In this paper I discuss two objections raised against von Fintel’s (1994) and Stanley and Szabó’s (2000a) hidden variable approach to quantifier domain restriction (QDR). One of them concerns utterances of sentences involving quantifiers for which no contextual domain restriction is needed, and the other concerns multiple quantified contexts. I look at various ways in which the approaches could be amended to avoid these problems, and I argue that they fail. I conclude that we need a more flexible account (...) of QDR, one that allows for the hidden variables in the LF responsible for QDR to vary in number. Recanati’s (2002; 2004) approach to QDR, which makes use of the apparatus of “variadic functions”, is flexible enough to account successfully for the two phenomena discussed. I end with a few comments on what I take to be the most promising way to construe variadic functions. (shrink)

We are concerned with showing how ‘labelled’ Natural Deduction presentation systems based on an extension of the so-called Curry-Howard functional interpretation can help us understand and generalise most of the deduction calculi designed to deal with the logical notion of existential quantification. We present the labelling mechanism for ‘’ using what we call ‘ɛ-terms’, which have the form of ‘a’) in a dual form to the ‘Ax.f’ terms of in the sense that the ‘witness’ is chosen at the time of (...) assertion/introduction.With the intention of demonstrating the generality of our framework, we provide a brief comparison with some other calculi of existentials, including the model-theoretic interpretations of Skolem and Herbrand, indicating the way in which some of the classical results on prenex normal forms can be extended to a wide range of logics. The essential ingredient is to look at the conceptual framework of labelled natural deduction as an attempted reconstruction of Frege's functional calculus where the devices of the Begriffsschrift work separately from, yet in harmony with, those of the Grundgesetze 1. (shrink)

* This work has been evolving for a while now. Some parts trace back to the few pages on the context-dependency of quantifiers in my dissertation. Reading Recanati’s paper on domains of discourse made me rethink some of my earlier conclusions without in the end actually changing them much. Other parts formed the material for several discussions in my seminar on context-dependency at MIT in the fall of 1995, which included several sessions exploring the issues raised in an early version (...) of Kratzer. Connecting the choice function approach to indefinites with a general analysis of contextual domain restriction was an idea that I considered then without however working out any details. Conversations with Lisa Matthewson and studying her paper inspired me to tackle the problem again. Connections to questions about the context-dependency of conditional sentences also tickled my fancy, since I have been working on conditionals for a while now. Having the opportunity to present this material at the Vilem Mathesius Series provided the impetus to flesh out some of my vague thoughts. Major influences come from the work done in recent years at MIT by Lisa Matthewson, Philippe Schlenker, Julie Legate, and Uli Sauerland. I thank Irene Heim, Lisa Matthewson, Orin Percus, Philippe Schlenker, and Jason Stanley for illuminating discussions about this material. At this point, I see this talk as a report on the state of the art, exploring an intriguing possible connection, but I do not claim substantial intellectual property rights. Mistakes are certainly mine, achievements are probably not. (shrink)

We introduce new first-order languages for the elementary n-dimensional geometry and elementary n-dimensional affine geometry , based on extending equation image and equation image, respectively, with new function symbols. Here, β stands for the betweenness relation and ≡ for the congruence relation. We show that the associated theories admit effective quantifier elimination.

Some philosophers, notably Professors Quine and Geach, have stressed the analogies they see between pronouns of the vernacular and the bound variables of quantification theory. Geach, indeed, once maintained that ‘for a philosophical theory of reference, then, it is all one whether we consider bound variables or pronouns of the vernacular'. This slightly overstates Geach's positition since he recognizes that some pronouns of ordinary language do function differently from bound variables; he calls such pronouns ‘pronouns of laziness'. Geach's characterisation (...) of pronouns of laziness has varied from time to time, but the general idea should be clear from a paradigm example: A man who sometimes beats his wife has more sense than one who always gives in to her.The pronouns ‘one’ and ‘her’ go proxy for a noun or a noun phrase in the sense that the pronoun is replaceable in paraphrase by simple repetition of its antecedent. (shrink)

