philosophy of logicby hilary putnam

Canadian Journal of Philosophy

Philosophy of Logic by Hilary PutnamReview by: Bas C. van FraassenCanadian Journal of Philosophy, Vol. 4, No. 4 (Jun., 1975), pp. 731-743Published by: Canadian Journal of PhilosophyStable URL: http://www.jstor.org/stable/40230549 .

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CANADIAN JOURNAL OF PHILOSOPHY Volume IV, Number 4, June 1975

CRITICAL NOTICE

Hilary Putnam Philosophy of Logic (New York, Harper & Row, 1971).

This is a delightful, stimulating, and especially provocative book. As stated in the preface, Putnam limits his discussion almost entirely to the

ontological issue, that is, the existence of abstract entities. Preliminary skirmishes occupy the first four chapters; the next four are devoted to a sustained and effective argument for realism in the philosophy of mathematics. (Because Putnam does so, I shall use "nominalism" in the wide sense in which it stands for any position opposed to realism.) I do not agree with a word of it, and shall do my best to counter Putnam's

arguments, but they have my unqualified admiration.

1. Preliminary Skirmishes In the first two chapters Putnam considers the question whether a

nominalistic account of logic is possible. It seems to me that he puts his

finger on the crucial issue in demanding, of the nominalist, an account of

validity. Perhaps more debatable is his insistence that we consider the

validity of schemata, such as

(1) A; therefore, A or

rather than of specific arguments or statements. Presumably, what we wish to say is that if the letters "A" and "B" are replaced in (1) by any English sentences, then if the resulting premise is true, so is the resulting conclusion. The problem Putnam sees here for the nominalist is that "any English sentence" cannot be restricted to those concrete entities which are

English sentences actually spoken or written down. We want to say much more than that. What more we want to say one is tempted to express by speaking of all possible English sentences.

And so Putnam has brought out the crucial task nominalists have faced at least since Ockham: to give an account of possibility, of what might be meant by statements about what might or could be, but is not, actual. In our century, the most sustained attempt to give a nominalistic account of

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logic is that of Wilfrid Sellars, and he recognized this task explicitly, and even in the very context in which Putnam poses it. Putnam, however, only raises the problem, and goes on to present arguments for realism.

In the third and fourth chapters, Putnam considers whether we can draw a line between logic and mathematics. He really considers only one attempt: to draw the line between first and higher order logic. That way of drawing the distinction has what he calls "the awkward consequence that the notions of validity and implication turn out to belong to mathematics and not to logic

" (p. 33).

I quite agree with Putnam's discussion of this attempt. But surely there is an obvious alternative? Existence has traditionally been the paradigm fact outside the reach of logic. A theory is substantial rather than part of logic if it has any existential implications. Of course, a substantial theory may be an applied logic, and specifically mathematical theories form one kind of applied logic. As do indeed many putative systems of logic, purifying logic is the task of eliminating existential import. Syllogistic with the revised square of opposition, and quantification theory valid for the empty domain, are pure logic. There exists pure logics of higher order also, and by Putnam's criteria they suffice to define the concepts of validity and implication. Mathematics cannot be developed as part of pure logic; it requires axioms with existential implication, such as that the null-set exists.

Nothing hinges on these questions, of course. Who cares where the line is drawn? But if we are to discuss the subject, I do see an ambiguity just where Putnam sees an awkward consequence. I would like to distinguish logic in the sense of the most elementary part of mathematics from logic in the sense of the science pursued by the logician. The first is a subject for the second; the second is itself an application of pure logic, and indeed of mathematics. The logician constructs, inter alia, models for language games, and often for language games actually played (consider "logic of singular terms", "logic of adverbs"). When he does this he is surely even less pure than the professional topologist. It is logic in this second sense that Brouwer correctly classified as applied mathematics.

2. Heart of the Matter 2/ Putnam's Strategy

The heart of Putnam's book is the Indispensability Argument. In Chapter V, Putnam argues that it is impossible to develop mathematics - even predicative set theory - within the confines of a nominalistic language. I don't think I need to describe what such a language is like, that is familiar from the early (mid-forties) work of Goodman and Quine. It shouid be added that it is assumed here too that the nominalist cannot reasonably postulate the existence of more than finitely many objects. The specific arguments which Putnam gives (for example, pages 38-39) are a bit opaque,

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but we need not stop to examine them. As Henkin pointed out in his "Some Notes on Nominalism" (JSL 18 (1953), 19-29; expecially page 27), a successful reconstruction of mathematics of the sort envisaged would constitute at once a finitist consistency proof for mathematics, and that is

impossible. By the end of Chapter VII, Putnam can say:

the set theoretic "needs" of physics are surprisingly similar to the set theoretic needs of pure logic. Both disciplines need some set theory to function at all. Both disciplines can "live" -- but live badly - on the meager diet of only predicative sets. Both can live extremely happily on the rich diet of

impredicative sets.

