Modifying the descriptive and theoretical generalizations of Relativized Minimality, we argue that a significant subset of weak island violations arise when an extracted phrase should scope over some intervener but is unable to. Harmless interveners seem harmless because they can support an alternative reading. This paper focuses on why certain wh-phrases are poor wide scope takers, and offers an algebraic perspective on scope interaction. Each scopal element SE is associated with certain operations (e.g., not with complements). When a wh-phrase scopes (...) over some SE, the operations associated with that SE are performed in its denotation domain. The requisite operations may or may not be available in a domain, however. We present an empirical analysis of a variety of wh-phrases. It is argued that the wh-phrases that escape all weak islands (i.e., can scope over any intervener) are those that range over individuals, the reason being that all Boolean operations are defined for their domain. Collectives, manners, amounts, numbers, etc. all denote in domains with fewer operations and are thus selectively sensitive to scopal interveners—a “semantic relativized minimality effect”. (shrink)

Ways of Scope Taking is concerned with syntactic, semantic and computational aspects of scope. Its starting point is the well-known but often neglected fact that different types of quantifiers interact differently with each other and other operators. The theoretical examination of significant bodies of data, both old and novel, leads to two central claims. (1) Scope is a by-product of a set of distinct Logical Form processes; each quantifier participates in those that suit its particular features. (2) Scope interaction is (...) further constrained by the semantics of the interacting operators. The arguments are developed using Minimalist syntax, Generalized Quantify theory, Discourse Representation Theory, and algebraic semantics. The contributors (Beghelli, Ben-Shalom, Doetjes, Farkas, Gutiérrez Rexach, Honcoop, Stabler, Stowell, Szabolcsi and Zwarts) make tightly related theoretical assumptions and focus on related empirical phenomena, which include the direct and inverse scope of quantifiers, distributivity, negation, modal and intensional contexts, weak islands, event-related readings, interrogatives, wh/quantifier interactions, and Hungarian syntax. An introduction to the formal semantics background is provided. Audience: Linguists, philosophers, computational and psycholinguists; advanced undergraduates, graduate students and researchers in these fields. (shrink)

Necessitism is the view that necessarily everything is necessarily something; contingentism is the negation of necessitism. The dispute between them is reminiscent of, but clearer than, the more familiar one between possibilism and actualism. A mapping often used to ‘translate’ actualist discourse into possibilist discourse is adapted to map every sentence of a first-order modal language to a sentence the contingentist (but not the necessitist) may regard as equivalent to it but which is neutral in the dispute. This mapping enables (...) the necessitist to extract a ‘cash value’ from what the contingentist says. Similarly, a mapping often used to ‘translate’ possibilist discourse into actualist discourse is adapted to map every sentence of the language to a sentence the necessitist (but not the contingentist) may regard as equivalent to it but which is neutral in the dispute. This mapping enables the contingentist to extract a ‘cash value’ from what the necessitist says. Neither mapping is a translation in the usual sense, since necessitists and contingentists use the same language with the same meanings. The former mapping is extended to a second-order modal language under a plural interpretation of the second-order variables. It is proved that the latter mapping cannot be. Thus although the necessitist can extract a ‘cash value’ from what the contingentist says in the second-order language, the contingentist cannot extract a ‘cash value’ from some of what the necessitist says, even when it raises significant questions. This poses contingentism a serious challenge. (shrink)

The present paper provides novel results on the model theory of Independence friendly modal logic. We concentrate on its particularly well-behaved fragment that was introduced in Tulenheimo and Sevenster (Advances in Modal Logic, 2006). Here we refer to this fragment as ‘Simple IF modal logic’ (IFML s ). A model-theoretic criterion is presented which serves to tell when a formula of IFML s is not equivalent to any formula of basic modal logic (ML). We generalize the notion of bisimulation familiar (...) from ML; the resulting asymmetric simulation concept is used to prove that IFML s is not closed under complementation. In fact we obtain a much stronger result: the only IFML s formulas admitting their classical negation to be expressed in IFML s itself are those whose truth-condition is in fact expressible in ML. (shrink)

