The interpretation of if -clauses in the scope of ordinary quantifiers has provoked semanticists into extraordinary measures, such as abandoning compositionality or claiming that if has no meaning. We argue that if -clauses have a normal conditional meaning, even in the scope of ordinary quantifiers, and that the trick is to have the right semantics for conditionals.

This dissertation investigates the ways in which natural language restricts the domains of quantifiers. Adverbs of quantification are analyzed as quantifying over situations. The domain of quantifiers is pragmatically constrained: apparent processes of "semantic partition" are treated as pragmatic epiphenomena. The introductory Chapter 1 sketches some of the background of work on natural language quantification and begins the analysis of adverbial quantification over situations. Chapter 2 develops the central picture of "semantic partition" as a side-effect of pragmatic processes (...) of anaphora resolution. I argue that the apparent effects of topic/focus articulation and presuppositional information on the interpretation of quantifiers are not the result of a direct and local mechanism of sentence grammar. Instead, I develop an analysis where the link is established via the anaphoric dependence of quantifier domains on the discourse context. Chapter 3 discusses the analysis of conditional clauses as quantifier restrictors, concentrating on the question whether conditional clauses restrict quantifiers directly or indirectly. A treatment is explored which has if-clauses constrain the value of the hidden domain variable of the restricted quantifier. Chapter 4, on unless-clauses, and Chapter 5, on only if- and even if-clauses, present some issues in the compositional analysis of complex conditional clauses. These chapters significantly expand the data coverage of the theory of A-quantification. Building on previous work of mine on exceptives, I analyze unless-clauses as exceptive operators on A-quantifiers. The analysis of only if-clauses, treated as conditional clauses that combine if with the focus adverb only, unearthes some interesting new properties. Chapter 6, finally, examines the phenomenon of donkey-anaphora in the light of the results of the previous chapters. I show that a solution to the proportion problem may become possible once we combine the situation-semantic approach to adverbial quantification with the pragmatic theory developed in Chapter 2 and further elaborated in the analysis of donkey anaphora in complex conditionals. (shrink)

In this paper, we provide a comprehensive survey of the space of possible analyses of the phenomenon of quantifier domain restriction, together with a set of considerations which militate against all but our own proposal. Among the many accounts we consider and reject are the ‘explicit’ approach to quantifier domain restric‐tion discussed, for example, by Stephen Neale, and the pragmatic approach to quantifier domain restriction proposed by Kent Bach. Our hope is that the exhaustive discussion of this special case of (...) context‐dependence will provide guidelines for how to decide, for an arbitrary case of context‐dependent discourse, whether it should be treated syntactically, semantically, or pragmatically. (shrink)

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)

This is a reply to H. Ben-Yami, 'Generalized quantifiers, and beyond' (this journal, 2009), where he argues that standard GQ theory does not explain why natural language quantifiers have a restricted domain of quantification. I argue, on the other hand, that although GQ theory gives no deep explanation of this fact, it does give a sort of explanation, whereas Ben-Yami's suggested alternative is no improvement.

Temporal logic as a modern discipline is separate from classical logic; it is seen as an addition or expansion of the more basic propositional and predicate logics. This approach is in contrast with logic in the Middle Ages, which was primarily intended as a tool for the analysis of natural language. Because all natural language sentences have tensed verbs, medieval logic is inherently a temporal logic. This fact is most clearly exemplified in medieval theories of supposition. As a case study, (...) we look at the supposition theory of Lambert of Lagny, extracting from it a temporal logic and providing a formalization of that logic. (shrink)

Section 1 provides a brief summary of the pair-list literature singling out some points that are particularly relevant for the coming discussion. -/- Section 2 shows that the dilemma of quantifi cation versus domain restriction arises only in extensional complement interrogatives. In matrix questions and in intensional complements only universals support pairlist readings, whence the simplest domain restriction treatment suffices. Related data including conjunction, disjunction, and cumulative readings are discussed -/- Section 3 argues that in the case of extensional complements (...) the domain restriction treatment is inadequate for at least two independent reasons. One has to do with the fact that not only upward monotonic quantifi ers support pairlist readings, and the other with the derivation of apparent scope out readings. The reasoning is supplemented with some discussion of the semantic properties of layered quantifi ers.  The above will establish the need for quantifi cation, so the question arises how the objections explicitly enlisted in the literature against quantifi cation can be answered. Section 4 considers the de dicto reading of the quantifi er s restriction, quanti cational variability, and the absence of pairlist readings with whether questions, and argues that they need not militate against the quanti ficational analysis. -/- Section 5 summarizes the emergent proposal -/- Finally, section 6 discusses the signifi cance of the above findings for the behavior of weak islands. (shrink)

