Tuesday, December 30, 2008

Definitions and content

Reading the Revision Theory of Truth has given me an idea that I'm trying to work out. This post is a sketch of a rough version of it. The idea is that circular definitions motivate the need for a sharper conception of content on an inferentialist picture, possibly other pictures of content too. It might have a connection to harmony as well, although that thread sort of drops out. The conclusion is somewhat programmatic, owing to the holidays hitting. Read more

In Articulating Reasons, there is a short discussion of harmony. It begins with a discussion of Dummett on slurs. Brandom says, "If Dummett is suggesting that what is wrong with the concept Boche is that its addition represents a nonconservative extension of the rest of the language, he is mistaken. Its non-conservativeness just shows that it has a substantive content, in that it implicitly involves a material inference that is not already implicit in th contents of other concepts being employed." (p. 71)
Being a non-conservative addition to a language means that the addition has a substantive content. It licences inferences to conclusions in which it does not appear that were not previously licenced. I want to point out something that doesn't seem to fit happily within this picture.

In the Revision Theory of Truth, Gupta and Belnap present a theory of circular definitions and several semantical systems for them. I will focus on the weakest system, S0. According to S0, the addition of circular definitions to a language constitutes a conservative extension of the language. That being said, it seems like the introduction of circular definitions brings with it a substantive content, given by the revision sequences and the set of definitions. From the quote, it seems that Brandom is saying that non-conservativeness is sufficient for substantive content, not necessary. We would have a nice counterexample otherwise. If we look at stronger systems, the Sn systems (n>0), then it turns out that the same set of definitions may not yield a conservative extension of the language.

It might be misleading to cast things in terms of the semantical systems since Brandom casts things in terms of inferential role. Gupta and Belnap offer proof systems for the S0 and Sn systems. These proof systems are sound and complete with respect to the appropriate semantical system. In the system C0, the addition of a set of definitions to a language yields a conservative extension, as would be expected.

The fact that circular definitions are conservative in S0 doesn't upset Brandom's claim above. It doesn't seem like we want to say that all circular definitions lack substantive content. Unlike non-circular definitions that are conservative over the base language and eliminable, they are not mere abbreviations. The addition of circular definitions has semantical consequences in the form of new validities. Circular definitions point out the need for necessary conditions on the notion of substantive content, since one would expect that there are circular definitions that aren't substantive, e.g. one with a definiens that is tautological. Alternatively, a sharper notion of content, and so substantive content, would help clarify what is going on with circular definitions of different stripes.

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