Branching quantifiers were first introduced by L. Henkin in his 1959 paper ‘Some Remarks on Infmitely Long Formulas’. By ‘branching quantifiers’ Henkin meant a new, non-linearly structured quantiiier-prefix whose discovery was triggered by the problem of interpreting infinitistic formulas of a certain form} The branching (or partially-ordered) quantifier-prefix is, however, not essentially infinitistic, and the issues it raises have largely been discussed in the literature in the context of finitistic logic, as they will be here. Our discussion transcends, however, the (...) resources of standard lst-order languages and we will consider the new form in the context of 1st-order logic with 1- and 2-place ‘Mostowskian` generalized quantifiers.2.. (shrink)

In this paper I examine a cluster of concepts relevant to the methodology of truth theories: 'informative definition', 'recursive method', 'semantic structure', 'logical form', 'compositionality', etc. The interrelations between these concepts, I will try to show, are more intricate and multi-dimensional than commonly assumed.

Following Henkin's discovery of partially-ordered (branching) quantification (POQ) with standard quantifiers in 1959, philosophers of language have attempted to extend his definition to POQ with generalized quantifiers. In this paper I propose a general definition of POQ with 1-place generalized quantifiers of the simplest kind: namely, predicative, or "cardinality" quantifiers, e.g., "most", "few", "finitely many", "exactly α", where α is any cardinal, etc. The definition is obtained in a series of generalizations, extending the original, Henkin definition first to a general (...) definition of monotone-increasing (M↑) POQ and then to a general definition of generalized POQ, regardless of monotonicity. The extension is based on (i) Barwise's 1979 analysis of the basic case of M↑ POQ and (ii) my 1990 analysis of the basic case of generalized POQ. POQ is a non-compositional Ist-order structure, hence the problem of extending the definition of the basic case to a general definition is not trivial. The paper concludes with a sample of applications to natural and mathematical languages. (shrink)

The paper offers a new analysis of the difficulties involved in the construction of a general and substantive correspondence theory of truth and delineates a solution to these difficulties in the form of a new methodology. The central argument is inspired by Kant, and the proposed methodology is explained and justified both in general philosophical terms and by reference to a particular variant of Tarski's theory. The paper begins with general considerations on truth and correspondence and concludes with a brief (...) outlook on the "family" of theories of truth generated by the new methodology. (shrink)

Sandu and Pietarinen [Partiality and Games: Propositional Logic. Logic J. IGPL 9 (2001) 101] study independence friendly propositional logics. That is, traditional propositional logic extended by means of syntax that allow connectives to be independent of each other, although the one may be subordinate to the other. Sandu and Pietarinen observe that the IF propositional logics have exotic properties, like functional completeness for three-valued functions. In this paper we focus on one of their IF propositional logics and study its properties, (...) by means of notions from computational complexity. This approach enables us to compare propositional logic before and after the IF make-over. We observe that all but one of the best-known decision problems experience a complexity jump, provided that the complexity classes at hand are not equal. Our results concern every discipline that incorporates some notion of independence such as computer science, natural language semantics, and game theory. A corollary of one of our theorems illustrates this claim with respect to the latter discipline. (shrink)

We study the expressive power of independence-friendly quantifier prefixes composed of universal$\left$, existential$\left$, and majority quantifiers$\left$. We provide four quantifier prefixes that can express NP hard properties and show that all quantifier prefixes capable of expressing NP-hard properties embed at least one of these four quantifier prefixes. As for the quantifier prefixes that do not embed any of these four quantifier prefixes, we show that they are equivalent to a first-order quantifier prefix composed of$\forall x$,$\exists x$, and Mx. In unison, (...) our results imply a dichotomy result: every independence-friendly quantifier prefix is either decidable in LOGSPACE or NP hard. (shrink)

