1999 What Anti-Realist Intuitionism Could Not Be

Abstract:

One of the two major parts of Dummett’s defense of intuitionismis the rejection of classical in favor of intuitionistic reasoning in mathematics,given that mathematical discourse is anti-realist. While there have been illu-minating discussions of what Dummett’s argument for this might be, no con-sensus seems to have emerged about its overall form. In this paper I give anaccount of this form, starting by investigating a fundamental, but littlediscussed question: to what view of the relation between deductive principlesand meaning is anti-realism committed? The result of this investigation is aconstraint on meaning theoretic assessments of logical laws. Given this con-straint, I show that, surprisingly, a consistent anti-realist critique of classicallogic could not rely on the rejection of bivalence. Moreover, a consistentanti-realist defense of intuitionism must begin with a radical rejection of thevery conception of logical consequence that underlies realist classical logic.It follows from these conclusions that anti-realist intuitionism seems commit-ted to proceeding by proof theoretic means.

Since its initial airing, in “The Philosophical Basis of Intuitionistic Logic”(1973a), Michael Dummett’s proposal for the rehabilitation ofmathematical intuitionism has become one of the more familiar positionsin the philosophy of logic. Given the amount of critical attention that ithas received, and given that Dummett has recently elaborated the project,in

The Logical Basis of Metaphysics (1991), one would think that surelyby now there could not be many aspects of it that is not fully understood.

There is, indeed, hardly any dispute over the general outlines ofDummett’s argument. As presented in “Philosophical Basis,” the defenseof intuitionism falls into two principal parts. First it is argued that theconcept of truth applying to mathematical discourse cannot be realist,i.e., cannot be independent of human beings’ in principle abilities to recog-

77

WHAT ANTI-REALISTINTUITIONISM COULD

NOT BE

BY

SANFORD SHIEH

Pacific Philosophical Quarterly 80 (1999) 77–102 0031–5621/99/0100–0000© 1999 University of Southern California and Blackwell Publishers Ltd. Published by

Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and350 Main Street, Malden, MA 02148, USA.

nize its application to mathematical sentences. This argument turns ontwo things: the requirement that the theory of meaning account for themanifestation of meaning, and the claim that if the meanings of sentencesare identified with realist truth conditions, then certain ‘undecidable’mathematical sentences do not satisfy this requirement. Second, Dummettargues that the rejection of a realist conception of truth for mathematicsimplies that the manifestation requirement can be satisfied only if thetruth conditions of mathematical statements are their conditions of proof.From this claim Dummett appears to infer two others: that the principleof bivalence does not hold in mathematical discourse, and that the logicalconstants must be explained in terms of proof rather than truth. Dummettthen concludes, without further elaboration, “as soon as we construe thelogical constants in terms of this conception of meaning, we become awarethat certain forms of reasoning which are conventionally accepted aredevoid of justification” (1973a, p. 226, emphasis mine).

The first part of this defense, centering on Dummett’s anti-realist posi-tion in the philosophy of language and metaphysics, has received the bulkof critical scrutiny. Although it would be rash to say that there is nodispute over its interpretation and assessment, there is a reasonable degreeof consensus on the identity of the premises and inferential transition thatit comprises, i.e., on the overall form of this argument.

Unfortunately the same cannot, it seems to me, be said of the secondpart of Dummett’s defense. It is not that this part has been entirelyneglected. There are a number of illuminating discussions of how,assuming that mathematical discourse is anti-realist, one might argue forthe rejection of classical in favor of intuitionistic reasoning in mathe-matics.1 But, while these discussions have certainly improved the prospectsof understanding this part of Dummett’s defense of intuitionism, they donot provide unambiguous answers to the most elementary questions aboutit. As stated in “Philosophical Basis,” the argument appears to involvethree ideas about mathematical discourse: the identity of truth with proof,the rejection of bivalence, and the necessity for a revision in the explana-tion of the logical constants. The first of these is constitutive of anti-realismabout mathematical discourse. About the other two, it is natural to ask:

● Are they premises of the argument?● In what sense, and for what reason, does bivalence fail to apply to

mathematics?● What exactly is an explanation of the logical constants, and how

should it be revised?● What is the relation between the failure of bivalence and the revision

in the explanation of the constants?● Are these independent of one another?● Are they both required for the argument?

78 PACIFIC PHILOSOPHICAL QUARTERLY

© 1999 University of Southern California and Blackwell Publishers Ltd.

● What exactly, according to this argument, is wrong with classicalreasoning?

● How exactly does intuitionistic reasoning avoid these problems?

This situation has, alas, not improved much with the appearance of TheLogical Basis of Metaphysics; for, in that work Dummett presents theidea of a proof theoretic justification of logical laws, but without sayingmuch about how it is related to the argument(s) suggested in the earlieressay. In particular, it is not clear how to answer the following questions.

● Is the proof theoretic justification procedure (usable as) part of thedefense of intuitionism?

● If so, is such a defense a new argument, or does it extend or modifythe argument of “Philosophical Basis”?

● How is the proof theoretic justification procedure related to therejection of bivalence in mathematics?

● How is it related to the explanation of the logical constants?● What are the relations among the proof theoretic justification of

logical laws, the rejection of bivalence, and the new explanation of the logical constants?

Thus the precise details of how anti-realism is supposed to lead to intu-itionistic reasoning in mathematics remain elusive.

It is not my intention to supply a complete account of the second partof Dummett’s defense of intuitionism. I intend, instead, to clarify the ele-mentary questions posed above, and, thereby, to indicate what the overallform of Dummett’s argument might be. I approach this task in a some-what indirect way. I will begin by discussing a question that has hardlybeen raised at all in extant studies of Dummett’s argument: to what viewof the relation between deductive principles and meaning is anti-realismcommitted? This relation surely underlies any attempt to assess the statusof deductive principles on the basis of theories of meaning, and hencemust constitute the general framework within which a meaning theoreticdefense of intuitionism can proceed. Thus, the clarification of this issueseems to me compulsory, before we can see our way to how Dummett’sargument is supposed to work.

The discussion of this preliminary issue occupies the first three sectionsof the paper. In section 1, I give an account of Dummett’s general andprogrammatic conception of the relation between meaning and logic. Insection 2 this conception is fleshed out through an account of Dummett’sinterpretation of Frege’s justification of classical logic. Section 3 reflectson how the general conception operates in the Fregean justification ofclassical reasoning, in order to extract a constraint on any meaning theo-retic assessment of logical laws.

WHAT ANTI-REALIST INTUITIONISM COULD NOT BE 79


The interaction of this constraint with anti-realism is the basis of myaccount of the overall structure of the anti-realist argument. I begin, insection 4, with a proposal for the rejection of classical logic based on thefailure of bivalence in mathematics. This proposal is criticized in section5, where I argue that a consistent anti-realist critique of classical logiccould not have much use for the rejection of the principle of bivalence.This is perhaps a surprising conclusion. Dummett has at times seemed toidentify realism with respect to an area of discourse with the adoption ofbivalence for it; thus it might seem that the rejection of bivalence is manda-tory for anti-realism about an area of discourse.2 Hence, it is hard to seehow anti-realist intuitionism could dispense with this rejection. However,in section 6 I generalize the argument against the relevance of a rejectionof bivalence to reach the principal thesis of this paper: a consistent anti-realist defense of intuitionism must rest on a conception of logical conse-quence distinct from that which underlies a realist view of classicalreasoning. The necessity of abandoning a realist view of logical conse-quence, I argue, is what makes the rejection of bivalence a dispensablepart of the critique of classical mathematical reasoning. I conclude insection 7 with a corollary of this thesis, namely, that the use of a prooftheoretic justification procedure is the only route available, so far, arrivingat a coherent anti-realist intuitionism.

