Gonzalo Rodriguez-Pereyra (2006) argues against attempts to preserve the entailment principle (or a restriction of it) while avoiding the explosion of truthmakers for necessities and truthmaker triviality. In doing so, he both defends the disjunction thesis--if something makes true a disjunctive truth, then it makes true one of its disjuncts--, and rejects the conjunction thesis--if something makes tue a conjunctive truth, then it makes true each of its conjuncts. In my discussion, I provide plausible counterexamples to the disjunction (...) thesis, and contend that Rodriguez-Pereyra's general defence of it fails. Then I defend the conjunction thesis from Rodriguez-Pereyra's case against it. (shrink)

In this paper, I offer reasons to conclude that either belief impermissivism or credal impermissivism is false. That is to say, I argue against the conjunction of belief impermissivism and credal impermissivism. I defend this conclusion in three ways. First, I show what I take to be an implausible consequence of holding that for any rational credence in p, there is only one correlating rational belief-attitude toward p, given a body of evidence. Second, I provide thought experiments designed to support (...) the intuition that there are at least a few credences in some cases for which more than one belief-attitude is rationally permissible. Third, I provide one possible theoretical grounding for my position by appeal to Jamesian values. (shrink)

By a well-known result of Kotlarski et al., first-order Peano arithmetic \ can be conservatively extended to the theory \ of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quantifiers. This result motivates the general question of determining natural axioms concerning the truth predicate that can be added to \ while maintaining conservativity over \. Our main result shows that conservativity fails even (...) for the extension of \ obtained by the seemingly weak axiom of disjunctive correctness \ that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, \ implies \\). Our main result states that the theory \ coincides with the theory \ obtained by adding \-induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski and Cieśliński. For our proof we develop a new general form of Visser’s theorem on non-existence of infinite descending chains of truth definitions and prove it by reduction to Gödel’s second incompleteness theorem, rather than by using the Visser–Yablo paradox, as in Visser’s original proof. (shrink)

The relation between ex falso and disjunctive syllogism, or even the justification of ex falso based on disjunctive syllogism, is an old topic in the history of logic. This old topic reappears in contemporary logic since the introduction of minimal logic by Johansson. The disjunctive syllogism seems to be part of our general non-problematic inferential practices and superficially it does not seem to be related to or to depend on our acceptance of the frequently disputable ex falso rule. We know (...) that the acceptance of the ex falso is a sufficient condition for the acceptance of the disjunctive syllogism, but the interesting question is: is the ex falso a necessary condition for the acceptance of the disjunctive syllogism? The aim of the present paper is to discuss some possible ways to define systems that combines the preservation of the disjunctive syllogism with the rejection of the ex falso. (shrink)

In Minker and Rajasekar (J Log Program 9(1):45–74, 1990), Minker proposed a semantics for negation-free disjunctive logic programs that offers a natural generalisation of the fixed point semantics for definite logic programs. We show that this semantics can be further generalised for disjunctive logic programs with classical negation, in a constructive modal-theoretic framework where rules are built from _claims_ and _hypotheses_, namely, formulas of the form \(\Box \varphi \) and \(\Diamond \Box \varphi \) where \(\varphi \) is a literal, respectively, (...) yielding a “base semantics” for general disjunctive logic programs. Model-theoretically, this base semantics is expressed in terms of a classical notion of logical consequence. It has a complete proof procedure based on a general form of the cut rule. Usually, alternative semantics of logic programs amount to a particular interpretation of nonclassical negation as “failure to derive.” The counterpart in our framework is to complement the original program with a set of hypotheses required to satisfy specific conditions, and apply the base semantics to the resulting set. We demonstrate the approach for the answer set semantics. The proposed framework is purely classical in mainly three ways. First, it uses classical negation as unique form of negation. Second, it advocates the computation of logical consequences rather than of particular models. Third, it makes no reference to a notion of preferred or minimal interpretation. (shrink)

John McDowell's conception of perceptual knowledge commits him to the claim that if I perceive that P then I am in a position to know that I perceive that P. In the first part of this essay, I present some reasons to be suspicious of this claim - reasons which derive from a general argument against 'luminosity' - and suggest that McDowell can reject this claim, while holding on to almost all of the rest of his conception of perceptual knowledge, (...) by supplementing his existing disjunctive conception of experience with a new disjunctive conception of perceiving. In the second part of the essay, I present some reasons for thinking that one's justification, in cases of perceptual knowledge, consists not in the fact that one perceives that P but in the fact that one perceives such-and-such. I end by suggesting that the disjunctive conception of perceiving should be understood as a disjunctive conception of perceiving such-and-such. (shrink)

A partial solution to Ono’s problem P54 is given. Here Ono’s problem P54 is whether Harrop disjunction property is equivalent to disjunction property or not in intermediate predicate logics. As an application of this result it is shown that some intermediate predicate logics satisfy Harrop disjunction property.

