Combining Philosophers

All the ideas for Anaxarchus, A.George / D.J.Velleman and R Keefe / P Smith

expand these ideas     |    start again     |     specify just one area for these philosophers


59 ideas

2. Reason / D. Definition / 7. Contextual Definition
Contextual definitions replace a complete sentence containing the expression [George/Velleman]
2. Reason / D. Definition / 8. Impredicative Definition
Impredicative definitions quantify over the thing being defined [George/Velleman]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
S5 collapses iterated modalities (◊□P→□P, and ◊◊P→◊P) [Keefe/Smith]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
The 'power set' of A is all the subsets of A [George/Velleman]
The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman]
Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
Grouping by property is common in mathematics, usually using equivalence [George/Velleman]
'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
The Axiom of Reducibility made impredicative definitions possible [George/Velleman]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman]
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
Differences between isomorphic structures seem unimportant [George/Velleman]
5. Theory of Logic / K. Features of Logics / 2. Consistency
Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman]
A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman]
5. Theory of Logic / K. Features of Logics / 3. Soundness
Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman]
5. Theory of Logic / K. Features of Logics / 4. Completeness
A 'complete' theory contains either any sentence or its negation [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Rational numbers give answers to division problems with integers [George/Velleman]
The integers are answers to subtraction problems involving natural numbers [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Real numbers provide answers to square root problems [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
Logicists say mathematics is applicable because it is totally general [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
The classical mathematician believes the real numbers form an actual set [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
A successor is the union of a set with its singleton [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set theory can prove the Peano Postulates [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman]
The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman]
Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman]
Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 8. Finitism
Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman]
Much infinite mathematics can still be justified finitely [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
The intuitionists are the idealists of mathematics [George/Velleman]
Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman]
7. Existence / D. Theories of Reality / 10. Vagueness / b. Vagueness of reality
Objects such as a cloud or Mount Everest seem to have fuzzy boundaries in nature [Keefe/Smith]
7. Existence / D. Theories of Reality / 10. Vagueness / c. Vagueness as ignorance
If someone is borderline tall, no further information is likely to resolve the question [Keefe/Smith]
The simplest approach, that vagueness is just ignorance, retains classical logic and semantics [Keefe/Smith]
The epistemic view of vagueness must explain why we don't know the predicate boundary [Keefe/Smith]
7. Existence / D. Theories of Reality / 10. Vagueness / f. Supervaluation for vagueness
Supervaluationism keeps true-or-false where precision can be produced, but not otherwise [Keefe/Smith]
Vague statements lack truth value if attempts to make them precise fail [Keefe/Smith]
Some of the principles of classical logic still fail with supervaluationism [Keefe/Smith]
The semantics of supervaluation (e.g. disjunction and quantification) is not classical [Keefe/Smith]
Supervaluation misunderstands vagueness, treating it as a failure to make things precise [Keefe/Smith]
7. Existence / D. Theories of Reality / 10. Vagueness / g. Degrees of vagueness
A third truth-value at borderlines might be 'indeterminate', or a value somewhere between 0 and 1 [Keefe/Smith]
People can't be placed in a precise order according to how 'nice' they are [Keefe/Smith]
If truth-values for vagueness range from 0 to 1, there must be someone who is 'completely tall' [Keefe/Smith]
How do we decide if my coat is red to degree 0.322 or 0.321? [Keefe/Smith]
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
Vague predicates involve uncertain properties, uncertain objects, and paradoxes of gradual change [Keefe/Smith]
Many vague predicates are multi-dimensional; 'big' involves height and volume; heaps include arrangement [Keefe/Smith]
If there is a precise borderline area, that is not a case of vagueness [Keefe/Smith]
13. Knowledge Criteria / D. Scepticism / 1. Scepticism
Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius]
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
Corresponding to every concept there is a class (some of them sets) [George/Velleman]