Re: completeness what is it exactly

On Sun, 20 Jul 2008 12:18:50 -0700 (PDT), translogi
<wilemien@xxxxxxxxxxxxxx> said:
Ahh

At least two different kinds of completeness.

In this post I stick to (modal) propositional logic

A (Uppercase) is a well formed formula
p (lowercase) is a propositional variable

1 the opposite of soundness

Better: the converse.

|= A -> |-A

if (using a truthtable, or other mechanical means) a formula is always
true

Whether one has a mechanical means of determining validity is completely
irrelevant to the definition. We do have such a means, of course, in
classical propositional logic and some modal logics, but we don't in
predicate logic, yet the definition of (semantic) completeness still
applies.

then that formula is provable .

also
if a formula A is unsatisfiable (never true) then |- ~A

(or is this a step to far)

No, it is exactly equivalent in classical logic.

2 negation completeness

All well formed formula's are (provable) true or false.

No, negation completeness has nothing whatever to do with truth. It is
a purely proof theoretic notion.

for every A |-A or |- ~A

this is more complicated.

No, this is wrong. Negation completeness is a property of theories, not
of logics. Perhaps you missed my earlier post, but a theory T is
negation complete if, for any formula A in the language of T, T |- A or
T |- ~A.

for this all propositional variables need to have a fixed truth value
(otherwise neither |-p or |- ~p can be deduced)

Again, the concept of negation completeness has absolutely nothing to do
with truth.

am i forgetting something?

Forgetting isn't exactly the right concept.

.

Relevant Pages

  • Re: Godels Incompleteness and Nonmonotonic Logic
    ... The two incompleteness theorems of Godel refer to what ... > cannot have both completeness and consistency. ... For non-monotonic logics with a perfect weak negation, ... The incompleteness that he probably means is 'the logics are not ...
    (comp.lang.prolog)
  • Re: Godels Incompleteness and Nonmonotonic Logic
    ... The two incompleteness theorems of Godel refer to what ... > cannot have both completeness and consistency. ... For non-monotonic logics with a perfect weak negation, ... The incompleteness that he probably means is 'the logics are not ...
    (sci.logic)
  • Re: completeness what is it exactly
    ... (I remember long ago having read somewhere about strong completeness ... and weak completeness but I  forgot what the difference was.) ... let's distinguish for logics between 'logical truth ... completeness' and 'logical consequence completeness' (LCC). ...
    (sci.logic)
  • Re: completeness what is it exactly
    ... What is completeness exactly? ... and weak completeness but I forgot what the difference was.) ... -- that applies to logics rather than theories. ... language L together with a semantics and a proof theory. ...
    (sci.logic)