In this paper we extend some techniques from the theory of conse- quences operations and logical matrices into the wider area of entailment relations.

Entailment relations, introduced by Scott in the early 1970s, provide an abstract generalisation of Gentzen’s multi-conclusion logical inference. Originally applied to the study of multi-valued logics, this notion has then found plenty of applications, ranging from computer science to abstract algebra. In particular, an entailment relation can be regarded as a constructive presentation of a distributive lattice and in this guise it has proven to be a useful tool for the constructive reformulation of several classical theorems in (...) commutative algebra. In this paper, motivated by these concrete applications, we state and prove a cut-elimination result for inductively generated entailment relations. We analyse some of its consequences and describe the existing connections with analogous results in the literature. (shrink)

The classesMatr( ) of all matrices (models) for structural finitistic entailments are investigated. The purpose of the paper is to prove three theorems: Theorem I.7, being the counterpart of the main theorem from Czelakowski [3], and Theorems II.2 and III.2 being the entailment counterparts of Bloom's results [1]. Theorem I.7 states that if a classK of matrices is adequate for , thenMatr( ) is the least class of matrices containingK and closed under the formation of ultraproducts, submatrices, strict homomorphisms (...) and strict homomorphic pre-images. Theorem II.2 in Section II gives sufficient and necessary conditions for a structural entailment to be finitistic. Section III contains theorems which characterize finitely based entailments. (shrink)

We extend classical work by Janusz Czelakowski on the closure properties of the class of matrix models of entailment relations—nowadays more commonly called multiple-conclusion logics—to the setting of non-deterministic matrices (Nmatrices), characterizing the Nmatrix models of an arbitrary logic through a generalization of the standard class operators to the non-deterministic setting. We highlight the main differences that appear in this more general setting, in particular: the possibility to obtain Nmatrix quotients using any compatible equivalence relation (not necessarily a (...) congruence); the problem of determining when strict homomorphisms preserve the logic of a given Nmatrix; the fact that the operations of taking images and preimages cannot be swapped, which determines the exact sequence of operators that generates, from any complete semantics, the class of all Nmatrix models of a logic. Many results, on the other hand, generalize smoothly to the non-deterministic setting: we show for instance that a logic is finitely based if and only if both the class of its Nmatrix models and its complement are closed under ultraproducts. We conclude by mentioning possible developments in adapting the Abstract Algebraic Logic approach to logics induced by Nmatrices and the associated equational reasoning over non-deterministic algebras. (shrink)

Some argue, following Bertrand Russell, that because general truths are not entailed by particular truths, general facts must be posited to exist in addition to particular facts. I argue on the contrary that because general truths (globally) supervene on particular truths, general facts are not needed in addition to particular facts; indeed, if one accepts the Humean denial of necessary connections between distinct existents, one can further conclude that there are no general facts. When entailment and supervenience do not (...) coincide it is only failure of supervenience, not failure of entailment, that carries ontological import. (shrink)

I explicate the crucial role played by efficient cause in Robert Rosen’s characterization of life, by elaborating on the topic of Aristotelian causality, and exploring the many alternate descriptions of causal and inferential entailments. In particular, I discuss the concepts of functional entailment and immanent causation, and examine how they fit into Robert Rosen’s relational-biology universe of living, anticipatory, and complex systems.

I explicate how various relational interactions between (M,R)-systems may have realizations in pathophysiology, and how the possible reversals of the effects of these interactions then become therapeutic models. Functional entailment receives a rigorous category-theoretic treatment, and plays a crucial role in this continuing saga of relational biology.

This paper presents a formal account of how to determine the discourse relations between propositions introduced in a text, and the relations between the events they describe. The distinct natural interpretations of texts with similar syntax are explained in terms of defeasible rules. These characterise the effects of causal knowledge and knowledge of language use on interpretation. Patterns of defeasible entailment that are supported by the logic in which the theory is expressed are shown to underly temporal interpretation.

