Combining Texts

Ideas for 'works', 'The Logical Basis of Metaphysics' and 'Against Coherence'

unexpand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts

4 ideas

5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / c. not
18801 Classical negation is circular, if it relies on knowing negation-conditions from truth-conditions [Dummett]
     Full Idea: Explanations of classical negation assume that knowing what it is for the truth-condition of some statement to obtain, independently of recognising it to obtain, we thereby know what it is for it NOT to obtain; but this presupposes classical negation.
     From: Michael Dummett (The Logical Basis of Metaphysics [1991], p.299), quoted by Ian Rumfitt - The Boundary Stones of Thought 1.1
     A reaction: [compressed wording] This is Dummett explaining why he prefers intuitionistic logic, with its doubts about double negation.
5. Theory of Logic / K. Features of Logics / 8. Enumerability
10082 There are infinite sets that are not enumerable [Cantor, by Smith,P]
     Full Idea: Cantor's Theorem (1874) says there are infinite sets that are not enumerable. This is proved by his 1891 'diagonal argument'.
     From: report of George Cantor (works [1880]) by Peter Smith - Intro to Gödel's Theorems 2.3
     A reaction: [Smith summarises the diagonal argument]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / b. Cantor's paradox
13483 Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD]
     Full Idea: The problem of Cantor's Paradox is that the power set of the universe has to be both bigger than the universe (by Cantor's theorem) and not bigger (since it is a subset of the universe).
     From: report of George Cantor (works [1880]) by William D. Hart - The Evolution of Logic 3
     A reaction: Russell eliminates the 'universe' in his theory of types. I don't see why you can't just say that the members of the set are hypothetical rather than real, and that hypothetically the universe might contain more things than it does.
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / e. Mirimanoff's paradox
8710 The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend]
     Full Idea: Cantor's Paradox says that the powerset of a set has a cardinal number strictly greater than the original set, but that means that the powerset of the set of all the cardinal numbers is greater than itself.
     From: report of George Cantor (works [1880]) by Michčle Friend - Introducing the Philosophy of Mathematics
     A reaction: Friend cites this with the Burali-Forti paradox and the Russell paradox as the best examples of the problems of set theory in the early twentieth century. Did this mean that sets misdescribe reality, or that we had constructed them wrongly?