truth as an epistemic notion - dag prawitz
TRANSCRIPT
Truth as an Epistemic Notion
Dag Prawitz
Published online: 24 September 2011
� Springer Science+Business Media B.V. 2011
Abstract What is the appropriate notion of truth for
sentences whose meanings are understood in epistemic
terms such as proof or ground for an assertion? It seems
that the truth of such sentences has to be identified with the
existence of proofs or grounds, and the main issue is
whether this existence is to be understood in a temporal
sense as meaning that we have actually found a proof or a
ground, or if it could be taken in an abstract, tenseless
sense. Would the latter alternative amount to realism with
respect to proofs or grounds in a way that would be con-
trary to the supposedly anti-realistic standpoint underlying
the epistemic understanding of linguistic expressions?
Before discussing this question, I shall consider reasons for
construing linguistic meaning epistemically and relations
between such reasons and reasons for taking an anti-realist
point of view towards the discourse in question.
Keywords Truth � Intuitionism � Anti-realism �Meaning-theory
1 Why Anti-Realism?
Notwithstanding that in most cases one needs strong rea-
sons to depart from our natural inclinations towards real-
ism, for some discourses it just seems obvious that what
one is talking about does not constitute an objective reality
existing independently of us. For instance, one naturally
takes an anti-realist view of fictional characters on that
ground. This needs not refrain one from speaking of a
domain of discourse, a fictional world, and of facts con-
cerning its individuals or entities. But one cannot explain
what it is for a statement about fictional characters to be
true by referring to these facts. In particular, one cannot
take such a statement to be always either true or false
because of given facts of the fictional world. It is rather the
other way around: the facts are determined by which
statements are true, and truth has to be explained in some
other way than by referring to facts. A natural suggestion is
that a statement is true if it follows from how the fictional
world has been given, in other words, if the statement
either belongs explicitly to the story told or can be inferred
from it. A fictional character has then only those properties
that she gets in this way.
Similarly, one has based anti-realism in mathematics on
a view of mathematical objects as our constructions. This is
how Arend Heyting begins his philosophical account of
mathematical intuitionism in the thirties, saying for
instance: ‘‘Mathematical objects […] are to their nature
depending on human thinking. Their existence is secured
only in so far as it can be determined by thinking; they
have properties only in so far as by thinking they can be
acknowledged to have them’’ (Heyting 1931, 240–241). As
Heyting points out, one has then grounds to doubt the use
of the law of the excluded third in mathematical reasoning.
However, Heyting also gives semantic explanations and
takes in this connection an equally pronounced anti-realist
position on behalf of intuitionism. According to intuition-
ism, a mathematical proposition expresses the intention of
a specific construction, and an assertion announces the
realization of the construction intended by the asserted
proposition. Heyting is keen to contrast this with the
classical position. A classical assertion may state ‘‘a fact of
transcendental nature’’, while the intuitionistic assertion of
a proposition p ‘‘states an empirical fact’’: ‘‘one knows how
D. Prawitz (&)
Department of Philosophy, Stockholm University,
106 91 Stockholm, Sweden
e-mail: [email protected]
123
Topoi (2012) 31:9–16
DOI 10.1007/s11245-011-9107-6
to prove p’’ (Heyting 1930, 958–959). After having also
explained the intuitionistic meaning of disjunction and
negation in particular, the rejection of the law of the
excluded third is plain: one cannot assert a proposition of
the form p _ : p intuitionistically as long as one is not able
either to find a realization of the intention expressed by the
proposition p or to derive a contradiction from the
assumption p.
The idea of bringing in meaning explanations to clarify
logical and ontological matters is taken up by Michael
Dummett, and is turned into a general philosophical pro-
gram for how to settle various disputes between realism
and anti-realism. He suggests that to take a stand on such a
dispute concerning a specific discourse we should first
investigate how one can develop a meaning theory for the
part of language in question. Such a theory should account
for all features of the use of linguistic expressions that
depend on knowing their meaning. If it turns out to be
impossible to develop a meaning theory in which the
understanding of sentences is explained in terms of truth
conditions such that, for every sentence, either its truth
condition or the truth condition of its negation obtains, then
we have given substance to the idea that we are not dealing
with an external reality that exists independently of us.
