Combining Texts

Ideas for 'The Evolution of Logic', 'A Future for Presentism' and 'The Foundations of Mathematics'

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4 ideas

5. Theory of Logic / L. Paradox / 1. Paradox
13334 Contradictions are either purely logical or mathematical, or they involved thought and language [Ramsey]
     Full Idea: Group A consists of contradictions which would occur in a logical or mathematical system, involving terms such as class or number. Group B contradictions are not purely logical, and contain some reference to thought, language or symbolism.
     From: Frank P. Ramsey (The Foundations of Mathematics [1925], p.171), quoted by Graham Priest - The Structure of Paradoxes of Self-Reference 1
     A reaction: This has become the orthodox division of all paradoxes, but the division is challenged by Priest (Idea 13373). He suggests that we now realise (post-Tarski?) that language is more involved in logic and mathematics than we thought.
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
13484 Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that [Hart,WD]
     Full Idea: Berry's Paradox: by the least number principle 'the least number denoted by no description of fewer than 79 letters' exists, but we just referred to it using a description of 77 letters.
     From: William D. Hart (The Evolution of Logic [2010], 3)
     A reaction: I struggle with this. If I refer to 'an object to which no human being could possibly refer', have I just referred to something? Graham Priest likes this sort of idea.
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox
13482 The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD]
     Full Idea: The Burali-Forti Paradox was a crisis for Cantor's theory of ordinal numbers.
     From: William D. Hart (The Evolution of Logic [2010], 3)
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
13507 The machinery used to solve the Liar can be rejigged to produce a new Liar [Hart,WD]
     Full Idea: In effect, the machinery introduced to solve the liar can always be rejigged to yield another version the liar.
     From: William D. Hart (The Evolution of Logic [2010], 4)
     A reaction: [He cites Hans Herzberger 1980-81] The machinery is Tarski's device of only talking about sentences of a language by using a 'metalanguage'.