Last week:
This week:
Did Tarski show how to define the concept of truth, i.e. define predicates that have the same meaning as the English predicate 'true'?
One problem: 'true' seems to apply to PROPOSITIONS (what we use sentences to say) where Tarski's predicates apply to SENTENCES.
The question here concerns the problem of TRUTH BEARERS: what are the bearers of truth? Some options:
But even if we agree that propositions are the bearers of truth, we can say that propositions are what sentences EXPRESS, and ask the question:
Does Tarski show to define predicates that have the same meaning as the English predicate 'expresses a truth'?
Another problem: 'true' (or 'expresses a truth') applies more widely than Tarski's predicates. It applies to the sentences of all languages — including English!
As we saw, Tarski thinks that this is the source of the Liar Paradox.
If he is right, his predicates cannot be co-extensive with 'true' (or 'expresses a truth').
Did Tarski show how to explicate truth, i.e. define predicates that can replace 'true' (or 'expresses a truth') in all legitimate theoretical contexts, but that lack its defects?
Here are four problems for this idea...
First, a problem raised by Michael Dummett: the PROBLEM of TRUTH CONDITIONAL SEMANTICS.
Many philosophers and linguists think that the notion of truth has an important theoretical role to play in an account of what the sentences of a language mean: "to know what a sentence means is to know its truth conditions".
But if a theory that tells us what the truth conditions of the sentences of a language is to serve as an account of what they mean, we have to already know the point of describing sentences as true.
Compare: being told their peng conditions.
So a Tarskian truth definition cannot both tell us what the sentences of the object language mean and tell us the meaning of the predicate being defined.
A possible response: reject the assumption that to know the meaning of a sentence is to know its truth condition.
Second, a problem raised by Hilary Putnam: the MODAL PROBLEM.
Remember our definition of 'trueL1':
∀s (s is trueL1 IFF: either
Let TWO be a name of '1 + 1 = 2'. Then, replacing 'trueL1' in 'TWO is trueL1' with its definiens, we get:
'TWO = TWO and one plus one is two or TWO = '1 + 1 = 3' and one plus one is three'
The problem is that the latter is true in all possible worlds in which one plus one is two (i.e. in all possible worlds.)
But isn't TWO false in some possible worlds — e.g. certain worlds in which '2' refers to the number three?
A possible response: deny that TWO is false in some possible worlds by insisting that sentences are individuated by their semantics as well as their syntax.
Third, a problem raised by Max Black: the PROBLEM OF NON-PROJECTABILITY.
The definitions that Tarski gives us don't project. That is, they don't tell us under what conditions other truth predicates, for other languages, hold of the sentences of those languages.
Fourth, a problem stressed by Scott Soames: the PROBLEM OF EPISTEMIC DIFFERENCE.
Knowing the conditions under which a sentence is true, whether or not it is sufficient for knowing what it means, at the very least provides information about what it does not mean.
For example, if one knows that 'la neige est blanche' is true IFF snow is white, one is at least in a position to know that it does not mean that snow is not white.
But knowing the conditions under which a sentence belongs to the extension of a Tarskian truth predicate does not provide even this sort of negative information about its meaning.
Suppose I tell you that 'yksi plus yksi on kaksi' is totta IFF one plus one is two. Are you able to determine whether 'yksi plus yksi on kaksi' means that one plus one is not two?
It seems not. Even if I give you a Tarski-style definition of 'totta', this doesn't rule out the possibility that 'yksi plus yksi on kaksi' means that one plus one is not two.
Tarski does give us predicates that apply to all and only the true sentences of various languages without giving rise to the Liar Paradox. But his approach is very restrictive.
In place of Tarski's hierarchy of languages, we might introduce a hierarchy of restricted PREDICATES:
This way, we can distinguish a hierarchy of syntactically individuated levels within a single language.
But this is still very restrictive, as is nicely brought out by Kripke's Watergate examples.
PROBLEM 1: How do we determine the appropriate subscript for truth predicates?
John Dean: 'Nothing Nixon said about Watergate up to the time of his resignation was true'
To assign a subscript, we would need to know the highest level to which sentences uttered by Nixon belonged.
