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DUMMETT’S ARGUMENT FOR LOGICAL REVISION
For Michael Dummett, classical inferences such as the law of excluded middle, double
negation elimination, and reductio ad absurdum presuppose a reality in-principle
unknowable. If this is right, then chagrin about the intelligibility of such an epistemically
unfriendly reality entails chagrin about classical logic. Far better, argues Dummett, for
our default logic to be intuitionistic, one that does not sanction use of strictly classical
inferences.
Most of the literature on this dialectic falls into two somewhat overlapping
batches. The first consists of philosophical and mathematical treatises on intuitionistic
logic and constructivism,1 and the second of discussions about the intelligibility of an
unknowable reality.2 Only recently has a rigorous investigation been undertaken into
Dummett’s arguments concerning the metaphysical and epistemic presuppositions of
classical logic. As a result of this investigation, a number of philosophers have professed
inability to discern even prima facie valid arguments for logical revision in Dummett’s
oevre.
For example, while discussing Dummett’s argument Neil Tennant says that one
would be forgiven for viewing it as “a non-sequitur of numbing grossness” (Tennant
1997, p. 160). Like Tennant, Crispin Wright has recently offered his own new argument
for intuitionism (Wright 1992). Unlike Tennant, he does not couple this argument with
criticism of Dummett, though the very fact that he does not compare his argument with
Dummett’s is telling.
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Finally, in a recent article discussing Dummett and Wright’s arguments for logical
revision, Joe Salerno states
It seems that important attempts to make the [intuitionist] revisionist’s point fail.
These are attempts made by Michael Dummett and Crispin Wright. The negative
thesis is that, given the resources provided by either Dummett or Wright, choice of
logic is not a realism-relevant feature—i.e, logical revision is not a consideration
that is enjoined by one’s stance on the possibility of verification transcendent truth.
(Salerno 2000, p. 212)
If Salerno and Tennant are right about the nonexistence of an even prima facie valid
argument for logical revision in Dummett’s oevre, then much of the literature on
Dummettian anti-realism should be likened to the crowds’ discussion of the naked
emperor’s fine garments. This would be completely outrageous, indeed scandalous.
But Tennant and Salerno are wrong. While Tennant has done as much as anyone
to advance Dummett’s anti-realist program (e.g. Tennant 1987, 1997), when explicating
Dummett’s argument for logical revision, he (as well as Salerno) completely ignores
important modal distinctions at the very heart of nearly all of Dummett’s writings on the
theory of meaning. A close and careful reading of even a few of Dummett’s canonical
works reveals an argument for logical revision that is prima facie valid and (once
formalized) extraordinarily clear. So clear that it can be presented perspicuously in a
Fitch style proof system.3
In what follows I begin by discussing the arguments that Salerno and Tennant
mistakenly attribute to Dummett. I am able to use this discussion, as well as attention to
some of Dummett’s canonical texts, to motivate and clarify Dummett’s actual argument.
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My purpose is not to settle any issues between Dummett and his opponents, but rather to
make clear what these issues are. Independent of considerations about “setting the record
straight,” this clarifying of the issues is the most important accomplishment of my
demonstration.
By correctly formulating Dummett’s proof, I am able to precisely
enumerate the four possible options for the defender of classical logic.
I. SALERNO’S INTEPRETATION OF DUMMETT’S ARGUMENT
“The Philosophical Basis of Intuitionistic Logic” is an early paper in which Dummett
makes his case for verificationism . For example, he writes
This [the realists’] conception violates the principle that use exhaustively
determines meaning; or, at least, if it does not, a strong case can be put up that it
does, and it is this case which constitutes the first type of ground which appears to
exist for repudiating classical in favor of intuitionistic logic for mathematics. For,
if the knowledge that constitutes a grasp of the meaning of a sentence has to be
capable of being manifested in actual linguistic practice, it is quite obscure in what
the knowledge of the condition under which a sentence is true can consist, when
that condition is not one which is always capable of being recognized as obtaining.
(Dummett 1975a, 224)
This is Dummett’s meaning-theoretic challenge to specify in what knowledge of meaning
consists, in terms of practical abilities that can be correctly attributed to a competent
language user. Dummett holds that a verificationist can provide such an account.