This paper investigates propositional hyperintensionality in counterfactuals. It starts with a scenario describing two children playing on a seesaw and studies the truth-value predictions for counterfactuals by four different semantic theories. The theories in question are Kit Fine’s truthmaker semantics, Luis Alonso-Ovalle’s alternative semantics, inquisitive semantics and Paolo Santorio’s syntactic truthmaker semantics. These predictions suggest that the theories that distinguish more of a given set of intensionally equivalent sentences (Fine and Alonso-Ovalle’s) fare better than those that do not (inquisitive semantics (...) and Santorio’s). Then we investigate how inquisitive semantics and Santorio can respond to these results. They can respond to them by helping themselves to considerations from Hurford disjunctions, disjunctions whose disjuncts stand in an entailment relation to one another. I argue that considerations from Hurford disjunctions are ad hoc modifications merely to predict the expected results. I conclude that the scenarios suggest a need for more fine-grained theories of sentential meaning in general. (shrink)

This paper offers an explanation of the fact that sentences of the form (1) ‘X may A or B’ may be construed as implying (2) ‘X may A and X may B’, especially if they are used to grant permission. It is suggested that the effect arises because disjunctions are conjunctive lists of epistemic possibilities. Consequently, if the modal may is itself epistemic, (1) comes out as equivalent to (2), due to general laws of epistemic logic. On the other hand, (...) on a deontic reading of may, (2) is only implied under exceptional circumstances – which usually obtain when (1) is used performatively. (shrink)

The validity of argument by disjunctive syllogism has been denied by proponents of relevant and paraconsistent logic. DS is stigmatised for its role in inferences — most notably C.I. Lewis's derivation of that fallacy of irrelevance ex falso quodlibet — that involve both it and other rules of inference governing disjunction, or, to speak more precisely, other rules of inference taken to apply to the very same disjunction that obeys DS. In avoiding these inferences the road less travelled (...) is to deny the identity rather than to deny DS: what follows is, then, an exercise in disjoining disjunctions. (shrink)

In this dissertation, I argue that all extant theories of prudential value are either a) enumeratively deficient, in that they are unable to accommodate everything that, intuitively, is a basic constituent of prudential value, b) explanatorily deficient, in that they are at least sometimes unable to offer a plausible story about what makes a given thing prudentially valuable, or c) both. In response to the unsatisfactory state of the literature, I present my own account, the Disjunctive Hybrid Theory or DHT. (...) DHT answers to and remedies each of the above inadequacies in a way that no other approach can. This account has the following general structure:Disjunctive Hybrid Theory (DHT): Thing x is basically good for person P if and only if x is either a) cared about (sufficiently and in the right way) by P, b) a bearer of (the right kind of) attitude-independent value, or c) both.Although it follows other recent accounts in combining elements from objective and subjective theories, DHT is a hybrid theory of a quite new kind. This is because it denies both subjective necessity (the constraint that, if thing x is to be basically good for person P, P must have some pro-attitude toward x) and objective necessity (the constraint that, if thing x is to be basically good for person P, x must have some attitude-independent value). I argue that the rejection of both necessity claims is called for if we are to move beyond the enumerative and explanatory limitations of existing accounts.I begin by outlining the general structure of DHT. I then argue, against various recent authors, that desire-satisfactionism remains the most appealing subjectivist approach to prudential value, in that it is best able to capture the central subjectivist insight. This insight is that a person can confer prudential value upon things by caring about them (sufficiently and in the right way). The subjectivist strand of DHT will thus be a version of desire-satisfactionism, which must be interpreted in line with what I call the object, as opposed to the combo, view. I move on to further motivate and develop the second, objectivist strand of DHT. This part of the theory involves a commitment to robustly attitude-independent prudential goods. I close by addressing some puzzles for the theory, and considering some of its more specific applications. (shrink)

This paper is a contribution to the general study of consequence relations which contain (definable) connective of “disjunction”. Our work is centered around the “proof by cases property”, we present several of its equivalent definitions, and show some interesting applications, namely in constructing axiomatic systems for intersections of logics and recognizing weakly implicative fuzzy logics among the weakly implicative ones.