In this paper, we present a theory of interaction between definiteness and quantifier structure, where the definite determiner (D) performs the function of contextually restricting the domain of quantificational determiners (Qs). Our motivating data come from Greek and Basque, where D appears to compose with the Q itself. Similar compositions are found in Hungarian and Bulgarian. Following earlier work (Giannakidou 2004, Etxeberria 2005, Etxeberria and Giannakidou 2009) we define a domain restricting function DDR, in which D modifies the Q and (...) supplies the contextual variable C to intersect with Q’s domain. DDR can also modify the NP in which case it works like an intersective modfier— and we suggest that this is the case in St’át’imcets Salish (drawing on data from Matthewson’s work). The result of DDR will be restricting the domain of Q by C, a weakly familiar property (in the sense of Roberts 2003, 2010), i.e. it is entailed in the common ground of the context. As a result of intersecting with this property, the Q that undergoes DDR becomes presuppositional, and we hypothesize that all presuppositional Qs (each-like Qs) have an underlying derivation of the domain restriction via D that we see in Greek and Basque overtly. Our analysis provides support for the program that domain restriction is grammatically represented, but we propose an important refinement: domain restriction can affect the Q itself (in agreement with von Fintel 1998, Martí 2003, Giannakidou 2004; pace Stanley 2002), and in fact quite systematically. We also show that the Q that is affected by DDR is a strong one, and explain the inability of weak Qs to undergo DDR as following from their status as adjectival. (shrink)

In this paper, I introduce a quantifier-pluralist theory of time, temporal quantifier relativism. Temporal quantifier relativism includes a restricted quantifier for every instantaneous moment of time. Though it flies in the face of orthodoxy, it compares favorably to rival theories of time. To demonstrate this, I first develop the basic syntax and semantics of temporal quantifier relativism. I then compare the theory to its rivals on three issues: the passage of time, the analysis of change, and temporal ontology.

This paper extends the investigations into logical properties of the quantified argument calculus (Quarc) by suggesting a series of proper subsystems which, although retaining the entire vocabulary of Quarc, restrict quantification in such a way as to make the result decidable. The proof of decidability is via a procedure that prunes the infinite branches of a derivation tree in what is a syntactic counterpart of semantic filtration. We demonstrate an application of one of these systems by showing that Aristotle’s assertoric (...) syllogistic is embeddable within, thus also providing another method of showing its decidability. (shrink)

I show that the contemporary dominant analysis of natural language quantifiers that are one-place determiners by means of binary generalized quantifiers has failed to explain why they are, according to it, conservative. I then present an alternative, Geachean analysis, according to which common nouns in the grammatical subject position are plural logical subject-terms, and show how it does explain that fact and other features of natural language quantification.

Many systems of quantified modal logic cannot be characterised by Kripke's well-known possible worlds semantic analysis. This book shows how they can be characterised by a more general 'admissible semantics', using models in which there is a restriction on which sets of worlds count as propositions. This requires a new interpretation of quantifiers that takes into account the admissibility of propositions. The author sheds new light on the celebrated Barcan Formula, whose role becomes that of legitimising the Kripkean interpretation (...) of quantification. The theory is worked out for systems with quantifiers ranging over actual objects, and over all possibilia, and for logics with existence and identity predicates and definite descriptions. The final chapter develops a new admissible 'cover semantics' for propositional and quantified relevant logic, adapting ideas from the Kripke-Joyal semantics for intuitionistic logic in topos theory. This book is for mathematical or philosophical logicians, computer scientists and linguists. (shrink)

This paper extends Kripke’s theory of truth to a language with a variably strict conditional operator, of the kind that Stalnaker and others have used to represent ordinary indicative conditionals of English. It then shows how to combine this with a different and independently motivated conditional operator, to get a substantial logic of restricted quantification within naive truth theory.

