Re: if the barber cuts everyones beard who doesn't shave himself....

On 25 May 2005 17:39:03 -0700, ken quirici <kquirici@xxxxxxxxx> said:
> Chris Menzel wrote:
> ...
>> >> 1. Ax(Rxy <-> ~Rxx) -> (Ryy <-> ~Ryy) Universal Instatiation
>> >> 2. ~(Ryy <-> ~Ryy) Theorem of propositional logic
>> >> 3. ~Ax(Rxy <-> ~Rxx) 1,2 Modus Tollens
>> >> 4. Ay~Ax(Rxy <-> ~Rxx) Universal Generalization
>> >> 5. ~EyAx(Rxy <-> ~Rxx) Negation/Quantifier Exchange
>>>
>> Yes, that is correct, though I am not really *concluding* this, I am
>> rather simply *reporting* to you that it is legitimate to apply UG here.
>> The *reason* it is legitimate is that, first, line 4 is a correct
>> application of the rule and, second, it is rather trivial to prove in
>> the model theory of predicate logic that UG is a sound (i.e.,
>> truth-preserving) rule of inference.
>>
>> Model theory provides a mathematically rigorous account of meaning for
>> logical languages, and is a standard part of an intermediate-level
>> course on predicate logic. I suspect you'd find the subject matter
>> illuminating.
>>
>> Regards,
>>
>> Chris Menzel
>
> Thanks. I'll look it up. I can only hope I'll be illuminated.
>
> Anyway, back to the matter at hand:
>
> http://www.wwnorton.com/college/phil/logic3/ch14/univgen.htm
>
> pls look at this ref. It indicates that you can't universally
> generalize (at least I take it to mean this) a letter in a wff,
> when the letter is an existential instantiation of a wff from
> which the current one is derived. And that's what the (1) way
> up above is - an existential instantiation of the wff you really
> want to prove.

No it isn't. Why do you think it is? What we're trying to prove is
Line 5, which isn't even existentially quantified.

> How would you respond to this? Am I misunderstanding?

Yes. Line 1 stands on its own as an axiom, one that is easily seen to
be logically valid.

> I could summarize my objections very simply as, if all you know is,
> 'there exists a y', how can you conclude 'for all y'?

You can't. But the point is irrelevant, as it has nothing to do with
the matter at hand.

> To be honest, I'm really trying to poke holes in the proof because I
> resent something so simple coming along and with a nonchalant wave of
> the hand dismissing the strenuous efforts of the ZF gang.

Who's doing that? No one has raised any problems for ZF in this thread.

> I still hew to my objections until I'm totally crushed.

You have yet to voice a coherent objection to anything.

Cheers.

Chris Menzel

.

Relevant Pages

  • Re: if the barber cuts everyones beard who doesnt shave himself....
    ... > rather simply *reporting* to you that it is legitimate to apply UG here. ... > the model theory of predicate logic that UG is a sound (i.e., ... generalize a letter in a wff, ... I still hew to my objections until I'm totally crushed. ...
    (sci.logic)
  • Re: Two questions on First order Model Theory
    ... _If_ all the notation means what I'm guessing it does then the ... any sense, because is not even a wff (can't be, because ... Usually in model theory one does not distinguish between a and c_a. ...
    (sci.logic)