Quantifiers, Misc

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)

So far, we have focused on discourse reference to atomic individuals and specific times, events, and states. The basic point of the argument was that all types of discourse reference involve attention-guided anaphora (in the sense of Bittner 2012: Ch. 2). We now turn to discourses involving anaphora to and by quantificational expressions. Today, we focus on quantification over individuals but the analysis we develop will directly generalize to other semantic types. The basic idea is that quantification is one more (...) species of top-level anaphora--to wit, anaphora to top-ranked sets. (shrink)

Kit Fine has reawakened a puzzle about variables with a long history in analytic philosophy, labeling it “the antinomy of the variable”. Fine suggests that the antinomy demands a reconceptualization of the role of variables in mathematics, natural language semantics, and first-order logic. The difficulty arises because: (i) the variables ‘x’ and ‘y’ cannot be synonymous, since they make different contributions when they jointly occur within a sentence, but (ii) there is a strong temptation to say that distinct variables ‘x’ (...) and ‘y’ are synonymous, since sentences differing by the total, proper substitution of ‘x’ for ‘y’ always agree in meaning. We offer a precise interpretation of the challenge posed by (i) and (ii). We then develop some neglected passages of Tarski to show that his semantics for variables has the resources to resolve the antinomy without abandoning standard compositional semantics. (shrink)

Featured course on "Dynamic Semantics" at NASSLLI 2016. Day 5: Quantification. Abstract: In discourse, quantifiers can function as antecedents or anaphors. We analyze a sample discourse in Dynamic Plural Logic (DPlL, van den Berg 1993, 1994), which represents not only current discourse referents, but also current relations by means of plural information states. This makes it possible to analyze quantification as structured discourse reference. Finally, the DPlL analysis is transposed into Update with Centering, to simplify the formalism and relate quantification (...) to earlier discussion in the course. (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)

This paper seeks to illustrate the advantages of not treating phonological words as distinguished building blocks in compositional semantics. Following Bobaljik 2012, we derive the relative readings of amount superlatives in two steps, [[[d-many] comparative] superlative]. The existence of two comparative constructions is revealed, involving more vs. the more. Each builds a different superlative construction, explaining the conflicting intuitions about superlatives in the literature, as well as puzzles relating to the definite article in superlatives.

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

This work gives an extended presentation of the treatment of variable-binding operators adumbrated in [3:1993d]. Illustrative examples include elementary languages with quantifiers and lambda-equipped categorial languages. Some remarks are also offered to illustrate the philosophical import of the resulting picture. Particularly, a certain conception of logic emerges from the account: the view that logics are true theories in the model-theoretic sense, i.e. the result of selecting a certain class of models as the only “admissible” interpretation structures (for a given language).

Italian translation of "On Logical Relativity" (2002), by Luca Morena.

Quantification over individuals, times, and worlds can in principle be made explicit in the syntax of the object language, or left to the semantics and spelled out in the meta-language. The traditional view is that quantification over individuals is syntactically explicit, whereas quantification over times and worlds is not. But a growing body of literature proposes a uniform treatment. This paper examines the scopal interaction of aspectual raising verbs (begin), modals (can), and intensional raising verbs (threaten) with quantificational subjects in (...) Shupamem, Dutch, and English. It appears that aspectual raising verbs and at least modals may undergo the same kind of overt or covert scope-changing operations as nominal quantifiers; the case of intensional raising verbs is less clear. Scope interaction is thus shown to be a new potential diagnostic of object-linguistic quantification, and the similarity in the scope behavior of nominal and verbal quantifiers supports the grammatical plausibility of ontological symmetry, explored in Schlenker (2006). (shrink)

I here argue that Ted Sider's indeterminacy argument against vagueness in quantifiers fails. Sider claims that vagueness entails precisifications, but holds that precisifications of quantifiers cannot be coherently described: they will either deliver the wrong logical form to quantified sentences, or involve a presupposition that contradicts the claim that the quantifier is vague. Assuming (as does Sider) that the “connectedness” of objects can be precisely defined, I present a counter-example to Sider's contention, consisting of a partial, implicit definition of the (...) existential quantifier that in effect sets a given degree of connectedness among the putative parts of an object as a condition upon there being something (in the sense in question) with those parts. I then argue that such an implicit definition, taken together with an “auxiliary logic” (e.g., introduction and elimination rules), proves to function as a precisification in just the same way as paradigmatic precisifications of, e.g., “red”. I also argue that with a quantifier that is stipulated as maximally tolerant as to what mereological sums there are, precisifications can be given in the form of truth-conditions of quantified sentences, rather than by implicit definition. (shrink)

This paper deals with the logical form of quantified sentences. Its purpose is to elucidate one plausible sense in which quantified sentences can adequately be represented in the language of first-order logic. Section 1 introduces some basic notions drawn from general quantification theory. Section 2 outlines a crucial assumption, namely, that logical form is a matter of truth-conditions. Section 3 shows how the truth-conditions of quantified sentences can be represented in the language of first-order logic consistently with some established undefinability (...) results. Section 4 sketches an account of vague quantifier expressions along the lines suggested. Finally, section 5 addresses the vexed issue of logicality. (shrink)