One final note. The ensuing is meant as a clarification of an argument;hence it should not be taken as a critique, or an endorsement, of the anti-realist defense of intuitionism.

1.

Dummett distinguishes between logical laws, such as the law of excludedmiddle, which states “the truth of every instance of [the] schema … ‘A ornot A’” and semantic principles, such as the thesis that “every statementis either true or false … called the principle of bivalence”. (1978, p. xix)Here is Dummett’s reasons for this distinction:

The importance of distinguishing the semantic principles from the logical laws lies in thefact, generally acknowledged in the case of [the law of excluded middle and the principleof bivalence], that, while acceptance of the semantic principle normally entails acceptanceof the corresponding logical law, the converse does not hold. (ibid.)

More generally, the relation between logical laws and semantic principlesis this:

The standard practice of logicans … is to seek to define two parallel notions of logicalconsequence, one syntactic and the other semantic, and then attempt to establish a relation



between them. The ideal is to establish their extensional equivalence. Proof of such equiv-

alence falls into two parts, a soundness theorem showing that, whenever the syntactic rela-

tion obtains, so does the semantic one, and a completeness theorem, showing the converse

inclusion. Failure of soundness … must be remedied.

… [T]he semantic notion always has a certain priority: the definition of the syntactic

relation is required to be responsible to the semantic relation, rather than the other way

about. The syntactic relation is defined by devising a set of primitive rules of inference, and

a corresponding notion of a formal deduction. If a semantic notion can be defined with respect

to which a soundness proof can be given, we then have a reason for regarding the primitive

rules of inference as valid … (1973b, p. 290; emphasis added)

These passages suggest the following picture of the relation betweendeductive and semantic principles. Logical laws are schemata, syntac-tically specified forms of statement or of argument. A semantic definitionor analysis of logical consequence is an account of this relation given interms of semantic principles. Since Dummett’s critique of classical logicis supposed to be based on considerations in the theory of meaning, it isnatural to conclude that semantic principles govern the meanings ofexpressions, and are determined by, or part of, the theory of meaning.3

The semantic analysis of consequence captures a notion of logical conse-quence conceptually prior to, or more fundamental than, syntacticanalyses of the consequence relation. Hence it yields a standard or crite-rion that determines whether or not any syntactically specified form ofinference is genuinely valid. The assessment of a logical law is then amatter of determining whether a syntactically specified form of inferenceconforms to the semantic criterion: as Dummett puts it, when there is asoundness proof for a system of logical laws (or rules of inference) wehave grounds for regarding these laws as truly valid; and, conversely,without a soundness proof we do not have such grounds, which is whyfailure of soundness must be remedied.4

So, we have here a general program for the assessment of logicalprinciples:

● the theory of meaning determines the semantic properties of lin-guistic expressions;

● these semantic properties determine the semantic analysis of logicalconsequence;

● and finally, semantic consequence determines the validity of logicallaws.

Our goal, ultimately, is to give an account of what kind of criticism ofclassical logic can be given, on the basis of this program. In order to doso, I will address first a question about the program:



● In what way is a semantic analysis of consequence prior to thesyntactic one?

This question may also be posed in two alternative ways. In what sensedoes a semantic analysis determine whether a given form of inference isgenuinely valid? Or, why must failure of soundness be remedied? I willapproach these questions by considering an instance of how the programmay be carried out: Dummett’s interpretation of Frege’s commitment toclassical logic.

2.

Dummett sees Frege’s philosophy of logic as a decisive advance over, butstill concerned with the same fundamental problem as, Aristotle’s accountof logic:

Logic began with Aristotle’s discovery that the validity of an argument could be charac-terised by its being an instance of a valid argument-schema … and is valid if every instancewith true premisses has a true conclusion … This presemantic notion of an interpretationof a schema by replacement was the only one that logic had to operate with until Frege.Frege supplied us for the first time with a semantics, that is to say, an analysis of the wayin which a sentence is determined as true or otherwise in accordance with its compositionout of its constituent words …

Once we have such a semantics, we can substitute for [the] notion of an interpretationby replacement that of a semantic interpretation, under which make a direct assignment tothe schematic letters of the semantic values of expressions of the appropriate categories,bypassing the expressions themselves. (1975, pp. 118–21; emphasis in original)

It is important to see how Dummett conceives of the problem thatAristotle, and subsequently Frege, faced. The problem is to give an expla-nation of the distinction between deductively correct, i.e., valid, argu-ments, and incorrect (invalid) ones. The Aristotelian explanation derivesthe validity of an individual argument from its conforming to a schematicpattern, in all the particular instances of which truth is preserved frompremises to conclusions. The limitation of this account, according toDummett, is that it provides no explanation of why there is truth preser-vation in instances of the argument forms. Since the account is silent onthe reasons underlying this truth preservation, it does not rule out thepossibility that it is accidental: our sentences’ relation to the world justhappened to work out in such a way that the former have the requiredtruth values. Thus, what at bottom motivates the thought that theAristotelian explanation is insufficient is the sense that deductive validityinvolves necessity.



Frege’s advance over Aristotle’s view of logic is his explanation ofdeductive validity given in terms of his semantic theory, or, more precisely,the part of his semantic theory Dummett calls the theory of reference.

This semantics rests on a syntactic analysis that provides an articulationof sentences into logical and non-logical expressions. Given this syntax,the semantics is formulated in accordance with two fundamental ideas:

The first fundamental idea … is that the condition for a complex sentence to be true dependssolely upon its composition out of atomic sentences … Let us now define the semantic valueof an atomic sentence to be whatever feature of it it is both necessary and sufficient that itpossess if every complex sentence is to be determined as true or otherwise in accordancewith its composition out of atomic sentences. Then the second fundamental principle ofFregean … semantics is that the semantic value of an atomic sentence (and hence of anysentence) is just its truth-value – its being true or not … (Dummett 1975, p. 120)

Starting from these two ideas Frege first extends the notion of semanticvalue to sub-sentential expressions:

The semantic value of any expression is … that feature of it which must be ascribed to itif every sentence in which it occurs is to be determined as true or otherwise. (ibid.)

The theory of reference that he then constructs has two familiarcomponents:

(R1) an account of the truth conditions of logically complex sentencesin terms of the truth values of its sub-sentences;

and

(R2) an account of the truth conditions of atomic sentences in termsof the semantic values of its constituent expressions.

