9 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. |
13827 | Logical consequence isn't a black box (Tarski's approach); we should explain how arguments work [Prawitz] |
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 [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) |
13826 | Model theory looks at valid sentences and consequence, but not how we know these things [Prawitz] |
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) |
21554 | Sets always exceed terms, so all the sets must exceed all the sets [Lackey] |
Full Idea: Cantor proved that the number of sets in a collection of terms is larger than the number of terms. Hence Cantor's Paradox says the number of sets in the collection of all sets must be larger than the number of sets in the collection of all sets. | |
From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127) | |
A reaction: The sets must count as terms in the next iteration, but that is a normal application of the Power Set axiom. |
21553 | It seems that the ordinal number of all the ordinals must be bigger than itself [Lackey] |
Full Idea: The ordinal series is well-ordered and thus has an ordinal number, and a series of ordinals to a given ordinal exceeds that ordinal by 1. So the series of all ordinals has an ordinal number that exceeds its own ordinal number by 1. | |
From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127) | |
A reaction: Formulated by Burali-Forti in 1897. |
16597 | Quantity is the capacity to be divided [Digby] |
Full Idea: Quantity …is divisibility, or a capacity to be divided into parts. | |
From: Kenelm Digby (Two treatises [1644], I.2.8), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 04.1 | |
A reaction: 'Quantity' is scholastic philosophy is a concept we no longer possess. Without quantity, a thing might potentially exist at a spaceless point. Quantity is what spreads things out. See Pasnau Ch. 4. |
16731 | Colours arise from the rarity, density and mixture of matter [Digby] |
Full Idea: The origin of all colours in bodies is plainly deduced out of the various degrees of rarity and density, variously mixed and compounded. | |
From: Kenelm Digby (Two treatises [1644], I.29.4), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 22.5 | |
A reaction: We are still struggling with this question, though I think the picture is gradually become clear, once you get the hang of the brain. Easy! See Idea 17396. |