Warren D. Goldfarb

The undecidability of the second-order unification problem

Abstract It is shown that there is no effective procedure for determining whether or not two terms of the language of second-order logic have a common instance.

Logic in the Twenties: The Nature of the Quantifier

We are often told, correctly, that modern logic originated with Frege. For Frege clearly depicted polyadic predication, negation, the conditional, and the quantifier as the bases of logic; moreover, he introduced the idea of a formal system, and argued that mathematical demonstrations, to be fully precise, must be carried out within a formal language by means of explicitly formulated syntactic rules. Consequently Frege has often been read as providing all the central notions that constitute our current understanding of quantification. For example, in his recent book on Frege [1973], Michael Dummett speaks of ”the semantics which [Frege] introduced for formulas of the language of predicate logic.” That is, “An interpretation of such a formula … is obtained by assigning entities of suitable kinds to the primitive nonlogical constants occurring in the formula … [T]his procedure is exactly the same as the modern semantic treatment of predicate logic” (pp. 89–90). Indeed, “Frege would therefore have had within his grasp the concepts necessary to frame the notion of the completeness of a formalization of logic as well as its soundness … but he did not do so” (p. 82). This common appraisal of Freges work is, I think, quite misleading. Even given Freges tremendous achievements, the road to an understanding of quantification theory was an arduous one. Obtaining such understanding and formulating those notions which are now common coin in the discussion of logical systems were the tasks of much of the work in logic during the nineteen-twenties.

Frege’s conception of logic

This short essay offers a brief account of Frege’s conception of logic from two main points of view: the novelty of his view on logic and the normative status of logic in his writings. I analyze Frege’s position with regard to logic by comparing it to the views of Mill and Kant. I also argue against a normative reading of Frege’s writings on the nature of logic, a reading which is not uncommon in contemporary literature.

The Unsolvability of the Gödel Class with Identity

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The decision problem for formulas with a small number of atomic subformulas

In this paper we consider classes of quantificational formulas specified by restrictions on the number of atomic subformulas appearing in a formula. Little seems to be known about the decision problem for such classes, except that the class whose members contain at most two distinct atomic subformulas is decidable [2]. (We use “decidable” and “undecidable” throughout with respect to satisfiability rather than validity. All undecidable problems to which we refer are of maximal r.e. degree.) The principal result of this paper is the undecidability of the class of those formulas containing five atomic subformulas and with prefixes of the form ∀∃∀…∀. In fact, we show the undecidability of two sub-classes of this class: one (Theorem 1) consists of formulas whose matrices are in disjunctive normal form with two disjuncts; the other (Corollary 1) consists of formulas whose matrices are in conjunctive normal form with three conjuncts. (Theorem 1 sharpens Orevkovs result [8] that the class of formulas in disjunctive normal form with two disjuncts is undecidable.) A second corollary of Theorem 1 shows the undecidability of the class of formulas with prefixes of the form ∀…∀∃, containing six atomic subformulas, and in conjunctive normal form with three conjuncts. These restrictions to prefixes ∀∃∀…∀ and ∀…∀∃ are optimal. For by a result of the first author [5], any class of prenex formulas obtained by restricting both the number of atomic formulas and the number of universal quantifiers is reducible to a finite class of formulas, and so each such class is decidable; and the class of formulas with prefixes ∃…∃∀…∀ is, of course, decidable.

The Consistency of Number Theory Via Herbrand's Theorem

In this sequel to [7] the method of the consistency proof presented there is extended to provide a proof of the ω -consistency of the systems of number theory which were there shown consistent. This proof yields sharp bounds on the ordinal recursions required to establish the κ -consistency of these systems. The main technical innovation of this proof is the extension of what are essentially the methods of Ackermann [1] for handling finite sets of critical formulae of the first and second kinds to apply as well to sets of critical formulae in which the rank ordering is transfinite. The notation, definitions, and results of [7] will be presupposed throughout; we suggest the reader keep a copy of that paper at hand.