Dummett does not simply say that a meaning-theoretical
approach is one way to settle ontological disputes. He
maintains that it is the only one available, or, at least, that
in the case of mathematics one cannot argue cogently in
favour of intuitionism from the premiss that mathematical
objects are our constructions. This premiss must be given a
content that is not only metaphorical, and the proper way to
do so is to formulate it as a thesis about the truth of basic
statements about mathematical objects. In the case of nat-
ural numbers, the premiss that these objects are our con-
structions could be taken as the thesis that what makes
numerical equations and inequalities true are our compu-
tations. The question is whether this settles how the truth of
undecidable sentences is to be understood.
Now, an intuitionist must also take a sentence such as
‘‘for any natural number n, n is either prime or not prime’’,
proved by induction, to be true. Hence by distributivity of
truth over universal quantification and disjunction, it fol-
lows that for any natural number n, one of the two sen-
tences ‘‘n is prime’’ and ‘‘n is not prime’’ is true. For a large
number n we may not know which one is true and which
one is false, yet it would be unreasonable (Dummett’s term
is ‘‘hard-headed’’) to deny that each one of these sentences
has a determinate truth-value. In spite of holding the nat-
ural numbers to be creations of the human mind, an intu-
itionist must thus admit that there is a group of sentences
about them, embracing all decidable ones, that have
determinate truth-values, are true or false, independently of
our knowledge.
In view of this, when coming to the undecidable sen-
tences and the crucial question about their truth, an intui-
tionist cannot support the claim that it is different with
them by simply invoking the premiss about the nature of
the natural numbers. If all the instances of a universally
quantified sentence have determinate truth-values that may
be unknown to us, why could not also this universal sen-
tence have a determinate truth-value unknown to us? It
cannot depend on the nature of natural numbers, since their
nature was not a hindrance for all the instances to have
determinate truth-values unknown to us. If there is a reason
for thinking that the truth of undecidable sentences is a
quite different matter, it must depend on the operations by
which we get such sentences, which in mathematics consist
of quantifications over infinite domains. We must therefore
turn to the meaning of these quantifiers.
This summarizes Dummett’s (1975) argument for say-
ing that the idea of natural numbers being our creations is
in itself compatible with an essentially realist view, and for
holding that therefore we must take a meaning-theoretical
approach to settle the issue that really matters in the dispute
between realism versus anti-realism. We may note that the
main point of the argument is that the thesis about the truth
of a numerical equation being determined by our compu-
tations should not be taken as saying that the equation is
true only if it has been shown to be so by our computations.
Rather, it must be understood as saying that the equation is
true only if it could in principle be proved to be true by
computation; in other words, if the relevant calculations
were carried out, their outcome would show the equation to
be true. All decidable sentences have therefore determinate
truth-values determined by our possible computations, but
as far as this argument goes, it is an open question how it is
for the undecidable ones in this respect.
The meaning-theoretical argument that Dummett then
gives for rejecting the classical explanation of the meaning
of quantifications over infinite domains is well known and
need not be rehearsed here (see for instance Dummett 1976).
Let us just recall its main starting point: we cannot explain
what it is to understand undecidable sentences, if their
meaning is taken to be given in terms of truth-conditions
that always obtain or do not obtain. To be able to ascribe
knowledge of the meaning of a sentence to a person there
must be some way in which her knowledge can be shown in
how she uses the sentence. But if the sentence is undecid-
able, it may be beyond our abilities to know whether the
truth-condition obtains or does not obtain, and then it is
difficult to see how a person’s knowledge of what the truth-
condition is can be exhibited fully in her linguistic behav-
iour. Therefore, there must be something else than tran-
scendent truth-conditions in terms of which the meaning of
sentences is given. The way in which we recognize the truth
of a sentence then suggests itself as a natural alternative.
10 D. Prawitz
123
The argument carried out so far is not an argument for
anti-realism. It remains to see what exactly can replace
truth as the central notion of a meaning theory, and whether
the theory will then support realism or anti-realism. This
will essentially depend on the notion of truth that this
meaning theory leads to, which is what I am mainly
interested in here. The question is thus not what truth can
amount to in anti-realism, but what notion of truth is
appropriate when meaning is explained in epistemic terms
and whether that notion leads to realism or anti-realism.
