The question of the origin of polyadic expressivity is explored and the results are brought to bear on Bertrand Russell's 1903 theory of denoting concepts, which is the main object of criticism in his 1905 "On Denoting". It is shown that, appearances to the contrary notwithstanding, the background ontology of the earlier theory of denoting enables the full-blown expressive power of first-order polyadic quantification theory without any syntactic accommodation of scopal differences among denoting phrases such as 'all (...) φ', 'every φ', and 'any φ' on the one hand, and 'some φ' and 'a φ' on the other. The case provides an especially vivid illustration of the general point that structural (or ideological) austerity can be paid for in the coin of ontological extravagance. (shrink)

In this paper, I have proposed a compositional semantic analysis of exception NPs from which three core properties of exception constructions could be derived. I have shown that this analysis overcomes various empirical and conceptual shortcomings of prior proposals of the semantics of exception sentences. The analysis was first formulated for simple exception NPs, where the EP-complement was considered a set-denoting term and the EP-associate was a monadic quantifier. It was then generalized in two steps: first, in order to account (...) for quantified EP-complements, and second, in order to account for polyadic quantifiers as the EP-associates. An additional assumption that was made in several places was that EPs may operate at the level of implications. The consequences of this assumption, though, still have to be investigated. (shrink)

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

The aim of this paper is to study the n-variable fragment of first order logic from a modal perspective. We define a modal formalism called cylindric mirror modal logic, and show how it is a modal version of first order logic with substitution. In this approach, we can define a semantics for the language which is closely related to algebraic logic, as we find Polyadic Equality Algebras as the modal or complex algebras of our system. The main contribution of (...) the paper is a characterization of the intended ‘mirror cubic’ frames of the formalisms and, a consequence of the special form of this characterization, a completeness theorem for these intended frames. As a consequence, we find complete finite yet unorthodox derivation systems for the equational theory of finite-dimensional representable polyadic equality algebras. (shrink)

There are enormous differences between quantifying name-variables only, quantifying verb-variables only, and quantifying both. These differences are found only in the logic of polyadic predication; and this presumably is why Richard Gaskin thinks that they distinguish names from transitive verbs only, and not from verbs generally. But that thought is mistaken: these differences also distinguish names from intransitive verbs. They thus vindicate the common idea that on the difference between names and verbs we may base grandiose metaphysical distinctions, and (...) undermine Gaskin's idea that both names and verbs may be said to designate objects. (shrink)

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

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

avant propos This paper is basically Keenan (1992) augmented by some new types of properly polyadic quantification in natural language drawn from Moltmann (1992), Nam (1991) and Srivastav (1990). In addition I would draw the reader's attention to recent mathematical studies of polyadic quantiicationz Ben-Shalom (1992), Spaan (1992) and Westerstahl (1992). The first and third of these extend and generalize (in some cases considerably) the techniques and results in Keenan (1992). Finally I would like to acknowledge the (...) stimulating and constructive discussions ofthe earlier paper with many scholars, notably Dorit Ben-Shalom, Jaap van der Does, Hans Kamp, Uwe Mormich, Arnim von Stechow, Mats Rooth, and Ede Zimmermann. And I repeat here the acknowledgment in the earlier paper to Jim Lambek, Ed Stabler and two anonymous referees for Linguistics and Philosophy (the latter responsible for substantial improvements in the proofs - see footnote 10). (shrink)

Many arguments for flexible type assignment to syntactic categories have to do with the need to account for the various scopings resulting from the interaction of quantified DPs with other quantified DPs or with intensional or negated verb contexts. We will define a type for arbitrary arity relations in polymorphic type theory. In terms of this, we develop the Boolean algebra of relations as far as needed for natural language semantics. The type for relations is flexible: it can do duty (...) for the whole family of types t, e → t, e → e → t, and so on. If we use this flexible type for the interpretations of verbs, we can perform a ‘flexible lift’ on DP interpretations, so that DPs get interpreted as operations on verb meanings. This leads to an elegant implementation of Keenan’s proposal for polyadic quantification in natural language (‘Beyond the Frege Boundary’). It turns out that scope reorderings are particularly easy to implement in this framework. (shrink)

