Re: completeness what is it exactly

Chris Menzel schrieb:

That's often referred to as "negation" completeness. There is also a
notion of completeness -- often referred to as "semantic" completeness
-- that applies to logics rather than theories. Say that a logic is a
language L together with a semantics and a proof theory. A logic is
*complete* if every logical truth of L (i.e., every sentence that is
true in every interpretation of L according to the semantics) is a
theorem of the proof theory. This is sometimes referred to as "weak"
(semantic) completeness. A logic is strongly complete if, whenever a
set S of sentences of L entails a sentence A of L, there is a proof of A
from S. (S entails A if every interpretation that makes all the
sentences in S true also makes A true.)

Yep,

And roughly if deduction theorem
and compactness holds, then weak = strong.

Proof:

=>: S |- A
<=>
S'|- A with S' finite
<=>
|- S' -> A
<=>
|= S' -> A
<=>
S' |= A with S' finite
<=>
S |= A

<=: |- A
<=>
{} |- A
<=>
{} |= A
<=>
|= A

Yes, maybe translogic was refering to this completness
of the |- in the other thread.

But it would be an add on to consider |=, as we
defined con() via |- in the other thread.

Bye
.

Relevant Pages