Are there disjunctive properties? This question is important for at least two reasons. First, disjunctive properties are invoked in defense of certain philosophical theories, especially in the philosophy of mind. Second, the question raises the prior issue of what counts as a genuine property, a central concern in the metaphysics of properties. I argue here, on the basis of general considerations in the metaphysics of properties, that there are no disjunctive properties. Specifically, I argue that genuine properties must guarantee similarity-in-a-respect (...) among their instances, and must inhere in their bearers (two notions to be clarified). Disjunctive properties would fail both requirements. I compare the case of disjunctive properties with “mere Cambridge properties,” determinable properties, and functional properties, and show how my conception of properties remains compatible with determinables and functional properties while ruling out disjunctive and mere Cambridge properties. (shrink)

This book is less about disjunction than about the English word ‘or’, and it is less for than against formal logicians—more exactly, against those who maintain that formal logic can be applied in certain ways to the evaluation of reasoning formulated in ordinary English. Nevertheless, there are many things to interest such of those persons who are willing to overlook the frequent animadversions directed against their kind in the book, and this review will concentrate on them.

We present evidence that preschool children oftentimes understand disjunctive sentences as if they were conjunctive. The result holds for matrix disjunctions as well as disjunctions embedded under every. At the same time, there is evidence in the literature that children understand or as inclusive disjunction in downward-entailing contexts. We propose to explain this seemingly conflicting pattern of results by assuming that the child knows the inclusive disjunction semantics of or, and that the conjunctive inference is a scalar implicature. (...) We make two assumptions about implicature computation in the child: that children access only a proper subset of the adult alternatives, and that children possess the adult capacity to strengthen sentences with implicatures. As a consequence, children are expected to sometimes not compute any implicatures at all, but in other cases they are expected to compute an implicature that is different from the adult implicature. We argue that the child’s conjunctive strengthening of disjunctive sentences realizes the latter possibility: the adult infers that the conjunction is false but the child infers that the conjunction is true. This behaviour is predicted when our assumptions about child development are coupled with the assumption that a covert exhaustive operator is responsible for strengthening in both the child and the adult. Specifically, children’s conjunctive strengthening is predicted to follow from the same mechanism used by adults to compute conjunctive free choice implicatures in response to disjunctive permission sentences. We furthermore argue that this parallel between the child and the adult extends to disambiguation preferences. In particular, we present evidence that children prefer to strengthen disjunctions to conjunctions, in matrix and embedded positions ; this result mirrors previous findings that adults prefer to compute free choice, at the root and under every. We propose a disambiguation strategy that explains the preference for conjunctive strengthening – by both the child and the adult – even though there is no general preference for exhaustification. Specifically, we propose that the preference for a conjunctive strengthening follows from a pragmatic preference for a complete answer to the Question Under Discussion. (shrink)

With any structural inference rule A/B, we associate the rule $${(A \lor p)/(B \lor p)}$$, providing that formulas A and B do not contain the variable p. We call the latter rule a join-extension ( $${\lor}$$ -extension, for short) of the former. Obviously, for any intermediate logic with disjunction property, a $${\lor}$$ -extension of any admissible rule is also admissible in this logic. We investigate intermediate logics, in which the $${\lor}$$ -extension of each admissible rule is admissible. We prove (...) that any structural finitary consequence operator (for intermediate logic) can be defined by a set of $${\lor}$$ -extended rules if and only if it can be defined through a set of well-connected Heyting algebras of a corresponding quasivariety. As we exemplify, the latter condition is satisfied for a broad class of algebraizable logics. (shrink)

The main result of this paper is the following theorem: a closure space X has an , , Q-regular base of the power iff X is Q-embeddable in It is a generalization of the following theorems:(i) Stone representation theorem for distributive lattices ( = 0, = , Q = ), (ii) universality of the Alexandroff's cube for T 0-topological spaces ( = , = , Q = 0), (iii) universality of the closure space of filters in the lattice of all (...) subsets for , -closure spaces (Q = 0). By this theorem we obtain some characterizations of the closure space given by the consequence operator for the classical propositional calculus over a formalized language of the zero order with the set of propositional variables of the power . In particular we prove that a countable closure space X is embeddable with finite disjunctions preserved into F iff X is a consistent closure space satisfying the compactness theorem and X contains a 0, -base consisting of -prime sets. (shrink)

