[FOM] Frege's Error
Hartley Slater
slaterbh at cyllene.uwa.edu.au
Thu Aug 11 21:08:29 EDT 2005
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
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