truth-tellers
in bradwardine’s
theory of truth
Greg Restall∗
Philosophy Department
The University of Melbourne
restall@unimelb.edu.au
september 29, 2008
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Abstract: Stephen Read’s work on Bradwardine’s theory of truth is some
of the most exciting work on truth and insolubilia in recent years [5, 6].
Read brings together modern tools of formal logic and Bradwardine’s the-
ory of signification to show that medieval distinctions can give great in-
sight into the behaviour of semantic concepts such as truth. In a number
of papers, I have developed a model theory for Bradwardine’s account of
truth [7, 8]. This model theory has distinctive features: it serves up models
in which every declarative object (any object signifying anything) signi-
fies its own truth. This leads to a puzzle: there is a good argument to
the effect that if anything signifies its own truth—if anything is a ‘truth-
teller’—it is false [6], and that this feature of a theory of truth—that not
every declarative need signify its own truth—is what distinguishes Brad-
wardine’s account from Buridan’s [5, 6]. What are we to make of this? If
the argument fails, what distinguishes problematic truth-tellers (such as
a sentence that explicitly says of itself that it is true) from benign truth
tellers? It is my task in this paper to explain this distinction, and to clarify
the behaviour of truth-tellers, given my the contemporary formal treat-
ment of Bradwardine’s account of signification.
1
bradwardine’s theory
Bradwardine’s theory of truth, as set out by Stephen Read in a series of pa-
pers [5, 6], defines truth in terms of a prior notion of signification. This notion
∗Thanks to Catarina Dutilh Novaes and to the audience at the 1st gpmr Conference in Me-
dieval and Applied Logic for feedback on some of the ideas in this paper, and especially to Stephen
Read for introducing me to Bradwardine’s theory of truth, and for confronting me with the puzzle
in this paper. The comments of anonymous referees also helped me clarify my thinking and ex-
pression of this material. Any remaining errors and infelicities are, of course, mine. This research
is supported by the Australian Research Council, through grant dp0343388, and the Penguin Café
Orchestra.
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is expressed grammatically by neither a predicate nor an operator. The concept
is expressed in claims of the form ‘x says that p’ or ‘x signifies that p’. The
phrase ‘says that’ is syntactically and semantically a hybrid. To form a sentence,
we can substitute a name or referring expression in place of ‘x’ and another sen-
tence in place of ‘p.’ Expressed by a ‘connecticate’ uniting a singular term with a
sentence, signification is the core notion from which truth is defined. I will write
the connecticate of signification with the simple infix colon “:”. Whenever t is a
singular term and p is a sentence,
t : p
is another sentence, to be read ‘t signifies that p’, or simply just that ‘t says that
p.’ According to Read’s analysis, Bradwardine uses signification to define the
truth predicate: t is true if and only if it signifies something, and everything it
signifies is the case. It is false if it signifies something that is not the case. So
truth and falsity are defined notions, where the definitions utilise signification
and what we have come to call propositional quantification.
At this point we must clarify a distinction between Bradwardine’s terminol-
ogy and modern vocabulary: quoting Read [5, page 191] we have
definition 1: A true proposition is an utterance signifying only as
things are.
definition 2: A false proposition is an utterance signifying other
than things are.
For Read and for Bradwardine, what is called a ‘proposition’ is what is denoted
by the referring expression in the first part of the claim of the form ‘x : p.’ The
proposition is the utterance or sentence or other object of which we are pred-
icating truth. To confuse matters, we now use ‘proposition’ to describe quan-
tification into the second position, in which the sentence is mentioned but not
used. To keep matters as clear and precise as I can, I will describe the objects
of which we will predicate truth and falsity sentences (thinking primarily of to-
kens, and not types, though we may sometimes predicate truth of types if the
sentences do not vary too much in significance from context to context), or ut-
terances or other such things. I will avoid talking about propositions as much as
possible in what follows, except when talking of what is now called propositional
quantification. Using this vocabulary, then, definitions 1 and 2 become:
• T s is defined as (∃p)(s : p) & (∀p)(s : p → p)
• Fs is defined as (∃p)(s : p & ¬p)
The object s is true when it signifies something, and all it signifies is the case,
and s is false, when it signifies something that is not the case. (In the formal
syntax ‘:’ binds more tightly than ‘&’ so ‘(∃p)(s : p & ¬p)’ is not to be read
as ‘(∃p)s : (p & ¬p)’.) Note that these formal definitions do not utilise any
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notion of being “the case”—there is no separate notion of truth playing a role
in Bradwardine’s own definitions of the notion of truth, except for any notion of
truth implicit in treating these marks as making assertions. When rendered into
English prose, to be sure, we use predications such as ‘is the case’ or ‘holds’ when
reading propositionally quantified expressions, but this need not mean that we
must use a second notion of truth to define the first.1
According to Read, Bradwardine’s account of truth and signification relies on
what we call Bradwardine’s axiom:
definition [bradwardine’s axiom] Every proposition signifies or means con-
tingently or necessarily everything which follows from it contingently or nec-
essarily [5, p. 209].