Mr B. H. Slater distinguishes between various pairs of questions: 1 Did they agree either that they would go to the pictures or that they would go out for a meal? 1 Did they agree that either they would go to the pictures or they would go out for a meal? 2 Is it known either that Jones was guilty or that Smith was guilty? 2 Is it known that either Jones was guilty or Smith was guilty? 3 Did he (...) say either that he would get the puzzle solved or that he would eat his hat? 3 Did he say that either he would get the puzzle solved or he would eat his hat? According to Slater, if we can only say ‘Yes’ to 1 or 3), then we immediately know what was agreed, namely, that either they would go to the pictures or they would go out for a meal; but if we can just say ‘Yes’ to 1 or 3), then we cannot immediately say what was agreed; all we know is that either they agreed to go to the pictures or they agreed to go out for a meal. Or, generalizing somewhat, a ‘Yes’ answer to the question ‘Did they agree that either p or q?’ tells us that they agreed that either p or q; but, with the question ‘Did they agree either that p or that q?’, while ‘Yes, they agreed that p’, and ‘Yes they agreed that q’ are possible answers , there is no answer, ‘Yes they agreed that r’, which is invariably correct, when ‘Yes’ is correct. In particular, the answer certainly cannot be ‘Yes, they agreed that p or q’. (shrink)

We investigate an information based generalization of the incompatibility-frame treatment of logics with non-classical negation connectives. Our framework can be viewed as an alternative to the neighbourhood semantics for extensions of lattice logic by various negation connectives, investigated by Hartonas. We set out the basic semantic framework, along with some correspondence results for extensions. We describe three kinds of constructions of canonical models and show that double negation law is not canonical with respect to any of these constructions. We also (...) compare our semantics to Hartonas’. (shrink)

Fodor (1990) argues that the theory of evolution by natural selection will not help to save naturalistic accounts of representation from the disjunction problem. This is because, he claims, the context 'was selected for representing things as F' is transparent to the substitution of predicates coextensive with F. But, I respond, from an evolutionary perspective representational contexts cannot be transparent: only under particular descriptions will a representational state appear as a "solution" to a selection "problem" and so be adaptive. (...) Only when we construe representational states as opaque in this manner are the generalizations of branches of evolutionary theory, like foraging theory, possible. (shrink)

Fodor argues that the theory of evolution by natural selection will not help to save naturalistic accounts of representation from the disjunction problem. This is because, he claims, the context 'was selected for representing things as F' is transparent to the substitution of predicates coextensive with F. But, I respond, from an evolutionary perspective representational contexts cannot be transparent: only under particular descriptions will a representational state appear as a "solution" to a selection "problem" and so be adaptive. Only (...) when we construe representational states as opaque in this manner are the generalizations of branches of evolutionary theory, like foraging theory, possible. (shrink)

Denotational semantics of logic programming and its extensions have been studied thoroughly for many years. In 1998, a game semantics was given to definite logic programs by Di Cosmo, Loddo, and Nicolet, and a few years later it was extended to deal with negation by Rondogiannis and Wadge. Both approaches were proven equivalent to the traditional semantics. In this paper we define a game semantics for disjunctive logic programs and prove soundness and completeness with respect to the minimal model semantics (...) of Minker. The overall development has been influenced by the games studied for PCF and functional programming in general, in the styles of Abramsky–Jagadeesan–Malacaria and Hyland–Ong–Nickau. (shrink)

According to the Multiple Realizability Argument, a higher-level property typically has many physical realizers, so it cannot be type-identical to any one of them. This enables the non-reductive physicalist to claim that some higher-level properties are type-distinct from physical properties. The reductive physicalist can counter with the Disjunctive Strategy: nothing prevents us from type-identifying the higher-level property with the disjunction of its realizers. Developing a powers-based ontology of properties, Shoemaker and Wilson present responses to the Disjunctive Strategy, wherein higher-level (...) property instances are token-distinct from their realizers, while instances of disjunctions are not. In this paper, I argue that such responses to the Disjunctive Strategy can be reasonably resisted, either by denying token-distinctness, or by insisting that exhaustively overlapping disjunctive properties are also token-distinct from the realizers. This secures the Disjunctive Strategy as a core component of the argument for reductive physicalism. (shrink)

Baratgin and colleagues have questioned certain aspects of the mental models theory related to disjunction. It is truth that, from this last theory, the paper authored by Baratgin et al. has already been responded. However, I try to further develop that response here by insisting in two important points of the theory: the role that modulation plays in it and the clear differences between its framework and standard logic. In this way, my main aim is to support to a (...) larger extent, by means of theoretical arguments based on its general approach, the idea that the objections presented in the mentioned paper do not really impact the mental models theory. (shrink)

