170 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] |
| 21394 | Philosophy is knowing each logos, how they fit together, and what follows from them [Epictetus] |
| 23367 | Even pointing a finger should only be done for a reason [Epictetus] |
| 20876 | Philosophy investigates the causes of disagreements, and seeks a standard for settling them [Epictetus] |
| 1887 | You cannot divide anything into many parts, because after the first division you are no longer dividing the original [Sext.Empiricus] |
| 22764 | Ordinary speech is not exact about what is true; we say we are digging a well before the well exists [Sext.Empiricus] |
| 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] |
| 22752 | Reasoning is impossible without a preconception [Sext.Empiricus] |
| 1885 | Proof moves from agreed premises to a non-evident inference [Sext.Empiricus] |
| 6021 | It is only when we say a proposition that we speak truly or falsely [Sext.Empiricus] |
| 9535 | 'Contradictory' propositions always differ in truth-value [Lemmon] |
| 9509 | That proposition that both P and Q is their 'conjunction', written P∧Q [Lemmon] |
| 9511 | We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q [Lemmon] |
| 9512 | We write the 'negation' of P (not-P) as ¬ [Lemmon] |
| 9510 | That proposition that either P or Q is their 'disjunction', written P∨Q [Lemmon] |
| 9513 | We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P) [Lemmon] |
| 9508 | The sign |- may be read as 'therefore' [Lemmon] |
| 9514 | If A and B are 'interderivable' from one another we may write A -||- B [Lemmon] |
| 9519 | A 'substitution-instance' is a wff formed by consistent replacing variables with wffs [Lemmon] |
| 9529 | A wff is 'inconsistent' if all assignments to variables result in the value F [Lemmon] |
| 9534 | Two propositions are 'equivalent' if they mirror one another's truth-value [Lemmon] |
| 9532 | 'Subcontrary' propositions are never both false, so that A∨B is a tautology [Lemmon] |
| 9531 | 'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology [Lemmon] |
| 9528 | A wff is a 'tautology' if all assignments to variables result in the value T [Lemmon] |
| 9530 | A wff is 'contingent' if produces at least one T and at least one F [Lemmon] |
| 9518 | A 'theorem' is the conclusion of a provable sequent with zero assumptions [Lemmon] |
| 9533 | A 'implies' B if B is true whenever A is true (so that A→B is tautologous) [Lemmon] |
| 9516 | A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔ [Lemmon] |
| 9517 | The 'scope' of a connective is the connective, the linked formulae, and the brackets [Lemmon] |
| 9396 | DN: Given A, we may derive ¬¬A [Lemmon] |
| 9398 | ∧I: Given A and B, we may derive A∧B [Lemmon] |
| 9394 | MPP: Given A and A→B, we may derive B [Lemmon] |
| 9399 | ∧E: Given A∧B, we may derive either A or B separately [Lemmon] |
| 9401 | ∨E: Derive C from A∨B, if C can be derived both from A and from B [Lemmon] |
| 9395 | MTT: Given ¬B and A→B, we derive ¬A [Lemmon] |
| 9393 | A: we may assume any proposition at any stage [Lemmon] |
| 9400 | ∨I: Given either A or B separately, we may derive A∨B [Lemmon] |
| 9402 | RAA: If assuming A will prove B∧¬B, then derive ¬A [Lemmon] |
| 9397 | CP: Given a proof of B from A as assumption, we may derive A→B [Lemmon] |
| 9522 | 'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q [Lemmon] |
| 9526 | We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q) [Lemmon] |
| 9527 | The Distributive Laws can rearrange a pair of conjunctions or disjunctions [Lemmon] |
| 9523 | De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions [Lemmon] |
| 9524 | We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q [Lemmon] |
| 9525 | We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q) [Lemmon] |
| 9521 | 'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q [Lemmon] |
| 9537 | Truth-tables are good for showing invalidity [Lemmon] |
| 9538 | A truth-table test is entirely mechanical, but this won't work for more complex logic [Lemmon] |
| 9536 | If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology [Lemmon] |
| 9539 | Propositional logic is complete, since all of its tautologous sequents are derivable [Lemmon] |