The Symbolic Logic Study Guide is designed to accompany the widely used symbolic logic textbook Language, Proof and Logic (LPL), by Jon Barwise and John Etchemendy (CSLI Publications 2003). The guide has two parts. The first part contains condensed, essential lecture notes, which streamline and systematize the first fourteen chapters of the book into seven teaching sections, and thus provide a clear, well-designed roadmap for the understanding of the text. The second part consists of twelve sample quizzes and solutions. The (...) Symbolic Logic Study Guide is essential for all instructors and students who use LPL in their symbolic logic classes. (shrink)

Our paper presents a novel theory of weak crossover effects, based entirely on quantifier scope preferences and their consequences for variable binding. The structural notion of 'crossover' play no role. We develop a theory of scope preferences which ascribes a central role to the AGR-P System.

Aristotle's syllogistic is extended to include denumerably many quantifiers such as 'more than 2/3' and 'exactly 2/3.' Syntactic and semantic decision procedures determine the validity, or invalidity, of syllogisms with any finite number of premises. One of the syntactic procedures uses a natural deduction account of deducibility, which is sound and complete. The semantics for the system is non-classical since sentences may be assigned a value other than true or false. Results about symmetric systems are given. And reasons are given (...) for claiming that syllogistic validity is relevant validity. (shrink)

This paper describes quantificational structures in Greenlandic Eskimo (Kalaallisut), a language where familiar quantificational meanings are expressed in ways that are quite different from English. Evidence from this language thus poses some formidable challenges for cross-linguistic theories of compositional semantics.

: This study describes a new field method, suited for investigating scope relations — and other aspects of truth conditional meaning — with native speaker consultants who may speak no other language and have no background in linguistics or logic. This method revealed a surprising scope contrast between the antipassive and the ergative construction in Greenlandic Eskimo. The results of this field work are described in detail and a crosslinguistic scope generalization is proposed based on Greenlandic Eskimo, Basque, Polish, Russian, (...) Finnish and English. (shrink)

Quantification, Negation, and Focus: Challenges at the Conceptual-Intentional Semantic Interface Tista Bagchi National Institute of Science, Technology, and Development Studies (NISTADS) and the University of Delhi Since the proposal of Logical Form (LF) was put forward by Robert May in his 1977 MIT doctoral dissertation and was subsequently adopted into the overall architecture of language as conceived under Government-Binding Theory (Chomsky 1981), there has been a steady research effort to determine the nature of LF in language in light of structurally (...) diverse languages around the world, which has ultimately contributed to the reinterpretation of LF as a Conceptual-Intentional (C-I) interface level between the computational syntactic component of the faculty of language and one or more interpretive faculties of the human mind. While this has opened up further possibilities of research in phenomena such as quantifier scope and scope interactions between negation, quantification, and focus, it has also given rise to a few real challenges to linguistic theory as well. Some of these are: (i) the split between lexical meaning – a matter supposedly belonging to the phase-wise selection of lexical arrays – and issues of semantic interpretation that arise purely from binding and scope phenomena (Mukherji 2010); (ii) partially relatedly, the level at which theta role assignment can be argued to take place, an issue that is taken up by me in Bagchi (2007); and (iii) how supposedly “pure” scopal phenomena relating to quantifiers, negation, and emphasizing expressions such as only and even (comparable to, e.g., Urdu/Hindi hii and bhii, Bangla –i and –o) also have dimensions of both focus and discourse reference. While recognizing all of these challenges, this talk aims to highlight particularly challenge (iii), both in terms of scholarship in the past and for the rich prospects for research on languages of south Asia with the semantics of quantification, negation, and focus in view. The scholarship of the past that I seek to relate this issue to is where, parallel to (and largely independently of) the research on LF that had been happening, Barwise and Cooper were developing their influential view of noun phrases as generalized quantifiers, culminating in their key 1981 article “Generalized Quantifiers and Natural Language” while, independently, McCawley, in his 1981 book Everything that Linguists have Always Wanted to Know about Logic, established through argumentation that all noun phrases semantically behave like generalized quantified expressions (further elaborated by him in the second – 1994 – revised edition of his book). I seek to demonstrate, based on limited data analysis from selected languages of south Asia, that our current understanding of quantification, negation, and focus under the Minimalist view owes something significant to the two major, but now largely marginalized, works of scholarship, and that for the way forward it is essential to adopt a more formal-semantic approach as adopted by them and also by later works such as Denis Bouchard’s (1995) The Semantics of Syntax, Mats Rooth’s work on focus (e.g., Rooth 1996, “Focus” in Shalom Lappin’s Handbook of Contemporary Semantic Theory), Heim and Kratzer’s Semantics in Generative Grammar (1998), and Yoad Winter’s (2002) Linguistic Inquiry article on semantic number, to cite just a few instances. (shrink)