Dag Westerståhl

Stockholm University | SU · Department of Philosophy

We show that intuitionistic propositional logic is Carnap categorical: the only interpretation of the connectives consistent with the intuitionistic consequence relation is the standard interpretation. This holds with respect to the most well-known semantics relative to which intuitionistic logic is sound and complete; among them Kripke semantics,...

Compositionality is a principle used in logic, philosophy, mathematics, linguistics, and computer science for assigning meanings to language expressions in a systematic manner following syntactic construction, thereby allowing for a perspicuous algebraic view of the syntax-semantics interface. Yet the status of the principle remains under debate, w...

We show that intuitionistic propositional logic is \emph{Carnap categorical}: the only interpretation of the connectives consistent with the intuitionistic consequence relation is the standard interpretation. This holds relative to the most well-known semantics with respect to which intuitionistic logic is sound and complete; among them Kripke sema...

This paper gives a uniform account of the meaning of generalizations with explicit exceptions that employ the prepositions “but”, “except”, and “except for”. Our theory is that exceptives depend on generalizations, which can but need not be universal, whose generality they limit, and some of whose exceptions they comment on. Every generalization in...

In a mathematical perspective, neighborhood models for modal logic are generalized quantifiers, parametrized to points in the domain of objects/worlds. We explore this analogy further, connecting generalized quantifier theory and modal neighborhood logic. In particular, we find interesting analogies between conservativity for linguistic quantifiers...

There are two main routes to a concept of (generalized) quantifier. The first starts from first‐order logic, FO, and generalizes from the familiar ∀ and ∃ occurring there. The second route begins with real languages, and notes that many so‐called noun phrases, a kind of phrase which occurs abundantly in most languages, can be interpreted in a natur...

I attempt an explication of what it means for an operation across domains to be the same on all domains, an issue that (Feferman, S.: Logic, logics and logicism. Notre Dame J. Form. Log. 40, 31–54 (1999)) took to be central for a successful delimitation of the logical operations. Some properties that seem strongly related to sameness are examined,...

The standard relation of logical consequence allows for non-standard interpretations of logical constants, as was shown early on by Carnap. But then how can we learn the interpretations of logical constants, if not from the rules which govern their use? Answers in the literature have mostly consisted in devising clever rule formats going beyond the...

The standard relation of logical consequence allows for non-standard interpretations of logical constants, as was shown early on by Carnap. But then how can we learn the interpretations of logical constants, if not from the rules which govern their use? Answers in the literature have mostly consisted in devising clever rule formats going beyond the...

SherG. Y.. Partially-ordered (branching) generalized quantifiers: a general definition. Journal of philosophical logic, vol. 26 (1997), pp. 1–43. - Volume 63 Issue 4 - Dag Westerståhl

The shift of interest in logic from just reasoning to all forms of information flow has considerably widened the scope of the discipline, as amply illustrated in Johan van Benthem’s recent book Logical Dynamics of Information and Interaction. But how much does this change when it comes to the study of traditional logical notions such as logical con...

We investigate what possessives mean by examining a wide range of English examples, pre-and postnominal, quantified and nonquantified, to arrive at general, systematic truth conditions for them. In the process, we delineate a rich class of paradigmatic possessives having crosslinguistic interest, exploiting characteristic semantic properties. One i...

As far as we know, the phenomenon of narrowing for possessives was discussed for the first time in Barker (1995), with examples such as 1. 1. a. Most planets’ rings are made of ice. b. Not every school’s linguistics program is as good as that one. Barker also proposed a general scheme for semantic interpretation of quantified (non-expanded) prenom...

The goal of the study of dependence and independence in logic is to establish a basic theory of dependence and independence phenomena underlying seemingly unrelated subjects such as game theory, random variables, database theory, scientific experiments, and probably many others. The monograph Dependence Logic (J. Vaananen, Cambridge UP, 2007) stimu...

The standard semantic definition of consequence with respect to a selected set X of symbols, in terms of truth preservation under replacement (Bolzano) or reinterpretation (Tarski) of symbols outside X, yields a function mapping X to a consequence relation \(\Rightarrow_X\). We investigate a function going in the other direction, thus extracting th...

The new linguistic understanding has prompted novel mathematical questions whose interest derives from their natural interpretation in a nonmathematical domain. This chapter surveys these linguistic results and synthesizes the mathematical observations they have given rise to, relating them to independent work from logic. The work illustrates how t...

Starting from the familiar observation that no straightforward treatment of pure quotation can be compositional in the standard (homomorphism) sense, we introduce general compositionality, which can be described as compositionality that takes linguistic context into account. A formal notion of linguistic context type is developed, allowing the cont...

This is the second part of a two-part article on compositionality, i.e. the principle that the meaning of a complex expression is determined by the meanings of its parts and the way they are put together. In the first, Pagin and Westerståhl (2010), we provide a general historical background, a formal framework, definitions, and a survey of variants...

This is the first part of a two-part article on semantic compositionality, that is, the principle that the meaning of a complex expression is determined by the meanings of its parts and the way they are put together. Here we provide a brief historical background, a formal framework for syntax and semantics, precise definitions, and a survey of vari...

This is a review of Herman Cappelen and John Hawthorne’s book Relativism and Monadic Truth (Oxford University Press, Oxford, 2009). KeywordsRelativism-Truth-Proposition

Bolzano’s definition of consequence in effect associates with each set X of symbols (in a given interpreted language) a consequence relation ÞX{\Rightarrow_X} . We present this in a precise and abstract form, in particular studying minimal sets of symbols generating ÞX{\Rightarrow_X} . Then we present a method for going in the other direction: extr...