| 13909 | Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....' [Lemmon] |
| 13902 | 'Gm' says m has property G, and 'Pmn' says m has relation P to n [Lemmon] |
| 13911 | The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E [Lemmon] |
| 13910 | Our notation uses 'predicate-letters' (for 'properties'), 'variables', 'proper names', 'connectives' and 'quantifiers' [Lemmon] |
| 13904 | Universal Elimination (UE) lets us infer that an object has F, from all things having F [Lemmon] |
| 13901 | Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules [Lemmon] |
| 13903 | Universal elimination if you start with the universal, introduction if you want to end with it [Lemmon] |
| 13906 | With finite named objects, we can generalise with &-Intro, but otherwise we need ∀-Intro [Lemmon] |
| 13908 | UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one [Lemmon] |
| 13905 | If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers [Lemmon] |
| 13900 | 'Some Frenchmen are generous' is rendered by (∃x)(Fx→Gx), and not with the conditional → [Lemmon] |
| 12196 | A valid hypothetical syllogism is 'that which does not begin with a truth and end with a falsehood' [Sext.Empiricus] |
| 9520 | The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q [Lemmon] |
| 6020 | 'Man is a rational mortal animal' is equivalent to 'if something is a man, that thing is a rational mortal animal' [Sext.Empiricus] |
| 1902 | Since Socrates either died when he was alive (a contradiction) or died when he was dead (meaningless), he didn't die [Sext.Empiricus] |
| 22744 | Parts are not parts if their whole is nothing more than the parts [Sext.Empiricus] |
| 22762 | Some properties are inseparable from a thing, such as the length, breadth and depth of a body [Sext.Empiricus] |
| 23359 | We can't believe apparent falsehoods, or deny apparent truths [Epictetus] |
| 1889 | If an argument has an absurd conclusion, we should not assent to the absurdity, but avoid the absurd argument [Sext.Empiricus] |
| 1871 | Whether honey is essentially sweet may be doubted, as it is a matter of judgement rather than appearance [Sext.Empiricus] |
| 23356 | Self-evidence is most obvious when people who deny a proposition still have to use it [Epictetus] |
| 1883 | How can the intellect know if sensation is reliable if it doesn't directly see external objects? [Sext.Empiricus] |
| 22748 | Some say motion is perceived by sense, but others say it is by intellect [Sext.Empiricus] |
| 1890 | We distinguish ambiguities by seeing what is useful [Sext.Empiricus] |
| 22759 | Fools, infants and madmen may speak truly, but do not know [Sext.Empiricus] |
| 20795 | Some things are their own criterion, such as straightness, a set of scales, or light [Sext.Empiricus] |
| 22760 | Madmen are reliable reporters of what appears to them [Sext.Empiricus] |
| 1870 | The basis of scepticism is the claim that every proposition has an equal opposing proposition [Sext.Empiricus] |
| 1882 | The necks of doves appear different in colour depending on the angle of viewing [Sext.Empiricus] |
| 1881 | The same oar seems bent in water and straight when out of it [Sext.Empiricus] |
| 1872 | The same tower appears round from a distance, but square close at hand [Sext.Empiricus] |
| 1873 | If we press the side of an eyeball, objects appear a different shape [Sext.Empiricus] |
| 20794 | How can sceptics show there is no criterion? Weak without, contradiction with [Sext.Empiricus] |
| 1874 | How can we judge between our impressions and those of other animals, when we ourselves are involved? [Sext.Empiricus] |
| 1876 | If we enjoy different things, presumably we receive different impressions [Sext.Empiricus] |
| 1880 | Some actions seem shameful when sober but not when drunk [Sext.Empiricus] |
| 1879 | Sickness is perfectly natural to the sick, so their natural perceptions should carry some weight [Sext.Empiricus] |
| 1877 | If we had no hearing or sight, we would assume no sound or sight exists, so there may be unsensed qualities [Sext.Empiricus] |
| 1878 | Water that seems lukewarm can seem very hot on inflamed skin [Sext.Empiricus] |
| 1910 | With us it is shameful for men to wear earrings, but among Syrians it is considered noble [Sext.Empiricus] |
