[FOM] Frege's error

[Richard Heck]

Slater's diagnosis of what leads to the contradiction in Frege's system 
would be more convincing if it were accompanied by an explanation of how 
this error manifests itself in the derivation of Russell's paradox. 
That, however, would prove difficult, since the remark reported in 
Carnap's notes---roughly, that, if there is a function F(x,y), then 
there is also a function F(x,x)---has no correlate in Frege's formal 
system.

I take it that Tennant was making essentially this point. There is a 
sense in which no expression that "denotes a function" ever occurs in 
Frege's system, so there can be no formal rule corresponding to this 
informal remark. That is why imposing the framework of the lambda 
calculus onto Frege's system can only be misleading and, indeed, 
anachronistic.

So what did Frege mean by the remark Carnap recorded? It is, in effect, 
a remark about the syntax of the system: If F(,) is a two-place 
function-symbol, then we can form both the expression F(x,y) and the 
expression F(x,x). Not too much controversial there.

That is not to say that there might not be a way of massaging Frege's 
system into consistency in accord with some of the ideas of the lambda 
calculus. At this point, umpteen ways of accomplishing the task are 
known, so that wouldn't be at all surprising, and maybe it would even be 
interesting. We'd have to see the system to be able to tell. (Maybe some 
of Nino Cocchiarella's modifications of comprehension could be 
understood in this light?) But it's hard to see why any one of these 
ways should have any special claim to reveal "the error" Frege made. 
That case would have to be made on independent philosophical grounds.

Richard Heck

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