Winter (2000) argues that so-called co-distributive or cumulative readings do not involve polyadic quantification (contra proposals by Krifka, Schwarzschild, Sternefeld, and others). Instead, he proposes that all such readings involve a hidden anaphoric dependency or a lexical mechanism. We show that Winter's proposal is insufficient for a number of cases of cumulative readings, and that Krifka's and Sternefeld's polyadic **-operator is needed in addition to dependent definites. Our arguments come from new observations concerning dependent plurals and clause-boundedness (...) effects with cumulative readings. (shrink)

Deductive Logic is an introductory textbook in formal logic. The book is divided into four parts covering (i) truth-functional logic, (ii) monadic quantifi- cation, (iii) polyadic quantification and (iv) names and identity, and there are exercises for all these topics at the end of the book. In the truth-functional logic part, the reader learns to produce paraphrases of English statements and arguments in logical notation (this subsection is called “analysis”), then about the semantic properties of such paraphrased statements (...) and arguments, such as satisfiability, implication and equivalence (“logical assessment”) and finally (“reflection”) there is an axiomatic proof method and some important extras such as disjunctive normal form and expressive adequacy. Parts two and three mirror this analysis/assessment/reflection structure for monadic and polyadic quantification, though this time the proof system is a natural deduction one, and part three contains a completeness proof for that system. The fourth part of the book introduces names, the identity predicate and descriptions and examines the additional expressive power which these provide. (shrink)

Sentences with multiple occurrences of plural definites give rise to certain effects suggesting that distributivity should be modeled by polyadic operations. Yet in this paper it is argued that the simpler treatment of distributivity using unary universal quantification should be retained. Seemingly polyadic effects are claimed to be restricted to definite NPs. This fact is accounted for by the special anaphoric (dependent) use of definites. Further evidence concerning various plurals, island constraints, and cumulative quantification is shown (...) to support this claim. In addition, it is shown that the evidence against a simple atomic version of unary distributivity is not decisive either. In the (uncommon) cases where distributivity with definites is not strictly atomic, they can be analyzed as dependent on implicit quantifiers. (shrink)

