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

Chris Menzel wrote:
> > >> A little better, I think, as it avoids Owen's metatheoretic step 2:
> >>
> >> 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. How would you respond to this? Am I misunderstanding?
Or am I understanding, but the point is incorrect.

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

Ken

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.
On the other hand, that's where the beauty in math (and I guess logic)
comes from - the sudden unification and simplification. However,
all that aside, and in the absence of a store of understanding from
model theory, I still hew to my objections until I'm totally crushed.
I estimate in about 15 mins.

.

Relevant Pages

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