Saturday, November 08, 2008

A note on expressive problems

In chapter 3 of the Revision Theory of Truth (RTT), Gupta and Belnap argue against the fixed point theory of truth. They say that fixed points can’t represent truth, generally, because languages with fixed point models are expressively incomplete. This means that there are truth-functions in a logical scheme, say a strong Kleene language, that can not be expressed in the language on pain of eliminating fixed points in the Kripke construction. An example of this is the Lukasiewicz biconditional. Another example is the exclusion negation. The exclusion negation of A, ¬A , is false when A is true, and true otherwise. The Lukasiewicz biconditional,A≡B , is true when A and B agree on truth value, false when they differ classically, and n otherwise. Read more

The shape of this argument seems to be the following. The languages we use appear to express all these constructions. If they don’t, we can surely add them or start using them. The descriptive problem of truth demands that our theories of truth work in languages that are as expressive. (Briefly, the descriptive problem is the problem of characterizing the behavior of truth as a concept we use, giving patterns of reasoning that are acceptable and such.) The fixed point theories prevent this, therefore they cannot be adequate theories of truth.

I’m not sure how forceful this argument is. I’m also not quite sure how damaging expressive incompleteness is. The expressive incompleteness at issue in the argument is truth-functional expressive incompleteness. There is lots of expressive incompleteness present in the languages under consideration. This is distinct from the language expressing its own semantics. The semantic concepts required for that need not be present since there will presumably be non-truth-functional notions used. It also isn’t part of the claim with respect to the languages we use. I will stop to discuss this point for a moment since I find it interesting.

The languages we use may or may not be able to express their own semantics. As Gupta says, rightly I think, one should be suspicious of anyone who claims that we must be able to express our semantic theories in the languages they are theories for. The primary reason for this is that we don’t know what a semantic theory for the complete language would be. The extant semantic theories we have work for small fragments that are regimented highly. Further, these theories are only defined on static languages, whereas the ones we use appear to be extensible. Additionally, these theories tend to be coupled to a syntactic theory that provides the structure of sentences on which the semantics recurs. There is no such syntactic theory for the languages we use either. The shape a semantic theory for our used language might be very different than the smaller models currently studied. It might not even contain truth. The requirement that a language be able to express its own semantic theory seems to stem from an idealization based on current semantic theories that, if the above is right, is illicit. The question of expressive completeness is distinct from this question of semantics. The question of what it is to give a semantics for a language in that language is interesting, and is raised in criticisms by both McGee and Martin. I hope to post on that soon.

One question that strikes me is how central to the descriptive problem is this expressive power? Expressive power itself is a notion that is somewhat obscure until one moves to a formal context in which one can tease apart distinctions. For example, it isn’t at all apparent that ‘until’ isn’t expressible using the standard tense operators or constructions, even though all these are, arguably, readily apparent in the languages we use. It isn’t clear, then, that the notion of expressibility is even workable until we move to a more theoretical setting from the less theoretic setting of language in use.

If we move to a more theoretical setting and discover that what we thought was vast expressive power has to be curtailed, then it isn’t clear that our earlier intuition is what must be preserved. One could hold out for a theory of truth that preserved it. Gupta clearly thinks this is one to hold on to. Perhaps this is what a detailed statement of the descriptive problem demands.

Something else that I wonder about this line of thought is how common expressive incompleteness, of the truth-functional kind, is among the most prominent logical systems. We have it in the classical case. In limited circumstances, we have it even with the addition of the T-predicate. In any case, we probably don’t want to stop with just logics that treat only truth-functions and T-predicates. We might want to add modal operators of some kind, and these are not truth-functional. What sort of expressive problems are generated, or not, then? I'm not sure, although there is an excellent chapter in RTT on comparing the expressive power of necessity as a predicate and as a sentence operator.

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