I have two main objectives. The first is to get a better understanding of what is at issue between friends and foes of higher-order quantification, and of what it would mean to extend a Boolos-style treatment of second-order quantification to third- and higherorder quantification. The second objective is to argue that in the presence of absolutely general quantification, proper semantic theorizing is essentially unstable: it is impossible to provide a suitably general semantics for a given language (...) in a language of the same logical type. I claim that this leads to a trilemma: one must choose between giving up absolutely general quantification, settling for the view that adequate semantic theorizing about certain languages is essentially beyond our reach, and countenancing an open-ended hierarchy of languages of ever ascending logical type. I conclude by suggesting that the hierarchy may be the least unattractive of the options on the table. (shrink)

This extended collection of papers is the result of putting recent ideas on quantification to work on a wide variety of languages.

This paper gives an overview of the common approach to quantification and generalised quantification in formal linguistics and philosophy of language. We point out how this usual general framework represents a departure from empirical linguistic data. We briefly sketch a different idea for proof theory which is closer to the language itself than standard approaches in many aspects. We stress the importance of Hilbert’s operators—the epsilon-operator for existential and tau-operator for universal quantifications. Indeed, these operators are helpful in (...) the construction of a semantic representation which is close to natural language in particular with quantified noun phrases as individual terms. We also define guidelines for the design of proof rules corresponding to generalized quantifiers. (shrink)

This paper gives an overview of the common approach to quantification and generalised quantification in formal linguistics and philosophy of language. We point out how this usual general framework represents a departure from empirical linguistic data. We briefly sketch a different idea for proof theory which is closer to the language itself than standard approaches in many aspects. We stress the importance of Hilbert’s operators—the epsilon-operator for existential and tau-operator for universal quantifications. Indeed, these operators are helpful in (...) the construction of a semantic representation which is close to natural language in particular with quantified noun phrases as individual terms. We also define guidelines for the design of proof rules corresponding to generalized quantifiers. (shrink)

We develop a variant of Least Fixed Point logic based on First Orderlogic with a relaxed version of guarded quantification. We develop aGame Theoretic Semantics of this logic, and find that under reasonableconditions, guarding quantification does not reduce the expressibilityof Least Fixed Point logic. But we also find that the guarded version ofa least fixed point algorithm may have a greater time complexity thanthe unguarded version, by a linear factor.

Propositional quantifiers are added to a propositional modal language with two modal operators. The resulting language is interpreted over so-called products of Kripke frames whose accessibility relations are equivalence relations, letting propositional quantifiers range over the powerset of the set of worlds of the frame. It is first shown that full second-order logic can be recursively embedded in the resulting logic, which entails that the two logics are recursively isomorphic. The embedding is then extended to all sublogics containing the logic (...) of so-called fusions of frames with equivalence relations. This generalizes a result due to Antonelli and Thomason, who construct such an embedding for the logic of such fusions. (shrink)

There is a growing movement towards construing some classic debates in ontology as meaningless, either because the answers seem obvious or the debates seem intractable. In this paper, I respond to this movement. The response has three components: First, the members of the two sides of the ontological debates that dismissivists have targeted are using different quantifiers. Second, the austere ontologist is using a more fundamental quantifier than her opponent. Third, the austere ontologist’s more fundamental quantifier is a restriction of (...) her opponent’s quantifier. This response takes seriously the intuition that there is something wrong with the ontological disputes, but does not entail dismissivism. (shrink)

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

According to the interpretational theory of logical validity (IR), logical validity is preservation of truth in all interpretations compatible with the intended meaning of logical expressions. IR suffers from a seemingly defeating objection, the so-called cardinality problem: any instance of the statement ‘There are n things’ is true under all interpretations, since it can be written down using only logical expressions that are not to be reinterpreted; yet ‘There are n things’ is not logically true. I argue that the cardinality (...) problem is indeed a serious problem for IR, when understood in terms of ‘asymmetry of information’. I then argue that IR can be rehabilitated by making quantifiers context-sensitive: what we do not reinterpret is the Kaplanian character of a quantifier, rather than its content. ‘There are n things’ is false in a context where fewer than n things are relevant, so it is not logically true in IR. I finally discuss some objections and ramifications of my account: I discuss how to make space for the possibility of an explicitly absolutely general quantifier in my framework, how terms can be logical even though context-sensitive, and how to recapture classical logic within my framework. (shrink)

