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Canonical models/Bigelow/Pargetter: deal with maximally consistent sets of sentences to provide completeness proofs.
Canonical models were discovered only after Hughes/Cresswell 1968, they were described in the later work (Hughes/Cresswell 1984).
Definition completeness theorem/Bigelow/Pargetter: is a theorem that proves that if a proposition in a certain semantics is guaranteed 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.
Then we prove this by finding an interpretation according to which it is false.
Definition canonical model/Bigelow/Pargetter: provides an interpretation which guarantees that every non-theorem is made wrong in at least one possible world.
We begin that there will be a sentence a, for which either a or ~a is a theorem. This can be added to the axioms to give another consistent set of sentences.
Maximum consistent set of sentences/Bigelow/Pargetter: it can be proved that for the axiom systems which we deal with, there is always a maximally consistent set of sentences.
That is, a consistent set of sentences to which no further sentence can be added without making the set inconsistent.
That is, for each sentence g is either γ in the set or ~ γ.
W: be the set of all maximally consistent extensions of the axiom system with which we have begun.
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990