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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
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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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