35 ideas
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 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] |
| 13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS] |
| 13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS] |
| 13526 | Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS] |
| 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] |
| 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] |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| 13539 | The LST Theorem is a serious limitation of first-order logic [Wolf,RS] |
| 13538 | If a theory is complete, only a more powerful language can strengthen it [Wolf,RS] |
| 13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS] |
| 13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
| 13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
| 7949 | Varied descriptions of an event will explain varied behaviour relating to it [Davidson, by Macdonald,C] |
| 21833 | Research suggest that we overrate conscious experience [Flanagan] |
| 21834 | Sensations may be identical to brain events, but complex mental events don't seem to be [Flanagan] |
| 20020 | If one action leads directly to another, they are all one action [Davidson, by Wilson/Schpall] |
| 20072 | We explain an intention by giving an account of acting with an intention [Davidson, by Stout,R] |
| 20045 | Acting for a reason is a combination of a pro attitude, and a belief that the action is appropriate [Davidson] |
| 23734 | The best explanation of reasons as purposes for actions is that they are causal [Davidson, by Smith,M] |
| 23737 | Reasons can give purposes to actions, without actually causing them [Smith,M on Davidson] |
| 20075 | Early Davidson says intentional action is caused by reasons [Davidson, by Stout,R] |
| 6664 | Reasons must be causes when agents act 'for' reasons [Davidson, by Lowe] |
| 3395 | Davidson claims that what causes an action is the reason for doing it [Davidson, by Kim] |
| 21837 | Morality is normative because it identifies best practices among the normal practices [Flanagan] |
| 21830 | For Darwinians, altruism is either contracts or genetics [Flanagan] |
| 21835 | We need Eudaimonics - the empirical study of how we should flourish [Flanagan] |
| 21831 | Alienation is not finding what one wants, or being unable to achieve it [Flanagan] |
| 21832 | Buddhists reject God and the self, and accept suffering as key, and liberation through wisdom [Flanagan] |