2 Another Reason for Seeking an Epistemic Theory
of Meaning
Before entering a discussion of the notion of truth, I want
to consider another reason for being unsatisfied with the
classical truth-conditional theory of meaning, namely that
it appears unable to give an account of the phenomenon of
deduction. The practice of deductive inference is an inte-
gral part of our use of language, and the legitimacy of a
specific deductive inference depends on the meaning of the
sentences involved. Hence, a meaning theory should have
something to say on this legitimacy. The classical theory of
meaning does indeed address itself to the concept of valid
inference, which it traditionally defines in a well-known
manner in terms of its central concept truth, namely, as
truth preservation under all assignments of meaning to the
non-logical terms of the sentences involved.
However, when the validity of an inference is defined in
this way, it says little about the legitimacy of a deductive
inference. An inference may be called legitimate if it can
be used to get a ground for an assertion. Although truth
preservation is a necessary condition for an inference to be
legitimate, it is obviously far from sufficient. If the epi-
stemic gap between the premisses and the conclusion is
sufficiently wide, the inference cannot be used legitimately
in a proof in spite of being valid; otherwise one-step proofs
would always suffice.
The point of a deductive inference is to get knowledge
or, what comes to the same here, a conclusive ground for
an assertion. The conclusion of an inference is a judgement
or an assertion made ‘‘because of’’, as one says, some
premisses, which are again judgements or assertions; this
general formulation is meant to include the case where the
premisses are assumptions or are assertions made under
certain assumptions and that the conclusion too is an
assertion made under assumptions. We make an inference
from warranted assertions thinking that the conclusion will
thereby be warranted too; this is the point of an inference,
and the expectation that the inference will achieve this
should be realized if the inference is legitimate, as I have
called it.
The central task in an account of deductive inference is
to explain how and why a deductive inference can be
legitimate in this sense. The first task is to find a condition
on inferences such that when it is satisfied the assertion
occurring as conclusion is conclusively justified given the
further obvious condition that the assertions occurring as
premisses are conclusively justified. Secondly, we have to
explain why one is so justified when these conditions are
satisfied. The normal form of such an explanation is a
meta-logical inference, in this case showing that one is
justified in making the assertion occurring as conclusion
given that the conditions in question are satisfied—nota
bene, this is a meta-logical inference that the philosopher
has to make, not the person who makes the inference; she
should be justified by just making the inferences, given that
the conditions are satisfied and that she knows the meaning
of the sentences involved.
The validity of an inference as commonly defined is
clearly not the right condition. Thus the question arises
whether a truth-conditional meaning theory has sufficient
resources to define an adequate condition and to prove that
it is adequate. The question of how to account for the
phenomenon of deduction is often overlooked, perhaps
because it is taken for granted that an assertion is justified
by proving it. It is indeed reasonable to say of something
deserving to be called a proof that it gives a justification of
its last assertion. But the problem is that we cannot invoke
here a general concept of proof agreed upon for which this
holds. I would suggest that the natural way to define the
concept of proof is to say that a proof is a chain of legit-
imate inferences, in other words, that the notion of proof
depends conceptually on the notion of legitimate inference.
One may respond to the challenge to account for the
phenomenon of deduction by pointing out that although we
have no general notion of proof, we have for various lan-
guages specific notions of formal proofs that have been
proved to be sound in the sense that provable sentences are
true. One may suggest that this amounts to an account of
our deductive practice, since a proof of a sentence A in a
formal system established to be sound constitutes a justi-
fication of the assertion of A. But this is in effect to propose
that a valid inference becomes legitimate because it has
been proved to be valid; in other words, that a conclusion
inferred from warranted premisses becomes warranted
because the inference has been proved to be truth pre-
serving. Since, as has already been argued, the validity of
an inference does not in itself imply its legitimacy, the
suggestion must be understood as saying that it is the
inference together with the soundness proof that justifies
the conclusion of the inference, and this obviously threat-
ens to lead to a regress.
