Hilary Putnam

A Computing Procedure for Quantification Theory

The hope that mathematical methods employed in the investigation of formal logic would lead to purely computational methods for obtaining mathematical theorems goes back to Leibniz and has been revived by Peano around the turn of the century and by Hilberts school in the 1920s. Hilbert, noting that all of classical mathematics could be formalized within quantification theory, declared that the problem of finding an algorithm for determining whether or not a given formula of quantification theory is valid was the central problem of mathematical logic. And indeed, at one time it seemed as if investigations of this “decision” problem were on the verge of success. However, it was shown by Church and by Turing that such an algorithm can not exist. This result led to considerable pessimism regarding the possibility of using modern digital computers in deciding significant mathematical questions. However, recently there has been a revival of interest in the whole question. Specifically, it has been realized that while no decision procedure exists for quantification theory there are many proof procedures available—that is, uniform procedures which will ultimately locate a proof for any formula of quantification theory which is valid but which will usually involve seeking “forever” in the case of a formula which is not valid—and that some of these proof procedures could well turn out to be feasible for use with modern computing machinery. Hao Wang [9] and P. C. Gilmore [3] have each produced working programs which employ proof procedures in quantification theory. Gilmores program employs a form of a basic theorem of mathematical logic due to Herbrand, and Wangs makes use of a formulation of quantification theory related to those studied by Gentzen. However, both programs encounter decisive difficulties with any but the simplest formulas of quantification theory, in connection with methods of doing propositional calculus. Wangs program, because of its use of Gentzen-like methods, involves exponentiation on the total number of truth-functional connectives, whereas Gilmores program, using normal forms, involves exponentiation on the number of clauses present. Both methods are superior in many cases to truth table methods which involve exponentiation on the total number of variables present, and represent important initial contributions, but both run into difficulty with some fairly simple examples. In the present paper, a uniform proof procedure for quantification theory is given which is feasible for use with some rather complicated formulas and which does not ordinarily lead to exponentiation. The superiority of the present procedure over those previously available is indicated in part by the fact that a formula on which Gilmores routine for the IBM 704 causes the machine to computer for 21 minutes without obtaining a result was worked successfully by hand computation using the present method in 30 minutes. Cf. §6, below. It should be mentioned that, before it can be hoped to employ proof procedures for quantification theory in obtaining proofs of theorems belonging to “genuine” mathematics, finite axiomatizations, which are “short,” must be obtained for various branches of mathematics. This last question will not be pursued further here; cf., however, Davis and Putnam [2], where one solution to this problem is given for ele

Meaning and Reference

the sigmcl derivative j [SCC(x)] w ill contain a component of the input \vhltc noise. To estimate this comimncnt and to incorporate the re\ults into the ddTcrcntlator would have no si:nific:incc for practictd situutlons. The rmrlts are readily cxteudcd to cover the cases when infinite time intervals are considered and for the cxrscs when higher deriv:]tivcs of the signals itrc to be estimated. As might be expected, the condition (9) is ncccssary and sufficient for the covariancc of j to contain no cfclt:i function. Note that the resulting continuity of COY~(f), j(?)] at every point on /=T means that the stochastic process ~({) is dlffercnti:ibic in tht me:in sqti:irc sense.[z 1 J. B. MOORE Ekpt. of Elec. Engrg. University of Ncwc:istlc New South Wales 2308. Australia

Models and Reality

In 1922 Skolem delivered an address before the Fifth Congress of Scandinavian Mathematicians in which he pointed out what he called a “relativity of set-theoretic notions”. This “relativity” has frequently been regarded as paradoxical; but today, although one hears the expression “the Lowenheim-Skolem Paradox”, it seems to be thought of as only an apparent paradox, something the cognoscenti enjoy but are not seriously troubled by. Thus van Heijenoort writes, “The existence of such a ‘relativity’ is sometimes referred to as the Lowenheim-Skolem Paradox. But, of course, it is not a paradox in the sense of an antinomy; it is a novel and unexpected feature of formal systems.” In this address I want to take up Skolems arguments, not with the aim of refuting them but with the aim of extending them in somewhat the direction he seemed to be indicating. It is not my claim that the “Lowenheim-Skolem Paradox” is an antinomy in formal logic ; but I shall argue that it is an antinomy, or something close to it, in philosophy of language . Moreover, I shall argue that the resolution of the antinomy—the only resolution that I myself can see as making sense—has profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language. The structure of my argument will be as follows: I shall point out that in many different areas there are three main positions on reference and truth: there is the extreme Platonist position, which posits nonnatural mental powers of directly “grasping” forms (it is characteristic of this position that “understanding” or “grasping” is itself an irreducible and unexplicated notion); there is the verificationist position which replaces the classical notion of truth with the notion of verification or proof, at least when it comes to describing how the language is understood; and there is the moderate realist position which seeks to preserve the centrality of the classical notions of truth and reference without postulating nonnatural mental powers.

Is Logic Empirical

I want to begin by considering a case in which ‘necessary’ truths (or rather ‘truths’, turned out to be falsehoods: the case of Euclidean geometry. I then want to raise the question: could some of the ‘necessary truths’ of logic ever turn out to be false for empirical reasons? I shall argue that the answer to this question is in the affirmative, and that logic is, in a certain sense, a natural science.