Since Fodor 1970, negation has worn a Homogeneity Condition to the effect that homogeneous predicates, ) denote homogeneously—all or nothing —to characterize the meaning of – when uttered out-of-the blue, in contrast to –:The mirrors are smooth. The mirrors are not smooth. The mirrors circle the telescope’s reflector. The mirrors do not circle the telescope’s reflector. It has been a problem for philosophical logic and for the semantics of natural language that – appear to defy the Principle of Excluded Middle (...) while – do not—Smooth ¬Smooth Circle ¬Circle. An impoverished logical form – has been the occasion to embellish all else—Boolean algebra, lexical presuppositions, Strongest Meaning Hypothesis, trivalence, supervaluation, double strengthening, etc., enriching the semantics and pragmatics with what remains a special theory of negation, which may be dismissed when the logical syntax and semantics of negation reflects that negated sentences are also tensed sentences. (shrink)

I introduce a formal language called the language of informational independence (IL-language, for short) that extends an ordinary first-order language in a natural way. This language is interpreted in terms of semantical games of imperfect information. In this language, one can define two negations: (i) strong or dual negation, and (ii) weak or contradictory negation. The latter negation, unlike the former, can occur only sentence-initially. Then I argue that, to a certain extent, the two negations match the distinction existing in (...) natural languages between sentential and constituent negation. As a corollary, I derive the fact that there are no mechanical rules for forming the contradictory negation of an English sentence. (shrink)

We shall introduce in this paper a language whose formulas will be interpreted by games of imperfect information. Such games will be defined in the same way as the games for first-order formulas except that the players do not have complete information of the earlier course of the game. Some simple logical properties of these games will be stated together with the relation of such games of imperfect information to higher-order logic. Finally, a set of applications will be outlined.

In this paper I shall attempt to compare the dialogue approach originally advocated by Lorenz and Lorenzen and the game-theoretical approach of Hintikka with each other. I shall not try to present any survey of either one of the approaches and will assume that the reader is familiar with the basic ideas of these theories.The original works of Lorenzen and Lorenz have been reprinted in their while Stegmüller contains a survey (in English) of the basic ideas and results. Works in (...) this tradition which apply the approach to quantum logic include Denecke, Mittelstaedt Mittelstaedt and Stachow, Stachow. The reader will also find van Dunn, Krabbe, Lorenz and Stachow to be relevant.The literature on Hintikka’s game-theoretical semantics as applied both to different formal languages (including infinitary languages and intensional logic) and to natural language is rapidly growing. As for the latter, the reader is referred to papers by Hintikka, Carlson and myself reprinted in Saarinen, to Carlson and ter Meulen, and to Saarinen. (shrink)

Several authors proposed to devise logical structures for Natural Language (NL) semantics in which noun phrases yield referential terms rather than standard Generalized Quantifiers. In this view, two main problems arise: the need to refer to the maximal sets of entities involved in the predications and the need to cope with Independent Set (IS) readings, where two or more sets of entities are introduced in parallel. The article illustrates these problems and their consequences, then presents an extension of the proposal (...) made in Sher (J Philos Logic 26:1–43, 1997 ) in order to properly represent the meaning of IS readings involving NL quantifiers. The solution proposed here allows to uniformly deal with both standard linear and IS readings, regardless of their actual existence in NL or quantifiers’ monotonicity. Sentences featuring nested quantifications are particularly problematic. By avoiding parallel nested quantification in the formulae, the proper true values are achieved. (shrink)

The problem of intentional identity, as originally offered by Peter Geach, says that there can be an anaphoric link between an indefinite term and a pronoun across a sentential boundary and across propositional attitude contexts, where the actual existence of an individual for the indefinite term is not presupposed. In this paper, a semantic resolution to this elusive puzzle is suggested, based on a new quantified intensional logic and game-theoretic semantics (GTS) of imperfect information. This constellation leads to an expressive (...) intensional language with a property of informational independence, argued to produce a purely semantic explication to intentional identity statements. One consequence is that various extra-logical and pragmatic factors become of secondary concern; it is possible to solve the puzzle by logico-semantic methods, albeit somewhat radically renewed ones. (shrink)

