Download The Logic of Logical Revision

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THE LOGIC OF LOGICAL REVISION:
FORMALIZING DUMMETT’S ARGUMENT
ABSTRACT: Neil Tennant and Joseph Salerno have recently attempted to rigorously
formalize Michael Dummett’s argument for logical revision. Surprisingly, both conclude
that Dummett commits elementary logical errors, and hence fails to offer an argument that
is even prima facie valid. After explicating the arguments Salerno and Tennant attribute
to Dummett, I show how broader attention to Dummett’s writings on the theory of
meaning allows one to discern, and formalize, a valid argument for logical revision. Then,
after correctly providing a rigorous statement of the argument, I am able to delineate four
possible anti-Dummettian responses. Following recent work by Stewart Shapiro and
Crispin Wright, I conclude that progress in the anti-realist’s dialectic requires greater
clarity about the key modal notions used in Dummett’s proof.
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 such epistemic unfriendliness 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 surrounding this dialectic falls into two overlapping groups.
The first consists of philosophical and mathematical treatises on intuitionistic logic and
constructivism (e.g. [Prawitz 1965], [Dragalin 1980]), and the second of discussions
about the intelligibility of an unknowable reality (e.g. [Appiah 1986], [Wright 1993]). It
is only recently that attempts have been made to rigorously formalize Dummett’s own
1
arguments concerning the metaphysical and epistemic presuppositions of classical logic.
One result of these investigations is that some philosophers have professed inability to
discern even prima facie valid arguments for logical revision in Dummett’s oevre.1
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: 160]. Like Tennant, Crispin Wright has offered his own new argument for
intuitionism [1992].2 Unlike Tennant, he does not couple this argument with criticism of
Dummett, though the very fact that he does not is telling.
Finally, in a recent article discussing Dummett and Wright’s arguments for logical
revision, Joseph 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
1
It is very important to distinguish the kind of argument under consideration from distinct
inferentialist considerations for intuitionism. The non-inferentialist, realist, order of explanation
involves explicating meaning in terms of truth conditions and valid inference in terms of truth
preservation. Inferentialists such as Brandom, Dummett, Prawitz, and Tennant seek to reverse
this order of explanation, regarding meaning to be given primarily by inferential role. From this
perspective, the nice proof-theoretic properties of natural deduction formalizations of intuitionist
logic can be considered evidence for intuitionism. For the clearest presentation of these
considerations, see Chapter 10 of [Tennant 1997]. Chapters 9-13 of [Dummett 1991] are also
central to this dialectic. Note that for this family of arguments to be plausible, one must already
accept Dummett’s inferentialism, rejecting the primacy of bivalent, truth conditional semantics.
Thus, as will become clear from my discussion, these inferentialist reasons for intuitionism (or,
in Tennant’s case, intuitionist relevant logic) presuppose the validity of the argument I locate in
Dummett.
2
In [Cogburn 2002a] and [Cogburn 2003] I critique, respectively, Wright and Tennant’s new
arguments.
2
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.
[2000: 212]
If Salerno and Tennant are right about the non-existence 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 clothes.
But Tennant and Salerno are wrong. While Tennant [1987; 1997] has done as
much as anyone to advance Dummett’s anti-realist program, when explicating Dummett’s
argument for logical revision he and Salerno ignore Dummett’s writings on the theory of
meaning. By attending to the role a logical semantics is to play in the theory of meaning,
one can discern an argument for logical revision that is both prima facie valid and, once
formalized, extraordinarily clear.
I. ON THE VERY IDEA OF FORMALIZING A LOGIC OF LOGICAL REVISION
Prior to considering Tennant and Salerno’s attempts to formalize Dummett’s arguments, a
few general considerations must be raised. Most importantly, one might hold that there is
a sense in which one cannot fully formalize arguments for logical revision. All of the
arguments under consideration in this paper (as well as Tennant, Salerno, and Wright’s
own arguments) involve extensive logical resources, including quantification over
propositions and modal and epistemic operators. Fully formalizing languages with these
resources involves recursively specifying a syntax and defining (either proof or model
theoretically) a complete consequence relation. Quinean animadversions against
3
languages more expressive than first order logic notwithstanding, for the classical
logician this should not be too worrying. The logics for languages with quantification
over propositions, epistemic, and modal operators have been rigorously formalized, and
such formalizations have found widespread applications in linguistics and computer
science.
