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Single Idea 21720
[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
]
Full Idea
The axiom of Reducibility ...is crucial in the reduction of classes to logic, ...and seems to be a quite legitimate logical notion for Russell.
Gist of Idea
Russell saw Reducibility as legitimate for reducing classes to logic
Source
comment on B Russell/AN Whitehead (Principia Mathematica [1913]) by Bernard Linsky - Russell's Metaphysical Logic 6.4
Book Ref
Linsky,Bernard: 'Russell's Metaphysical Logic' [CSLI 1999], p.101
A Reaction
This is an unusual defence of the axiom, which is usually presumed to have been kicked into the long grass by Quine. If one could reduce classes to logic, that would destroy the opposition to logicism in a single neat coup.
The
26 ideas
from 'Principia Mathematica'
8204
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Lewis's 'strict implication' preserved Russell's confusion of 'if...then' with implication
[Quine on Russell/Whitehead]
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9359
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Russell's implication means that random sentences imply one another
[Lewis,CI on Russell/Whitehead]
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21707
|
Russell unusually saw logic as 'interpreted' (though very general, and neutral)
[Russell/Whitehead, by Linsky,B]
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10036
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In 'Principia' a new abstract theory of relations appeared, and was applied
[Russell/Whitehead, by Gödel]
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18248
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A real number is the class of rationals less than the number
[Russell/Whitehead, by Shapiro]
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10037
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'Principia' lacks a precise statement of the syntax
[Gödel on Russell/Whitehead]
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18152
|
Russell takes numbers to be classes, but then reduces the classes to numerical quantifiers
[Russell/Whitehead, by Bostock]
|
10025
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Russell and Whitehead took arithmetic to be higher-order logic
[Russell/Whitehead, by Hodes]
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8683
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Russell and Whitehead were not realists, but embraced nearly all of maths in logic
[Russell/Whitehead, by Friend]
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10093
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The ramified theory of types used propositional functions, and covered bound variables
[Russell/Whitehead, by George/Velleman]
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8691
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The Russell/Whitehead type theory was limited, and was not really logic
[Friend on Russell/Whitehead]
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9542
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The best known axiomatization of PL is Whitehead/Russell, with four axioms and two rules
[Russell/Whitehead, by Hughes/Cresswell]
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21720
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Russell saw Reducibility as legitimate for reducing classes to logic
[Linsky,B on Russell/Whitehead]
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10044
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Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element
[Russell/Whitehead, by Shapiro]
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10040
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Russell showed, through the paradoxes, that our basic logical intuitions are self-contradictory
[Russell/Whitehead, by Gödel]
|
21725
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The multiple relations theory says assertions about propositions are about their ingredients
[Russell/Whitehead, by Linsky,B]
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8684
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Russell and Whitehead consider the paradoxes to indicate that we create mathematical reality
[Russell/Whitehead, by Friend]
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8746
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To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it
[Russell/Whitehead, by Shapiro]
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12033
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An object is identical with itself, and no different indiscernible object can share that
[Russell/Whitehead, by Adams,RM]
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10305
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In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains
[Bernays on Russell/Whitehead]
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23474
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A judgement is a complex entity, of mind and various objects
[Russell/Whitehead]
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23455
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The meaning of 'Socrates is human' is completed by a judgement
[Russell/Whitehead]
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23480
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The multiple relation theory of judgement couldn't explain the unity of sentences
[Morris,M on Russell/Whitehead]
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18275
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Only the act of judging completes the meaning of a statement
[Russell/Whitehead]
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23453
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Propositions as objects of judgement don't exist, because we judge several objects, not one
[Russell/Whitehead]
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18208
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We regard classes as mere symbolic or linguistic conveniences
[Russell/Whitehead]
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