We could formalise the condition like this:
If t : p then if (if p then q) then t : q
Rearranging, we might have another formulation
If (if p then q) then (if t : p then t : q).
The crucial issue in understanding Bradwardine’s axiom is what form of con-
ditional expression might be used in formulating it. We can consider material
conditionals
If (p ⊃ q) then (t : p ⊃ t : q)
where we can infer p ⊃ q either from ¬p or from q . These conditionals
are very weak, and it is most likely that they are not the right way to get to the
core of Bradwardine’s notion of truth. After all, we have
If ¬p then (t : p ⊃ t : q).
If q then (t : p ⊃ t : q).
which together almost trivialise signification. First, if t : p and ¬p, then t : q—so
if t signifies something (namely p ) such that ¬p, then t signifies everything.
So anything signifying something not the case signifies everything and any-
thing. Second, if q, then if t signifies that p (if t signifies something) then t
signifies that q. That is, anything that signifies something signifies everything
that is the case. We therefore would have three options: (1) an object can sig-
nify nothing, or (2) if it signifies something, it signifies everything that is the
case—and (3) if it also signifies something that is not the case, it signifies ev-
erything. An objects status with regard to signification can be reduced to two
pieces of information. Step 1: Is it declarative?—does it signify anything? If
1 Whether this is a mark against propositional quantification is a bone of contention. I think
that a coherent story—which neither takes a notion of propositional truth as primitive, nor con-
fuses use and mention of propositional expressions—is possible, but it is not my place to argue
this here [1, 2].
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not, it signifies nothing. If so, then it signifies at least everything that is the
case. Step 2: Does it signify anything that is not the case? If so, it is false, and
it signifies everything. If not, it is true. So, everything is either nondeclarative,
or if it is declarative, it is either true or false. If Bradwardine’s axiom is read in
a material fashion, signification collapses into three cases: nondeclarativeness,
truth and falsity.
So, if we want a richer analysis of signification, we must explore different
options. One different option is to consider modal conditionals:
If (p ⇒ q) then (t : p ⇒ t : q)
where p ⇒ q means that p entails q . This kind of reading is considered in
“Modal Models for Bradwardine’s Theory of Truth” [8]. I showed there that you
could model Bradwardine’s theory in a ‘possible worlds’ model by representing
taking signification as another modal operator—we treat t : p in the same vein
as
p , where the necessity is relatvised to the object doing the signifying. In
this way, if p entails q , then anything signifying p signifies q . If p is
t-necessary, then q is also t-necessary.
The details of the model theory are not important, for the puzzle I will dis-
cuss here is a problem even in the simpler non-modal case, and its diagnosis is
independent of modal matters. For more detail on the modal reading of Brad-
wardine’s axiom, see my earlier papers [7, 8].
The crucial feature of Bradwardine’s theory of truth is found in its analysis of
the insolubilia: a liar statement, for example, is something that signifies its own
untruth. It is an object l such that l : ¬Tl. What can we say about l?
Bradwardine’s reasoning concerning liars is straightforward. We may start
by asking: is l true? That is, do we have T l? T l is, by definition, (∃p)(l : p) &
(∀p)(l : p → p). For the first conjunct, (∃p)(l : p), we can point to l : ¬T l. Here
is something that l signifies, by assumption. What about the second conjunct?