It should thus be clear that the legitimacy of an inference
must depend on the inference itself and the agent’s
Truth as an Epistemic Notion 11
123
understanding of the sentences involved in the inference. It
cannot depend on a proof that provides the agent with
explicit knowledge of the validity of the inference. Instead
of soundness one must show for each valid immediate
inference—by which I mean an inference that cannot be
replaced by a sequence of simpler ones—that (1) a person
who performs the inference, (2) knows the meaning of the
sentences appearing in the inference, and (3) is justified in
asserting the premisses is thereby, without further inference
(from her side), justified in asserting the conclusion.1 Can
one show this when meaning is given in terms of truth-
conditions and the validity of inference is defined in the
traditional way? A positive answer may of course be based
on whatever meta-inferences one finds useful, but the
person who is shown to be justified is not to be assumed to
make any additional inferences than the one in question;
otherwise it is not in virtue of making that inference and
satisfying the conditions (2) and (3) that she is justified in
asserting the conclusion, and we get a regress when asking
why these additional inferences are legitimate.
If the knowledge assumed in condition (2) is explicit
knowledge of what the truth-conditions of the sentences
are, then, obviously, an agent for whom (1)–(3) hold needs
not know the sentence asserted by the conclusion to be
true—she will in general have to make additional infer-
ences to get to know this. However, the knowledge referred
to in condition (2) must in general be assumed to be
implicit, and it is far from clear what such knowledge of
classical truth-conditions amounts to. It would carry too far
to discuss this in detail here. But even if one suggests that
such knowledge consists in the ability to make certain
legitimate inferences, one cannot reasonably claim that it
can be equated with abilities of that kind. The classical
truth-conditional meaning theory is certainly not formu-
lated so as to see how the conditions (1)–(3) above imply
that the person in question is justified in making the
assertion that occurs as conclusion, and it is not easy to see
how it could be so formulated. We have here another
reason to consider a truth-conditional meaning theory
inadequate. The problem is not in itself that truth is biva-
lent or that truth-conditions are knowledge transcendent,
which were features of the truth-conditional meaning the-
ory that Dummett’s argument depended on. The fault that
the present argument finds in a truth-conditional theory of
meaning is rather that the truth-conditions contain too little
information to allow us to infer that a person who knows
the meaning of a sentence also knows what counts as
ground for asserting the sentence.
3 Epistemic Theories of Meaning
The argument sketched in the preceding section, like Dum-
mett’s argument at the end of Sect. 1, makes it natural to look
for a theory of meaning where the sense of a sentence is given
in terms of how it is established as true, in other words, in
terms of what is required to be justified in asserting the
sentence or to have a ground for the assertion. The first
question is then what is to be exactly the central notion, as
Dummett calls it, of such an epistemic theory of meaning.
One may suggest that the proofs appearing in the so-
called BHK-interpretation, where Heyting’s meaning
explanations are reworked in the form of a definition of
what it is to be a proof of a sentence,2 already offer such a
notion. However, it is questionable whether this is the
notion we want. One well-known problem was already
illustrated above: even from an intuitionistic point of view
there are perfectly cogent proofs of disjunctions that do not
contain explicitly a proof of any of the disjuncts, but
something is a proof of a disjunction according to the
BHK-interpretation only if it is built up explicitly from a
proof of one of the disjuncts. For this and many other
reasons, one must distinguish in mathematics between what
has been called canonical and non-canonical proofs3; a
distinction similar to the one between direct and indirect
evidence that is commonly made in empirical discourse.4
One should note that it is not enough to substitute
‘canonical proof’ for ‘proof’ in the BHK-definition of proof.
A canonical proof of a compound sentence must in some
cases be defined in terms of what counts as non-canonical
proofs of the constituents. The two notions of canonical and
non-canonical proof must therefore be defined by a simulta-
neous recursion over the built up of sentences.5 It should also
be noted that the definition proceeds in a way that does not
conform to what I suggested as the natural way of defining the
concept of proof, namely, as a chain of inferences required to
satisfy some condition that makes them legitimate.
In fact, canonical and non-canonical proofs may not be
built up of inferences at all. For instance, a ‘proof’ of an
implication A ? B is simply a function that applied to
proofs of A yields a proof of B, and the ‘proofs’ of A and
B may again be just functions, which may make one doubt
that the notion of proof is really an epistemic one.6
1 It is somewhat inappropriate to say that we assert the conclusion or
a premiss of an inference, since the premisses and the conclusion areassertions (or judgements). This way of speaking is nevertheless often
convenient and is used here; it is even appropriate if we think of the
premisses and conclusion as represented by sentences.