For example, Montague’s original Intensional Logic allows quantification over
expressions of any logical type, including propositions, and includes modal operators and
epistemic propositional attitudes verbs [Dowty et. al. 1992]. If we understand the
knowability operator in proofs for logical revision as shorthand for the claim that it is
possible that there exists someone who knows that P, then it is trivial to demonstrate that
any of the proofs in this paper are sound with respect to Montague’s semantics.3
Unfortunately, such formalization can beg the question against either the classicist
or the intuitionist.4 For example, on the possible worlds understanding of modal
operators, ‘It is necessarily the case that P,’ ([]P) means that P is true in all possible
worlds and ‘It is possible that P,’ (<>P) means that there exists a possible world where P
is true. But then the standard modal entailment from, ‘It is not the case it is necessarily
3
Actually, Montague’s original Intensional Logic would need to be amended slightly by adding
nested possible worlds to express different strengths of modality, for example, such that the
empirically possible worlds are a subset of the metaphysically possible worlds, which are a
subset of the logically possible worlds. For modal logicians and those interested in modal
metaphysics, this is old hat, and in the context of the demonstration, it is clear that this is not
logically problematic. However, the substantive conclusion of this paper is that the key modal
notions I am able to isolate are in desperate need of philosophical clarification by both the
intuitionist and classicist.
4
the case that P,’ ([]P) to, ‘It is possible that it is not the case that P,’ (<>P) is
equivalent to saying that if it is not the case that for every possible world w, P is true in w,
then it is the case that there exists a world w, such that P is not true in w. But the form of
this inference (from xA[x] to xA[x]) is notoriously intuitionistically invalid. So
any argument defending classical logic that invoked such understandings of the modal
operators should be rejected as question begging by the intuitionist.
One might think that an intuitionist cannot face such a problem in crafting
arguments for intuitionism, since every intuitionistically valid argument is also classically
valid. Interestingly, this is not the case. For example, A.G Dragalin [1988] and P.
Oddifreddi [1996] have shown formulations of Church’s Thesis put forward by
intuitionists to be intuitionistically consistent with the Peano Axioms yet classically
inconsistent with them. For the intuitionist who subscribes to Church’s Thesis this is
clearly an argument for revision: assuming Church’s Thesis and the Peano Axioms are
true, then classical logic can’t be correct.
Such an argument begs the question against the classicist, though. The formal
language statements of Church’s Thesis under consideration can only fairly be thought of
as representations of Church’s Thesis if one interprets the quantifiers intuitionistically.
Consider the following, from Kreisel [1965].
xyR(x, y)  exz[T1(e, x, z)  R(x, U(z))]
Given the intuitionist construal of the quantifiers, any choice function associated with the
antecedent will be intuitively computable, as for the intuitionist to assert ‘xyR(x, y)’ is
4
I thank an anonymous reviewer for raising this important issue and encouraging me to expand
on this point.
5
to assert that for any x there is an intuitively computable procedure by which a y can be
found such that R(x, y) holds. However, a classicist would never interpret the antecedent
in this manner! On the standard classical model-theoretic interpretation of the quantifiers
‘xyR(x, y)’ can be true even if there is no such procedure associated with it. Thus a
classicist can correctly hold that the intuitionist’s argument, as an argument against
classical logic, begs the question.5
One solution to this is, I think, to utilize what might be considered a partial,
implicit, formalization of arguments for logical revision. That is, as with full
formalization, regiment all premises and conclusions in a familiar recursively specified
syntax, and then explicitly state canonical inference rules that are: (1) sufficient to get
from the premises to the conclusion, and (2) such that they can be justified by semantic
(model or proof theoretic) frameworks sound with respect to either classical or intuitionist
reasoning. In this case, neither an intuitionistic nor a classically problematic principle is
allowed, and thus each side will see the argument as valid by their own lights.
An easier solution that works just as well is to require that arguments for
intuitionistic revision be valid by the classicist’s own lights, and require that arguments
for classical logic be intuitionistically valid. If this requirement is in place, then neither
side can be rightfully accused of begging the question against the other.
5
In [Cogburn 2002b] I discuss this issue as well as reasons why Dummettian anti-realists must
reject Church’s Thesis. In [Cogburn 2003] I show Tennant’s new argument to beg the question
in precisely the manner Dragalin’s does.
