fom
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Re: Ordinals describable by a finite string of symbols
Posted:
Jun 30, 2013 11:43 AM


On 6/30/2013 9:05 AM, apoorv wrote:
> Well, my sense is that the infinite enters mathematics or logic through the universal > Sentence..The universal sentence attempts to use ' a finite string ' to represent or define > Something infinite. To me, it is an appealing proposition, that 'Infinity cannot be finitely represented'. I made some posts earlier along those lines, but obviously, the mainstream > View is different. > Apoorv
I saw those posts and had been appreciative of the remarks.
You are basically correct. Consider this remark by Kant:
"Logicians are justified in saying that, in the employment of judgements in syllogisms, singular judgements can be treated like those that are universal. For, since they have no extension at all, the predicate cannot relate to part only of that which is contained in the concept of the subject and be excluded from the rest. The predicate is valid of that concept without any such exception, just as if it were a general concept and had an extension to the whole of which the predicate applied. If, on the other hand, we compare a singular with a universal judgement, merely as knowledge, in respect of quantity, the singular stands to the universal as unity to infinity"
Kant "Critique of Pure Reason" B96
So, what is he talking about? Consider the two sentences:
All men are mortal.
Plato is mortal.
The first is a universal judgement and the second is an existential judgement. Kant is saying precisely what you are saying. Indeed, how is "... the singular stands to the universal as unity to infinity" that different from "... the infinite enters mathematics or logic through the universal"
But, the intent of Leibniz' principle of identity of indiscernibles is to discern objects through differences expressible by logical predicates. Hence, Leibnizian identity in this regard is arrived at via the inability to prove a difference. Hence, identity within a domain of discourse is merely the complement of diversity.
So, where the problem arises is with definition.
Consider the sentences:
Definition: Ax(x=V() <> Ay((ycx <> y=x)))
Assumption: ExAy((ycx <> y=x))
where 'c' is an irreflexive transitive predicate.
The definition is a universal statement that establishes the denotation for an object.
The sign of equality in the definiendum "warrants" the use of the denotation as a singular term . That warranting is invalid if the definiens is not satisfied uniquely within the domain of discourse. This is the relational aspect that identity is warranted if diversity cannot be proved.
It is the existence statement that substantiates the domain (more precisely, "substantiates the essence of the definition"). Thus, it is the existence statement that forces the substantiation of what is presupposed in the definition.
These relationships are expressed in Aristotle:
"If a thesis asserts one or the other part of a contradiction  for example, that something is or that something is not  it is an assumption; otherwise it is a definition. For a definition is a thesis (since the arithmetician, for example, lays it down that a unit is what is indivisible in quantity), but it is not an assumption (since what it is to be a unit and that a unit is are not the same)."
Aristotle "Posterior Analytics" 20
To make sense of the earlier part of Kant's remarks, note that "species" partition "genera" in the syllogistic hierarchy. So, taking an example of Leibniz,
A rational man is a man
AB is B
one sees that the genus 'man' has a species 'rational man' so that the predicate 'is rational' has application in the logical system.
Kant is saying that the concept of a singular judgement cannot be partitioned in the sense of a general concept. Nevertheless, the universal denoted by 'all' applies uniformly; that is, one may treat a singleton as a domain of discourse.
You are correct in saying that the mainstream view is different. Whereas the remarks I have given above distinguish between the warranted use of a denotation and its existence, you will find that in "classical" logic, there is a presupposition whereby denotation is bound with existence.
This is the very first sentence in the link:
http://plato.stanford.edu/entries/logicfree/