Is it the case that (∀p)(l : p → p)? Well, if this is he case, then in particular since
l : ¬T l, we could conclude ¬T l. So, we have shown that if T l, it follows that ¬T l.
By reductio ad absurdum, we may conclude ¬Tl.
So, we have proved that a liar statement is not true. Does it follow that
it is also true? The fact that we find the reasoning to T l compelling is what
makes liar paradox a genuine paradox. We are tempted to reason as follows: we
have proved that the l is not true. This is just what the liar says, so it must
be true. This last move is the step that Bradwardine resists, according to Read.
Yes, something that the liar says is not the case. It does not follow that l is true,
for this would require concluding that everything that the liar says is the case:
T l entails (∀p)(l : p → p). Can we prove this? No, we cannot. Only a further
assumption, such that the liar signifies exactly one thing, or that everything that
the liar signifies is entailed by ¬Tl, or some other such conclusion would help
us derive the claim that the liar is also true. In the absence of this assumption,
we may not conclude that the liar is true. It is simply false. It must signify
something that is not the case.
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What does the liar signify that is not the case? According to Read, Brad-
wardine’s response is that the liar signified its own truth: that l : T l. As far as
I can see, this does not follow from his explicit assumptions, strictly speaking.
I have argued elsewhere [7] that, on Bradwardine’s account—as understood by
Read—some liar statements might signify their own truth, others might not.
What they have in common is that they signify their own untruth, and because
of that, each liar statement signifies something that is not the case, and hence,
are untrue. However, that ‘something else’ can vary from one liar statement to
another.
Nonetheless, it is quite possible to construct models of this formalisation of
Bradwardine’s theory in which not only do insolubilia such as liar statements
say of themselves that they are true, but in which all declarative objects declare
their own truth. The construction of the “Modal Models” paper [8] delivers
such models. The fact that Bradwardine’s theory has models in which every
declarative object declares its own truth gives rise to the puzzle to be considered
in this paper.
2
the puzzle of truth-tellers
Consider the following statement:
2 + 2 = 4 and (r) is true
(r)
Taking the signification of this sentence at face value, we may conclude that
r : (2 + 2 = 4 & T r), and hence, since (r) signifies anything that follows from
what (r) signifies, we have
r : 2 + 2 = 4
r : T r
The statement (r) signifies that 2 + 2 = 4 and it also signifies its own truth. Now,
what can we say about (r)? Is it true, or is it false? It seems that the theory we
have seen so far does not tell us one way or another. When we inquire after (r)’s
truth, we are forced to ask: is 2 + 2 = 4? Hopefully our arithmetical knowledge
extends that far, and we can answer in the affirmative. Now we ask: is (r) true?
To answer this, we are left back where we started. It seems that it could be
true — at least, the assumption that (r) is true does not conflict with anything
that we have seen so far. If (r) is true, then everything that (r) says is the case.
(r) says that 2 + 2 = 4, and that (r) is true. This is consistent and coherent. But
then, it seems that (r) could equally well be false — at least, the assumption that
(r) is false does not conflict with anything that we have seen so far. If (r) is false,
then something that (r) says is not the case. (r) says that 2 + 2 = 4, and that (r)
is true. This is consistent and coherent — that (r) is true is something that (r)
says that is not the case.
So, for all we know, the truth of (r), and its falsity are both possible. Do
we have any reason to prefer one over another? Stephen Read thinks that we
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do [6]. Read takes (r) to be false, since there is nothing to make it be true. The
cycle of checking continues for ever, and there is no way to ground the truth of
(r) in anything outside it. So, it is false.
Now consider another sentence
2 + 2 = 4
(s)
The sentence (s) says that 2 + 2 = 4. Does the sentence (s) also signify its
own truth? In my model theory for Bradwardine’s account of truth, everything
declarative says of itself that it is true. Could this be the case? If it were the case,
for (s) we would have
s : 2 + 2 = 4
s : T s
This looks familiar. We have said the same sort thing concerning what (s) says
that we have said concerning what (r) says. Does not the same reasoning apply?
