[FOM] Godel numbers, use, and mention

Hartley Slater slaterbh at cyllene.uwa.edu.au
Thu Jun 5 22:18:36 EDT 2003

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

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