Recent work on the acquisition of number words has emphasized the importance of integrating linguistic and developmental perspectives [Musolino, J. (2004). The semantics and acquisition of number words: Integrating linguistic and developmental perspectives. Cognition93, 1-41; Papafragou, A., Musolino, J. (2003). Scalar implicatures: Scalar implicatures: Experiments at the semantics-pragmatics interface. Cognition, 86, 253-282; Hurewitz, F., Papafragou, A., Gleitman, L., Gelman, R. (2006). Asymmetries in the acquisition of numbers and quantifiers. Language Learning and Development, 2, 76-97; Huang, Y. T., Snedeker, J., Spelke, (...) L. (submitted for publication). What exactly do numbers mean?]. Specifically, these studies have shown that data from experimental investigations of child language can be used to illuminate core theoretical issues in the semantic and pragmatic analysis of number terms. In this article, I extend this approach to the logico-syntactic properties of number words, focusing on the way numerals interact with each other (e.g. Three boys are holding two balloons) as well as with other quantified expressions (e.g. Three boys are holding each balloon). On the basis of their intuitions, linguists have claimed that such sentences give rise to at least four different interpretations, reflecting the complexity of the linguistic structure and syntactic operations involved. Using psycholinguistic experimentation with preschoolers (n=32) and adult speakers of English (n=32), I show that (a) for adults, the intuitions of linguists can be verified experimentally, (b) by the age of 5, children have knowledge of the core aspects of the logical syntax of number words, (c) in spite of this knowledge, children nevertheless differ from adults in systematic ways, (d) the differences observed between children and adults can be accounted for on the basis of an independently motivated, linguistically-based processing model [Geurts, B. (2003). Quantifying kids. Language Acquisition, 11(4), 197-218]. In doing so, this work ties together research on the acquisition of the number vocabulary with a growing body of work on the development of quantification and sentence processing abilities in young children [Geurts, 2003; Lidz, J., Musolino, J. (2002). Children's command of quantification. Cognition, 84, 113-154; Musolino, J., Lidz, J. (2003). The scope of isomorphism: Turning adults into children. Language Acquisition, 11(4), 277-291; Trueswell, J., Sekerina, I., Hilland, N., Logrip, M. (1999). The kindergarten-path effect: Studying on-line sentence processing in young children. Cognition, 73, 89-134; Noveck, I. (2001). When children are more logical than adults: Experimental investigations of scalar implicature. Cognition, 78, 165-188; Noveck, I., Guelminger, R., Georgieff, N., & Labruyere, N. (2007). What autism can tell us about every. . . not sentences. Journal of Semantics,24(1), 73-90. On a more general level, this work confirms the importance of integrating formal and developmental perspectives [Musolino, 2004], this time by highlighting the explanatory power of linguistically-based models of language acquisition and by showing that the complex structure postulated by linguists has important implications for developmental accounts of the number vocabulary. (shrink)

We consider the notion of everyday language. We claim that everyday language is semantically bounded by the properties expressible in the existential fragment of second–order logic. Two arguments for this thesis are formulated. Firstly, we show that so–called Barwise's test of negation normality works properly only when assuming our main thesis. Secondly, we discuss the argument from practical computability for finite universes. Everyday language sentences are directly or indirectly verifiable. We show that in both cases they are bounded by second–order (...) existential properties. Moreover, there are known examples of everyday language sentences which are the most difficult in this class (NPTIME–complete). (shrink)