A typical approach to semantics for relevance (and other) logics: specify a class of algebraic structures and take amodelto be one of these structures, α, together with some function or relation which associates with every formulaAa subset ofα. (This is the approach of, among others, Urquhart, Routley and Meyer and Fine.) In some cases there are restrictions on the class of subsets of α with which a formula can be associated: for example, in the semantics of Routley and Meyer (...) [1973], a formula can only be associated with subsets which are closed upwards. It is natural to take apropositionof α to be such a subset of α, and, further, to take the propositional quantifiers to range over these propositions. (Routley and Meyer [1973] explicitly consider this interpretation.) Given such an algebraic semantics, we call (following Routley and Meyer [1973], who follow Henkin [1950]) the above-described interpretation of the quantifiers theprimaryinterpretation associated with the semantics. (shrink)

Hintikka’s semantic approach to meaning, a development of Wittgenstein’s view of meaning as use, is the general theme of this chapter. We will focus on the analysis of quantified sentences and on the scope of the principle of compositionality and compare Hintikka’s take on these issues with that of Frege. The aim of this paper is to show that Hintikka’s analysis of quantified expressions as choice functions, in spite of its obvious dissimilarities with respect to the higher-order approach, is actually (...) very close to the Fregean stance on compositionality and context dependence. In particular, we will defend that the Fregean approach to quantifiers is unavoidably linked to the idea that quantified expressions are context-dependent, and therefore should not be conceived under the traditional inside-out model for analysis. (shrink)

There is growing interest in the effects of sports-related repetitive head impacts (RHIs) on athletes’ cognitive capabilities. This study examines the effect of RHIs in data collected from adolescent athletes to estimate the magnitude and longevity of RHIs on sensorimotor and cognitive performance. A non-linear regression model estimated the longevity of RHI effects by adding a half-life parameter embedded in an exponential decay function. A model estimate of this parameter allows the possibility of RHI effects to attenuate over time (...) and introduces a mechanism to study the cumulative effect of RHIs. The posterior distribution of the half-life parameter associated with short-distance headers (<30 m) is centered around 6 days, whereas the posterior distribution of the half-life parameter associated with long-distance headers extends beyond a month. Additionally, the magnitude of the effect of each short header is around 3 times smaller than that of a long header. The results indicate that, on both tasks, response time (RT) changes after long headers are bigger in magnitude and last longer compared to the effects of short headers. Most important, we demonstrate that deleterious effects of long headers extend beyond 1 month. Although estimates are based on data from a relatively short-duration study with a relatively small sample size, the proposed model provides a mechanism to estimate long-term behavioral slowing from RHIs, which may be helpful to reduce the risk of additional injury. Finally, differences in the longevity of the effects of short and long RHIs may help to explain the large variance found between biomechanical input and clinical outcome in studies of concussion tolerance. (shrink)

In “Elusive Knowledge” (1996), David Lewis deduces contextualism about 'knowledge' from an analysis of the nature of knowledge. For Lewis, the context relativity of 'knowledge' depends upon the fact that knowledge that p implies the elimination of all the possibilities in which ~p. But since 'all' is context relative, 'knowledge' is also context relative. In contrast to Lewis, in Knowledge and Practical Interests (2005), Jason Stanley argues that since all context sensitive expressions can have different interpretations within the same discourse, (...) contextualists cannot consistently embrace the following two claims: (i) 'knowledge' functions like a quantifier and (ii) distinct occurrences of 'knowledge' within the same discourse must be associated with the same standard. In response to Stanley, in my paper, I argue that (i) and (ii) are both true. More specifically, I argue that with the help of global domains, we can overcome Stanley’s objections to Lewis and, accordingly, provide the linguistic basis that epistemic contextualism needs. (shrink)

This paper studies the distribution of ‘list readings’ in questions like who does everyone like? vs. who likes everyone?. More generally, it focuses on the interaction between wh-words and quantified NPs. It is argued that, contrary to widespread belief, the pattern of available readings of constituent questions can be explained as a consequence of Weak Crossover, a well-known property of grammar. In particular, list readings are claimed to be a special case of ‘functional readings’, rather than arising from quantifying into (...) questions. Functional readings are argued to be encoded in the syntax as doubly indexed traces, which straightforwardly leads to a Crossover account of the absence of list readings in who likes everyone?. Empirical and theoretical consequences of this idea for the syntax and semantics of questions are considered. (shrink)

A quantified universe is a set M equipped with a Riesz space equation image of real functions on Mn, for each n, and a second order operation equation image. Metric structures 4, graded probability structures 9 and many other structures in analysis are examples of such universes. We define ultraproduct of quantified universes and study properties preserved by this construction. We then discuss logics defined on the basis of classes of quantified universes which are closed under this construction.