We generalize, by a progressive procedure, the notions of conjunction and disjunction of two conditional events to the case of n conditional events. In our coherence-based approach, conjunctions and disjunctions are suitable conditional random quantities. We define the notion of negation, by verifying De Morgan’s Laws. We also show that conjunction and disjunction satisfy the associative and commutative properties, and a monotonicity property. Then, we give some results on coherence of prevision assessments for some families of compounded conditionals; (...) in particular we examine the Fréchet-Hoeffding bounds. Moreover, we study the reverse probabilistic inference from the conjunction Cn+1 of n + 1 conditional events to the family {Cn,En+1|Hn+1}. We consider the relation with the notion of quasi-conjunction and we examine in detail the coherence of the prevision assessments related with the conjunction of three conditional events. Based on conjunction, we also give a characterization of p-consistency and of p-entailment, with applications to several inference rules in probabilistic nonmonotonic reasoning. Finally, we examine some non p-valid inference rules; then, we illustrate by an example two methods which allow to suitably modify non p-valid inference rules in order to get inferences which are p-valid. (shrink)

This paper introduces a generalized version of inquisitive semantics, denoted as GIS, and concentrates especially on the role of disjunction in this general framework. Two alternative semantic conditions for disjunction are compared: the first one corresponds to the so-called tensor operator of dependence logic, and the second one is the standard condition for inquisitive disjunction. It is shown that GIS is intimately related to intuitionistic logic and its Kripke semantics. Using this framework, it is shown that (...) the main results concerning inquisitive semantics, especially the axiomatization of inquisitive logic, can be viewed as particular cases of more general phenomena. In this connection, a class of non-standard superintuitionistic logics is introduced and studied. These logics share many interesting features with inquisitive logic, which is the strongest logic of this class. (shrink)

The existence of disjunctive properties has been put in question by many philosophers. In this article, I shall offer an argument for the existence of such properties. I shall show that they are required in order to ground certain objective and unique resemblances between things. In Section 1, I develop the argument by presenting a toy example involving three entities: a red fish (Bill), a glass and round fish bowl (Bowl), and a red ball (Ball). In Section 2, I deal (...) with two strategies to rebut my argument. Such strategies respectively consist in denying that there is any objective and unique similarity between Bill, Bowl and Ball, and in holding that such a similarity is based on non-disjunctive properties. In Section 3, I examine and reject two further strategies, that consist in holding that grounding objective and unique similarities is not enough in order for something to exist as a property and in pointing to some more general concerns about disjunctive properties. (shrink)

In the paper we discuss criticisms against David Armstrong’s general theory of truthmaking by Gonzalo Rodriguez-Pereyra, Peter Schulte and Benjamin Schnieder, and conclude that Armstrong’s theory survives these criticisms. Special attention is given to the problems concerning Entailment Principle, Conjunction Thesis, Disjunction Thesis and to the notion of explanation.

Causal theories of content, a popular family of approaches to defining the content of mental states, commonly run afoul of two related and serious problems that prevent them from providing an adequate theory of mental content—the misrepresentation problem and the disjunction problem. In this paper, I present a causal theory of content, built on information theoretic tools, that solves these problems and provides a viable model of mental content. This is the greatest surprise reduction theory of content, which identifies (...) the content of a signal as the event the surprisal of which is most reduced by that signal. Conceptually, this amounts to the claim that the content of a signal is the event the probability of which has increased by the largest proportion, or the event that the signal makes the most less surprising to us. I develop the greatest surprise reduction theory of content in four stages. First, I introduce the general project of causal theories of content, and the challenges presented to this project by the misrepresentation and disjunction problems. Next, I review two recent and prominent causal theories of content and demonstrate the serious challenges faced by these approaches, both clarifying the need for a solution to the misrepresentation and disjunction problems and providing a conceptual background for the greatest surprise reduction theory. Then, I develop the greatest surprise reduction theory of content, demonstrate its ability to resolve the misrepresentation and disjunction problems, and explore some additional applications it may have. Finally, I conclude with a discussion of a particularly difficult challenge that remains to be addressed—the partition problem—and sketch a path to a potential solution. (shrink)

Herbrand disjunctions are a means for reducing the problem ofwhether a first-oder formula is valid in an open theory T or not to theproblem whether an open formula, one of the so called Herbrand disjunctions,is T -valid or not. Nevertheless, the set of Herbrand disjunctions, which hasto be examined, is undecidable in general. Fore this reason the practicalvalue of Herbrand disjunctions has been estimated negatively .Relying on completeness proofs which are based on the algebraizationtechnique presented in [30], but taking a (...) more optimistic view, we show howHerbrand disjunctions become the base of a method for building in theoriesinto automatic theorem provers [26]. Surveying newer results we discusshow to treat heterogeneous theories [29] as well as practical implications ofdifferent normal form transformations. (shrink)

