We consider connections between number sense—the ability to judge number—and the interpretation of natural language quantifiers. In particular, we present empirical evidence concerning the neuroanatomical underpinnings of number sense and quantifier interpretation. We show, further, that impairment of number sense in patients can result in the impairment of the ability to interpret sentences containing quantifiers. This result demonstrates that number sense supports some aspects of the language faculty.

This paper establishes model-theoretic properties of \, a variation of monadic first-order logic that features the generalised quantifier \. We will also prove analogous versions of these results in the simpler setting of monadic first-order logic with and without equality and \, respectively). For each logic \ we will show the following. We provide syntactically defined fragments of \ characterising four different semantic properties of \-sentences: being monotone and continuous in a given set of monadic predicates; having truth preserved under (...) taking submodels or being truth invariant under taking quotients. In each case, we produce an effectively defined map that translates an arbitrary sentence \ to a sentence \ belonging to the corresponding syntactic fragment, with the property that \ is equivalent to \ precisely when it has the associated semantic property. As a corollary of our developments, we obtain that the four semantic properties above are decidable for \-sentences. (shrink)

The standard relation of logical consequence allows for non-standard interpretations of logical constants, as was shown early on by Carnap. But then how can we learn the interpretations of logical constants, if not from the rules which govern their use? Answers in the literature have mostly consisted in devising clever rule formats going beyond the familiar what follows from what. A more conservative answer is possible. We may be able to learn the correct interpretations from the standard rules, because the (...) space of possible interpretations is a priori restricted by universal semantic principles. We show that this is indeed the case. The principles are familiar from modern formal semantics: compositionality, supplemented, for quantifiers, with topic-neutrality. (shrink)

We try to explain Tarski's conception of logical notions, as it emerges from alecture of his, delivered in 1966 and published posthumously in 1986 (Historyand Philosophy of Logic 7, 143–154), a conception based on the idea ofinvariance. The evaluation of Tarski's proposal leads us to consider an interesting(and neglected) reply to Skolem in which Tarski hints at his own point of view onthe foundations of set theory. Then, comparing the lecture of 1966 with Tarski'slast work and with an earlier paper (...) written with Lindenbaum, it is shown thatTarski's conception of logical notions, with its essentially type-theoretic character,did not undergo any significant modifications throughout his life. A remark onTarski's prudential attitude on the topic in the famous paper on the concept oflogical consequence (and elsewhere) concludes our paper. (shrink)

This paper investigates the “general” semantics for first-order logic introduced to Antonelli, 637–58, 2013): a sound and complete axiom system is given, and the satisfiability problem for the general semantics is reduced to the satisfiability of formulas in the Guarded Fragment of Andréka et al. :217–274, 1998), thereby showing the former decidable. A truth-tree method is presented in the Appendix.

Although the invariance criterion of logicality first emerged as a criterion of a purely mathematical interest, it has developed into a criterion of considerable linguistic and philosophical interest. In this paper I compare two different perspectives on this criterion. The first is the perspective of natural language. Here, the invariance criterion is measured by its success in capturing our linguistic intuitions about logicality and explaining our logical behavior in natural-linguistic settings. The second perspective is more theoretical. Here, the invariance criterion (...) is used as a tool for developing a theoretical foundation of logic, focused on a critical examination, explanation, and justification of its veridicality and modal force. (shrink)

