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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 Natural deduction shows the heart of reasoning (and sequent calculus is just a tool) [Gentzen, by Hacking]
13753 Natural deduction takes proof from assumptions (with its rules) as basic, and axioms play no part [Bostock]
13755 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 In natural deduction we work from the premisses and the conclusion, hoping to meet in the middle [Bostock]
18120 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 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]
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]