Van Lambalgen (1990) proposed a translation from a language containing a generalized quantifierQ into a first-order language enriched with a family of predicatesR i, for every arityi (or an infinitary predicateR) which takesQxg(x, y1,..., yn) to x(R(x, y1,..., y1) (x,y1,...,yn)) (y 1,...,yn are precisely the free variables ofQx). The logic ofQ (without ordinary quantifiers) corresponds therefore to the fragment of first-order logic which contains only specially restricted quantification. We prove that it is decidable using the method of analytic (...) tableaux. Related results were obtained by Andréka and Németi (1994) using the methods of algebraic logic. (shrink)

In this paper, we present a prenex form theorem for a version of Independence Friendly logic, a logic with imperfect information. Lifting classical results to such logics turns out not to be straightforward, because independence conditions make the formulas sensitive to signalling phenomena. In particular, nested quantification over the same variable is shown to cause problems. For instance, renaming of bound variables may change the interpretations of a formula, there are only restricted quantifier extraction theorems, and slashed connectives cannot (...) be so easily removed. Thus we correct some claims from Hintikka [8], Caicedo & Krynicki [3] and Hodges [11]. We refine definitions, in particular the notion of equivalence, and sharpen preconditions, allowing us to restore those claims, including the prenex form theorem of Caicedo & Krynicki [3], and, as a side result, we obtain an application to Skolem forms of classical formulas. It is a known fact that a complete calculus for IF-logic is impossible, but with our results we establish several quantifier rules that form a partial calculus of equivalence for a general version of IF-logic reflecting general properties of information flow in games. (shrink)

The cofinality quantifiers were introduced by Shelah as an example of a compact logic stronger than first-order logic. We show that the classes of models axiomatized by these quantifiers can be turned into an Abstract Elementary Class by restricting to positive and deliberate uses. Rather than using an ad hoc proof, we give a general framework of abstract Skolemizations. This method gives a uniform proof that a wide rang of classes are Abstract Elementary Classes.

In this paper I argue for a restriction on certain types of LF movement, which I call ‘wh-related LF movement’. Evidence comes from a number of wh-in-situ constructions in German, such as the scope-marking construction and multiple questions. For semantic reasons, the in situ element in those constructions has to move at LF to either a position reserved for wh-phrases, or even higher up in the structure. The restriction (the Minimal Quantified Structure Constraint, MQSC) is that an intervening quantified expression (...) blocks this movement. In the case of every, the MQSC leads to an unambiguously distributive interpretation of the question. In the case of all other intervening operators, including negation, it leads to ungrammaticality. (shrink)

Although first-order Kripke semantics has become a well established branch of modal logic, very little - almost nothing - is written about logics with a weaker modal fragment. We try to help the situation by isolating principles determining the interaction between quantifiers and modalities in minimal semantics. First, we let the standard-model properties of monotonic and anti-monotonic domains clue us in on how to do this – i. e. we try to articulate, in terms of the inclusiveness of the (...) domains of a certain set of worlds, a set of semantical restrictions that will validate the Barcan and converse Barcan formulae respectively. As it turns out, this can indeed by done, but only by adding assumptions strong enough to make the models virtually normal. Since the whole point of switching to a minimal framework would be to generalise the logic, we therefore abandon the worlds-objects thinking altogether, and switch to a much simpler and more direct validation strategy in which the propositions we are after are simply picked out as such. (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)

Stephen Barker (The Philosophical Quarterly, 47 (1997), pp. 195–211) has presented a new argument for a pure material implication analysis of indicative conditionals. His argument relies crucially on the assumption that general indicatives such as ‘Every girl, if she gets a chance, bungee‐jumps’ are correctly analysed as having the formal structure (for all x)(if x gets a chance, x bungee‐jumps). This paper argues that an approach first proposed by David Lewis must be pursued: the ‘if’‐clause in these sentences restricts the (...) quantifier. Only the Lewis‐style analysis can deal with sentences involving non‐universal quantifiers such as ‘Most letters are answered if they are shorter than 5 pages’. I show that Barker's reasons for rejecting the restrictor analysis are not cogent and that the restrictor analysis connects widely with recent work in natural language semantics. (shrink)