Graham Priest has asked whether the consequence relation associated with the Anderson–Belnap system of Tautological Entailment,1 in the language with connectives ¬, ∧, ∨, and countably many propositional variables as tomic formulas, maximal amongst the substitution-invariant relevant consequence relations on this language. Here a consequence relation is said to be relevant just in case whenever for a set of formulas Γ and formula B, we have Γ B only if some propositional variable occurring in B occurs in (...) at least one formula in Γ. (It follows that relevant consequence relations are atheorematic in the sense that whenever Γ B for some such consequence relation , Γ = ∅.) Here I write up in more detail the upshot of the conversation – returning an affirmative answer to Priest’s question – about this in the common room that Greg Restall and I were participating in last Friday [ = October 6, 2006], dotting some “i”s and crossing some “t”s (and adding the odd further reflection). (shrink)

Saul Kripke’s work on the semantics of modal logics is well known, unlike his work on Anderson and Belnap’s system E of Entailment (a modal relevance logic), which included his proof of the decidability of its implicational fragment E_>, and also a counterexample to the conjecture of Belnap that E_> is the intersection of the implicational fragments of the relevance logic R and the modal logic S4. This led to Storrs McCall’s suggesting that the “mingle” axiom might be added (...) to E_>, and that this system might be used instead of S4_>. We give a counterexample to this conjecture. We also disprove the conjecture Anderson and Belnap attribute to McCall, in which “restricted mingle” is added to E_>. We use a modal extension of the lattice M_0 and follow ideas of Meyer. In the 1970s, Dunn showed that RM (R with the mingle axiom) can be modeled using a binary accessibility relation, like the accessibility relation in Kripke’s semantics for normal modal logics and his semantics for intuitionistic logic. Dunn also gave a variant interpretation for EM (E with the mingle axiom). We explore some other variations of these binary relational semantics. (shrink)

We investigate an approach for drawing logical inference from inconsistent premisses. The main idea in this approach is that the inconsistencies in the premisses should be interpreted as uncertainty of the information. We propose a mechanism, based on Kinght’s [14] study of inconsistency, for revising an inconsistent set of premisses to a minimally uncertain, probabilistically consistent one. We will then generalise the probabilistic entailment relation introduced in [15] for propositional languages to the first order case to draw logical (...) inference from a probabilistic set of premisses. We will show how this combination can allow us to limit the effect of uncertainty introduced by inconsistent premisses to only the reasoning on the part of the premise set that is relevant to the inconsistency. (shrink)

I argue that there are Leibnizian-style cosmological arguments for the existence of God which start from very mild premises which affirm the mere possibility of a principle of sufficient reason. The utilization of such premises gives a great deal of plausibility to such types of argumentation. I spend the majority of the paper defending three major objections to such mild premises viz., a reductio argument from Peter van Inwagen and William Rowe, which proffers and defends the idea that a necessary (...) proposition cannot explain a contingent one. I, then, turn to an amelioration of the Rowe/van Inwagen argument which attempts to appeal to an entailment relation between explanans and explanandum that is fettered out in terms of relevance logic. Subsequent to dispelling with that worry, I tackle objections to the utilization of weak principles of sufficient reason that depend essentially upon agglomerative accounts of explanation. (shrink)

This paper presents a neighborhood semantics for logics of entailment. It begins with a minimal system Min that expresses the most fundamental assumptions about the entailment relation, and continues by examining various extensions that reflect further assumptions that might be made about entailment. This leads first to the logic B that is the basic relevant logic, and then to more powerful systems. All of these logics are proved to be sound and strongly complete. With B the (...) neighborhood semantics meets the Routley-Meyer relational semantics for relevant logic; these connections are examined. The minimal and basic entailment logics are shown to have the finite model property, and hence to be decidable. (shrink)

Do children understand how different numbers are related before they associate them with specific cardinalities? We explored how children rely on two abstract relations – contrast and entailment – to reason about the meanings of ‘unknown’ number words. Previous studies argue that, because children give variable amounts when asked to give an unknown number, all unknown numbers begin with an existential meaning akin to some. In Experiment 1, we tested an alternative hypothesis, that because numbers belong to a scale (...) of contrasting alternatives, children assign them a meaning distinct from some. In the “Don’t Give-a-Number task”, children were shown three kinds of fruit (apples, bananas, strawberries), and asked to not give either some or a number of one kind (e.g. Give everything, but not [some/five] bananas). While children tended to give zero bananas when asked to not give some, they gave positive amounts when asked to not give numbers. This suggests that contrast – plus knowledge of a number’s membership in a count list – enables children to differentiate the meanings of unknown number words from the meaning of some. Experiment 2 tested whether children’s interpretation of unknown numbers is further constrained by understanding numerical entailment relations – that if someone, e.g. has three, they thereby also have two, but if they do not have three, they also do not have four. On critical trials, children saw two characters with different quantities of fish, two apart (e.g. 2 vs. 4), and were asked about the number inbetween – who either has or doesn’t have, e.g. three. Children picked the larger quantity for the affirmative, and the smaller for the negative prompts even when all the numbers were unknown, suggesting that they understood that, whatever three means, a larger quantity is more likely to contain that many, and a smaller quantity is more likely not to. We conclude by discussing how contrast and entailment could help children scaffold the exact meanings of unknown number words. (shrink)