In the dissertation we study the complexity of generalized quantifiers in natural language. Our perspective is interdisciplinary: we combine philosophical insights with theoretical computer science, experimental cognitive science and linguistic theories. -/- In Chapter 1 we argue for identifying a part of meaning, the so-called referential meaning (model-checking), with algorithms. Moreover, we discuss the influence of computational complexity theory on cognitive tasks. We give some arguments to treat as cognitively tractable only those problems which can be computed in polynomial time. (...) Additionally, we suggest that plausible semantic theories of the everyday fragment of natural language can be formulated in the existential fragment of second-order logic. -/- In Chapter 2 we give an overview of the basic notions of generalized quantifier theory, computability theory, and descriptive complexity theory. -/- In Chapter 3 we prove that PTIME quantifiers are closed under iteration, cumulation and resumption. Next, we discuss the NP-completeness of branching quantifiers. Finally, we show that some Ramsey quantifiers define NP-complete classes of finite models while others stay in PTIME. We also give a sufficient condition for a Ramsey quantifier to be computable in polynomial time. -/- In Chapter 4 we investigate the computational complexity of polyadic lifts expressing various readings of reciprocal sentences with quantified antecedents. We show a dichotomy between these readings: the strong reciprocal reading can create NP-complete constructions, while the weak and the intermediate reciprocal readings do not. Additionally, we argue that this difference should be acknowledged in the Strong Meaning hypothesis. -/- In Chapter 5 we study the definability and complexity of the type-shifting approach to collective quantification in natural language. We show that under reasonable complexity assumptions it is not general enough to cover the semantics of all collective quantifiers in natural language. The type-shifting approach cannot lead outside second-order logic and arguably some collective quantifiers are not expressible in second-order logic. As a result, we argue that algebraic (many-sorted) formalisms dealing with collectivity are more plausible than the type-shifting approach. Moreover, we suggest that some collective quantifiers might not be realized in everyday language due to their high computational complexity. Additionally, we introduce the so-called second-order generalized quantifiers to the study of collective semantics. -/- In Chapter 6 we study the statement known as Hintikka's thesis: that the semantics of sentences like ``Most boys and most girls hate each other'' is not expressible by linear formulae and one needs to use branching quantification. We discuss possible readings of such sentences and come to the conclusion that they are expressible by linear formulae, as opposed to what Hintikka states. Next, we propose empirical evidence confirming our theoretical predictions that these sentences are sometimes interpreted by people as having the conjunctional reading. -/- In Chapter 7 we discuss a computational semantics for monadic quantifiers in natural language. We recall that it can be expressed in terms of finite-state and push-down automata. Then we present and criticize the neurological research building on this model. The discussion leads to a new experimental set-up which provides empirical evidence confirming the complexity predictions of the computational model. We show that the differences in reaction time needed for comprehension of sentences with monadic quantifiers are consistent with the complexity differences predicted by the model. -/- In Chapter 8 we discuss some general open questions and possible directions for future research, e.g., using different measures of complexity, involving game-theory and so on. -/- In general, our research explores, from different perspectives, the advantages of identifying meaning with algorithms and applying computational complexity analysis to semantic issues. It shows the fruitfulness of such an abstract computational approach for linguistics and cognitive science. (shrink)

Recent technical developments in the logic of nominalism make it possible to improve and extend significantly the approach to mathematics developed in Mathematics without Numbers. After reviewing the intuitive ideas behind structuralism in general, the modal-structuralist approach as potentially class-free is contrasted broadly with other leading approaches. The machinery of nominalistic ordered pairing (Burgess-Hazen-Lewis) and plural quantification (Boolos) can then be utilized to extend the core systems of modal-structural arithmetic and analysis respectively to full, classical, polyadic third- and (...) fourthorder number theory. The mathenatics of many structures of central importance in functional analysis, measure theory, and topology can be recovered within essentially these frameworks. (shrink)

We argue that a comprehensive theory of reciprocals must rely on a general taxonomy of restrictions on the interpretation of relational expressions. Developing such a taxonomy, we propose a new principle for interpreting reciprocals that relies on the interpretation of the relation in their scope. This principle, the Maximal Interpretation Hypothesis (MIH), analyzes reciprocals as partial polyadic quantifiers. According to the MIH, the partial quantifier denoted by a reciprocal requires the relational expression REL in its scope to denote a (...) maximal relation in REL’s interpretation domain. In this way the MIH avoids a priori assumptions on the available readings of reciprocal expressions, which are necessary in previous accounts. Relying extensively on the work of Dalrymple et al. (Ling Philos 21:159–210, 1998), we show that the MIH also exhibits some observational improvements over Dalrymple et al.’s Strongest Meaning Hypothesis (SMH). In addition to deriving some attested reciprocal interpretations that are not expected by the SMH, the MIH offers a more restrictive account of the way context affects the interpretation of reciprocals through its influence on relational domains. Further, the MIH generates a reciprocal interpretation at the predicate level, which is argued to be advantageous to Dalrymple et al.’s propositional selection of reciprocal meanings. More generally, we argue that by focusing on restrictions on relational domains, the MIH opens the way for a more systematic study of the ways in which lexical meaning, world knowledge and contextual information interact with the interpretation of quantificational expressions. (shrink)