There are three points to note about this semantics. First, on Dummett’sinterpretation, Frege’s account of quantification is part of (R1): the sub-sentences of a qualified sentence are all its substitution instances. Second,the notion of semantic value is an abstraction from the specific meetingsof non-logical expressions. Finally, (R1) states the contributions made tothe truth conditions of logically complex sentences by the logical con-stants, and hence can be thought of as an account of the meanings orsenses of the constants.

With this semantics, we have completed the first step of the program forthe semantic assessment of logic described in section 1: we have now anaccount of the requisite semantic properties of linguistic expressions. Wecan now move to the second step, Frege’s analysis of logical consequence.



This analysis comes from an interpretation, based on the theory of refer-ence, of a truism about what constitutes a valid form of argument: everyinstance with true premises has a true conclusion. The first step in theinterpretation turns this truism into a no less truistic claim about validity:

(V) If the truth conditions of any set of instances of the premises arefulfilled, then so is the truth condition of the correspondinginstance of the conclusion.

Now it is possible to bring to bear the account of truth conditions thatconstitutes Frege’s theory of reference.

The non-logical expressions of any instance of an argument schemawill have some, among all possible, semantic values. And these semanticvalues are sufficient to determine the truth or falsity of all the premisesand of the conclusion in this instance of the argument schema. So, inorder to determine whether truth is preserved from premises to conclusionin any one instance of the argument schema, all that is required are thesemantic values of the non-logical expressions occurring in that instance.It follows that, in order to determine whether truth preservation holdsfor all instances of the argument schema, i.e., whether (V) holds, what isrequired are all the semantic values that the non-logical expressions mighthave in any instance of the schema.5 Now, a set of semantic values thatit is possible for the non-logical expressions to have is what Dummettcalls a semantic interpretation of the argument schema. So, whether theschema satisfies (V) is determined by whether the conclusion schema istrue under every semantic interpretation in which the premise schemataare true.6

Let us call the foregoing analysis the classical analysis or criterion oflogical consequence. It completes the second step of the program for thesemantic assessment of logic.

It is important, for the ensuing argument, to see that this analysis con-tains two independent and separable ideas. The first, and more funda-mental, idea underlying the analysis is an interpretation of the truism (V)that distinguishes two factors determining deductive validity:

(a) the meanings of the logical constants;(b) the possible semantic values of linguistic expressions.

Let us call this idea the semantic value interpretation of validity. Thesecond idea concerns the role that the principle of bivalence plays in theclassical criterion. Bivalence enters the analysis in the guise of the secondfundamental principle of Fregean semantics. It is thus a constraint on thepossible semantic values of atomic sentences, and, therefore, as Dummett



notes, of all sentences; hence it imposes a restriction on what semanticinterpretations are possible.

That these two ideas are independent follows from the fact that is possi-ble to define the notion of semantic value for all expressions, includingsub-sentential ones, even if one abandons the second fundamental prin-ciple of Fregean semantics.7 Hence, an essentially Fregean analysis of log-ical consequence, based on the semantic value interpretation of validity,does not need to adopt the principle of bivalence.

Given the restriction imposed by bivalence, one can move to the thirdstep of the program. Here the details are straightforward and familiar. I reproduce a version of the justification of the law of excluded middlehere, for reference in the ensuing argument.

1. Given the meaning of ‘or’, a statement of the form

p or q is truejust in case at least one of p and q is true; and it is false just in caseboth p and q are false.

2. Given the meaning of ‘not’, a statement of the form not p is truejust in case p is false, and false just in case p is true.

3. Every statement is either true or false.4. Hence there are only two possible interpretations of the schema p

or not p.5. If p is assigned the value true, then, by 1, p or not p is true.6. If, on the other hand, p is assigned the value false, then, by 2, not

p is true.7. Hence, by 1 again, p or not p is true.8. So, p or not p must be true.

3.

I turn now to discuss the sense in which a semantic analysis of consequencedetermines whether a given form of inference is genuinely valid.

One possible answer is this. Recall that Dummett claims that “accep-tance of the semantic principle normally entails acceptance of the corre-sponding logical law” (1978, p. xix). This appears to suggest that if one didnot accept the semantic principle, one would have a prima facie reason notto accept the logical law.8 And so the claim seems to be that the semanticjustification is required to convince one who did not accept the logical lawin question to reason with it. Since the criterion of validity is meant to becompletely general, it appears that the justification might be intended toconvince a skeptic about all of logic to accept deductive reasoning.

But this is a view explicitly rejected by Dummett. For instance, in TheLogical Basis of Metaphysics he writes, “there is no sceptic who denies thevalidity of all principles of deductive reasoning, and, if there were, there



would obviously be no reasoning with him” (1991, p. 204). Moreover, in“The Justification of Deduction,” Dummett suggests a different accountof the purpose of a philosophical justification of logical laws:

The validity of a particular form of inference is not a premise for the semantic proof of itssoundness; at worst, that form of inference is employed in the course of the proof. Now, clearly,a circularity of this form would be fatal if our task were to convince someone, who hesitatesto accept inferences of this form, that it is in order to do so. But to conceive the problem ofjustification in this way is to misrepresent the position that we are in. Our problem is not topersuade anyone, not even ourselves, to employ deductive arguments: it is to find a satisfactoryexplanation of the role of such arguments in our use of language … A philosopher who asksfor a justification of the process of deductive reasoning is not seeking to be persuaded of itsjustifiability, but to be given an explanation of it. (1975, pp. 295–6)

The suggestion made in this passage may be spelt out as follows: thesemantic justification of logical laws is not intended to convince us thata form of argument is in fact truth preserving; rather, it provides an expla-nation for why it is truth preserving. The underlying idea is that if anargument schema is truth preserving for the right reasons, then it isgenuinely logical.9

In the case of the Fregean analysis of consequence, what these “rightreasons” might be has already been suggested. This analysis, as notedabove, abstracts from the meanings of the non-logical expressions, butdepends on an account of the meanings of the logical constants. Hence,we may take it to be an explanation of deductive validity in terms of themeanings of the logical constants. More precisely, the explanation is thatany instance of a valid schematic form preserves truth because, given themeanings of the logical constants, no matter what the non-logical expres-sions meant, the truth condition of the conclusion is fulfilled providedthat those of the premises are. Thus, any principle of inference (or anytruth) is determined as logical if it is truth preserving (or true) in virtueof the meanings of the logical constants. On this construal the Fregeananalysis, to put it bluntly, determines what is logic and what is not, onthe basis of semantic principles. One might say that, on this interpretation,a semantic justification of logical laws provide a criterion of logicality.

This interpretation appears to be general. Prima facie, it is independentof the particular details of the classical analysis of consequence. So it canbe extended to any attempt to assess logical laws on semantic grounds.That is, any such attempt tacitly relies on the idea that the semantic factsused in the assessment determine whether the principles being assessed arecorrect for the right reasons, and thus constitutes a criterion of logicality.

This interpretation of the semantic analysis of consequence as a crite-rion of logicality has certain implications for the kind of criticism ofclassical reasoning in mathematics that can be given, on the basis of



Dummett’s general view of the relation between logical and semantic prin-ciples. It seems that there are two ways in which a set of putative logicallaws can fail to be justified with respect to a semantic theory:

● the theory shows that some of these laws are invalid,● the theory does not show that some of these laws are valid.

