Re: completeness what is it exactly

On Jul 9, 6:59 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Jul 9, 10:50 am, translogi <wilem...@xxxxxxxxxxxxxx> wrote:

.so i guess
A theory T is complete :<=> forall "A" (T |- A v T |- ~A)

 is false

if a is an variable/ contingent neither T |- A nor  T |- ~A

Do you mean where A is a formula that may have free variables in it?

If so, then, yes, "T is negation complete" does not say that for every
formula (even if it has free variables) A in the language of T, we
have T |- A or T |- ~A.

Rather, "T is negation complete" says that for every SENTENCE A in the
language of T, we have T |- A or T |- ~A.

MoeBlee

But then what is then a sentence?
Not every wff (well formed formula )

Or more precisely
what makes

1) P -> P a sentence
and
2) (P-> P) -> P not a sentence?
or should i read it as

1a) for every P [P->P}
but then the second one should be read as
2a) for every P [(P->P)->P]
and so it is a sentence (although a false one)

Or maybe if a sentence defind as a well formed formula that is allways
true or always false in every model.
(Given that we are talking about completeness we can talk about true
and false setences i guess, it isn't obviously circular, but i would
prefer a definition of what a sentence is without referencing to Truth
or falsehood)

Maybe it is just painfully obvious for all of you
But I am just having a bit of philosophical struggle with this at the
moment.

Thanks




.

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