Ideas from 'Gentzen's Analysis of First-Order Proofs' by Dag Prawitz [1974], by Theme Structure

[found in 'A Philosophical Companion to First-Order Logic' (ed/tr Hughes,R.I.G.) [Hackett 1993,0-87220-181-3]].

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5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
Logic is based on transitions between sentences
                        Full Idea: I agree entirely with Dummett that the right way to answer the question 'what is logic?' is to consider transitions between sentences.
                        From: Dag Prawitz (Gentzen's Analysis of First-Order Proofs [1974], §04)
                        A reaction: I always protest at this point that reliance on sentences is speciesism against animals, who are thereby debarred from reasoning. See the wonderful Idea 1875 of Chrysippus. Hacking's basic suggestion seems right. Transition between thoughts.
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
Natural deduction introduction rules may represent 'definitions' of logical connectives
                        Full Idea: With Gentzen's natural deduction, we may say that the introductions represent, as it were, the 'definitions' of the logical constants. The introductions are not literally understood as 'definitions'.
                        From: Dag Prawitz (Gentzen's Analysis of First-Order Proofs [1974], 2.2.2)
                        A reaction: [Hacking, in 'What is Logic? §9' says Gentzen had the idea that his rules actually define the constants; not sure if Prawitz and Hacking are disagreeing]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
In natural deduction, inferences are atomic steps involving just one logical constant
                        Full Idea: In Gentzen's natural deduction, the inferences are broken down into atomic steps in such a way that each step involves only one logical constant. The steps are the introduction or elimination of the logical constants.
                        From: Dag Prawitz (Gentzen's Analysis of First-Order Proofs [1974], 1.1)