunctionality of a particular subset of computers, namely personal devices from a user-centred perspective. The paper also discusses the concept of “cognitive task”, as recently employed in some definitions of cognitive artifacts, and investigates the metaphysical properties of such artifacts. From a representational point (...) of view, the article introduces the concepts of artifactual meta-representation and of semi-transparency, two features that personal devices share with some cognitive and non-cognitive artifacts. Recognising the meta-representational nature of personal devices and of other cognitive artifacts, thus overcoming semi-transparency, is important for the understanding of why different artifacts offer us different cognitive affordances as well as different cognitive advantages. In this sense, it is not simply a theoretical achievement, but has some important practical consequences. In our highly technological world we can use different kinds of computers and artifacts for solving the same tasks, and we need to understand why some artifacts are better suited for some tasks than others. The ultimate characterisation of personal devices that emerges from this work is that of a sort of super-artifact. This special status is given to personal devices because of their distinctive features. They are in fact intrinsically multifunctional and meta-representational artifacts, with extremely variable structures. As super-artifacts, personal devices are characterised by macro-functionality and can be easily used as both cognitive artifacts and tools for other functions, depending on the kind of representations they instantiate. (shrink)

The two ``relevance'' criteria set out by Anderson and Belnap are discussed. It is argued that the motivation backing the variable sharing property is far weaker than it is commonly made out to be, and that the use-criterion does not distinguish between relevant logics such as E and R and ``irrelevant'' logics such as S4, intuitionistic and classical logic. In short, then, the paper argues that Anderson and Belnap's two criteria of relevance are both motivationally unsound, and do (...) not accomplish what they are commonly taken to accomplish. (shrink)

The Routley star, an involutive function between possible worlds or set-ups against which negation is evaluated, is a hallmark feature of Richard Sylvan and Val Plumwood's set-up semantics for the logic of first-degree entailment. Less frequently acknowledged is the weaker mate function described by Sylvan and his collaborators, which results from stripping the requirement of involutivity from the Routley star. Between the mate function and the Routley star, however, lies an broad field of intermediate semantical conditions characterizing an infinite number (...) of consequence relations closely related to first-degree entailment. In this paper, we consider the semantics and proof theory for deductive systems corresponding to set-up models in which the mate function is cyclical. We describe modifications to Anderson and Belnap's consecution calculus LE_fde2 that correspond to these constraints, for which we prove soundness and completeness with respect to the set-up semantics. Finally, we show that a number of familiar metalogical properties are coordinated with the parity of a mate function's period, including refined versions of the variable-sharing property and the property of gentle explosiveness. (shrink)

Priest argued in Fusion and Confusion (Priest in Topoi 34(1):55–61, 2015a) for a new concept of logical consequence over the relevant logic B, one where premises my be “confused” together. This paper develops Priest’s idea. Whereas Priest uses a substructural proof calculus, this paper provides a Hilbert proof calculus for it. Using this it is shown that Priest’s consequence relation is weaker than the standard Hilbert consequence relation for B, but strictly stronger than Anderson and Belnap’s original relevant notion of (...) consequence. Unlike the latter, however, Priest’s consequence relation does not satisfy a variant of the variable sharing property. This paper shows that how it can be modified so as to do so. Priest’s consequence relation turns out to be surprisingly weak in some respects. The prospects of strengthening it is raised and discussed in a broader philosophical context. (shrink)

The purpose of this paper is to introduce justification logics based on conditional logics. We introduce a new family of logics, called conditional justification logics, which incorporates a counterfactual conditional in its language. For the semantics, we offer relational models that merge the selection-function semantics of conditional logics with the relational semantics of justification logics. As an application, we formalize Nozick’s counterfactual conditions in his analysis of knowledge and investigate their connection to Aumann’s concepts of knowledge. Additionally, we explore Gettier’s (...) counterexamples to the justified true belief analysis, as well as McGinn’s counterexamples to Nozick’s analysis of knowledge. Furthermore, we introduce a justification logic that includes a relevant counterfactual conditional and we demonstrate the variable-sharing property for this conditional. We also develop a tableau system for this logic and establish its completeness with respect to ternary relational models. Finally, we formalize Nozick’s counterfactual conditions using this relevant counterfactual conditional and represent the sheep in the field example of Chisholm within this logic. (shrink)

The "depth relevance condition" (drc) is a strengthening of the "variable-sharing property" (vsp). Deep relevant logics are logics fulfilling the drc, and Brady's DR is a key item in this class. The "qualified factorization principles" (qfp) are strong distribution principles. The qfp can be added to Relevance logic R without the result collapsing in a logic lacking the vsp. The aim of this paper is to show that DR (and any logic included in it) can be extended with (...) the qfp, the drc being preserved. © 2016 Elsevier B.V., All rights reserved. (shrink)

