On 2/17/2013 10:03 AM, Shmuel (Seymour J.) Metz wrote: > In <88qdnU1ZNsB9j4LMnZ2dnUVZ_uudnZ2d@giganews.com>, on 02/15/2013 > at 11:02 PM, fom <fomJUNK@nyms.net> said: > >> When one invokes the axiom, > >> Ax(x=x) > >> by > >> a=a > >> there is an ontological interpretation of the >> sign of equality corresponding with the sense >> of indecomposability. > > I don't see how it is either omtological or indecomposable. The > inference is valid regardless of how you model "=". >
The validity of the inference being justified by what stance?
A number of years ago, there was a John Correy on these newsgroups who formulated a set of axioms that could be syntactically manipulated to "prove" the existence of non-self-identicals. He had been hammered on the basis of the ontological import of x=x.
In the first edition of "Principia Mathematica" distinction was made between bare quantification ("real variables") and scoped quantification ("apparent variables"). Then, in the second edition, the position changed.
I agree that the step above, within a derivation, is merely a transformation rule. Originally, Frege treated universal quantification as arbitrary substitution with all senseless substitutions being interpreted as false. That was found to be objectionable by those such as Russell.
Outside of a derivation, there is more ambiguity.
>> Of course, mathematicians generally do not know >> of description theory. > > Is that true? I'd buy the claim that for most mathematicians it is not > relevant to their sphere of interest.
That would have been a better way of completing that phrase.
> >> That is, when one presupposes >> the ontological interpretation that gives >> rise to the necessity of >> |- (x=y -> Az(zex <-> zey)) > > Isn't that a special case of a more general axiom schema? For any > propositional function P of two variables, |- (x=y -> Az(P)z,x) >-> > P(z,y)) >
Correct. That schema is generally referred to as Leibniz' law. This is Leibniz' actual statement when introducing the sense of it:
"What St. Thomas affirms on this point about angels or intelligences ("that here every individual is a lowest species") is true of all substances, provided one takes the specific difference in the way that geometers take it with regard to their figures."
Although the value of investigations into formal grammars cannot be denied, to call that schema Leibniz' law is historically inaccurate.
The quote above is made with respect to an intensional syllogistic logic and geometric rigidity. So, one is looking at a nesting of part-whole relations relative to a metric. Cantor's intersection theorem is closer to Leibniz' law than the schema above.
It had been Frege who gave Leibniz' law its logical form. Even if one disregards the fact that he retracted his logicism at the end of his career in favor of a geometric foundation for mathematics, the naive use of that schema is, at least, questionable for set theory. Frege, himself, made the distinction,
"But although the relation of equality can only be thought of as holding for objects, there is an analogous relation for concepts."
It is simplest to express the difference in his statements which follow with the formulas:
AxAy(x=y <-> Az(xez <-> yez))
AxAy(x=y <-> Az(zex <-> zey))
The first is object identity and the second is concept identity.
Moreover, Cantor certainly objected to the treatment of sets as extensions. He insisted that sets had a prior criterion of identity. In his critique of Frege he wrote:
"... fails utterly to see that quantitatively the "extension of a concept" is something wholly indeterminate; only in certain cases is the "extension of a concept" quantitatively determined; and then, to be sure, if it is finite it has a determinate number; if it is infinite, a determinate power. But for such a quantitative determination of the "extension of a concept" the concepts "number" and "power" must previously be given from the other side. It is a twisting of procedure if one attempts to ground the latter concepts on the concept of the "extension of a concept".
One should additionally note that Cantorian set theory had been predicated on a theory of ones, or units, that had been argued against by Frege.
The two formulas given above may be paraphrased as:
"A set is a collection taken as an object"
"A set is determined by its elements"
The first is asserting a prior identity criterion whereas, for pure set theory, the second is not.
What the situation comes down to is this. The predicate symbols 'c', 'e', and '=' are like the fundamental constants in physics. The history of "undefined language primitives" has led to treating a system of relations whose relationship to one another can be made clear as if it cannot.
The real distinction of the situation is found in epistemic logic. For the practical purposes of mathematics, it comes down to the construction of language through definition, because a definition is not the same as a stipulation. A stipulation is impermanent and used to parametrize a problem domain. A definition is expected to have a structural relation to the theoretical coherence of the science.