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)

Proponents of the subset account of property realization commonly make the assumption that the summing of properties entails the summing of their forward-looking causal features. This paper seeks to establish that this assumption is false. Moreover, it aims to demonstrate that without this assumption the fact that the subset account captures an entailment relation—which it must if it is to be of any use to non-reductive physicalism—becomes questionable.

An assertion of high conditional probability or, more briefly, an HCP assertion is a statement of the type: The conditional probability of B given A is close to one. The goal of this paper is to construct logics of HCP assertions whose conclusions are highly likely to be correct rather than certain to be correct. Such logics would allow useful conclusions to be drawn when the premises are not strong enough to allow conclusions to be reached with certainty. This goal (...) is achieved by taking Adams' (1966) logic, changing its intended application from conditionals to HCP assertions, and then weakening its criterion for entailment. According to the weakened entailment criterion, called the Criterion of Near Surety and which may be loosely interpreted as a Bayesian criterion, a conclusion is entailed if and only if nearly every model of the premises is a model of the conclusion. The resulting logic, called NSL, is nonmonotonic. Entailment in this logic, although not as strict as entailment in Adams' logic, is more strict than entailment in the propositional logic of material conditionals. Next, NSL was modified by requiring that each HCP assertion be scaled; this means that to each HCP assertion was associated a bound on the deviation from 1 of the conditional probability that is the subject of the assertion. Scaling of HCP assertions is useful for breaking entailment deadlocks. For example, it it is known that the conditional probabilities of C given A and of - C given B are both close to one but the bound on the former's deviation from 1 is much smaller than the latter's, then it may be concluded that in all likelihood the conditional probability of C given A ∧ B is close to one. The resulting logic, called NSL-S, is also nonmonotonic. Despite great differences in their definitions of entailment, entailment in NSL is equivalent to Lehmann and Magidor's rational closure and, disregarding minor differences concerning which premise sets are considered consistent, entailment in NSL-S is equivalent to entailment in Goldszmidt and Pearl's System-Z⁺. Bacchus, Grove, Halpern, and Koller proposed two methods of developing a predicate calculus based on the Criterion of Near Surety. In their randomstructures method, which assumed a prior distribution similar to that of NSL, it appears possible to define an entailment relation equivalent to that of NSL. In their random-worlds method, which assumed a prior distribution dramatically different from that of NSL, it is known that the entailment relation is different from that of NSL. (shrink)

The paper is concerned with the rational requirements for suspended judgment, or what suspending judgment about a question rationally commits one to. It shows that two purported rational requirements for suspended judgment cannot both be true at the same time, at least when the entailment relation between questions is understood a certain way. The first one says that one is rationally required to suspend judgment about those questions that are entailed by the questions that one already suspends judgment (...) about. The second one says that one is rationally required to believe the presuppositions of the questions one suspends judgment about. Two plausible solutions to the problem are discussed. One of them explicates the relation of entailment between questions in an alternative manner, and the other one rejects the presupposition requirement. Either way, a closure requirement for suspended judgment can be maintained which is analogous to the closure requirement for belief. Whereas belief is ideally closed under entailment between propositions, suspended judgment is ideally closed under entailment between questions. (shrink)

In Categories 12 (14b11–22), Aristotle famously claims that [1] true sentences and reality stand in a mutually implicative relationship, and that [2] reality causes the truth of sentences but not vice versa. In this paper, I first argue that Boethius’ reading of the above passage led medieval logicians to assess [1] and [2] within the framework of a theory of consequence. Then, I consider two important questions raised by Boethius and later logicians in relation to [1] and [2], and, (...) namely, whether the implication described in [1] really satisfies the requirement for consequence, and whether the truthmaking relation posited in [2] can be boiled down to the relation of consequence. Boethius himself, I argue, answered affirmatively to both questions and thus (much like some contemporary truthmaker theorists) committed himself to an account of truthmaking as entailment. I then consider criticisms to Boethius’ account found in the writings of William of Champeaux, two anonymous authors connected to the school of Jocelyn of Soissons, Peter Abelard, and the author of the Ars Meliduna. Here, I argue that the emergence of a concern for relevance drives the need for abandoning Boethius’ account as well as his conception of truthmaking as entailment. (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.

