>
> Can anyone think of any significant mathematical difference between the
> following two arguments?
>
> 1. s Premise
> 2. {x: x = d & s} = {x: x = d} From 1., given substitution salva
> veritate of logical equivalents
> 3. {x: x = d & t} = {x: x = d} From 2., given substitution salva
> veritate of co-referring terms
> 4. t From 3., given substitution salva
> veritate of logical equivalents
>
> And (with i is the iota/definite-description operator)
>
> 1. s Premise
> 2. ix(x = d & s) = ix(x = d) From 1., given substitution salva
> veritate of logical equivalents
> 3. ix(x = d & t) = ix(x = d) From 2., given substitution salva
> veritate of co-referring terms
> 4. t From 3., given substitution salva
> veritate of logical equivalents
>
I think the argument you wanted might be better stated this way.
Consider the following terms:
1. The fact that p
2. The fact that: {x: x = a} = {x: x = a & p}
3. The fact that: {x: x = a} = {x: x = a & q}
4. The fact that q
Then (1) = (2) and (3) = (4), by logical equivalence, and (2) = (3), by
substitution of identicals.
How are we to understand the notation "{x: ...}"? In formal treatments
of set theory, we don't have such notation. One can introduce it, to be
sure, but the usual way of doing so would be via Russell's theory of
descriptions: Take "{x: A(x)}" to abbreviate "iy[(x)(x \in y iff A(x)]".
Now, of course, one will have all the usual problems about scope, since
not all such descriptions are proper. For example, the term "{x: x=x}"
or "iy[(x)(x \in y iff x = x)]" is improper. And, in any event, we've
just re-introduced descriptions.
In order to make this work, then you would need to treat terms of the
form "{x: A(x)}" as primitive denoting expressions. There are consistent
systems in which this can be done. An example dear to my own heart would
be the predicative fragment of Frege's /Grundgesetze/. Changing notation
to something like Frege's, we can then write the foregoing argument thus:
1. The fact that p
2'. The fact that: ^x(x = a) = ^x(x = a & p)
3'. The fact that: ^x(x = a) = ^x(x = a & q)
4. The fact that q
But now one can question the claim that (1) = (2'). If 'p' abbreviates,
say, "Pigs fly", then it is not at all clear that "Pigs fly" is
LOGICALLY equivalent to: ^x(x = a) = ^x(x = a & p). Perhaps they are
equivalent in some strong sense, but the existence of ^x(x = a) is not
obviously a fact of LOGIC. Even if we waive that point, one will need to
appeal to some principles governing these terms to show that (1) and (2)
are logically equivalent. The obvious principle here would be Frege's
Basic Law V:
^x[A(x)] = ^x[B(x)] iff (x)[A(x) = B(x)]
But it is very unclear indeed that Law V is a truth of logic, even if
the background logic is predicative.
So, yes, there does seem to be a significant difference between this
formulation and the usual one in terms of descriptions.
Richard Heck
--
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Richard G Heck, Jr
Professor of Philosophy
Brown University
http://bobjweil.com/heck/
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