I presented a sub-classical relating logic based on a relating via an NL-inspired relating relation Rcss. The relation Rcss is motivated by the NL-phenomenon of phrasal (subsentential) coordination, exhibiting an important aspect of contents relating among the arguments of binary connectives. The resulting logic Lcss can be viewed as a relevance logic exhibiting a contents related relevance, stronger than the variable-sharing property of other relevance logics like R. Note that relating here is not “tailored” to justify some predetermined (...) logic; rather, the relating relation is independently justified, and induces a logic not previously investigated. (shrink)

We show that the elimination rule for the multiplicative (or intensional) conjunction $\wedge$ is admissible in many important multiplicative substructural logics. These include LL m (the multiplicative fragment of Linear Logic) and RMI m (the system obtained from LL m by adding the contraction axiom and its converse, the mingle axiom.) An exception is R m (the intensional fragment of the relevance logic R, which is LL m together with the contraction axiom). Let SLL m and SR m be, respectively, (...) the systems which are obtained from LL m and R m by adding this rule as a new rule of inference. The set of theorems of SR m is a proper extension of that of R m, but a proper subset of the set of theorems of RMI m. Hence it still has the variable-sharing property. SR m has also the interesting property that classical logic has a strong translation into it. We next introduce general algebraic structures, called strong multiplicative structures, and prove strong soundness and completeness of SLL m relative to them. We show that in the framework of these structures, the addition of the weakening axiom to SLL m corresponds to the condition that there will be exactly one designated element, while the addition of the contraction axiom corresponds to the condition that there will be exactly one nondesignated element (in the first case we get the system BCK m, in the second - the system SR m ). Various other systems in which multiplicative conjunction functions as a true conjunction are studied, together with their algebraic counterparts. (shrink)

In this paper we elaborate a conception of entailment based on what we call the Ackermann principle, which explicates valid entailment through a logical connection between sentences depending on their informational content. We reconstruct Dunn’s informational semantics for entailment on the basis of Restall’s approach, with assertion and denial as two independent speech acts, by introducing the notion of a ‘position description’. We show how the machinery of position descriptions can effectively be used to define the positive and the negative (...) information carried by sentences of a given language and to present a formalized version of the Ackermann principle as an inclusion relationship between the informational contents of the conclusions and the premises of a valid entailment. Being so interpreted, the entailment relation exhibits certain properties, including the property of transitivity. Whereas properties such as Anderson and Belnap’s variable sharing property or Parry’s proscriptive principle are normally presented as imposing a relevance requirement on valid entailment, the suggested formalization of the Ackermann principle supports all of Gentzen’s structural rules, including weakening, a rule that is normally given up in sequent-style proof systems for relevance logics. In this way we propose an Ackermann-inspired explication of the nature of entailment as a relation between the informational contents of sentences. (shrink)

In Anderson and Belnap (1976), Meyer stated that logics which hold a weak version of the variable sharing property –referred to here as quasi-relevant property, QRP– are good enough when some degree of relevance is needed. In this article, research regarding the relation between the QRP and 4-valued matrices is conducted. The aim is threefold: (1) to define the class of 4-valued quasi-relevant matrices and to determine which of those hold a natural conditional; (2) to specify which matrices (...) from the set established in 1 verify Routley and Meyer basic logic B; (3) to strengthen these logics with truth-functional modal operators following a Łukasiewiczean approach to modal notions. (shrink)

Let R+ be the positive fragment of Anderson and Belnap’s Logic of Relevance, R. And let RMO+ be the result of adding the Mingle principle ) to R+. We have shown in [2] that either a minimal negation or else a semiclassical one can be added to RMO+ preserving the variable-sharing property. Moreover, each of there systems is given a semantics in the Routley-Meyer style. In describing in [2] the models for RMO+ plus minimal negation, we noted that (...) a similar strategy would give us a semantics for R+ with minimal negation; that is, constructive R. The aim of this paper is to prove this claim. (shrink)

A situation, in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (point-valued) solution for TU-games assigns a payoff distribution to every TU-game. In this article we discuss a class of equal surplus sharing solutions consisting of all convex combinations of the CIS-value, the ENSC-value and the equal division solution. We provide several characterizations of this class of solutions on variable (...) and fixed player set. Specifications of several properties characterize specific solutions in this class. (shrink)

The logic of analytic implication $\mathbf{PAI}$ developed by William T. Parry is part of a family of relevance logics that exhibits a strong variable-sharing property: $\phi\to\psi$ is a theorem only if every variable occurring in $\psi$ also occurs in $\phi$. A demodalized version of $\mathbf{PAI}$, known as $\mathbf{DAI}$, was formulated in \citet{dunn1972modification}. \citet{urquhart1973semantical} further introduced a modal extension of $\mathbf{DAI}$, called the logic of analytic implication with necessity, $\mathbf{AIN}$. Urquhart conjectured that $\mathbf{PAI}$ can be embedded into $\mathbf{AIN}$ (...) by a G\"{o}del-McKinsey-Tarski style translation. This paper provides a proof of this previously unverified conjecture and extends the result to other logics in the neighbourhood of $\mathbf{PAI}$. In particular, we show that intuitionistic $\mathbf{PAI}$ can be embedded into $\mathbf{AIN}$ by the G\"{o}del-McKinsey-Tarski translation. (shrink)

