Frege’s system of first-order logic is presented in a contemporary framework. The system described is distinguished by economy of expression and an unusual syntax.

Standard first-order logic plus quantifiers of all finite orders ("SFOLω") faces four well-known difficulties when used to characterize the behavior of certain English quantifier phrases. All four difficulties seem to stem from the typed structure of SFOLω models. The typed structure of SFOLω models is in turn a product of an asymmetry between the meaning of names and the meaning of predicates, the element-set asymmetry. In this paper we examine a class of models in which this asymmetry of meaning is (...) removed. The models of this class permit definitions of the quantifiers which allow a desirable flexibility in fixing the domain of quantification. Certain SFOLω type restrictions are thereby avoided. The resulting models of English validate all of the standard first-order logical truths and are free of the four deficiencies of SFOLω models. (shrink)

This paper will attempt to integrate (1) some new reflections on the implications for ontology of Monistic interpretations of formulae in quantification theory, with (2) a review of earlier material that I have published on such implications, and with (3) a sketch of several points made by others which bear on related issues. I hope that a review of these moves and issues might indirectly help the reader in dealing with two strange questions. (A) Does the development of (...) modern quantification theory with its ways of systematically formalising relational inferences throwany light on Ontology, on the question of What There Is for scientists and applied mathematicians to study? (B) Are there any analogies between Monist interpretations of quantification theory and the attraction of holding down the number of basic entities in physics by viewing some as mere modes of other more ‘really basic’ ones? More directly I try to throw sympathetic light on more traditional philosophers of science like Leibniz and Spinoza. (shrink)

The second printing of Principia Mathematica in 1925 offered Russell an occasion to assess some criticisms of the Principia and make some suggestions for possible improvements. In Appendix A, Russell offered *8 as a new quantification theory to replace *9 of the original text. As Russell explained in the new introduction to the second edition, the system of *8 sets out quantification theory without free variables. Unfortunately, the system has not been well understood. This paper shows (...) that Russell successfully antedates Quine's system of quantification theory without free variables. It is shown as well, that as with Quine's system, a slight modification yields a quantification theory inclusive of the empty domain. (shrink)

This paper presents a number of objections to Jeffrey King's quantificational theory of complex demonstratives. Some of these objections have to do with modality, whereas others concern attitude ascriptions. Various possible replies are considered. The debate between quantificational theorists and direct reference theorists over complex demonstratives is compared with recent debates concerning definite descriptions.

I criticized Jeffrey King’s theory of complex demonstratives in “Problems for a Quantificational Theory of Complex Demonstratives.” King replied in “Complex Demonstratives as Quantifiers: Objections and Replies.” I here comment on some of King’s replies.

This paper examines the quantification theory of *9 of Principia Mathematica. The focus of the discussion is not the philosophical role that section *9 plays in Principia's full ramified type-theory. Rather, the paper assesses the system of *9 as a quantificational theory for the ordinary predicate calculus. The quantifier-free part of the system of *9 is examined and some misunderstandings of it are corrected. A flaw in the system of *9 is discovered, but it is shown (...) that with a minor repair the system is semantically complete. Finally, the system is contrasted with the system of *8 of Principia's second edition. (shrink)

Never shall this be proved, that things that are not are. ParmenidesTo say that something does not exist, or that there is something which is not, is clearly a contradiction in terms; hence “ ” must be true. Moreover, we should certainly expect leave to put any primitive name of our language for the “x” of any matrix “ … x … ”, and to infer the resulting singular statement from “ ”; it is difficult to contemplate any alternative logical (...) rule for reasoning with names. (shrink)

In earlier papers, I have sketched some aspects of a semantical theory of quantification, developed in terms of the concept of a model set. Furthermore, I have indicated a method of relating this semantical, or half-semantical, theory to the usual deductive methods. This method turned on considering proofs of logical truth as frustrated attempts to describe a counter-example to the sentence to be proved. In this paper, I propose to relate some of the observations that were made (...) in the earlier papers to the doctrines which are generally known as picture theories of language. My main thesis is that these theories can be related rather closely to the logic of quantification by developing them in a suitable direction, and by using the concept of a model set as a mediating link. (shrink)