Generalized quantifiers are functions from pairs of properties to truth-values; these functions can be used to interpret natural language quantifiers. The space of such functions is vast and a great deal of research has sought to find natural constraints on the functions that interpret determiners and create quantifiers. These constraints have demonstrated that quantifiers rest on number and number sense. In the first part of the paper, we turn to developing this argument. In the remainder, we report on work in (...) neurobiology that test the empirical claims of the theory. In particular, we look at fMRI experiments and behavioral experiments with various patient populations that support the intimate connection between natural language quantification and number sense. (shrink)

Questions with a quantificational subject have readings that seemingly involve quantification into questions (called ‘QiQ’ for short). In particular, in single-wh questions with a universal quantifier, QiQ-readings call for pair-list answers, similar to pair-list readings of multiple-wh questions. This paper unifies the derivation of QiQ-readings and distinguishes QiQ-readings from pair-list readings of multiple-wh questions. I propose that pair-list multiple-wh questions and QiQ-questions both involve a wh-dependency, namely, that the wh-/quantificational subject stands in a functional dependency with the trace of (...) the wh-object. In particular, in a pair-list multiple-wh question, the wh-subject binds into the trace of the wh-object across an identity operator; in a QiQ-question, the quantificational subject binds into the trace of the wh-object across a predication operator. These operations give rise to distinct definedness requirements, which vary with the quantificational force of the wh-/quantificational subject. The proposed analysis explains a contrast in domain exhaustivity between the pair-list readings of multiple-wh questions and questions with a universal quantifier, while also doing justice to the intuitive similarities between these two types of questions. I further propose that the observed QiQ-effect in a QiQ-question is derived by extracting one of the minimal proposition sets that satisfy the aforementioned quantificational predication condition. The values of these sets determine whether the QiQ-reading is available and whether a QiQ-question admits pair-list answers and/or has a choice flavor. (shrink)

A general meta-logical theory is developed by considering ontological disputes in the systems of metaphysics. The usefulness of this general meta-logical theory is demonstrated by considering the case of the ontological dispute between the metaphysical systems of Lewis’ Modal Realism and Terence Parsons’ Meinongianism. Using Quine’s criterion of ontological commitments and his views on ontological disagreement, three principles of metalogic is formulated. Based on the three principles of metalogic, the notions of independent variable and dependent variable are introduced. Then, the (...) ontological dispute between Lewis’ Modal Realism and Terence Parsons’ Meinongianism are restated in the light of the principles of metalogic. After the restatement, Independent variable and dependent variables are fixed in both Lewis’ Modal Realism and Terence Parsons’ Meinongianism to resolve the dispute. Subsequently, a new variety of quantifiers are introduced which is known as functionally isomorphic quantifiers to provide a formal representation of the resolution of the dispute. The specific functionally isomorphic quantifier which is developed in this work is known as st-quantifier. It is indicated that how st-quantifier which is one of the functionally isomorphic quantifiers can function like existential quantifier. It is also shown that there is some kind of inconsistency which is unavoidable in stating the ontological disagreement and therefore, paraconsistent logic is a requirement in stating the ontological disputes. (shrink)

By supplying propositional calculus with a probability semantics we showed, in our 1996, that finite stochastic problems can be treated by logic-theoretic means equally as well as by the usual set-theoretic ones. In the present paper we continue the investigation to further the use of logical notions in probability theory. It is shown that quantifier logic, when supplied with a probability semantics, is capable of treating stochastic problems involving countably many trials.