I argue that Universalism (the thesis that mereological composition is unrestricted) entails Extensionalism (the thesis that sameness of composition is sufficient for identity) as long as the parthood relation is transitive and satisfies the Weak Supplementation principle (to the effect that whenever a thing has a proper part, it has another part disjoint from the first).

Following Moore, I use ‘P entails Q’ as a convenient shorthand for ‘Q can be deduced logically from P’, ‘From P, Q follows logically’, ‘There is a logically valid argument with P as sole premise and Q as conclusion’, and the like.1 Apart from a minor point to be raised in Section XVI, distinctions within this cluster do not matter for present purposes. An analysis of the concept of entailment is answerable to careful, educated uses of expressions such as (...) those. An analysis which condemned nearly everything we say about what follows from what simply would not be an analysis of the common concept of entailment. If the concept were inconsistent, some common uses of it would be condemned; but only by standards established by the others. C. I. Lewis maintained this: to say that P entails Q is to say that it is logically impossible that (P & ¬Q).2 If Quine is right, then ‘entails’ and ‘impossible’ are as suspect as all other intensional terms. So perhaps they are; but their uses are not wholly without structure, and there are wrong ways of interrelating them. Lewis’s contention is about the internal geography of the intensional area, not its relations to the surrounding conceptual territory: it is an attempted analysis of one intensional expression in terms of another. I shall argue that Lewis was right, and also - by implication - that his thesis is helpful and clarifying - that is, that it is a genuine analysis. As is well known, Lewis’s analysis implies that each impossible proposition entails every proposition. Accepting the analysis, I accept this result. For one thing, Lewis has an argument for it (I use ‘→’ to abbreviate ‘entails’): (1) P & ¬P.. (shrink)

According to the principle of conditional power aggregation (CPA), conditional powers conjoin when the properties that bestow them conjoin. Sophie Gibb has argued that CPA is false given Shoemaker’s account of conditional powers and that this leads to a problem for his account of subset realization. In short: If CPA is rejected, subset realization fails to be an entailment relation, in which case it cannot provide a basis for non-reductive physicalism. I defend the subset account against this argument (...) by denying that CPA fails. I argue that (i) Shoemaker’s account of conditional powers does not warrant a rejection of CPA, (ii) his account is incomplete and should be supplemented with a further sufficient condition for when a property bestows a conditional power, and (iii) this further sufficient condition supports CPA. (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)

According to moral error theory, moral discourse is error-ridden. Establishing error theory requires establishing two claims. These are that moral discourse carries a non-negotiable commitment to there being a moral reality and that there is no such reality. This paper concerns the first and so-called non-negotiable commitment claim. It starts by identifying the two existing argumentative strategies for settling that claim. The standard strategy is to argue for a relation of conceptual entailment between the moral statements that comprise (...) moral discourse and the statement that there is a moral reality. The non-standard strategy is to argue for a presupposition relation instead. Error theorists have so far failed to consider a third strategy, which uses a general entailment relation that doesn’t require intricate relations between concepts. The paper argues that both entailment claims struggle to meet a new explanatory challenge and that since the presupposition option doesn’t we have prima facie reason to prefer it over the entailment options. The paper then argues that suitably amending the entailment claims enables them to meet this challenge. With all three options back on the table the paper closes by arguing that error theorists should consider developing the currently unrecognised, non-conceptual entailment claim. (shrink)

We introduce and explore the notion of duality for entailment relations induced by preference orderings on states. We discuss the relationship between these preferential entailment relations from the perspectives of Boolean algebra, inference rules, and modal axiomatisation. Interpreting the preference relations as accessibility relations establishes modular Gödel-Löb logic as a suitable modal framework for rational preferential reasoning.

