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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
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Logic is the study of sound argument, or of certain artificial languages (or applying the latter to the former)
[Hodges,W]
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10288
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Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model
[Hodges,W]
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10289
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Up Löwenheim-Skolem: if infinite models, then arbitrarily large models
[Hodges,W]
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10287
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If a first-order theory entails a sentence, there is a finite subset of the theory which entails it
[Hodges,W]
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10283
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A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables
[Hodges,W]
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10284
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There are three different standard presentations of semantics
[Hodges,W]
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10285
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I |= φ means that the formula φ is true in the interpretation I
[Hodges,W]
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10286
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A 'set' is a mathematically well-behaved class
[Hodges,W]
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10473
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Model theory studies formal or natural language-interpretation using set-theory
[Hodges,W]
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10475
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A 'structure' is an interpretation specifying objects and classes of quantification
[Hodges,W]
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10474
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|= should be read as 'is a model for' or 'satisfies'
[Hodges,W]
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10476
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The idea that groups of concepts could be 'implicitly defined' was abandoned
[Hodges,W]
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10478
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Since first-order languages are complete, |= and |- have the same meaning
[Hodges,W]
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10477
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|= in model-theory means 'logical consequence' - it holds in all models
[Hodges,W]
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10480
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First-order logic can't discriminate between one infinite cardinal and another
[Hodges,W]
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10481
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Models in model theory are structures, not sets of descriptions
[Hodges,W]
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