Combining Texts

All the ideas for 'poems', 'A Tour through Mathematical Logic' and 'Enquiry concerning Principles of Morals'

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33 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]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / j. Ethics by convention
6017 Nomos is king [Pindar]
22. Metaethics / B. Value / 2. Values / f. Altruism
3926 The human heart has a natural concern for public good [Hume]
23. Ethics / A. Egoism / 1. Ethical Egoism
3929 No moral theory is of any use if it doesn't serve the interests of the individual concerned [Hume]
23. Ethics / C. Virtue Theory / 1. Virtue Theory / a. Nature of virtue
3925 Personal Merit is the possession of useful or agreeable mental qualities [Hume]
23. Ethics / C. Virtue Theory / 3. Virtues / c. Justice
3922 Justice only exists to support society [Hume]
23560 If we all naturally had everything we could ever desire, the virtue of justice would be irrelevant [Hume]
23. Ethics / D. Deontological Ethics / 2. Duty
3918 Moral philosophy aims to show us our duty [Hume]
23. Ethics / D. Deontological Ethics / 6. Motivation for Duty
3919 Conclusions of reason do not affect our emotions or decisions to act [Hume]
23. Ethics / E. Utilitarianism / 1. Utilitarianism
3928 Virtue just requires careful calculation and a preference for the greater happiness [Hume]
23. Ethics / E. Utilitarianism / 3. Motivation for Altruism
3923 No one would cause pain to a complete stranger who happened to be passing [Hume]
3924 Nature makes private affections come first, because public concerns are spread too thinly [Hume]
24. Political Theory / B. Nature of a State / 1. Purpose of a State
3921 The safety of the people is the supreme law [Hume]
24. Political Theory / C. Ruling a State / 3. Government / a. Government
3927 Society prefers helpful lies to harmful truth [Hume]
25. Social Practice / B. Equalities / 4. Economic equality
3920 If you equalise possessions, people's talents will make them unequal again [Hume]