We present a relational proof system in the style of dual tableaux for a multimodal propositional logic for order of magnitude qualitative reasoning to deal with relations of negligibility, non-closeness, and distance. This logic enables us to introduce the operation of qualitative sum for some classes of numbers. A relational formalization of the modal logic in question is introduced in this paper, i.e., we show how to construct a relational logic associated with the logic for order-of-magnitude reasoning and its dual (...) tableau system which is a validity checker for the modal logic. For that purpose, we define a validity preserving translation of the modal language into relational language. Then we prove that the system is sound and complete with respect to the relational logic defined as well as with respect to the logic for order of magnitude reasoning. Finally, we show that in fact relational dual tableau does more. It can be used for performing the four major reasoning tasks: verification of validity, proving entailment of a formula from a finite set of formulas, model checking, and verification of satisfaction of a formula in a finite model by a given object. (shrink)

This paper has two goals. First, we develop frameworks for logical systems which are able to reflect not only non-monotonic patterns of reasoning, but also paraconsistent reasoning. Our second goal is to have a better understanding of the conditions that a useful relation for nonmonotonic reasoning should satisfy. For this we consider a sequence of generalizations of the pioneering works of Gabbay, Kraus, Lehmann, Magidor and Makinson. These generalizations allow the use of monotonic nonclassical logics as the underlying logic (...) upon which nonmonotonic reasoning may be based. Our sequence of frameworks culminates in what we call plausible, nonmonotonic, multiple-conclusion consequences relations . Our study yields intuitive justifications for conditions that have been proposed in previous frameworks and also clarifies the connections among some of these systems. In addition, we present a general method for constructing plausible nonmonotonic relations. This method is based on a multiple-valued semantics, and on Shoham's idea of preferential models. (shrink)

Several philosophers have recently claimed that if a proposition is cancellable from an uttered sentence then that proposition is not entailed by that uttered sentence. The claim should be a familiar one. It has become a standard device in the philosopher's tool-kit. I argue that this claim is false. There is a kind of entailment—which I call “modal entailment”—that is context-sensitive and, because of this, cancellable. So cancellability does not show that a proposition is not entailed by an (...) uttered sentence. I close the paper by describing an implication this has for a disagreement between J. L. Austin and Grice concerning the relation between felicity and truth. (shrink)

In Categories 12 (14b11–22), Aristotle famously claims that [1] true sentences and reality stand in a mutually implicative relationship, and that [2] reality causes the truth of sentences but not vice versa. In this paper, I first argue that Boethius’ reading of the above passage led medieval logicians to assess [1] and [2] within the framework of a theory of consequence. Then, I consider two important questions raised by Boethius and later logicians in relation to [1] and [2], and, (...) namely, whether the implication described in [1] really satisfies the requirement for consequence, and whether the truthmaking relation posited in [2] can be boiled down to the relation of consequence. Boethius himself, I argue, answered affirmatively to both questions and thus (much like some contemporary truthmaker theorists) committed himself to an account of truthmaking as entailment. I then consider criticisms to Boethius’ account found in the writings of William of Champeaux, two anonymous authors connected to the school of Jocelyn of Soissons, Peter Abelard, and the author of the Ars Meliduna. Here, I argue that the emergence of a concern for relevance drives the need for abandoning Boethius’ account as well as his conception of truthmaking as entailment. (shrink)

1. Anderson and Belnap's most explicit characterization of the fallacy of modality is as follows: "Modal fallacies arise when it is claimed that entailments follow from, or are entailed by, contingent propositions." The view which Nelson attributes to Anderson and Belnap, on the other hand, is "that necessary propositions are entailed only by necessary ones, never by contingent ones." Anderson and Belnap speak of "entailments," whereas Nelson generalizes to "necessary propostitions." The move is far from innocent, as we shall see. (...) For though Anderson and Belnap hold that every true entailment is a necessary proposition, not every necessary proposition is an entailment. In order to grasp the implications of this distinction more clearly, let us see how Anderson and Belnap's statement of the modal fallacy might legitimately be restated in terms of necessity. To that end we must draw a related distinction between necessary propositions and necessity propositions. A expresses a necessary proposition iff A → A → A holds. A expresses a necessity proposition iff A is of the form B → C. The particular form of these definitions is dependent on the fact that entailment rather than necessity is primitive in Anderson and Belnap's system; the forms in parentheses are the equivalents in terms of necessity. Now it follows from these definitions that every entailment is a necessity proposition and that every necessity proposition is an entailment. The modal fallacy may thus be equivalently characterized as the claim that a necessity proposition is entailed by a contingent proposition. That this is different from Nelson's formulation is clear from the following considerations. First, not every necessary proposition is a necessity proposition; example: "Either it is raining or it is not raining." Second, not every necessity proposition is a necessary proposition; example: any false necessity proposition. Thus the possibility is left open that some necessary proposition might be entailed by a contingent one. This is exactly the case with Nelson's example 2, "That every polychromatic surface is red entails that every polychromatic surface is colored." This might well be a theorem in a suitable extension of E. Such cases do not, however, arise among the theorems of EI. In that limited system, all that a contingent proposition can entail is itself. (shrink)

