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Single Idea 13544
[filed under theme 5. Theory of Logic / K. Features of Logics / 6. Compactness
]
Full Idea
Being 'compact' means that if we have an inconsistency or an entailment which holds just because of the truth-functors and quantifiers involved, then it is always due to a finite number of the propositions in question.
Gist of Idea
Inconsistency or entailment just from functors and quantifiers is finitely based, if compact
Source
David Bostock (Intermediate Logic [1997], 4.8)
Book Ref
Bostock,David: 'Intermediate Logic' [OUP 1997], p.183
A Reaction
Bostock says this is surprising, given the examples 'a is not a parent of a parent of b...' etc, where an infinity seems to establish 'a is not an ancestor of b'. The point, though, is that this truth doesn't just depend on truth-functors and quantifiers.
The
17 ideas
with the same theme
[satisfaction by satisfying the finite subsets]:
9995
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Proof in finite subsets is sufficient for proof in an infinite set
[Enderton]
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10771
|
Compactness is important for major theories which have infinitely many axioms
[Tharp]
|
10772
|
Compactness blocks infinite expansion, and admits non-standard models
[Tharp]
|
13544
|
Inconsistency or entailment just from functors and quantifiers is finitely based, if compact
[Bostock]
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13618
|
Compactness means an infinity of sequents on the left will add nothing new
[Bostock]
|
13841
|
Why should compactness be definitive of logic?
[Boolos, by Hacking]
|
10287
|
If a first-order theory entails a sentence, there is a finite subset of the theory which entails it
[Hodges,W]
|
13496
|
First-order logic is 'compact': consequences of a set are consequences of a finite subset
[Hart,WD]
|
17789
|
No logic which can axiomatise arithmetic can be compact or complete
[Mayberry]
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13630
|
Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures
[Shapiro]
|
13646
|
Compactness is derived from soundness and completeness
[Shapiro]
|
13699
|
Compactness surprisingly says that no contradictions can emerge when the set goes infinite
[Sider]
|
10975
|
Compactness does not deny that an inference can have infinitely many premisses
[Read]
|
10977
|
Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite)
[Read]
|
10976
|
Compactness makes consequence manageable, but restricts expressive power
[Read]
|
10974
|
Compactness is when any consequence of infinite propositions is the consequence of a finite subset
[Read]
|
17867
|
If a concept is not compact, it will not be presentable to finite minds
[Almog]
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