What should a theory do with semantic paradoxes? Barker and Kripke on paradoxes
Federico Matías Pailos-C.O.N.I.C.E.T./U.B.A.
Stephen Barker maintains that the best semantic theory available is his expresivist
speech acts theory [STA]. One of the main reasons in support of this claim is that this theory (in
particular with the aid of one of its sub-theories, the expresivist theory of truth) [ET], leads to
the solution of many semantic paradoxes. The classic solutions to them are all unsatisfactory,
because they are all based in the same traditional presuppositions that lead to the paradoxes: the
sense/force distinction, the defense of propositions as contents of propositional attitudes, the
defense of propositions or beliefs as primary truth-bearers. Another benefit of STA is that, being
successful, shows how a natural language such as English could formulate its own semantic.
One can give the general meaning conditions of potentially all of the English assertions. The
meaning of any English assertion could be made explicit. And one can do so in English, and not
in any metalanguage that includes English (as its object language). One can give the semantics
of all English assertion; not just the semantics of a part of it. And this is also a point in which
STA advantages traditional theories, such as some semantic traditional theory, some tarskian
truth-conditions semantics. This kind of theory is forced to give the semantics for just a part of
the whole English, and to do that from some more powerful metalanguage that includes the
(previously mentioned) part, a metalanguage that could say more things about the object
language that the things that could be said from inside the object language. In the next pages, I
will summarize some aspects of how STA deals with the semantic paradoxes. I will do very
briefly later some comments about the presumed ability that STA says English has of being a
universal language.
Let’s start with the semantic paradox’s issue. Barker provides and incredible simple
answer to deal with a whole lot of misfortune consequences of the expressive capacity of natural
language. The semantic paradoxes are a bunch of assertions (each of which could be, by itself, a
bunch of related assertions) such that we cannot conclude that they are true –because falsehood
derives from truth-, nor false –because truth derives from falsehood). It’s a common belief that
it is a problem to defend that the same assertion could be both true and false at the same time –a
problem as dramatic as that contradictions are admissible.1 Those who investigate the semantic
properties are devoted to construct meaning theories that avoid those consequences. Main
notion of Barker’s solution is that of *indeterminacy. Primary truth bearers are assertions. An
assertion S is such because it expresses a well determined mental -property. If a presumed
assertion does not express a genuine mental -property, then it is not an authentic assertion at
all, and the sentence that presumably, when asserted, expresses that property, is *indeterminate.
When is it the case that a presumed assertion fails to express a genuine mental -property? It
fails when one cannot assert neither that it is true nor false, and neither non true nor non false,
nor that it should be one of them. (If it is judged that a sentence S is *indeterminate, then, in
particular, it cannot be correctly judged that S is not true.) And one cannot assert that (that they
are true, nor false, nor non true, etcetera) of some particular sentences: those that don’t satisfy el
principle of Non Relationality.
Non Relationality. If S really fixes a -property , then there is in principle an
adequate description of ’s nature that is non-relational in this sense: it features no description
of the form -[R] for any sentence R.2
There are certain sentences that, despite being grammatically correct and not suffering
from any reference failure, cannot be genuinely asserted because they cannot satisfy this
principle. If this is so, then these assertions don’t express a genuine mental -property. Some of
1
Priest, in In Contradictio (Dordrecht, 1987) defends something like this.
“Non-Relationality is consistent with some thoughts featuring the concept of -property, as in a
sentence of the form N is a -property. Possessing as a constituent the concept of a -property does not
mean that the content is partially relationally specified by being the -property of a sentence”. Stephen
Barker, Expressivism about Truth Meets the Semantic Paradoxes, page 21 (unpublished).
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these sentences lead to the so called ‘semantic paradoxes’, one of them being the Liar paradox.
Think about the following sentence:
L: L is not true.
If L is true, then L is not true. But if L is not true, then L is true. As we don’t want
contradictions, we should make some adjustments in our semantic theory or in our truth theory
to avoid the occurrence of the paradox. A possible solution, different from Barker’s, is to state
that L is not meaningful, and thus that it cannot have truth conditions –at least if one adopts a
truth-conditions semantic. But, why cannot L be meaningful, if it is grammatical and does not
suffer from reference failure? Why, if we accept that self reference isn’t per se something
incorrect (because there are genuinely self reference sentences)? Barker states that these
solutions are inadmissible, because all of them are victims of kinds of revenge towards them: if
L isn’t truth-apt, then it isn’t true. But if it isn’t true, then it is true, and that’s a contradiction.
