Philosophy of Logic (Routledge Revivals)

Hilary Putnam

• Publisher: Taylor & Francis
• Year: 2014
• Print ISBN: 9780367475147
• eBook ISBN: 9781317832577
• Edition: 1
• Topic: Philosophy
• Subtopic: Logic in Philosophy

First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.

II. The Nominalism-Realism Issue

DOI: 10.4324/9781315823676-2

At this stage it is already clear that there are philosophical problems connected with logic, and at least one reason for this is also clear: namely, the difficulty of getting any universally acceptable statement of the general principles that all logicians somehow seem to recognize. If we explore this difficulty, further philosophical problems connected with logic will become clearer.

Philosophers and logicians who regard classes, numbers, and similar “mathematical entities” as somehow make-believe are usually referred to as “nominalists”. A nominalist is not likely to say:

(A) “For all classes S, M, P: if all S are M and all M are P, then all S are P”.

He is more likely to write:

(B) “The following turns into a true sentence no matter what words or phrases of the appropriate kind one may substitute for the letters S, M, P: ‘if all S are M and all M are P, then all S are P’.”

The reason is clear if not cogent: the nominalist does not really believe that classes exist; so he avoids formulation (A). In contrast to classes, “sentences” and “words” seem relatively “concrete”, so he employs formulation (B).

It is thus apparent that part of the disagreement over the “correct” formulation of the most general logical principles is simply a reflection of philosophical disagreement over the existence or nonexistence of “mathematical entities” such as classes.

Independently of the merits of this or that position on the “nominalism-realism” issue, it is clear, however, that (B) cannot really be preferable to (A). For what can be meant by a “word or phrase of the appropriate kind” in (B)? Even if we waive the problem of just what constitutes the “appropriate kind” of word or phrase, we must face the fact that what is meant is all possible words and phrases of some kind or other, and that possible words and phrases are no more “concrete” than classes are.

This issue is sometimes dodged in various ways. One way is to say that the appropriate “phrases” that one may substitute for S, M, P are all the “one-place predicates” in a certain “formalized language.” A formalized language is given by completely specifying a grammar together with the meanings of the basic expressions. Which expressions in such a language are one-place predicates (i.e., class-names, although a nominalist wouldn’t be caught dead calling them that) is specified by a formal grammatical rule. In fact, given a formalized language L, the class of permissible substitutions for the dummy letters S, M, P in can be defined with great precision, so that the task of telling whether a certain string of letters is or is not a “substitution-instance” of (1) (as the result of a permissible substitution is called) can even be performed purely mechanically, say, by a computing machine.

1. (5) If all S are M and all M are P, then all S are P

This comes close to satisfying nominalistic scruples, for then it seems that to assert the validity of (5) is not to talk about “classes” at all, but merely to say that all substitution-instances of (5) (in some definite L) are true; i.e., that all the strings of letters that conform to a certain formal criterion (being a substitution-instance of (5) in the formalized language L) are true. And surely “strings of letters” are perfectly concrete—or are they?

Unfortunately for the nominalist, difficulties come thick and fast. By a logical schema is meant an expression like (5) which is built up out of “dummy letters”, such as S, M, P, and the logical words if-then, all, some, or, not, identical, is (are), etc. Such schemata have been used by logicians, from Aristotle to the present day, for the purpose of setting out logical principles (although Aristotle confined his attention to a very restricted class of schemata, while modern logicians investigate all possible schemata of the kind just described). A schema may be, like (5), a “valid” schema—that is, it may express a “correct” logical principle (what correctness or validity is, we still have to see), or it may be “invalid”. For example, is an example of an invalid schema—one that fails to express a correct logical principle. Ancient and medieval logicians already classified a great many schemata as valid or invalid.

1. If some S are P, then all S are P

Now, defining valid is obviously going to pose deep philosophical issues. But the definition of valid we attributed to the nominalist a moment ago, viz., a schema S is valid just in case all substitution-instances of S in some particular formalised language L are true—is unsatisfactory on the face of it. For surely when I say that (5) is valid, I mean that it is correct no matter what class-names may be substituted for S, M, P. If some formalized language L contained names for all the classes of things that could be formed, then this might come to the same as saying “all substitution-instances of (5) in L are true”. But it is a theorem of set theory that no language L can contain names for all the collections of things that could be formed, at least not if the number of things is infinite.

