Re: FOL and Infinity



On Apr 12, 11:24 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Apr 12, 1:10 pm, sudhir <sudhir...@xxxxxxxxxxx> wrote:

On Apr 12, 12:39 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:> On Apr 11, 1:27 pm, sudhir <sudhir...@xxxxxxxxxxx> wrote:
.
How does the sentence AxPhi(x) imply Phi(1) ,
unless some intuitive semantic content
is read into AxPhi(x)?

No matter what the interpretation of the language, AxPhi(x) implies
Phi(c) for any constant 'c' in the language. That is, for any constant
'c' in the language, there is no model in which AxPhi(x) is true but
Phi(c) is false.

This is precisely what I said.

No, it's not.

Your definition of AxPhi(x) is "For any
constant c
 in the language, AxPhi(x) implies Phi(c)'.

No, I didn't give a "definition" of a formula AxPhi(x). Indeed, I said
that I don't know what you mean by a "definition" of such a formula.
You skipped what I wrote.
I know you do not have a definition for AxPhi(x). All you have is a
intuition
captured in your rule UI-'For all constants c, AxPhi(x) implies
Phi(c).'


..
Phi(i) is not even a formula of sentential logic.
I thought the meaning was clear -if Phi(1), Phi(2) and so on are
sentence symbols, then no
wff of sentential logic could imply each of them.

Precisely such a sentence is introduced  in FOL , namely  AxPhi(x) ..In
a way, AxPhi(x) is
a limit to the sequence
Phi(1), Phi(1)&Phi(2), Phi(1)&Phi(2)&Phi(3), and so on.

You like to use words like "limit" in your own personal way, it seems.

The big assumption is that such a limit exists;

No, there's no notion of such a limit when we formulate a language so
that "AxPhi(x)" is a formula of the language.
..Of course you are not formally introducing the notion of a limit.You
are informally
looking for a sentence from which you can infer each of the sentences
Phi(1),Phi(2) and so on.
In sentential logic, you do not find such a sentence. So you introduce
the sentence AxPhi(x)
(you could use any other symbol ) and claim that from it you can infer
each of Phi(1) ,Phi(2) and so on.
It is nothing but a finite shorthand for an infinite amount of
information.

.