[FOM] historical question about the axiomatisation of identity

[Hartley Slater]

Richard Heck writes:

>A reconstruction of Frege's system that does without the identification
>of sentences as names could thus simply take identity to be axiomatized
>by (IIIa) and (III*), that is, basically, by (III'). Of course, this
>treatment is ineliminably second-order.

He is too kind, or at least he does not see what an immense amount of 
further work would need to be done to make plausible such a 
reconstruction, even supposing it could be done.

If one separates out identity from equivalence, as Heck suggests, 
then what is to become of the idea that concepts are functions, for a 
start?  This idea arises from the attempt to see, for instance, what 
might be put 'Fa <-> T', where 'T" is a tautology, as like 'f (a)=T' 
where 'T' is a referring phrase.  In addition, truth is a property of 
thoughts so in connection with *truth* one has properly neither an 
equivalence, nor an identity but a simple predication 'T---Fa' where 
'T' is 'is true', and '---' is (a bit like) Frege's horizontal. 
Frege was also confused in this latter area too - see my recent 
paper 'Choice and Logic' in JPL (2205) 34, 207-216, especially p214f. 
- so all in all it is much the best to trace accurate historical 
definitions of identity elsewhere.
-- 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 6488 1246 (W), 9386 4812 (H)
Fax: (08) 6488 1057
Url: http://www.philosophy.uwa.edu.au/staff/slater