To put it another way, what we get, if we adopt the nominalist’s suggestion, is not one notion of validity, but an infinite series of such notions: validity in L ₁, validity in L ₂, validity in L ₃, … where each notion amounts simply to “truth of all substitution-instances” in the appropriate L_i .

We might try to avoid this by saying that a schema S is valid just in case all of its substitution-instances in every L are true; but then we need the notion of all possible formalized languages—a notion which is, if anything, less “concrete” than the notion of a “class”.

Secondly, the proposed nominalistic definition of validity requires the notion of “truth.” But this is a problematic notion for a nominalist. Normally we do not think of material objects—e.g., strings of actually inscribed letters (construed as little mounds of ink on paper) as “true or “false”; it is rather what the strings of letters express that is true or false. But the meaning of a string of letters, or what the string of letters “expresses”, is just the sort of entity the nominalist wants to get rid of.

Thirdly, when we speak of all substitution-instances of (5), even in one particular language L, we mean all possible substitution-instances—not just the ones that happen to “exist” in the nominalistic sense (as little mounds of ink on paper). To merely say that those instances of (5) which happen to be written down are true would not be to say that (5) is valid; for it might be that there is a false substitution-instance of (5) which just does not happen to have been written down. But possible substitution-instances of (5)—possible strings of letters—are not really physical objects any more than classes are.

One problem seems to be solved by the foregoing reflections. There is no reason in stating logical principles to be more puristic, or more compulsive about avoiding reference to “nonphysical entities”, than in scientific discourse generally. Reference to classes of things, and not just to things, is a commonplace and useful mode of speech. If the nominalist wishes us to give it up, he must provide us with an alternative mode of speech which works just as well, not just in pure logic, but also in such empirical sciences as physics (which is full of references to such “nonphysical” entities as state-vectors, Hamiltonians, Hilbert space, etc.). If he ever succeeds, this will affect how we formulate all scientific principles—not just logical ones. But in the meantime, there is no reason not to stick with such formulations as (A), in view of the serious problems with such formulations as (B). [And, as we have just seen, (B), in addition to being inadequate, is not even really nominalistic.]

To put it another way, the fact that (A) is “objectionable” on nominalistic grounds is not really a difficulty with the science of logic, but a difficulty with the philosophy of nominalism. It is not up to logic, any more than any other science, to conform its mode of speech to the philosophic demands of nominalism; it is rather up to the nominalist to provide a satisfactory reinterpretation of such assertions as (5), and of any other statements that logicians (and physicists, biologists, and just plain men on the street) actually make.

Even if we reject nominalism as a demand that we here and now strip our scientific language of all reference to “nonphysical entities”, we are not committed to rejecting nominalism as a philosophy, however. Those who believe that in truth there is nothing answering to such notions as class, number, possible string of letters, or that what does answer to such notions is some highly derived way of talking about ordinary material objects, are free to go on arguing for their view, and our unwillingness to conform our ordinary scientific language to their demands is in no way an unwillingness to discuss the philosophical issues raised by their view. And this we shall now proceed to do.

We may begin by considering the various difficulties we just raised with formulation (B), and by seeing what rejoinder the nominalist can make to these various difficulties.

First, one or two general comments. Nelson Goodman, who is the best-known nominalist philosopher, has never adopted the definition of validity as “truth of all substitution-instances”. (It comes from Hugues Leblanc and Richard Martin.) However, Goodman has never tackled the problem of defining logical validity at all, so I have taken the liberty of discussing the one quasi-nominalistic attempt I have seen. Secondly, Goodman denies that nominalism is a restriction to “physical” entities. However, while the view that only physical entities (...

Table of contents

• Cover Page
• Half Title Page
• Title Page
• Copyright Page
• Original Title Page
• Original Copyright Page
• Table of Contents
• Preface
• I. What Logic Is
• II. The Nominalism-Realism Issue
• III. The Nominalism-Realism Issue and Logic
• IV. Logic vs. Mathematics
• V. The Inadequacy of Nominalistic Language
• VI. Predicative vs. Impredicative Conceptions of "Set"
• VII. How Much Set Theory Is Really Indispensable for Science?
• VIII. Indispensability Arguments
• IX. Unconsidered Complications