Disjunction

ABSTRACT ‘You may have beer or wine’ suggests that you may have beer and you may have wine. Following Klinedinst, I argue that this ‘free choice’ effect is a special kind of scalar implicature, arising from the application of an unspecific predicate to a plurality. I show that the implicature can be derived from general norms of cooperative communication, without postulating new grammatical rules or hidden lexical items. The derivation calls for an extension to the classical neo-Gricean model. I give (...) independent arguments for this extension. (shrink)

Free Choice is the principle that possibly p or q implies and is implied by possibly p and possibly q. A variety of recent attempts to validate Free Choice rely on a nonclassical semantics for disjunction, where the meaning of p or q is not a set of possible worlds. This paper begins with a battery of impossibility results, showing that some kind of nonclassical semantics for disjunction is required in order to validate Free Choice. The paper then provides a (...) positive account of Free Choice, by identifying a family of dynamic semantics for disjunction that can validate the inference. On all such theories, the meaning of p or q has two parts. First, p or q requires that our information is consistent with each of p and q. Second, p or q narrows down our information by eliminating some worlds. It turns out that this second component of or is well behaved: there is a strongest such meaning that p or q can express, consistent with validating Free Choice. The strongest such meaning is the classical one, on which p or q eliminates any world where both p and q are false. In this way, the classical meaning of disjunction turns out to be intimately related to the validity of Free Choice. (shrink)

We argue that, if taken seriously, Kripke's view that a language for science can dispense with a negation operator is to be rejected. Part of the argument is a proof that positive logic, i.e., classical propositional logic without negation, is not categorical.

This paper investigates the prospects for a semantic theory that treats disjunction as a modal operator. Potential motivation for such a theory comes from the way in which modals embed within disjunctions. After reviewing some of the relevant data, I go on to distinguish a variety of modal theories of disjunction. I analyze these theories by considering pairs of conflicting desiderata, highlighting some of the tradeoffs they must face.

In this paper, I argue that the disjunction elimination rule presupposes the principle that a true disjunction contains at least one true disjunct. However, in some contexts such as supervaluationism or quantum logic, we have good reasons to reject this principle. Hence, disjunction elimination is restricted in at least one respect: it is not applicable to disjunctions for which this principle does not hold. The insight that disjunction elimination presupposes the principle that a true disjunction contains at least one true (...) disjunct is applied to two arguments which argue for this very principle. I show that these arguments are rule-circular since they rest on disjunction elimination. I claim that rule-circularity better explains why the arguments fail than the explanations provided by Rumfitt (2015), which, for instance, rely on controversial principles about truth. (shrink)

In this paper we discuss the extent to which conjunction and disjunction can be rightfully regarded as such, in the context of infectious logics. Infectious logics are peculiar many-valued logics whose underlying algebra has an absorbing or infectious element, which is assigned to a compound formula whenever it is assigned to one of its components. To discuss these matters, we review the philosophical motivations for infectious logics due to Bochvar, Halldén, Fitting, Ferguson and Beall, noticing that none of them discusses (...) our main question. This is why we finally turn to the analysis of the truth-conditions for conjunction and disjunction in infectious logics, employing the framework of plurivalent logics, as discussed by Priest. In doing so, we arrive at the interesting conclusion that —in the context of infectious logics— conjunction is conjunction, whereas disjunction is not disjunction. (shrink)

This paper proposes a semantics for free choice permission that explains both the non-classical behavior of modals and disjunction in sentences used to grant permission, and their classical behavior under negation. It also explains why permissions can expire when new information comes in and why free choice arises even when modals scope under disjunction. On the proposed approach, deontic modals update preference orderings, and connectives operate on these updates rather than propositions. The success of this approach stems from its capacity (...) to capture the difference between expressing the preferences that give rise to permissions and conveying propositions about those preferences. (shrink)

In many languages, the same particles that form quantifier words also serve as connectives, additive and scalar particles, question markers, roots of existential verbs, and so on. Do these have a unified semantics, or do they merely bear a family resemblance? Are they aided by silent operators in their varied roles―if yes, what operators? I dub the particles “quantifier particles” and refer to them generically with capitalized versions of the Japanese morphemes. I argue that both MO and KA can be (...) assigned a stable semantics across their various roles. The specific analysis I offer is motivated by the fact that MO and KA often combine with just one argument; I propose that this is their characteristic behavior. Their role is to impose semantic requirements that are satisfied when the immediately larger context is interpreted as the meet/join of their host’s semantic contribution with something else. They do not perform meet/join themselves. The obligatory vs. optional appearance of the particles depends on whether the meet/join interpretations arise by default in the given constellation. I explicate the proposal using the toolkit of basic Inquisitive Semantics. (shrink)