This paper addresses the two interpretations that a combination ofnegative indefinites can get in concord languages like French:a concord reading, which amounts to a single negation, and a doublenegation reading. We develop an analysis within a polyadic framework,where a sequence of negative indefinites can be interpreted as aniteration of quantifiers or via resumption. The first option leadsto a scopal relation, interpreted as double negation. The secondoption leads to the construction of a polyadic negative quantifiercorresponding to the concord reading. (...) Given that sentential negationparticipates in negative concord, we develop an extension of thepolyadic approach which can deal with non-variable binding operators,treating the contribution of negation in a concord context assemantically empty. Our semantic analysis, incorporated into agrammatical analysis formulated in HPSG, crucially relies on theassumption that quantifiers can be combined in more than one wayupon retrieval from the quantifier store. We also considercross-linguistic variation regarding the participation ofsentential negation in negative concord. (shrink)

Theabstract variable binding calculus (VB-calculus) provides a formal frame-work encompassing such diverse variable-binding phenomena as lambda abstraction, Riemann integration, existential and universal quantification (in both classical and nonclassical logic), and various notions of generalized quantification that have been studied in abstract model theory. All axioms of the VB-calculus are in the form of equations, but like the lambda calculus it is not a true equational theory since substitution of terms for variables is restricted. A similar problem with the (...) standard formalism of the first-order predicate logic led to the development of the theory of cylindric and polyadic Boolean algebras. We take the same course here and introduce the variety of polyadic VB-algebras as a pure equational form of the VB-calculus. In one of the main results of the paper we show that every locally finite polyadic VB-algebra of infinite dimension is isomorphic to a functional polyadic VB-algebra that is obtained from a model of the VB-calculus by a natural coordinatization process. This theorem is a generalization of the functional representation theorem for polyadic Boolean algebras given by P. Halmos. As an application of this theorem we present a strong completeness theorem for the VB-calculus. More precisely, we prove that, for every VB-theory T that is obtained by adjoining new equations to the axioms of the VB-calculus, there exists a model D such that T s=t iff D s=t. This result specializes to a completeness theorem for a number of familiar systems that can be formalized as VB-calculi. For example, the lambda calculus, the classical first-order predicate calculus, the theory of the generalized quantifierexists uncountably many and a fragment of Riemann integration. (shrink)

In recent papers Fred Sommers and Michael Lockwood have independently argued that the distinction between the ‘is’ of predication and the ‘is’ of identity is not well-founded. This claim is somewhat obscure since, on the theories they advocate, it is not only still possible to distinguish between the ‘is’ of predication and the ‘is’ of identity, but important to do so on pain of turning good arguments bad. Sommers' way of putting it, namely that we don't need identity, is no (...) better. Of course we don't need identity since, in second-order quantification logic, identity is definable in terms of predication. Perhaps the best way of putting their point is by means of Sommers' claim that identity statements can be formalized by monadic relations. We can therefore mark the IP-distinction by the essentially polyadic nature of identity. (shrink)

This chapter offers a logical, linguistic, and philosophical account of modern quantification theory. Contrasting the standard approach to quantifiers (according to which logical quantifiers are defined by enumeration) with the generalized approach (according to which quantifiers are defined systematically), the chapter begins with a brief history of standard quantifier theory and identifies some of its logical, linguistic, and philosophical strengths and weaknesses. It then proceeds to a brief history of generalized quantifier theory and explains how it overcomes the weaknesses (...) of the standard theory. One of the main philosophical advantages of the generalized theory is its philosophically informative criterion of logicality. The chapter describes the work done so far in this theory, highlights some of its central logical results, offers an overview of its main linguistic contributions, and discusses its philosophical significance. (shrink)

This edited volume explores the intersection of medicine and philosophy throughout history, calling attention to the role of quantification in understanding the medical body. Retracing current trends and debates to examine the quantification of the body throughout the early modern, modern and early contemporary age, the authors contextualise important issues of both medical and philosophical significance, with chapters focusing on the quantification of temperaments and fluids, complexions, functions of the living body, embryology, and the impact of quantified (...) reasoning on the concepts of health and illness. With insights spanning from the sixteenth century to the nineteenth century, this book provides a wide-ranging overview of attempts to ‘quantify’ the human body at various points. Arguing that medicine and philosophy have been constantly in dialogue with each other, the authors discuss how this provided a strategic opportunity both for medical thought and philosophy to refine and further develop. Given today’s fascination with the quantification of the body, represented by the growing profusion of self-tracking devices logging one’s sleep, diet or mood, this collection offers an important and timely contribution to an emerging and interdisciplinary field of study. -/- . (shrink)