Another version of the same thing: telling that L is not a possible truth bearer implies denying
that it could be true, false, not true nor not false. Those who affirm this thesis believe, in most
cases, that truth is a property. They also understand properties as sets. Having a property A is
being an element of A. If something is not an element of A, then it doesn’t have the property A,
but L is not an element of the set of true sentences. So L is not true. Therefore, L is true.
Contradiction. Barker classifies Kripke’s solution to semantic paradoxes as a kind of strategy
based on treating them as sentences that don’t really have meaning. Barker tells us that Kripke
defends the idea that a sentence S is not grounded if and only if S hasn’t a specification of its
truth conditions that does not use the notion of truth. If S is grounded, then it doesn’t have any
truth value. L isn’t grounded, so it doesn’t have any truth value. Then S is not true. But if S is
not true, then there is a sentence like ‘S is not true’ that is true but not grounded, because it
doesn’t have any specifications of its truth conditions that don’t use the truth predicate. So
Barker concludes: ‘To avoid contradiction Kripke needs to re-introduce an objectlanguage/metalanguage distinction. As we shall see, STA’s *indeterminacy solution does not
require that distinction’. 3 STA’s solution to the Liar’s paradox follows immediately: L is
*indeterminate. Any attempt to specify the -property picked up by L leads either to a regress –
with the characteristic that every new step has the form -[L]  n-reject: -[L]- or to a loop
–with the characteristic that the specification of -[L] is Reject: -[L is true], which
specification is: -[L]. So -[L] doesn’t satisfy Non Relationality. Then L is not grounded,
and cannot be asserted.
This answer seems legitimate. I wish, non the less, explore a little bit more the details of
Kripke’s solution.
Kripke, in ‘Outline of a theory of truth’,4 presents a method for constructing a language
that contains its own truth predicate. The basic language, the language based on which the other
languages will be construed, allows ‘gaps’ of truth values. Some sentences could be neither true
nor false. So a sentence could be meaningful, but at the same time it could lack a truth value.
Kripke thinks that a sentence is meaningful if there could be circumstances that make it true or
false, even though there were other circumstances in which they were nor true nor false. In order
to prove this, he begins by a language L0 that doesn’t have a truth predicate. Nevertheless, we
can add a truth predicate to the language, and assign to it an extension and an antiextension that
are partial sets. That’s how we get L1. Kripke shows that there an extension of this original
language could be produced, so as to include more and more sentences to the extension or to the
antiextension of the truth predicate. This way, in L1, the extension includes all truth sentences
that don’t include occurrences of the truth predicate, and the antiextension all false sentences of
the same kind. In L2, the truth predicate extension extends to include at least all the sentences
that predicate truth of the elements of the extension of the truth predicate in L1, and so on in the
next levels. The process of extending the languages continues up to an ordinal level . In that
3
Stephen Barker, Expressivism about Truth Meets the Semantic Paradoxes, page 27, note 24.
Kripke, Saul, ‘Outline of a theory of truth’, The Journal of Philosophy, vol. 72, number 19, pages 690716.
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level there is no other possible process of ‘enlargement’ that includes more elements to the
extension or the antiextension of the truth-predicate. Kripke called this two entities together,
extension and antiextension, a ‘minimal fixed point’. The language that includes this minimal
fix point includes its own truth-predicate. With the support of these results, Kripke defines the
notion of ‘grounding’: a sentence A of any of the languages construed by this method is
grounded if it has a truth value in a minimal fixed point. If A doesn’t satisfy this requirement, it
is ungrounded. Intuitively: a sentence of any of these languages is grounded if its truth value can
be explained in terms of the truth values of a set of sentences of L1 that are part of the extension
or the antiextension of the truth predicate in L1. A consequence of this way of stating things is
that sentences whose truth value is problematic, become ungrounded. They are neither true nor
false. One example of this kind of sentences is the Liar Paradox. But another example of that
kind of sentences is the sentence numbered (3) in Kripke’s article:
(3): (3) is true.
