Re: completeness what is it exactly

On Jul 11, 9:29 am, translogi <wilem...@xxxxxxxxxxxxxx> wrote:
On Jul 9, 6:59 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

"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.

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

A well formed formula with no free variables.

Or more precisely
what makes

1) P -> P a sentence
and
2) (P-> P) -> P not a sentence?

If 'P' is a sentence, then they're both sentences (of some suitable
language). Who said one is a sentence but not the other?

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)

I don't know what difficulty you find in the simple rule:

If P is a sentence and Q is a sentence, then (P -> Q) is a sentence,
which entails (with informal convention of dropping outer parentheses)
that if P is a sentence then P -> P is a sentence and (P -> P) -> P is
a sentence.

Or maybe if a sentence defind as a well formed formula that is allways
true or always false in every model.

No, simply a sentence is a well formed formula with no free variables.

(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)

Such a definition is in just about any textbook in mathematical logic.
A sentence is a well formed formula with no free varialbes.

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.

It's hardly philosophical. Just look at the definitions of the syntax.

MoeBlee

.

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