The argument we are concerned with attempts to show that such verificationism
entails intuitionist revision. This requires, in turn, understanding the logical structure of
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Dummett’s verificationist claim. Then, where “[]” means “it is necessarily the case that,”
we can formalize Dummett’s verificationist belief as
Verificationism
[]X((X k1X)  (~X k1~X)).
Given some interpretations of the necessity operator, Dummett clearly intends this. A
verificationist meaning theorist does not think that verificationism is a parochial truth
about us, our language, and the world. Rather, it is supposed to reflect some deep truth
about the nature of language, thought, and any possible world. Dummett holds that it is
necessary that all truths are knowable.
In “Revising the Logic of Logical Revision,” J. Salerno presents Dummett as
holding that verificationism somehow renders classical logic inconsistent with the
existence of sentences that can neither be proven nor refuted. This is an understandable
interpretation of Dummett’s argument, as Dummett often seems to say that we all agree
that there are such undecidable sentences in a language, and that the existence of such
undecidable sentences, in conjunction with bivalence and verificationism, leads to a
contradiction. For example,
It is when the principle of bivalence is applied to undecidable statements that we
find ourselves in the position of being unable to equate an ability to recognize when
a statement has been established as true or as false with a knowledge of its truthcondition, since it may be true in cases when we lack the means to recognize it as
true or false when we lack the means to recognize it as false. (Dummett 1976b, p.
63)
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So the thought seems to be that since Verificationism, the existence of an undecidable
sentence, and bivalence entail a contradiction, we should eschew bivalence.
More
formally, where the premises are,
Verificationism
[]X((X k1X)  (~X k1~X))
Undecidability
Y(~k2Y  ~k2~Y)
Bivalence
X(X  ~X)
and “” denotes absurdity, Dummett’s main claim seems to be:
Salerno’s Version of Dummett’s Argument:
Verificationism, Undecidability, Bivalence |- 
Using the law of excluded middle instead of Bivalence, and calling Verificationism “the
knowability principle,” Salerno gives the argument for this in the following manner.
Let us suppose that some indicative of the given class is undecidable. By accepting
the law of excluded middle, one accepts the truth or falsity of every sentence, so the
undecidable sentence is either true or false. First, suppose it is false. Then it
follows from the knowability principle that we could prove it false. But we cannot
prove it false, since it is undecidable. Second, suppose the undecidable sentence is
true. Then it follows from the knowability principle that we could recognize it as
true. Again, this is in contradiction with its undecidability! But then we have
absurdity resting on excluded middle, the knowability principle, and the
undecidability thesis. Something must go. (Salerno 2000, 214)
If this is right then the debate between Dummett and his opponent is clear.
The
Dummettian anti-realist concludes that Bivalence caused the contradiction, and eschews
Bivalence while asserting Verificationism and Undecidability. The Dummettian realist
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concludes that Verificationism is the root of the evil and continues to assert Bivalence
and Undecidability.
Unfortunately (as Salerno points out), if this is the dialectic, then the Dummettian
anti-realist loses the debate.
Verificationism and Undecidability are themselves
intuitionistically inconsistent!
Reason why Salerno’s Version of Dummett’s Argument Fails:
Verificationism, Undecidability |- 
As with the proof Salerno attributes to Dummett, the necessity operator in front of
Verificationism isn’t needed to establish this. That is, one can intuitionistically prove the
following.
Important Lemma:
|- X((X k1X)  (~X k1~X)) ~Y(~k1Y  ~k1~Y) 4
Given that “X((X k1X)  (~X k1~X))” is just Verificationism without the modal,
and that “~Y(~k1Y  ~k1~Y)” is just the denial of Undecidability, our Important
Lemma entails the Reason why Salerno’s Version of Dummett’s Argument Fails.
Independently of undermining what one might take to be Dummett’s argument,
this result is deeply problematic for the Dummettian.
It seems to show that
Verificationism and Undecidability are already jointly contradictory, without the
assumption of Bivalence. One might think that, since no one would deny the existence of
undecidable sentences, we have shown Dummett’s verificationism to be false.