(pages 55-56)

And Chapter VIM begins:

So far I have been developing an argument for realism along roughly the

following lines: quantification over mathematical entities is indispensable for

science, both formal and physical; therefore we should accept such

quantification; but this commits us to accepting the existence of the mathematical entities in question.

(page 57)

Contrary to appearances, there is here no essential appeal to Quine's debatable "criterion of ontic commitment." The weakest set theory on which science can live, to use Putnam's term, asserts (i.e. has as a theorem) that some mathematical entities exist (for example, the null-set, and its unit

set, and so on). There is a minor equivocation introduced by the metaphor of "need"

and "diet to live on". Do we need to accept the existence of these entities to do science or mathematics, or to make sense of these subjects? We have the testimony of Abraham Robinson and Paul Cohen that one can do mathematics without believing in the existence of mathematical entities.

Surely Putnam's retort must be that they failed, in his opinion, to make sense of the subject. Many physicists, though acutely aware of the need for

mathematics, must surely be at the very least agnostic about the reality of mathematical entities - more damagingly, they are most likely unaware of the issue altogether.

Putnam's book is addressed to a philosophical problem. The reference to physics establishes that we must make sense at least of predicative set

theory. No philosophy of mathematics has succeeded unless it makes sense at least of that much mathematics. This is how we should read Putnam's intermediate conclusion. The Indispensability Argument is meant to carry us from that intermediate conclusion to the acceptance of the existence of mathematical entities.

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In the remaining three sections of this part, I shall discuss in detail the three parts I discern in Putnam's Chapter VIII (roughly: page 57, pages 59-62, and pages 63-74).

22 Indispensability Arguments: Aim and Strategy The aim of an Indispensability Argument is to establish, first of all, a

conclusion of form

(1) In order to , we need to postulate B. I shall leave it open, for

now, whether. has a more specific form such as

(a) do X (engage in X) (b) make sense of X

where X is a discipline such as physics or mathematics. After (1) has been

established, there are several subordinate arguments: methodological, ontological, and mixed:

(2) We need to . Hence, we need to postulate B.

(3) It is possible to . Hence, B.

(4) We need to believe that it is possible to . Hence, B.

The last, mixed subordinate argument is a "practical inference." The

premise does not imply that it is possible to , and even granting (1), it does not imply more than that we need to postulate B. But the premise is followed immediately by the assertion that - a leap of faith.

I have two fundamental objections to Indispensability Arguments. The first is that, even if we grant (1), the subordinate arguments have no force or cogency. The second is that (1) cannot be established, in the crucial cases, without circularity.

For the first objection, consider (2) and (4). It might be thought, pace Moore's Paradox, that the validity of (2) leads us immediately to the practical inference (4). How could we say, defensibly, that must be postulated - and not do so? Such problems will be recognized, this late in the day, as philosophically spurious, though existentially excruciating. The situation is exactly that of the nineteenth century clergyman who is losing his faith. In his Ibsen world it is unquestionably accepted that life is impossible without faith, that the existence of God alone can ground morality and give meaning to life. (Premise of form (1).) But how can he make the inference to the existence of God? Following our philosophical conscience, we must counsel suicide sooner than an invalid inference.

Turning now to the second objection, note that (1) and (3) are not very different: (3) is valid if and only if (1) is. [This is perhaps a bit

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charitable; I am reading both as saying that we cannot unless is true.] But to give them any import at all, we must go beyond validity, and ask about the truth of the premise. And since all other parts of the

Indispensability Argument are without cogency or force, we have now

gotten to the crux of the matter. At this point I must consider both forms (a) and (b) of . Even if the

realist is right, it cannot be denied that mathematics can be done without

subscribing to realism, by "double-think" or by steadfast refusal to face the issues. Hence if in (1), has form (a), then "doing X" must be construed as "doing X without double-think and without refusing to try and make sense of X" - if, that is, (1) has any chance of being established at all.

