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Completeness/Bigelow/Pargetter: occurs when our explicit semantics guarantees all and only the extroverted asserted theorems. That is, our semantics does not read anything into our language, which is not already there.
Definition "extroverted axiomatics"/Terminology/Bigelow/Pargetter: an axiomatics that is developed in an already existing language.
Completeness/correspondence theory/Bigelow/Pargetter: the existence of completeness proofs provides a kind of correspondence theory.
Completeness: for us, we can show that all the propositions that are true to our semantics in all possible worlds can be derived.
Definition completeness theorem/Bigelow/Pargetter: is a theorem that proves that if a proposition in a certain semantics is assuredly true, this proposition can be proved as a theorem. How can we prove this? How can we prove that each such proposition is a theorem?
Solution: we prove the contraposition of the theorem: Instead:
If a is assuredly true in semantics, a is a theorem.
If a is not a theorem, it is not assuredly true in semantics.
We prove this by finding an interpretation according to which it is false.
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990