Observe that complement questions can be either directly or indirectly conjoined, but they can only be indirectly disjoined. • What theories of questions and coordination predict this difference? • Look at Partition theory (Groenendijk & Stokhof 1984) and Inquisitive Semantics (Groenendijk & Roelofsen 2009, Ciardelli et al. 2012).

Crosslinguistically, questions frequently make crucial use of morphosyntactic elements which also occur outside of questions. Chief among these are focus, disjunctions, and wh-words with indefinite semantics. This paper provides a compositional account of the semantics of wh-, alternative, and polar questions in Yucatec Maya (YM), which are composed primarily of these elements. Key to the account is a theory of disjunctions and indefinites (extending work by others) which recognizes the inherently inquisitive nature of these elements. While disjunctions and indefinites are (...) inquisitive, they differ from questions since they are also truth-conditionally informative. Compositionally, then, the role of focus in YM questions is to presuppose the informative component of an indefinite wh-word or disjunction, rendering the inquisitive component the question’s only new contribution to the discourse. In addition to deriving question denotations compositionally, the account also captures a potentially surprising fact: focused disjunctions in YM can function as either questions or assertions, depending solely on the discourse context. (shrink)

There are certain uses of and and or that cannot be explained by their normal meanings as truth-functional connectives, even with sophisticated pragmatic resources. These include examples such as The cops show up, and a fight will break out (‘If the cops show up, a fight will break out’), and I have no friends, or I would throw a party (‘I have no friends. If I did have friends, I would throw a party.’). We argue that these uses are indeed (...) distinct from the more ordinary uses of and and or, but that they are nonetheless related in a principled way. To explain them we give an analysis of what we call the dynamic effects of connectives, which arise in all their uses. The special uses at issue are then argued to be instances where the connectives exhibit their dynamic effects without their truth-conditional meaning. (shrink)

I show that the model-theoretic meaning that can be read off the natural deduction rules for disjunction fails to have certain desirable properties. I use this result to argue against a modest form of inferentialism which uses natural deduction rules to fix model-theoretic truth-conditions for logical connectives.

We present a series of arguments for logical nativism, focusing mainly on the meaning of disjunction in human languages. We propose that all human languages are logical in the sense that the meaning of linguistic expressions corresponding to disjunction (e.g. English or , Chinese huozhe, Japanese ka ) conform to the meaning of the logical operator in classical logic, inclusive- or . It is highly implausible, we argue, that children acquire the (logical) meaning of disjunction by observing how adults use (...) disjunction. Findings from studies of child language acquisition and from cross-linguistic research invite the conclusion that children do not learn to be logical—it comes naturally to them. (shrink)

Imagine a dog tracing a scent to a crossroads, sniffing all but one of the exits, and then proceeding down the last without further examination. According to Sextus Empiricus, Chrysippus argued that the dog effectively employs disjunctive syllogism, concluding that since the quarry left no trace on the other paths, it must have taken the last. The story has been retold many times, with at least four different morals: (1) dogs use logic, so they are as clever as humans; (2) (...) dogs use logic, so using logic is nothing special; (3) dogs reason well enough without logic; (4) dogs reason better for not having logic. This paper traces the history of Chrysippus's dog, from antiquity up to its discussion by relevance logicians in the twentieth century. (shrink)

Hurford’s Constraint (Hurford, Foundations of Language, 11, 409–411, 1974) states that a disjunction is infelicitous if its disjuncts stand in an entailment relation: #John was born in Paris or in France. Gazdar (Pragmatics, Academic Press, NY, 1979) observed that scalar implicatures can obviate the constraint. For instance, sentences of the form (A or B) or (Both Aand B) are felicitous due to the exclusivity implicature of the first disjunct: A or B implicates ‘not (A and B)’. Chierchia, Fox, and Spector (...) (Handbook of semantics, 2008) use the obviation of Hurford’s Constraint in these cases to argue for a theory of local implicature. I present evidence indicating that the constraint needs to be modified in two ways. First, implicatures can obviate Hurford’s Constraint only in earlier disjuncts, not later ones: #(Both A and B) or (A or B). Second, the constraint rules out not only disjuncts that stand in an entailment relation, but also disjuncts that are even mutually consistent: #John is from Russia or Asia. I propose to make sense of these facts by providing an incremental evaluation procedure which checks that each new disjunct to the right is inconsistent with the information to its left, before the disjunct can be strengthened by local implicature. (shrink)

