Combining Texts

All the ideas for 'The Fixation of Belief', 'Investigations in the Foundations of Set Theory I' and 'Necessity and Non-Existence'

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45 ideas

1. Philosophy / E. Nature of Metaphysics / 3. Metaphysical Systems
Metaphysics does not rest on facts, but on what we are inclined to believe [Peirce]
2. Reason / A. Nature of Reason / 4. Aims of Reason
Reason aims to discover the unknown by thinking about the known [Peirce]
2. Reason / D. Definition / 8. Impredicative Definition
Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine]
3. Truth / B. Truthmakers / 5. What Makes Truths / a. What makes truths
Some sentences depend for their truth on worldly circumstances, and others do not [Fine,K]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo]
Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg]
Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy]
Set theory can be reduced to a few definitions and seven independent axioms [Zermelo]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation
The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy]
Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR]
7. Existence / A. Nature of Existence / 2. Types of Existence
There are levels of existence, as well as reality; objects exist at the lowest level in which they can function [Fine,K]
7. Existence / D. Theories of Reality / 2. Realism
Realism is basic to the scientific method [Peirce]
7. Existence / D. Theories of Reality / 3. Reality
Bottom level facts are subject to time and world, middle to world but not time, and top to neither [Fine,K]
7. Existence / D. Theories of Reality / 4. Anti-realism
If someone doubted reality, they would not actually feel dissatisfaction [Peirce]
7. Existence / D. Theories of Reality / 8. Facts / b. Types of fact
Tensed and tenseless sentences state two sorts of fact, which belong to two different 'realms' of reality [Fine,K]
9. Objects / B. Unity of Objects / 1. Unifying an Object / a. Intrinsic unification
Modal features are not part of entities, because they are accounted for by the entity [Fine,K]
9. Objects / D. Essence of Objects / 6. Essence as Unifier
What it is is fixed prior to existence or the object's worldly features [Fine,K]
9. Objects / D. Essence of Objects / 9. Essence and Properties
Essential features of an object have no relation to how things actually are [Fine,K]
9. Objects / F. Identity among Objects / 5. Self-Identity
Self-identity should have two components, its existence, and its neutral identity with itself [Fine,K]
9. Objects / F. Identity among Objects / 6. Identity between Objects
We would understand identity between objects, even if their existence was impossible [Fine,K]
10. Modality / A. Necessity / 8. Transcendental Necessity
Proper necessary truths hold whatever the circumstances; transcendent truths regardless of circumstances [Fine,K]
10. Modality / C. Sources of Modality / 6. Necessity from Essence
It is the nature of Socrates to be a man, so necessarily he is a man [Fine,K]
10. Modality / E. Possible worlds / 2. Nature of Possible Worlds / a. Nature of possible worlds
Possible worlds may be more limited, to how things might actually turn out [Fine,K]
The actual world is a totality of facts, so we also think of possible worlds as totalities [Fine,K]
11. Knowledge Aims / A. Knowledge / 4. Belief / c. Aim of beliefs
The feeling of belief shows a habit which will determine our actions [Peirce]
We are entirely satisfied with a firm belief, even if it is false [Peirce]
We want true beliefs, but obviously we think our beliefs are true [Peirce]
A mere question does not stimulate a struggle for belief; there must be a real doubt [Peirce]
13. Knowledge Criteria / B. Internal Justification / 2. Pragmatic justification
We need our beliefs to be determined by some external inhuman permanency [Peirce]
13. Knowledge Criteria / B. Internal Justification / 4. Foundationalism / b. Basic beliefs
Demonstration does not rest on first principles of reason or sensation, but on freedom from actual doubt [Peirce]
13. Knowledge Criteria / C. External Justification / 1. External Justification
Doubts should be satisfied by some external permanency upon which thinking has no effect [Peirce]
13. Knowledge Criteria / D. Scepticism / 6. Scepticism Critique
Once doubt ceases, there is no point in continuing to argue [Peirce]
26. Natural Theory / B. Natural Kinds / 2. Defining Kinds
What is true of one piece of copper is true of another (unlike brass) [Peirce]
27. Natural Reality / D. Time / 2. Passage of Time / c. Tenses and time
It is said that in the A-theory, all existents and objects must be tensed, as well as the sentences [Fine,K]
A-theorists tend to reject the tensed/tenseless distinction [Fine,K]
27. Natural Reality / D. Time / 2. Passage of Time / f. Tenseless (B) series
B-theorists say tensed sentences have an unfilled argument-place for a time [Fine,K]
27. Natural Reality / G. Biology / 3. Evolution
Natural selection might well fill an animal's mind with pleasing thoughts rather than true ones [Peirce]
28. God / B. Proving God / 2. Proofs of Reason / d. Pascal's Wager
If death is annihilation, belief in heaven is a cheap pleasure with no disappointment [Peirce]