This chapter was written in 2013 and was posted in the Semantics Archive in January 2014. The preprint of the published version has been in the Semantics Archive since 2016. The Semantics Archive is an electronic preprint archive hosted by the Linguistics Society of America. -/- The chapter looks at indicative conditionals embedded under quantifiers, with a special emphasis on ‘one-case’, episodic, conditionals as in "No query was answered if it came from a doubtful address." It agrees with earlier assessments (...) that a complete conditional (with antecedent and consequent) is embedded under a quantifier in those constructions, but then proceeds to create a dilemma by showing that we can’t always find the right interpretation for that conditional. Contrary to earlier assessments, Stalnaker’s conditional won’t always do. The paper concludes that the embedded conditional in the sentence above is a material implication, but the "if"-clause also plays a pragmatic role in restricting the domain of the embedding quantifier. That an appeal to pragmatics should be necessary at all goes with Edgington’s verdict that “we do not have a satisfactory general account of sentences with conditional constituents.” -/- One of the conclusions drawn at the end of the chapter is that the English modal "will" should be given a Stalnaker selection function analysis. The idea is implemented by attaching to "will" an accessibility function variable that receives its value from the context of use and can be dynamically updated by preceding "if"-clauses, as proposed in von Fintel (1994). The argument for a Stalnaker selection function analysis of "will" provided by the paper is that, together with the restrictor analysis of "if"-clauses, it yields the correct truth-conditions for "will"-conditionals (as opposed to episodic, 'one case', conditionals) embedded under quantifiers. The proposal that the English modal "will" should be given a Stalnaker selection function analysis has since been taken up and elaborated considerably in Cariani & Santorio's (2018) paper on "will" in Mind, and is a central idea in Cariani's (2021) book The Modal Future. (shrink)

One well known problem regarding quantifiers, in particular the 1storder quantifiers, is connected with their syntactic categories and denotations. The unsatisfactory efforts to establish the syntactic and ontological categories of quantifiers in formalized first-order languages can be solved by means of the so called principle of categorial compatibility formulated by Roman Suszko, referring to some innovative ideas of Gottlob Frege and visible in syntactic and semantic compatibility of language expressions. In the paper the principle is introduced for categorial languages generated (...) by the Ajdukiewicz’s classical categorial grammar. The 1st-order quantifiers are typically ambiguous. Every 1st-order quantifier of the type k > 0 is treated as a two-argument functorfunction defined on the variable standing at this quantifier and its scope (the sentential function with exactly k free variables, including the variable bound by this quantifier); a binary function defined on denotations of its two arguments is its denotation. Denotations of sentential functions, and hence also quantifiers, are defined separately in Fregean and in situational semantics. They belong to the ontological categories that correspond to the syntactic categories of these sentential functions and the considered quantifiers. The main result of the paper is a solution of the problem of categories of the 1st-order quantifiers based on the principle of categorial compatibility. (shrink)

The relationship between Lexical-Functional Grammar (LFG) functional structures (f-structures) for sentences and their semanticinterpretations can be formalized in linear logic in a way thatcorrectly explains the observed interactions between quantifier scopeambiguity, bound anaphora and intensionality.Our linear-logic formalization of the compositional properties ofquantifying expressions in natural language obviates the need forspecial mechanisms, such as Cooper storage, in representing thescoping possibilities of quantifying expressions. Instead, thesemantic contribution of a quantifier is recorded as a linear-logicformula whose use in a proof will (...) establish the scope of thequantifier. Different proofs can lead to different scopes. In eachcomplete proof, the properties of linear logic ensure thatquantifiers are properly scoped. (shrink)

Functions of type n are characteristic functions on n-ary relations. In Beyond the Frege Boundary [6], Keenan established their importance for natural language semantics, by showing that natural language has many examples of irreducible type n functions, where he called a function of type n reducible if it can be represented as a composition of functions of type 1 . We will give a normal form theorem for functions of type n , and use this to show that natural (...) language has many examples of irreducible type n functions in a much stronger sense, where we take a function to be reducible if it can be represented as a composition of functions of lower types. (shrink)