On Godel's Way In: The Influence of Rudolf Carnap

The philosopher Rudolf Carnap (1891–1970), although not himself an originator of mathematical advances in logic, was much involved in the development of the subject. He was the most important and deepest philosopher of the Vienna Circle of logical positivists, or, to use the label Carnap later preferred, logical empiricists. It was Carnap who gave the most fully developed and sophisticated form to the linguistic doctrine of logical and mathematical truth: the view that the truths of mathematics and logic do not describe some Platonistic realm, but rather are artifacts of the way we establish a language in which to speak of the factual, empirical world, fallouts of the representational capacity of language. (This view has its roots in Wittgensteins Tractatus , but Wittgensteins remarks on mathematics beyond first-order logic are notoriously sparse and cryptic.) Carnap was also the thinker who, after Russell, most emphasized the importance of modern logic, and the distinctive advances it enables in the foundations of mathematics, to contemporary philosophy. It was through Carnaps urgings, abetted by Hans Hahn, once Carnap arrived in Vienna as Privatdozent in philosophy in 1926, that the Vienna Circle began to take logic seriously and that positivist philosophy began to grapple with the question of how an account of mathematics compatible with empiricism can be given (see Goldfarb 1996 ). A particular facet of Carnaps influence is not widely appreciated: it was Carnap who introduced Kurt Godel to logic, in the serious sense. Although Godel seems to have attended a course of Schlicks on philosophy of mathematics in 1925–26, his second year at the University, he did not at that time pursue logic further, nor did the seminar leave much of a trace on him. In the early summer of 1928, however, Carnap gave two lectures to the Circle which Godel attended, or so I surmise. At these occasions, Carnap presented material from his manuscript treatise, Untersuchungen zur allgemeinen Axiomatik , that is, “Investigations into general axiomatics”, which dealt with questions of consistency, completeness and categoricity. Carnap later circulated this material to various people including Godel.

First-Order Frege Theory is Undecidable

The system whose only predicate is identity, whose only nonlogical vocabulary is the abstraction operator, and whose axioms are all first-order instances of Freges Axiom V is shown to be undecidable.

On Dummett’s “Proof-Theoretic Justifications of Logical Laws”

This paper deals with Michael Dummett’s attempts at a proof-theoretic justification of the laws of (intuitionistic) logic, pointing to several critical problems inherent in this approach. It discusses in particular the role played by “boundary rules” in Dummett’s semantics. For a revised approach based on schematic validity it is shown that the rules of intuitionistic logic can indeed be justified, but it is argued that a schematic conception of validity is problematic for Dummett’s philosophy of logic.

Skolem Reduction Classes

Among the earliest and best-known theorems on the decision problem is Skolems result [7] that the class of all closed formulas with prefixes of the form V. V] . is a reduction class for satisfiability for the whole of quantification theory. This result can be refined in various ways. If the Skolem prefix alone is considered, the best result [8] is that the VVV3 class is a reduction class, for Gbdel [3], Kalmair [4], and Schiitte [6] showed the VV3 3 class to be solvable. The purpose of this paper is to describe the more complex situation that arises when (Skolem) formulas are restricted with respect to the arguments of their atomic subformulas. Before stating our theorem, we must introduce some notation. Let x, Y1, Y2, . be distinct variables (we shall use v1, v2, . and w1, w2, as metavariables ranging over these variables), and for each i ? 1 let Y(i) be the set {Y15 , yi}. An atomic formula Pv1... Vk will be said to be {V1,, V. } -based. For any n > 1, p 2 1, and any subsets Y1, , Yp of y(n), let C(n, Y1, , Yp) be the class of all those closed formulas with prefix Vy... Vyn3x such that each atomic subformula not containing the variable x is Yi-based for some i, 1 2 among Y1, ..., Y,. We prove our theorem by means of five lemmas. Lemma 1 reduces the originconstrained domino problem to the decision problem for a certain class B of Skolem formulas. Wang [9] showed that the halting problem for Turing machines is reducible to the origin-constrained domino problem; hence B is a reduction class. Lemmas 2 and 3 exploit Lemma 1 to show that C(3, {Y1, Y2}, {Y2, y3}) and C(3, y(2), y(3)) are reduction classes. Lemmas 4 and 5 show that the unsolvability of these two classes implies the unsolvability of the classes specified in the theorem. Let n > 1; let F be a formula VY1 ... Vyn3xM, with M quantifier-free; let ff be an n-adic function sign; and let e be a 0-adic function sign. Then D(F) is the set of all terms constructible from the function signsfn and e. An instance of F is any of the formulas M[y1/t1, ... , yn/tn, x/fn(tl .. t)] where t1, ... , tn are terms in D(F). (Our substitution notation: G[vl/tl, ..., Vkltk] is the result of replacing all occurrences of v1 in G, if any, by occurrences of t1, .., Vk by tk.) It follows from the