The Introduction outlines, in a concise way, the history of the Lvov-Warsaw School – a most unique Polish school of worldwide renown, which pioneered trends combining philosophy, logic, mathematics and language. The author accepts that the beginnings of the School fall on the year 1895, when its founder Kazimierz Twardowski, a disciple of Franz Brentano, came to Lvov on his mission to organize a scientific circle. Soon, among the characteristic features of the School was its serious approach towards philosophical studies (...) and teaching of philosophy, dealing with philosophy and propagation of it as an intellectual and moral mission, passion for clarity and precision, as well as exchange of thoughts, and cooperation with representatives of other disciplines.The genesis is followed by a chronological presentation of the development of the School in the successive years. The author mentions all the key representatives of the School (among others, Ajdukiewicz, Lesniewski, Łukasiewicz,Tarski), accompanying the names with short descriptions of their achievements. The development of the School after Poland’s regaining independence in 1918 meant part of the members moving from Lvov to Warsaw, thus providing the other segment to the name – Warsaw School of Logic. The author dwells longer on the activity of the School during the Interwar period – the time of its greatest prosperity, which ended along with the outbreak of World War 2. Attempts made after the War to recreate the spirit of the School are also outlined and the names of followers are listed accordingly. The presentation ends with some concluding remarks on the contribution of the School to contemporary developments in the fields of philosophy, mathematical logic or computer science in Poland. (shrink)

This collection of papers, celebrating the contributions of Swedish logician Dag Prawitz to Proof Theory, has been assembled from those presented at the Natural Deduction conference organized in Rio de Janeiro to honour his seminal research. Dag Prawitz’s work forms the basis of intuitionistic type theory and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics in Logic, Linguistics and Theoretical Computer Science. The range of contributions includes material on the extension of natural deduction with higher-order (...) rules, as opposed to higher-order connectives, and a paper discussing the application of natural deduction rules to dealing with equality in predicate calculus. The volume continues with a key chapter summarizing work on the extension of the Curry-Howard isomorphism, via methods of category theory that have been successfully applied to linear logic, as well as many other contributions from highly regarded authorities. With an illustrious group of contributors addressing a wealth of topics and applications, this volume is a valuable addition to the libraries of academics in the multiple disciplines whose development has been given added scope by the methodologies supplied by natural deduction. The volume is representative of the rich and varied directions that Prawitz work has inspired in the area of natural deduction. (shrink)

Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) como questões acerca das conexões destes com a realidade não mental e extralinguística. A razão daquela qualificação é a seguinte: por um lado, a investigação em questão é qualificada como filosófica em virtude do elevado grau de generalidade e abstracção das questões examinadas (entre outras coisas); por outro, a investigação é qualificada como lógica em virtude de ser uma investigação logicamente disciplinada, no sentido de nela se fazer um uso intenso de conceitos, técnicas e métodos provenientes da disciplina de lógica. O agregado de tópicos que constitui a área de estudos lógico-filosóficos é já visível, pelo menos em parte, no Tractatus Logico-Philosophicus de Ludwig Wittgenstein, uma obra publicada em 1921. E uma boa maneira de ter uma ideia sinóptica do território disciplinar abrangido por esta enciclopédia, ou pelo menos de uma porção substancial dele, é extrair do Tractatus uma lista dos tópicos mais salientes aí discutidos; a lista incluirá certamente tópicos do seguinte género, muitos dos quais se podem encontrar ao longo desta enciclopédia: factos e estados de coisas; objectos; representação; crenças e estados mentais; pensamentos; a proposição; nomes próprios; valores de verdade e bivalência; quantificação; funções de verdade; verdade lógica; identidade; tautologia; o raciocínio matemático; a natureza da inferência; o cepticismo e o solipsismo; a indução; as constantes lógicas; a negação; a forma lógica; as leis da ciência; o número. (shrink)

The article studies two related issues. First, it introduces the notion of the contraposition of quantifiers which is a “dual” notion of symmetry and has similar relations to co-intersectivity as symmetry has to intersectivity. Second, it shows how symmetry and contraposition can be generalised to higher order type quantifiers, while preserving their relations with other notions from generalized quantifiers theory.

The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying “natural,” robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the range of such logics is small.