The result should not be interpreted in this manner. The proof just shows us that
the modal “k2” in the statement of Undecidability (Y(~ k2Y  ~ k2~Y)) needs to be
sufficiently idealized in some manner. For example, one might interpret it as saying that
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there exists a sentence such that it is not possible, given any finite extension of our
conceptual resources, to know whether the sentence is true or false. Thus, depending
upon the strength of the modal “k1” in Verificationism ([]X((X k1X)
 (~X
k1~X))), a verificationist like Dummett need not be committed to anything clearly
absurd.
While this shows that the Dummettian anti-realist’s position is not obviously
absurd, we still have no argument for logical revision.
II. TENNANT’S INTERPRETATION OF DUMMETT’S ARGUMENT
Strangely, the very argument that Salerno takes to be Dummett’s undoing is a key lemma
in the argument Tennant attributes to Dummett. An immediate consequence of the
Reason why Salerno’s Version of Dummett’s Argument Fails, is the
1st Lemma Tennant Attributes to Dummett:5
Verificationism |- ~(Undecidability)
It should be noted that Tennant seems aware of the fact that one might interpret Dummett
in the manner of Salerno. Where “” is the name of the 1st Lemma Tennant Attributes to
Dummett, Tennant explicitly states the following.
Note that , on Dummett’s own understanding of the course of argument, would
not itself, rely on the principle of bivalence as a premise. . . [Dummett’s] argument
must incorporate the principle of bivalence as a premise of its other subargument. . .
(Tennant 1997, p. 181)
One might think this is a more charitable reading of Dummett. However, Tennant’s
reading is no more charitable, as the entire argument hinges on the “other subargument,”
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2nd Lemma Tennant Attributes to Dummett:
Bivalence |- Undecidability
Were the 2nd lemma plausible, it and the 1st together would establish the inconsistency
of Verificationism and Bivalence. (Tennnant 1997, pp. 180-181) Even more so than the
argument Salerno attributes to Dummett, this forces a stark choice, one must either (with
Dummett) give up Bivalence, or (with the classicist) give up Verificationism.
The bulk of Chapter 6 of Tennant’s The Taming of the True (“The Manifestation
Argument is Dead”) concerns itself with attempting to discern an argument from
Bivalence to Undecidability. After a discussion involving, among other things, provably
undecidable discourses such as Peano Arithmetic and new, unpublished, independence
results attributed to Harvey Friedman, Tennant concludes the chapter by stating
All that, however, still fails to make the desired logical transition available to the
Dummettian: the transition, that is, from bivalence to the existence of recognition
transcendent truths. (Tennant 1997, p. 194)
Thus, like Salerno, Tennant concludes that Dummett never provided a prima facie valid
argument for logical revision.
III. DUMMETT’S ARGUMENT
A careful study of Dummett’s texts on the theory of meaning reveals that, pace Salerno
and Tennant, the principle of Bivalence by itself plays no role in Dummett’s argument.
Rather, Dummett takes classical model-theoretic semantics, when used as part of a
semantics for natural language or mathematics, to imply the possibility of the kind of
undecidable sentences intuitionistically inconsistent with verificationism. As central as
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this point is in all of Dummett’s writings on the “theory of meaning,” its dialectical
relevance for Dummet’s revisionary arguments has been consistently overlooked.
I
formalize the principle thus.
Dummett’s Insight
TCS  2X(~k2X  ~k2~X)
where “TCS” means, roughly, “Classical model-theoretic semantics is the correct
semantics for the logical operators, and is a part of the correct semantics for natural
language and mathematics.”6
What is constitutive of TCS for Dummett’s purposes is that the truth predicate
attaches correctly to a sentence in virtue of the referential relations of the subsentential
units of the sentence. Thus, Dummett writes,
To have a realistic view, it is not enough to suppose that statements of the given
class are determined, by the reality to which they relate, either as true or as false;
one has also to have a certain conception of the manner in which they are so
determined.
This conception consists essentially in the classical two-valued
semantics: and this, in turn, embodies an appeal to the notion of reference as
indispensable. (Dummett 1982, p. 231))
On this conception a simple atomic sentence like “Fred is envious” is true if, and only if,
the entity referred to by “Fred” is in the extension of the set of entities referred to by
“envious.”