6
As noted, the argument I attribute to Dummett is clearly valid by the classicist’s
own lights. Cursory examination shows the inferences to be sound with respect to the
model theoretic framework canonical for those engaged in linguistic semantics.6
Unfortunately, it is less clear whether the argument is sound for an intuitionist,
because there is no intuitionistic treatment of modal logic canonical in the same way
Montagovian type theory is for semanticists in the truth-conditional tradition. This being
said, there is also nothing obviously wrong with the argument from an intuitionistic point
of view, and the resources it uses are no more expressive than those used by other
intuitionists such as Salerno, Tennant, and Wright. In addition, it is enough for the
intuitionist that the argument be sound with respect to classical inference. For the
intuitionist is then in a position to argue that classicists are hoisted by their own petards.
Of course, the formal validity of the argument needn’t convince one, as the issue
of the plausibility of the argument’s premises must also be settled. In this light, I think
6
Philosophers of language sometimes fail to appreciate fully the influence within linguistics and
computer science of Montague’s, and David Lewis’ [1970] for that matter, revolutionary early
work in developing categorial grammar and extending Church’s lambda calculus. Theories
stemming from the Montagovian approach are the only extant theories to successfully recursively
correlate non-trivial fragments of natural languages with interpreted formal languages. To
realize the extent of the cross-disciplinary misunderstanding on this point, one need only
consider recent treatment of compositionality and the syntax-semantics interface in the respective
disciplines. For example, every essay in the canonical (for linguists) [Lappin 1996] and [Partner
& Partee 2002] and nearly every paper in the top linguistics journals Linguistics and Philosophy
and Semantics build upon Montague’s achievements. On the other hand, none of the essays in
otherwise excellent [Wright and Hale 1997] exhibit awareness of either post-Montague
semantics or semantics friendly, computationally tractable, non-transformational approaches to
syntax such as Tree Adjoining Grammar, Head Driven Phrase Structure Grammar, and
Categorial Grammar.
7
that my formalization of Dummett’s argument is a major contribution to the anti-realist
dialectic. Thus, independent of considerations about ‘setting the record straight,’
(defending Dummett from Salerno and Tennant’s criticisms) I take this clarifying of the
issues to be the most important accomplishment of my demonstration. By correctly
formulating Dummett’s proof, I am able to precisely enumerate the four possible options
Dummett’s argument forces upon the defender of classical logic.
II. SALERNO’S INTEPRETATION
‘The Philosophical Basis of Intuitionistic Logic’ is an early paper in which Dummett
makes his case for verificationism. In contrasting his own verificationist view with his
opponents’ realist position, 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.
[1975a; 224]
This is Dummett’s meaning-theoretic challenge, to specify that in which 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.
8
Dummett’s argument for logical revision is an attempt to show that such
verificationism undermines classical logic. Formalizing this requires explicating the
logical structure of Dummett’s verificationist claim. Thus, where ‘[]1’ means ‘it is
necessarily the case that,’ and ‘k1X’ means that X is knowable, we can present Dummett’s
verificationist belief as
Verificationism
[]1X((X k1X)  (¬X k1¬X)).
Given some interpretations of the necessity operator, Dummett clearly intends this. For a
verificationist meaning theorist, verificationism is supposed to reflect a 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,’ Salerno presents Dummett as holding
that verificationism 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 their existence, 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: 63]
9
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, with ‘k2Y’ denoting that Y is knowable, where the premises are,
Verificationism
[]1X((X k1X)  (¬X k1¬X))
Undecidability
Y(¬k2Y  ¬k2¬Y)
Bivalence
X(X  ¬X)
and ‘’ refers to absurdity, Dummett’s main claim seems to be:
Salerno’s Version of Dummett’s Argument:
Verificationism, Undecidability, Bivalence |- 
Using the law of excluded middle rather than 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, 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 concludes
that Verificationism is the problem and continues to assert Bivalence and Undecidability.
10
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! That is, one can intuitionistically prove the following.
