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Single Idea 10478

[filed under theme 5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic ]

Full Idea

In first-order languages the completeness theorem tells us that T |= φ holds if and only if there is a proof of φ from T (T |- φ). Since the two symbols express the same relationship, theorist often just use |- (but only for first-order!).

Gist of Idea

Since first-order languages are complete, |= and |- have the same meaning

Source

Wilfrid Hodges (Model Theory [2005], 3)

Book Ref

'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.8

A Reaction

[actually no spaces in the symbols] If you are going to study this kind of theory of logic, the first thing you need to do is sort out these symbols, which isn't easy!

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]