Nevertheless Kripke wants to distinguish (and I think he has good reasons to do it)
sentences that raise paradoxes –such as the sentence we call ‘the Liar Paradox’- from other
sentences like (3). These last sentences have the following characteristic: their truth value is
complicated to fix, but doesn’t raise paradoxes. If one begins the hierarchy of languages with
one that includes the sentence (3) in the truth predicate extension, then one gets as a result that
(3) is true in some minimal fixed point. The sentence (3), that isn’t true or false in any fixed
point of the hierarchy of languages that Kripke originally constructed, is now true in some fixed
point. Kripke, then, distinguishes paradoxical sentences from sentences like (3) in the following
way: a sentence is paradoxical if it doesn’t have a truth value in any fixed point. This explains
the intuition that (3) isn’t a paradoxical sentence, but that even so there is something strange
with it. What is strange is that it is ungrounded. Sentence like the Liar Paradox, besides being
ungrounded, cannot have a truth value in any minimal fixed point.
Another subtlety of Kripke’s theory is that a disjunction of a true sentence and an
ungrounded sentence is a true sentence (because of the way in which Kleene’s strong trivalent
logic –the one Kripke employs in this work- treats logical constants). Many of us have the
intuition that a disjunction such as ‘Cervantes wrote El Quijote or everything I say is false’ is
true. On STA, *indeterminacy spreads: if an assertion includes a *indeterminate part, the hole
assertion is *indeterminate. Now assume that the previous sentence is asserted. In that assertion,
the second disjunct is *indeterminate, so the hole assertion is *indeterminate. If one rejects this
intuition, Kripke’s theory still has the following merit: it rescues the idea that a disjunction is
true if and only if at least one of its disjuncts is true. A theory like STA, that admits that
*indeterminacy spreads, cannot do justice to this intuition.
Let’s analyze now the following two sentences:
(4) All Nixon’s utterances about Watergate are false.
(5) All Dean’s utterances about Watergate are false.
Where (4) is uttered by Dean. With these sentences we have two problems. The first one
is the following. Given the hierarchy of languages stated by Tarski’s theory, were each level
contains a truth predicate that doesn’t apply to sentences in their own level nor to sentences of
languages of a superior level, sentences like (4) and (5) should belong to some particular level.
But it seems to be the case that what level (4) belongs to depends on empirical facts: which
were the sentences about Watergate that Nixon have uttered, and to what level belongs the
‘higher’ sentence uttered by Nixon (about Watergate). Besides, the most frequent intention of
those who make assertions as (4) is that, as Kripke says, it ‘seek its own level’. 5 The other
problem is this. Let’s suppose that Dean pretends to include (5) within the utterances that (4)
talks about. Moreover, let’s suppose that Nixon pretends to also talk about (4) when he utters
(5). If sentences had fixed language levels, these two things couldn’t be made at once. From
5
‘Outline…’, page 696.
within a tarskiana theory, one cannot assign truth values to sentences that don’t have their own
level already fixed. But it seems that we can make it with sentences as (4) and (5) even before
their level is fixed. For example, if Dean had made at least one true assertion about Watergate,
then (5) would be false. A theory that doesn’t explain how this is possible should give at least
some explanation. (It should explain why this cannot be accomplished, or why it doesn’t matter
to accomplish it.) Kripke’s theory explains why these intuitions are right, showing at the same
time which one is the level that corresponds to each assertion of these sentences. For example, if
some true assertion is made by Dean (about Watergate), and the lowest level those assertions is
, then (5) is grounded and false, and its level is  + 1. If some other assertion of level ,
besides (5), uttered by Nixon, is true, then (4) would be grounded and false, and it would belong
at least to  + 1 level. That’s how Kripke explains these situations. Could Barker’s STA do
something analogous? Maybe, but Barker hasn’t explained how. The way STA treats sentences
as (4) and (5), seems to condemn them to *indeterminacy, because any attempt to fix a property as their mental property defended, seems to fail to satisfy Non Relationality. This
makes this approach less attractive than Kripke’s, because it doesn’t allow certain potential
assertions that seem to be made with the sincere intention of uttering a true thing –and therefore,
to have a define truth value. Therefore, STA separates the semantic of the utterances away from
the competent speakers’ intentions –and so, becomes less attractive. But even if this wasn’t a
good reason for preferring Kripke’s solution, there would still be a good reason that Kripke’s
theory can determine on some occasions, the truth value of sentences like (4) and (5). Barker’s
theory cannot do the same thing. On STA, (4) and (5) are *indeterminate, and so are not
genuine truth bearers. Is there any disadvantage in Kripke’s theory to compare to Barker’s?
