8 ideas
| 13831 | Logic is based on transitions between sentences [Prawitz] |
| 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. |
| 13825 | Natural deduction introduction rules may represent 'definitions' of logical connectives [Prawitz] |
| 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 [Prawitz] |
| 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) |
| 15923 | Poincaré rejected the actual infinite, claiming definitions gave apparent infinity to finite objects [Poincaré, by Lavine] |
| Full Idea: Poincaré rejected the actual infinite. He viewed mathematics that is apparently concerned with the actual infinite as actually concerning the finite linguistic definitions the putatively describe actually infinite objects. | |
| From: report of Henri Poincaré (On the Nature of Mathematical Reasoning [1894]) by Shaughan Lavine - Understanding the Infinite |
| 13768 | Validity can preserve certainty in mathematics, but conditionals about contingents are another matter [Edgington] |
| Full Idea: If your interest in logic is confined to applications to mathematics or other a priori matters, it is fine for validity to preserve certainty, ..but if you use conditionals when arguing about contingent matters, then great caution will be required. | |
| From: Dorothy Edgington (Conditionals [2001], 17.2.1) |
| 13770 | There are many different conditional mental states, and different conditional speech acts [Edgington] |
| Full Idea: As well as conditional beliefs, there are conditional desires, hopes, fears etc. As well as conditional statements, there are conditional commands, questions, offers, promises, bets etc. | |
| From: Dorothy Edgington (Conditionals [2001], 17.3.4) |
| 13764 | Are conditionals truth-functional - do the truth values of A and B determine the truth value of 'If A, B'? [Edgington] |
| Full Idea: Are conditionals truth-functional - do the truth values of A and B determine the truth value of 'If A, B'? Are they non-truth-functional, like 'because' or 'before'? Do the values of A and B, in some cases, leave open the value of 'If A,B'? | |
| From: Dorothy Edgington (Conditionals [2001], 17.1) | |
| A reaction: I would say they are not truth-functional, because the 'if' asserts some further dependency relation that goes beyond the truth or falsity of A and B. Logical ifs, causal ifs, psychological ifs... The material conditional ⊃ is truth-functional. |
| 13765 | 'If A,B' must entail ¬(A & ¬B); otherwise we could have A true, B false, and If A,B true, invalidating modus ponens [Edgington] |
| Full Idea: If it were possible to have A true, B false, and If A,B true, it would be unsafe to infer B from A and If A,B: modus ponens would thus be invalid. Hence 'If A,B' must entail ¬(A & ¬B). | |
| From: Dorothy Edgington (Conditionals [2001], 17.1) | |
| A reaction: This is a firm defence of part of the truth-functional view of conditionals, and seems unassailable. The other parts of the truth table are open to question, though, if A is false, or they are both true. |