In a series of publications I have claimed that by contrast to standard formal languages, quantifiers in natural language combine with a general term to form a quantified argument, in which the general term's role is to determine the domain or plurality over which the quantifier ranges. In a recent paper Zoltán Gendler Szabó tried to provide a counterexample to this analysis and derived from it various conclusions concerning quantification in natural language, claiming it is often ‘bare’. I show (...) that Szabó's example fails, and that even if it were successful his conclusions would not be supported by it. (shrink)

We use imperatives to refute a naïve analysis of update potentials (force-operators attaching to sentences), arguing for a dynamic analysis of imperative force as restrictable, directed, and embeddable. We propose a dynamic, non-modal analysis of conditional imperatives, as a counterpoint to static, modal analyses. Our analysis retains Kratzer's analysis of if-clauses as restrictors of some operator, but avoids typing it as a generalized quantifier over worlds (against her), instead as a dynamic force operator. Arguments for a restrictor treatment (but against (...) a quantificational treatment) are mustered, and we propose a novel analysis of update on conditional imperatives (and an independently motivated revision of the standard ordering-semantics for root modals that makes use of it). Finally, we argue that imperative force is embeddable under an operation much like dynamic conjunction. (shrink)

Let H be a proof system for quantified propositional calculus (QPC). We define the Σqj-witnessing problem for H to be: given a prenex Σqj-formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the Σq1-witnessing problems for the systems G*1and G1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce and study the systems G*0 and (...) G0, in which cuts are restricted to quantifier-free formulas, and prove that the Σqj-witnessing problem for each is complete for NC1. Our proof involves proving a polynomial time version of Gentzen’s midsequent theorem for G*0 and proving that G0-proofs are TC0-recognizable. We also introduce QPC systems for TC0 and prove witnessing theorems for them. We introduce a finitely axiomatizable second-order system VNC1 of bounded arithmetic which we prove isomorphic to Arai’s first order theory AID + Σb0-CA for uniform NC1. We describe simple translations of VNC1 proofs of all bounded theorems to polynomial size families of G*0 proofs. From this and the above theorem we get alternative proofs of the NC1 witnessing theorems for VNC1 and AID. (shrink)

It is often possible to know what a speaker intends to communicate without knowing what they intend to say. In such cases, speakers need not intend to say anything at all. Stanley and Szabó's influential survey of possible analysis of quantifier domain restriction is, therefore, incomplete and the arguments made by Clapp and Buchanan against Truth Conditional Compositionality and propositional speaker-meaning are flawed. Two theories should not always be viewed as incompatible when they associate the same utterance with different propositions, (...) as there may be many ways to interpret speakers that are compatible with their intentions. (shrink)

Abstract -/- Quantifier domain restriction (QDR) and two versions of nominal restriction (NR) are implemented as restrictions that depend on a previously introduced interpreter and interpretation time in a two-dimensional semantic framework on the basis of simple type theory and categorial grammar. Against Stanley (2002) it is argued that a suitable version of QDR can deal with superlatives like tallest. However, it is shown that NR is needed to account for utterances when the speaker intends to convey different restrictions for (...) multiple uses of the same quantifying determiner. We argue that NR generally fares better with such examples but also observe that examples like Every sailor waves at every sailor might be pragmatically anomalous. An account of contextual domain restriction is proposed that (i) excludes these anomalous readings (but it is shown how they could be included), (ii) makes it possible to express different contextual domain restrictions as long-range dependencies on an interpreter and an interpretation time, and (iii) additionally models restrictions based on locative constructions as general mereological constraints introduced by shifting the index. (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)

Classical generalized quantifier (GQ) theory posits that quantificational determiners (Q-dets) combine with a nominal argument of type et, a first order predicate, to form a GQ. In a recent paper, Matthewson (2001) challenges this position by arguing that the domain of a Q-det is not of type et, but e, an entity. In this paper, I defend the classical GQ view, and argue that the data that motivated Matthewson’s revision actually suggest that the domain set can, and indeed in certain (...) languages must, be contextually restricted overtly. (shrink)

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)

