Richard Montague

Formal Philosophy. Selected Papers of Richard Montague

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Logical necessity, physical necessity, ethics, and quantifiers

Some philosophers, for example Quine, doubt the possibility of jointly using modalities and quantification. Simple model‐theoretic considerations, however, lead to a reconciliation of quantifiers with such modal concepts as logical, physical, and ethical necessity, and suggest a general class of modalities of which these are instances. A simple axiom system, analogous to the Lewis systems S1 —S5, is considered in connection with this class of modalities. The system proves to be complete, and its class of theorems decidable.

Theories Incomparable with Respect to Relative Interpretability

The present paper concerns the relation of relative interpretability introduced in [8], and arises from a question posed by Tarski: are there two finitely axiomatizable subtheories of the arithmetic of natural numbers neither of which is relatively interpretable in the other? The question was answered affirmatively (without proof) in [3], and the answer was generalized in [4]: for any positive integer n , there exist n finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. A further generalization was announced in [5] and is proved here: there is an infinite set of finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. Several lemmas concerning the existence of self-referential and mutually referential formulas are given in Section 1, and will perhaps be of interest on their own account.

Recursion Theory as a Branch of Model Theory1

Publisher Summary The desirability of generalizing the theory of recursive functions and relations has for some time been widely appreciated. The goal is to obtain the notions of recursiveness and recursive innumerability, which will have interest in connection not only with the natural numbers but also with structures of an arbitrary or almost arbitrary sort, non-denumerable, and denumerable. To some logicians, it has also appeared desirable to unify two of the dominant subfields of contemporary logic, model theory, and recursion theory. It is possible by routine methods to translate the notions of recursion theory into the language of model theory. But to do this in such a way that the translations will have a natural and simple model-theoretic content, and the methods of proof having a common character with those of general model theory have not heretofore been fully accomplished. This chapter seeks to fulfill both objectives. It contains a theory of recursiveness applicable to any model (or structure) whatsoever, and forms a natural branch of the general theory of definability within a model.

Set Theory and Higher-Order Logic

Publisher Summary This chapter explains set theory and higher-order logic. Several mutual applications of set theory and higher-order logic are developed. Second-order logic is used to discover the standard models of Zermelo–Fraenkel set theory. Consideration of standard models leads to the introduction of new systems of set theory; one of these systems is applied in finding a definition of truth for higher-order sentences and Zerrnelo–Fraenkel set theory with individuals is given a philosophical justification as logically true within higher-order logic. The chapter considers three well-known first-order theories. The first, called “Peanos arithmetic,” has the non-logical constants. The second theory, called “the theory of real closed fields” has the non-logical constants. The last principle is called the “continuity schema” and plays a role analogous to that of the induction schema.

The Proper Treatment of Mass Terms in English

The basic problem is to determine what concrete mass nouns denote and how one is to give truth conditions for sentences containing them. I ignore for the most part ‘abstract mass nouns’ and ‘mass adjectives’, but see below.

REDUCTIONS OF HIGHER-ORDER LOGIC

A higher-order logic is considered which contains variables of all finite and transfinite types, together with a more restricted logic in which the types of variables are all in a sense describable ordinals. It is shown that several important problems connected with the restricted higherorder logic, pertaining, for instance, to the determination of spectra and the characterization of logical truth, are reducible to the corresponding problems for second-order logic; the principal results of this sort extend earlier results of Zykov and Hintikka. It is obtained as a corollary that the set of logical truths of second-order logic does not appear in any reasonable extension of the Kleene hierarchy.

Two Contributions to the Foundations of Set Theory

Publisher Summary A simple alternative formulation is given in the chapter for an axiom that is recently proposed by Levy as an addition to the axioms of Zermelo–Fraenkel set theory. If T is a theory obtained from Zermelo set theory by adding all instances with no more than a given number of alternations of quantifiers of a schema valid in Zermelo–Fraenkel set theory, then the consistency of T is demonstrable in Zermelo–Fraenkel set theory (a slightly stronger result is obtained). The chapter also mentions the term expressions that are to be identified with natural numbers. The chapter distinguishes an infinite set of natural numbers that are to be known as variables and—for each positive integer n—an infinite set of natural numbers to be known as n place predicate.

Ein Ausschnitt des Englischen

Als zweites Beispiel legen wir eine naturliche Sprache — namlich einen ausdrucklich beschrankten Ausschnitt E1 des Englischen — vor.

Logic: Techniques of Formal Reasoning.

Logic: Techniques of Formal Reasoning, 2/e is an introductory volume that teaches students to recognize and construct correct deductions. It takes students through all logical steps--from premise to conclusion--and presents appropriate symbols and terms, while giving examples to clarify principles. Logic, 2/e uses models to establish the invalidity of arguments, and includes exercise sets throughout, ranging from easy to challenging. Solutions are provided to selected exercises, and historical remarks discuss major contributions to the theories covered.