Then, for Dummett, it is the way the truth of logically complex statements are
determined in classical model theory that is important. He writes,
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The truth value of a quantified statement is, on this conception, determined by the
truth-values of its instances, so that the instances stand to the quantified statement
just as the constituent subsentences of a complete sentence whose principal
operator is a sentential connective stand to the complex sentence: the truth-value of
the quantified statement is a truth-function of the truth-values of its instances, albeit
an infinitary one if the domain is infinite. The truth-value of a universally
quantified statement is the logical product of the truth-values of its instances, that
of an existentially quantified statement the logical sum of the truth-values of its
instances. These operations, these possibly infinitary truth-functions, are conceived
of as being everywhere defined, that is, as having a value in every case: in other
words, the application of the operation of universal or of existential quantification
to any predicate that is determinately true or false of each object in the domain will
always yield a sentence that is itself determinately either true or false,
independently of whether we are able to come to know its truth-value or not.
(Dummett 1982, pp. 231-232)
To grasp Dummett’s point, consider a possible world containing an infinite sequence of
red and green objects.
Now consider the sentence, “There are infinitely many red
objects.” If classical model theory gave the correct interpretation of this sentence, then
the sentence would be made true or false in the way Dummett describes above. Yet there
would be no way for the inhabitants of such a world to tell whether or not the sentence
were true or false, since they could not survey the infinite domain. Thus in the described
possible world, there does exist an in-principle undecidable sentence. In the case of
decidable, yet contingent, empirical predicates such as “red” it is easy to generate such
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scenarios given the quantificational apparatus of classical logic, and the attendant modeltheoretic interpretation of those quantifiers.7
Dummett’s Insight does allow one to deduce absurdity from Verificationism and
TCS. This, then is the form of Dummett’s real argument for logical revision.
Dummett’s Argument:
Verificationism, Dummett’s Insight |- ~TCS
Proving this will require greater clarity about the inferential roles of key modal notions
occurring in the premises.
Premises:
Verificationism
Dummett’s Insight
[]1X((X  k1X)  (~X  k1~X))
TCS  X(~k2X  ~ k2~X)
Then, our demonstration necessarily involves the following modal inferences.
Transformation rules:
K.
|- P, therefore |- []L P.
Rules of inference:
[]~ dist.
[]x~P, therefore ~xP
[] dist.
[]x(P Q), therefore []xP []xQ
[] >.
[]LP, therefore []yP (If P is logically necessary, then P is
necessary in any other modality)}
With these rules we can establish Dummett’s Argument. Since the modal reasoning is
somewhat complicated, I present this as a formal proof.
Demonstration of Dummett’s Argument:
1. X((X k1X)  (~X k1~X))
11
~Y(~k1Y  ~k1~Y)
by Important Lemma
2. []L(X((X k1X)  (~X k1~X))
~Y(~k1Y  ~k1~Y))
1 K.
3. []1(X((X k1X)  (~X k1~X))
~Y(~k1Y  ~k1~Y))
2 [] >.
4. []1X((X k1X)  (~X k1~X))
[]1~Y(~k1Y  ~k1~Y)
3 [] dist.
5. []1X((X  k1X)  (~X  k1~X))
Verificationism
6. []1~Y(~k1Y  ~k1~Y)
4, 5 elimination
7. ~1Y(~k1Y  ~k1~Y)
6 []~ dist.
8. TCS  X(~k2X  ~ k2~X)
Dummett’s Insight
9. | TCS
For ~ introduction
10.| X(~k2X  ~ k2~X)
8, 9  elimination
11. | X(~k2X  ~ k2~X)  ~1Y(~k1Y  ~k1~Y)
7, 10  introduction
12.| 
11 ~ elimination
13. ~TCS
9-12 ~ introduction
Q.E.D.
III. Q.E.D.?
The main purpose of this paper is to show that Dummett himself provided the resources
for a prima facie valid argument for logical revision. However the point of this is not
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merely to give Dummett his due. Like all good philosophical demonstrations, Dummett’s
carves the logical space of possible positions in terms of how one reacts to his proof. My
presentation of Dummett’s argument sets in bold relief several morals about how debate
about intuitionistic logical revision should continue. The proof shows there to be exactly
four strategies for disagreeing with Dummett. One may: (1) argue that Verificationism is
false, (2) argue that Dummett’s Insight is false, (3) argue that the proof equivocates on
line 9 due to incomparable notions of possibility (1 and 2) and (4) argue that the
proof equivocates on line 9 due to incomparable notions of knowability (k1 and k2).
