49 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] |
13479 | Given that thinking aims at truth, logic gives universal rules for how to do it [Burge] |
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] |
8132 | We now have a much more sophisticated understanding of logical form in language [Burge] |
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] |
17622 | We come to believe mathematical propositions via their grounding in the structure [Burge] |
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] |
16901 | The equivalent algebra model of geometry loses some essential spatial meaning [Burge] |
9159 | You can't simply convert geometry into algebra, as some spatial content is lost [Burge] |
13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS] |
16902 | Peano arithmetic requires grasping 0 as a primitive number [Burge] |
13518 | Modern mathematics has unified all of its objects within set theory [Wolf,RS] |
16892 | Is apriority predicated mainly of truths and proofs, or of human cognition? [Burge] |
9382 | Subjects may be unaware of their epistemic 'entitlements', unlike their 'justifications' [Burge] |
8126 | Anti-individualism says the environment is involved in the individuation of some mental states [Burge] |
8127 | Broad concepts suggest an extension of the mind into the environment (less computer-like) [Burge] |
8129 | Anti-individualism may be incompatible with some sorts of self-knowledge [Burge] |
8131 | Some qualities of experience, like blurred vision, have no function at all [Burge] |
3115 | Are meaning and expressed concept the same thing? [Burge, by Segal] |
20289 | Ethics is universalisable - it must involve an impartial and universal view of things [Singer] |
20286 | Following an inner voice for morality is irresponsible in a rational agent [Singer] |
20282 | The sanctity of a human life depends either on being of our species, or on being a person [Singer] |
20278 | 'Marginal utility' says something is more useful if it is in short supply [Singer] |
20281 | Why should I do anything for posterity? What has posterity ever done for me? [Singer] |
20276 | Conflict of rules might be avoided by greater complexity, or by a hierarchy of rules [Singer] |
20290 | Psychopaths may just be bored, because they cannot participate in normal emotional life [Singer] |
20288 | You can't condemn violent revolution without assessing the evils it prevents [Singer] |
21997 | In Marxism the state will be superseded [Singer] |
20287 | If 49% of the population can be wrong, so can 51% [Singer] |
21993 | Materialist history says we are subject to incomprehensible forces [Singer] |
20277 | Equality of interests is a minimal principle, not implying equal treatment [Singer] |
20279 | Equality of opportunity unfairly rewards those lucky enough to have great ability [Singer] |
20285 | If a right entails having the relevant desire, many creatures might have no right to life [Singer] |
20284 | Why should a potential person have the rights of an actual person? [Singer] |
20283 | Killing a chimp is worse than killing a human too defective to be a person [Singer] |
14349 | If there are no finks or antidotes at the fundamental level, the laws can't be ceteris paribus [Burge, by Corry] |