[FOM] Frege's Error

[Hartley Slater]

At 12:00 PM -0400 11/8/05, TMV Janssen wrote:
>Let be given f defined by f(x,y) = x>y.
>Then we may define g by g(x) = f(x,0) and h  by h(x)= f(x,x).
>
>There is nothing wrong or problematic with Frege's remarks one finds in
>Carnap's lecture notes.

The question is whether one can 'let f be defined by f(x,y) = x>y'. 
Here is the abstract of my talk at the coming AAL05 conference in 
Perth this September, and notice in particular the points about 
identity and equivalence.  A copy of the full paper is available on 
request.

Frege's Error Identified.

It is well known that it was Russell's Paradox that alerted Frege to 
the trouble with his system.  It is less well known that there is no 
trouble when 'x is not a member of x' is analysed relationally, i.e. 
as saying that <x,x> is a member of {<y,z> | y is not a member of z}. 
For substitution of that set abstract for 'x' does not produce a 
contradiction.  What Russell's Paradox shows, therefore, is merely 
that not all relations between a thing and itself can be a matter of 
that thing falling under a concept, i.e.  -(R)(EP)(x)(Rxx iff Px). 
Taking off from this, it is what might have led Frege to think 
otherwise which is the concern of the present paper.
Clearly, given a two-valued function f(x,y), one can invariably 
obtain a function of one variable f(x,x)=g(x).  So it was Frege's 
analogy between functions and predicates, in 'Function and Concept', 
which led him astray.  Predicates are not functions in the required 
way.  First, if anything like 'Pa=T', or 'Rab=F' holds it is with '=' 
as material equivalence, 'T' a tautology, and 'F' a contradiction 
(thus sentences are not referential terms with the same reference as 
'The true' or 'The false').  But then one has that 'Pa iff T' and 
'Rab iff F' are equivalent to 'Pa' and '-Rab' respectively, making 
'Pa' and 'Rab' quite unlike mathematical functions, and 'T' and 'F' 
nothing like their values.  Indeed, the values of predicative 
expressions are thoughts, not truth values.  But such inaccuracies in 
the parallel between predicates and functions get dramatically 
enlarged upon the introduction of reflexive expressions. A relational 
expression like 'Rxy' generates a thought about x and y, and so the 
diagonal expression 'Rxx' generates a thought about x and itself. 
That thought still has two subjects, and is not expressible by a 
one-place constant predicate with x as subject.  If each of A, B, and 
C shaves D they do the same thing - shave D - but if each shaves 
himself, or say, in a ring, shaves his neighbour on his left, then 
they only do the same kind of thing, i.e. what they do merely has a 
common functional expression: shave f(s) where s is the subject.
-- 
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