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Single Idea 13626
[filed under theme 5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
]
Full Idea
It follows from Gödel's incompleteness theorem that the semantic consequence relation of second-order logic is not effective. For example, the set of logical truths of any second-order logic is not recursively enumerable. It is not even arithmetic.
Gist of Idea
Semantic consequence is ineffective in second-order logic
Source
Stewart Shapiro (Foundations without Foundationalism [1991], Pref)
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.-16
A Reaction
I don't fully understand this, but it sounds rather major, and a good reason to avoid second-order logic (despite Shapiro's proselytising). See Peter Smith on 'effectively enumerable'.
Related Idea
Idea 10083
A set is 'effectively enumerable' if a computer could eventually list every member [Smith,P]
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
|
Validity is a conclusion following for premises, even if there is no proof
[Bostock]
|
13348
|
It seems more natural to express |= as 'therefore', rather than 'entails'
[Bostock]
|
13349
|
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]
|
10899
|
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]
|
10695
|
Logical consequence is either necessary truth preservation, or preservation based on interpretation
[Beall/Restall]
|
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]
|