Concepts stand at the centre of human cognition. We use concepts in categorizing objects and events in the world, in reasoning and action, and in social interaction. It is therefore not surprising that the study of concepts constitutes a central area of research in philosophy and psychology, yet only recently have the two disciplines developed greater interaction. Recent experiments in psychology that test the role of concepts in categorizing and reasoning have found a great deal of variation, across individuals and (...) cultures, in categorization behaviour. Meanwhile, philosophers of language and mind have investigated the semantic properties of concepts, and how concepts are related to linguistic meaning and linguistic communication. A key motivation behind this was the idea that concepts must be shared across individuals and cultures. With the dawn of experimental philosophy, the proposal that the experimental data from psychology lacks relevance to semantics is increasingly difficult to defend. -/- This volume brings together leading psychologists and philosophers to advance the interdisciplinary debate on the role of concepts in categorizing and reasoning, the relationship between concepts and linguistic meaning and communication, the challenges conceptual variation poses to communication, and the social and political effects of conceptual change. (shrink)

Different directed acyclic graphs may be Markov equivalent in the sense that they entail the same conditional independence relations among the observed variables. Chickering provided a transformational characterization of Markov equivalence for DAGs, which is useful in deriving properties shared by Markov equivalent DAGs, and, with certain generalization, is needed to prove the asymptotic correctness of a search procedure over Markov equivalence classes, known as the GES algorithm. For DAG models with latent variables, maximal ancestral graphs provide a neat (...) representation that facilitates model search. However, no transformational characterization -- analogous to Chickering's -- of Markov equivalent MAGs is yet available. This paper establishes such a characterization for directed MAGs, which we expect will have similar uses as it does for DAGs. (shrink)

A Bayesian network consists of two distinct parts: a directed acyclic graph (DAG or belief-network structure) and a set of parameters for the DAG. The DAG in a Bayesian network can be used to represent both causal hypotheses and sets of probability distributions. Under the causal interpretation, a DAG represents the causal relations in a given population with a set of vertices V when there is an edge from A to B if and only if A is a direct cause (...) of B relative to V. (We adopt the convention that sets of variables are capitalized and boldfaced, and individual variables are capitalized and italicized.) Under the statistical interpretation a DAG G can be taken to represent a set of all distributions all of which share a set of conditional independence relations that are entailed by satisfying a local directed Markov property (defined below). (shrink)

Ironically Anderson and Belnap argue for the rejection of Disjunctive Syllogism by means of an argument that appears to employ it. We aim to establish a ‘variable-sharing’ result for Classical Core Logic that is stronger than any such result for any other system. We define an exigent relevance condition R(X,A) on the premise-set X and the conclusion A of any proof, exploiting positive and negative occurrences of subformulae. This treatment includes first-order proofs. Our main result on relevance is (...) that for every proof of A from X in Classical Core Logic, we have R(X,A). R(X,A) is a best possible explication of the sought notion of relevance. Our result is optimal, and challenges relevantists in the Anderson–Belnap tradition to identify any strengthening of the relation R(X,A) that can be shown to hold for some subsystem of Anderson–Belnap _R_ but that can be shown to fail for Classical Core Logic. (shrink)

This paper concerns the interface between relevant logics and recent developments in semantics for hyperintensional operators. In the latter area, the notion of topicality or aboutness has begun to play an explicit modeling role, with classes of models being studied which incorporate representations of topics, and of operations on topics. The idea is to use these topics in the evaluations of formulas including hyperintensional operators in order to explain why classically equivalent formulas may not be intersubstitutable in contexts using those (...) operators—though classically equivalent, if a pair of formulas can be interpreted to concern totally unrelated topics, then we should not be able to substitute one for the other in contexts which are sensitive to topics. Relevant logics, and the central relevance property of Variable Sharing, have been read as concerning something like topics in the past, and recently this angle on relevance has seen renewed interest. In this paper, I’ll develop and motivate a method for representing topics in the matrix models of logics, and apply this method to _R_ and its matrix models (given by De Morgan monoids) to argue that there is a clear semantic perspective on topicality according to which _R_ is topic-sensitive as a logic. (shrink)

The relationship between relevant logic and mathematical practice has often been twofold: while some relevant logicians have focused on the descriptive project of capturing (in some sense) informal mathematical reasoning better than classical logic does, others have engaged in the normative project of suggesting how mathematics should be done as opposed to how it is done now. Both projects have received heavy criticism. In this paper, I argue that the intuitive idea of variable sharing does point to a (...) real aspect of mainstream mathematical practices, and this fact can be used to make sense of both relevantist projects: on one hand, relevant logics may be used to provide a more accurate model of this particular aspect of current practices, while on the other, they can suggest new practices which deviate from the standard precisely in their approach to variable sharing. (shrink)