Correspondence analysis is Kooi and Tamminga’s universal approach which generates in one go sound and complete natural deduction systems with independent inference rules for tabular extensions of many-valued functionally incomplete logics. Originally, this method was applied to Asenjo–Priest’s paraconsistent logic of paradox LP. As a result, one has natural deduction systems for all the logics obtainable from the basic three-valued connectives of LP -language) by the addition of unary and binary connectives. Tamminga has also applied this technique to the paracomplete (...) analogue of LP, strong Kleene logic \. In this paper, we generalize these results for the negative fragments of LP and \, respectively. Thus, the method of correspondence analysis works for the logics which have the same negations as LP or \, but either have different conjunctions or disjunctions or even don’t have them as well at all. Besides, we show that correspondence analyses for the negative fragments of \ and LP, respectively, are also suitable without any changes for the negative fragments of Heyting’s logic \ and its dual \ and LP). (shrink)

This paper proves the strong normalization of classical natural deduction with disjunction and permutative conversions, by using CPS-translation and augmentations. Using them, this paper also proves the strong normalization of classical natural deduction with general elimination rules for implication and conjunction, and their permutative conversions. This paper also proves that natural deduction can be embedded into natural deduction with general elimination rules, strictly preserving proof normalization.

The motivation of this study is that Reiter’s default theory as well as assumption-based argumentation frameworks corresponding to default theories have difficulties in handling disjunctive information, while a disjunctive default theory (ddt) avoids them. This paper presents the semantic correspondence between generalized assumption-based argumentation (ABA) and extended disjunctive logic programming as well as the correspondence between ABA and nonmonotonic reasoning approaches such as disjunctive default logic and prioritized circumscription. To overcome the above-mentioned difficulties of ABA frameworks corresponding to default (...) theories, we propose an assumption-based framework (ABF) translated from an extended disjunctive logic program (EDLP) since an EDLP can be translated into a ddt. Our ABF incorporates explicit negation and the connective of disjunction “|” to Heyninck and Arieli’s ABF induced by a disjunctive logic program. In this paper, first, we show how arguments are constructed from disjunctive rules in our proposed ABF. Then, we show the correspondence between answer sets of an EDLP P and stable extensions of the ABF translated from P with trivialization rules. After defining rationality postulates, we show answer sets of a consistent EDLP are captured by consistent stable extensions of the translated ABF with no trivialization rules. Finally, we show the correspondence between ABA and disjunctive default logic (resp. prioritized circumscription). The relation between ABA and possible model semantics of EDLPs is also discussed. (shrink)

Causal Modeling Semantics (CMS, e.g., Galles and Pearl 1998; Pearl 2000; Halpern 2000) is a powerful framework for evaluating counterfactuals whose antecedent is a conjunction of atomic formulas. We extend CMS to an evaluation of the probability of counterfactuals with disjunctive antecedents, and more generally, to counterfactuals whose antecedent is an arbitrary Boolean combination of atomic formulas. Our main idea is to assign a probability to a counterfactual (A ∨ B) > C at a causal model M as a weighted (...) average of the probability of C in those submodels that truthmake A∨ B (Briggs 2012; Fine 2016, 2017). The weights of the submodels are given by the inverse distance to the original model M, based on a distance metric proposed by Eva, Stern, and Hartmann (2019). Apart from solving a major problem in the epistemology of counterfactuals, our paper shows how work in semantics, causal inference and formal epistemology can be fruitfully combined. (shrink)

This is the second part of a paper devoted to the study of the maximal intermediate propositional logics with the disjunction property , whose first part has appeared in this journal with the title “A method to single out maximal propositional logics with the disjunction property I”. In the first part we have explained the general results upon which a method to single out maximal constructive logics is based and have illustrated such a method by exhibiting the Kripke (...) semantics of maximal constructive logics extending the logic ST of Scott, for which, in turn, a semantical characterization in terms of Kripke frames has been given. In the present part we complete the illustration of the method of the first part, having in mind some aspects which might be interesting for a classification of the maximal constructive logics, and an application of the heuristic content of the method to detect the nonmaximality of apparently maximal constructive logics. Thus, on the one hand we introduce the logic AST , which is compared with ST and is seen as a logic “alternative” to it, in a sense which will be precisely explained. We provide a Kripke semantics for AST and show that there are maximal constructive logics which neither are extensions of ST nor are extensions of AST. Finally, we give a further application of the results of the first part by exhibiting the Kripke semantics of a maximal constructive logic extending AST. On the other hand, we compare the maximal constructive logics presented in both parts of the paper with a constructive logic introduced by Maksimova , which has been conjectured to be maximal by Chagrov and Zacharyashchev ; from this comparison a disproof of the conjecture arises. (shrink)