2 Troelstra (1977) and Troelstra and van Dalen (1988). For a recent
comment, see Prawitz (2012).3 Prawitz (1974), Dummett (1977), and Martin-Lof (1984).4 Prawitz (1995).5 Martin-Lof (1987) and Prawitz (1973 and 1987).6 Per Martin-Lof (1998) has drawn the conclusion that it is not. He
and Goran Sundholm (1998) indicate this by referring to proofs so
conceived as ‘‘proof-objects’’, where proof-objects are just truth-
makers in terms of which the meaning of propositions is explained,
much like in realist theories of meaning.
12 D. Prawitz
123
A suggested alternative, inspired by Gentzen’s idea that
his introduction rules can be seen as giving the meaning of
the logical constants, is to define a notion of valid argu-
ment.7 The idea is that an argument is a chain of inferences
and, for it to be valid, there must be functions assigned to
inferences other than introduction inferences by the help of
which the argument can be transformed into one in
canonical form, i.e., ending with an introduction inference.
Although an argument is built up by inferences, the validity
of an argument is again defined by recursion over how
sentences are built up. Given the notion of a valid argument
or canonical proof, a notion of valid inference may be
defined, but this is to reverse the usual conceptual order.
If the notion of valid argument is taken as the central
notion of a theory of meaning, it has to be discussed
whether it satisfies Dummett’s requirement that knowledge
of the meaning of a sentence is manifest in linguistic
behaviour.8 When it comes to the problem raised in the
previous section concerning the legitimacy of inferences, it
is quite clear that none of the approaches considered so far
is satisfactory.
Although what are called proofs in the BHK-interpre-
tation are not proofs as usually conceived, the interpreta-
tion succeeds in some way to explain the constructive
meaning of a sentence. I have suggested (Prawitz 2009,
2012) that since a BHK-proof of a sentence A can be seen
as a construction that one has to be in possession of in order
to be justified in asserting A, it may be looked upon as a
ground for asserting A in terms of which the meaning of
A is explained. If we accept that to be justified in asserting
a sentence A is to know the meaning of A and to be in
possession of such a ground for asserting A, we can fur-
thermore see an inference as not only a speech act but as
containing also an operation by which we aim to transform
given grounds for the premisses to a ground for the con-
clusion. As I have developed in more detail elsewhere
(Prawitz 2009, 2011), an inference can then be defined to
be valid if it does yield a ground for the conclusion when
applied to given grounds for the premisses.
In contrast to the traditional notion of validity in the
sense of truth preservation, the new notion of validity
implies legitimacy. In particular, for any valid inference, it
can be inferred that a person becomes justified in asserting
the conclusion by performing the inference, given that the
conditions (2) to (3) of the previous Sect. 2 are satisfied.
Finally, a proof may now be defined as chain of inferences
that are valid in the new sense.
4 Truth
The epistemic meaning theories sketched in the previous
section do not employ a notion of truth when explaining the
use of sentences for making assertions. Nevertheless, the
notion of truth is needed for several reasons, in particular to
tell what the content of an assertion is. A sentence can be
used with different forces even in mathematics. For
instance, we may make an assumption or a conjecture, or
wonder whether something is the case, in addition to make
an assertion. In a reasonable theory of meaning, the content
should stay the same for the different uses, and should be
possible to equate with what it is for the sentence to be true.
When the meaning of a sentence is explained in terms of
a relation such as P is a proof of the sentence A or a is a
ground for asserting A, we are told when it is right to assert
a sentence. But do we get to know the content of the
assertion, what is asserted, or what it is for the asserted
sentence to be true? Naturally, one wants to say in such
meaning theories that a sentence A is true if, and only if,
there is a proof of A or a ground for asserting A. But the
problem is how this existence is to be understood. Does it
mean that one has a proof or ground at hand, or does it
mean merely that there exists one in a non-temporal sense,
a proof which one may perhaps find one day, but which
may also remain unknown forever?
