# User:IssaRice/Computability and logic/Models symbol

From Machinelearning

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 .^{[1]}The formula is true in . ^{[1]}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.^{[2]}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 .
- 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)
- https://math.stackexchange.com/a/2506938/35525

## See also

## References

- ↑
^{1.0}^{1.1}Derek Goldrei.*Propositional and Predicate Calculus*. p. 134. - ↑ Boolos; Burgess; Jeffrey.
*Computability and Logic*. p. 168.