Significance range theory

Philosophical Studies, 2022

Call a quantifier 'unrestricted' if it ranges over absolutely all objects. Arguably, unrestricted quantification is often presupposed in philosophical inquiry. However, developing a semantic theory that vindicates unrestricted quantification proves rather difficult, at least as long as we formulate our semantic theory within a classical first-order language. It has been argued that using a type theory as framework for our semantic theory provides a resolution of this problem, at least if a broadly Fregean interpretation of type theory is assumed. However, the intelligibility of this interpretation has been questioned. In this paper I introduce a type-free theory of properties that can also be used to vindicate unrestricted quantification. This alternative emerges very naturally by reflecting on the features on which the type-theoretic solution of the problem of unrestricted quantification relies. Although this alternative theory is formulated in a non-classical logic, it preserves the deductive strength of classical strict type theory in a natural way. The ideas developed in this paper make crucial use of Russell's notion of range of significance.

Bulletin of Symbolic Logic, 2004

Commun Pure Appl Math, 1987

Journal of Logic and Computation, 2011

We show that for each n and m, there is an existential first order sentence which is NOT logically equivalent to a sentence of quantifier rank at most m in infinitary logic augmented with all generalized quantifiers of arity at most n. We use this to show the strictness of the quantifier rank hierarchies for various logics ranging from existential (or universal) fragments of first order logic to infinitary logics augmented with arbitrary classes of generalized quantifiers of bounded arity.

2013

This paper surveys the common approach to quantification and generalised quantification in formal linguistics and philosophy of language. We point out how this general setting departs from empirical linguistic data, and give some hints for a different view based on proof theory, which on many aspects gets closer to the language itself. We stress the importance of Hilbert's operator epsilon and tau for, respectively, existential and universal quantifications. Indeed, these operators help a lot to construct semantic representation close to natural language, in particular with quantified noun phrases as individual terms. We also define guidelines for the design of the proof rules corresponding to generalised quantifiers. Résumé Cet article dresse un rapide panorama de l'approche commune de la quantification généralisée ou non en linguistique formelle et en philosophie du langage. Nous montrons que ce cadre général est va parfoisà l'encontre des données linguistiques, et nous donnons quelques indications pour une approche différente basée sur la théorie de la démonstration, qui sur bien des points s'avre plus proche de la langue. Nous soulignons l'importance des opérateurs tau et epsilon de Hilbert qui rendent respectivement compte de la qualification universelle et existentielle. En effet ces opérateurs permettent de construire des des représentations sémantiques qui suivent la lange avec, en particulier des groupes nominaux quantifiées qui soient des termes individuels. Nous donnons aussi des principes pour définir des rgles de déduction qui correspondent aux quantificateurs généralisés.

Annals of Mathematical Logic, 1980

A theory ~,f Boolean ~alued models for generalized quanliliels i:. dexelopcd ~ilh a ~pccial emphasis on he Hfirtig-quantilier. As an application a Boolean exicnsion is obt;dned in xxhich the decision problem of the H~irlig-quantilim is ..~. lntroductie n Second-c, rder logic L n (with quantification over setsl is reputedly lacking model theoretic re ;ults. This situation had led to a search for well-behaved axiomatizablc fragments of L n, and indeed, many such interesting fragments have been found (see e.g. [8_' and [13]L The purpose of the present paper is to investigate those fragments o>: L n which fall well outside the category of these nice axiomatizable fragments. The simplest non-axiomatizable logic is LQ~,. where O.;A(x)<-~{aIAIo}} is infnite. There is, ho'vevcr, a straightforxvard h~tinitary formal system for l.O,+. E\cn the much strong,:r logic I.W.v.here Wx,:Atx. y)<-~ {~a. l'li .411~. hi} well-orders its field. permits a syntactical characlerization of validity ~sce [5]k No such results are known for LI. where lxy/dxIB(y~ "~+ card{t~ i A{a~} = caldII, ] B(bl}. This is tile Hiirtig-quantifier and is also denoted by O~. Ill the unix'erse of constructible sets LI is as powerful as L u itself, but there are models of set * The paper is based on a part of the author's Ph.D.-lhesis a~ Manche:.tcr t.J~fi'.er:,it). P-)77, The author wishes tc e~press here his gratitude to his stlpervisor. P. H. G. Aczel. for the help mtd encotlragenteill g'~.'ell during lhc preparation of the thesis, The ll,,esis and paper x~crc prepared v, hil¢ the aulhor was ti:lancially supporled b*, (.)sk. I:llltlltllC]l l~'olmdat!on.

chapter 1 of ‘Absolute Generality’ (eds. A. Rayo and G. Uzquiano), Oxford: Oxford University Press (2006).

There are four broad grounds upon which the intelligibility of quantification over absolutely everything has been questioned-one based upon the existence of semantic indeterminacy, another on the relativity of ontology to a conceptual scheme, a third upon the necessity of sortal restriction, and the last upon the possibility of indefinite extendibility. The argument from semantic indeterminacy derives from general philosophical considerations concerning our understanding of language. For the Skolem-Lowenheim Theorem appears to show that an understanding of quantification over absolutely everything (assuming a suitably infinite domain) is semantically indistinguishable from the understanding of quantification over something less than absolutely everything; the same first-order sentences are true and even the same first-order conditions will be satisfied by objects from the narrower domain. From this it is then argued that the two kinds of understanding are indistinguishable tout court and that nothing could count as having the one kind of understanding as opposed to the other.

Communications on Pure and Applied Mathematics, 1980

The Journal of Symbolic Logic, 1996

The concept of a generalized quanti er of a given similarity type was de ned in Lin66]. Our main result says that on nite structures di erent similarity types give rise to di erent classes of generalized quanti ers. More exactly, for every similarity type t there is a generalized quanti er of type t which is not de nable in the extension of rst order logic by all generalized quanti ers of type smaller than t. This was proved for unary similarity types by Per Lindstr om Wes] with a counting argument. We extend his method to arbitrary similarity types.