In either case, the natural assessment of the situation is that there is aconflict between our conception of the meanings of mathematical state-ments and the forms of reasoning we adopt in mathematics. In the firstcase, to the extent that we take our conception of meaning to be morefundamental, it seems that we have no choice but to give up these formsof reasoning which are not truth preserving. But, in the second case it isnot clear what we should do, even if we accept that our conception ofmeaning is more fundamental. According to the interpretation of semanticjustifications as criteria of logicality, in this case we have a form of reason-ing that is not logical (or that we cannot certify as logical). But this assess-ment is consistent with the claim that the validity of the form of reasoningis peculiar to mathematics, in the sense of being truth-preserving onlywithin mathematical contexts. In this case it is unclear whether there isany reason to give up using the principle, although, clearly, this situationraises the question why there is such a restriction on the domain of validityof the form of inference.10

Finally, the interpretation of the classical criterion as a criterion of logi-cality also yields a constraint on meaning theoretic assessments of logicallaws. Since semantic consequence is supposed to constitute the standardfor distinguishing the logical from the non-logical, applicable to all (syn-tactically specified) forms of inference, it would seem that the properties(of forms of inference) on which it is based cannot be logical ones. For,otherwise, the criterion presupposes that these properties are indeedlogical, and hence cannot be the basis for determining whether they arelogical or not. Specifically, in the present case, the classical criterion mustpresuppose that a statement’s having the truth conditions it has is inde-pendent of its being valid, or its being implied by certain other statements,etc., that is of its having the logical properties it has. And so deductivevalidity is not, as it were, a primitive property that one simply recognizesin certain arguments or forms of argument; rather, one recognizes thisproperty in virtue of recognizing certain semantic properties of the argu-ments or forms in question.11

4.

Since the realist justification of classical logic bases the validity of the lawof excluded middle on the principle of bivalence, a natural idea for the crit-



icism of classical logic is that the invalidity of this law might be establishedfrom the failure of bivalence. This idea is reinforced by two facts, both notedabove. First, in a number of places, Dummett gives an apparently fairlystraightforward argument against the applicability of bivalence to ‘unde-cidable’ sentences. Second, Dummett’s apparent identification of realismwith bivalence, together with the central role that bivalence plays in Frege’sjustification of classical logic, makes it hard to see how an anti-realist critiqueof classical reasoning can fail to involve the rejection of bivalence.

In order to construct such an argument one has to do two things: givean account of precisely how ‘undecidable’ mathematical sentences ‘fail’to satisfy bivalence, and show how the invalidity of excluded middle fol-lows from this ‘failure.’

The first of these tasks would seem to require an explanation of theconcept of undecidability. But I will here omit a full discussion of thisconcept, because many of the complexities involved in explicating it arenot directly relevant to the present argument.12 Rather, I will simply adoptthe following interpretation of the failure of bivalence by undecidable state-ments: we have no justification for the claim that such statements (whateverthey might be) have one of the two semantic values, true or false.13

Dummett’s writings provide no explicit accounts of an argument fromthe failure of bivalence to the invalidity of excluded middle. But it is easyto sketch such an argument.

Begin with the observation that, in the realist justification of classicallogic, the principle of bivalence is required to insure that there are exactlytwo possible semantic interpretations of the schema p or not p, corre-sponding, naturally, to the case where p is assigned the semantic valuetrue, and the one in which it is assigned the value false. By appeal to themeanings of the logical constants ‘or’ and ‘not’, the schema is shown tobe true under these two interpretations. Since exactly these two interpre-tations constitute all the possible ones, the schema is valid.

But now we are assuming that we have no justification for acceptingthat undecidable mathematical statements have one of the two semanticvalues. So, if instances of the schema include undecidable statements, wedo not know that the schema is in all cases correctly interpreted by eitherof the two semantic interpretations described above. Hence we have nojustification for concluding that the schema is true under all possiblesemantic interpretations, and therefore no argument ensuring that the lawof excluded middle is semantically valid, if its instances include undecid-able mathematical statements.

Let us call this argument ‘the bivalence argument.’14 Note that it doesnot demonstrate the invalidity of excluded middle. Hence, in accordancewith the second point made above about the classical criterion, the crit-icism of classical logic it licenses is that the law of excluded middle is nota logical principle.



The basic structure of the bivalence argument is this. To begin with,the anti-realist view of meaning, which is a doctrine about the truth con-ditions of statements, leads to a thesis about the possible semantic valuesof undecidable mathematical statements. This thesis, secondly, conflictswith the realist constraint on the semantic value of sentences that is theprinciple of bivalence. And this, finally, supports the judgement that wehave no reason to accept the putative logical law of excluded middle asin fact logically valid.

It follows from this description of the structure of the bivalence argu-ment that it accepts the semantic value interpretation of deductive validitythat underlies the classical analysis of consequence – deductive validity isdetermined by:

(a) the meanings of the logical constants; and(b) the possible semantic values of linguistic expressions.

The bivalence argument disagrees with the Fregean realist justification ofclassical logic only in rejecting Frege’s account of the possible semanticvalues of sentences, an account justified by his realism. But it does notreject Frege’s account of the meanings of the logical constants. That is,the argument rejects Frege’s account of (b), but not of (a).

From this last point it follows that, according to the conclusion of thebivalence argument, the only feature or consequence of the realist concep-tion of truth incompatible with the anti-realist view of meaning is biva-lence. Thus, as far as the bivalence argument is concerned, it is consistentwith anti-realism to adopt an essentially Fregean analysis of logical conse-quence for mathematical discourse, provided that the underlying semantictheory does not employ a notion of truth implying bivalence. There are,as is well-known, a number of types of semantic theories that do notimply bivalence. In particular, two principal ones are those allowingpartial truth value assignments and those employing supervaluations. Thelatter however, as Dummett himself points out, validates classical logic.15

Hence, it is unclear that the bivalence argument successfully rejectsclassical logic. This, however, is hardly conclusive, since it has also beenargued that supervaluational semantic theories are not consistent withanti-realism.16

5.

I turn now to a more fundamental worry about the bivalence argument.Let us begin with a question: suppose we accept the anti-realist conceptionof meaning, as applied to mathematics, so that mathematical truth is iden-



tical to provability – is it clear that we understand what the bivalenceargument is claiming?

The main issue is this. The bivalence argument starts from the realistjustification of the law of excluded middle. And, in particular, it acceptsthe following premise in that argument: given the meaning of ‘or’, a state-ment of the form p or q is true just in case at least one of p and q is true;and it is false just in case both p and q are false. If truth is identified withproof, this claim surely means:

We have (there is) a proof of a statement of the form p or q just incase we have (there is) a proof of at least one of p and q; and we have(there is) a refutation of it just in case we have (there is) a refutationof both p and q.

