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Single Idea 10602
[filed under theme 5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
]
Full Idea
A 'natural deduction system' will have no logical axioms but may rules of inference.
Gist of Idea
A 'natural deduction system' has no axioms but many rules
Source
Peter Smith (Intro to Gödel's Theorems [2007], 09.1)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.59
A Reaction
He contrasts this with 'Hilbert-style systems', which have many axioms but few rules. Natural deduction uses many assumptions which are then discharged, and so tree-systems are good for representing it.
The
14 ideas
with the same theme
[proofs built from introduction and elimination rules]:
13832
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Natural deduction shows the heart of reasoning (and sequent calculus is just a tool)
[Gentzen, by Hacking]
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13753
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Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part
[Bostock]
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13755
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Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it
[Bostock]
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13754
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Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E)
[Bostock]
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13758
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In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle
[Bostock]
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18120
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The Deduction Theorem is what licenses a system of natural deduction
[Bostock]
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13823
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In natural deduction, inferences are atomic steps involving just one logical constant
[Prawitz]
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10602
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A 'natural deduction system' has no axioms but many rules
[Smith,P]
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21612
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Or-elimination is 'Argument by Cases'; it shows how to derive C from 'A or B'
[Williamson]
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13685
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Natural deduction helpfully allows reasoning with assumptions
[Sider]
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19298
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Unlike axiom proofs, natural deduction proofs needn't focus on logical truths and theorems
[Hale]
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18783
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Many-valued logics lack a natural deduction system
[Mares]
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15001
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'Tonk' is supposed to follow the elimination and introduction rules, but it can't be so interpreted
[Sider]
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18800
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Introduction rules give deduction conditions, and Elimination says what can be deduced
[Rumfitt]
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