Combining Texts

All the ideas for 'Lectures on Ethics', 'A Tour through Mathematical Logic' and 'Convergence'

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31 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 ∀
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 / C. Capacities of Minds / 1. Faculties
20657 There are 23 core brain functions, with known circuit, transmitters, genes and behaviour [Watson]
17. Mind and Body / E. Mind as Physical / 1. Physical Mind
20656 Traditional ideas of the mind were weakened in the 1950s by mind-influencing drugs [Watson]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / e. Human nature
20655 Humans have been hunter-gatherers for 99.5% of their existence [Watson]
25. Social Practice / F. Life Issues / 4. Suicide
19739 The maxim for suicide is committed to the value of life, and is thus contradictory [Kant]
26. Natural Theory / C. Causation / 7. Eliminating causation
20650 The Uncertainty Principle implies that cause and effect can't be measured [Watson]
27. Natural Reality / B. Modern Physics / 2. Electrodynamics / a. Electrodynamics
20649 The interference of light through two slits confirmed that it is waves [Watson]
27. Natural Reality / B. Modern Physics / 2. Electrodynamics / c. Electrons
20661 Electrons rotate in hyrogen atoms 10^13 times per second [Watson]
27. Natural Reality / B. Modern Physics / 2. Electrodynamics / d. Quantum mechanics
20647 Quantum theory explains why nature is made up of units, such as elements [Watson]
27. Natural Reality / B. Modern Physics / 4. Standard Model / a. Concept of matter
20654 Only four particles are needed for matter: up and down quark, electron, electron-neutrino [Watson]
27. Natural Reality / F. Chemistry / 1. Chemistry
20651 The shape of molecules is important, as well as the atoms and their bonds [Watson]
27. Natural Reality / G. Biology / 2. Life
20652 In 1828 the animal substance urea was manufactured from inorganic ingredients [Watson]
20658 Information is physical, and living can be seen as replicating and preserving information [Watson]