In this paper Grice's requirements for assertability are imposed on the disjunction of Classical Logic. Defining material implication in terms of negation and disjunction supplemented by assertability conditions, results in the disappearance of the most important paradoxes of material implication. The resulting consequence relation displays a very strong resemblance to Schurz's conclusion-relevant consequence relation.

In the tradition of substructural logics, it has been claimed for a long time that conjunction and inclusive disjunction are ambiguous:we should, in fact, distinguish between ‘lattice’ connectives (also called additive or extensional) and ‘group’ connectives (also called multiplicative or intensional). We argue that an analogous ambiguity affects the quantifiers. Moreover, we show how such a perspective could yield solutions for two well-known logical puzzles: McGee’s counterexample to modus ponens and the lottery paradox.

This paper is a contribution to the program of constructing formal representations of pragmatic aspects of human reasoning. We propose a formalization within the framework of Adaptive Logics of the exclusivity implicature governing the connective 'or'.

The connective or can be treated as an inclusive disjunction or else as an exclusive disjunction. Although researchers are aware of this distinction, few have examined the conditions under which each interpretation should be anticipated. Based on linguistic-pragmatic analyses, we assume that interpretations are initially inclusive before either (a) remaining so, or (b) becoming exclusive by way of an implicature ( but not both ). We point to a class of situations that ought to predispose disjunctions to inclusive interpretations and (...) to situations that encourage exclusive interpretations. A disjunction's ultimate interpretation is based on its potential informativeness, where the interpretation of the disjunctive utterance having the smallest number of true conditions is considered most informative. Our investigation leads to five experiments employing arbitrary materials. Among the problems expected to encourage inclusive interpretations are those that present disjunctions in the antecedents of conditionals and in question forms. The best candidates to produce implicatures are those disjunctions that underdetermine an expected conjunctive conclusion, although other disjunctive utterances that are more informative as exclusive are discussed and tested. (shrink)

The de Morgan laws characterize how negation, conjunction, and disjunction interact with each other. They are fundamental in any semantics that bases itself on the propositional calculus/Boolean algebra. This paper is primarily concerned with the second law. In English, its validity is easy to demonstrate using linguistic examples. Consider the following: (3) Why is it so cold in here? We didn’t close the door or the window. The second sentence is ambiguous. It may mean that I suppose we did not (...) close the door or did not close the window, but I am not sure which. This `I am not sure which’ reading is irrelevant to us because it has disjunction scoping over negation. But the sentence may equally well mean (and indeed this is the preferred reading) that we didn’t close the door and did not close the window. This `neither’ reading bears out de Morgan law (2). (shrink)

The quantum logical `or' is analyzed from a physical perspective. We show that it is the existence of EPR-like correlation states for the quantum mechanical entity under consideration that make it nonequivalent to the classical situation. Specifically, the presence of potentiality in these correlation states gives rise to the quantum deviation from the classical logical `or'. We show how this arises not only in the microworld, but also in macroscopic situations where EPR-like correlation states are present. We investigate how application (...) of this analysis to concepts could alleviate some well known problems in cognitive science. (shrink)

T?s?, a thirteenth century logician writing in Arabic, uses two logical connectives to build up molecular propositions: ?if-then?, and ?either-or?. By referring to a dichotomous Tree, T?s? shows how to choose the proper disjunction relative to the terms in the disjuncts. He also discusses the disjunctive propositions which follow from a conditional proposition.