Should we not also conclude that (s) is false, for in the process to evaluate the
truth of (s) we can ask: does 2 + 2 = 4? Yes. Is (s) true? Well, for that we need
to check everything that (s) says. What does it say? Well, it says that 2 + 2 = 4
(which is the case), and it says that (s) is true. Well, is that the case? Is (s) true?
. . . The process never stops.
It seems that the reasoning, if it is good in the case of (r), is also good in the
case of (s). But (s) was nothing special. Any true claim would have done just as
well. In my models for Bradwardine’s theory, every declarative object is a truth
teller, and it seems as if—if Read is right—we should say that they are all false.
This is the puzzle: on the one hand, Read’s argument in ‘Symmetry and
Paradox’ [6] explains why truth-tellers should be taken to be false. On the other
hand, there is a model construction that gives us interpretations in which every
declarative tells its own truth. Is this view of declaratives consistent with Brad-
wardine’s position? It would seem to follow from these premises that declara-
tives are all false. But that is absurd, since there are some truths. So, some as-
sumption must go: either we are to reject the thesis that non-paradoxical declar-
atives can declare their own truth (and hence, reject my model construction), or
we are to reject Read’s reasoning, or finally, we are to reject our formalisation of
Bradwardine’s theory of truth. Which of these is to be rejected?2
3
what goes on in the model?
In this section I will examine the model construction in a little more detail, so
we can understand how this construction treats sentences like (s) and those like
(r) differently. In this way, we will be able to use that construction to provide
some insight into the different ways that sentences can act as truth-tellers. This
2 Thanks to Stephen Read for presenting this puzzle over a long lunch in Bonn after the close
of the gpmr conference.
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will require attention to a little more technical detail, in order to clearly see the
difference between these different truth-tellers. I will attempt to involve just the
amount of technicality required to make the philosophical point, and no more.
For the full details of the model construction, I refer you elsewhere [8]. Then in
the final section of the paper I will attempt to apply what we have learned from
that construction, and to point to further philosophical questions concerning
signification and Bradwardine’s account of truth.
To make the reasoning precise, I will fix on particular examples of the sen-
tences with the features of our (r) and (s) from the previous section. The model
construction takes any formal theory with a possible worlds model and propo-
sitional quantification, and adds the logic of signification in such a way as to
satisfy Bradwardine’s axiom. It also ensures that, if the theory in question pro-
vides a means of quotation, then the object A says that A. The condition that
A : A ensures that the truth-elimination axiom holds
T A ⊃ A
since T A entails (∀p)( A : p ⊃ p), and A : A delivers A as a conclusion.
The converse—A ⊃ T A —does not, in general hold. A liar sentence l is not
true. It doesn’t follow that it’s true that it’s not true.
Now, how can we construct sentences with the features of (r) and (s)? Since
(s) is an arithmetic sentence, let our base model for our construction be a model
of arithmetic, and let our language be the language of arithmetic, including +,
×, 0, 1 and the usual language of first-order predicate logic, supplemented with
signification and propositional quantification (and hence, Bradwardine’s truth
and falsity predicates which are definable in terms of signification and proposi-
tional quantification).
The model construction is iterative, in the manner familiar in current model-
theoretic treatments of the paradoxes [3, 4, 9]. At every stage of the model con-
struction, let the model interpret arithmetic in a completely standard fashion.
The domain of the model is the collection of natural numbers, and addition and
multiplication are given their natural interpretation. For signification, we need
to just choose an encoding of the sentences of the language, in the manner of
Gödel. For each sentence, we need some effective and natural way to choose a
number which encodes it—its Gödel number. If the Gödel number of a formula
A is the number n, then we will let A be a term in the language that denotes
that number (1 + 1 + · · · + 1, the n-fold addition of 1 will do). The only thing to
vary from one stage to the next is the interpretation of signification.
To interpret signification, at stage 0 of the construction, we will say that the
numbers that are not Gödel numbers of any formula signify nothing. (They are
non-declarative). The numbers that are Gödel numbers—at the start—signify
everything (they are false). At the start of the construction, nothing is true,
according to the model’s ‘internal’ account of truth.