I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit (...) quantification, but on the logical conception of set which motivates naive set theory. The accepted solution is to replace this with the iterative conception of set. -/- I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2--3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected. (shrink)

Rough polyadic modal logics, introduced in the paper, contain modal operators of many arguments with a relational semantics, based on the Pawlak's rough set theory. Rough set approach is developed as an alternative to the fuzzy set philosophy, and has many applications in different branches in Artificial Intelligence and theoretical computer science.

I think it would be fair to say that, until about 1900, philosophers were generally reluctant to admit the existence of what are nowadays called polyadic properties.1 It is important to recognize, however, that this reluctance on the part of pre-twentieth-century philosophers did not prevent them from theorizing about relations. On the contrary, philosophers from the ancient through the modern period have had much to say about both the nature and the ontological status of relations. In this paper I (...) examine the views of one such philosopher, namely, Albert the Great. (shrink)

Ordinary English contains different forms of quantification over objects. In addition to the usual singular quantification, as in 'There is an apple on the table', there is plural quantification, as in 'There are some apples on the table'. Ever since Frege, formal logic has favored the two singular quantifiers ∀x and ∃x over their plural counterparts ∀xx and ∃xx (to be read as for any things xx and there are some things xx). But in recent decades it (...) has been argued that we have good reason to admit among our primitive logical notions also the plural quantifiers ∀xx and ∃xx. More controversially, it has been argued that the resulting formal system with plural as well as singular quantification qualifies as ‘pure logic’; in particular, that it is universally applicable, ontologically innocent, and perfectly well understood. In addition to being interesting in its own right, this thesis will, if correct, make plural quantification available as an innocent but extremely powerful tool in metaphysics, philosophy of mathematics, and philosophical logic. For instance, George Boolos has used plural quantification to interpret monadic second-order logic and has argued on this basis that monadic second-order logic qualifies as “pure logic.” Plural quantification has also been used in attempts to defend logicist ideas, to account for set theory, and to eliminate ontological commitments to mathematical objects and complex objects. (shrink)

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction rules for (...) the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement. (shrink)

Relative to an ordinary context, an utterance of the sentence ‘Everything is in the car’ communicates a proposition about a restricted domain. But how does this work? One possibility is that quantifier expressions like 'everything' are context sensitive and range over different domains in different contexts. Another possibility is that quantifier expressions are not context sensitive, but have a fixed, absolutely general meaning, and ordinary utterances communicate a restricted content via Gricean mechanisms. I argue that, contrary to received opinion, the (...) latter view has both a number of methodological and also intuitive advantages over the former. I then reply to three objections to the latter view: the binding argument (due to Stanley and Szabo), the availability-based attack (due to Recanati), and an argument based on Recanati’s scope principle. (shrink)

It is known that every α-dimensional quasi polyadic equality algebra (QPEA α ) can be considered as an α-dimensional cylindric algebra satisfying the merrygo- round properties . The converse of this proposition fails to be true. It is investigated in the paper how to get algebras in QPEA from algebras in CA. Instead of QPEA the class of the finitary polyadic equality algebras (FPEA) is investigated, this class is definitionally equivalent to QPEA. It is shown, among others, that (...) from every algebra in a β-dimensional algebra can be obtained in QPEA β where , moreover the algebra obtained is representable in a sense. (shrink)