In his account of intuitionism referred to in the above,
Heyting used the term ‘‘true’’ only in connection with the
classical conception of an assertion, when contrasted with
the intuitionistic one. But Heyting does not leave the reader
in doubt about how he understands an intuitionistic asser-
tion: its content is that the construction intended by the
asserted proposition is realized, or, as he also puts it, that
one knows how to prove the proposition. He explicitly
remarks that an intuitionistic assertion has empirical con-
tent: a certain construction or proof has been found. He
could as well have said that this is what it is for a sentence
to be true; that he does not use the term true in this context
is a terminological matter.
Dummett holds the notion of truth to be important even
in a theory of meaning that takes verification or proof as its
central notion, but says that ‘‘it is far from being a trivial
matter how the notion of truth… should be explained’’
(Dummett 1976, 116); in fact, his position on the issue has
varied. He sometimes seems to assign a much weaker
content to assertions than Heyting, saying for instance:
‘‘the content of an assertion is that the statement asserted
has been, or is capable of being, verified’’ (ibid., 117; my
italics). Similarly, he has said that it is possible for a
7 Prawitz (1973), Dummett (1991, ch. 10) and Prawitz (2006).8 Doubts concerning this have been expressed for instance by Peter
Pagin (2009). As argued by Williamson (2003), one has to beware of
identifying a person’s implicit knowledge of the meaning of a
sentence with her actual use of an inference rule or of a form of
argument or proof; the knowledge is rather a question of knowing the
rule or knowing what counts as a valid argument or proof in the
language in question, as pointed out by Cozzo (2008).
Truth as an Epistemic Notion 13
123
constructivist to equate truth with provable and ‘‘to agree
with a platonist that a mathematical statement, if true, is
timelessly true’’. But he immediately qualifies this by
saying that to remain faithful to the basic principles of
intuitionism, one must not interpret the provability of a
sentence A as ‘‘independently of our knowledge, there
exists a proof of A’’. To this he adds: ‘‘We can prove A’
must be understood as being rendered true only by our
actually proving A.’’ (Dummett 1977, 19). Thus, after all,
Dummett takes here the same position as Heyting.
Everyone agrees that to be right in asserting a sentence
one should know a proof or ground. What Heyting and
Dummett do here is to identify the content of an assertion,
or the truth of a sentence, with the condition for asserting
the sentence. This has obviously strange consequences,
especially when a sentence is used for other purposes than
making assertions.9 To illustrate one such consequence,
consider a knowledgeable mathematician who wonders
whether every even number greater than 2 is the sum of
two primes, or conjectures, like Goldbach, that it is so. He
does not wonder whether this has been proved, nor does he
conjecture that it has been proved—he knows that it has
not. Obviously, the wonder or conjecture concerns the truth
of the sentence in some other sense of truth.
Dummett pays attention to one strange consequence of
identifying truth with the actual existence of a proof,
namely that it is in conflict with his position presented in
Sect. 1, according to which numerical equations or, more
generally, decidable arithmetical sentences have determi-
nate truth-values even when we do not know them. Such a
sentence may thus be true although no proof of it is at hand,
contrary to the identification of truth with the actual exis-
tence of a proof. To cope with this problem he suggests a
somewhat weaker notion of truth, according to which one
allows as true a sentence for which we have either a proof
or a method that will in fact yield a proof, if applied, ‘‘even
if we do not know this’’ (Dummett 1998, 123). Although
this ad hoc proposal solves the particular problems it is
intended to solve such as compatibility with the distribu-
tivity of truth over disjunction, it leaves the notion of truth
tensed with its strange consequences.
The fatal flaw in the identification of truth with the
actual existence of a proof (or method for finding one)
shows itself already in the fact that it is incoherent with any
reasonable account of the validity of inference. This comes
out most clearly if we consider what it is to make an
assumption in a proof or draw a conclusion from the
assertion of a sentence held to be true under assumptions.
In any reasonable account of this, one cannot take truth to
mean something stronger than the existence of a proof in
the weakest possible sense of existence. If we strengthened
this to actual existence of a proof, we would put more
content into the assumption or premiss than it should have.
In pure mathematics, we get the strange result that the
content would include empirical information of a temporal
kind that we cannot make any use of, but as soon as we go
outside of pure mathematics, the absurd result would be
that given the truth of what is normally taken to be a purely
mathematical sentence, a lot of empirical statements about
our present knowledge and abilities would also be true. It is
then hard to avoid that the theory would accept inferences
that everyone would reject in practice.