At stage n + 1, the interpretation of signification is defined in terms of the
behaviour of the model at stage n. The crucial notion is that of a variant of
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a stage. At stage 1 we know, for example, that no matter how we interpret
signification, the sentence 2 + 2 = 4 is true, and the sentence (∃x)(x + 1 = 0)
is false. All purely arithmetic sentences get their appropriate interpretation at
stage 0 no matter how signification is interpreted. So, we can add, for example
2 + 2 = 4
: 2 + 2 = 4 and rule out 2 + 2 = 4
: q for any arithmetical
statement q that is false. However, signification is not yet completely fixed for
other sentences. The construction discused in ‘Modal Models’ [8] defines the
inductive procedure by which we find, at each stage the class of variants of a
model at stage n (which allow signification to vary wherever it is not yet fixed
by what occurs at the previous stage), and then define significantion at stage
n + 1 to be defined by what holds in every variant at stage n. Signification
for arithmetic statements is settled at stage 1, signification for those statements
expressing signification of arithmetic statements is settled at stage 2, and so on.3
The model construction ensures that if, at some stage of the development, A
is settled (A is the case, and furthermore, it will remain the case in each of the
other models we construct, no matter how we further refine signification), then
at the next stage, T A is be settled too. The process will eventually come to a
halt (a fixed point, at which no more is added to signification), and at this stage,
many claims of the form A ⊃ T A are true. For sentences A that are grounded,
A ⊃ T A holds.4
Now we can choose our sentences to do the job of (r) and (s). The sentence S is
straightforward. We will let it be the sentence5
2 + 2 = 4
For R, on the other hand, we must do more work. Gödel has shown us that
in any theory of arithmetic strong enough, predicates have fixed points in the
following way. If φ is a predicate, then there is some sentence φ such that we
can prove in an arithmetical theory6 that Q ↔ φ Q . If you like, it is a sentence
which “says of itself” that it is P. I write “says of itself” in scare quotes, as
this does not feature signification, which is our official account of what things
say. However, given Bradwardine’s axiom, this untutored notion of signification
becomes the literal truth. If it is a theorem that Q ↔ φ Q , then since we
have Q : Q by Bradwardine’s axiom we can conclude that Q : φ Q , so our
official theory of signification indeed tells us that Q says of itself that it is φ.
3 We then define limit stages as settling whatever is settled at any stage up to that limit, and a
fixed point is found as usual, at some countable stage [8].
4 See “Modal Models” [8] for a precise definition of the sense of groundedness in play here. It is
not the same as being settled. Groundedness as defined there is a syntactic property of sentences.
Settledness is relative to a model and a stage.
5 Actually, in the vocabulary I chose, there is no sentence 2 + 2 = 4, literally speaking, since
there are no terms 2 and 4. Literally speaking, the sentence can be (1+1)+(1+1) = ((1+1)+1)+1,
taking ‘2’ as shorthand for ‘1 + 1’, ‘3’ as shorthand for ‘2 + 1’ which is shorthand for ‘(1 + 1) + 1,’
‘4’ as shorthand for ‘3 + 1’, which is shorthand for . . . etc.
6 Which theory? Robinson’s Arithmetic will do. It suffices to represent recursive functions: in
particular, the diagonal function.
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So, to do the job of (r) in our intuitive reflections, we want something that
says that 2 + 2 = 4 and that it is true. What will do this? We apply Gödel’s trick
to find a sentence R such that in an arithmetic theory we can prove
R ↔ (2 + 2 = 4 & T R )
using the general ‘predicate’ (2 + 2 = 4 & T x) as the context for the fixed point.
Gödel proves for us that there is a fixed point and a sentence with this property.
In the naïve sense, R says that 2 + 2 = 4, and that T R . In our model all
truths of arithmetic are true, so Gödel’s reasoning applies, and so in our model,
R ↔ (2 + 2 = 4 & T R ) holds. And so, the construction, as signification is
refined and made more precise, R ↔ (2 + 2 = 4 & T R ) will continue to hold,
as this process merely refines signification, and leaves the arithmetic part of
the model untouched. As our reasoning above has shown, it follows that R :
(2 + 2 = 4 & T R ) too.
Let us pause here: we have a formal theory (arithmetic, augmented by
propositional quantification and signification), and sentences R and S exhibit-
ing the behaviour that ought to give rise to the puzzle of truth-tellers. Is there
any difference between a sentence S which merely happens to signify its own
truth, and a sentence R which does so explicitly?
