[FOM] Frege and the Truth-Values [Was: historical question about the axiomatisation of identity]

[Richard Heck]

Hartley Slater wrote:

> 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 [Leibniz's Laws]. 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.

So far as I can see, the reconstruction is formally trivial: It's 
ordinary second-order logic, and the translation between Frege's system 
and the ordinary one is also trivial, *as far as the formulae that are 
actually used in the construction of arithmetic are concerned*. Of 
course there are some things that can be expressed and even proven in 
Frege's system that cannot be expressed or proven in the standard 
system. For example, Frege claims in section 10 of "Grundgesetze" that 
each of the two truth-values is its own unit class. That has to go. It 
is notable, however, that Frege does not actually establish an axiom 
that states or even implies this claim: It is not provable in the formal 
system (or wouldn't be, if the system were consistent, that is, it 
isn't, when the system is made consistent).

> 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?

I don't see that there is any problem there. For one thing, the idea 
that concepts are functions is present already in "Begriffsschrift", 
whereas the idea that sentences are of the same logical type as names is 
not: Sentences do not have the same grammatical distribution as names in 
that system. They do both occur flanking the sign for identity of 
content, but there is no indication that Frege would have regarded 
"(F)(x)(Fx -> Fx) = 0" as well-formed, where "=" should really be the 
triple bar (\equiv). For another, as Dummett pointed out thirty years 
ago, if there are truth-values, then there is no reason there can't be 
functions that have them as values. It just doesn't matter whether 
truth-values are objects. Lots of people recognize functions that take 
other functions as their values, and you don't have to think that 
functions are objects to hold that view.

> This idea [that concepts are functions] 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. 

If what is meant here is that the idea aries in this way FOR FREGE, then 
the claim is demonstrably false. As noted, the idea that concepts are 
functions is present in Begriffsschrift. (In fact, every example of a 
"function" that Frege gives in Begriffsschrift is actually of what we 
would call a predicate: See section 9.) As Frege himself notes in the 
introduction to Grundgesetze, however, the idea that sentences denote 
truth-values is not present there. It does not appear explicitly until 
twelve years later, in "Function and Concept".

If the claim is that one can only really make sense of the idea that 
concepts are functions by seeing "Fa <--> T" as 'like' "f(a) = T", then 
I have no objection, but I don't see why that should be any problem. Why 
can't the former be relevantly and illuminatingly LIKE the latter even 
if truth-values are not objects?

My own view is that Frege was led to regard truth-values as objects 
because he was overly impressed by certain sorts of simplifications the 
treatment of truth-values as objects made possible. These are connected 
with his treatment of "double value-ranges".

> In addition, truth is a property of thoughts[...]

Frege famously denied that claim, and I'd deny it, too, in the same 
sense and for much the same reasons. But that is a very long story.

> [...]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. 

I won't comment on the substantive issues here. I would note, however, 
that the horizontal is the easiest thing of all to reconstruct: It's 
just the unary truth-function that corresponds to identity. Of course, 
it's a lot less useful in the reconstructed system than it was in 
Frege's own.

> 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. 

As I mentioned in the earlier note, Frege axiomatizes identity using 
just reflexivity and substitution in "Begriffsschrift", which is, for 
the reasons mentioned above, immune from any concerns along the lines 
expressed here. I therefore don't see why one shouldn't regard it as an 
"accurate" axiomatization of identity.

Richard Heck