Soundness of proof procedure p:

forall Alpha, forall Beta( (Alpha turnstile-p Beta --> Alpha double-turnstile Beta) -->

p is sound)

Completenessof proof procedure p:

forall Alpha, forall Beta( (Alpha double-turnstile Beta --> Alpha turnstile-p Beta) -->

p is complete)

Example: P could be the application of modus ponens.

Monotonic logic:

A logic is monotonic if the following implication is true:

forall (KB1, KB2,Alpha) (KB1 double-turnstile Alpha) --> (KB1 and KB2 double-turnstile Alpha)

Adding knowledge never makes something already true false.

First-order predicate calculus and propositional logic are monotonic.

What happens in a FOPC reasoning system when a contridiction is entered into the KB?

Anything can be inferred to be true. (See later, when resolution. Any idea now?)

(Suppose P and not P are in the knowledge base. I want to prove S.

And-introduction: P and S. Resolve P and S with not P. Conclusion is S!

So, all bets are off if contradictions are introduced into the KB.