[FOM] historical question about the axiomatisation of identity

[Richard Heck]

Muller F.A. wrote:

>Two historical questions.
>
>Q1: Were Hilbert & Ackermann the first to [axiomatize identity as an equivalence relation that obeys substitutivity]?
>
>Q2: Was Godel the first to see that reflexivity and substitutivity suffice?
>
>F.A. Muller
>Utrecht University & Erasmus University Rotterdam 
>
The exact answers to these questions depend somewhat upon the background
logic, but I think the answer is "No" in both cases.

In "Principia", Russell axiomatizes identity using just substitutivity,
in the form:
a = b iff (F)(Fa --> Fb)
Of course Russell then goes on to prove that identity is an equivalence
relation. Note the use of the conditional instead of the otherwise
expected biconditional. Russell may have gotten this treatment from
Frege, as I'll explain.

Frege's axiomatization of identity in "Begriffsschrift" uses just
reflexivity (54) and substitutivity (52)., the latter in the form:
c = d --> (Fc --> Fd)
One would suppose G\"{o}del knew of this. Frege proves symmetry (55) but
not, so far as I know, transitivity, though it's hard to imagine that he
didn't know it could be proven. (Frege knew what equivalence relations
were, and he knew that identity was one.) That suggests that it was
Frege who was the first to realize that reflexivity and substitutivity
sufficed.

For the occasionally over-sensitive: I'm not claiming that G\"{o}del
plagarized anything. I doubt a reference would have been required there.
The possibility of the sort of axiomatization G\"{o}del uses may have
been well known by 1930. Anyone know?

Of course, Frege's system is second-order, but there's nothing Frege
does with identity that needs the second-order stuff.

The history following "Begrifffsschrift" is messy but interesting.

There is a little noticed remark in the 1881 paper on Boole in which
Frege says that he would then prefer to *define* identity in other
terms. He doesn't say how he proposes to do so. However, there is little
option but to suppose that he intended to do so in terms of Leibniz's
Laws. This suggestion is reinforced by remarks in "Die Grundlagen":
Frege also says there that he intends to adopt Leibniz's suggestion as
an Erkl\"{a}rung of identity, which Austin mistranslates "definition".
Still, it seems to me Frege still holds the view from 1881. In any
event, he claims a little later that "[I]n universal substitutivity all
the laws of identity are contained" (section 65), a claim one would
naturally take to be based upon his possession of the relevant proofs.

By "Grundgesetze", Frege has changed his mind about defining identity,
but the formal treatment of it remains unchanged. One has to be a little
careful here. Frege's identification of sentences as names causes
identity to play (what we would see as) a double role: It's identity,
but it's also, in effect, the biconditional. There are thus two basic
laws that govern identity:
(III) g[a=b] --> g[(F)(Fa --> Fb)]
(IV) (---a = -+- b) v (--- a = ---b)
Here, I am using "---" for Frege's "horizontal", the function that takes
the True to the True and everything else to the False and "-+-" for his
negation. What (IV) says is thus roughly equivalent to what we would
write as:
(IV') (p <--> q) v (p <--> ~q)
It's thus a form of excluded middle, so it doesn't really have anything
to do with identity. So it may be set aside.

(The identity sign also occurs in basic laws V and VI, by the way, which
we may again set aside, as they are non-logical, by our lights. The
former concerns value-ranges; the latter, Frege's form of the
description operator, which operates on value-ranges.)

As far as (III) is concerned, it takes the form it does because of
Frege's identification of sentences as names. What Frege wants to write is:
(III') a = b <--> (F)(Fa --> Fb),
but he can't. He could only write:
(III'') (a = b) = (F)(Fa --> Fb),
and that won't do the work he wants. So he gives us (III), which is
really only applied twice, as far as the theory of identity goes. In the
first application, g() is taken to be the horizontal, so that we have:
(III''') a = b --> (F)(Fa --> Fb),
whence UI (basic law IIb) delivers:
(IIIa) a = b --> (Fa --> Fb),
In the other application, g() is taken to be negation, whence we have:
~(a = b) --> ~(F)(Fa --> Fb),
and contraposition yields:
(III*) (F)(Fa --> Fb) --> a = b.
Frege doesn't state this general fact, but he uses a form of it to prove
reflexivity, (IIIe).

There is one other application of (III), but it is not really to
identity as such. It's used in the proof of:
(IIIi) (--- a = b) = (a = b),
which says that (a = b) is always a truth-value. (A form of (III*) is
also used here, by the way.)

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.

I don't know if Frege ever proves transitivity as a theorem in
"Grundgesetze", either. Perhaps he just doesn't need it. Again, however,
it's hard to imagine he didn't know it could be proven.

Richard Heck