If semantic consequence is analyzed with extensive games, logical reasoning can be accounted for by looking at how players solve entailment games. However, earlier approaches to game semantics cannot achieve this reduction, by want of explicitly dened preferences for players. Moreover, although entailment games can naturally translate the idea of argumentation about a common ground, a cognitive interpretation is undermined by the complexity of strategic reasoning. We thus describe a class of semantic extensive entailment game with algorithmic (...) players, who have preferences for parsimonious spending of computational resources and thus compute partial strategies under qualitative uncertainty about future histories. We prove the existence of local preferences for moves and of strategic fixpoints that allow to map game-trees to tableaux proofs, and exhibit a strategy prole that solves the fixpoint selection problem, and can be mapped to systematic constructions of semantic trees, yielding a completeness result by translation. We then discuss the correspondence between proof heuristics and strategies in our games, the relations of our games to gts, and possible extensions to other entailment relations. We conclude that the main interest of our result lies in the possibility to bridge argumentative and cognitive models of logical reasoning, rather than in new meta-theoretic results. All proofs are given in appendix. (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)

What is the relation between entailment and reasons for belief? In this paper, I discuss several answers to this question, and I argue that these answers all face problems. I then propose the following answer: for all propositions p1,...,pn and q, if the conjunction of p1,..., and pn entails q, then there is a reason against a person's both believing that p1,..., and that pn and believing the negation of q. I argue that this answer avoids the problems (...) that the other answers to this question face, and that it does not face any other problems either. I end by showing what the relation between deductive logic, reasons for belief and reasoning is if this answer is correct. (shrink)

In the literature on mereology it is often accepted that mereological universalism entails extensionalism. More precisely, many accept that, if parthood is assumed to be a partial order, the thesis that every plurality of entities has a mereological fusion entails the thesis that different composite entities have different proper parts. Central to this idea is the principle known as ‘Weak Supplementation’ which many take to impose an important constraint on the relation of proper parthood. In this paper I argue (...) that this claim is false as the principle that I will call ‘Minimal Supplementation’ appears to be capable of doing all the work done by Weak Supplementation but without entailing extensionalism if conjoined with universalism and core mereology. (shrink)

We study the EPR-type correlations from the perspective of the relational interpretation of quantum mechanics. We argue that these correlations do not entail any form of “non-locality”, when viewed in the context of this interpretation. The abandonment of strict Einstein realism implied by the relational stance permits to reconcile quantum mechanics, completeness, (operationally defined) separability, and locality.

Care is at the heart of the Maori values system, which calls for humans to be kaitiaki, caretakers of the maun y the life-force, in each other and in nature. The relational Five Well-beings approach, based on four case studies of Maori businesses, demonstrates how business can create spiritual, cultural, social, environmental and economic well-being. A Well-beings approach entails praxis, which brings values and practice together with the purpose of consciously creating well-being and, in so doing, creates multi-dimensional wealth. Underlying (...) the Well-beings approach is an ethic of care and an intrinsic stakeholder view of business. (shrink)

The Trek Separation Theorem states necessary and sufficient conditions for a linear directed acyclic graphical model to entail for all possible values of its linear coefficients that the rank of various sub-matrices of the covariance matrix is less than or equal to n, for any given n. In this paper, I extend the Trek Separation Theorem in two ways: I prove that the same necessary and sufficient conditions apply even when the generating model is partially non-linear and contains some cycles. (...) This justifies application of constraint-based causal search algorithms to data generated by a wider class of causal models that may contain non-linear and cyclic relations among the latent variables. (shrink)