Linguists and philosophers of language have shown increasing interest in the expressions that refer to quantifiers: determiners like ‘every’ and ‘many’, in addition to determiner phrases like ‘some king’ and ‘no cat’. This thesis addresses several puzzles where the way we understand quantifier expressions depends on features that go beyond standard truth conditional semantic meaning. One puzzle concerns the fact that it is often natural to understand ‘Every king is in the yard’ as being true if all of the (...) kings at the party are in the yard, even though the standard truth conditions predict it to be true if and only if every king in the universe is in the yard. Another puzzle emerges from the observation that ‘Every American king is in the yard’ sounds odd relative to contexts where there are no American kings, even though the standard truth conditions predict it to be trivially true. These puzzles have been widely discussed within linguistics and philosophy of language, and have implications for topics as diverse as the distinction between semantics and pragmatics and the ontological commitments of ordinary individuals. Yet few attempts have been made to incorporate discussions from the linguistics literature into the philosophical literature. This thesis argues that attending to the linguistics literature helps to address these puzzles. In particular, my solutions to these puzzles rely on notions from work on information structure, an often overlooked area of linguistics. I will use these notions to develop a new theory of the pragmatics of ordinary discourse, in the process of resolving the puzzles. In the first two chapters, I provide accessible overviews of key notions from the literature on quantifier expressions and information structure. In the third chapter, I discuss the problem of contextual domain restriction. In the fourth chapter, I consider the problems posed by empty restrictors. In the final chapter, I tackle the issue of category mistakes. (shrink)

The simplest quantified modal logic combines classical quantification theory with the propositional modal logic K. The models of simple QML relativize predication to possible worlds and treat the quantifier as ranging over a single fixed domain of objects. But this simple QML has features that are objectionable to actualists. By contrast, Kripke-models, with their varying domains and restricted quantifiers, seem to eliminate these features. But in fact, Kripke-models also have features to which actualists object. Though these philosophers have (...) introduced variations on Kripke-models to eliminate their objectionable features, the most well-known variations all have difficulties of their own. The present authors reexamine simple QML and discover that, in addition to having a possibilist interpretation, it has an actualist interpretation as well. By introducing a new sort of existing abstract entity, the contingently nonconcrete, they show that the seeming drawbacks of the simplest QML are not drawbacks at all. Thus, simple QML is independent of certain metaphysical questions. (shrink)

In the semantics of natural language, quantification may have received more attention than any other subject, and one of the main topics in psychological studies on deductive reasoning is syllogistic inference, which is just a restricted form of reasoning with quantifiers. But thus far the semantical and psychological enterprises have remained disconnected. This paper aims to show how our understanding of syllogistic reasoning may benefit from semantical research on quantification. I present a very simple logic that pivots on (...) the monotonicity properties of quantified statements - properties that are known to be crucial not only to quantification but to a much wider range of semantical phenomena. This logic is shown to account for the experimental evidence available in the literature as well as for the data from a new experiment with cardinal quantifiers ("at least n" and "at most n"), which cannot be explained by any other theory of syllogistic reasoning. (shrink)

Stephen Barker (The Philosophical Quarterly, 47 (1997), pp. 195–211) has presented a new argument for a pure material implication analysis of indicative conditionals. His argument relies crucially on the assumption that general indicatives such as ‘Every girl, if she gets a chance, bungee‐jumps’ are correctly analysed as having the formal structure (for all x)(if x gets a chance, x bungee‐jumps). This paper argues that an approach first proposed by David Lewis must be pursued: the ‘if’‐clause in these sentences restricts the (...) quantifier. Only the Lewis‐style analysis can deal with sentences involving non‐universal quantifiers such as ‘Most letters are answered if they are shorter than 5 pages’. I show that Barker's reasons for rejecting the restrictor analysis are not cogent and that the restrictor analysis connects widely with recent work in natural language semantics. (shrink)

We prove some results about the limitations of the expressive power of quantifiers on finite structures. We define the concept of a bounded quantifier and prove that every relativizing quantifier which is bounded is already first-order definable (Theorem 3.8). We weaken the concept of congruence closed (see [6]) to weakly congruence closed by restricting to congruence relations where all classes have the same size. Adapting the concept of a thin quantifier (Caicedo [1]) to the framework of finite structures, we (...) define the concept of a meager quantifier. We show that no proper extension of first-order logic by means of meager quantifiers is weakly congruence closed (Theorem 4.9). We prove the failure of the full congruence closure property for logics which extend first-order logic by means of meager quantifiers, arbitrary monadic quantifiers, and the Härtig quantifier (Theorem 6.1). (shrink)