Semantic theories based on a hierarchy of types have prominently been used to defend the possibility of unrestricted quantification. However, they also pose a prima facie problem for it: each quantifier ranges over at most one level of the hierarchy and is therefore not unrestricted. It is difficult to evaluate this problem without a principled account of what it is for a quantifier to be unrestricted. Drawing on an insight of Russell’s about the relationship between quantification and the (...) structure of predication, we offer such an account. We use this account to examine the problem in three different type-theoretic settings, which are increasingly permissive with respect to predication. We conclude that unrestricted quantification is available in all but the most permissive kind of type theory. (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)

If we wish to formulate an axiomatic truth-theory interpreting a modal language and treat the symbol of necessity as a sentential operator and not as a quantifier over possible worlds, there arise various problems. These are due partly to the fact that words could have meant something other than what they actually mean and partly to certain principles of modal metaphysics. One of those principles is existentialism about propositions: a proposition that is expressed in a sentence containing a non-empty (...) name could not exist if the referent of the name did not exist. The paper explains how the problems arise. It also explains how we can avoid them using substitutional quantification in the metalanguage. The truth-theory we can construct in that way is compositional and homophonic, and its theorems interpreting the various sentences of the modal language are derived in a straightforward way. The paper develops one such theory for a propositional modal language and one for a first-order modal language. The first-order case involves a number of intricacies that are discussed. (shrink)

Summary While historians have dealt with the origins of the concept of drug receptors in the work of Paul Ehrlich (1854–1915) and John N. Langley (1852–1925) as well as with some of its applications in modern pharmaceutical research, the history of the receptor theory as such has been neglected. Discussing major developments and conceptual changes in receptor theory between about 1910 and 1960 (including relevant contributions by A. V. Hill, A. J. Clark, J. H. Gaddum, E. J. Ariëns (...) and others), this paper attempts to fill this gap in historiography. It provides a case study of the unfolding of research under a new paradigm, but it considers also contemporary criticism and scepticism. By the early 1960s, quantitative investigations of drug action and interpretations of the experimental findings in terms of the receptor concept had become constitutive of the emerging field of ‘molecular pharmacology’. Even then, however, receptors were still hypothetical entities. (shrink)

The format of this book is unusual, especially for a book about linguistics. The book is meant primarily as a research monograph aimed at linguists who have some background in formal semantics, e. g. Montague Grammar. However, I have two other audiences in mind. Linguists who have little or no experience of formal semantics, but who have worked through a basic mathematics for linguists course (e. g. using Wall, 1972, or Partee, 1978), should, perhaps with the help of a sympathetic (...) Montague gramma rian, be able to discover enough of how I have adapted some of the basic ideas in formal semantics to make the developments that I undertake in the rest of the book accessible. Logicians and computer scientists who know about model theoretic semantics and formal systems should be able to glean enough from Chapters I and II about linguistic concerns and techniques to be able to read the remainder of the book, again possibly with the help of a sympathetic Montague grammarian. However, readers should beware. Chapter II is not meant as a general introduction either to formal semantics or to linguistics and while much of the presentation there is going over ground that is already well covered in the literature, the particular formulation and the emphases are very much oriented to the developments to be undertaken later in the book. (shrink)

Bruno Whittle (2019) has recently extended Kripke’s semantical theory of truth to languages containing generalized quantifiers. There are reasons for axiomatizing semantical theories, and for regarding Halbach and Horsten’s PKF as a good axiomatization of Kripke’s. PKF is a theory in Partial Logic. The present paper complements Whittle’s by showing how Partial Logic, and then PKF, may be extended to cover binary quantifiers meaning ‘every’, ‘some’, and ‘most’.