Re: completeness what is it exactly

On 09 Jul 2008 19:46:33 +0300, Aatu Koskensilta <aatu.koskensilta@xxxxxx> said:
Rupert <rupertmccallum@xxxxxxxxx> writes:

One speaks of "strong completeness" and "weak completeness" for the
propositional calculus and for the first-order predicate calculus.
These are both forms of semantic completeness. The "weak completeness
theorem" for the first-order predicate calculus says that a finite
set of sentences is consistent if and only if it has a model. The
"strong completeness theorem" generalises this to sets of sentences
which are not necessarily finite. This is related to the compactness
theorem.

The more usual definition is that a logic is weakly complete if there
is a deductive system in which all of its validities are provable and
strongly complete if there is a deductive system such that if A is a
logical consequence of B then B is derivable from A.

A from B, of course.

.

Relevant Pages

  • Re: completeness what is it exactly
    ... propositional calculus and for the first-order predicate calculus. ... These are both forms of semantic completeness. ... theorem" for the first-order predicate calculus says that a finite set ... is a deductive system in which all of its validities are provable and ...
    (sci.logic)
  • Re: completeness what is it exactly
    ... What is completeness exactly? ... A theory is descriptively complete with respect to a certain semantics ... propositional calculus and for the first-order predicate calculus. ... The "weak completeness ...
    (sci.logic)