Monday, 22 April 2013

Further thoughts on Priest's Inclosure Schema

After publishing my post on Priest’s Inclosure Schema (IS) a few days ago, I’ve had a number of interesting exchanges on the content of the post, including with Priest himself. So here are a few additional thoughts, in case anyone is interested.

Regarding the charge of extensional inadequacy (over- and undergeneration), I think it had been made sufficiently clear by others before me that the fact that the Curry paradox does not fit into IS is a big blow if IS claims to be a formal explanans for the informal concept of paradoxes of self-reference. However, while Priest’s original claim seemed to pertain to paradoxes of self-reference specifically, he seems to have changed a bit the intended scope of IS, and now tends to talk about ‘Inclosure paradoxes’. I don’t think there is anything wrong with this ‘change of heart’, but it does have consequences for how we should conceive the role of IS in debates on paradoxes. To make sense of this development, let me turn to a distinction introduced by S. Shapiro (in the words of L. Horsten in his SEP entry on philosophy of mathematics):
Shapiro draws a useful distinction between algebraic and non-algebraic mathematical theories (Shapiro 1997). Roughly, non-algebraic theories are theories which appear at first sight to be about a unique model: the intended model of the theory. We have seen examples of such theories: arithmetic, mathematical analysis… Algebraic theories, in contrast, do not carry a prima facie claim to be about a unique model. Examples are group theory, topology, graph theory, ….
By analogy, I would submit that IS was first introduced as a ‘non-algebraic theory’, intended to capture one very precise class of arguments, namely paradoxes of self-reference. But as things moved along, it became clear to Priest and others that IS in fact determines a different but possibly equally interesting class of arguments, which he refers to as Inclosure paradoxes. From this point of view, IS is now an ‘algebraic theory’: rather than starting with a given target-phenomenon and trying to formulate a formal account which would capture all of (and only) this phenomenon, IS is now a freestanding formal account, and it is a non-trivial question as to which class(es) of entities it accurately describes. (In non-algebraic theories, you start with the phenomenon and look for the theory; in algebraic theories, you start with the theory and look for the phenomenon.)

From this angle, it becomes a noteworthy observation, rather than an extensional failure, to notice that Curry does not fit into IS, and that the sorites paradoxes and some reductio arguments do fit into IS, thus unveiling some (surprising) structural similarities. In other words, if IS is intended as an ‘algebraic theory’, then the charges of over- and –undergeneration do not get off the ground.

But it seems to me that this would represent a significant departure from how IS was originally presented in Priest’s 1994 paper, namely as a formal explanans for the class of self-referential paradoxes. I would suggest that proponents of IS could give us a clearer account of how exactly they see the role of IS in research on paradoxes (in particular, as a non-algebraic or as an algebraic theory, in Shapiro's sense). Priest has already been moving in this direction, for example when he claims that Inclosure paradoxes are those that have to do with contradiction and with the limits of thought as such. However, it is not yet clear to me why Curry does not concern the limits of thought as such (apart from the fact that it is not captured by IS…), so I look forward to the continuation of this debate.


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