What does the model construction say? What happens to R and S in the
model construction? Do they share the same fate, or do they differ? At this
point I will not go through all of the technical details, but give the gist of the
process. In the original model S holds, since it is a purely arithmetic sentence
(it does not feature signification or anything defined in terms of signification,
such as T or F). So, not only does S hold: it is settled. It would continue to
hold however signification is refined. This means that at the next stage of the
construction T S will be settled as true too.
R fares very differently. At the first stage of the construction T fails of every
object whatsoever. So, T R does not hold, and in the model (2 + 2 = 4 & T R )
does not hold. Since, in the first model R ↔ (2 + 2 = 4 & T R ) holds, we can
conclude that R does not hold either.
However, R is not settled, for if we vary signification in the model a little bit
to make T R hold, then R would hold too (the other conjunct 2 + 2 = 4 is no
problem—it is satisfied at every stage). So, R does not hold at the first stage,
but it is not settled as false either. So, we do not vary signification with respect
to R at this stage, and T R is not made true at the next stage. But as with the
first stage, so with the second. The situation here is no different. T R does not
hold, so R does not hold. But it would hold if we refined signification for R
a little. But since R is not settled, we do not add T R yet. And so on. In the
process of model refinement, S comes out as true, but R does not.
But look! From stage two of the process, not only do S and T S hold: since
T S is now the case in our model from this point on, we have S → T S , and
since S : S and we have Bradwardine’s axiom, we can conclude S : T S . In
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our construction S is also a truth-teller. Yet S came out to be true and R did not.
The model treats S and R very differently.
4
what does this mean?
What can we say about signification and truth-telling with what we have learned?
The first consequence is that you cannot tell simply say that if x : T x then x is
paradoxical. Truth-telling need not be a sign of paradox, even for a proponent
of Bradwardine’s axiom. It is true that if x is true, then everything that x says is
the case, and hence, that x is true, and so, if x is true, it must be the case that x is
true. But that fact is unsurprising. It is always the case that if x is true, it must
be the case that x is true. (In general, if p then p, no matter what p you choose!)
It is another matter to say that if x : T x then if x is true you must first convince
yourself that x is true, and that it is true that x is true, etc. Here is an analogy: if
y : p then since p implies p & p, then y : (p & p) too. Now, to check whether y is
true or not, do we need to check p and then p & p, and then (p & p) & (p & p)
. . . ad infinitum? No, we do not. Not all unfoldings of consequences such as
these lead to the type of ‘ungroundedness’ found in sentences that explicitly say
of themselves that they are true.
What can we say about the difference between benign and benighted re-
gresses? This must be a matter for further reflection: my contribution to the
discussion from this work is that—at least in the cases I have been discussing—
ungroundedness can sometimes be a syntactic matter. More precisely, you can-
not tell that something is grounded or not just by being told some of the things
that this thing says. If I tell you that z : 2 + 2 = 4 and that z : T z , I cannot
conclude that z suffers from a benighted regress, for we have seen two objects—
(r) and (s)—both satisfying this condition, one of which is is true, and the other
not. What matters, at least as far as the models we have been discussing is con-
cerned, is not simply whether or not something says of itself that it is true, but
the way it says of itself that it is true. We can verify to our satisfaction that (s)
2 + 2 = 4
(s)
is true by verifying that 2 and 2 add to 4. Even though (s) says that (s) is true,
we do not need to independently verify that before concluding that (s) is true.
But the same cannot be said for (r).
2 + 2 = 4 and (r) is true
(r)
Here, the syntax demands that we consider the matter of the truth of (r) in the
process of evaluating it.
If this analysis is right, then it seems that we cannot read off whether or
not a sentence is paradoxical simply in terms of what it says. What can we say
about benighted regress? If we are to say that some sentences are properly ‘un-
grounded’, in terms of what is groundedness defined? Is it purely syntactical?
Greg Restall, restall@unimelb.edu.au
september 29, 2008
Version 0.99

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Document Outline

• bradwardine's theory
• the puzzle of truth-tellers
• what goes on in the model?
• what does this mean?

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