We generalize Shelah's analysis of cardinality quantifiers for a superstable theory from Chapter V of Classification Theory and the Number of Nonisomorphic Models. We start with a set of bounds for the cardinality of each formula in some general invariant family of formulas in a superstable theory (in Classification Theory, a uniform family of formulas is considered) and find a set of derived bounds for all formulas. The set of derived bounds is sharp: up to a technical restriction, every (...) model that satisfies the original bounds has a sufficiently saturated elementary extension that satisfies the original bounds and such that for each formula the set of its realizations in the extension has arbitrarily large cardinality below the corresponding derived bound of the formula. (shrink)

Conditional sentences with quantifying expressions are systematically ambigous. In one reading, the if -clause restricts the domain of the overt quantifier; in the other, the if -clause restricts the domain of a covert quantifier, which defaults to epistemic necessity. Although the ambiguity follows directly from the Lewis- Kratzer line on if, it is not generally acknowledged, which has led to pseudoproblems and spurious arguments.

This work adopts the perspective of plural logic and measurement theory in order first to focus on the microstructure of comparative determiners; and second, to derive the properties of comparative determiners as these are studied in Generalized Quantifier Theory, locus of the most sophisticated semantic analysis of natural language determiners. The work here appears to be the first to examine comparatives within plural logic, a step which appears necessary, but which also harbors specific analytical problems examined here.Since nominal comparatives involve (...) plural and mass reference, we begin with a domain of discourse upon which a lattice structure (Link's) is imposed, and which maps (via abstract dimensions such asweight in kilograms) to concrete measures (in N,R+). The mapping must be homomorphic and Archimedean. Comparisons begin as simple predicates on dimensions or measures; from these we derive classes of predicates on the domain, i.e., generalized determiners (quantifiers), and show, e.g., how monotonicity properties follow in the derivation. This results in a proposal for a logical language which includes derived determiners, and which is an attractive target for semantics interpretation; it also turns out that some interesting comparative determiners are first order, at least when restricted to nonparametric and noncollective predications. (shrink)

Quantified propositional intuitionistic logic is obtained from propositional intuitionistic logic by adding quantifiers ∀p, ∃p, where the propositional variables range over upward-closed subsets of the set of worlds in a Kripke structure. If the permitted accessibility relations are arbitrary partial orders, the resulting logic is known to be recursively isomorphic to full second-order logic (Kremer, 1997). It is shown that if the Kripke structures are restricted to trees of at height and width at most ω, the resulting logics (...) are decidable. This provides a partial answer to a question by Kremer. The result also transfers to modal S4 and some Gödel-Dummett logics with quantifiers over propositions. (shrink)

Following Panaccio, 'John believes that p' is analysed as 'For some x such that x is true if and only if p, John believes x'. On this view the complement clause 'that p' acts as a restricted existential quantifier and it contributes a higher-order property.

We introduce a restricted second-order logic $\textrm{SO}^{\textit{plog}}$ for finite structures where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. We demonstrate the relevance of this logic and complexity class by several problems in database theory. We then prove a Fagin’s style theorem showing that the Boolean queries which can be expressed in the existential fragment of $\textrm{SO}^{\textit{plog}}$ correspond exactly to the class of decision problems that can be computed by a non-deterministic (...) Turing machine with random access to the input in time $O^k)$ for some $k \ge 0$, i.e. to the class of problems computable in non-deterministic poly-logarithmic time. It should be noted that unlike Fagin’s theorem which proves that the existential fragment of second-order logic captures NP over arbitrary finite structures, our result only holds over ordered finite structures, since $\textrm{SO}^{\textit{plog}}$ is too weak as to define a total order of the domain. Nevertheless, $\textrm{SO}^{\textit{plog}}$ provides natural levels of expressibility within poly-logarithmic space in a way which is closely related to how second-order logic provides natural levels of expressibility within polynomial space. Indeed, we show an exact correspondence between the quantifier prefix classes of $\textrm{SO}^{\textit{plog}}$ and the levels of the non-deterministic poly-logarithmic time hierarchy, analogous to the correspondence between the quantifier prefix classes of second-order logic and the polynomial-time hierarchy. Our work closely relates to the constant depth quasipolynomial size AND/OR circuits and corresponding restricted second-order logic defined by David A. Mix Barrington in 1992. We explore this relationship in detail. (shrink)

