User:IssaRice/Computability and logic/Models symbol
The "models" symbol, , is used for several purposes in mathematical logic. Roughly, there are two basic purposes:
- When the symbol that comes before "" is a structure/interpretation, then it says something about truth in that structure/interpretation.
- When the symbol that comes before "" is a sentence or set of sentences, then it says something about semantic consequence (also called logical consequence, logical implication, semantic implication). In this case, we are talking about all possible structures/interpretations.
In either of the above two purposes, we are talking about the semantics (rather than syntax) of a logical system.
- If is a structure/interpretation and is a set of sentences, then means ...
- If is a theory and is a sentence, then means ...
- If is a theory and is a set of sentences, then means ...
- If is a set of axioms for a theory , and is a sentence, then means ...
- If is a set of axioms for a theory , and is a set of sentences, then ...
- if is a formula (or wff), then ...
- also the variant without anything in front, e.g.,
|Part before ""||Part after ""||Possible pronunciations||Meaning|
|A structure/interpretation||A sentence or formula|| The structure satisfies the formula .
The formula is true in .
The structure makes true the formula .
|A set of sentences or formulas||A sentence or formula|| is a logical consequence of .
logically implies .
is a semantic consequence of .
is true in every model of .
|Nothing||A sentence or formula|| is valid.
is a tautology (especially in the case of propositional logic).
Other tricky things:
- Some books only use the models symbol for one of the two use cases. E.g. Boolos/Burgess/Jeffrey only uses the symbol for truth in an interpretation. Therefore you might be really confused when you start reading other books and you start seeing stuff like .
- In the Boolos/Burgess/Jeffrey book, if a set of sentences comes after , then the sentences are taken disjunctively rather than conjunctively. (see p. 168)
- Derek Goldrei. Propositional and Predicate Calculus. p. 134.
- Boolos; Burgess; Jeffrey. Computability and Logic. p. 168.