In this paper (except in Section 5) all quantifiers are assumedto be so called simple unaryquantifiers, and all models are assumedto be finite. We give a necessary and sufficientcondition for a quantifier to be definablein terms of monotone quantifiers. For amonotone quantifier we give a necessaryand sufficient condition for beingdefinable in terms of a given set of bounded monotonequantifiers. Finally, we give a necessaryand sufficient condition for a monotonequantifier to be definable in terms of agiven monotone quantifier.Our analysis shows that (...) the quantifierat least one half and its relatives behavedifferently than other monotone quantifiers. (shrink)

One logic or many? I say—many. Or rather, I say there is one logic for each way of specifying the class of all possible circumstances, or models, i.e., all ways of interpreting a given language. But because there is no unique way of doing this, I say there is no unique logic except in a relative sense. Indeed, given any two competing logical theories T1 and T2 (in the same language) one could always consider their common core, T, and settle (...) on that theory. So, given any language L, one could settle on the minimal logic T0 corresponding to the common core shared by all competitors. That would be a way of resisting relativism, as long as one is willing to redraw the bounds of logic accordingly. However, such a minimal theory T0 may be empty if the syntax of L contains no special ingredients the interpretation of which is independent of the specification of the relevant L-models. And generally—I argue—this is indeed the case. (shrink)

We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multi-quantifier sentences. First, we show that the standard constructions that turn simple determiners into complex quantifiers, namely Boolean operations, iteration, cumulation, and resumption, are tractable. Then, we provide an insight into branching operation yielding intractable natural language multi-quantifier expressions. Next, we focus on a linguistic case study. We use computational complexity results to investigate semantic (...) distinctions between quantified reciprocal sentences. We show a computational dichotomy<br>between different readings of reciprocity. Finally, we go more into philosophical speculation on meaning, ambiguity and computational complexity. In particular, we investigate a possibility to<br>revise the Strong Meaning Hypothesis with complexity aspects to better account for meaning shifts in the domain of multi-quantifier sentences. The paper not only contributes to the field of the formal<br>semantics but also illustrates how the tools of computational complexity theory might be successfully used in linguistics and philosophy with an eye towards cognitive science. (shrink)

A special kind of substitution on languages called iteration is presented and studied. These languages arise in the application of semantic automata to iterations of generalized quantifiers. We show that each of the star-free, regular, and deterministic context-free languages are closed under iteration and that it is decidable whether a given regular or determinstic context-free language is an iteration of two such languages. This result can be read as saying that the van Benthem/Keenan ‘Frege Boundary’ is decidable for large subclasses (...) of natural language quantifiers. We also determine the state complexity of iteration of regular languages. (shrink)

The semantic automata framework, developed originally in the 1980s, provides computational interpretations of generalized quantifiers. While recent experimental results have associated structural features of these automata with neuroanatomical demands in processing sentences with quantifiers, the theoretical framework has remained largely unexplored. In this paper, after presenting some classic results on semantic automata in a modern style, we present the first application of semantic automata to polyadic quantification, exhibiting automata for iterated quantifiers. We also discuss the role of semantic automata in (...) linguistic theory and offer new empirical predictions for sentence processing with embedded quantifiers. (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.

In a recent paper, “The Concept of Logical Consequence,” W. H. Hanson criticizes a formal-structural characterization of logical consequence in Tarski and Sher. Hanson accepts many principles of the formal-structural view. Relating to Sher 1991 and 1996a, he says.

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)

Philosophers are divided on whether the proof- or truth-theoretic approach to logic is more fruitful. The paper demonstrates the considerable explanatory power of a truth-based approach to logic by showing that and how it can provide (i) an explanatory characterization —both semantic and proof-theoretical—of logical inference, (ii) an explanatory criterion for logical constants and operators, (iii) an explanatory account of logic’s role (function) in knowledge, as well as explanations of (iv) the characteristic features of logic —formality, strong modal force, generality, (...) topic neutrality, basicness, and (quasi-)apriority, (v) the veridicality of logic and its applicability to science, (v) the normativity of logic, (vi) error, revision, and expansion in/of logic, and (vii) the relation between logic and mathematics. The high explanatory power of the truth-theoretic approach does not rule out an equal or even higher explanatory power of the proof-theoretic approach. But to the extent that the truth-theoretic approach is shown to be highly explanatory, it sets a standard for other approaches to logic, including the proof-theoretic approach. (shrink)