Important Lemma:
|- X((X k1X)  (¬X k1¬X)) ¬Y(¬k1Y  ¬k1¬Y) 7
Given that ‘X((X k1X)  (¬X k1¬X))’ is just Verificationism without the
necessity operator in front, and that ‘¬Y(¬k1Y  ¬k1¬Y)’ is just the denial of
Undecidability, our Important Lemma entails inconsistency of Verificationism and
Undecidability. Bivalence is a complete non-sequitur in Salerno’s Version of
7
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
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.| | 
2, 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
11
Dummett’s Argument. But then, if Salerno’s interpretation is correct, Dummett has
committed an egregiously basic logical error.8
Independently of undermining what one might take to be Dummett’s argument,
this result is deeply problematic for the Dummettian. 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. For example, one might interpret it as saying that there exists a
sentence such that it is not possible, with means currently at our disposal, to know
whether the sentence is true or false. Thus, depending upon the strength of the modal ‘k1’
in Verificationism ([]1X((X k1X)  (¬X k1¬X))), a verificationist like Dummett
need not be committed to a contradiction. For example, if ‘k1X’ means that X is
knowable with arbitrarily large finite extensions of our conceptual resources, there would
be no problem. This doesn’t contradict the claim that there is some sentence we can’t
presently verify.
While this shows that the Dummettian anti-realist’s position is not obviously
absurd, we still have no argument for logical revision.
8
As further discussion will reveal, Tennant [1997] seemed to be aware that one might read
Dummett this way. In [Cogburn 1999], the author: (a) explicitly considers, though only to reject,
this interpretation of Dummett, and (b) attributes what I have here called, ‘Important Lemma,’ to
a 1996 personal communication from Neil Tennant. Tennant credits me [Tennant 1997: 160],
with coming up with the label (‘single sentence argument’) of the revisionary argument he
attributes to Dummett.
12
III. TENNANT’S INTERPRETATION
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
Important Lemma, is
Tennant’s 1st Lemma:9
Verificationism |- ¬(Undecidability)
It should be noted that Tennant is aware of the fact that one might interpret Dummett in
the manner of Salerno. Where ‘’ is the name of Tennant’s 1st Lemma, he 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. . .
[1997: 181]
One might think this is a more charitable reading of Dummett. However, Tennant’s
reading is no more charitable as the entire case then hinges on the ‘other subargument,’
Tennant’s 2nd Lemma:
Bivalence |- Undecidability
Were the 2nd lemma plausible, it and the 1st together would establish the inconsistency
of Verificationism and Bivalence. [Tennnant 1997: 180-181] Even more than with 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.
9
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.
13
The bulk of Chapter 6 of Tennant’s The Taming of the True (‘The Manifestation
Argument is Dead’) is concerned 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. [1997: 194]
Thus, like Salerno, Tennant concludes that Dummett never even provided a prima facie
valid argument for logical revision.
IV. DUMMETT’S ARGUMENT
At this point, one must ask if it is plausible that the author of Elements of Intuitionism,
and The Logical Basis of Metaphysics could perpetuate such a (in Tennant’s words) ‘nonsequitur of numbing grossness,’ or whether he would commit the elementary logical error
that Salerno attributes to him. That is, the principle of charity alone provides strong
evidence for a formalization of Dummett’s argument free from Salerno and Tennant’s
criticisms.
In addition, further textual evidence does support an argument different from the
kind Salerno and Tennant attribute to Dummett. While Salerno and Tennant both focus
on the principle of bivalence, a close reading of Dummett’s writings on the theory of
meaning show bivalence not to have a central role in Dummett’s argument. Instead,
Dummett focuses primarily on the role classical model theoretic semantics plays in the
14
theory of meaning. This only yields a criticism of classical logic indirectly, as a result of
Dummett’s core belief that inference is in need of justification by the theory of meaning
[Dummett 1973 ; Dummett, 1975c; Dummett, 1976b]. It is the failure of classical model
theoretic semantics (sound with respect to classical logic) that leads Dummett to suggest
using a Heyting style constructivist semantics (sound with respect to intuitionist
deduction) instead.
Two things are essential to understanding the role truth conditional semantics
plays in a Dummettian theory of meaning. First, he presents truth conditional semantics
as having the truth predicate attach to a sentence in virtue of the referential relations of
the subsentential units of that sentence. Thus, in a later article, almost as if in rebuke to
Salerno and Tennant’s focus on bivalence, 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: 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.’