PROBLEM 2: In some cases, it is not even possible in principle to assign a subscript.
John Dean: 'Most of Nixon's Watergate-related statements are not true'
Nixon: 'Most of John Dean's Watergate-related statements are true'
The subscript on Dean's 'true' will have to be higher than any subscript on any truth predicate uttered by Nixon.
But so too must the subscript on Nixon's 'true' be higher than any on any truth predicate uttered by John Dean.
The subscripting approach treats this pair as paradoxical. But it's perfectly conceivable that both sentences are true!
Suppose neither uttered any other sentence containing the word 'true'. And that 90% of the sentences Nixon uttered were false, while 90% of those Dean uttered were true.
There is a RISK of paradox, however.
Suppose that, apart from these, Nixon and Dean both uttered an even number of sentences, exactly half of which are true and half false.
The lesson: strategies that employ SYNTACTIC criteria to screen off paradoxical sentences will rule out sentences for which there is only a RISK of paradox.
Kripke's own theory promises to do better. Let's take a look.
One way to avoid the Liar Paradox is by denying, or at least refusing to accept, that the liar sentence is either true or false. (We'll see next week how exactly this helps.)
But if this to be anything more than an ad hoc response, we need some principled reason for thinking that it is correct.
Can we provide one?
There is an intuitive sense in which the truth of some but not all sentences depends on the truth of other sentences. Compare:
Whether ''Snow is white' is true' is true depends on whether 'Snow is white' is true. But the truth of 'Snow is white' doesn't depend on the truth of another sentence in the same way. It depends on whether snow is white.
These sentences form the initial segment of a certain hierarchy of sentences (the SNOW HIERARCHY) each successive member of which depends for its truth, in the relevant sense, on whether the previous member was true:
Since the truth of the first member does not depend on the truth of any other sentence, there is reasonably clear sense in which each member of the hierarchy is GROUNDED.
This suggests: the principled reason for refusing to accept that the liar sentence is true or false is that it is UNGROUNDED. Kripke spells this out more precisely.
First, we construct a definition of 'trueL' that applies to those sentences of L that do not contain the word 'trueL'.
These are assigned to either the EXTENSION of 'trueL', the set of sentences to which it applies, or to its ANTI-EXTENSION, the set of sentences to which it does not apply.
We then extend the assignment so that it covers sentences that DO contain the word 'trueL', treating sentences that do not (yet) belong to either its extension or its anti-extension as neither true nor false.
For this extension, we need a logic in which sentences can be neither true nor false. Kripke uses the Kleene strong logic, in which sentences are true, false, or indeterminate.
Thus, at this stage:
(The choice of K3 here is not essential — any logic that allows sentences to be in some sense neither true nor false will do. Kripke also considers a SUPERVALUATIONIST approach.)
This gives us a level 0 of sentences: those whose truth does not depend on the truth values of other sentences.
In other words: the sentences that are assigned to either the extension or the anti-extension of 'trueL' at this initial stage.
The English sentences at level 0 will include:
But not:
Notice that whether or not a sentence belongs to level 0 is not determined by its syntax, and in particular by whether or not it contains the word 'true'!
The process can then be repeated on through subsequent levels:
Level 1: sentences whose truth values are determined by those of the sentences at level 0. For example:
Level 2: sentences that do not belong to lower levels whose truth values are determined by the truth values of sentences at those lower levels. For example:
The process goes on beyond all finite levels, accommodating examples like:
Eventually, Kripke shows, we will reach a MINIMUM FIXED POINT at which no new sentences get added to the extension and anti-extension of 'trueL'.
The liar sentence turns out to be ungrounded in the precise sense that it is not assigned to either the extension or anti-extension of 'trueL' at any level in this process.
Neither are non-paradoxical but otherwise defective sentences such as:
On the other hand, if most of Dean's Watergate-related statements are true, and most of Nixon's are false, both of the following will be found in the extension of 'true':
Since the defectiveness of liar sentences is not treated as syntactic, this is a far less restrictive approach.