User:IssaRice/Computability and logic/Models symbol

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The "models" symbol, \models, is used for several purposes in mathematical logic. Roughly, there are two basic purposes:

  1. When the symbol that comes before "\models" is a structure/interpretation, then it says something about truth in that structure/interpretation.
  2. When the symbol that comes before "\models" 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 \mathfrak A is a structure/interpretation and \Gamma is a set of sentences, then \mathfrak A \models \Gamma means ...
  • If T is a theory and \phi is a sentence, then T \models \phi means ...
  • If T is a theory and \Gamma is a set of sentences, then T \models \Gamma means ...
  • If \Sigma is a set of axioms for a theory T, and \phi is a sentence, then \Sigma \models \phi means ...
  • If \Sigma is a set of axioms for a theory T, and \Gamma is a set of sentences, then ...
  • if \phi is a formula (or wff), then ...
  • also the variant without anything in front, e.g., \models \phi
Part before "\models" \models Part after "\models" Possible pronunciations Meaning
A structure/interpretation \mathfrak A \models A sentence or formula \phi The structure \mathfrak A satisfies the formula \phi.[1]
The formula \phi is true in \mathfrak A.[1]
The structure \mathfrak A makes true the formula \phi.
A set of sentences or formulas \Gamma \models A sentence or formula \phi \phi is a logical consequence of \Gamma.
\Gamma logically implies \phi.
\phi is a semantic consequence of \Gamma.
\phi is true in every model of \Gamma.
Nothing \models A sentence or formula \phi \phi is valid.[2]
\phi is a tautology (especially in the case of propositional logic).

Notes

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 \Gamma \models \phi.
  • In the Boolos/Burgess/Jeffrey book, if a set of sentences comes after \models, then the sentences are taken disjunctively rather than conjunctively. (see p. 168)
  • https://math.stackexchange.com/a/2506938/35525

See also

References

  1. 1.0 1.1 Derek Goldrei. Propositional and Predicate Calculus. p. 134.
  2. Boolos; Burgess; Jeffrey. Computability and Logic. p. 168.