I argue that the conjunction of perdurantism (the view that objects are temporally extended) and universalism (the thesis that any old class of things has a mereological fusion) gives rise to undesired complications when combined with certain plausible assumptions concerning the semantics of tensed statements.

This chapter argues that special quantifiers such as 'something' when occurring in argument position are not ordinary or substitutional quantifiers; rather they have a reifying force introducing a domain of tropes or kinds of tropes to quantify over.

Some philosophers, for example Quine, doubt the possibility of jointly using modalities and quantification. Simple model-theoretic considerations, however, lead to a reconciliation of quantifiers with such modal concepts as logical, physical, and ethical necessity, and suggest a general class of modalities of which these are instances. A simple axiom system, analogous to the Lewis systems S1 —S5, is considered in connection with this class of modalities. The system proves to be complete, and its class of theorems decidable.

I am interested in extending modal calculi by adding propositional quantifiers, given by the rules for quantifier introduction: provided that p does not occur free in A.

I am a friend of supervaluationism. A statement lacks a definite truth value if, and only if, it comes out true on some admissible ways of precisifying the semantics of the relevant vocabulary and false on others. In this paper, I focus on the special case of identity statements. I take it that such statements, too, may occasionally suffer a truth-value gap, including philosophically significant instances. Yet there is a potentially devastating objection that can be raised against the supervaluationist treatment (...) of such cases—in fact two objections. Luckily, both can be resisted. But seeing how requires that we take a closer look at the ontological presuppositions of supervaluationism, allowing for more leeway than is usually supposed. (shrink)

In this paper I revisit the main arguments for a predicate analysis of descriptions in order to determine whether they do in fact undermine Russell's theory. I argue that while the arguments without doubt provide powerful evidence against Russell's original theory, it is far from clear that they tell against a quantificational account of descriptions.

We study the non-Fregean propositional logic with propositional quantifiers, denoted by $\mathsf{SCI}_{\mathsf{Q}}$. We prove that $\mathsf{SCI}_{\mathsf{Q}}$ does not have the finite model property and that it is undecidable. We also present examples of how to interpret in $\mathsf{SCI}_{\mathsf{Q}}$ various mathematical theories, such as the theory of groups, rings, and fields, and we characterize the spectra of $\mathsf{SCI}_{\mathsf{Q}}$-sentences. Finally, we present a translation of $\mathsf{SCI}_{\mathsf{Q}}$ into a classical two-sorted first-order logic, and we use the translation to prove some model-theoretic properties (...) of $\mathsf{SCI}_{\mathsf{Q}}$. (shrink)

§1. Introduction. After the pioneering work of Mostowski [29] and Lindström [23] it was Jon Barwise's papers [2] and [3] that brought abstract model theory and generalized quantifiers to the attention of logicians in the early seventies. These papers were greeted with enthusiasm at the prospect that model theory could be developed by introducing a multitude of extensions of first order logic, and by proving abstract results about relationships holding between properties of these logics. Examples of such properties areκ-compactness.Any (...) set of sentences of cardinality ≤ κ, every finite subset of which has a model, has itself a model.Löwenheim-Skolem Theorem down toκ.If a sentence of the logic has a model, it has a model of cardinality at most κ.Interpolation Property.If ϕ and ψ are sentences such that ⊨ ϕ → Ψ, then there is θ such that ⊨ ϕ → θ, ⊨ θ → Ψ and the vocabulary of θ is the intersection of the vocabularies of ϕ and Ψ.Lindstrom's famous theorem characterized first order logic as the maximal ℵ0-compact logic with Downward Löwenheim-Skolem Theorem down to ℵ0. With his new concept of absolute logics Barwise was able to get similar characterizations of infinitary languagesLκω. But hopes were quickly frustrated by difficulties arising left and right, and other areas of model theory came into focus, mainly stability theory. No new characterizations of logics comparable to the early characterization of first order logic given by Lindström and of infinitary logic by Barwise emerged. What was first called soft model theory turned out to be as hard as hard model theory. (shrink)

