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4 ideas
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
Full Idea: The distinctive feature of second-order logic is that it presupposes that, given a domain, there is a fact of the matter about what the relations on it are, so that the range of the second-order quantifiers is fixed as soon as the domain is fixed. | |
From: Shaughan Lavine (Understanding the Infinite [1994], V.3) | |
A reaction: This sounds like a rather large assumption, which is open to challenge. I am not sure whether it was the basis of Quine's challenge to second-order logic. He seems to have disliked its vagueness, because it didn't stick with 'objects'. |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
Full Idea: The Law of Excluded Middle is (part of) the foundation of the mathematical practice of employing proofs by contradiction. | |
From: Shaughan Lavine (Understanding the Infinite [1994], VI.1) | |
A reaction: This applies in a lot of logic, as well as in mathematics. Come to think of it, it applies in Sudoku. |
13986 | Plato found antinomies in ideas, Kant in space and time, and Bradley in relations [Plato, by Ryle] |
Full Idea: Plato (in 'Parmenides') shows that the theory that 'Eide' are substances, and Kant that space and time are substances, and Bradley that relations are substances, all lead to aninomies. | |
From: report of Plato (Parmenides [c.364 BCE]) by Gilbert Ryle - Are there propositions? 'Objections' |
14150 | Plato's 'Parmenides' is perhaps the best collection of antinomies ever made [Russell on Plato] |
Full Idea: Plato's 'Parmenides' is perhaps the best collection of antinomies ever made. | |
From: comment on Plato (Parmenides [c.364 BCE]) by Bertrand Russell - The Principles of Mathematics §337 |