Combining Texts

All the ideas for 'The Sayings of Confucius', 'Remarks on the forces of inorganic Nature' and 'A Tour through Mathematical Logic'

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28 ideas

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 ∀
13521 Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
13522 Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [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
13519 Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
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]
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]
19. Language / F. Communication / 1. Rhetoric
7357 People who control others with fluent language often end up being hated [Kongzi (Confucius)]
22. Metaethics / A. Ethics Foundations / 1. Nature of Ethics / h. Against ethics
7358 All men prefer outward appearance to true excellence [Kongzi (Confucius)]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / e. Human nature
7362 Humans are similar, but social conventions drive us apart (sages and idiots being the exceptions) [Kongzi (Confucius)]
23. Ethics / B. Contract Ethics / 2. Golden Rule
7360 Do not do to others what you would not desire yourself [Kongzi (Confucius)]
23. Ethics / C. Virtue Theory / 2. Elements of Virtue Theory / f. The Mean
7359 Excess and deficiency are equally at fault [Kongzi (Confucius)]
23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues
7363 The virtues of the best people are humility, maganimity, sincerity, diligence, and graciousness [Kongzi (Confucius)]
24. Political Theory / C. Ruling a State / 2. Leaders / d. Elites
7361 Men of the highest calibre avoid political life completely [Kongzi (Confucius)]
24. Political Theory / D. Ideologies / 3. Conservatism
23393 Confucianism assumes that all good developments have happened, and there is only one Way [Norden on Kongzi (Confucius)]
27. Natural Reality / A. Classical Physics / 1. Mechanics / d. Gravity
13605 Gravity isn't a force, because it produces effects without diminishing [Mayer]