Second, for Dummett, what is important is the way the truth of logically complex
statements are determined in classical model theory. He writes,
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
15
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: 231-232]
Thus, for Dummett, the combination of how reference and quantification are handled
renders truth conditional semantics problematic.10 The quantificational apparatus alone
allows us to refer to infinite totalities, and the semantic clauses of truth conditional
semantics then state how such claims are determined to be true or false by the reality to
which they refer.11 Dummett wishes to replace this realist meaning-theory with a
10
In addition to the infinite, in this context Dummett (in [Dummett 1976b], and [Dummett 1991:
314-316]) also mentions subjunctive conditionals as well as our capacity to refer to inaccessible
regions of space-time.
11
A very plausible objection at this point concerns the possibility of deflationary interpretations
of truth conditional semantics. Couldn’t one utilize truth conditional semantics without being
committed to Dummettian realism? Dummett is aware of this possible criticism, and argues that
a deflationary interpretation of the classical model theoretic apparatus prohibits that apparatus
from playing the explanatory role demanded by the theory of meaning. For example,
On the one hand, it can be the expression of adherence to the redundancy theory of
truth, or to the similar view that the whole explanation of the notion of truth is given by
a Tarskian truth-definition. As we have seen, these views exclude the possibility of
taking the notion of truth to have a significant role in a meaning-theory, and certainly
of its being the central notion in the strong sense. [Dummett 1991: 331].
16
constructivist theory that correlates verification (as opposed to truth) conditions with
statements.
So it is clear from Dummett’s later work that he takes truth conditional semantics,
when used as a core part of a theory of meaning, to be epistemically problematic. But
what is this problem? How should we cash out the claim that a sentence is true
‘independently of whether we come to know it or not’? As we have seen, Tennant cashes
this independence out as the existence of an undecidable sentence, but then is unable to
find an argument from bivalence to the existence of an undecidable sentence.
I interpret Dummett’s independence claim in a much weaker manner than
Tennant, as merely affirming the possibility of the existence of an undecidable sentence.
Thus, while Tennant takes Dummett to hold that bivalence on its own entails the
existence of an undecidable sentence, I read Dummett as taking 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. We can formalize the principle in this manner.
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
Of those that interpret Tarski style definitions of truth as somehow dispensing with reference,
Dummett again argues that, insofar as such definitions play a role in the theory of meaning, the
notion of reference is still ‘surreptitiously appealed to’ [Dummett 1982: 234] in the way that
causes truth to be epistemically unconstrained.
17
language and mathematics.’ If we cash out the independence claim in this way, it
becomes much less controversial, since: (a) the antecedent is logically stronger, and (b)
the consequent is logically weaker. Thus, a sound argument with Dummett’s Insight as a
premise should be much easier to make.
Dummett’s Insight is very easy to defend, given realistic interpretations of the
consequent’s modal (licensed by the antecedent!). For example, consider a possible
world consisting of an infinite number of red objects and denizens of that world that can
talk to one another. Now consider the sentence, ‘There is no green object.’ 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 that the sentence was true, since they could not
survey the infinite domain. Thus, in the described possible world there does exist an inprinciple undecidable sentence.
In the case of decidable, yet contingent, empirical predicates such as ‘red’ it is
easy to generate such scenarios given the quantificational apparatus of classical logic, and
the attendant model-theoretic interpretation of those quantifiers. For example, consider a
world containing an infinite number of red and green objects and the sentence
(expressible in second order logic), ‘There are an infinite number of green objects.’
Unlike the sentence, ‘there is no green object’ (which is falsifiable in worlds where it is
false), this new sentence is neither verifiable nor falsifiable if it is true or false, and by
classical model theory it must be one or the other.
Dummett takes the classical mathematician’s structures to be exactly like the redgreen world I describe. For Dummett, this is why the classical mathematician is
18
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. [Dummett 1967]).12
To fully grasp Dummett’s point, it must be noted that Dummett is not stating that
infinity by itself yields undecidable sentences.13 This would be to read Dummett
uncharitably in exactly the manner Salerno has. As Dummett’s own Elements of
Intuitionism shows, it is precisely the constructivist account of all infinity as potential
infinity that renders intuitionist mathematics so radically different from its classical
counterpart. For Dummett, what gives rise to the potential unknowability is the way in
which infinity is modeled in classical model theory.
Realist semantics are not problematic when applied to finite domains and nonmodal sentences, precisely because such uses of realist semantics do not in any way give
12
This is the key to understanding why Dummett sees intuitionism as the main competing
position to Platonism in the philosophy of mathematics. It is also key to understanding why one
who 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) interpret Dummett as holding the view that bivalence itself
entails such a sentence.