In this paper, we investigate (1) what can be salvaged from the original project of "logicism" and (2) what is the best that can be done if we lower our sights a bit. Logicism is the view that "mathematics is reducible to logic alone", and there are a variety of reasons why it was a non-starter. We consider the various ways of weakening this claim so as to produce a "neologicism". Three ways are discussed: (1) expand the conception of logic (...) used in the reduction, (2) allow the addition of analytic-sounding principles to logic so that the reduction is not to "logic alone" but to logic and truths knowable a priori, and (3) revise the conception of "reducible". We show how the current versions of neologicism fit into this classification scheme, and then focus on a kind of neologicism which we take to have the most potential for achieving the epistemological goals of the original logicist project. We argue that that the "weaker" the form of neologicism, the more likely it is to be a new form of logicism, and show how our preferred system, though mathematically weak, is metaphysically and epistemogically strong, and can "reduce" arbitrary mathematical theories to logic and analytic truths, if given a legitimate new sense of "reduction". (shrink)

Jaakko Hintikka points out the power of Skolem functions to affect both what there is and what we know. There is a tension in his presupposition that these functions actually extend the realm of logic. He claims to have resolved the tension by “reconstructing constructivism” along epistemological lines, instead of by a typical ontological construction; however, after the collapse of the distinction between first and second order, that resolution is not entirely satisfactory. Still, it does throw light on the conceptual (...) analysis Hintikka proposes. (shrink)

The essays collected in this volume are all concerned with the connection between fiction and truth. This question is of utmost importance to metaphysics, philosophy of language, philosophical logic and epistemology, raising in each of these areas and at their intersections a large number of issues related to creation, existence, reference, identity, modality, belief, assertion, imagination, pretense, etc. All these topics and many more are addressed in this collection, which brings together original essays written from various points of view by (...) philosophers of diverse trends. These essays constitute major contributions to the current debates that the connection between truth and fiction continually enlivens, and give a sense of the directions in which research on this question is heading. Contributors: Fred Adams, Frederick Kroon, Robert Howell, Brendan Murday, Terence Parsons, Graham Priest, Erich Rast, Manuel Rebuschi, Marion Renauld, R.M. Sainsbury, Grant Tavinor, Alberto Voltolini. (shrink)

We study the computational complexity of the model checking problem for quantifier-free dependence logic ${(\mathcal{D})}$ formulas. We characterize three thresholds in the complexity: logarithmic space (LOGSPACE), non-deterministic logarithmic space (NL) and non-deterministic polynomial time (NP).

Quantifiers in Language and Logic (QLL) is a major contribution to natural language semantics, specifically to quantification. It integrates the extensive recent work on quantifiers in logic and linguistics. It also presents new observations and results. QLL should help linguists understand the mathematical generalizations we can make about natural language quantification, and it should interest logicians by presenting an extensive array of quantifiers that lie beyond the pale of classical logic. Here we focus on those aspects of QLL we judge (...) to be of specific interest to linguists, and we contribute a few musings of our own, as one mark of a worthy publication is whether it stimulates the reader to seek out new observations, and QLL does. QLL is long and fairly dense, so we make no attempt to cover all the points it makes. But QLL has a topic index, a special symbols index and two tables of contents, a detailed one and an overview one, all of which help make it user friendly. QLL is presented in four parts: I, The Logical Conception of Quantifiers and Quantification with an introductory section Quantification . II, Quantifiers of Natural Language , the most extensive section in the book and of the most direct interest to linguists. III, Beginnings of a Theory of Expressiveness, Translation, and Formalization introduces notions of expressive power and definability, and IV, presents recent work and techniques concerning quantifier definability over finite domains, making accessible to linguists recent work in finite model theory. (shrink)