So we may as well go on to form (b) of . To make sense of

mathematics, you need to postulate the existence of mathematical entities. But what are the criteria of making sense of mathematics? If the criterion is that set theory be exhibited as a true story, literally understood, then (1) is well-nigh tautological. But since no one is denying that philosophy of mathematics has the task of making sense of mathematics, the criterion I have just mentioned must be peculiar to the realist. Brouwer saw the task

quite differently: the facts about mathematics are facts about a certain human activity. That this human activity should go on even if there be no abstract entities is certainly conceivable. Construing mathematical activity as an intelligible human activity, is also a candidate for the criterion of making sense of mathematics. But on such a criterion, () cannot be established.

Briefly then, (1) cannot be established on pain of circularity in the choice of criteria for making sense of mathematics.

I conclude that Indispensability Arguments, the aurora borealis of the

realist firmament, have no cogency or force, and are mistaken and

misguided in every possible way. Putnam does not wholly disregard the

possibility of such a reaction. His answer is oblique and comes in his

discussion of fictionalism; it is summarized on pages 73/74, and I shall deal

with it below.

23 Dismissive Positions Concerning Realism Between his development of the Indispensability Argument and his

detailed discussion of Fictionalism, Putnam considers two views on which

the whole debate is based on a mistake. The first of these views has no

name, but Putnam refers for illustrations to Austin's Sense and Sensibi/ia; the second is conventionalism.

On the first view, such sentences as "numbers exist", "sets exist", "the null-set exists", though they look like assertions (of fact), are not genuine assertions at all. This may be elaborated in various ways: Those sentences are not intelligible, or neither true nor false, or linguistically deviant. The second view is that these sentences are genuine assertions, and indeed true

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assertions; but they are true by convention, and not, say, because of what the facts are.

It is not easy to make sense of either view. For example, the conventionalist view seems to be that "sets exist" is true whether or not there are sets. Perhaps it would be more charitable to regard "true by convention" as a status sentences may have if they are really neither true nor false. Putnam gives a number of ways in which these two positions might be understood, and various arguments against them. It is unfortunate that the discussion remains very abstract. It seems for example that, in

discussing the first view, Putnam has his sights on the followers of the later

Wittgenstein. But if he did, it would have been more worth his while to focus on the able and acute essays of Goodstein on mathematics.

Be that as it may, I do not see much room for such dismissive

positions either. The ground rules for the ontological debate we learned from Quine's "On What There Is." Those ground rules imply that the

predicate "exists" is unequivocal and sharp. To say that it is unequivocal means that "exists" has the same meaning in all contexts; needless confusion is created by the idea that persons exist in one way and

numbers, if they exist at all, in another way. To say that the predicate is

sharp means that it is not vague; there are no borderline cases. I am afraid that is not very precise. But perhaps I can limit myself to giving an

example. After we have entirely understood what "hairy men in room 114" refers to, it is a question of fact whether there are any: that question is

truly answered by yes or no and has no fallible presuppositions beyond those due to the vagueness of "hairy". In this the question differs from "Are the hairy men in room 114 short?" which suffers in addition from

vagueness in the predicate "short." There seems to be no such vagueness in the term "set". But if there is

no vagueness in the term "set" then, accordingly, the sentence "sets exist" must definitely be true or false. Existence assertions have, as such, no

presuppositions at all; any presupposition would have to come, it seems, from vagueness in the subject term. Hence dismissive anti-realist positions are not really feasible. The only feasible anti-realist position is the position that a philosophy of mathematics, an account that makes sense of

mathematics, need not involve the assertion that mathematical entities exist. There may perhaps come a point where Quine's ground rules for the

debate have to be challenged. Perhaps we shall come to a point at which the whole debate will seem misguided. Whether or not Putnam's arguments in these pages succeed, I do not think the debate has reached such a point yet.

24 Fictionalism, the Serious Rival On page 63 we come to a version of nominalism which explicitly

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rejects the Indispensability Argument without dismissing the debate. Here is Putnam's statement of the position, which he credits to Duhem and Vai hinger:

the fictional ist says, in substance "Yes, certain concepts ... are indispensable, but no, that has no tendency to show that entities corresponding to those

concepts actually exist. It only shows that those 'entities' are useful fictions.**

(page 63)

This is, besides the Quine-Goodman program of the forties, the main rival, which Putnam tackles, and he is eminently fair to it Indeed, most of his discussion purports to be occupied with refuting bad reasons for rejecting Fictionalism. Nevertheless, enough good is espied in those bad reasons to make the final refutation of Fictionalism a matter of summing up.