§1. Introduction. After the pioneering work of Mostowski [29] and Lindström [23] it was Jon Barwise's papers [2] and [3] that brought abstract model theory and generalized quantifiers to the attention of logicians in the early seventies. These papers were greeted with enthusiasm at the prospect that model theory could be developed by introducing a multitude of extensions of first order logic, and by proving abstract results about relationships holding between properties of these logics. Examples of such properties areκ-compactness.Any (...) set of sentences of cardinality ≤ κ, every finite subset of which has a model, has itself a model.Löwenheim-Skolem Theorem down toκ.If a sentence of the logic has a model, it has a model of cardinality at most κ.Interpolation Property.If ϕ and ψ are sentences such that ⊨ ϕ → Ψ, then there is θ such that ⊨ ϕ → θ, ⊨ θ → Ψ and the vocabulary of θ is the intersection of the vocabularies of ϕ and Ψ.Lindstrom's famous theorem characterized first order logic as the maximal ℵ0-compact logic with Downward Löwenheim-Skolem Theorem down to ℵ0. With his new concept of absolute logics Barwise was able to get similar characterizations of infinitary languagesLκω. But hopes were quickly frustrated by difficulties arising left and right, and other areas of model theory came into focus, mainly stability theory. No new characterizations of logics comparable to the early characterization of first order logic given by Lindström and of infinitary logic by Barwise emerged. What was first called soft model theory turned out to be as hard as hard model theory. (shrink)

In this paper, we show how the problem of accounting for the semanticsof temporal preposition phrases (tPPs) leads us to some surprisinginsights into the semantics of temporal expressions ingeneral. Specifically, we argue that a systematic treatment of EnglishtPPs is greatly facilitated if we endow our meaning assignments with context variables, a device which allows a tPP to restrict domainsof quantification arising elsewhere in a sentence. We observe that theuse of context variables implies that tPPs can modify expressions intwo ways, and (...) we use this observation to predict the behaviour of tPPswhose complements are themselves modified by other tPPs. (shrink)

It is argued that the usual proposals for dealing with mass-Quantification--All x is f--Are inadequate with the predicate is complex or when multiple quantification is considered. Mass-Quantification is seen as a generalisation of ordinary (thing) quantification in that the specialising assumption that the domain of quantification is atomic is not made. It is suggested that the semantic values of predicates are complete ideals of the boolean algebra consisting of the quantity which is the domain of quantification and all its sub-Quantities, (...) While logical connectives are represented by operations on complete ideals. (shrink)

Previous theorists have claimed that Russell’s theory of definite descriptions gives the wrong truth conditions to sentences in which definite descriptions are embedded under certain other operators; but the other operators used, such as conditionals and propositional attitude verbs, have introduced intensional and hyperintensional complications that might be thought to obscure the point against Russell. This paper shows that the same kind of problem arises when the operator in question allows the context to be extensional. It is further argued that (...) presuppositional theories of definite descriptions give intuitively satisfying analyses of the novel data. (shrink)

We prove a local normal form theorem of the Gaifman type for the infinitary logic L∞ωω whose formulas involve arbitrary unary quantifiers but finite quantifier rank. We use a local Ehrenfeucht-Fraïssé type game similar to the one in [9]. A consequence is that every sentence of L∞ωω of quantifier rank n is equivalent to an infinite Boolean combination of sentences of the form ψ, where ψ has counting quantifiers restricted to the -neighborhood of y.

Here, I combine the semantics of Mares and Goldblatt [20] and Seki [29, 30] to develop a semantics for quantified modal relevant logics extending ${\bf B}$. The combination requires demonstrating that the Mares–Goldblatt approach is apt for quantified extensions of ${\bf B}$ and other relevant logics, but no significant bridging principles are needed. The result is a single semantic approach for quantified modal relevant logics. Within this framework, I discuss the requirements a quantified modal relevant logic must satisfy to be (...) “sufficiently classical” in its modal fragment, where frame conditions are given that work for positive fragments of logics. The roles of the Barcan formula and its converse are also investigated. (shrink)

