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Single Idea 13835
[filed under theme 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
]
Full Idea
Only the cut rule can have a conclusion that is less complex than its premises. Hence when cut is not used, a derivation is quite literally constructive, building up from components. Any theorem obtained by cut can be obtained without it.
Gist of Idea
Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with
Source
Ian Hacking (What is Logic? [1979], §08)
Book Ref
'A Philosophical Companion to First-Order Logic', ed/tr. Hughes,R.I.G. [Hackett 1993], p.235
Related Ideas
Idea 13834
Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C' [Hacking]
Idea 13352
'Cutting' allows that if x is proved, and adding y then proves z, you can go straight to z [Bostock]
The
19 ideas
from Ian Hacking
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7447
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Probability was fully explained between 1654 and 1812
[Hacking]
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7448
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Probability is statistical (behaviour of chance devices) or epistemological (belief based on evidence)
[Hacking]
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7459
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Follow maths for necessary truths, and jurisprudence for contingent truths
[Hacking]
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7449
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Epistemological probability based either on logical implications or coherent judgments
[Hacking]
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7450
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In the medieval view, only deduction counted as true evidence
[Hacking]
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7451
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Formerly evidence came from people; the new idea was that things provided evidence
[Hacking]
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7452
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An experiment is a test, or an adventure, or a diagnosis, or a dissection
[Hacking, by PG]
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7454
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Gassendi is the first great empiricist philosopher
[Hacking]
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13833
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'Thinning' ('dilution') is the key difference between deduction (which allows it) and induction
[Hacking]
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13834
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Gentzen's Cut Rule (or transitivity of deduction) is 'If A |- B and B |- C, then A |- C'
[Hacking]
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13835
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Only Cut reduces complexity, so logic is constructive without it, and it can be dispensed with
[Hacking]
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13837
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With a pure notion of truth and consequence, the meanings of connectives are fixed syntactically
[Hacking]
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13838
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A decent modern definition should always imply a semantics
[Hacking]
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13839
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Perhaps variables could be dispensed with, by arrows joining places in the scope of quantifiers
[Hacking]
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13843
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If it is a logic, the Löwenheim-Skolem theorem holds for it
[Hacking]
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13840
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First-order logic is the strongest complete compact theory with Löwenheim-Skolem
[Hacking]
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13844
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A limitation of first-order logic is that it cannot handle branching quantifiers
[Hacking]
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13842
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Second-order completeness seems to need intensional entities and possible worlds
[Hacking]
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13845
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The various logics are abstractions made from terms like 'if...then' in English
[Hacking]
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