Thursday, October 30, 2008

A note on the revision theory

In the Revision Theory of Truth, Gupta says (p. 125) that a circular definition does not give an extension for a concept. It gives a rule of revision that yields better candidate hypotheses when given a hypothesis. More and more revisions via this rule are supposed to yield better and better hypotheses for the extension of the concept. This sounds like there is, in some way, some monotony given by the revision rule. What this amounts to is unclear though.

For a contrast case, consider Kripke's fixed point theory of truth. It builds a fixed point interpretation of truth by claiming more sentences in the extension and anti-extension of truth at each stage. This process is monotonic in an obvious way. The extension and anti-extension only get bigger. If we look at the hypotheses generated by the revision rules, they do not solely expand. They can shrink. They also do not solely shrink. Revision rules are non-monotonic functions. They are eventually well-behaved, but that doesn't mean monotonicity. One idea would be that the set of elements that are stable under revision monotonically increases. This isn't the case either. Elements can be stable in initial segments of the revision sequence and then become unstable once a limit stage has been passed. This isn't the case for all sorts of revision sequences, but the claim in RTT seemed to be for all revision sequences. Eventually hypotheses settle down and some become periodic, but it is hard to say that periodicity indicates that the revisions result in better hypotheses.

The claim that a rule of revision gives better candidate extensions for a concept is used primarily for motivating the idea of circular definitions. It doesn't seem to figure in the subsequent development. The theory of circular definitions is nice enough that it can stand without that motivation. Nothing important hinges on the claim that revision rules yield better definitions, so abandoning it doesn't seem like a problem. I'd like to make sense of it though.

henri galinon said...

A tentative suggestion.
Why not just say that the revision rule is a rule of improvement, but that it may or may not converge ?
I can't remember if "rule of improvement" is a phrase that G&B also use, but I think it could be used.
Perhaps we could say that the phrasing in term of "better and better extension" is just a loose one that motivates/explains/enforces/clarifies the intuition that the revision rule is one of improvement (one that works in most cases yielding "better and better extensions"). To repeat, the payoff of insisting on the fact we have an improvement rule at work in the revision process rather than directly say that we get "better and better extension" lies in the fact that it is admitted that a rule can be deceiving and fail to acheive the desired result ("failure to converge") in some circumstances.

Justin said...

I too was baffled by the claim that revision sequences give you better extensions when I read it. Like Henri, I wonder if perhaps it's just a way of motivating the project.

It does seem to be true ceteris paribus that the rule takes you to better and better extensions.

ansten said...

At least in the case of Herzberger-sequences (lim inf limit rule), I think there is quite a clear sense in which the stages gets "better": they approach the ultimate stability-point, the point at which all sentences that eventually stabilize stabilizes.
I am not sure how things stand with sequences using other, I would say less natural, limit rules.