The first bad reason for rejecting Fictionalism is provided by verification ism. There are no observational consequences that distinguish the

thesis that these are electrons (or whatever) from the thesis that all our

experience is just what it would be if there were electrons (or whatever). Hence, these two theses mean the same or have the same (objective,

cognitive) content. Putnam quickly runs through some of the standard

arguments against verificationism. I don't suppose that there are any verificationists today. If there are

they should read Putnam's objections, here and elsewhere. What is more

disturbing is that the example of à disputed hypothesis has been changed from 'There are no sets" and 'There are no electrons" to "There are

demons." It is only to be expected that Putnam will have an easier time

convincing us not to accept demons than that it is necessary to accept sets

or electrons! I do not wish to accuse Putnam of a biassed selection of

examples; perhaps, on the contrary, his selection should have been a bit less

random. The worst verification ist argument of all, according to Putnam, ran as

follows:

"If you do admit the demon hypothesis as a logical possibility, then you will be doomed to utter scepticism; for you will never be able to offer any reason to

say that it is false." (page 66)

It is very important to see how Putnam parries this move, for his answer

becomes part also of his refutation of Fictionalism. On Pages 66-68 he gives us a capsule introduction to a current fashion on induction. Rationality

requires that if two hypotheses lead to the same testable consequences, and

one is a priori more plausible than the other, then we should not accept the other. Where do a priori plausibility ratings come from? These we

supply ourselves, either each his own or as a community.

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to accept a plausibility ordering is neither to make a judgment of empirical fact nor to state a theorem of deductive logic; it is to take a methodological stand. One can only say whether the demon hypothesis is "crazy" or not if one has taken such a stand; I report the stand I have taken (and, speaking as one who has taken this stand, I add: and the stand all rational men take, implicitly or

explicitly). (page 67)

Does the difference between Putnam and Duhem or Vaihinger about electrons also reduce to a stand on what is a priori the more plausible? How disappointing.

I am also a bit puzzled by how an initial plausibility rating is not a

judgment but a stand. Has it no empirical consequences? Does it never involve any judgment about what is more likely to happen? Or are statements of likelihood not statements at all, bearing to judgments of

empirical fact a deceptive grammatical similarity? And when the Bayesians explained that there is no justification for the initial plausibility ordering of

hypotheses I was mollified. After all they showed that, as evidence is

gathered, posterior probability judgments tend to agreement no matter what initial probabilities were assigned. This was very comforting, for it seemed that the subjective element would never play a crucial role. But here, to my consternation, Putnam shows that the initial plausibility ordering, which is a

quasi-ideological stand, can play a decisive role. I am almost inclined not to be mollified any more!

Be that as it may, I am afraid that Putnam has been played a trick by his own example. When the two competing hypotheses say that there are --

respectively, are no - demons, the situation is surely as Putnam describes it. But consider the more relevant pair of competing hypotheses, which we

may call "Vaihinger" and "Rutherford". The two agree on what the observable facts are like, but the latter adds that there are electrons.

Vaihinger, who is by Putnam's account as agnostic about electrons as about

God, adds nothing. Hence Rutherford entails Vaihinger, and by all accounts of a priori plausibility orderings, Vaihinger is therefore no less plausible than Rutherford.

The second reason for rejecting Fictionalism is called "instrumentalism" by Putnam. The instrumentalist, in Putnam's sense, defends realism against Fictionalism. His argument is a clever one, and Putnam subscribes to it, with the qualification that the "instrumentalist" places unwarranted emphasis on prediction.

The argument in question, stripped of that emphasis, is this: "we cannot separate the grounds which make it rational to accept a proposition from the grounds which make it rational to accept is true" (page 69). This overstates the case rather: the Fictionalist does not say "I accept that there are electrons but I do not accept that it is true that there are

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electrons", nor anything approximating that. He says "I accept the hypothesis that all observable or testable consequences of the electron theory are true; and I use that theory because it is a convenient systemization of its own observable consequences, as you might expect." (This also answers the point about the sun rise tomorrow on page 71.) Putnam adds to the argument a species of what on page 65 he called "open question argument": after the Fictionalist has praised the virtues of some theory or other, Putnam asks him "But then... what further reasons could one want before one regarded it as rational to believe a theory?" Unfortunately, since verificationism has had its Requiem, the Fictionalist can answer "Nothing that can be had." And he may take a quasi-ideological stand of the sort Putnam takes concerning plausibility, attesting that he

places enormous value on economy of belief, especially regarding what there is.

