|Model Theory: The model theory investigates whether an axiom system is fulfilled and thus provides one (or more) models. Model theory belongs to the semantics because it uses the concept of truth, while the proof theory belongs to the domain of syntax by asking for the existence of finite character string (of proofs). One problem is the exclusion of unintended models._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
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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. - Moreover ist is an attribution function. - Then the truth conditions can be recursively defined.
Def 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 "actual 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 set theory)._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution.
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