But, now, what does this claim say? We are, of course, focussing onmathematical statements, so the claim can be understood in two ways.First, it can be understood as a claim about outright provability, asopposed to provability in a theory, i.e., about deductive justification fromno premises or proof from logic alone. On this interpretation, the claimis: p or q is provable from logic alone if and only if p is or q is. But whatthis amounts to is the claim that the system of purely logical inferenceapplicable to mathematical statements satisfies the disjunction property.17

And that implies that this system of inferences cannot be classical logic.Second, we can interpret premise 1 to be about provability in a mathe-matical theory. In this case the claim it makes is: p or q is provable froma theory if and only if p is or q is. This claim is satisfied by such theoriesas Heyting arithmetic, and by most intuitionistic mathematical theories.But it is not, in general, satisfied by classical mathematical theories.18

Thus, if anti-realism is true of mathematics, then to accept the premisethat:

(*) the intuitive truth conditions of disjunctions correctly capture themeaning of ‘or’ in mathematical discourse,

is already to accept that no legitimate system of purely deductive inferenceapplying to mathematics is classical. Put in this way, it would seem thatno classical logician should accept this premise; she should ask; whatreason have we been given for assuming that the disjunction propertyought to hold for mathematical reasoning? Hence it would appear thatthe bivalence argument begs the question against the validity of classicallogic. So, regardless of whether Dummett himself advances this argument,a consistent anti-realist should not do so.

The argument just given faces a natural objection. I claimed that noclassical logician should have reason to accept premise (*), and this claim



formed my basis for concluding that the bivalence argument is question-begging. But why should we accept this claim? Premise (*), after all, isbased simply on our intuitions about the truth conditions of disjunctivestatements. Hence, once it is granted that anti-realism is true, we mustaccept that the disjunction property holds of our system of deductiveinference, on pain of giving up our intuitions about disjunctive statements.Perhaps this accounts for Dummett’s remark, quoted above, that “as soonas we construe the logical constants in terms of the[e anti-realist] concep-tion of meaning, we become aware that certain forms of reasoning whichare conventionally accepted are devoid of justification.”

What this objection, if correct, shows, however, is that, although thebivalence argument is not question-begging, it is unnecessary for the anti-realist rejection of classical mathematical reasoning. If correct, the objec-tion shows that, once anti-realism is accepted, we do not need the claimthat undecidable mathematical statements are not guaranteed to be eithertrue or false, in order to show that the validity of classical reasoning isunjustified. All we need is the intuitive claim that statements of the formp or q are true just in case at least one of p and q is.

So, on the assumption that the objection is correct, the upshot of theforegoing is this. The bivalence argument is either question-begging orunnecessary. If the former, then, clearly, the rejection of bivalence failsto yield a cogent criticism of classical logic in mathematics. If the latter,then, again, the rejection of bivalence fails as the basis for the anti-realistcriticism of classical logic. For, here the claim is that classical mathematicsfails because of the truth conditions of disjunctions, and not because offacts about the semantic values of instances of the law of excluded middle.

But it is also not entirely clear whether the objection is correct. Theobjection depends on claiming that the proof conditions interpretation ofour intuitions about disjunctive statements commits us to accepting non-classical reasoning. But is that really so? Certainly this way of understand-ing our intuitions is inconsistent with classical mathematical practice. Butwhy should we think that the right response to this incoherence is to rejectclassical mathematical practice? Why not conclude that we should nolonger think that our intuitions about disjunctive statements is correct?

If we accept anti-realism about mathematics, we are committed toidentifying truth conditions with proof conditions. My argument so farshows that this identification implies that we have to think of our intu-ition about the truth conditions of disjunctions as a claim about thepractice of mathematical justification. Now, it turns out that this claimconflicts with classical mathematical practice. So it is clear that somethinghas to be done in order to bring our intuitions in line with our practice.However, what is not yet clear on the basis of these arguments is whatexactly has to be done. In particular, it is not clear that anything in anti-realism about mathematics commits us to abandoning classical logic



rather than our intuitions. Anti-realism implies that to understand amathematical sentence, i.e., to grasp its truth condition, is to know whatcounts as a proof of it. But, at first sight, this by itself is consistent withthe claim that, in grasping the sense of any sentence of the form, p ornot p, we count any condition whatsoever as one in which there is aproof of it, so that, in particular, we count it as proved even when neitherp nor not p is.

One objection to this claim is that it is inconsistent with Dummett’srejection of holism, for the following reason. Dummett infers from therejection of holism that theories of meaning must respect a constraint ofcompositionality.19 However difficult it may be to spell out what therequirement of compositionality amounts to, it seems to imply that theconditions under which there is (we have) a proof of disjunction must bein some way specifiable in terms of the proof conditions of its disjuncts.To specify any condition whatsoever as one in which there is a proof ofp or not p certainly does not give the proof conditions of this disjunctionin terms of its disjuncts.20

However, it is not clear that a compositionality constraint of the sortjust outlined suffices for a rejection of classical reasoning. Consider thefollowing account of the proof conditions of disjunctions:

We have (there is) a proof of a statement of the form p or q just in casewe have (there is) a proof of at least one of p and q, or,p is the negation of q, or,q is the negation of p.21

Call this account ‘the classical proof conditions of disjunctions.’ It appearsto state the proof conditions of disjunctions in terms of the proof condi-tions of its disjuncts. Yet, clearly, it would validate the law of excludedmiddle. Thus, if this account correctly represents the meaning of the con-stant ‘or’, then anti-realism, even an anti-realism incorporating compo-sitionality, is consistent with the use of classical logic in mathematics.

Note that the dependence of truth on proof in anti-realism is absolutelycritical for the argument just given. If one takes truth and proof to beindependent, the rejection of bivalence does not amount to a claim aboutthe deductive system of mathematics, and so can be used non-circularlyto reject classical logic.22

6.

I will now generalize last section’s criticism of the bivalence argument.This generalization will point to some difficulties for using the semantic



value interpretation of validity as the basis for an anti-realist evaluationof deductive inference in mathematics.

I begin by recalling the basic idea underlying the semantic value inter-pretation of validity: the standard of correctness for a form of inferenceis that it satisfies condition (V). Now consider how (V) must be under-stood, given anti-realism about mathematics, i.e., given the identificationof truth conditions with proof conditions. Surely, it must be understoodas the following condition on forms of inference:

(VAR) If we have (there are) proofs of all statements that are instancesof the premises, then we have (there is) a proof of the statementthat is the corresponding instance of the conclusion.

So, what follows from this? To begin with, it seems relatively uncontro-versial that, in general, in giving a proof of a mathematical statement, wegive a deductive argument for it. But surely it is no less controversial thatit is not coherent to take a deductive argument to justify a sentence, unlessone acknowledges that the forms of inference employed in the argumentare valid.

Hence it would seem that, in general, we cannot coherently conceiveof a form of inference as satisfying (VAR), without acknowledging thevalidity of at least some forms of inference.

But now we have a problem, for according to the classical criterion,satisfying (VAR) is the justification for the validity of any form of inference.Suppose we are trying to determine where a form of argument, A, is validor not. In order to determine this, we have to be able to decide whetherthe rules of inference used in any set of arguments for the premises andthe conclusion are valid. If A itself is one of these rules, then we haveargued in a circle; otherwise we have to use (VAR) to determine whetherthese rules are valid. Hence, the use of the classical criterion in determiningwhether we have reason to accept a form of inference will lead either toa circle or to an infinite regress.

