Pro globalization

Writing the last post reminded me of something that came up when I was last up in St Andrews visiting the lovely people at Arche (doubly lovely that time since they gave me a phD the same week). While thinking about stuff presented by (among others) Achille Varzi, Greg Restall and Dominic Hyde, I suddenly realized something disturbing about super and sub-valuationists notions of "local validity". (Local validity, by the way, is important because everyone accepts that *its* not revisionary. The substantial question is whether *global* validity is revisionary. Lots of people think it is, and I'm inclined to think not). Below the fold, I explain why....

It's easiest to appreciate the worry in the dual "subvaluationist" setting. Take a standard sorites argument, taking you from Fa, through loads of conditional premises, to the repugnant conclusion Fz. Now the standard subvaluationist line is that though every premise is (sub-)true, the reasoning is invalid (*global* subvaluational consequence departs from classical consequence on multi-premise reasoning of just this sort.). But local validity matches classical validity even on multi-premise reasoning (details are e.g. in the paper Achille Varzi presented to Arche).

Problem! We've got a valid argument with true premises, whose conclusion is absurd (and in particular, it's not true: even a dialethist can't accept it). It really doesn't come much worse than that.

You can reconstruct the same problem for a supervaluationist using local validity, if you take multi-conclusion logic seriously. And you should. It addresses this question: if you've established that a load of propositions fail to be true, what can you conclude? If the conclusions C follow from the premises A, then if each of the conclusions are "rejectable" (fails to be true) one of the premises is rejectable (fails to be true).

Take a sorites series a, b, c,....,z and consider the following set of formulae: {Fa&~Fb; Fb&~Fc; ....;Fy&~Fz}. In a classical multi-conclusion setting, the premises {Fa, ~Fz} entail this set of conclusions. The result therefore carries over to a supervaluationist setting under local validity (but - crucially - not with global validity).

Now, each of the conclusions is really bad (only an epistemicist could buy into one of them). For the supervaluationist, they're each rejectable. So one of the premises must be rejectable too. But of course, neither is.

Either way, this seems to me pretty devastating for "local validity" fans. (NB: I chatted about this to Achille Varzi, and he's put forward a response in the footnotes of the paper cited above. I don't think it works, but it raises some really nice questions about what we want a notion of consequence for.)