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Single Idea 13618
[filed under theme 5. Theory of Logic / K. Features of Logics / 6. Compactness
]
Full Idea
The logic of truth-functions is compact, which means that sequents with infinitely many formulae on the left introduce nothing new. Hence we can confine our attention to finite sequents.
Gist of Idea
Compactness means an infinity of sequents on the left will add nothing new
Source
David Bostock (Intermediate Logic [1997], 5.5)
Book Ref
Bostock,David: 'Intermediate Logic' [OUP 1997], p.217
A Reaction
This makes it clear why compactness is a limitation in logic. If you want the logic to be unlimited in scope, it isn't; it only proves things from finite numbers of sequents. This makes it easier to prove completeness for the system.
Related Idea
Idea 13630
Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures [Shapiro]
The
17 ideas
with the same theme
[satisfaction by satisfying the finite subsets]:
9995
|
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]
|
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]
|
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]
|
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]
|
10975
|
Compactness does not deny that an inference can have infinitely many premisses
[Read]
|
17867
|
If a concept is not compact, it will not be presentable to finite minds
[Almog]
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