This chapter discusses ontological commitment to properties, understood as ontological correlates of predicates. We examine the issue in four metaontological settings, beginning with an influential Quinean paradigm on which ontology concerns what there is. We argue that this naturally but not inevitably avoids ontological commitment to properties. Our remaining three settings correspond to the most prominent departures from the Quinean paradigm. Firstly, we enrich the Quinean paradigm with a primitive, non-quantificational notion of existence. Ontology then concerns what exists. We argue (...) that this strengthens the Quinean case against ontological commitment to properties while also newly distinguishing between stronger and weaker forms of nominalism. Secondly, we enrich the Quinean paradigm with the ideology of fundamentality. Ontology then centrally concerns what’s fundamental. We argue that this leaves ontological commitment to properties wide open although Bradleyan regress threatens. Thirdly, we enrich the Quinean paradigm with primitive higher-order quantifiers. Ontology then expands to concern what there higher-order is and what there first-order is. We argue that this naturally but not inevitably incurs ontological commitment to properties. (shrink)

In this note we give an interpretation of cylindric algebras as algebras of sentences of first order logic. We show that the isomorphism types of such algebras of sentences coincide with the class of neat reducts of cylindric algebras. Also we show how this interpretation sheds light on some recent results. This is done by likening Henkin's Neat Embedding Theorem to his celebrated completeness proof.

We define an interpretation of modal languages with polyadic operators in modal languages that use monadic operators (diamonds) only. We also define a simulation operator which associates a logic $\Lambda^{sim}$ in the diamond language with each logic Λ in the language with polyadic modal connectives. We prove that this simulation operator transfers several useful properties of modal logics, such as finite/recursive axiomatizability, frame completeness and the finite model property, canonicity and first-order definability.

The question whether natural language permits quantification over intentional objects as the ‘nonexistent’ objects of thought is the topic of a major philosophical controversy, as is the status of intentional objects as such. This paper will argue that natural language does reflect a particular notion of intentional object and in particular that certain types of natural language constructions (generally disregarded in the philosophical literature) cannot be analysed without positing intentional objects. At the same time, those intentional objects do not (...) come for free; rather they are strictly dependent on intentional acts that generally need to have a presence, in one way or another, in the semantic structure of the sentence. (shrink)

Semantic interpretations of both natural and formal languages are usually taken to involve the specification of a domain of entities with respect to which the sentences of the language are to be evaluated. A question that has received much attention of late is whether there is unrestricted quantification, quantification over a domain comprising absolutely everything there is. Is there a discourse or inquiry that has absolute generality? After framing the debate, this article provides an overview of the main (...) arguments for and against the possibility of unrestricted quantification, highlighting some of the broader implications of the debate. (shrink)

This essay introduces a puzzle about the interaction between quantifiers and epistemic modals. The puzzle motivates the idea that whether an object satisfies an epistemically modalized predicate depends on the mode of presentation of the domain of quantification. I compare two ways of implementing this idea, one using counterpart theory, the other using Aloni's 'conceptual covers' theory, and then provides some evidence in favor of the former.

This paper criticizes George Boolos's famous use of plural quantification to argue that monadic second-order logic is pure logic. I deny that plural quantification qualifies as pure logic and express serious misgivings about its alleged ontological innocence. My argument is based on an examination of what is involved in our understanding of the impredicative plural comprehension schema.

This book surveys research in quantification starting with the foundational work in the 1970s. It paints a vivid picture of generalized quantifiers and Boolean semantics. It explains how the discovery of diverse scope behavior in the 1990s transformed the view of quantification, and how the study of the internal composition of quantifiers has become central in recent years. It presents different approaches to the same problems, and links modern logic and formal semantics to advances in generative syntax. A (...) unique feature of the book is that it systematically brings cross-linguistic data to bear on the theoretical issues, discussing French, German, Dutch, Hungarian, Russian, Japanese, Telugu (Dravidian), and Shupamem (Grassfield Bantu), and pointing to formal semantic literature involving quantification in around thirty languages. -- -/- 1. What this book is about and how to use it; 2. Generalized quantifiers and their elements: operators and their scopes; 3. Generalized quantifiers in non-nominal domains; 4. Some empirically significant properties of quantifiers and determiners; 5. Potential challenges for generalized quantifiers; 6. Scope is not uniform and not a primitive; 7. Existential scope versus distributive scope; 8. Distributivity and scope; 9. Bare numeral indefinites; 10. Modified numerals; 11. Clause-internal scopal diversity; 12. Towards a compositional semantics of quantifier words. (shrink)

Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the categoricity (...) problem is generated by the finite nature of the standard calculi and one direction in which it can be solved is to strengthen the deductive systems by adding infinitary rules (such as the ω-rule), i.e., to construct a full formalization. Another main direction is to provide a natural semantics for the standard rules of inference, i.e., a semantics for which these rules are categorical. My aim in this paper is to analyze some recent approaches for solving the categoricity problem and to argue that a logical inferentialist should accept the infinitary rules of inference for the first order quantifiers, since our use of the expressions “all” and “there is” leads us beyond the concrete and finite reasoning, and human beings do sometimes employ infinitary rules of inference in their reasoning. (shrink)

In 2015, A.V. Figallo and G. Pelaitay introduced tense n×m-valued Łukasiewicz–Moisil algebras, as a common generalization of tense Boolean algebras and tense n-valued Łukasiewicz–Moisil algebras. Here we initiate an investigation into the class tpLMn×m of tense polyadic n × m-valued Łukasiewicz–Moisil algebras. These algebras constitute a generalization of tense polyadic Boolean algebras introduced by Georgescu in 1979, as well as the tense polyadic n-valued Łukasiewicz–Moisil algebras studied by Chiriţă in 2012. Our main result is a representation theorem (...) for tense polyadic n × m-valued Łukasiewicz–Moisil algebras. (shrink)

Semantic theories based on a hierarchy of types have prominently been used to defend the possibility of unrestricted quantification. However, they also pose a prima facie problem for it: each quantifier ranges over at most one level of the hierarchy and is therefore not unrestricted. It is difficult to evaluate this problem without a principled account of what it is for a quantifier to be unrestricted. Drawing on an insight of Russell’s about the relationship between quantification and the (...) structure of predication, we offer such an account. We use this account to examine the problem in three different type-theoretic settings, which are increasingly permissive with respect to predication. We conclude that unrestricted quantification is available in all but the most permissive kind of type theory. (shrink)

I provide a classification of varieties of pantheism. I argue that there are two different kinds of commitments that pantheists have. On the one hand, there is an ontological commitment to the existence of a sum of all things. On the other hand, there is an ideological commitment: either collectively or distributively, the sum of all things is divine.

We propose a generalization of Sahlqvist formulas to polyadic modal languages by representing such languages in a combinatorial PDL style and thus, in particular, developing what we believe to be the right syntactic approach to Sahlqvist formulas at all. The class of polyadic Sahlqvist formulas PSF defined here expands essentially the so far known one. We prove first-order definability and canonicity for the class PSF.

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)

We propose a generalization of Sahlqvist formulae to polyadic modal languages by representing modal polyadic languages in a combinatorial style and thus, in particular, developing what we believe to be the right approach to Sahlqvist formulae at all. The class of polyadic Sahlqvist formulae PSF defined here expands essentially the so far known one. We prove first-order definability and canonicity for the class PSF.

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)

Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the categoricity (...) problem is generated by the finite nature of the standard calculi and one direction in which it can be solved is to strengthen the deductive systems by adding infinite rules (such as the ω-rule), i.e., to construct a full formalization. Another main direction is to provide a natural semantics for the standard rules of inference, i.e., a semantics for which these rules are categorical. My aim in this paper is to analyze some recent approaches for solving the categoricity problem and to argue that a logical inferentialist should accept the infinite rules of inference for the first order quantifiers, since our use of the expressions “all” and “there is” leads us beyond the concrete and finite reasoning, and human beings do sometimes employ infinite rules of inference in their reasoning. (shrink)