Many philosophers are baffled by necessity. Humeans, in particular, are deeply disturbed by the idea of necessary laws of nature. In this paper I offer a systematic yet down to earth explanation of necessity and laws in terms of invariance. The type of invariance I employ for this purpose generalizes an invariance used in meta-logic. The main idea is that properties and relations in general have certain degrees of invariance, and some properties/relations have a stronger degree of invariance than others. (...) The degrees of invariance of highly-invariant properties are associated with high degrees of necessity of laws governing/describing these properties, and this explains the necessity of such laws both in logic and in science. This non-mysterious explanation has rich ramifications for both fields, including the formality of logic and mathematics, the apparent conflict between the contingency of science and the necessity of its laws, the difference between logical-mathematical, physical, and biological laws/principles, the abstract character of laws, the applicability of logic and mathematics to science, scientific realism, and logical-mathematical realism. (shrink)

Contrary to received opinion, the Aristotelian Square of Opposition (square) is logically sound, differing from standard modern predicate logic (SMPL) only in that it restricts the universe U of cognitively constructible situations by banning null predicates, making it less unnatural than SMPL. U-restriction strengthens the logic without making it unsound. It also invites a cognitive approach to logic. Humans are endowed with a cognitive predicate logic (CPL), which checks the process of cognitive modelling (world construal) for consistency. The square is (...) considered a first approximation to CPL, with a cognitive set-theoretic semantics. Not being cognitively real, the null set Ø is eliminated from the semantics of CPL. Still rudimentary in Aristotle’s On Interpretation (Int), the square was implicitly completed in his Prior Analytics (PrAn), thereby introducing U-restriction. Abelard’s reconstruction of the logic of Int is logically and historically correct; the loca (Leaking O-Corner Analysis) interpretation of the square, defended by some modern logicians, is logically faulty and historically untenable. Generally, U-restriction, not redefining the universal quantifier, as in Abelard and loca, is the correct path to a reconstruction of CPL. Valuation Space modelling is used to compute the effects of U-restriction. (shrink)

In standard model-theoretic semantics, the meaning of logical terms is said to be fixed in the system while that of nonlogical terms remains variable. Much effort has been devoted to characterizing logical terms, those terms that should be fixed, but little has been said on their role in logical systems: on what fixing their meaning precisely amounts to. My proposal is that when a term is considered logical in model theory, what gets fixed is its intension rather than its extension. (...) I provide a rigorous way of spelling out this idea, and show that it leads to a graded account of logicality: the less structure a term requires in order for its intension to be fixed, the more logical it is. Finally, I focus on the class of terms that are invariant under isomorphisms, as they render themselves more easily to mathematical treatment. I propose a mathematical measure for the logicality of such terms based on their associated Löwenheim numbers. (shrink)

Besides their cardinal and proportional readings, many and few have been argued to allow for a ‘reverse’ proportional reading that defies the conservativity universal. Recently, an analysis has been developed that derives the correct truth conditions for this reading while preserving conservativity. The present paper investigates two predictions of this analysis, based on two key ingredients. First, many is decomposed into a determiner stem many and the degree operator POS. This predicts that other elements may scopally intervene between the two (...) parts. Second, non-reverse and reverse readings arise from the versatile association possibilities of POS, as independently witnessed in the grammar. In languages in which the same versatile association profile is granted to the superlative degree operator -est, reverse readings are expected to arise for most as well. Both predictions are borne out. (shrink)

In this paper we consider the modal logic with both \ and \ arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Gödel algebra \. We provide an axiomatic system extending the one from Caicedo and Rodriguez :37–55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most usual frame restrictions are given (...) too. We also prove that in the studied logic it is not possible to get \ as an abbreviation of \, nor vice-versa, showing that indeed the axiomatic system we present does not coincide with any of the mono-modal fragments previously axiomatized in the literature. (shrink)

