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Single Idea 10977
[filed under theme 5. Theory of Logic / K. Features of Logics / 6. Compactness
]
Full Idea
Compact consequence undergenerates - there are intuitively valid consequences which it marks as invalid, such as the ω-rule, that if A holds of the natural numbers, then 'for every n, A(n)', but the proof of that would be infinite, for each number.
Gist of Idea
Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite)
Source
Stephen Read (Thinking About Logic [1995], Ch.2)
Book Ref
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.44
The
42 ideas
from 'Thinking About Logic'
10966
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A proposition objectifies what a sentence says, as indicative, with secure references
[Read]
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10983
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Knowledge of possible worlds is not causal, but is an ontology entailed by semantics
[Read]
|
10982
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How can modal Platonists know the truth of a modal proposition?
[Read]
|
10987
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Three traditional names of rules are 'Simplification', 'Addition' and 'Disjunctive Syllogism'
[Read]
|
10985
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We should exclude second-order logic, precisely because it captures arithmetic
[Read]
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10970
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A theory of logical consequence is a conceptual analysis, and a set of validity techniques
[Read]
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10984
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Logical consequence isn't just a matter of form; it depends on connections like round-square
[Read]
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10973
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A theory is logically closed, which means infinite premisses
[Read]
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10978
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In second-order logic the higher-order variables range over all the properties of the objects
[Read]
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10971
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A logical truth is the conclusion of a valid inference with no premisses
[Read]
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10988
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Any first-order theory of sets is inadequate
[Read]
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10974
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Compactness is when any consequence of infinite propositions is the consequence of a finite subset
[Read]
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10975
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Compactness does not deny that an inference can have infinitely many premisses
[Read]
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10977
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Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite)
[Read]
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10976
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Compactness makes consequence manageable, but restricts expressive power
[Read]
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10972
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The non-emptiness of the domain is characteristic of classical logic
[Read]
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10986
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Not all validity is captured in first-order logic
[Read]
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10979
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Although second-order arithmetic is incomplete, it can fully model normal arithmetic
[Read]
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10980
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Second-order arithmetic covers all properties, ensuring categoricity
[Read]
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10981
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A possible world is a determination of the truth-values of all propositions of a domain
[Read]
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10989
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The standard view of conditionals is that they are truth-functional
[Read]
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10992
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The point of conditionals is to show that one will accept modus ponens
[Read]
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10997
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Von Neumann numbers are helpful, but don't correctly describe numbers
[Read]
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11001
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Equating necessity with truth in every possible world is the S5 conception of necessity
[Read]
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11000
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If worlds are concrete, objects can't be present in more than one, and can only have counterparts
[Read]
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10998
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The mind abstracts ways things might be, which are nonetheless real
[Read]
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11004
|
Necessity is provability in S4, and true in all worlds in S5
[Read]
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10996
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Actualism is reductionist (to parts of actuality), or moderate realist (accepting real abstractions)
[Read]
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10995
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A haecceity is a set of individual properties, essential to each thing
[Read]
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11011
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Same say there are positive, negative and neuter free logics
[Read]
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11007
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Quantifiers are second-order predicates
[Read]
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11005
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Negative existentials with compositionality make the whole sentence meaningless
[Read]
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11012
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A 'supervaluation' gives a proposition consistent truth-value for classical assignments
[Read]
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11013
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Identities and the Indiscernibility of Identicals don't work with supervaluations
[Read]
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11014
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Self-reference paradoxes seem to arise only when falsity is involved
[Read]
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11018
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There are fuzzy predicates (and sets), and fuzzy quantifiers and modifiers
[Read]
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11016
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Would a language without vagueness be usable at all?
[Read]
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11019
|
Supervaluations say there is a cut-off somewhere, but at no particular place
[Read]
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11017
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Some people even claim that conditionals do not express propositions
[Read]
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11025
|
Infinite cuts and successors seems to suggest an actual infinity there waiting for us
[Read]
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11020
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Realisms like the full Comprehension Principle, that all good concepts determine sets
[Read]
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11024
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Semantics must precede proof in higher-order logics, since they are incomplete
[Read]
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