Ideas from 'works' by Gerhard Gentzen [1938], by Theme Structure

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5. Theory of Logic / A. Overview of Logic / 2. History of Logic

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

5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives

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?

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.

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.

6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic

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.