Sometimes it is instead against the identification of truth
with the tenseless existence of a proof that a charge of
incoherence is brought. It is held to abandon the anti-
realistic position that is supposed to underlie an epistemic
theory of meaning, and is said to be ‘‘totally unfaithful to
the whole spirit of intuitionism’’ (Raatikainen 2004, 141).
As for the unfaithfulness, it should first be noted that to
make such a charge in the present context is to misunder-
stand completely the dialectical situation.
An epistemic theory of meaning developed in response
to Dummett’s argument against a truth-conditional theory
of meaning referred to in Sect. 1 has in no way an under-
lying anti-realistic assumption. On the contrary, as we saw,
Dummett’s meaning-theoretical investigation is meant to
settle the debate between realism and anti-realism, and
accordingly, its starting point is required explicitly to be
neutral with respect to realism versus anti-realism. When it
turns out, following his argument, that one has to replace a
classical truth-functional theory of meaning with a theory
that takes proof as its central notion, it still remains to
discuss how the notion of truth is to be defined, and first
then, when such a notion has been found, it is appropriate
to ask, without commitments, whether the theory supports
realism or anti-realism.10 As for the epistemic theory of the
previous section that took a notion of ground as central, it
has not an underlying anti-realistic starting point either; its
motivation was that classical truth-conditions contain too
little information.
As said in the introduction, there are of course meaning
explanations in epistemic terms that have as starting point a
rejection of a realist picture of the domain of discourse. In
such cases it might seem inconsistent to adopt the view that
proofs, verifications, or grounds for assertions concerning
objects in this domain exist in a tenseless sense indepen-
dently of whether they have been constructed or are known
by us. However, we should then recall again Dummett’s
argument that in spite of having started from such an anti-
9 I have discussed some of them elsewhere (e.g. Prawitz 1998). For a
recent survey, see Raatikainen (2004).
10 Dummett (1987, 286) says that he is happy for any outcome of the
investigation: if it supports realism, ‘‘we shall have discovered the
true justification of realism’’.
14 D. Prawitz
123
realist premiss concerning the natural numbers, one must
be very hard-headed not to admit that a decidable arith-
metical sentence A has a truth-value even if we have not
performed the relevant computations that settles which
truth-value it is. Hence, for such a sentence A, ‘‘A is true’’
must be equated, not with ‘‘A has been proved’’, but with
‘‘A is provable’’, or ‘‘there is a proof of A’’. If one wants to
maintain that for undecidable sentences it is nevertheless
inconsistent with intuitionism to take their truth to consist
in the mere tenseless existence of a proof, a new argument
is again needed; one cannot just refer to the view of the
natural numbers as our constructions.
There may be such an argument, and this is what
Dummett maintains. His argument may be put simply as
this: if it is assumed that a proof of an undecidable sentence
A exists already before it has been found, then either there
is a proof of A or there is not, and since the non-existence is
the same as a proof of the negation of A, the law of
bivalence and realism follow (cf. Dummett 1987, 285–286
and 1998). The argument seems to rely on the idea that if
the assumption that proofs exist already before they have
been found is to have any substance, it must be given a
realist interpretation, which amounts to saying that either
there is a proof of A or there is not.
However, the criticism against the equation of truth with
the actual existence of a proof or ground is first of all that
truth then becomes tensed with the consequence that
A becomes true when A is proved or a ground is found. To
raise this criticism is not to say that when a proof is found it
existed already. The point is rather that tense should be
dropped when speaking about truth and the existence of
proofs or grounds, as it usually is when we speak about the
existence of numbers. Even a constructivist can use ‘is’
without tense when saying that there is a number with a
certain property. Such a use does not bring with it a
commitment to holding that for any property, either there is
a number with the property or there is not.
A constructivist or intuitionist thinks that the system of
natural numbers and the logical constants are our inventions,
and he or she may perhaps say, if tense is to be used at all in
this connection, that numbers and proofs, and thereby
properties of natural numbers, come into existence when we
lay down the rules of computation and the interpretations of
the logical constants. In that sense it could be said that a proof
had already an existence before it was hit upon, but this is of
course not to be interpreted realistically.
