60 ideas
| 8014 | Resolve a complex into simple elements, then reconstruct the complex by using them [Hobbes, by MacIntyre] |
| 13520 | A 'tautology' must include connectives [Wolf,RS] |
| 13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS] |
| 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] |
| 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] |
| 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] |
| 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] |
| 7559 | Every part of the universe is body, and non-body is not part of it [Hobbes] |
| 2356 | Appearance and reality can be separated by mirrors and echoes [Hobbes] |
| 2357 | Dreams must be false because they seem absurd, but dreams don't see waking as absurd [Hobbes] |
| 2358 | Freedom is absence of opposition to action; the idea of 'free will' is absurd [Hobbes] |
| 6214 | Liberty and necessity are consistent, as when water freely flows, by necessity [Hobbes] |
| 23987 | The 'simple passions' are appetite, desire, love, aversion, hate, joy, and grief [Hobbes, by Goldie] |
| 2362 | The will is just the last appetite before action [Hobbes] |
| 2363 | Reason is usually general, but deliberation is of particulars [Hobbes] |
| 2360 | 'Good' is just what we desire, and 'Evil' what we hate [Hobbes] |
| 2368 | Men's natural desires are no sin, and neither are their actions, until law makes it so [Hobbes] |
| 2359 | Desire and love are the same, but in the desire the object is absent, and in love it is present [Hobbes] |
| 2370 | All voluntary acts aim at some good for the doer [Hobbes] |
| 2371 | A contract is a mutual transfer of rights [Hobbes] |
| 2372 | The person who performs first in a contract is said to 'merit' the return, and is owed it [Hobbes] |
| 8015 | Hobbes wants a contract to found morality, but shared values are needed to make a contract [MacIntyre on Hobbes] |
| 5337 | For Hobbes the Golden Rule concerns not doing things, whereas Jesus encourages active love [Hobbes, by Flanagan] |
| 2374 | In the violent state of nature, the merest suspicion is enough to justify breaking a contract [Hobbes] |
| 2375 | Suspicion will not destroy a contract, if there is a common power to enforce it [Hobbes] |
| 8016 | Fear of sanctions is the only motive for acceptance of authority that Hobbes can think of [MacIntyre on Hobbes] |
| 2377 | No one who admitted to not keeping contracts could ever be accepted as a citizen [Hobbes] |
| 2379 | If there is a good reason for breaking a contract, the same reason should have stopped the making of it [Hobbes] |
| 2373 | The first performer in a contract is handing himself over to an enemy [Hobbes] |
| 2382 | Someone who keeps all his contracts when others are breaking them is making himself a prey to others [Hobbes] |
| 2383 | Virtues are a means to peaceful, sociable and comfortable living [Hobbes] |
| 2376 | Injustice is the failure to keep a contract, and justice is the constant will to give what is owed [Hobbes] |
| 2367 | In time of war the life of man is solitary, poor, nasty, brutish and short [Hobbes] |
| 19764 | Hobbes attributed to savages the passions which arise in a law-bound society [Hobbes, by Rousseau] |
| 20566 | Hobbes says the people voluntarily give up their sovereignty, in a contract with a ruler [Hobbes, by Oksala] |
| 2366 | There is not enough difference between people for one to claim more benefit than another [Hobbes] |
| 20485 | Hobbes says people are roughly equal; Locke says there is no right to impose inequality [Hobbes, by Wolff,J] |
| 2369 | If we seek peace and defend ourselves, we must compromise on our rights [Hobbes] |
| 20484 | We should obey the laws of nature, provided other people are also obeying them [Hobbes, by Wolff,J] |
| 7573 | The legal positivism of Hobbes said law is just formal or procedural [Hobbes, by Jolley] |
| 2380 | Punishment should only be for reform or deterrence [Hobbes] |
| 2361 | If fear of unknown powers is legal it is religion, if it is illegal it is superstition [Hobbes] |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| 2364 | Causation is only observation of similar events following each other, with nothing visible in between [Hobbes] |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| 2365 | Religion is built on ignorance and misinterpretation of what is unknown or frightening [Hobbes] |
| 2378 | Belief in an afterlife is based on poorly founded gossip [Hobbes] |