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Single Idea 13842
[filed under theme 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
]
Full Idea
Second-order logic has no chance of a completeness theorem unless one ventures into intensional entities and possible worlds.
Gist of Idea
Second-order completeness seems to need intensional entities and possible worlds
Source
Ian Hacking (What is Logic? [1979], §13)
Book Ref
'A Philosophical Companion to First-Order Logic', ed/tr. Hughes,R.I.G. [Hackett 1993], p.246
The
19 ideas
from Ian Hacking
7448
|
Probability is statistical (behaviour of chance devices) or epistemological (belief based on evidence)
[Hacking]
|
7447
|
Probability was fully explained between 1654 and 1812
[Hacking]
|
7459
|
Follow maths for necessary truths, and jurisprudence for contingent truths
[Hacking]
|
7449
|
Epistemological probability based either on logical implications or coherent judgments
[Hacking]
|
7450
|
In the medieval view, only deduction counted as true evidence
[Hacking]
|
7451
|
Formerly evidence came from people; the new idea was that things provided evidence
[Hacking]
|
7452
|
An experiment is a test, or an adventure, or a diagnosis, or a dissection
[Hacking, by PG]
|
7454
|
Gassendi is the first great empiricist philosopher
[Hacking]
|
13833
|
'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction
[Hacking]
|
13834
|
Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C'
[Hacking]
|
13835
|
Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with
[Hacking]
|
13837
|
With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically
[Hacking]
|
13838
|
A decent modern definition should always imply a semantics
[Hacking]
|
13839
|
Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers
[Hacking]
|
13843
|
If it is a logic, the Löwenheim-Skolem theorem holds for it
[Hacking]
|
13842
|
Second-order completeness seems to need intensional entities and possible worlds
[Hacking]
|
13840
|
First-order logic is the strongest complete compact theory with Löwenheim-Skolem
[Hacking]
|
13844
|
A limitation of first-order logic is that it cannot handle branching quantifiers
[Hacking]
|
13845
|
The various logics are abstractions made from terms like 'if...then' in English
[Hacking]
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