The quantified relevant logic RQ is given a new semantics in which a formula for all xA is true when there is some true proposition that implies all x-instantiations of A. Formulae are modelled as functions from variable-assignments to propositions, where a proposition is a set of worlds in a relevant model structure. A completeness proof is given for a basic quantificational system QR from which RQ is obtained by adding the axiom EC of 'extensional confinement': for all x(A V (...) B) -> (A V for all xB). with x not free in A. Validity of EC requires an additional model condition involving the boolean difference of propositions. A QR-model falsifying EC is constructed by forming the disjoint union of two natural arithmetical structures in which negation is interpreted by the minus operation. (shrink)

Extreme events, which are usually characterized by generalized extreme value models, can exhibit long-term memory, whose impact needs to be quantified. It was known that extreme recurrence intervals can better characterize the significant influence of long-term memory than using the GEV model. Our statistical analyses based on time series datasets following the Lévy stable distribution confirm that the stretched exponential distribution can describe a wide spectrum of memory behavior transition from exponentially distributed intervals to power-law distributed ones, extending the previous (...) evaluation of the stretched exponential function using Gaussian/exponential distributed random data. Further deviation and discussion of a historical paradox are also provided, based on the theoretical analysis of the Bayesian law and the stretched exponential distribution. (shrink)

We present an algebraic-style of semantics, which we call a content semantics, for quantified relevant logics based on the weak system BBQ. We show soundness and completeness for all quantificational logics extending BBQ and also treat reduced modelling for all systems containing BB d Q. The key idea of content semantics is that true entailments AB are represented under interpretation I as content containments, i.e. I(A)I(B) (or, the content of A contains that of B). This is opposed to the truth-functional (...) way which represents true entailments as truth-preservations over all set-ups (or worlds), i.e. (VaK) (if I(A, a) = T then I(B, a)= T). (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. (shrink)

This paper continues my discussion with Michael Dummett on Frege’s senses, published in The Philosophy of Michael Dummett and further developed in Diametros. In his reply to my original paper, Dummett came to agree with me that senses are neither objects nor functions, since they have a categorially different kind of linguistico-metaphysical function to perform. He then asks how we might quantify over senses, if they are neither objects nor functions. He discusses two main options, and finds one unviable (...) and the other “very un-Fregean.” I then offer a Fregean or neo-Fregean option in my rejoinder. And I still hold that this way out will do the job, or is at least plausible enough that the burden of persuasion is on those who disagree. But I hope to show in this paper that on a more complete examination of Frege, there are at least twenty Fregean or neo-Fregean ways out, with the one I proposed being option. (shrink)

The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete model theory for many propositionally quantified relevant logics (and some non-relevant ones). This involves a restriction on which sets of worlds are admissible as propositions, and an interpretation of propositional quantification that makes ∀ pA true when there is some true admissible proposition that entails all p -instantiations of A . It is also shown that without the admissibility qualification many of the systems considered are (...) semantically incomplete, including all those that are sub-logics of the quantified version of Anderson and Belnap’s system E of entailment, extended by the mingle axiom and the Ackermann constant t . The incompleteness proof involves an algebraic semantics based on atomless complete Boolean algebras. (shrink)

We define a theory of two-sort bounded arithmetic whose provably total functions are exactly those in ${\mathcal{F}_{LOGCFL}}$ by way of a generalized quantifier that expresses computations of SAC 1 circuits. The proof depends on Kolokolova’s conditions for the connection between the provable capture in two-sort theories and descriptive complexity.

Exhumation and study of the 1945 paradox of confirmation brings out the defect of its formulation. In the context of quantifier conditional-probability logic it is shown that a repair can be accomplished if the truth-functional conditional used in the statement of the paradox is replaced with a connective that is appropriate to the probabilistic context. Description of the quantifier probability logic involved in the resolution of the paradox is presented in stages. Careful distinction is maintained between a formal (...) logic language and its semantics, as the same language may be outfitted with different semantics. An acquaintance with sections 1? 5 of Hailperin (2006) covering the sentential aspects of probability logic is assumed as background information for quantifier probability logic. (shrink)