In this work we define contingency logic with arbitrary announcement. In contingency logic, the primitive modality contingency formalises that a proposition may be true but also may be false, so that if it is non-contingent then it is necessarily true or necessarily false. To this logic one can add dynamic operators to describe change of contingency. Our logic has operators for public announcement and operators for arbitrary public announcement, as in the dynamic epistemic logic called arbitrary public announcement logic. However, (...) our language primitive is the more suitable notion of public announcement whether, instead of the public announcement that. We compare the expressive power of our logic and its various fragments to related dynamic epistemic logics. We further present an axiomatisation and show its completeness by adapting a method to demonstrate completeness of arbitrary public announcement logic. Various extensions are also shown to be complete with respect to the corresponding frame classes. (shrink)

This paper offers a semantic study in multi-relational semantics of quantified N-Monotonic modal logics with varying domains with and without the identity symbol. We identify conditions on frames to characterise Barcan and Ghilardi schemata and present some related completeness results. The characterisation of Barcan schemata in multi-relational frames with varying domains shows the independence of BF and CBF from well-known propositional modal schemata, an independence that does not hold with constant domains. This fact was firstly suggested for classical modal systems (...) by Stolpe, 557–575, 2003), but unfortunately that work used only models and not frames. (shrink)

Within Linguistics the semantic analysis of natural languages (English, Swahili, for example) has drawn extensively on semantical concepts first formulated and studied within classical logic, principally first order logic. Nowhere has this contribution been more substantive than in the domain of quantification and variable binding. As studies of these notions in natural language have developed they have taken on a life of their own, resulting in refinements and generalizations of the classical quantifiers as well as the discovery of new (...) types of quantification which exceed the expressive capacity of the classical quantifiers. We refer the reader to Keenan and Westerståhl (1997) for an overview of results in this area. Here, we focus on one property of quantification in natural language—its inherently sortal nature—which distinguishes it from quantification in classical logic. (shrink)

Our position is critical of the dominant method in psychology, and critical of the social use of psychological models. We assert that the scientific approach in psychology must break with modernist claim, but without sinking into post-modernist relativism. We consider that the epistemology associated with experimental psychology should be a specific epistemology associated with the particular objects studied. By calling on pragmatism and realism, psychology can find the resources to use quantitative studies as an action deployed within a complex, interactive (...) and dynamic social space, and thus break with its normative practice and function. (shrink)

Although Ludwik Fleck is today recognized as one of the pioneers of the historical sociology of science, his historical and epistemological writings, most of them dating from the 1930s, long remained practically unknown. They were rediscovered following the mention of Fleck's principal work, the monographGenesis and Development of a Scientific Fact(1935) in the preface of Kuhn'sThe Structure of Scientific Revolutions(1962), and thanks to the efforts of W. Baldamus (1977) and his student T. Schnelle (1982) and of the editors of the (...) English translation of Fleck's book, R. K. Merton and T. J. Trenn. Fleck's work was studied by several scholars and was the subject of two meetings: “Colloquium Ludwik Fleck” (Hamburg, 1981) organized by L. Schaffer and T. Schnelle, and a symposium organized by R. S. Cohen (Berlin, 1984). Some of the papers presented at these meetings were published in Cohen and Schnelle 1986. Today, more than fifty years after the publication of his principal study, Fleck is on his way to becoming a “classic” of the sociology of science and of epistemology. (shrink)

Fine and Kripke extended S5, S4, S4.2 and such to produce propositionally quantified systems , , : given a Kripke frame, the quantifiers range over all the sets of possible worlds. is decidable and, as Fine and Kripke showed, many of the other systems are recursively isomorphic to second-order logic. In the present paper I consider the propositionally quantified system that arises from the topological semantics for S4, rather than from the Kripke semantics. The topological system, which I dub , (...) is strictly weaker than its Kripkean counterpart. I prove here that second-order arithmetic can be recursively embedded in . In the course of the investigation, I also sketch a proof of Fine's and Kripke's results that the Kripkean system is recursively isomorphic to second-order logic. (shrink)

Scientific realism holds that the terms in our scientific theories refer and that we should believe in their existence. This presupposes a certain understanding of quantification, namely that it is ontologically committing, which I challenge in this paper. I argue that the ontological loading of the quantifiers is smuggled in through restricting the domains of quantification, without which it is clear to see that quantifiers are ontologically neutral. Once we remove domain restrictions, domains of quantification can include (...) non-existent things, as they do in scientific theorizing. Scientific realism would therefore require redefining without presupposing a view of ontologically committing quantification. (shrink)

