Combining Texts

All the ideas for 'Mahaprajnaparamitashastra', 'A Short History of Ethics' and 'A Tour through Mathematical Logic'

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31 ideas

1. Philosophy / B. History of Ideas / 4. Early European Thought
8013 In the Reformation, morality became unconditional but irrational, individually autonomous, and secular [MacIntyre]
1. Philosophy / B. History of Ideas / 5. Later European Thought
8021 The Levellers and the Diggers mark a turning point in the history of morality [MacIntyre]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
13520 A 'tautology' must include connectives [Wolf,RS]
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
13524 Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
13522 Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS]
13521 Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
13523 Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / e. Axiom of the Empty Set IV
13529 Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / n. Axiom of Comprehension
13526 Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
13534 In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
13535 First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13531 Model theory reveals the structures of mathematics [Wolf,RS]
13532 Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
13519 Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
13533 First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
13537 An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
13539 The LST Theorem is a serious limitation of first-order logic [Wolf,RS]
5. Theory of Logic / K. Features of Logics / 4. Completeness
13538 If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
5. Theory of Logic / K. Features of Logics / 10. Monotonicity
13525 Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13530 An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
13518 Modern mathematics has unified all of its objects within set theory [Wolf,RS]
15. Nature of Minds / A. Nature of Mind / 2. Psuche
8006 When Aristotle speaks of soul he means something like personality [MacIntyre]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / j. Ethics by convention
8002 Sophists don't distinguish a person outside one social order from someone outside all order [MacIntyre]
22. Metaethics / B. Value / 1. Nature of Value / b. Fact and value
8012 The value/fact logical gulf is misleading, because social facts involve values [MacIntyre]
22. Metaethics / C. The Good / 2. Happiness / b. Eudaimonia
8005 'Happiness' is a bad translation of 'eudaimonia', which includes both behaving and faring well [MacIntyre]
23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues
7903 The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]
23. Ethics / C. Virtue Theory / 3. Virtues / c. Justice
8001 'Dikaiosune' is justice, but also fairness and personal integrity [MacIntyre]
23. Ethics / D. Deontological Ethics / 2. Duty
8023 My duties depend on my identity, which depends on my social relations [MacIntyre]
24. Political Theory / A. Basis of a State / 3. Natural Values / a. Natural freedom
8022 I am naturally free if I am not tied to anyone by a contract [MacIntyre]
25. Social Practice / C. Rights / 1. Basis of Rights
8031 Fans of natural rights or laws can't agree on what the actual rights or laws are [MacIntyre]
29. Religion / B. Monotheistic Religion / 5. Bible
8008 The Bible is a story about God in which humans are incidental characters [MacIntyre]