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11023
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The logical connectives are 'defined' by their introduction rules [Gentzen]
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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.
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From:
Gerhard Gentzen (works [1938]), quoted by Stephen Read - Thinking About Logic Ch.8
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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?
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11213
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Each logical symbol has an 'introduction' rule to define it, and hence an 'elimination' rule [Gentzen]
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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.
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From:
Gerhard Gentzen (works [1938], II.5.13), quoted by Ian Rumfitt - "Yes" and "No" III
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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.
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18810
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Aristotle's proofs give understanding, so it can't be otherwise, so consequence is necessary [Smiley, by Rumfitt]
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Full Idea:
The ingredient of necessity [in Aristotle's account of consequence] is required by his demand that proof should produce 'understanding' [episteme], coupled with his claim that understanding something involves seeing that it cannot be otherwise.
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From:
report of Timothy Smiley (Conceptions of Consequence [1998], p.599) by Ian Rumfitt - The Boundary Stones of Thought 3.2
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A reaction:
An intriguing reverse of the normal order. Not 'necessity in logic delivers understanding', but 'reaching understanding shows the logic was necessary'.
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