Generalizing Cooper’s method of quantifier elimination for Presburger arithmetic, we give a new proof that all parametric Presburger families \ [as defined by Woods ] are definable by formulas with polynomially bounded quantifiers in an expanded language with predicates for divisibility by f for every polynomial \. In fact, this quantifier bounding method works more generally in expansions of Presburger arithmetic by multiplication by scalars \: \alpha \in R, t \in X\}\) where R is any ring of functions from X (...) into \. (shrink)

The Quantified argument calculus (Quarc) has received a lot of attention recently as an interesting system of quantified logic which eschews the use of variables and unrestricted quantification, but nonetheless achieves results similar to the Predicate calculus (PC) by employing quantifiers applied directly to predicates instead. Despite this noted similarity, the issue of the relationship between Quarc and PC has so far not been definitively resolved. We address this question in the present paper, and then expand upon that result. (...) Utilizing recent developments in structural proof theory, we develop a G3-style sequent calculus for Quarc and briefly demonstrate its structural properties. We put these properties to use immediately to construct direct proofs of the meta-theoretical properties of the system. We then incorporate an abstract (and, as we shall see, logical) predicate into the system in a way that preserves all the structural properties. This allows us to identify a system of Quarc which is deductively equivalent to PC, and also yields a constructive method of demonstrating the Craig interpolation theorem (which speaks in favor of the aforementioned predicate being logical). We further generalize this extension to develop a bivalent system of Quarc with defining clauses that still maintains all the desirable properties of a good proof system. (shrink)

This paper examines the question of how so-called indeterminate phrases in Japanese (Kuroda 1965) associate with relevant particles higher in the structure. In the universal construction in Japanese, the restrictor (provided by an indeterminate phrase) sometimes appears to be separate from the universal particle mo. It is proposed that quantification at a distance is only apparent, and that the restriction is in fact provided locally by the sister constituent of mo as a whole. The proposal leads us to a (...) straightforward uniform picture of the syntax-semantics mapping of the universal construction and wh-questions, building upon Hamblin’s (1973) semantics for wh-phrases as sets of alternatives. It allows for a switch of perspective on a long-standing puzzle regarding locality effects in the indeterminate–particle association by deriving the locality pattern from the way indeterminate phrases are interpreted and associated with particles, without any stipulations. (shrink)

A principal goal of this book is to develop and apply the Situation Semantics framework. Jean Mark Gawron and Stanley Peters adopt a version of the theory in which meanings are built up via syntactically driven semantic composition rules. They provide a substantial treatment of English incorporating treatments of pronomial anaphora, quantification, donkey anaphora, and tense. The book focuses on the semantics of pronomial anaphora and quantification. The authors argue that the ambiguities of sentences with pronouns cannot be (...) adequately accounted for with a theory that represents anaphoric relations only syntactically; their relational framework uniformly deals with anaphoric relations as relations between utterances in context. They argue that there is no need for a syntactic representation of anaphoric relations, or for a theory that accounts for anaphoric ambiguities by resorting to two or more kinds of anaphora. Quantifier scope ambiguities are handled analogously to anaphoric ambiguities. This treatment integrates the Cooper Store mechanism with a theory of meaning that provides both a natural setting for it and a convincing account of what, semantically, is going on. Jean Mark Gawron is a researcher for Hewlett Packard Laboratories, Palo Alto. Stanley Peters is professor of linguistics and symbolic systems at Stanford University and is director of the Center for the Study of Language and Information. (shrink)

Noun phrases with overt determiners, such as <i>some apples</i> or <i>a quantity of milk</i>, differ from bare noun phrases like <i>apples</i> or <i>milk</i> in their contribution to aspectual composition. While this has been attributed to syntactic or algebraic properties of these noun phrases, such accounts have explanatory shortcomings. We suggest instead that the relevant property that distinguishes between the two classes of noun phrases derives from two modes of existential quantification, one of which holds the values of a variable (...) fixed throughout a quantificational context while the other allows them to vary. Inspired by Dynamic Plural Logic and Dependence Logic, we propose Plural Predicate Logic as an extension of Predicate Logic to formalize this difference. We suggest that temporal <i>for</i>-adverbials are sensitive to aspect because of the way they manipulate quantificational contexts, and that analogous manipulations occur with spatial <i>for</i>-adverbials, habituals, and the quantifier <i>all</i>. (shrink)