We continue our work [5] on the logic of multisets (or on the multiset semantics of linear logic), by interpreting further the additive disjunction . To this purpose we employ a more general class of processes, called free, the axiomatization of which requires a new rule (not compatible with the full LL), the cancellation rule. Disjunctive multisets are modeled as finite sets of multisets. The -Horn fragment of linear logic, with the cut rule slightly restricted, is sound with respect (...) to this semantics. Another rule, which is a slight modification of cancellation, added to HF makes the system sound and complete. (shrink)

We develop an axiomatic theory of “generalized Routley-Meyer logics.” These are first-order logics which are can be characterized by model theories in a certain generalization of Routley-Meyer semantics. We show that all GRM logics are subclassical, have recursively enumerable consequence relations, satisfy the compactness theorem, and satisfy the standard structural rules and conjunction and disjunction introduction/elimination rules. We also show that the GRM logics include classical logic, intuitionistic logic, LP/K3/FDE, and the relevant logics.

We extend our general approach to characterizing information to multi-agent systems. In particular, we provide a formal description of an agent''s knowledge containing exactly the information conveyed by some (honest) formula . Only knowing is important for dynamic agent systems in two ways. First of all, one wants to compare different states of knowledge of an agent and, secondly, for agent a''s decisions, it may be relevant that (he knows that) agent b does not know more than . There are (...) three ways to study the question whether a formula can be interpreted as minimal information. The first method is semantic and inspects minimal models for (with respect to some information order on states). The second one is syntactic and searches for stable expansions, minimal with respect to some language *. The third method is a deductive test, known as the disjunction property. We present a condition under which the three methods are equivalent. Then, we show how to construct the order by collecting layered orders. Focusing on the multi-agent case we identify languages * for various orders , and show how they yield different notions of honesty for different multi-modal systems. We then provide several tools for studying honesty types and illustrate their usefulness on a number of examples, for three modal systems of particular interest. Finally, we relate the different notions of minimal knowledge, and describe possible patterns of honesty for these systems. (shrink)

New conjunctionlike and disjunctionlike operations on orthomodular lattices are defined with the aid of formal Mackey decompositions of not necessarily compatible elements. Various properties of these operations are studied. It is shown that the new operations coincide with the lattice operations of join and meet on compatible elements of a lattice but they necessarily differ from the latter on all elements that are not compatible. Nevertheless, they define on an underlying set the partial order relation that coincides with the original (...) one. The new operations are in general nonassociative: if they are associative, a lattice is necessarily Boolean. However, they satisfy the Foulis-Holland-type theorem concerning associativity instead of distributivity. (shrink)

I advance a new theory of causal relevance, according to which causal claims convey information about conditional probability functions. This theory is motivated by the problem of disjunctive factors, which haunts existing probabilistic theories of causation. After some introductory remarks, I present in Section 3 a sketch of Eells's (1991) probabilistic theory of causation, which provides the framework for much of the discussion. Section 4 explains how the problem of disjunctive factors arises within this framework. After rejecting three proposed solutions, (...) I offer in Section 6 a new approach to causation that avoids the problem. Decision-theoretic considerations also support the new approach. Section 8 develops the consequences of the new theory for causal explanation. The resulting theory of causal explanation incorporates the new insights while respecting important work on scientific explanation by Salmon (1971), Railton (1981), and Humphreys (1989). My conclusions are enumerated in Section 9. (shrink)

Relevantism, as a matter of definition, rejects classical logic as incorrect and adopts instead a relevant logic as encapsulating correct inference. It rejects classical logic on the grounds that the rule of Material Detachment, from A and not A or B to infer B,, sometimes leads from truth to falsity. Relevantism — although promoted by some relevant logicians, and an integral part of ultralogic — has recently encountered heavy, but interesting, criticism from relevance logicians themselves.