This article proposes a new logical framework for NL quantification. The framework is based on Generalized Quantifiers, Skolem-like functional dependencies, and Maximality of the involved sets of entities. Among the readings available for NL sentences, those where two or more sets of entities are independent of one another are particularly challenging. In the literature, examples of those readings are known as Collective and Cumulative readings. This article briefly analyzes previous approaches to Cumulativity and Collectivity, and indicates (Schwarzschild in Pluralities. Kluwer, (...) Dordrecht, 1996) as the best proposal so far to deal with these readings. Then, it incorporates its insights in the logical framework defined in Robaldo (J Philos Log 39(1):23–58, 2009a), leading to a scalable logical account for NL quantification. (shrink)

In this paper I discuss Ernst Zermelo’s ideas concerning the possibility of developing a system of infinitary logic that, in his opinion, should be suitable for mathematical inferences. The presentation of Zermelo’s ideas is accompanied with some remarks concerning the development of infinitary logic. I also stress the fact that the second axiomatization of set theory provided by Zermelo in 1930 involved the use of extremal axioms of a very specific sort.1.

We discuss O&C's probabilistic approach from a probability logical point of view. Specifically, we comment on subjective probability, the indispensability of logic, the Ramsey test, the consequence relation, human nonmonotonic reasoning, intervals, generalized quantifiers, and rational analysis.

Permutation invariance is often presented as the correct criterion for logicality. The basic idea is that one can demarcate the realm of logic by isolating specific entities—logical notions or constants—and that permutation invariance would provide a philosophically motivated and technically sophisticated criterion for what counts as a logical notion. The thesis of permutation invariance as a criterion for logicality has received considerable attention in the literature in recent decades, and much of the debate is developed against the background of ideas (...) put forth by Tarski in a 1966 lecture (Tarski 1966/1986). But as noted by Tarski himself in the lecture, the permutation invariance criterion yields a class of putative ‘logical constants’ that are essentially only sensitive to the number of elements in classes of individuals. Thus, to hold the permutation invariance thesis essentially amounts to limiting the scope of logic to quantificational phenomena, which is controversial at best and possibly simply wrong. In this paper, I argue that permutation invariance is a misguided approach to the nature of logic because it is not an adequate formal explanans for the informal notion of the generality of logic. In particular, I discuss some cases of undergeneration of the criterion, i.e. the fact that it excludes from the realm of logic operators that we have good reason to regard as logical, especially some modal operators. (shrink)

Adding certain cardinality quantifiers into first-order language will give substantially more expressive languages. Thus, many mathematical concepts beyond first-order logic can be handled. Since basic modal logic can be seen as the bisimular invariant fragment of first-order logic on the level of models, it has no ability to handle modally these mathematical concepts beyond first-order logic. By adding modalities regarding the cardinalities of successor states, we can, in principle, investigate modal logics of all cardinalities. Thus ways of exploring model-theoretic logics (...) can be transferred to modal logics. (shrink)

Let us sum up. We began with the question, “What is the interest of a model-theoretic definition of validity?” Model theoretic validity consists in truth under all reinterpretations of non-logical constants. In this paper, we have described for each necessity concept a corresponding modal invariance property. Exemplification of that property by the logical constants of a language leads to an explanation of the necessity, in the corresponding sense, of its valid sentences. I have fixed upon the epistemic modalities in characterizing (...) the logical constants: to be a logical constant in the language of a population is to be invariant over a modality describing complete possible epistemic states of that population (or an idealized analogue thereof). The grounds for this characterization are these: (1) It leads, I believe, to an extensionally reasonable demarcation of the logical constants, including clear cases and excluding clear non-cases. It gives a principled criterion for deciding unclear cases. (2) It provides an analysis of the topic-neutrality of logic. (3) It leads to an explanation of the epistemic necessity of the logical truths in terms of the topic-neutrality of the logical constants.All the same, it is reasonable to ask, even if the suggested demarcation of logic is extensionally correct, whether it can reasonably be expected to be fundamental. The epistemic invariance of an expression is a rather striking property, one which we should want to explain. What is missing, then, is an explanation of the distinguishing epistemic properties of the constants in terms of more fundamental properties involving their understanding and use. It would be these that properly define the nature, not just the extent, of logic. (shrink)