There are philosophers, notably Wilfrid Sellars, who are not realists in

philosophy of mathematics, but are scientific realists, that is, realists with

respect to the theoretical entities of science. I have here simply followed Putnam's suit in assuming sufficient parallelism between the two kinds of debate to make examples about electrons relevant. In any case, Fictionalism

presents itself to Putnam as a challenge to the Indispensability Argument strategy. It will be clear that in my opinion, Putnam has not succeeded in

meeting that challenge. Nevertheless, Putnam has given a clear and

provocative exposition of the realist position. It is no less creditable that he has brought back to our attention what may well be the most powerful rival realism has.

3. Concluding Remarks In this last part, I shall not give further critical attention to the details

of Putnam's arguments. Instead, I shall address myself to two problems which his book raised for me, as it were, in passing. The first is whether there does not remain some hope for an anti-realist position which is different from Fictionalism. The second is whether there is not some

significant extent to which all can agree, the nominalists as well, that realism has got it right.

First then, about anti-realist positions. The Fictionalist, you will have

noticed, insists just as much as does the realist, on the literal construal of what mathematics (and scientific theory) says. It is a theorem that there is a unique set which has no members; this must not be construed in any way except as saying that the null set exists. That, according to the Fictionalist, may well be false. Of the whole edifice of propositions presented by mathematics and science, the Fictionalist accepts as true only those which are solely about observable phenomena. (Different Fictionalists may presumably differ on what is observable.) The Fictionalist goes beyond a

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similarly minded Instrumentalist by presenting a further account of mathematics and science, in which it is likened to literature and myth. (That part of Fictionalism is not presented by Putnam; suffice it to say that realists have not been the only ones to attempt to give a positive account of mathematics and science.)

In all this, the position of Fictionalism in philosophy of mathematics is

quite different from the anti-realist positions of recent decades. Much as I

dislike the use of "nominalism", with its connotations of flatus vocis, to stand for all anti-realist positions, I shall classify these recent positions as Reconstructionist Nominalism. An example is the Goodman-Quine program: they tried to exhibit mathematics as indeed a true story, but a true story in disguise. They tried to give mathematical propositions a construal, different of course from their literal import, upon which they are true

propositions. Upon its literal import, mathematics implies, indeed says that sets exist; not so on a Reconstructionist Nominalist construal.

Such a switch in construal cannot be called a translation, except in a technical, syntactic sense. Whether a translation in the syntactic sense is also a translation in the semantic, or more ordinary sense, depends on whether it preserves truth-values. Now the Reconstructionist Nominalist is not at all concerned with the truth-values of mathematical assertions literally understood; he is only concerned that upon his construal they should clearly be true if they are theorems.

The simplest example I know is the paper "Some Notes on Nominalism" by Henkin, to which I have already referred above. To some extent Wilfrid Sellars' account of mathematics may be understood along the lines of that early paper by Henkin. The reconstrual is simple: all quantifiers are understood in the sense of substitution quantification. The problems later discussed for that reconstrual of quantificational locations were already seen by Henkin, and solved in a simple way: the substituends are not limited to expressions belonging to the extant language of mathematics, but may belong to another language or some extension of the language. When thus understood, the logic of quantification is the same whether we construe quantifiers referentially or substitutionally.

What are the objections to taking this figurative meaning of quantifiers in mathematical language as the right one, the only one that need concern us at all? The position runs into trouble over infinity. To make the reconstruction of mathematics successful, one needs an unlimited supply of substituends. These are expressions, and English or German expressions would do very well. Now, in one sense, English does have an unlimited number of expressions. Not however, in the sense of expressions actually written or spoken during the history of mankind, future and past - at least, not guaranteeably so. In what sense then? We are tempted to say: in the sense that, however many expressions of English, of distinct sorts, are

