Scottish Nous

Extensional Equivalence does not Symmetry Make

So we have two syntactically identical sentences which may be converted into one another via mere permutation of terms.  This appears to suggest that, prima facie, whatever is true of the terms in one sentence is true of the terms in the other.  Thus, if we have good reason to suppose that one of the terms is projectible (or suitable for use in inductive inference), then we automatically have just as good a reason for suppposing the same for the other.  (Likewise, if we have good reason to suppose that one of the terms contains ten letters, we have just as good reason for supposing the same of the other.)

If the above can be demonstrated false, then the alleged symmetry between the terms will be broken.  Breaking the symmetry is, I take it, the most appealing way out of the paradox, although John Norton has argued in his paper, forthcoming in Synthese and available here, that where the symmetry cannot be so broken (say as in a grueified total science), the formal equivalence of "grue" and "green" amounts to a mere notational difference.  Still, opting for localized inductive reasoning (rejecting the notion that there is ONE BIG THEORY OF INDUCTION), he thinks symmetry-breaking facts are available in most possible worlds.  How about something a bit more robust?  I'd prefer to show that there are symmetry-breaking facts in all possible worlds, and, in fact, I think that it's relatively easy to do.

My question is this: Why think, however, that the formal equivalence of "grue" and "green" is all that matters vis-a-vis their symmetry?  They are, in virtue of their apparent interdefinability, logically on a par, but it hardly follows that just because two terms are logically interdefinable, if we have good reason to suppose F holds of one, we are automatically entitled to suppose that F holds of the other.  That, it seems, depends upon what F is.  Perhaps the idea is this:  If induction were a demonstrative form of inference, then, as with deduction, logical interdefinability would suffice for showing equivalence in reasoning. 

But, of course, induction is non-demonstrative, and surely the goal of the inductive theorist is not to demonstrate that induction can be made demonstrative!  I think this is where Goodman went wrong.  He gave us the relation of cause-and-effect or the principle of the uniformity of nature, and then, upon the assumption that such a concession was reasonable, tried to make inductive inference demonstrative.  It's no wonder he had such a hard time.

Gruesome Predicates

I wonder how fair the following formulation of the definition of "grue" is to Goodman:

x is grue iffdef  EsAx[[ s = {z| EBzt}] & [[(x e s) & GREENx] V [(x !e s) & BLUEx]]],

where EBzt = the dyadic relation "examined before" which takes some object z and some future time t as arguments.  (If you can't read the above definition, the comments contain the pretty version.  Movable Type is hereby awarded fifty bonus points for making the insertion of logical symbols into comment's infinitely easier than Typepad. Open up a comment window here to see what I mean.  Can you easily use logical symbols in the main post with MT, or just in the comments?)

If the definition is fair, it requires, as a necessary condition upon anything being grue, the existence of some set z of objects such that those objects are examined before some future time t.  Personally, I'm not a fan of predicates which depend for their instantiation upon the existence of a set of observed objects, although I suppose that some quantum physicists and other "world-makers" (such as Goodman) do not share my adversion.  (Some of these folks are even fond of Schrodinger's cat!)  The perplexing philosophy of color aside, I suspect that predicates the instantiation of which is independent of human observation have a better claim to projectibility than their gruesome counterparts.  Spelling out the details of this argument is one of tonight's obligations, though I hope it goes relatively quickly since I'm eager to get to what I take to be more powerful symmetry breaking objections (based on observationality) to the whole gruesome family.