What Theories are Not

Publisher Summary The chapter discusses the “received view” on the role of theories in empirical science. It discusses those theories that are to be thought of as “partially interpreted calculi” in which only the “observation terms” are “directly interpreted”. The view divides the nonlogical vocabulary of science into two parts: observation terms and theoretical terms. This division of terms that belongs to two classes is allowed to generate a division of statements into two classes, such as observational statements and theoretical statements. A scientific theory is conceived of as an axiomatic system that may be thought of as initially uninterpreted and that gains empirical meaning as a result of a specification of meaning for the observation terms alone. It should be noted that the dichotomy under discussion in the chapter is intended as an explicative not a stipulative one. That is, the words “observational” and “theoretical” are not having arbitrary new meanings bestowed upon them; instead, pre-existing uses of these words (especially in the philosophy of science) are presumably being sharpened and made clear.

What is mathematical truth

In this paper I argue that mathematics should be interpreted realistically – that is, that mathematics makes assertions that are objectively true or false, independently of the human mind, and that something answers to such mathematical notions as ‘set’ and ‘function’. This is not to say that reality is somehow bifurcated – that there is one reality of material things, and then, over and above it, a second reality of ‘mathematical things’. A set of objects, for example, depends for its existence on those objects: if they are destroyed, then there is no longer such a set. (Of course, we may say that the set exists ‘tenselessly’, but we may also say the objects exist ‘tenselessly’: this is just to say that in pure mathematics we can sometimes ignore the important difference between ‘exists now’ and ‘did exist, exists now, or will exist’.) Not only are the ‘objects’ of pure mathematics conditional upon material objects; they are, in a sense, merely abstract possibilities. Studying how mathematical objects behave might better be described as studying what structures are abstractly possible and what structures are not abstractly possible. The important thing is that the mathematician is studying something objective, even if he is not studying an unconditional ‘reality’ of nonmaterial things, and that the physicist who states a law of nature with the aid of a mathematical formula is abstracting a real feature of a real material world, even if he has to speak of numbers, vectors, tensors, state-functions, or whatever to make the abstraction.

Explanation and Reference

In this paper I try to contrast Marxist (and more broadly realist) theories of meaning with what may be called ‘idealist’ theories of meaning. But a word of explanation is clearly in order.

For Ethics and Economics without the Dichotomies

Because Vivian Walsh’s fine essay reviews such a large part of the philosophy I have done in my life, especially (but by no means exclusively) insofar as that philosophy bears on questions of interest to economists as well as to value theorists generally, it is not feasible to comment on it in detail (and Walsh’s fascinating exposition of Pasinetti’s work, I am only competent to learn from, and not to comment on). What does seem feasible, and not only feasible but reasonable as well, is for me to review some of the principal topics that I treat in Putnam (2002a) and then to point out connections with some of the other work, for instance, that of Martha Nussbaum, to which he refers.

Three-valued logic

Let us make up a logic in which there are three truth-values, T, F, and “M,” instead of the two truth-values T and F. And, instead of the usual rules, let us adopt the following: (a) If either component in a disjunction is true (“T”), the disjunction is true; if both components are false, the disjunction is false (“F”); and in all other cases (both components middle, or one component middle and one false) the disjunction is middle (“M”). (b) If either component in a conjunction is false (“F”), the conjunction is false; if both components are true, the conjunction is true (“T”); and in all other cases (both components middle, or one component middle and one true) the conjunction is middle (“M”). (c) A conditional with true antecedent has the same truth-value as its consequent; one with false consequent has the same truth-value as the denial of its antecedent; one with true consequent or false antecedent is true; and one with both components middle (“M”) is true. (d) The denial of a true statement is false; of a false one true; of a middle one middle.

A Philosopher Looks at Quantum Mechanics (Again)

‘A Philosopher Looks at Quantum Mechanics’ (Putnam [1965]) explained why the interpretation of quantum mechanics is a philosophical problem in detail, but with only the necessary minimum of technicalities, in the hope of making the difficulties intelligible to as wide an audience as possible. When I wrote it, I had not seen Bell ([1964]), nor (of course) had I seen Ghirardi et al. ([1986]). And I did not discuss the ‘Many Worlds’ interpretation. For all these reasons, I have decided to make a similar attempt forty years later, taking account of additional interpretations and of our knowledge concerning non-locality. (The Quantum Logical interpretation proposed in Putnam [1968] is not considered in the present paper, however, because Putnam [1994b] concluded that it was unworkable.) Rather than advocate a particular interpretation, this paper classifies the possible kinds of interpretation, subject only to the constraints of a very broadly construed scientific realism. Section 7 does, however, argue that two sorts of interpretation—ones according to which a ‘collapse’ is brought about by the measurement (e.g. the traditional ‘Copenhagen’ interpretation), and the Many Worlds interpretation or interpretations—should be ruled out. The concluding section suggests some possible morals of a cosmological character. 1. Background2. Scientific realism is the premise of my discussion3. What ‘quantum mechanics’ says—and some problems4. Other interpretations of quantum mechanics5. The problem of Einsteins bed6. Classification of the possible kinds of interpretation7. Which interpretations I think we can rule out8. The ‘moral’ of this discussion Background Scientific realism is the premise of my discussion What ‘quantum mechanics’ says—and some problems Other interpretations of quantum mechanics The problem of Einsteins bed Classification of the possible kinds of interpretation Which interpretations I think we can rule out The ‘moral’ of this discussion