1 The problem A straightforward treatment of quotation cannot be compositional. Consider, as we shall in this paper, the simplest case: pure quotation in written language by means of quote marks, 1 as in (1) a. 'Cicero' has six letters b. 'farfalla' is Italian for butterfly c. 'str jd e' is not a sentence d. 'ℵ' is a Hebrew letter Although Cicero i...

Tarski’s definition of logical consequence for an interpreted language rests on the distinction between extra-logical symbols, whose interpretation is allowed to vary across models, and logical symbols, aka logical constants, whose interpretation remains fixed. In this perspective, logicality come first, and consequence is a by-product of the divis...

This note explains the circumstances under which a type < 1 > quantifier can be decomposed into a type < 1, 1 > quantifier and a set, by fixing the first argument of the former to the latter. The motivation comes from the semantics of Noun Phrases (also called Determiner Phrases) in natural languages, but in this article, I focus on the logical fac...

The conflict between classical and intuitionistic mathematics - henceforth, the C- I conflict — has been discussed at length and in depth by a number of famous scholars. Why an outside perspective? Is such a perspective interesting, or even possible?

For a long time, the word ‘quantifier’ in linguistics and philosophy simply stood for the universal and existential quantifiers of standard predicate logic. In fact, this use is still prevalent in elementary textbooks. It seems fair to say that the dominance of predicate logic in these fields has obscured the fact that the quantifier expressions fo...

Quantification is a topic which brings together linguistics, logic, and philosophy. Quantifiers are the essential tools with which, in language or logic, we refer to quantity of things or amount of stuff. In English they include such expressions as no, some, all, both, or many. This book presents the definitive interdisciplinary exploration of how...

A semantics may be compositional and yet partial, in the sense that not all well-formed expressions are assigned meanings by it. Examples come from both natural and formal languages. When can such a semantics be extended to a total one, preserving compositionality? This sort of extension problem was formulated by Hodges, and solved there in a parti...

We study definability in terms of monotone generalized quantifiers satisfying Isomorphism Closure, Conservativity and Extension. Among the quantifiers with the latter three properties – here called CE quantifiers – one finds the interpretations of determiner phrases in natural languages. The property of monotonicity is also linguistically ubiquitou...

A standard view is that idioms present problems for compositionality. The question of compositionality, however, should be posed for a semantics, not for individual phrases. The paper focuses on the idiom extension problem: If in a given language a certain phrase acquires the status of an idiom, how can the syntax and semantics be extended to accom...

We consider two formalisations of the notion of a compositional semantics for a language, and find some equivalent statements in terms of substitutions. We prove a theorem stating necessary and sufficient conditions for the existence of a “canonical” ...

The clause for existential formulas in the Tarskian definition of satisfaction for predicate logic can be written as follows: $${\text{A}},{\kern 1pt} \alpha \left| { = {\kern 1pt} \exists x\varphi {\kern 1pt} \Leftrightarrow {\kern 1pt} {\text{there}}{\kern 1pt} is{\kern 1pt} \beta {\kern 1pt} {\text{such that }}\alpha {\kern 1pt} {{\text{ = }}_x}...

We study generalized quantifiers on finite structures.With every function $$f$$ : ? ? ?we associate a quantifier Q $$_{\text{f}} $$ by letting Q $$_{\text{f}} $$ x? say “there are at least $$_{\text{f}} $$ (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...

We characterize the generalized quantifiers $Q$ which satisfy the scheme $QxQy\phi \leftrightarrow QyQx\phi$, the so-called self-commuting quantifiers, or quantifiers with the Fubini property.

We give a condensed survey of recent research on generalized quantifiers in logic, linguistics and computer science, under the following headings: Logical definability and expressive power, Polyadic quantifiers and linguistic definability, Weak semantics and axiomatizability, Computational semantics, Quantifiers in dynamic settings, Quantifiers and...

After background explanation of the links between generalized quantifier theory and natural language semantics, we survey some recent work on logics with generalized quantifiers, mainly concerned with problems of definability and expressive power in finite models and of axiomatizability, which is relevant to linguistic issues.

The International Congresses of Logic, Methodology and Philosophy of Science, which are held every fourth year, give a cross-section of ongoing research in logic and philosophy of science. Both the invited lectures and the many contributed papers are conductive to this end. At the 9th Congress held in Uppsala in 1991 there were 54 invited lectures...

A new formalism for predicate logic is introduced, with a non-standard method of binding variables, which allows a compositional formalization of certain anaphoric constructions, including donkey sentences and cross-sentential anaphora. A proof system in natural deduction format is provided, and the formalism is compared with other accounts of this...

The paper elaborates two points: i) There is no principal opposition between predicate logic and adherence to subject-predicate form, ii) Aristotle's treatment of quantifiers fits well into a modern study of generalized quantifiers.

The idea that branching of quantifiers occurs in natural languages originates with Hintikka (1973; reprinted in 1974). Hintikka gave several examples of English sentences whose logical analysis, he claimed, requires essential use of partially ordered quantifier prefixes, in particular the so-called Henkin prefix $$ \left. \begin{gathered} \forall x...

Introduction We shall study (generalized) quantifiers in the framework introduced by Barwise and Cooper in [ 1 ], the logical investigation of which has been continued in van Ben them [3] and Keenan and Stavi [2]. The present paper, although self-contained, is in the spirit of [3]. Its main characteristic is a systematic use of a method of proof in...

A common misunderstanding is that there is something logically amiss with the classical square of opposition, and that the problem is related to Aristotle's and medieval philosophers' rejection of empty terms. But [Parsons 2004] convincingly shows that most of these philosophers did not in fact reject empty terms, and that, when properly understood...