13
I thank an anonymous reviewer for encouraging me to expand upon this point.
19
rise to potential unknowability. The problem arises when we think of infinity as complete
in the way we would a finite set.
The most celebrated example of this way of thinking relates to quantification over
finite, surveyable domains by learning the procedure of conducting a complete
survey, establishing the truth-value of every instance of the quantified statement.
The assumption that the understanding so gained may be extended without further
explanation to quantification over infinite domains rests on the idea that it is only a
practical difficulty which impedes our determining the truth-values of sentences
involving such quantification in a similar way; and, when challenged is defended by
appeal to a hypothetical being who could survey infinite domains in the same
manner as we survey finite ones. [Dummett 1976b: 61]
Dummett himself goes to great pains to show this to be an illicit move. Even if God does
exist, it would
not vindicate the realist’s claim that he must know one or the other from knowing,
of each number, whether it is prime or not. That would follow only if we assume
that the infinitely many individual propositions to the effect that a particular
number is prime together determine the proposition about prime pairs as true or as
false. That, however, is just the question at issue. The realist wishes to attribute to
us an understanding of the quantifiers as operators yielding a statement whose
truth-value is jointly determined by the individual instances, independently of our
means for recognising it as true or false. When the domain is infinite, his opponent
denies that we can understand them in any such way: even should an angel inform
him that God understands them in that way, he would still deny that we can; this
would then really be a case of our being unable to understand the thoughts of the
Almighty. [Dummett 1991: 350]
For our purposes, the important point is that for Dummett the classical model theoretic
treatment of quantifiers yields the possibility of undecidable sentences. To one that might
respond that such sentences, while undecidable to us, may not be to God, Dummett has
20
argued that God’s ability to decide them would then be irrelevant to our understanding of
the sentences.
Moreover, Dummett goes on to argue that the existence of an omniscient being
does not, in fact, commit one to modeling infinity in the manner of classical model
theory.
The anti-realist may even be disposed to doubt the angel: if an infinite process is
one which it makes no sense to speak of as having been completed, then it makes
no sense to speak of God’s completing such a process either. Our objection to the
fantasy of the superhuman arithmetician was that he did not exist; a stronger
objection is that, since he completes infinite tasks and uses their outcome to
evaluate quantified propositions, he could not exist. [1991: 350-351]
There are many issues floating in these passages. Again, for our purposes the important
thing is to note how Dummett is arguing that, when we attend to infinity, the classical
model theoretic account of quantification yields sentences potentially undecidable both to
us and to God.
Dummett closes The Logical Basis of Metaphysics by remarking on the fact that
an omniscient God does not in any way entail realism.
It is a persistent illusion that, from the premiss that God knows everything, it can be
deduced that he knows whether any given proposition is true or false--that is, that
he either knows that it is true or knows that it is false, and that his omniscience
therefore entails that the proposition is either true or false. On the contrary, its
being either true or false is required as a further premiss in order to deduce from his
omniscience that he knows, in the sense stated, whether it is true or false. [1991:
351]
This is an important point because it blocks one from turning Dummett’s modus tollens
into a realist’s modus ponens. The realist might be tempted to argue that, since we do
21
have a clear conception of God, then the classical model theoretic conception of
quantification is perfectly reasonable. Dummett has shown that the conception used by
the realist in making any such argument already assumes that the law of bivalence (which,
for Dummett, stands in need of justification by classical model theory) is correct.
This is enough to illustrate at least the prima facie plausibility of Dummett’s
Insight, the key premise in what I take to be Dummett’s argument for logical revision.
This, then, is the argument’s form.
Dummett’s Argument:
Verificationism, Dummett’s Insight |- ¬TCS
Rigorously proving this will require greater clarity about the inferential role of key modal
notions occurring in the following premises.
Premises:
Verificationism
[]1X((X  k1X)  (¬X  k1¬X))
Dummett’s Insight
TCS  X(¬k2X  ¬k2¬X)
Our demonstration necessarily involves the following modal inferences.14
Transformation rules:
K.
|- P, therefore |- []L P (If P is provable from no premises, then
the logical necessity of P is provable from no premises.)