This argument, if right, takes us to the main conclusion of this paper:

Assuming the truth of anti-realism about mathematics, the semantic valueinterpretation of validity cannot, without modification, furnish the stan-dard for evaluating the validity of logical principles.

Therefore it cannot be the basis for an anti-realist rejection of classicallogic.

The problem here should have a familiar ring. Recall the argument forthe claim that Dummett’s program for the semantic assessment of logicalprinciples requires conceptual priority of semantic properties over logicalproperties, and hence their independence of one another. I said that,



without this independence, the semantic criterion of validity cannot bethe basis for determining the logical status of any statement. On an anti-realist conception of truth conditions, semantic properties are derivedfrom conditions of proof. So, unless proof is understood formalistically,these conditions cannot be independent of logical properties.

This conclusion allows us to address the following reply to the firstargument: Dummett’s criticism of classical logic is ad hominem, directedto demonstrate the failure of a realist justification of classical logic. Butthe point of Dummett’s criticism surely is that classical reasoning isunjustified when one rejects the realist conception of meaning in favor ofthe anti-realist conception. So the argument has to assume responsibilityfor all the consequences of the latter conception. Whether this objectionholds depends on what is meant by the failure of justification. What islost, when realism is given up, is not merely the realist justification ofclassical logic, it is, more importantly, the very standard or criterion ofvalidity on the basis of which there can be a realist justification.

The significance of these arguments might then be put like this: the factthat the principle of bivalence fails to apply to certain mathematicalsentences constitutes a criticism of classical reasoning, but only from arealist perspective. The failure of the bivalence argument is traceable toa failure to take anti-realism seriously enough; it does not go sufficientlyfar in tracing out the consequences of anti-realism. It considers what anti-realism implies about the applicability of the principle of bivalence to‘undecidable’ statements, but it fails to see the implications of anti-realismfor the very standard of deductive validity itself.

7.

The preceding discussion raises the following question for anti-realism:given the identification of truth with proof, what reason can be given forthinking that the disjunction property holds, or fails to hold? I.e., whatreason is there for thinking that the justification of any statement of theform p or q requires the possibility of justifying one of p or q?23

The conclusion we have reached at the end of the last section does notshow that (VAR) cannot be part of the basis for assessing the correctness offorms of inference. The reason is this. Suppose that there is a set of formsor rules of inference that can be determined as valid on some basis otherthan their satisfying (VAR). Call these fundamental rules. Suppose also that,assuming only that these rules are valid, together with general features ofthe concept of deductive argument, we can define a notion of valid deduc-tive argument. A valid argument in this sense is valid in virtue of the



fundamental rules already accepted as valid; call these canonical arguments.Now, restrict the justifications mentioned in (VAR) to canonical arguments:

(VCA) Given canonical arguments for all statements that constitute aninstance of the premise schemata, we can find (there exists) acanonical argument for the statement that is the correspondinginstance of the conclusion schema.

(VCA) can then be taken to be the condition for the validity of rules ofinference other than the fundamental ones.24 Hence, on the assumptionthat we can identify a set of fundamental rules, (VCA) provides a potentialanalysis of logical consequence.

But of course so far we have merely pushed the question back one step,for, at this point, we have to ask, what grounds, apart from satisfying(VCA), could there be for determining whether a rule of inference is valid?That is, what is the criterion for being a fundamental rule of inference?

An obvious suggestion is this: in arriving at the classical criterion ofvalidity, we used the meanings of the logical constants to give truth con-ditions of logically complex sentences in terms of the semantic values oftheir sub-sentences. If anti-realism identifies truth with proof, then thenatural transposition of this idea is the claim that the meanings of thelogical constants gives the proof conditions of logically complex sentencesin terms of the proof conditions of their sub-sentences. But these proofconditions state when a logically complex sentence can be proved fromits sub-sentences, and hence are claims about the validity of certain formsof argument. Hence, the fundamental rules may be identified preciselywith these forms of inference.

This furnishes a ground for identifying fundamental forms of inference:a rule of inference is fundament if its acceptance as valid is required forsomeone to count as knowing the meaning of a logical constant.

An analysis of logical consequence based on such a set of fundamentalrules and (VCA) is still a semantic analysis, for, ultimately, meaningremains the basis for accepting the validity of any form of inference.However, this conception of consequence differs from that which is basedon the semantic value interpretation of validity, since it is not the casethat the validity of every form of inference is determined by the satisfactionof condition (V). I will call the standard for evaluating deductive inferencebased on this analysis the anti-realist criterion of validity.

But, of course, what I have just described is in fact nothing other thanone type of proof-theoretic justification of logical laws that Dummettdescribes in Logical Basis. Thus, we have arrived at final conclusion ofthis paper:



The proof theoretic justification of logical laws, rather than the biva-lence argument, is a (perhaps the only) basis for the criticism of mathe-matical reasoning consistent with anti-realism.

The foregoing also shows just what a proof theoretic justification oflogical laws has to accomplish, in order that one might use it to providean anti-realist defense of intuitionism.

The notion of a fundamental rule is so far explained only in terms ofwhat we are committed to in virtue of knowing the meanings of thelogical constants. But nothing in this explanation tells us whether anycurrently accepted rule of inference is not fundamental. And we clearlyhave to show that not every currently accepted form of inference in math-ematics is fundamental; otherwise all classical forms of inference wouldbe judged valid. It is not clear how this is to be done, for the followingreason. Given anti-realism’s identification of meaning with proof condi-tions, to know the meaning of a mathematical statement is to know underwhat circumstances we have a proof of it. But this is possible, it wouldseem, only by knowing what forms of inference are valid. And this claimseems perfectly compatible with the claim that knowledge of the meaningof a mathematical sentence requires acceptance of the validity of all therules of inference generally accepted in mathematical reasoning. Thisclaim characterizes the position that Dummett calls mathematical holism;and its rejection is a requirement of a successful anti-realist defense of intuitionism.25

But the rejection of mathematical holism does not suffice to ensure thecoherence of anti-realist intuitionism. The reason lies in examples such asthe putatively compositional classical proof conditions of disjunctiongiven at the end of section 5. It suggests that the explanation of a funda-mental rule given so far does not rule out that the possibility that accep-tance of the law of excluded middle is required for someone to count asknowing the meanings of the logical constants ‘or’. So, the anti-realistclearly has to put more constraints on the notion of fundamental rule, inorder to show that the basic classical forms of reasoning are not funda-mental, and therefore not valid for that reason.26

Finally, if these two problems can be resolved, it remains for the anti-realist to show that classical reasoning is not valid by the standards ofthe proof theoretic account of logical consequence.

8.

I end with a summary of the account of anti-realist intuitionism that I have developed, thereby answering those questions from opening of thispaper that remain unanswered.



1. The failure of bivalence plays no role in an anti-realist argumentfor intuitionism.

2. Rather, anti-realism implies that a semantic analysis of logicalconsequence must be given in terms of conditions of proof, insteadof constraints on semantic values.

3. This analysis requires the identification of a set of fundamentalforms of inference which are not justified in terms of the semanticcriterion of validity.

4. Anti-realism also suggests that an account of the meanings of thelogical constants may be given in terms of their contributions tothe proof conditions of logically complex sentences.

