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Single Idea 13824
[filed under theme 5. Theory of Logic / H. Proof Systems / 1. Proof Systems
]
Full Idea
Frege's formal definition of derivability is perhaps the first investigation in general proof theory.
Gist of Idea
Proof theory began with Frege's definition of derivability
Source
report of Gottlob Frege (Begriffsschrift [1879]) by Dag Prawitz - Gentzen's Analysis of First-Order Proofs 2 n2
Book Ref
'A Philosophical Companion to First-Order Logic', ed/tr. Hughes,R.I.G. [Hackett 1993], p.207
A Reaction
In 'On General Proof Theory §1' Prawitz says "proof theory originated with Hilbert" in 1900. Presumably Frege offered a theory, and then Hilbert saw it as a general project.
The
20 ideas
from 'Begriffsschrift'
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7622
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In 1879 Frege developed second order logic
[Frege, by Putnam]
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7729
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Frege replaced Aristotle's subject/predicate form with function/argument form
[Frege, by Weiner]
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9991
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For Frege the variable ranges over all objects
[Frege, by Tait]
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10536
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Frege's domain for variables is all objects, but modern interpretations first fix the domain
[Dummett on Frege]
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9950
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A quantifier is a second-level predicate (which explains how it contributes to truth-conditions)
[Frege, by George/Velleman]
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13609
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Frege produced axioms for logic, though that does not now seem the natural basis for logic
[Frege, by Kaplan]
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7730
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Frege introduced quantifiers for generality
[Frege, by Weiner]
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7742
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Frege reduced most quantifiers to 'everything' combined with 'not'
[Frege, by McCullogh]
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13824
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Proof theory began with Frege's definition of derivability
[Frege, by Prawitz]
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17855
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It may be possible to define induction in terms of the ancestral relation
[Frege, by Wright,C]
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10607
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Frege's logic has a hierarchy of object, property, property-of-property etc.
[Frege, by Smith,P]
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11008
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Existence is not a first-order property, but the instantiation of a property
[Frege, by Read]
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22280
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Frege's account was top-down and decompositional, not bottom-up and compositional
[Frege, by Potter]
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7741
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The predicate 'exists' is actually a natural language expression for a quantifier
[Frege, by Weiner]
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22270
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Frege changed philosophy by extending logic's ability to check the grounds of thinking
[Potter on Frege]
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8939
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We should not describe human laws of thought, but how to correctly track truth
[Frege, by Fisher]
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17745
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For Frege, 'All A's are B's' means that the concept A implies the concept B
[Frege, by Walicki]
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7728
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Frege has a judgement stroke (vertical, asserting or judging) and a content stroke (horizontal, expressing)
[Frege, by Weiner]
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4971
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I don't use 'subject' and 'predicate' in my way of representing a judgement
[Frege]
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16881
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The laws of logic are boundless, so we want the few whose power contains the others
[Frege]
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