The language of quantification theory does not seem to adequately reflect the logic of mass terms in ordinary english. Mass terms are treated as though they are true of objects which can be counted. In this paper, It is argued that by placing certain restrictions upon formulas which contain the identity sign it is possible to arrive at a formalization of mass term sentences which avoids this difficulty. The proposed restrictions are defended against charges that certain mass term sentences seem (...) to be rendered inexpressible. (shrink)

This chapter examines Bunge’s distinction between the logical concept of existence and the ontological one. We introduce a new conceptual existence predicate in an intensional environment that depends on the evaluation world. So that we can investigate restricted areas where the different kinds of concepts might exist. We hope this new predicate would encompass Bunge’s philosophical position which he designates as conceptualist and fictional materialism. The basic hybridization acts as a bridge between intensions and extensions because @ works as (...) a useful rigidifier. In hybrid logic, the accessibility relation and many properties this relation might have can be easily expressed in the formal language. The initial hypothesis is that hybridization and intensionality can serve as unifying tools in the areas involved in this research; namely, Logic, Philosophy of Science and Linguistics. (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)

Two theories of implicit domain restriction have gained considerable prominence over the last two decades. According to von Fintel (Restrictions on quantifier domaines, Ph.D. thesis, University of Massachusetts at Amherst, 1994), quantifiers come with covert restrictors and, as a result of this, induce domain restriction; according to Stanley [in Gerhard and Peter (eds) Logical form and language, Clarendon Press, Oxford, 2002; Stanley and Szabó (Mind Lang 15(2–3):2192–2161, 2000)], by contrast, nouns, as opposed to quantifiers, come with covert restrictors. (...) In this article, I do three things. First, I assess the arguments that have been given for and against these two accounts and show that none of them is conclusive. Second, I advance a novel empirical argument based on the observed pragmatic behaviour of bare nouns, an argument that falsifies Stanley’s theory while providing clear evidence in support of von Fintel’s (1994). Finally, I discuss the relevance of the bare noun data in the context of another important debate—namely, whether domain restriction is a local mechanism only, or whether it can also be achieved by global means. (shrink)

This paper demonstrates the undecidability of a number of logics with quantification over public announcements: arbitrary public announcement logic, group announcement logic, and coalition announcement logic. In APAL we consider the informative consequences of any announcement, in GAL we consider the informative consequences of a group of agents all of which are simultaneously making known announcements. So this is more restrictive than APAL. Finally, CAL is as GAL except that we now quantify over anything the agents not in that group (...) may announce simultaneously as well. The logic CAL therefore has some features of game logic and of ATL. We show that when there are multiple agents in the language, the satisfiability problem is undecidable for APAL, GAL, and CAL. In the single agent case, the satisfiability problem is decidable for all three logics. (shrink)

Quantificational determiners are often said to be devices for expressing relations. For example, the meaning of _every_ is standardly described as the inclusion relation, with a sentence like _every frog is green_ meaning roughly that the green things include the frogs. Here, we consider an older, non-relational alternative: determiners are tools for creating restricted quantifiers. On this view, determiners specify how many elements of a restricted domain (e.g., the frogs) satisfy a given condition (e.g., being green). One (...) important difference concerns how the determiner treats its two grammatical arguments. On the relational view, the arguments are on a logical par as independent terms that specify the two relata. But on the restricted view, the arguments play distinct logical roles: specifying the limited domain versus supplying an additional condition on domain entities. We present psycholinguistic evidence suggesting that the restricted view better describes what speakers know when they know the meaning of a determiner. In particular, we find that when asked to evaluate sentences of the form _every F is G_, participants mentally group the Fs but not the Gs. Moreover, participants forego representing the group defined by the intersection of F and G. This tells against the idea that speakers understand _every F is G_ as implying that the Fs bear relation (e.g., inclusion) to a second group. (shrink)

It is well known that occurrences of sentences such as “Every bottle is empty” will sometimes be understood relative to a subset of the set of all bottles in the universe. Much has been written about what mechanism should be used to model this phenomenon of domain restriction. However, comparatively little attention has been paid to the question of when domain restriction is triggered. I will begin by challenging a recent partial answer to this question. I will then develop my (...) own partial answer based on observations pertaining to information structure. (shrink)