In closing I would like to suggest that these combine to produce a challenge to the
Dummettian. One could argue that insofar as Verificationism and Dummett’s Insight are
plausible they involve incomparable modal notions. Then the Dummettian would face a
dilemma; either one of Verificationism or Dummett’s Insight is false, or the proof
equivocates on line 11.
To see how this strategy might work, assume that “k1” meant “is knowable by
God” and “k2” means “is knowable by a person.” Then line 11 (X(~k2X  ~ k2~X)
 ~1Y(~k1Y  ~k1~Y)) would be the claim that: (1) it is possible there exists a
sentence unknowable by people, and (2) it is not possible that there exists a sentence
unknowable by God. Clearly these two claims do not contradict one another. But then
the conclusion that truth conditional semantics is mistaken would not follow.8
Now what is really interesting is that Verificationism becomes much less
controversial if one merely affirms that all truths are knowable by God, as long as this
claim isn’t taken to presuppose God’s existence. Likewise, Dummett’s Insight becomes
much less controversial if it merely affirms that the correctness of truth conditional
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semantics implies that it is possible that some sentences are unknowable to us. Now, for
the purpose of argument assume that any more controversial statements of
Verificationism and Dummett’s Insight are false. Then our dilemma would undermine
Dummett’s argument; either one of Verificationism or Dummett’s Insight are false, or the
proof equivocates on line 9.
Another strategy for establishing the same dilemma would be to argue that the
truth of Verificationism forces “k1P” to mean something like “defeasible evidence is
available for P,” and that the truth of Dummett’s Insight requires “k2” to mean “P can be
known with certainty.” Again, line 11 would equivocate, merely stating that (1) it is
possible that some sentence can’t be known with certainty to be true or to be false, and
(2) it is not possible for a sentence to be such that there is no defeasible evidence
available for its truth or falsity.
Of course these possible objections do not even mention the possibility operators
1 and 2 occuring in Verificationism and Dummett’s Insight. Given the indisputable
ambiguity of the modal “is possible” it would be surprising if one couldn’t argue for our
dilemma by focusing on them.
Finally, it should also be clear that the proper interpretations of the modals in
Dummett’s key claims will be in some sense discourse dependent. It might be the case,
for example that our dilemma undermines Dummett’s argument when applied to
empirical claims, but fails to undermine it when applied to mathematical claims. Or that
it fails for both kinds of claims, but for different reasons involving the modals.
I will not further explore these possible refutations.
I have come to praise
Dummett’s argument, not bury it. The reason I enumerate the classicist’s strategies is to
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establish that future anti-realist debate about logical revision should be a debate about the
proper modal notions occurring in Dummett’s argument. I take heart in the fact that
Dummett himself is aware of this. In “What is a Theory of Meaning (II)” he writes
In this way, we try to convince ourselves that our understanding of what it is for
undecidable sentences to be true consists in our grasp of what it would be to be
able to use such sentences to give direct reports of observation. We cannot do
this; but we know just what powers a superhuman observer would have to have
in order to be able to do it- a hypothetical being for whom the sentences in
question would not be undecidable. And we tacitly suppose that it is in our
conception of the powers which such a superhuman observer would have to
have, and how he would determine the truth-values of the sentences, that our
understanding of their truth-conditions consists. This line of thought is related
to a second regulative principle governing the notion of truth: If a statement is
true, it must be in principle possible to know that it is true. This principle is
closely connected with the first one [that if a statement is true, then there must
be something which makes it true]: for, if it were in principle impossible to
know the truth of some true statement, how could there be anything which made
that statement true? (Dummett, 1976b, p. 61)
Interestingly, Dummett takes this kind of verificationism to be one that he and the
defender of classical logic should agree about. Clearly then, for Dummett the important
question concerns how super the superhuman observer needs to be. But this is precisely
the question of how to construe the modal notions occurring in the premises of
Dummett’s Argument. Thus, if I am right (about Dummett’s argument), then Dummett is
15
right. Anti-realists and realists should conclude that significant progress in many of the
debates begun by Dummett now requires attaining much greater clarity and insight into
the relevant modal notions.9
REFERENCES
Appiah, Anthony. (1986) For Truth in Semantics, Basil Blackwell Press.
