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Topic: Ordinals describable by a finite string of symbols
Replies: 16   Last Post: Jul 9, 2013 7:59 AM

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Posts: 1,968
Registered: 12/4/12
Re: Ordinals describable by a finite string of symbols
Posted: Jun 30, 2013 11:43 AM
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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"

"Critique of Pure Reason"

So, what is he talking about? Consider the two

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:

Ax(x=V() <-> Ay(-(ycx <-> y=x)))

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)."

"Posterior Analytics"

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:

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