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|Model Theory||I 85
Model theory: semantically: "all models in which A is true are models in which B is also true": B follows from A - proof theory: syntactically "there is a formal derivation of B from A".
Model theory/Field: if one says that a logically true sentence is true in all models, a model exists in a set of objects plus the fixing which predicates (if any) of them are true in the model, which names (if any) denote these objects, etc. - "besides. attribution function" - then the truth conditions can be recursively defined - "Definition logically true: here: true for each model-".
Kripke: with him a non-empty set of possible worlds is called actual. Definition possible/Kripke: a sentence of the form "MA" (diamond) will then be true in a model if and only if A is in at least one possible world in a model true - Problem/Kripke: in order that "MA" is logically true, A itself has to be logically true. Solution/FieldVsKripke: we do not accept a possible world" - our model is the -"acutal world portion" of the Kripkean model.
Proof Theory: does not provide any results that could not be obtained otherwise.
Model theory/modal logic/FieldVsKripke: unlike Kripke: without possible world - "which sentences with the operator "logically possible" are logical true?" "N.B.: Both model theories are platonic - (pure quantity theory).
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980