In sentences like Every teacher laughed we think of every teacher as a unary (=type (1)) quantifier - it expresses a property of one place predicate denotations. In variable binding terms, unary quantifiers bind one variable. Two applications of unary quantifiers, as in the interpretation of No student likes every teacher, determine a binary (= type (2)) quantifier; they express properties of two place predicate denotations. In variable binding terms they bind two variables. We call a binary quantifier Fregean (or (...) reducible) if it can in principle be expressed by the iterated application of unary quantifiers. In this paper we present two mathematical properties which distinguish non-Fregean quantifiers from Fregean ones. Our results extend those of van Benthem (1989) and Keenan (1987a). We use them to show that English presents a large variety of non-Fregean quantifi ers. Some are new here, others are familiar (though the proofs that they are non-Fregean are not). The main point of our empirical work is to inform us regarding the types of quantification natural language presents - in particular (van Benthem, 1989) that it goes beyond the usual (Fregean) analysis which treats it as mere iterated application of unary quantifiers. Secondarily, our results challenge linguistic approaches to "Logical Form" which constrain variable binding operators to "locally" bind just one occurrence of a variable, e.g., the Bijection Principle (BP) of Koopman and Sportiche (1983). The BP (correctly) blocks analyses like For which x, x's mother kissed x? for Who did his mother kiss? since For which x would locally bind two occurrences of x. But some of our irreducible binary quantifiers are naturally represented by operators which do locally bind two variables. This paper is organized as follows: Section 1 provides an explicit formulation of our questions of concern. Section 2 classifies the English constructions which we show to be non-Fregean. Section 3 presents the mathematical properties which test for non-Fregean quantification and applies these tests to the constructions in Section 2. Proofs of the mathema tical properties are given in the Appendix. (shrink)

In this paper, we study natural language constructions which were first examined by Barwise: The richer the country, the more powerful some of its officials. Guided by Barwise’s observations, we suggest that conceivable interpretations of such constructions express the existence of various similarities between partial orders such as homomorphism or embedding. Semantically, we interpret the constructions as polyadic generalized quantifiers restricted to finite models. We extend the results obtained by Barwise by showing that similarity quantifiers are not expressible in elementary (...) logic over finite models. We also investigate whether the proposed readings are sound from the cognitive perspective. We prove that almost all similarity quantifiers are intractable. This leads us to first-order variants, which only approximate the strong readings, but are cognitively more plausible. Driven by the question of ambiguity, we recall Barwise’s argumentation in favour of strong readings, enriching it with some arguments of our own. Given that Barwise-like sentences are indeed ambiguous, we use a generalized Strong Meaning Hypothesis to derive predictions for their verification. Finally, we propose a hypothesis according to which conflicting pressures of communication and cognition might give rise to an ambiguous construction, provided that different semantic variants of the construction withstand different pressures involved in its usage. (shrink)

Let us assume that you are entrusted by UNESCO with an important task. You are asked to devise a universal logical language, a Begriffsschrift in Frege's sense, which is to serve the purposes of science, business and everyday life. What requirements should such a “conceptual notation” satisfy? There are undoubtedly many relevant desiderata, but here I am focusing on one unmistakable one. In order to be a viable lingua universalis, your language must in any case be capable of representing any (...) possible configuration of dependence and independence between different variables. For if such a configuration is possible in principle, there is no guarantee that it might not one day show up among the natural, human or social phenomena we have to study.But how are dependencies and independencies between variables expressed in our familiar logical notation? Every logician worth his or her truth-table knows the answer. Dependencies between two variables are expressed by dependencies between the quantifiers to which they are bound. For instance, inthe variable y depends on x, while inz depends on x but not on y, while u depends on both x and y.But how is the dependence of a quantifier on another one expressed in familiar logical languages? Obviously by occurring in its scope, indicated by the pair of parentheses following it. But the nesting of scopes is a transitive and antisymmetrical relation which allows branching only in one direction. Hence other kinds of structures of dependence and independence between variables are not representable in the received logical notation. Such previously inexpressible structures form the subject matter of what has been referred to as independence-friendly logic. (shrink)

