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Single Idea 10477

[filed under theme 5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |= ]

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]

The 16 ideas from Wilfrid Hodges

10282 Logic is the study of sound argument, or of certain artificial languages (or applying the latter to the former) [Hodges,W]
10287 If a first-order theory entails a sentence, there is a finite subset of the theory which entails it [Hodges,W]
10288 Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model [Hodges,W]
10289 Up Löwenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W]
10283 A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables [Hodges,W]
10284 There are three different standard presentations of semantics [Hodges,W]
10285 I |= φ means that the formula φ is true in the interpretation I [Hodges,W]
10286 A 'set' is a mathematically well-behaved class [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]