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Full Idea
If every structure which is a model of a set of sentences T is also a model of one of its sentences φ, then this is known as the model-theoretic consequence relation, and is written T |= φ. Not to be confused with |= meaning 'satisfies'.
Gist of Idea
|= in model-theory means 'logical consequence' - it holds in all models
Source
Wilfrid Hodges (Model Theory [2005], 3)
Book Ref
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.7
A Reaction
See also Idea 10474, which gives the other meaning of |=, as 'satisfies'. The symbol is ALSO used in propositional logical, to mean 'tautologically implies'! Sort your act out, logicians.
Related Idea
Idea 10474 |= should be read as 'is a model for' or 'satisfies' [Hodges,W]
10473 | Model theory studies formal or natural language-interpretation using set-theory [Hodges,W] |
10475 | A 'structure' is an interpretation specifying objects and classes of quantification [Hodges,W] |
10474 | |= should be read as 'is a model for' or 'satisfies' [Hodges,W] |
10476 | The idea that groups of concepts could be 'implicitly defined' was abandoned [Hodges,W] |
10478 | Since first-order languages are complete, |= and |- have the same meaning [Hodges,W] |
10477 | |= in model-theory means 'logical consequence' - it holds in all models [Hodges,W] |
10480 | First-order logic can't discriminate between one infinite cardinal and another [Hodges,W] |
10481 | Models in model theory are structures, not sets of descriptions [Hodges,W] |