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Single Idea 13347
[filed under theme 5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
]
Full Idea
The classical definition of validity counts an argument as valid if and only if the conclusion does in fact follow from the premises, whether or not the argument contains any demonstration of this fact.
Gist of Idea
Validity is a conclusion following for premises, even if there is no proof
Source
David Bostock (Intermediate Logic [1997], 1.2)
Book Ref
Bostock,David: 'Intermediate Logic' [OUP 1997], p.5
A Reaction
Hence validity is given by |= rather than by |-. A common example is 'it is red so it is coloured', which seems true but beyond proof. In the absence of formal proof, you wonder whether validity is merely a psychological notion.
The
16 ideas
with the same theme
[fitting with the truth of some formulae]:
19237
|
Deduction is true when the premises facts necessarily make the conclusion fact true
[Peirce]
|
13344
|
X follows from sentences K iff every model of K also models X
[Tarski]
|
10694
|
Logical consequence is when in any model in which the premises are true, the conclusion is true
[Tarski, by Beall/Restall]
|
10479
|
Logical consequence: true premises give true conclusions under all interpretations
[Tarski, by Hodges,W]
|
13347
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Validity is a conclusion following for premises, even if there is no proof
[Bostock]
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13348
|
It seems more natural to express |= as 'therefore', rather than 'entails'
[Bostock]
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13349
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Γ|=φ is 'entails'; Γ|= is 'is inconsistent'; |=φ is 'valid'
[Bostock]
|
10477
|
|= in model-theory means 'logical consequence' - it holds in all models
[Hodges,W]
|
13626
|
Semantic consequence is ineffective in second-order logic
[Shapiro]
|
13637
|
If a logic is incomplete, its semantic consequence relation is not effective
[Shapiro]
|
10893
|
Γ |= φ for sentences if φ is true when all of Γ is true
[Zalabardo]
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10899
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Γ |= φ if φ is true when all of Γ is true, for all structures and interpretations
[Zalabardo]
|
21611
|
Formal semantics defines validity as truth preserved in every model
[Williamson]
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10695
|
Logical consequence is either necessary truth preservation, or preservation based on interpretation
[Beall/Restall]
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13240
|
A sentence follows from others if they always model it
[Beall/Restall]
|
14506
|
'Roses are red; therefore, roses are colored' seems truth-preserving, but not valid in a system
[Koslicki]
|