65 ideas
| 23350 | A wise philosophers uses reason to cautiously judge each aspect of living [Epictetus] |
| 23355 | The task of philosophy is to establish standards, as occurs with weights and measures [Epictetus] |
| 23367 | Even pointing a finger should only be done for a reason [Epictetus] |
| 21394 | Philosophy is knowing each logos, how they fit together, and what follows from them [Epictetus] |
| 20876 | Philosophy investigates the causes of disagreements, and seeks a standard for settling them [Epictetus] |
| 23344 | Reason itself must be compounded from some of our impressions [Epictetus] |
| 23343 | Because reason performs all analysis, we should analyse reason - but how? [Epictetus] |
| 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] |
| 23359 | We can't believe apparent falsehoods, or deny apparent truths [Epictetus] |
| 23356 | Self-evidence is most obvious when people who deny a proposition still have to use it [Epictetus] |
| 23329 | We make progress when we improve and naturalise our choices, asserting their freedom [Epictetus] |
| 23342 | Freedom is acting by choice, with no constraint possible [Epictetus] |
| 23330 | Freedom is making all things happen by choice, without constraint [Epictetus] |
| 23332 | Zeus gave me a nature which is free (like himself) from all compulsion [Epictetus] |
| 23331 | Not even Zeus can control what I choose [Epictetus] |
| 23338 | You can fetter my leg, but not even Zeus can control my power of choice [Epictetus] |
| 20875 | If we could foresee the future, we should collaborate with disease and death [Epictetus] |
| 23347 | If I know I am fated to be ill, I should want to be ill [Epictetus] |
| 23325 | Epictetus developed a notion of will as the source of our responsibility [Epictetus, by Frede,M] |
| 20873 | Tragedies are versified sufferings of people impressed by externals [Epictetus] |
| 23364 | Homer wrote to show that the most blessed men can be ruined by poor judgement [Epictetus] |
| 23340 | We consist of animal bodies and god-like reason [Epictetus] |
| 23366 | We see nature's will in the ways all people are the same [Epictetus] |
| 23358 | Every species produces exceptional beings, and we must just accept their nature [Epictetus] |
| 23345 | Don't be frightened of pain or death; only be frightened of fearing them [Epictetus] |
| 23339 | I will die as becomes a person returning what he does not own [Epictetus] |
| 23357 | Knowledge of what is good leads to love; only the wise, who distinguish good from evil, can love [Epictetus] |
| 23363 | The evil for everything is what is contrary to its nature [Epictetus] |
| 23328 | The essences of good and evil are in dispositions to choose [Epictetus] |
| 23362 | All human ills result from failure to apply preconceptions to particular cases [Epictetus] |
| 23353 | We have a natural sense of honour [Epictetus] |
| 23354 | If someone harms themselves in harming me, then I harm myself by returning the harm [Epictetus] |
| 23324 | In the Discourses choice [prohairesis] defines our character and behaviour [Epictetus, by Frede,M] |
| 4022 | Epictetus says we should console others for misfortune, but not be moved by pity [Epictetus, by Taylor,C] |
| 23365 | If someone is weeping, you should sympathise and help, but not share his suffering [Epictetus] |
| 23361 | Health is only a good when it is used well [Epictetus] |
| 23346 | A person is as naturally a part of a city as a foot is part of the body [Epictetus] |
| 23351 | We are citizens of the universe, and principal parts of it [Epictetus] |
| 20874 | A citizen is committed to ignore private advantage, and seek communal good [Epictetus] |
| 23352 | A citizen should only consider what is good for the whole society [Epictetus] |
| 22604 | Punishing a criminal for moral ignorance is the same as punishing someone for being blind [Epictetus] |
| 23368 | Perhaps we should persuade culprits that their punishment is just? [Epictetus] |
| 23349 | Asses are born to carry human burdens, not as ends in themselves [Epictetus] |
| 23341 | God created humans as spectators and interpreters of God's works [Epictetus] |
| 23348 | Both god and the good bring benefits, so their true nature seems to be the same [Epictetus] |
| 16714 | The state should kill blasphemous heretics [Erasmus] |
| 23360 | Each of the four elements in you is entirely scattered after death [Epictetus] |