Kripke inverted. A transfinite approach to a theory of truth ...

Reading Kripke's proposal that there must be meta-language/object-language congruence at some indefinitely distant point in an indefintely dimensioned space of linguistic hierarchies, I wondered:  why not start from there, then?  (If we count backwards from infinity, it is zero that is out of reach ...)

Also, I have remarked before in this blog that the top of the meta-linguistic hierarchy must always be whatever language we are speaking now:  this conversation, in other words.

So:

In the conventional picture, a language can only be a meta-language if includes a T predicate that can be interpreted as a truth predicate for some 'object' language.  Only the bottom level object language can do without one.  We may be confused by the fact that we call this thing a 'truth' predicate, of course, since all it does is make a distinction between certain classes of statements in the object language.  The meta-language may also be used to articulate a theory of truth for the object language if it says how this distinction is made - although the theory may only comprise comprehensive lists of statements in each class.

The reason that the theory cannot be articulated within the object language is that it fails to ground 'liar' statements - Tarski regarded this as catastrophic, as he thought the liar must be interpreted as both true and false.  Since the T predicate can only be interpreted as a truth predicate if it avoids contradiction (among other things?), it must render at least two discrete classes of statements in the object language.  (As a minimum, one which can be interpreted as containing 'true' statements, and the other as containing 'false' ones).  Not all statements in the object language (formally) need to belong to one of the classes.

If the T predicate was part of the object language, we could construct these two statements:

(1)  'Statement (1) belongs in class B' and
(2)  'Statement (2) belongs in class A'

If membership of class A is interpreted as 'being True' then (1) generates a liar paradox, if B, then (2) is a liar.  The other statement of the pair, in each case, is normatively self-referential in a similar way to the liar, but does not generate an immediate contradiction.  (There are lots of variations on this representation - I can't guarantee that this one isn't flawed.)

The meta-linguistic solution is to make statements like (1) and (2) only available in a higher order language, and not in the 'object' language.  A meta-language, then is defined as a language which contains a truth predicate for some object language.

A question related to Kripke's project is whether there is a meta-language which must contain its own truth-predicate.  He seems to be saying that while this is ruled out in a simple 'enumerable' hierarchy of meta-languages, the need to be able to talk about the whole hierarchy indicates that the full set of meta-languages is not enumerable.  He suggests that (a) this creates some special problems which are not easy to address, but also that (b) it creates a space for a language with the property of being able to contain its own truth predicate.  (I may not have this completely right?)

It isn't clear how we get to this language, however, if we take the traditional starting point of an object language with no truth predicate.

What I suggest is that this is a mistake - and in fact that it is incoherent.  There is no unambiguous case of a language without a truth predicate:

If we are considering some behaviour (taken very generally) as a candidate for linguistic behaviour, then Davidson argues (correctly, I think) that we can only draw a positive conclusion if we can come up with a truth-preserving translation schema for the behaviour.  If we cannot produce such a schema for a pattern of behaviour, we do not have grounds for regarding the behaviour as a language at all.

On the other hand having such a proposed schema does not guarantee that we are dealing with a language.  Davidson shows that some Principle of Charity is needed to get started here, and, in addition, the Kripke/Goodman paradox renders all intentional interpretations of behaviour provisional.

The only thing that can ‘guarantee’ such a judgement is sharing it with a speaker of the object language.  In this case, failure to correctly interpret their behaviour is also, directly, failure to share their judgement.  This may not sound very reassuring, but it puts our shared judgment with speakers of the object language on the same footing as our shared judgements of each other as speakers of the meta-language.  We cannot say, within a shared conversation, that it is possible that we are radically failing to understand one another - this is a speculation which renders its own (apparent) articulation unintelligible.

The judgment that we are speaking a language properly (with one another) must include a judgement that we share a theory of truth (though not necessarily one we can articulate).  If we don’t generally know how to tell the truth, then we don’t know how to speak the language.  If we want to explicitly make these kinds of judgements, then we need the equivalent of a truth predicate for the language we are using (even if this is tacitly embedded in the statement that we are spreaking the language properly).

Asking someone whether you are speaking their language properly includes asking them whether you know how to tell the truth in it.  It is not possible to unambiguously identify behaviour as linguistic without this shared agreement, therefore it is not possible to unambiguously identify a language as a language unless it contains a truth predicate.

So, no unambiguous case of an L0.

(Interestingly, this renders the 'slab block' game from PI ambiguous in the relevant way.  We cannot know that the participants are talking to one another.)

When we converse with one another, we may explicitly negotiate specific articulated theories and facts, but we also – both explicitly and tacitly – negotiate the application of the truth predicate for our shared language.  If we have no truth predicate, we don't have a way of explicitly agreeing that we have a shared language.  Whether we also want to say that we negotiate the ‘theory of truth’ for this language depends on whether we think of this as fully articulable (in which case the answer is no) or only partly  (which allows the answer to be yes).

So, I think of this as inverting Kripke because my starting point is the necessity of a self-applicable truth predicate in any language within which a reliable judgement about linguistic status can be made.  This is, a fortiori, any language that can be used for philosophical discussion.

Our judgements about truth telling - our applications of the predicate - must generally be reliable in this language if we are not to be generally talking nonsense (which is ruled out by the incoherence of 'we are generally talking nonsense'.)

Finally, we can only unambiguously identify behaviour as language use if we can share this judgment with the users of the language.  We can't form this judgement on the basis of behaviour alone.  This means that the idea of a language without a truth predicate is actually more incoherent than the idea of one which contains its own truth predicate (so long as we don't conclude that the theory of truth for this language is fully articulable).