This is a comprehensive study of the English word 'or', and the logical operators variously proposed to present its meaning. Although there are indisputably disjunctive uses of or in English, it is a mistake to suppose that logical disjunction represents its core meaning. 'Or' is descended from the Anglo-Saxon word meaning second, a form which survives in such expressions as "every other day." Its disjunctive uses arise through metalinguistic applications of an intermediate adverbial meaning which is conjunctive rather than disjunctive (...) in character. These conjunctive uses have puzzled philosophers and logicians, and have been discussed extensively under such headings as "free choice permission." This study examines the textbook myths that have clouded our understanding of how or and other "logical" vocabulary comes to have something approaching its logical meaning in natural languages. It considers the various historical conceptions of disjunction and its place in logic from the Stoics to the present day. (shrink)

I argue that the conjunctive distribution of permissibility over or, which is a puzzling feature of free-choice permission is just one instance of a more general class of conjunctive occurrences of the word, and that these conjunctive uses are more directly explicable by the consideration that or is a descendant of oper than by reference to the disjunctive occurrences which logicalist prejudices may tempt us to regard as semantically more fundamental. I offer an account of how the disjunctive uses of (...) or may have come about through an intermediate discourse-adverbial use of or, drawing a parallel with but, which, etymologically, is disjunctive rather than conjunctive and whose conjunctive uses seem to represent just such a discourse-adverbial application. (shrink)

In this paper we study the relations between the fragment L of classical logic having just conjunction and disjunction and the variety D of distributive lattices, within the context of Algebraic Logic. We prove that these relations cannot be fully expressed either with the tools of Blok and Pigozzi's theory of algebraizable logics or with the use of reduced matrices for L. However, these relations can be naturally formulated when we introduce a new notion of model of a sequent calculus. (...) When applied to a certain natural calculus for L, the resulting models are equivalent to a class of abstract logics (in the sense of Brown and Suszko) which we call distributive. Among other results, we prove that D is exactly the class of the algebraic reducts of the reduced models of L, that there is an embedding of the theories of L into the theories of the equational consequence (in the sense of Blok and Pigozzi) relative to D, and that for any algebra A of type (2,2) there is an isomorphism between the D-congruences of A and the models of L over A. In the second part of this paper (which will be published separately) we will also apply some results to give proofs with a logical flavour for several new or well-known lattice-theoretical properties. (shrink)

Proves two simple identities relating the biconditional and exclusive disjunction. -/- The PDF has been made available gratis by the publisher.

In this paper we study the relations between the fragment L of classical logic having just conjunction and disjunction and the variety D of distributive lattices, within the context of Algebraic Logic. We prove that these relations cannot be fully expressed either with the tools of Blok and Pigozzi's theory of algebraizable logics or with the use of reduced matrices for L. However, these relations can be naturally formulated when we introduce a new notion of model of a sequent calculus. (...) When applied to a certain natural calculus for L, the resulting models are equivalent to a class of abstract logics (in the sense of Brown and Suszko) which we call distributive. Among other results, we prove that D is exactly the class of the algebraic reducts of the reduced models of L, that there is an embedding of the theories of L into the theories of the equational consequence (in the sense of Blok and Pigozzi) relative to D, and that for any algebra A of type (2,2) there is an isomorphism between the D-congruences of A and the models of L over A. In the second part of this paper (which will be published separately) we will also apply some results to give proofs with a logical flavour for several new or well-known lattice-theoretical properties. (shrink)

In relevant logics one can formally distinguish two logical operators, symbolised as ┌A ∨ B┐ and ┌A + B┐. Addition holds for ‘ ∨ ’ and Disjunctive Syllogism for ‘ + ’, but not vice versa. The question arises, whether this distinction between two different formal notions of disjunction can be found in natural reasoning. First it is necessary to rebut Quinean objections to any rival to classical logic, Crice's claim that an intensional disjunction is not needed to explain the (...) everyday uses of or, and Kempson and Cormack's argument that there can be no ambiguity between putative readings one of which entails the other (for A + B ⊢ A ∨ B. Finally Jackson's use of assertibility-conditions to defend the thesis that if ┐A then B┌ is equivalent to ┌A ⊃ B ┐ is rejected, but the notion of robustness which he introduces is usefully adapted to show why Disjunctive Syllogism must fail for ‘∨’, Having cleared the ground in this way, two uses of or are considered which have in everyday reasoning the inferential properties of ‘∨’ and ‘+’, thus defending the relevantist claim that or is ambiguous. (shrink)

In a letter David Makinson has pointed out to me a clear-cut coun- terexample to the formula T3: S = M in [ S()), propounded in [1] for objective of disjunction. As he says: let p; q be independent E- propositions and put = p ^ q, = p^ q. Since p is an E-proposition, S = fug for some u in SE. Then S = S _ ) = S = fug; but u 62 V () [ V () (...) S() [ S() M in [ S()). (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)