Whether it is right to say that each number either has a
certain property or lacks it, or that either there is a proof of
a sentence A or there is not, must depend on what is meant
by ‘there is’, ‘or’, and ‘not’. If we understand these locu-
tions in a constructivist manner, then it is not right to say
for any sentence that it has a proof or it has not. We have
been able to show for some sentences that either they have
a proof or they have not; but for other sentences we have
not, and for such a sentence we simply lack any justifica-
tion for asserting that either it has a proof or it has not.
Similarly, since we know no method for always finding a
proof or ground when there exists one, we have no justi-
fication for saying that if a sentence is true, then it is
possible in principle to find a proof or ground for asserting
the sentence. The truth of a sentence should therefore not
be equated with provability or the possibility of finding a
ground for asserting the sentence but simply with the
tenseless existence of a proof or ground.
Acknowledgments I am grateful to professor Cesare Cozzo for
constructive comments after his reading of an earlier version of the
paper.
References
Cozzo C (2008) On the copernican turn in semantics. Theoria
74:295–317
Dummett M (1975) The philosophical basis of intuitionistic logic. In:
Rose HE et al (eds) Logic colloquium ‘73. North-Holland
Publishing Company, Amsterdam, pp 5–40
Dummett M (1976) What is a theory of meaning? (II). In: Evans G,
McDowell J (eds) Truth and meaning. Clarendon Press, Oxford
Dummett M (1977) Elements of intuitionism. Clarendon Press,
Oxford
Dummett M (1987) Reply to Dag Prawitz. In: Taylor B (ed) Michael
Dummett—Contributions to philosophy. Martinus Nijhoff Pub-
lishers, Dordrecht, pp 281–286
Dummett M (1991) The logical basis of metaphysics. Duckworth,
London
Dummett M (1998) Truth from the constructive standpoint. Theoria
64:122–138
Heyting A (1930) Sur la logique intuitionniste. Academie Royale de
Belgique, Bulletin de la Classe des Sciences 16:957–963
Heyting A (1931) Die intuitionistische Grundlegung der Mathematik.
Erkenntnis 2:106–115
Martin-Lof P (1984) Intuitionistic type theory. Bibliopolis, Napoli
Martin-Lof P (1987) Truth of a proposition, evidence of a judgement,
validity of a proof. Synthese 73:407–420
Martin-Lof P (1998) Truth and knowability: on the principles C and K
of Michael Dummett. In: Dales HG, Oliveri G (eds) Truth in
mathematics. Clarendon Press, Oxford, pp 105–114
Pagin P (2009) Compositionality, understanding, and proofs. Mind
118:713–737
Prawitz D (1973) Towards a foundation of a general proof theory. In:
Suppes P (ed) Logic, methodology, and philosophy of science IV.
North-Holland Publishing Company, Amsterdam, pp 225–250
Prawitz D (1974) On the idea of a general proof theory. Synthese
27:63–77
Prawitz D (1987) Dummett on a theory of meaning and its impact on
logic. In: Taylor B (ed) Michael Dummett—Contributions to
philosophy. Nijhoff, Dordrecht, pp 117–165
Prawitz D (1995) Quine and verificationism. Inquiry 37:487–494
Prawitz D (1998) Comments on the papers. Theoria 64:283–337
Prawitz D (2006) Meaning approached via proofs. Synthese 148:
507–524
Prawitz D (2009) Inference and knowledge. In: Pelis M (ed) The
Logica yearbook 2008. College Publications, King’s College
London, London, pp 175–192
Truth as an Epistemic Notion 15
123
Prawitz D (2011) The epistemic significance of valid inference.
Synthese (forthcoming)
Prawitz D (2012) Truth and proof in intutionism (forthcoming)
Raatikainen P (2004) Conceptions of truth in intuitionism. Hist Philos
Log 25:131–145
Sundholm G (1998) Proofs as acts and proofs as objects. Some
questions for Dag Prawitz. Theoria 64:187–216
Troelstra AS (1977) Aspects of constructive mathematics. In: Barwise
J (ed) Handbook of mathematical logic. North-Holland Publish-
ing Company, Amsterdam, pp 973–1052
Troelstra, van Dalen (1988) Constructivism in mathematics. North-
Holland Publishing Company, Amsterdam
Williamson T (2003) Understanding and inference. Proc Aristot Soc
Suppl 77:249–292
16 D. Prawitz
123