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produced, we could always produce more. This is a modal statement, and Reconstructionist Nominalism runs headlong into the age-old problem: what account shall we give of possibility? And once we have given a

nominalistically acceptable account of possibility - as Sellars perhaps has -

can we go back and complete the account of mathematics successfully? Is Fictionalism perhaps in a better position with respect to this sort of

problem? I confess myself unable to answer any of these questions. The second problem is that of finding the core of truth in realism,

from a nominalist perspective. We might say that the realist proposes a semantic analysis of mathematical language. Indeed, he proposes the analysis which alone can claim the right to the title "literal construal." An

analogous case would occur if a philosopher analyzing the language of a

monograph on Hamlet were to say that, in any admissible interpretation,

(i) the domain of discourse is a non-empty set

(ii) a part of this domain forms the extension of "is a main character"

(iii) another part of this domain forms the extension of "is neurotic"

(iv) the extension of "is a main character" is part of the extension of

"is neurotic"

and then point with pride to the result that, on his analysis, the sentence

"All the main characters are neurotic" is true, just like it says in the

monograph. And we must grant this hypothetical philosopher that, in one way, he

has got the semantics right. He has got it right in the sense that all the

arguments which are accepted as valid in the monograph are also valid on

his analysis. But in another way, he has not got the semantics right. For

any domain of discourse is treated as a set of real things, and in every domain he has assigned a denotation to "Shakespeare" and also to

"Hamlet" - and that is make-believe. He has modelled talk "about"

non-existents by means of talk about existents: the author of the

monograph talks "about" non-existents just as if he were talking about a

special sort of existent. There is an important equivocation in "semantic analysis" which

Davidson has been pointing to for some time. If we are just being logicians, then the only purpose to which we put semantic analysis is to achieve a

systematic and manageable picture of the valid inference patterns in a

language. And this we can do by modelling - by means of makebelieve truth conditions. As long as ferreting out what is valid is our only criterion for a good semantic analysis, make-belief is perfectly alright. To give another analogous case, we get a very good picture of logical relations

among questions if we treat a question as identical with the disjunction of its direct answers. This is a semantic analysis of questions, and a good one

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if what we are after is inferential relationships. It is in no way a good analysis if we are trying to answer what a question is - questions are not statements - although we might get valuable clues that way.

The nominalist should grant that the realist's literal construal of mathematical language furnishes a good semantic analysis in the first sense. The realist is right about the inferential structure of classical mathematics. But this does not mean that the realist's truth-conditions for mathematical assertions are very important. An alternative semantic analysis, such as

Henkin's, may end up by displaying the same inferential structure, while

starting with different truth-conditions. What is important is that in both these analyses there is from the nominalist point of view, a large element of make-believe.

Semantic analysis need not proceed by means of truth-conditions,

although this is the usual procedure. A semantic analysis of questions, for

example, need not inevitably treat them as entities capable of being true or

false, i.e. statements. We have a once shining through now somewhat

tarnished example of a different sort of semantic analysis of declarative

sentences in the emotivist theory of ethics. Perhaps the usual semantic

analysis of modal locutions, which is so successful in ferreting out patterns of valid inference, but has given no clue at all to truth-conditions (in the sense of Davidson) for modal statements, will be replaced by something similar.

When we raise this issue to the nominalists, we see the Fictionalist and the Reconstruction ist parting ways. For the Reconstructionist wishes to

replace the realist semantic analysis by one of his own, yielding the same inference patterns, and proceeding by the route of providing new and

improved truth-conditions. But the Fictionalist seems to proceed differently. As far as ferreting out the inferential structure is concerned, we may imagine him saying, the .realist semantics is successful - why provide another one? But mathematics remains just as practically useful and

intellectually interesting if we stop thinking of it as true. The realist has

clearly described the picture that bewitches us - the picture that guides inference. Let us just add that this is a matter of make-believe. We speak "about" mathematical entities as if we are speaking about real things - it matters not at all whether they are real.

Are there truth-conditions, in the sense of Davidson, for mathematical

statements, upon which classical mathematics turns out to be true? The realist offers the literal construal as yeilding the truth-conditions in

question, and adds, as a postulate, that enough mathematical entities exist to make at least predicative set theory true. The nominalists reject the postulate. Among them, the Reconstructionist sets out to find new

truth-conditions, and the Fictionalist says there is no need. It is a lovely

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debate, which has persisted in various forms for centuries, and Putnam's book provides an eminently readable introduction to its current form.

BAS C. van FRAASSEN

University of Toronto

August 1974

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philosophy of logicby hilary putnam

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