Let us assume that you are entrusted by UNESCO with an important task. You are asked to devise a universal logical language, a Begriffsschrift in Frege's sense, which is to serve the purposes of science, business and everyday life. What requirements should such a “conceptual notation” satisfy? There are undoubtedly many relevant desiderata, but here I am focusing on one unmistakable one. In order to be a viable lingua universalis, your language must in any case be capable of representing any (...) possible configuration of dependence and independence between different variables. For if such a configuration is possible in principle, there is no guarantee that it might not one day show up among the natural, human or social phenomena we have to study.But how are dependencies and independencies between variables expressed in our familiar logical notation? Every logician worth his or her truth-table knows the answer. Dependencies between two variables are expressed by dependencies between the quantifiers to which they are bound. For instance, inthe variable y depends on x, while inz depends on x but not on y, while u depends on both x and y.But how is the dependence of a quantifier on another one expressed in familiar logical languages? Obviously by occurring in its scope, indicated by the pair of parentheses following it. But the nesting of scopes is a transitive and antisymmetrical relation which allows branching only in one direction. Hence other kinds of structures of dependence and independence between variables are not representable in the received logical notation. Such previously inexpressible structures form the subject matter of what has been referred to as independence-friendly logic. (shrink)

Motivated by constraint satisfaction problems, Feder and Vardi (SIAM Journal of Computing, 28, 57–104, 1998) set out to search for fragments of satisfying the dichotomy property: every problem definable in is either in P or else NP-complete. Feder and Vardi considered in this connection two logics, strict NP (or SNP) and monadic, monotone, strict NP without inequalities (or MMSNP). The former consists of formulas of the form , where is a quantifier-free formula in a relational vocabulary; and the latter is (...) the fragment of SNP whose formulas involve only negative occurrences of relation symbols, only monadic second-order quantifiers, and no occurrences of the equality symbol. It remains an open problem whether MMSNP enjoys the dichotomy property. In the present paper, SNP and MMSNP are characterized in terms of partially ordered connectives. More specifically, SNP is characterized using the logic D of partially ordered connectives introduced in Blass and Gurevich (Annals of Pure and Applied Logic, 32, 1–16, 1986), Sandu and Väänänen (Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 38, 361–372 1992), and MMSNP employing a generalization C of D introduced in the present paper. (shrink)

We study generalized quantifiers on finite structures.With every function : we associate a quantifier Q by letting Q x say there are at least (n) elementsx satisfying , where n is the sizeof the universe. This is the general form ofwhat is known as a monotone quantifier of type .We study so called polyadic liftsof such quantifiers. The particular lifts we considerare Ramseyfication, branching and resumption.In each case we get exact criteria fordefinability of the lift in terms of simpler quantifiers.

Adding branching quantification to a first-order language increases the expressive power of the language,without adding to its ontology. The present paper is a defense of this claim against Quine (1970) and Patton (1991).

Next SectionWe discuss the thesis formulated by Hintikka (1973) that certain natural language sentences require non-linear quantification to express their meaning. We investigate sentences with combinations of quantifiers similar to Hintikka's examples and propose a novel alternative reading expressible by linear formulae. This interpretation is based on linguistic and logical observations. We report on our experiments showing that people tend to interpret sentences similar to Hintikka sentence in a way consistent with our interpretation.

We introduce generalized quantifiers, as defined in Tarskian semantics by Mostowski and Lindström, in logics whose semantics is based on teams instead of assignments, e.g., IF-logic and Dependence logic. Both the monotone and the non-monotone case is considered. It is argued that to handle quantifier scope dependencies of generalized quantifiers in a satisfying way the dependence atom in Dependence logic is not well suited and that the multivalued dependence atom is a better choice. This atom is in fact definably equivalent (...) to the independence atom recently introduced by Väänänen and Grädel. (shrink)

