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### Ideas of Gerhard Gentzen, by Text

#### [German, 1909 - 1945, Pupil of Weyl. Professor at Prague. A Nazi, starved to death after the war.]

 1935 Investigations into Logical Deduction
 p.231 13832 Natural deduction shows the heart of reasoning (and sequent calculus is just a tool) Full Idea: Gentzen thought that his natural deduction gets at the heart of logical reasoning, and used the sequent calculus only as a convenient tool for proving his chief results. From: report of Gerhard Gentzen (Investigations into Logical Deduction [1935]) by Ian Hacking - What is Logic? §05
 1938 works
 p.63 11065 The inferential role of a logical constant constitutes its meaning Full Idea: Gentzen argued that the inferential role of a logical constant constitutes its meaning. From: report of Gerhard Gentzen (works [1938]) by Robert Hanna - Rationality and Logic 5.3 A reaction: Possibly inspired by Wittgenstein's theory of meaning as use? This idea was the target of Prior's famous connective 'tonk', which has the role of implying anything you like, proving sentences which are not logical consequences.
 p.126 10067 Gentzen proved the consistency of arithmetic from assumptions beyond arithmetic Full Idea: Gentzen proved the consistency of arithmetic from assumptions which transcend arithmetic. From: report of Gerhard Gentzen (works [1938]) by Alan Musgrave - Logicism Revisited §5 A reaction: This does not contradict Gödel's famous result, but reinforces it. The interesting question is what assumptions Gentzen felt he had to make.
 p.228 11022 Gentzen introduced a natural deduction calculus (NK) in 1934 Full Idea: Gentzen introduced a natural deduction calculus (NK) in 1934. From: report of Gerhard Gentzen (works [1938]) by Stephen Read - Thinking About Logic Ch.8
 p.229 11023 The logical connectives are 'defined' by their introduction rules Full Idea: The introduction rules represent, as it were, the 'definitions' of the symbols concerned, and the elimination rules are no more, in the final analysis, than the consequences of these definitions. From: Gerhard Gentzen (works [1938]), quoted by Stephen Read - Thinking About Logic Ch.8 A reaction: If an introduction-rule (or a truth table) were taken as fixed and beyond dispute, then it would have the status of a definition, since there would be nothing else to appeal to. So is there anything else to appeal to here?
 II.5.13 p.787 11213 Each logical symbol has an 'introduction' rule to define it, and hence an 'elimination' rule Full Idea: To every logical symbol there belongs precisely one inference figure which 'introduces' the symbol ..and one which 'eliminates' it. The introductions represent the 'definitions' of the symbols concerned, and eliminations are consequences of these. From: Gerhard Gentzen (works [1938], II.5.13), quoted by Ian Rumfitt - "Yes" and "No" III A reaction: [1935 paper] This passage is famous, in laying down the basics of natural deduction systems of logic (ones using only rules, and avoiding axioms). Rumfitt questions whether Gentzen's account gives the sense of the connectives.