Dragalin, A.G. (1980) Mathematical Intuitionism, Introduction to Proof Theory,
American Mathematical Society.
Dummett, M.
(1967) “Platonism”, in Dummett, Michael. (1978) Truth and other
Enigmas,
Harvard University Press, 202-214.
Dummett, M. (1975a) “The Philosophical Basis of Intuitionistic Logic”, in Rose, H. and
Shepherdson, J. (eds.), Logic Colloquium ’73, Amsterdam, Oxford and New
York,
reprinted in Dummett, Michael. (1978) Truth and other Enigmas, Harvard
University
Press, 215-247.
Dummett, M. (1975b) “Wang’s Paradox”, Synthese, 30, 301-24 reprinted in Dummett,
Michael. (1978) Truth and other Enigmas, Harvard University Press, 248-268.
Dummett, Michael. (1976a) “Is Logic Empirical,” Lewis, H. (ed.) Contemporary British
Philosophy, London, pp. 45-68, reprinted in Dummett, Michael. (1978) Truth and
other Enigmas, Harvard University Press, 269-289.
Dummett, M. (1976b) “What is a Theory of Meaning? (II)”, in G. Evans and J. McDowell
16
(eds.), Truth and Meaning, Oxford: Clarendon Press, reprinted in Dummett,
Michael.
(1993) The Seas of Language, Clarendon Press, 34-93.
Dummett, Michael. (1982) “Realism,” Synthese, 52, 55-112, reprinted in Dummett,
Michael.
(1993) The Seas of Language, Clarendon Press, 230-276.
Dummett, Michael. (1991) The Logical Basis of Metaphysics, Harvard University Press.
Prawitz, Dag. (1965) Natural Deduction: A Proof-Theoretical Study, Almqvst and
Wiksell.
Salerno, Joseph. (2000) “Revising the Logic of Logical Revision,” Philosophical Studies,
99, 211-227.
Shapiro, Stewart. (1993) “Anti-Realism and Modality,” in J. Czermak (ed.), Philosophy
of
Mathematics, Proceedings of the 15th International Wittgenstein-Symposium, Part
1, HolderpPichler-Tempsky, 269-287.
Tennant, Neil. (1987) Anti-Realism and Logic: Truth as Eternal, Clarendon Press.
Tennant, Neil. (1997) The Taming of the True, Clarendon Press.
Wright, Crispin. (1992) Truth and Objectivity, Harvard University Press.
Wright, Crispin. (1993) Realism, Meaning, and Truth, 2nd ed., Basil Blackwell Press.
17
ENDNOTES
1
For example, Prawitz’s Natural Deduction: A Proof Theoretical Study, and Dragalin’s
Mathematical Intuitionism: An Introduction to Proof Theory.
2
For example, Appiah’s For Truth in Semantics and many of the articles in Wright’s
Realism Meaning and Truth. Tennant Anti-realism and Logic: Truth as Eternal and The
Taming of the True combine both strains.
3
Of course the fact that one can formalize a proof does not entail that one should do so.
However, the dialectical situation we find ourselves in is special.
First, since this
argument has eluded those Dummett exegetes (e.g. Salerno, Tennant, and Wright)
sympathetic to intuitionism and verificationism, I chose to pay the price of a kind of
inelegance when what is purchased is logical unassailability. Second, it is very important
that Dummett not commit himself to the validity of classical logic in any part of his
argument. Unfortunately, it is much more difficult to discern whether a complicated nonformal argument involves resources unavailable to the intuitionist. For the reader moved
by this issue, the time wasted by trudging through a formal proof is gained back tenfold
by not going down the path of fruitlessly trying to discern a tu quoque argument against
Dummett. Third, and most important, by explicitly presenting Dummett’s argument, I am
able to show exactly where the defender of classical logic should demure.
4
Here is the proof.
