[FOM] Consistency of Robinson arithmetic

Keith Brian Johnson joyfuloctopus at yahoo.com
Thu May 19 01:07:57 EDT 2011

Richard Heck wrote:

>Perhaps the most intriguing possibility, to me, however, lies in work on
>what people call "transmission failure". These are cases where one knows
>that P, has a good (logical) argument from P to Q, and yet it seems as
>if one can't (come to?) know Q on that basis. A classic sort of
>(putative) example is: I know that that thing in the cage over there is
>a zebra; if it's a zebra, then it isn't a cleverly disguised mule; so I
>know that it isn't a cleverly disguised mule. Here, it seems as if one's
>belief (even knowledge) can't support the claim that the thing isn't a
>cleverly disguised mule. It's as if there's a sort of epistemic circularity.

While I have not yet published my account of this, I have given it some 
thought.  It seems to me (to put this very briefly) that there is no real 
problem here if one speaks carefully; the problem comes if one speaks sloppily 
in a way that obscures the possibility of error, so that paradoxical examples 
exploiting that very possibility can be created.  There are two sources of 
error:  First, our ordinary metaphysical and epistemic assumptions (such as that 
our basic thought processes are reasonably reliable, that what feel like 
memories of mental phenomena previously experienced really are, and that [for 
empirical claims] there really is an objectively existing reality of which our 
senses give us reasonably reliable information); second, our fallibility even 
given those basic assumptions.  (Let's conjoin those metaphysical and epistemic 
assumptions and call them "T.")  Because of this fallibility, we should usually 
say not that we are (fully) justified in believing that p, but that we are 
(fully) justified in believing that probably-p (or, alternatively, not that we 
are fully justified in believing that p but that we are only partially justified 
in believing that p, with some degree of epistemic likelihood); so, normally, we 
should say not "S knows that p" but rather "Given T, S knows that probably-p."  
Thus, when we say that S is justified in believing (read "knows") that that 
animal is a zebra, then should also say that S is justified in believing (read 
"knows") that that animal isn't a cleverly-disguised mule, for the simple reason 
that what we really mean--or *should* really mean--is that given T, S is 
justified in believing (read "knows") that that animal is probably a zebra, so 
that given T, S is justified in believing (read "knows") that that animal is 
probably not a cleverly-disguised mule.  The former epistemic likelihood 
("probability") can hardly be high if the latter isn't.

Therefore, I don't think that anyone "worried about some kind of circularity in 
consistency proofs," as Heck puts it, should appeal to the transmission failure 
problem above.

Keith Brian Johnson

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