Ideas from 'Principia Mathematica' by B Russell/AN Whitehead [1913], by Theme Structure
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4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
9542

The best known axiomatization of PL is Whitehead/Russell, with four axioms and two rules [Hughes/Cresswell]

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
21720

Russell saw Reducibility as legitimate for reducing classes to logic [Linsky,B]

4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
10044

Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element [Shapiro]

18208

We regard classes as mere symbolic or linguistic conveniences

5. Theory of Logic / B. Logical Consequence / 7. Strict Implication
8204

Lewis's 'strict implication' preserved Russell's confusion of 'if...then' with implication [Quine]

9359

Russell's implication means that random sentences imply one another [Lewis,CI]

5. Theory of Logic / C. Ontology of Logic / 1. Ontology of Logic
21707

Russell unusually saw logic as 'interpreted' (though very general, and neutral) [Linsky,B]

5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic
10036

In 'Principia' a new abstract theory of relations appeared, and was applied [Gödel]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
18248

A real number is the class of rationals less than the number [Shapiro]

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / a. Defining numbers
18152

Russell takes numbers to be classes, but then reduces the classes to numerical quantifiers [Bostock]

6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
10025

Russell and Whitehead took arithmetic to be higherorder logic [Hodes]

8683

Russell and Whitehead were not realists, but embraced nearly all of maths in logic [Friend]

10037

'Principia' lacks a precise statement of the syntax [Gödel]

6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
10093

The ramified theory of types used propositional functions, and covered bound variables [George/Velleman]

8691

The Russell/Whitehead type theory was limited, and was not really logic [Friend]

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
10305

In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains [Bernays]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
8684

Russell and Whitehead consider the paradoxes to indicate that we create mathematical reality [Friend]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
8746

To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it [Shapiro]

9. Objects / F. Identity among Objects / 7. Indiscernible Objects
12033

An object is identical with itself, and no different indiscernible object can share that [Adams,RM]

12. Knowledge Sources / E. Direct Knowledge / 2. Intuition
10040

Russell showed, through the paradoxes, that our basic logical intuitions are selfcontradictory [Gödel]

18. Thought / A. Modes of Thought / 6. Judgement / a. Nature of Judgement
21725

The multiple relations theory says assertions about propositions are about their ingredients [Linsky,B]

23474

A judgement is a complex entity, of mind and various objects

23455

The meaning of 'Socrates is human' is completed by a judgement

23480

The multiple relation theory of judgement couldn't explain the unity of sentences [Morris,M]

18275

Only the act of judging completes the meaning of a statement

19. Language / D. Propositions / 3. Concrete Propositions
23453

Propositions as objects of judgement don't exist, because we judge several objects, not one