This paper presents an extended version of the Quantified Argument Calculus (Quarc). Quarc is a logic comparable to the first-order predicate calculus. It employs several nonstandard syntactic and semantic devices, which bring it closer to natural language in several respects. Most notably, quantifiers in this logic are attached to one-place predicates; the resulting quantified constructions are then allowed to occupy the argument places of predicates. The version presented here is capable of straightforwardly translating natural-language sentences involving defining clauses. A three-valued, (...) model-theoretic semantics for Quarc is presented. Interpretations in this semantics are not equipped with domains of quantification: they are just interpretation functions. This reflects the analysis of natural-language quantification on which Quarc is based. A proof system is presented, and a completeness result is obtained. The logic presented here is capable of straightforward translation of the classical first-order predicate calculus, the translation preserving truth values as well as entailment. The first-order predicate calculus and its devices of quantification can be seen as resulting from Quarc on certain semantic and syntactic restrictions, akin to simplifying assumptions. An analogous, straightforward translation of Quarc into the first-order predicate calculus is impossible. (shrink)

Dummett's logic LC quantified, Q-LC, is shown to be characterized by the extended frame Q+, ,D, where Q+ is the set of non-negative rational numbers, is the numerical relation less or equal then and D is the domain function such that for all v, w Q+, Dv and if v w, then D v . D v D w . Moreover, simple completeness proofs of extensions of Q-LC are given.

The following result is an approximation to the answer of the question of Kokorin about decidability of a quantifier-free theory of field of rational numbers. Let Q0 be a subset of the set of all rational numbers which contains integers 1 and −1. Let be a set containing Q0 and closed by the functions of addition, subtraction and multiplication. For example coincides with Q0 if Q0 is the set of all binary rational numbers or the set of all decimal (...) rational numbers. It is clear that , where Z is the set of all integers and Q is the set of all rational numbers. A negationless theory uses only conjunction and disjunction as logical connectives. Let T be a quantifier-free negationless elementary theory of signature , where Pol is the list of all polynomials with rational coefficients from Q0 in which exponent of the polynomials and rational coefficients are written in binary number system. Theorem 1. Theory T is NP-hard and if is everywhere dense then the theory T is decidable by an algorithm which belongs to EXPTIME. The set of all rational numbers or the set of all binary rational numbers may be taken instead of . The quantifier-free negationless theory of the field of rational numbers may be regarded as a base of constructive mathematics. (shrink)

Most interpreting theories claim that different interpreting types should involve varied processing mechanisms and procedures. However, few studies have examined their underlying differences. Even though some previous results based on quantitative approaches show that different interpreting types yield outputs of varying lexical and syntactic features, the grammatical parsing approach is limited. Language sequences that form without relying on parsing or processing with a specific linguistic approach or grammar excel other quantitative approaches at revealing the sequential behavior of language production. As (...) a non-grammatically-bound unit of language sequences, frequency motif can visualize the local distribution of content and function words, and can also statistically classify languages and identify text types. Thus, the current research investigates the distribution, length and position-dependent properties of frequency motifs across different interpreting outputs in pursuit of the sequential generation behaviors. It is found that the distribution, the length and certain position-dependent properties of the specific language sequences differ significantly across simultaneous interpreting and consecutive interpreting output. The features of frequency motifs manifest that both interpreting output is produced in the manner that abides by the least effort principle. The current research suggests that interpreting types can be differentiated through this type of language sequential unit and offers evidence for how the different task features mediate the sequential organization of interpreting output under different demand to achieve cognitive load minimization. (shrink)

The present study examined the neural substrate of two classes of quantifiers: numerical quantifiers like ” at least three” which require magnitude processing, and logical quantifiers like ” some” which can be understood using a simple form of perceptual logic. We assessed these distinct classes of quantifiers with converging observations from two sources: functional imaging data from healthy adults, and behavioral and structural data from patients with corticobasal degeneration who have acalculia. Our findings are consistent with the claim that numerical (...) quantifier comprehension depends on a lateral parietal-dorsolateral prefrontal network, but logical quantifier comprehension depends instead on a rostral medial prefrontal-posterior cingulate network. These observations emphasize the important contribution of abstract number knowledge to the meaning of numerical quantifiers in semantic memory and the potential role of a logic-based evaluation in the service of non-numerical quantifiers. (shrink)