7 ideas
11022 | Gentzen introduced a natural deduction calculus (NK) in 1934 [Gentzen, by Read] |
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 |
10121 | Contradiction is not a sign of falsity, nor lack of contradiction a sign of truth [Pascal] |
Full Idea: Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth. | |
From: Blaise Pascal (works [1660]), quoted by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6 | |
A reaction: [Quoted in Auden and Kronenberger's Book of Aphorisms] Presumably we would now say that contradiction is a purely formal, syntactic notion, and not a semantic one. If you hit a contradiction, something has certainly gone wrong. |
11023 | The logical connectives are 'defined' by their introduction rules [Gentzen] |
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 [Gentzen] |
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 [Gentzen, by Hanna] |
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. |
10067 | Gentzen proved the consistency of arithmetic from assumptions beyond arithmetic [Gentzen, by Musgrave] |
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. |
3031 | The greatest good is not the achievement of desire, but to desire what is proper [Menedemus, by Diog. Laertius] |
Full Idea: Hearing someone assert that the greatest good was to succeed in everything that one desires, he said,"It is a much greater good to desire what is proper". | |
From: report of Menedemus (fragments/reports [c.310 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.16.12 |