[FOM] Godel numbers, use, and mention

[Hartley Slater]

Dean Buckner writes (FOM Digest Vol 6 Issue 8):

>This could revolutionise our approach to criminal detection!  It's certain
>that Jack the Ripper committed all the Whitechapel murders.  But it's
>uncertain whether any of the "usual suspects" committed them.  So, by
>Leibniz, we can rule them all out.

There is no trouble with Leibniz' Law in intensional contexts, since 
with a proper analysis of referential terms, like that provided in 
the epsilon calculus, it is easily seen that reference is still 
transparent in such contexts.  Nevertheless, if a is one of the 
suspects, and it is just known that there is a murderer, it is quite 
consistent that it is not known that a is that murderer.  K(Ex)Mx 
does entail KMexMx (where 'e' is epsilon), since (Ex)Mx=MexMx.  But 
K(Ex)Mx does not entail KMa, even if Ma, for there might be two 
murderers, and yet a not be known to be *that* one, since even in 
fact he needn't be *that* one.

One needs to have uniqueness before an identification can be made, 
i.e. one needs K(Ex)(y)(My <-> y=x) with Ma.  The former entails 
(Ex)(y)(My <-> y=x), and so (y)(My <-> y=ex(y)(My <-> y=x)), and Ma 
<-> a=ex(y)(My <-> y=x), giving a=ex(y)(My <-> y=x), and since 
KMex(y)(My <-> y=x)   {because K(y)(My <-> y=ex(y)(My <-> y=x)), and 
so K[Mex(y)(My <-> y=x) <-> ex(y)(My <-> y=x)=ex(y)(My <-> y=x)], and 
automatically K[ex(y)(My <-> y=x)=ex(y)(My <-> y=x)]}, we get KMa.

The point formally reinforces Gettier's well known conclusion about 
knowledge.  Someone justifiably believing there is a man in the room 
may thereby justifiably believe there is a person in the room, and 
this can be true, and yet they would not know that there was a person 
in the room, if the initial belief was mistaken, and in fact there 
was only a woman there.  Likewise, having the wrong suspect for the 
one and only murderer would not count as knowledge that there is such 
a murderer, even if there was such a murderer.  A conviction, or 
positive identification must be obtained, in other words, before 
knowledge can be claimed.  See my book 'Intensional Logic' (Avebury, 
Aldershot 1994, pp129-130), or my paper 'Routley's Formulation of 
Transparency'  in History and Philosophy of Logic 13 (1992) 
pp220-221).  There is a logic behind it all, even if many still want 
to believe it is one great big mess.
-- 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html