14
Again, these are sound with a Montagovian understanding of the logical operators. On the
standard possible worlds understanding of the modals they are sound with respect to
intuitionistic reasoning. The lack of a canonical semantics for intuitionist modal logic might
cause consternation here, but as argued earlier, it shouldn’t; the proof would only beg the
question if the principles were sound with respect to intuitionistic reasoning and unsound with
respect to classical reasoning, as with Tennant’s own proof [Cogburn, 2003]. This being said, I
will go on to show that progress in this debate requires more clarity about these modal and
epistemic notions.
22
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 now rigorously formalize Dummett’s Argument.
Demonstration of Dummett’s Argument:
1. X((X k1X)  (¬X k1¬X))
¬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.
Clearly, this argument does not suffer from the flaws Salerno and Tennant find
with Dummett. Also, unlike the arguments Salerno and Tennant discern, the main
23
conclusion is not that we should eschew classical inference, but rather that we should
eschew the use of classical model theory in the theory of meaning. It is only because of
Dummett’s further belief that classical inference needs the justification of classical model
theory that his argument can be applied against classical inference.15 As an exegetical
point, this provides more evidence for my interpretation of Dummett’s argument. In
addition to charity concerns, if Salerno and Tennant were right in their interpretations,
then Dummett’s voluminous writings about the justification of deduction as well as the
theory of meaning would have nothing to do with his writings on logical revision. Given
how intertwined these writings are in Dummett’s corpus, this would be extraordinarily
strange.16
15
To see how involved this issue is, consider whether or not TCS itself entails bivalence. 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 and 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 anti-realism) 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.
Thus, I have essentially assumed that TCS includes the assumption that, to borrow a phrase from
Stewart Shapiro, truth in a model is a good model of truth.
16
Also, see the discussion in footnote 1. By my interpretation of Dummett’s argument, Chapters
9-13 of The Logical Basis of Metaphysics make perfect sense. The charge of Dummett’s
argument is to shake loose one’s adherence to classical model theory. Then, separate arguments
are needed to show that intuitionism provides the resources for the best inferentialist alternative
to classical model theory.
24
V. 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
merely to give Dummett his due. Like many important philosophical demonstrations,
Dummett’s carves the logical space of possible positions in terms of how one reacts to the
proof. My presentation of Dummett’s argument sets in bold relief several morals about
how debate about intuitionistic logical revision should continue.
The proof itself shows 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 11 due to incomparable notions of possibility
(1 and 2) and (4) argue that the proof equivocates on line 11 due to incomparable
notions of knowability (k1 and k2).
In closing I would like to suggest that these combine to produce an important
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
25
that truth conditional semantics is mistaken would not follow. I conjecture that this is
exactly why, in the passages discussed above, Dummett is at great pains to show that
even if God exists, he (like us) does not have knowledge of a completed infinity.
The reason this is so pressing for Dummett 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 semantics implies that it is possible that some sentences are unknowable to
us. But then, if any more controversial statements of Verificationism and Dummett’s
Insight are false, it would follow that our dilemma undermines Dummett’s argument;
either one of Verificationism or Dummett’s Insight are false, or the proof equivocates on
line 11.
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 ‘k2P’ to mean ‘P can be
known with certainty.’ As before, Verificationism and Dummett’s Insight both become,
respectively, less controversial under these construals. But then 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
26
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. As
Crispin Wright’s Truth and Objectivity illustrates, a discourse-by-discourse study of the
various normative constraints on the truth predicate is both important and very
complicated. It is a consequence of my discussion that the kind of undertaking Wright
and others have begun will have repercussions for the correctness of patterns of
reasoning. This is, of course, the very project enunciated in Dummett’s The Logical
Basis of Metaphysics.
Rather than seeking to refute Dummett, the main reason that I have enumerated
the classicist’s strategies is to establish that future anti-realist debate about logical
revision should be about the proper modal notions occurring in Dummett’s argument as
applied to different discourses. 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
27
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? [1976b: 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
right. Anti-realists and realists should conclude that significant progress in the debate
over logical revision now requires attaining much greater clarity and insight into the
relevant modal notions.17
17
I must thank Stewart Shapiro for encouragement and feedback. It is a consequence of my
discussion that Shapiro’s ‘Anti-Realism and Modality,’ as well as Anthony Appiah’s For Truth
in Semantics, deserve much more critical attention from those interested in the issue of logical
revision. I would also like to thank an anonymous reviewer, Emily Beck Cogburn, Joseph
Salerno, William Taschek, Neil Tennant, and Crispin Wright for encouragement and helpful
comments on an earlier draft of this material.
28
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