For 30 years, Quine, a dominant figure in logical theory and philosophy of logic, has combined insights in methodology, language, epistemology, and ontology, to blur the boundaries of speculative metaphysics and natural sciences. This revised text contains two new essays with replies from Quine.

Today there appears to be a widespread impression that W. V. Quine's notorious critique of modal logic, based on certain ideas about reference, has been successfully answered. As one writer put it some years ago: “His objections have been dead for a while, even though they have not yet been completely buried.” What is supposed to have killed off the critique? Some would cite the development of a new ‘possible-worlds’ model theory for modal logics in the 1960s; others, the development (...) of new ‘direct’ theories of reference for names in the 1970s.These developments do suggest that Quine's unfriendliness towards any formal logics but the classical and indifference towards theories of reference for any singular terms but variables were unfortunate. (shrink)

Introduction. The system of axioms for set theory to be exhibited in this paper is a modification of the axiom system due to von Neumann. In particular it adopts the principal idea of von Neumann, that the elimination of the undefined notion of a property (“definite Eigenschaft”), which occurs in the original axiom system of Zermelo, can be accomplished in such a way as to make the resulting axiom system elementary, in the sense of being formalizable in the logical calculus (...) of first order, which contains no other bound variables than individual variables and no accessory rule of inference (as, for instance, a scheme of complete induction).The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and to utilize at the same time some of the set-theoretic concepts of the Schröder logic and of Principia mathematica which have become familiar to logicians. As will be seen, a considerable simplification results from this arrangement.The theory is not set up as a pure formalism, but rather in the usual manner of elementary axiom theory, where we have to deal with propositions which are understood to have a meaning, and where the reference to the domain of facts to be axiomatized is suggested by the names for the kinds of individuals and for the fundamental predicates.On the other hand, from the formulation of the axioms and the methods used in making inferences from them, it will be obvious that the theory can be formalized by means of the logical calculus of first order (“Prädikatenkalkul” or “engere Funktionenkalkül”) with the addition of the formalism of equality and the ι-symbol for “descriptions” (in the sense of Whitehead and Russell). (shrink)

Naturalism holds that there is no higher access to truth than empirically testable hypotheses. Still it does not repudiate untestable hypotheses. They fill out interstices of theory and lead to further hypotheses that are testable.A hypothesis is tested by deducing, from it and a background of accepted theory, some observation categorical that does not follow from the background alone. This categorical, a generalized conditional compounded of two observation sentences, admits in turn of a primitive experimental test.The observation sentences themselves, like (...) ape cries and bird calls, are in holophrastic association with ranges of neural intake. Denotation of determinate objects figures neither in this association nor in deducing the categorical from the scientific hypotheses. Hence the indeterminacy of reference; ontology is purely auxiliary to the structure of theory. Truth, however, is seen still as transcendent at least in this sense: we say of a superseded scientific theory not that it ceased to be true, but that it is found to have been false. (shrink)

In the middle of myMathematical logicI defined a certain class of formulae as “stratified,” and conjectured that exclusion from this class is a feature “shared, presumably, by all the untenable statements(p. 157). This ushered in a set of axioms of class-membership which Rosser has since shown to be inconsistent. Accordingly, inElement and numberI dropped the principle*200, in which had been assembled axioms to the effect, roughly, that “stratified functions of elements are elements.” In lieu of*200 I set forth alternatives in (...) which no appeal is made to stratification. The system ofMathematical logicexclusive of*200 carries over as an unchanging framework; and this framework admits, we know, of a simple consistency proof. My concern in the present paper is to draw attention to certain relationships between this framework and earlier theories. (shrink)

Systems TRC and TRCU of illative combinatory logic are introduced and shown to be equivalent in consistency strength and expressive power to Quine's set theory ‘New Foundations’ and the fragment NFU + Infinity of NF described by Jensen, respectively. Jensen demonstrated the consistency of NFU + Infinity relative to ZFC; the question of the consistency of NF remains open. TRC and TRCU are presented here as classical first-order theories, although they can be presented as equational theories; they are not constructive.

This paper extends the results of an earlier paper by the author . New subsystems of the combinatory logic TRC shown in that paper to be equivalent to NF are introduced; these systems are analogous to subsystems of NF with predicativity restrictions on set comprehension introduced and shown to be consistent by Crabbé. For one of these systems, an exact equivalence in consistency strength and expressive power with the analogous subsystem of NF is established.

The Interplay of Logic, Set Theory and Semantics in Quine's Philosophy L. Decock. In philosophy of science Quine's name is linked to the so-called Quine- Duhem thesis. The discussion of this thesis still continues even after several decades.9 ...