Nathan Salmon, in his paper Trans-World Identification and Stipulation (1996) purports to give a proof for the claim that facts concerning trans-world identity cannot be conceptually reduced to general facts. He calls this claim ‘Extreme Haecceitism.’ I argue that his proof is fallacious. However, I also contend that the analysis and ultimate rejection of his proof clarifies the fundamental issues that are at stake in the debate between the reductionist and haecceitist solutions to the problem of trans-world identity. These issues (...) hinge on the ability of modal logic and possible worlds semantics to draw a hard and fast distinction between the logic and the metaphysics of modal logic. I shall claim that the considerations in this paper call into question the viability of such a distinction. (shrink)

§1. Introduction. After the pioneering work of Mostowski [29] and Lindström [23] it was Jon Barwise's papers [2] and [3] that brought abstract model theory and generalized quantifiers to the attention of logicians in the early seventies. These papers were greeted with enthusiasm at the prospect that model theory could be developed by introducing a multitude of extensions of first order logic, and by proving abstract results about relationships holding between properties of these logics. Examples of such properties areκ-compactness.Any set (...) of sentences of cardinality ≤ κ, every finite subset of which has a model, has itself a model.Löwenheim-Skolem Theorem down toκ.If a sentence of the logic has a model, it has a model of cardinality at most κ.Interpolation Property.If ϕ and ψ are sentences such that ⊨ ϕ → Ψ, then there is θ such that ⊨ ϕ → θ, ⊨ θ → Ψ and the vocabulary of θ is the intersection of the vocabularies of ϕ and Ψ.Lindstrom's famous theorem characterized first order logic as the maximal ℵ0-compact logic with Downward Löwenheim-Skolem Theorem down to ℵ0. With his new concept of absolute logics Barwise was able to get similar characterizations of infinitary languagesLκω. But hopes were quickly frustrated by difficulties arising left and right, and other areas of model theory came into focus, mainly stability theory. No new characterizations of logics comparable to the early characterization of first order logic given by Lindström and of infinitary logic by Barwise emerged. What was first called soft model theory turned out to be as hard as hard model theory. (shrink)

We describe a knowledge representation and inference formalism, based on an intensional propositional semantic network, in which variables are structures terms consisting of quantifier, type, and other information. This has three important consequences for natural language processing. First, this leads to an extended, more natural formalism whose use and representations are consistent with the use of variables in natural language in two ways: the structure of representations mirrors the structure of the language and allows re-use phenomena such as pronouns and (...) ellipsis. Second, the formalism allows the specification of description subsumption as a partial ordering on related concepts (variable nodes in a semantic network) that relates more general concepts to more specific instances of that concept, as is done in language. Finally, this structured variable representation simplifies the resolution of some representational difficulties with certain classes of natural language sentences, namely, donkey sentences and sentences involving branching quantifiers. The implementation of this formalism is called ANALOG (A NAtural LOGIC) and its utility for natural language processing tasks is illustrated. (shrink)

We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of (...) structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation. (shrink)

The Bounds of Logic presents a new philosophical theory of the scope and nature of logic based on critical analysis of the principles underlying modern Tarskian logic and inspired by mathematical and linguistic development. Extracting central philosophical ideas from Tarski’s early work in semantics, Sher questions whether these are fully realized by the standard first-order system. The answer lays the foundation for a new, broader conception of logic. By generally characterizing logical terms, Sher establishes a fundamental result in semantics. Her (...) development of the notion of logicality for quantifiers and her work on branching are of great importance for linguistics. Sher outlines the boundaries of the new logic and points out some of the philosophical ramifications of the new view of logic for such issues as the logicist thesis, ontological commitment, the role of mathematics in logic, and the metaphysical underpinning of logic. She proposes a constructive definition of logical terms, reexamines and extends the notion of branching quantification, and discusses various linguistic issues and applications. (shrink)

Online collection of papers by Devitt, Dretske, Guarino, Hochberg, Jackson, Petitot, Searle, Tye, Varzi and other leading thinkers on philosophy and the foundations of cognitive Science. Topics dealt with include: Wittgenstein and Cognitive Science, Content and Object, Logic and Foundations, Language and Linguistics, and Ontology and Mereology.