5. Such a revised explanation of the logical constants could providethe fundamental forms of inference involved in the anti-realistsemantic analysis of consequence.

6. The availability of such a revised explanation for a defense of intu-itionism depends on the rejection of mathematical holism, and onan anti-realist account of fundamental rule of inference that deniesthat status to the key rules of classical reasoning.

7. If such a revised semantic analysis of consequence is available, themeaning theoretic assessment of logical laws with respect to it maybe accomplished by the proof theoretic justification procedure ofLogical Basis.

8. Thus, the term of criticism applying to classical mathematics is thatclassical reasoning fails to cohere with the fundamental forms ofreasoning that constitute our understanding of the logical constants.

9. By the same token, the basis for the defense of intuitionistic mathe-matical reasoning is that it does cohere with our understandingof the logical constants.

10. From this perspective, the proof theoretic procedure is a necessarypart of the anti-realist defense of intuitionism; it is thus not adifferent argument from the one proposed in “PhilosophicalBasis,” but rather completes that earlier argument.27

Department of PhilosophyWesleyan University

NOTES

1 See, for instance Cozzo (1994), Luntley (1988), Milne (1994), Prawitz (1977, 1978,1987a, 1987b, and 1994), Rasmussen and Ravnkilde (1982), Rasmussen (1990), Tennant(1987), Weir (1986), and Wright (1981 and 1982).

2 See Dummett (1978), p. xxxi, Dummett (1963), p. 146. But compare Dummett (1982),p. 269.

3 It should be noted that the conception of semantics figuring in the present account isa fairly narrow one. In particular, on this account, meaning is conceptually prior to semantics



in the sense that the properties and relations of expressions that figure in semantics aredetermined by the theory of meaning, so that semantic properties are properties that expres-sions have in virtue of their meanings. Thus, for example, the topological interpretation ofintuitionistic logic would not count as a semantics; Dummett calls the definition of conse-quence based on the topological interpretation an “algebraic” definition, to be distinguishedfrom a “semantic” one (see 1973b, pp. 293–4).

I am indebted to an anonymous referee for urging me to make this point explicitly.4 The attribution to Dummett of this view of the assessment of logical laws must be

qualified by the fact that in the last passage quoted, he writes of the “standard practice oflogicians,” without explicitly indicating agreement with that practice.

I am indebted to an anonymous referee for this point.5 Quantification, of course, introduces certain complications in this picture. Specifically,

the question is how to incorporate the notion of domain of quantification into this analysis.One possibility is to take the domain as a semantic value; this, however, is clearly inconsistentwith the definition of semantic value. The other possibility is to take the specification of a domain to be required in interpreting any quantification. The latter appears to have theconsequence that absolutely unrestricted quantification is not clearly intelligible. On thispoint see Parsons (1974) and Glanzberg (unpublished).

6 One well-known problem with accounts of this sort is that it is not clear whether talkof all possible combinations of semantic values is coherent. The problem derives, again,from the interpretation of quantification. What exactly is involved in talking about all logi-cally possible domains of quantification? For familiar reasons, such talk is not clearly mathe-matically coherent. So, it is equally unclear whether it is conceptually coherent. See, again,Parsons (1974).

One way in which this account of logical consequence differs from the contemporaryTarski–Quine analysis is that it does not dispense with modal notions. The Tarski–Quineanalysis defines consequence in terms of quantification over a domain of interpretationswhose existence is asserted by some background set or class theory (ZF, BNG, etc.). TheFregean analysis, by contrast, quantifies over “all (logically) possible combinations ofsemantic views,” without making this logical modality any more precise or mathematicallytractable. Thus, Frege’s analysis is not, and cannot be, an attempt to effect a reduction oflogical necessity to some non-modal notion; it explains the sense in which instances of validargument forms must be truth preserving in terms of the preservation of truth under allpossible combinations of semantic values.

It follows that the Fregean analysis is not (or not obviously) vulnerable to the kind of criticismthat has recently been made by Etchemendy (1990), of the Tarski–Quine analysis, namely, that,in reducing logical truth to plain truth it fails to capture the necessity of logical consequence.

It should be noted, on the other hand, that Etchemendy’s criticism also does not obviouslysucceed with respect to the Tarskian analysis. For the latter is intended, not to be faithfulto our intuitions of logical necessity, but to provide a mathematically tractable replacementof it. This last point is made in Goldfarb (forthcoming), which also contains a clear expo-sition of the Tarski–Quine analysis.

7 See Dummett (1991), pp. 34–5.8 I should note here that the attribution of this claim to Dummett will be rejected in

the remainder of section 3. I am indebted to an anonymous referee for comments displayinghow it is possible to misread the intention of this attribution.

9 The account Dummett himself gives in this paper, of what an explanation of deductivereasoning could be, is that it tries to show how deductive reasoning could be both fruitfuland correct. Unfortunately that account does not obviously apply to the question that con-cerns us here, namely, the status of the Fregean analysis of consequence.

My account of the philosophical role of semantic justifications of logic is suggested byHeck (forthcoming). Heck should not be held responsible for my development of the ideain terms of the meanings of the logical constants.



10 I am indebted to an anonymous referee for suggestions for clarifying this point.11 The same point holds of the contemporary Tarski–Quine analysis. See Goldfarb

(forthcoming) for an account.12 Dummett defines ‘undecidable’ as follows:

An [u]ndecidable sentence is simply one whose sense is such that, though in certain effectivelyrecognizable situations we acknowledge it as true, in other we acknowledge it as false, andin yet others no decision is possible, we possess no effective means for bringing about a sit-uation which is of one or other of the first two kinds. (1981, p. 466)

The problems with this definition are as follows.It is hard to see how this definition squares with the two familiar notions of decidability

and undecidability in logic. Gödel’s first incompleteness theorem states that in any systemof arithmetical axioms and rules of inference in which primitive recursion can be formalizedthere are (closed) formulas such that neither they nor their negations are derivable – theseare formally undecidable sentences. This notion of undecidability is defined with respect toa formal system, so a formula cannot be undecidable in this sense simpliciter, without refer-ence to a formal system, and, moreover, every formula is trivially decidable with respect toa set of axioms that includes it. Another notion is the recursive decidability of a class of for-mulas (or schemata), i.e., the existence of a recursive characteristic function for that class.Again, a single formula, as opposed to a class of formulas, cannot be undecidable in thissense. Moreover, a singleton class of formulas is again trivially decidable in this sense.

Decidability in either of these senses is a relational property of formulas. But in Dummett’sdefinition of an ‘undecidable’ sentence, there is no mention of a set of axioms or classes ofsentences, relative to which a given statement is ‘decidable’ or ‘undecidable.’ Of course, itmay be that in the definition reference to some set of axioms or class of sentences is meantto be tacitly understood. But which one? As Dummett surely cannot fail to know, intu-itionistic and classical mathematics allow different principles of proof. Moreover, evenwithin classical mathematics, what counts as an axiom for a mathematical theory is not ingeneral settled. Again, this is a point that Dummett is well aware of, since his account ofthe significance of Gödel’s first incompleteness theorem is that “for any definite character-isation of a class of grounds for making an assertion about all natural numbers, there willbe a natural extension of it” (1978b, p. 194). The basis of this claim is, of course, the factthat arithmetical truth is not recursively axiomatizable; in this sense, we cannot, even inprinciple, settle on all the axioms or principles of proof of arithmetic.

