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From: Aatu Koskensilta on 17 Mar 2010 06:25
marcos <marcos(a)tomasacarolina.e.telefonica.net> writes:

> Only one question: you invoke (*) (T' |- (x)(y)(B(x,f(x)) and B(x,y)
> --> y=f(x) to prove (Eu)B(x,u)&Q<-->R(t). Please, explain me the
> steps.

For simplicity, suppose the formula P under consideration is R(f(t))
where t is a term in which f does not occur. Applying the transformation
we obtain P*:

(Eu)(B(t,u) & R(u)) <--> R(f(t))

In the <-- direction, suppose that R(f(t)). We use the axiom B(x,f(x))
and existential generalization to get (Eu)(B(x,u) & R(u)):

1. R(f(t)), by assumption
2. B(t,f(t)), by the defining axiom for f
3. B(t,f(t)) & R(f(t)), combining 1. and 2.
4. (Eu)(B(t,u) & R(t)), from 3. by existential generalization

In the --> direction, assume that (Eu)(B(t,u) & R(u)). We use (*)

(x)(y)(B(x,f(x)) and B(x,y) --> y = f(x))

to get R(f(t)) as follows:

1. (Eu)(B(t,u) & R(u)), by assumption
2. B(t,a) & R(a), introducing the name a to stand for an
object asserted to exist in 1.
3. B(t,a), from 2.
4. R(a), again from 2.
5. a = f(t), from 3. and (*)
6. R(f(t)), from 5. and 4.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 17 Mar 2010 07:17
Aatu Koskensilta <aatu.koskensilta(a)uta.fi> writes:

> We use (*)
>
> (x)(y)(B(x,f(x)) and B(x,y) --> y = f(x))

Since we have the axiom (x)B(x,f(x)) (*) simplifies to

(x)(y)(B(x,y) --> y = f(x))

which is perhaps a bit more transparent.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: marcos on 17 Mar 2010 11:59
I understand it at last!!. Now I think that it's very simple, but the
book should have explained better. Thanks, Aatu Koskensilta and
William Elliot, thank you very much !!.

Marcos Castillo

From: Aatu Koskensilta on 17 Mar 2010 14:01
marcos <marcos(a)tomasacarolina.e.telefonica.net> writes:

> Now I think that it's very simple, but the book should have explained
> better.

In the preface to the second edition Mendelson explains the text "was
intended to be a simple, clear introduction to mathematical logic
unencumbered by excessive notation and terminology". But, as we know
from literary theory, authorial intent doesn't count for much.

I've developed quite a distaste for the text -- which is by no means
altogether deplorable, and does have significant redeeming qualities --
mainly owing to Mendelson's style. (His occasional abuse of the English
language can also be somewhat jarring.) I wonder, though, if those who
have taught logic using this text would be willing to share their
opinion?

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: marcos on 18 Mar 2010 06:43
On 17 mar, 19:01, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> marcos <mar...(a)tomasacarolina.e.telefonica.net> writes:
> > Now I think that it's very simple, but the book should have explained
> > better.
>
> In the preface to the second edition Mendelson explains the text "was
> intended to be a simple, clear introduction to mathematical logic
> unencumbered by excessive notation and terminology". But, as we know
> from literary theory, authorial intent doesn't count for much.
>
> I've developed quite a distaste for the text -- which is by no means
> altogether deplorable, and does have significant redeeming qualities --
> mainly owing to Mendelson's style. (His occasional abuse of the English
> language can also be somewhat jarring.) I wonder, though, if those who
> have taught logic using this text would be willing to share their
> opinion?
>
> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
>  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

I've got two different opinions about this book: until Godel's
completeness theorem I find the book readable, easy to understand; but
since the mentioned theorem it is not that easy, and it leaves to the
reader the comprehension of important steps in the proof of the
statements.
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