We consider a class of Lindström extensions of first-order logic which are susceptible to a natural Skolemization procedure. In these logics Ehrenfeucht Mostowski (EM) functors for theories with arbitrarily large models can be obtained under suitable restrictions. Characteristic dependencies between algebraic properties of the quantifiers and the maximal domains of EM functors are investigated.Results are applied to Magidor Malitz logic,L(Q <ω), showing e.g. its Hanf number to be equal to ℶω(ℵ1) in the countably compact case. Using results of Baumgartner, (...) the maximal number of isomorphism types of linearly ordered models of regular cardinality is shown to be achieved for theories that admit an EM functor on a typically restricted domain. (shrink)

THEOREM 1. (⋄ ℵ 1 ) If B is an infinite Boolean algebra (BA), then there is B 1 such that $|\operatorname{Aut} (B_1)| \leq B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut} (B_1)\rangle \equiv \langle B, \operatorname{Aut}(B)\rangle$ . THEOREM 2. (⋄ ℵ 1 ) There is a countably compact logic stronger than first-order logic even on finite models. This partially answers a question of H. Friedman. These theorems appear in §§ 1 and 2. THEOREM 3. (a) (⋄ ℵ 1 ) If (...) B is an atomic ℵ 1 -saturated infinite BA, ψ ε L ω 1ω and $\langle B, \operatorname{Aut} (B)\rangle \models\psi$ then there is B 1 such that $|\operatorname{Aut}(B_1)| \leq |B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut}(B_1)\rangle\models\psi$ . In particular if B is 1-homogeneous so is B 1 . (b) (a) holds for B = P(ω) even if we assume only CH. (shrink)

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)

We prove that the logics of Magidor-Malitz and their generalization by Rubin are distinct even for PC classes. Let $M \models Q^nx_1 \cdots x_n \varphi(x_1 \cdots x_n)$ mean that there is an uncountable subset A of |M| such that for every $a_1, \ldots, a_n \in A, M \models \varphi\lbrack a_1, \ldots, a_n\rbrack$ . Theorem 1.1 (Shelah) $(\diamond_{\aleph_1})$ . For every n ∈ ω the class $K_{n + 1} = \{\langle A, R\rangle \mid \langle A, R\rangle \models \neg Q^{n + 1} (...) x_1 \cdots x_{n + 1} R(x_1, \ldots, x_{n + 1})\}$ is not an ℵ 0 -PC-class in the logic L n , obtained by closing first order logic under Q 1 , ..., Q n . I.e. for no countable L n -theory T, is K n + 1 the class of reducts of the models of T. Theorem 1.2 (Rubin) $(\diamond_{\aleph_1}).^3$ . Let $M \models Q^E x y\varphi(x, y)$ mean that there is $A \subseteq |M|$ such that $E_{A, \varphi} = \{\langle a, b \rangle \mid a, b \in A$ and $M \models \varphi\lbrack a, b\rbrack\}$ is an equivalence relation on A with uncountably many equivalence classes, and such that each equivalence class is uncountable. Let $K^E = \{\langle A, R\rangle\mid \langle A, R\rangle\models \neg Q^Exy R(x, y)\}$ . Then K E is not an ℵ 0 -PC-class in the logic gotten by closing first order logic under the set of quantifiers {Q n ∣ n ∈ ω} which were defined in Theorem 1.1. (shrink)

This volume on the semantic complexity of natural language explores the question why some sentences are more difficult than others. While doing so, it lays the groundwork for extending semantic theory with computational and cognitive aspects by combining linguistics and logic with computations and cognition. -/- Quantifier expressions occur whenever we describe the world and communicate about it. Generalized quantifier theory is therefore one of the basic tools of linguistics today, studying the possible meanings and the inferential power of quantifier (...) expressions by logical means. The classic version was developed in the 1980s, at the interface of linguistics, mathematics and philosophy. Before this volume, advances in "classic" generalized quantifier theory mainly focused on logical questions and their applications to linguistics, this volume adds a computational component, the third pillar of language use and logical activity. This book is essential reading for researchers in linguistics, philosophy, cognitive science, logic, AI, and computer science. (shrink)