We extend our general approach to characterizing information to multi-agent systems. In particular, we provide a formal description of an agent's knowledge containing exactly the information conveyed by some (honest) formula φ. Only knowing is important for dynamic agent systems in two ways. First of all, one wants to compare different states of knowledge of an agent and, secondly, for agent a's decisions, it may be relevant that (he knows that) agent b does not know more than φ. There are (...) three ways to study the question whether a formula φ can be interpreted as minimal information. The first method is semantic and inspects 'minimal' models for φ (with respect to some information order ≤ on states). The second one is syntactic and searches for stable expansions, minimal with respect to some language $\scr{L}^{\ast}$ . The third method is a deductive test, known as the disjunction property. We present a condition under which the three methods are equivalent. Then, we show how to construct the order ≤ by collecting 'layered orders'. Focusing on the multi-agent case we identify languages $\scr{L}^{\ast}$ for various orders ≤, and show how they yield different notions of honesty for different multi-modal systems. We then provide several tools for studying honesty types and illustrate their usefulness on a number of examples, for three modal systems of particular interest. Finally, we relate the different notions of minimal knowledge, and describe possible patterns of honesty for these systems. (shrink)

Recently, Brass and Dix showed 143–165) that the well founded semantics WFS can be defined as a confluent calculus of transformation rules. This led not only to a simple extension to disjunctive programs 167–213), but also to a new computation of the well-founded semantics which is linear for a broad class of programs. We take this approach as a starting point and generalize it considerably by developing a general theory of Confluent LP-systems CS . Such a system CS is a (...) rewriting system on the set of all logic programs over a fixed signature L and it induces in a natural way a canonical semantics. Moreover, we show four important applications of this theory: most of the well-known semantics are induced by confluent LP-systems , there are many more transformation rules that lead to confluent LP-systems , semantics induced by such systems can be used to model aggregation , the new systems can be used to construct interesting counterexamples to some conjectures about the space of well-behaved semantics. (shrink)

Recent evidence suggests that experienced events are often mapped to too many episodic states, including those that are logically or experimentally incompatible with one another. For example, episodic over-distribution patterns show that the probability of accepting an item under different mutually exclusive conditions violates the disjunction rule. A related example, called subadditivity, occurs when the probability of accepting an item under mutually exclusive and exhaustive instruction conditions sums to a number >1. Both the over-distribution effect and subadditivity have been (...) widely observed in item and source-memory paradigms. These phenomena are difficult to explain using standard memory frameworks, such as signal-detection theory. A dual-trace model called the over-distribution model can explain the episodic over-distribution effect, but not subadditivity. Our goal is to develop a model that can explain both effects. In this paper, we propose the Generalized Quantum Episodic Memory model, which extends the Quantum Episodic Memory model developed by Brainerd, Wang, and Reyna. We test GQEM by comparing it to the OD model using data from a novel item-memory experiment and a previously published source-memory experiment examining the over-distribution effect. Using the best-fit parameters from the over-distribution experiments, we conclude by showing that the GQEM model can also account for subadditivity. Overall these results add to a growing body of evidence suggesting that quantum probability theory is a valuable tool in modeling recognition memory. (shrink)

This monograph presents a general theory of weakly implicative logics, a family covering a vast number of non-classical logics studied in the literature, concentrating mainly on the abstract study of the relationship between logics and their algebraic semantics. It can also serve as an introduction to algebraic logic, both propositional and first-order, with special attention paid to the role of implication, lattice and residuated connectives, and generalized disjunctions. Based on their recent work, the authors develop a powerful uniform framework (...) for the study of non-classical logics. In a self-contained and didactic style, starting from very elementary notions, they build a general theory with a substantial number of abstract results. The theory is then applied to obtain numerous results for prominent families of logics and their algebraic counterparts, in particular for superintuitionistic, modal, substructural, fuzzy, and relevant logics. The book may be of interest to a wide audience, especially students and scholars in the fields of mathematics, philosophy, computer science, or related areas, looking for an introduction to a general theory of non-classical logics and their algebraic semantics. (shrink)

Ambiguous terms are applicable to different kinds of things, but so are general terms, since a general kind may include various species. Thus a bank may be the side of a river or a certain kind of financial institution, and an animal may be a dog or a cat. Similarly, an ambiguous sentence is true in different kinds of situations, and so is a general sentence in that different specific situations may make the same general sentence true. Thus the sentence, (...) The bank collapsed yesterday,’ can be true of different varieties of banks, and the sentence, The animal is untrained,’ can be true of different varieties of animals. These similarities of ambiguity and generality have led many linguists and philosophers to confuse the two. Basic to this confusion is a failure to notice a difference in the scopes of disjunctions involved in ambiguity and generality. (shrink)