The problem here is of course not merely that tacit reference to a set of axioms and rulesfails to provide an unambiguous characterization of undecidability. The more fundamentalworry is that if undecidability is relative to which axioms and rules are accepted, then it wouldseem that what counts as undecidable in the case of classical mathematics presupposes theacceptance of classical reasoning. But then it is not clear how the concept of undecidabilitycould be the basis of a non-circular argument against classical reasoning in mathematics.

Note that these points are independent of the intuitionists’ insistence that mathematicalpractice is not fully captured by a formal system. The present difficulties arise simply fromthe fact that classical and intuitionist mathematical practices are distinct, and hence so mustbe the notions of proof determined by these practices.

13 One interpretation inconsistent with intuitionism is that ‘undecidable’ mathematicalstatements do not have one of these two semantic values. Given the intuitionist interpretationof negation, the claim that a mathematical statement does not have the semantic value truerequires a proof that no proof of it is possible, which then is a proof of its negation. SeeDummett (1977), p. 17.

The interpretation adopted in the text may also be stated as the claim that we have noguarantee that undecidable mathematical sentences have one of the two classical truth values.



Similar formulations have been adopted in Wright (1981) and Luntley (1988). I have arguedin Shieh (1997) that an epistemic reading of this formulation resolves a number of the prob-lems surrounding Dummett’s notion of undecidability that I described in note 13.

I am grateful to an anonymous referee for pointing out the similarity of my account ofDummett’s conception of undecidability to that of Luntley, as well as for comments thathelped me to formulate more clearly my account of that conception.

14 The argument just sketched constitutes the core of the arguments involved in the disputebetween Wright and Rasmussen and Ravnkilde in Wright (1981), Rasmussen and Ravnkilde(1982), Wright (1982), and Rasmussen (1990).

15 Dummett (1991), p. 319. See also Hugly and Sayward (1992), for an account of howclassical logic can be validated on the basis of a semantics employing partial truth valueassignments.

16 See Rasmussen and Ravnkilde (1982) and Rasmussen (1990).17 Note that the disjunction property is not the claim that, in a system of deductive infer-

ence, a disjunction has to be inferred from one of its disjuncts. Rather, it is the claim thatif, and only if, a disjunction can be deduced, one of its disjuncts also can.

18 I am grateful to an anonymous referee for the following objection to the two inter-pretations just canvassed. This is, that from the perspective of classical intuitionism, neitherinterpretation is an acceptable account of proof conditions. Brouwer and Heyting insistedthat mathematics is contentful (inhaltlich), and so must be sharply distinguished from itsvarious formalizations. For them, the practice of mathematical proof, in contrast to deriva-tion in formalized theories, cannot be circumscribed once and for all; it is, rather, alwaysopen to possible future expansions. Thus, theories such as Heyting arithmetic or first-orderintuitionist logic do not capture the notion of mathematical proof; at best they codify certainaspects of intuitionist practice. What counts as acceptable proof is not sharply prescribedbut rather constitutes a growing domain.

There are two points to be made in reply to this objection.First of all, the objection appears to rest on a conflation of ‘mathematical theory’ with

‘formalized mathematical theory.’ Intuitionist mathematics includes a number of theoriesthat are not formalized theories; for example, ‘Intuitionistic algebra’ (Troesltra and vanDalen 1988, p. 383) includes, among others, “group theory” (ibid., p. 389) and “[t]he theoryof polynomial rings” (ibid., p. 415). Both of these intuitionistic mathematical theories satisfythe disjunction property while their classical counterparts do not.

This leads to the second and more important point. The claim made in the text is noteffected by the observation about the intuitionistic conception of proof that the objectionvery correctly makes. The intuitionistic notion of proof requires the following interpretationof premise 1:

There is an acceptable mathematical proof of a statement of the form p or q just in casethere is an acceptable mathematical proof of at least one of p and q.

But it is hard to see how any mathematical practice that satisfies this description couldbe classical. Or, to look at the matter in another way, suppose a future extension of intu-itionistic mathematical practice violated this description. On what basis could it then bedistinguished from classical mathematics?

19 See, e.g., Dummett (1973a), pp. 218–20, and (1991), pp. 221–5.20 I am grateful to an anonymous referee for bringing to my attention, not only this possi-

ble conflict between my argument and compositionality, but also the following differentformulation of that conflict: if the standard natural deduction rule of disjunction introduc-tion and elimination are accepted, it seems to follow that the necessary and sufficientcondition for having a proof of a disjunction is a proof of either disjunct.



The reasoning underlying this formulation, however, is apparently this: if proof of eitherdisjunct is not necessary for a proof of the disjunction, then one could not establish a normal-ization result for disjunctive inferences. So this objection does not rest only on compositionality.

To the extent that this objection relies on compositionality, my reply in the text to myversion of the objection seems to me to apply to this objection as well. To the extent thatit relies on normalization, it needs, it seems to me, to be bolstered by some account of whynormalization is a philosophical requirement of anti-realism.

21 This account is adapted from Wright (1981), p. 449.I am grateful to an anonymous referee for pointing out that my definition might be

thought to satisfy the letter of the compositionality constraint without being in accord withits spirit. But, naturally, in order for the sense of unease expressed by this thought to befully discussed, the spirit of compositionality needs to be more exactly specified.

22 Most arguments for many-valued logic are of this form. See, e.g., the classic textLukasiewicz (1930).

23 Note that the question is not whether every proof p or q must reach that conclusionform a subsidiary proof of one of p or q. Rather, the issue is whether proofs of p or q froma subsidiary proof of one of p or of q are more fundamental in the sense that if they arenot available, then there are no proofs of p or q.

24 A complication in this account should be noted. If there are fundamental rules of infer-ence that involve subsidiary deductions and the discharge of hypotheses, such as the ruleof conditional introduction in natural deduction, then the definition of valid canonicalargument has to proceed in tandem with that of valid non-canonical argument, by simul-taneous recursion. See Dummett (1991), pp. 259–64.

25 Dummett (1991), pp. 225–7.26 I am grateful to an anonymous referee for suggesting that the classical proof conditions

of disjunction may be ruled out by proof-theoretic constraints such as Dummett’s notionof harmony. If this is the case, it would count against the classical proof conditions only ifharmony is a compositional constraint; the latter claim needs to be argued for. Moreover,whether the classical conditions are ruled out by harmony depends on how that notion isinterpreted and technically implemented. It can easily be shown that if harmony consists ofan upward proof-theoretic justification of elimination rules based on introduction rules (seeDummett 1991, chapters 11 and 12), then the definition in question can be justified withrespect to a set of introduction rules that justifies intuitionistic logic, provided that classicalreasoning is used in the meta-language.

27 I would like to express my gratitude to an anonymous referee for many helpful com-ments, criticisms and suggestions that considerably improved this paper.

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1999 What Anti-Realist Intuitionism Could Not Be

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