Adding certain cardinality quantifiers into first-order language will give substantially more expressive languages. Thus, many mathematical concepts beyond first-order logic can be handled. Since basic modal logic can be seen as the bisimular invariant fragment of first-order logic on the level of models, it has no ability to handle modally these mathematical concepts beyond first-order logic. By adding modalities regarding the cardinalities of successor states, we can, in principle, investigate modal logics of all cardinalities. Thus ways of exploring model-theoretic logics (...) can be transferred to modal logics. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

Let me start with a well-known story. Kant held that logic and conceptual analysis alone cannot account for our knowledge of arithmetic: “however we might turn and twist our concepts, we could never, by the mere analysis of them, and without the aid of intuition, discover what is the sum [7+5]” (KrV, B16). Frege took himself to have shown that Kant was wrong about this. According to Frege’s logicist thesis, every arithmetical concept can be defined in purely logical terms, and (...) every theorem of arithmetic can be proved using only the basic laws of logic. Hence, Kant was wrong to think that our grasp of arithmetical concepts and our knowledge of arithmetical truth depend on an extralogical source—the pure intuition of time (Frege 1884, §89, §109). Arithmetic, properly understood, is just a part of logic. (shrink)

This paper proposes an Interface Transparency Thesis concerning how linguistic meanings are related to the cognitive systems that are used to evaluate sentences for truth/falsity: a declarative sentence S is semantically associated with a canonical procedure for determining whether S is true; while this procedure need not be used as a verification strategy, competent speakers are biased towards strategies that directly reflect canonical specifications of truth conditions. Evidence in favor of this hypothesis comes from a psycholinguistic experiment examining adult judgments (...) concerning ‘Most of the dots are blue’. This sentence is true if and only if the number of blue dots exceeds the number of nonblue dots. But this leaves unsettled, e.g., how the second cardinality is specified for purposes of understanding and/or verification: via the nonblue things, given a restriction to the dots, as in ‘|{x: Dot(x) & ~Blue(x)}|’; via the blue things, given the same restriction, and subtraction from the number of dots, as in ‘|{x: Dot(x)}| – |{x: Dot(x) & Blue(x)}|’; or in some other way. Psycholinguistic evidence and psychophysical modeling support the second hypothesis. (shrink)

We shall try to defend two non-standard views that run counter to two well-entrenched familiar views. The standard views are the universal and existential quantifiers of first-order logic are not modal operators, and the quantifiers are extensional. If that is correct then the counterclaims create genuine problems for some traditional philosophical doctrines.

The objectives of this paper are twofold. The first is to present a differentiation between two kinds of deferred uses of indexicals: those in which indexical utterances express singular propositions (I term them deferred reference proper) and those where they express general propositions (called descriptive uses of indexicals). The second objective is the analysis of the descriptive uses of indexicals. In contrast to Nunberg, who treats descriptive uses as a special case of deferred reference in which a property contributes to (...) the proposition expressed, I argue that examples in which a general proposition is indeed expressed by an indexical cannot be treated by assuming that the property is a deferred referent of the pronoun. I propose an analysis of descriptive uses of indexicals by means of a pragmatic mechanism of ‘descriptive anaphora’, which attempts to explain the special kind of contribution of the property retrieved from the context to the proposition that is characteristic of the descriptive interpretation. (shrink)

(2012). Reasoning without believing: on the mechanisation of presuppositions and partiality. Journal of Applied Non-Classical Logics: Vol. 22, No. 4, pp. 295-317. doi: 10.1080/11663081.2012.705962.

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)