Re: FOL and Infinity



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. Your definition of AxPhi(x) is "For any
constant c
in the language, AxPhi(x) implies Phi(c)'.
You cannot explain or define AxPhi(x) without using 'For any(all)'
That is where you are into
an interminable regress.
The sentence Phi(1) &Phi(2) implies each of Phi(1) and Phi(2).
The sentence Phi(1) &Phi(2) &Phi(3) implies each of Phi(1) , Phi(20
and Phi(3).
One could go on like this, but there is no sentence of sentential
logic which will imply
Phi(i) for every i.
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.
The big assumption is that such a limit exists; that such a sentence
(comprising of a finite string of symbols)
can convey that each of the independent sentences Phi(1),Phi(2) and so
on is true. AxPhi(x) is a shorthand for
an infinite amount of information ; it is the first idol in the Temple
of Infinity.

..

.



Relevant Pages

  • Re: 3 basic metalogic questions.
    ... in B. But here B is a sentence, without any free variables. ... the language which does not specify an interpretation of the ... You meant to say that EyAxRimplies AxEyR. ... Suppose you have an interpretation M of your language containing the ...
    (sci.logic)
  • Re: Perl "expert"
    ... fact that these words are coming from my fingers implies it's an ... you stop and ask directions to Loch Ness. ... Perl gets an unjustified rap as a write-only language because people pass ...
    (perl.beginners)
  • Re: FOL and Infinity
    ...  in the language, AxPhiimplies Phi'. ... AxPhiis a wff. ... What do you mean by "sentence symbols"? ... that "AxPhi" is a formula of the language. ...
    (sci.logic)
  • Re: assertion redefinition
    ... I am trying to convince a non-Eiffel language supplier to add DBC to ... equivalent to no precondition at all. ... the general expression "a implies b" where a and b evaluate to True ... Because now the weaker precondition for B.mathfunc is ...
    (comp.lang.eiffel)
  • Re: strange getch+kbhit behaviour
    ... According to my understanding of the language, ... will'' implies ``not sure if I will''. ... I'm complaining about you continuing to dig when you should have said ...
    (comp.lang.c)