Maybe there is. Let’s look at that issue.
The main disadvantage of Kripke’s solution is that it cannot give a universal language,
in the sense of a language containing its own semantic theory. That is because Kripke assumes
truth-conditions semantics (of some kind), and despite the fact that he shows how to get a
language with its own truth and satisfaction predicate. Kripke should postulate a metalanguage.
Kripke himself shows that there are assertions that we cannot make in the object language he
constructed. They are assertions that we desire to make. Going to the details, one reason that
supports this thesis is this: ‘the induction defining the minimal fixed point is carried out in a settheoretic metalanguage, not in the object language itself’.6 A second is the following: there are
assertions about the object language that cannot be made in the object language itself. For
example, that the sentences like the Liar Paradox are not true, because the inductive process
(that leads to a construction of fixed points) never makes them true. Because of the way we have
interpreted negation and the truth predicate, one can never say that in the object language. If one
takes a minimal fixed point as a model of (certain stage) of natural language, as Kripke
suggests, then the possibility of asserting that the Liar Paradox isn’t true emerges only in a more
advanced language stage, ‘one in which speakers reflect on the generation process leading to a
minimal fixed point. It is not itself part of that process. The necessity to ascend to a
metalanguage may be one of the weaknesses of the present theory’.7 This language, in which the
theory is presented, may be considered as empty of truth value gaps. But the proof given by
Gödel and Tarski, that shows that a language cannot contain its own semantics (understood as a
truth-conditions semantics), do apply to this metalanguage. This metalanguage, then, cannot
contain its own semantics, and then it cannot be a universal language (in this sense).
What we can say in the metalanguage is that sentence like (the one that expresses) the
Liar Paradox is not true, ‘in the sense that the inductive process never makes them true’.8 There
is no fixed point (of those constructed beginning with L0) in which they are grounded and true.
Could we assert, in this sense, that they are false? No, because for being so they must be
grounded, and they are not. That avoids the Liar Paradox in the metalanguage. The truth
predicate in the metalanguage isn’t treated in the same way as the truth predicate that is part of
the (sentence that expresses the) Liar Paradox. As we previously announced, we don’t have
‘Outline…’, page 714.
‘Outline…’, page 714.
8
‘Outline…’, page 714.
6
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either the Liar Paradox in the object language, because that sentence that expresses it is not
grounded. It is never true or false, nor not true, nor not false, because it is not grounded. In the
object language, the extension of ‘being not true’ is the same as ‘being false’. So, contrary to
what Barker claims, Kripke’s solution is not victim of any form of revenge.
Let’s assume now that Barker’s theory does allow a language capable of expressing its
own semantics. The costs of accepting that theory are the followings: (i) one cannot distinguish
*indeterminate sentences simpliciter from those that, besides being *indeterminate, raise
paradoxical scenarios; (ii) the theory doesn’t respect the intuition that claims that sentences
‘seek its own level’; (iii) the theory doesn’t give space to the intuition that claims that a
disjunction is true if and only if one of the disjunts is true.9 We can do all of this in Kripke’s
theory. Is it worth giving up the (i)-(iii) desiderata and adopting STA? Is it worth having a
language capable of expressing its own semantics, even though it lets us without an explanation
of the previously mentioned multiplicity of phenomena? If the kind of assertions that STA
leaves without explanation of its semantic peculiarities were assertions about colours, would we
still think that it is worth paying the price? I don’t have a define position over this issue. My
only interest, here, was to show some of the limits that the benefits that STA seem to have.
There are a few other little advantages of Kripke’s solution over Barker’s. I don’t have the time
necessary for talking about them. So I won’t talk about them.
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