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Single Idea 13753
[filed under theme 5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
]
Full Idea
Natural deduction takes the notion of proof from assumptions as a basic notion, ...so it will use rules for use in proofs from assumptions, and axioms (as traditionally understood) will have no role to play.
Gist of Idea
Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part
Source
David Bostock (Intermediate Logic [1997], 6.1)
Book Ref
Bostock,David: 'Intermediate Logic' [OUP 1997], p.240
A Reaction
The main rules are those for introduction and elimination of truth functors.
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]
|
13753
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Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part
[Bostock]
|
13755
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Excluded middle is an introduction rule for negation, and ex falso quodlibet will eliminate it
[Bostock]
|
13754
|
Natural deduction rules for → are the Deduction Theorem (→I) and Modus Ponens (→E)
[Bostock]
|
13758
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In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle
[Bostock]
|
18120
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The Deduction Theorem is what licenses a system of natural deduction
[Bostock]
|
13823
|
In natural deduction, inferences are atomic steps involving just one logical constant
[Prawitz]
|
10602
|
A 'natural deduction system' has no axioms but many rules
[Smith,P]
|
21612
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Or-elimination is 'Argument by Cases'; it shows how to derive C from 'A or B'
[Williamson]
|
13685
|
Natural deduction helpfully allows reasoning with assumptions
[Sider]
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19298
|
Unlike axiom proofs, natural deduction proofs needn't focus on logical truths and theorems
[Hale]
|
18783
|
Many-valued logics lack a natural deduction system
[Mares]
|
15001
|
'Tonk' is supposed to follow the elimination and introduction rules, but it can't be so interpreted
[Sider]
|
18800
|
Introduction rules give deduction conditions, and Elimination says what can be deduced
[Rumfitt]
|