Interval temporal logics provide both an insight into a nature of time and a framework for temporal reasoning in various areas of computer science. In this paper we present sound and complete relational proof systems in the style of dual tableaux for relational logics associated with modal logics of temporal intervals and we prove that the systems enable us to verify validity and entailment of these temporal logics. We show how to incorporate in the systems various relations between intervals (...) and/or various time orderings. (shrink)

The early thirteenth century tract Ars Meliduna deals with the issue whether categorical propositions entail, or are entailed by, ‘continuing’ propositions, i.e. by implications. From the perspective of modern logic, with implication interpreted as a material, truth-functional connective, the first question has to be answered in the affirmative because, e.g. β entails (α ⊃ β). But conversely (α ⊃ β) ‘normally’ doesn’t entail the truth (or the falsity) of any of the components α, β; hence the second question should be (...) answered in the negative. These results cannot, however, be directly transferred to medieval logic because implication was there usually interpreted as a strict implication. Yet, within the framework of modal logic, one obtains parallel results. In particular, whenever β is necessary, then (α → β) must be true. Furthermore, if (α → β) is itself necessary, then it is entailed by α. Thus, in particular, ‘Socrates is a man’ entails the analytically true implication ‘If Socrates is a man, Socrates is an animal’, and the same premiss also entails the tautological consequent ‘If Socrates is white, Socrates is white’. (shrink)

I ate my broccoli, though my broccoli did not eat me. The eating relation, like many other relations, differentiates between its arguments. The fact that eating holds between a and b does not entail that it holds between b and a. How are we to make sense of this? The standard view is that relations are sensitive to the order of their arguments. As natural as this view is, it has been the target of a powerful objection from Kit (...) Fine. I examine Fine’s objection and defend the standard view. (shrink)

In the Tractatus Wittgenstein criticizes Frege and Russell's view that laws of inference (Schlussgesetze) "justify" logical inferences. What lies behind this criticism, I argue, is an attack on Frege and Russell's conceptions of logical entailment. In passing, I examine Russell's dispute with Bradley on the question whether all relations are "internal".

Deflationary positions have been defended in many areas of philosophy. Most prominent are semantic deflationism about truth and reference, and meta-ontological deflationism, according to which existence has no deep nature and the standard neo-Quinean approach to ontology is misguided. Although both kinds of views have generated much discussion, surprisingly little attention has been paid to the question of how they relate to each other. Are they independent, is it advisable to hold them all at once, or do they even entail (...) each other? One exception is Amie Thomasson, who has argued that semantic deflationism actually entails meta-ontological deflationism. This is unexpected, since semantic deflationism is usually regarded as much less controversial than meta-ontological deflationism. In our paper, we will argue that Thomasson’s argument fails though, and that the connection between the views is in fact weaker than she makes them out to be. (shrink)

An exact truthmaker for A is a state which, as well as guaranteeing A’s truth, is wholly relevant to it. States with parts irrelevant to whether A is true do not count as exact truthmakers for A. Giving semantics in this way produces a very unusual consequence relation, on which conjunctions do not entail their conjuncts. This feature makes the resulting logic highly unusual. In this paper, we set out formal semantics for exact truthmaking and characterise the resulting notion (...) of entailment, showing that it is compact and decidable. We then investigate the effect of various restrictions on the semantics. We also formulate a sequent-style proof system for exact entailment and give soundness and completeness results. (shrink)

The early thirteenth century tract Ars Meliduna deals with the issue whether categorical propositions entail, or are entailed by, ‘continuing’ propositions, i.e. by implications. From the perspective of modern logic, with implication interpreted as a material, truth-functional connective, the first question has to be answered in the affirmative because, e.g. β entails (α ⊃ β). But conversely (α ⊃ β) ‘normally’ doesn’t entail the truth (or the falsity) of any of the components α, β; hence the second question should be (...) answered in the negative. These results cannot, however, be directly transferred to medieval logic because implication was there usually interpreted as a strict implication. Yet, within the framework of modal logic, one obtains parallel results. In particular, whenever β is necessary, then (α → β) must be true. Furthermore, if (α → β) is itself necessary, then it is entailed by α. Thus, in particular, ‘Socrates is a man’ entails the analytically true implication ‘If Socrates is a man, Socrates is an animal’, and the same premiss also entails the tautological consequent ‘If Socrates is white, Socrates is white’. (shrink)