Claim: |- X((X k1X)  (~X k1~X)) ~Y(~k1Y  ~k1~Y)
Proof:
1. | X((X k1X)  (~X k1~X))
Assumption for introduction
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2. | | Y(~k1Y  ~k1~Y)
Assumption for ~ introduction
3. | | | (~k1P  ~k1~P)
Assumption for elimination
4. | | | (P k1P)  (~P k1~P)
1  elimination
5. | | | (P k1P)
4  elimination
6. | | | ~k1P
3  elimination
7. | | | | P
Assumption for ~ introduction
8. | | | | k1P
5, 7 elimination
9. | | | | k1P  ~k1P
6, 8  introduction
10.| | | | 
9 ~ elimination
11.| | | ~P
7-10 ~ introduction
12.| | | ~P k1~P
4  elimination
13.| | | k1~P
11, 12 elimination
14.| | | ~k1~P
3  elimination
15.| | | k1~P  ~k1~P
13, 14  introduction
16.| | | 
15 ~ elimination
17.| | 
3-16 elimination
18.| ~Y(~k1Y  ~k1~Y)
2-17 ~ introduction
19. X((X k1X)  (~X k1~X)) ~Y(~k1Y  ~k1~Y)
1-18 introduction
5
In his discussion Tennant does not modalize Verificationism either. As with Salerno,
presenting his argument with the modal causes no problem, as []P logically entails P.
19
6
For our purposes, we can assume that TCS itself entails bivalence, though this should
not be completely without controversy. As far as I know, nowhere in the secondary
literature on Dummett is this entailment questioned, though it does not follow without
further argumentation. Classical model-theoretic semantics of the kind utilized by
mathematical logicians as well as natural language semanticists defines truth in a model,
not truth simpliciter. Thus, classical semantics simply tells us that a sentence is either
true or false in any interpretation of the sentence. It is only if one takes truth in an
intended interpretation to be a good model of truth simpliciter, that one gets bivalence
from classical semantics. While Dummett has extraordinarily insightful things to say
about this issue in, for example, The Logical Basis of Metaphysics and “Wang’s
Paradox,” no one has yet systematically explored (from within the framework of antirealism) the prospects of using classical semantics to justify classical inference but not to
utilize truth in an intended interpretation as a notion of truth simpliciter. However, in
what follows I assume that TCS includes the assumption that, to borrow a phrase from
Stewart Shapiro, truth in a model is a good model of truth.
7
Dummett takes the structures in which the classical mathematician trades as being
exactly like the red-green world I describe. For Dummett, this is why the classical
mathematician is committed to the view that it could be the case that the natural number
structure is such that for example, every even number is the sum of two primes, while no
finite proof of such a claim exists (e.g. see Dummett’s “Platonism”). This is the key to
understanding why Dummett sees intutionism as the main competing position to
Platonism in the philosophy of mathematics. It is also key to understanding why one who
20
restricted his attention to Dummett’s “The Philosophical Basis of Intuitionistic Logic”
might read the argument in the way Tennant does.
Modal talk in mathematics is
somewhat dicey, since most of us accept both that (in some sense) mathematical truth
entails mathematical necessity and (in some sense) that an undecided mathematical claim
is possibly true and possibly false. So one who ignored the second modal intuition
(perhaps due to allegiance to the first) while restricting her attention to Dummett’s
discussion of mathematics (e.g. in “The Philosophical Basis. . .”) might collapse the
modal in Dummett’s Insight. On this view Dummett would believe that truth conditional
semantics entailed the actual existence of an undecidable sentence. Then, if one ignored
Dummett’s voluminous writings on the role of logical semantics in theory of meaning,
one might (like Tennant) foist upon Dummett the view that bivalence itself entails such a
sentence.
8
This is exactly why, in the concluding chapter of The Logical Basis of Metaphysics,
Dummett is at great pains to show that even if God exists, he does not have knowledge of
a completed infinity.
9
Stewart Shapiro first encouraged me to render explicit the modal commitments of
Dummettian anti-realists. It is a consequence of my discussion that Shapiro’s “Antirealism and Modality,” as well as Anthony Appiah’s For Truth and Semantics deserve
much more critical attention from Dummettians than they have received. I would also
like to thank Emily Beck Cogburn, Joseph Salerno, William Taschek, and Neil Tennant
for encouragement and helpful comments on an earlier draft of this material.
21