In this paper we address some central problems of combination of logics through the study of a very simple but highly informative case, the combination of the logics of disjunction and conjunction. At first it seems that it would be very easy to combine such logics, but the following problem arises: if we combine these logics in a straightforward way, distributivity holds. On the other hand, distributivity does not arise if we use the usual notion of extension between consequence relations. (...) A detailed discussion about this phenomenon, as well as some possible solutions for it, are given. (shrink)

This study is concerned with the properties of the disjunction operator, or, and the acquisition of these properties by English-speaking children. Previous research has concluded that adult truth conditions for logical connectives are acquired relatively late in the course of language development. With particular reference to disjunction, the results of several studies have led to two claims. First, it has been argued that the full range of truth-conditions associated with inclusive-or is not initially available to children; instead, children are supposed (...) to interpret disjunction using the truth conditions that are associated with exclusive-or (e.g., Beilin and Lust 1975; Braine and Rumain 1981, 1983; Paris 1973). A second, related claim is that even when children respond as if they have access to the complete range of truth conditions, their responses result from the failure to distinguish or from and (Paris, 1973). (shrink)

This paper investigates scalar implicatures and downward entailment in child English. In previous experimental work we have shown that adults’ computation of scalar implicatures is sensitive to entailment relations. For instance, when the disjunction operator or occurs in positive contexts, an implicature of exclusivity arises. By contrast when the disjunction operator occurs within the scope of a downward entailing linguistic expression, no implicature of exclusivity is computed. Investigations on children’s computation of scalar implicatures in the same contexts have led to (...) a slightly different picture. In particular it has proven difficult to demonstrate that children compute scalar implicatures (in non- downward entailing contexts) using the Truth Value Judgment task, a technique that has been used successfully in showing children’s sensitivity to other semantic phenomena. Adopting a different experimental technique called the Felicity Judgment task, however, we demonstrated children’s knowledge of the prerequisites to the computation of scalar implicatures (Chierchia, Crain, Guasti, Gualmini and Meroni, 2001). (shrink)

Poverty -of-stimulus arguments have taken new ground recently, augmented by experimental findings from th e study of child language. In this paper, we briefly review two variants of the poverty-of-stimulus argument that have received empirical support from studies of child language; then we examine a third argument of this kind in more detail. The case under discussion involves the structural notion of c-command as it pertains to children’s interpretation of disjunction in the scope of negation.

This paper investigates young children's knowledge of scalar implicatures and downward entailment. In previous experimental work, we have shown that young children access the full range of truth-conditions associated with logical words in classical logic, including the disjunction operator, as well as the indefinite article. The present study extends this research in three ways, taking disjunction as a case study. Experiment 1 draws upon the observation that scalar implicatures (SIs) are cancelled (or reversed) in downward entailing (DE) linguistic environments, e.g., (...) in the scope of negation (Chierchia, 2000). Experiment 2 was designed to determine if scalar implicatures are used by children, like adults, to influence the interpretation of disjunction in non-DE contexts, yielding an implicature of exclusivity for disjunction. Whereas adult controls always rejected assertions of the form A or B in positive (non-DE) contexts in which assertions of the form A and B were also true, many children accepted assertions with disjunction in such contexts. To provide an interpretation to the findings from Experiment 2, a new experimental technique was devised and used in Experiment 3. The new technique presents pairs of assertions to children, who are asked to judge which assertion is a ‘better’ description of the context. The findings from Experiment 3 demonstrated children's awareness that A and B is more informative than A or B in positive contexts, where both statements are true. Taken together, the findings of Experiments 2 and 3 are compatible with the view that some children lack the computational resources to apply scalar implicatures when a single assertion is presented alone (see Reinhart, 1999). (shrink)

This study is concerned with the acquisition of the disjunction operator, or, in English. Two mutually inconsistent claims have been made about the acquisition of disjunction. One claim is that the acquisition of the adult truth conditions for logical connectives, including disjunction, is a late and not fully universal, achievement. With particular reference to disjunction, the findings from several studies are interpreted as showing that only the truth conditions associated with exclusive-or are available to young children (e.g., Beilin and Lust (...) 1975; Braine and Rumain 1981, 1983; Paris 1973). There is a related claim, that even when children respond as if they have access to a broader range of truth conditions, namely those associated with inclusive-or, children's adult-like responses are the result of a failure to distinguish or from and (Paris 1973). (shrink)