13831 | 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. |
13827 | Logical consequence isn't a black box (Tarski's approach); we should explain how arguments work |
Full Idea: Defining logical consequence in the way Tarski does is a rather meagre result, treating an argument as a black box, observing input and output, while disregarding inner structure. We should define logical consequence on the basis of valid arguments. | |
From: Dag Prawitz (On the General Idea of Proof Theory [1974], §2) |
13825 | 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] |
13823 | 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) |
13826 | Model theory looks at valid sentences and consequence, but not how we know these things |
Full Idea: In model theory, which has dominated the last decades, one concentrates on logically valid sentences, and what follows logically from what, but one disregards questions concerning how we know these things. | |
From: Dag Prawitz (On the General Idea of Proof Theory [1974], §1) |