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)

We discuss O&C's probabilistic approach from a probability logical point of view. Specifically, we comment on subjective probability, the indispensability of logic, the Ramsey test, the consequence relation, human nonmonotonic reasoning, intervals, generalized quantifiers, and rational analysis.

We introduce two new belief revision axioms: partial monotonicity and consequence correctness. We show that partial monotonicity is consistent with but independent of the full set of axioms for a Gärdenfors belief revision sytem. In contrast to the Gärdenfors inconsistency results for certain monotonicity principles, we use partial monotonicity to inform a consistent formalization of the Ramsey test within a belief revision system extended by a conditional operator. We take this to be a technical dissolution of the well-known Gärdenfors dilemma.In (...) addition, we present the consequential correctness axiom as a new measure of minimal revision in terms of the deductive core of a proposition whose support we wish to excise. We survey several syntactic and semantic belief revision systems and evaluate them according to both the Gärdenfors axioms and our new axioms. Furthermore, our algebraic characterization of semantic revision systems provides a useful technical device for analysis and comparison, which we illustrate with several new proofs. (shrink)

Gareth Evans'' proposal, as amended by Steven Neale –that a definite pronoun with a quantifiedantecedent that does not bind it has the sense ofa definite description – has been challenged inthe singular case by appeal to counter-examplesinvolving failure of the uniqueness condition forthe legitimacy of a singular description. Thischallenge is here extended to the plural.Counter-examples are provided by cases in which aplural description `the Fs'' does not denote,despite the propriety of the use of `they'' or`them'' it is to replace, because (...) of failure ofcumulativeness. A noncumulative predicate isnot distributive, and conditions for thelegitimacy of `the Fs'' designed to accommodatenondistributive `Fs'' are given in the context ofa two sorted theory with generalized quantifiers.Failure of cumulativeness is not to be confusedwith failure of maximality as Neale and othersdefine it. If not all Fs are Gs, `The Fs are Gs''is false; but it does not follow that `the Fs'' isillegitimate; and if `Fs'' is distributive. it isso only if there are no Fs. These differencesgranted, I give a partial defense of theEvans–Neale proposal from deficiencies in analternative based on the views of P. T.Geach. (shrink)