| 1911 | Even if all known nations agree on a practice, there may be unknown nations which disagree [Sext.Empiricus] |
| 6026 | How can you investigate without some preconception of your object? [Sext.Empiricus] |
| 1886 | If you don't view every particular, you may miss the one which disproves your universal induction [Sext.Empiricus] |
| 22746 | If we try to conceive of a line with no breadth, it ceases to exist, and so has no length [Sext.Empiricus] |
| 23330 | Freedom is making all things happen by choice, without constraint [Epictetus] |
| 23342 | Freedom is acting by choice, with no constraint possible [Epictetus] |
| 23329 | We make progress when we improve and naturalise our choices, asserting their freedom [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] |
| 22741 | The incorporeal is not in the nature of body, and so could not emerge from it [Sext.Empiricus] |
| 1884 | If we utter three steps of a logical argument, they never exist together [Sext.Empiricus] |
| 22763 | We can only dream of a winged man if we have experienced men and some winged thing [Sext.Empiricus] |
| 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] |
| 23339 | I will die as becomes a person returning what he does not own [Epictetus] |
| 23345 | Don't be frightened of pain or death; only be frightened of fearing them [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] |
| 22754 | Saying the good is useful or choiceworth or happiness-creating is not the good, but a feature of it [Sext.Empiricus] |
| 22755 | Like a warming fire, what is good by nature should be good for everyone [Sext.Empiricus] |
| 23328 | The essences of good and evil are in dispositions to choose [Epictetus] |
| 22756 | If a desire is itself desirable, then we shouldn't desire it, as achieving it destroys it [Sext.Empiricus] |
| 6032 | Right actions, once done, are those with a reasonable justification [Sext.Empiricus] |
| 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] |
| 23352 | A citizen should only consider what is good for the whole society [Epictetus] |
| 20874 | A citizen is committed to ignore private advantage, and seek communal good [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] |
| 1517 | The tektraktys (1+2+3+4=10) is the 'fount of ever-flowing nature' [Sext.Empiricus] |
| 1894 | Some say that causes are physical, some say not [Sext.Empiricus] |
| 1896 | If there were no causes then everything would have been randomly produced by everything [Sext.Empiricus] |
| 1897 | Knowing an effect results from a cause means knowing that the cause belongs with the effect, which is circular [Sext.Empiricus] |
| 1898 | Cause can't exist before effect, or exist at the same time, so it doesn't exist [Sext.Empiricus] |
| 1895 | Causes are either equal to the effect, or they link equally with other causes, or they contribute slightly [Sext.Empiricus] |
| 1899 | Does the original self-mover push itself from behind, or pull itself from in front? [Sext.Empiricus] |
| 22747 | A man walking backwards on a forwards-moving ship is moving in a fixed place [Sext.Empiricus] |
| 1900 | If time and place are infinitely divided, it becomes impossible for movement ever to begin [Sext.Empiricus] |
| 1901 | If all atoms, times and places are the same, everything should move with equal velocity [Sext.Empiricus] |
| 1903 | If motion and rest are abolished, so is time [Sext.Empiricus] |
| 1904 | Time must be unlimited, but past and present can't be non-existent, and can't be now, so time does not exist [Sext.Empiricus] |
| 22749 | Time doesn't end with the Universe, because tensed statements about destruction remain true [Sext.Empiricus] |
| 22750 | Time is divisible, into past, present and future [Sext.Empiricus] |
| 1905 | How can time be divisible if we can't compare one length of time with another? [Sext.Empiricus] |
| 22742 | Socrates either dies when he exists (before his death) or when he doesn't (after his death) [Sext.Empiricus] |
| 22751 | If the present is just the limit of the past or the future, it can't exist because they don't exist [Sext.Empiricus] |
| 23341 | God created humans as spectators and interpreters of God's works [Epictetus] |
| 22730 | All men agree that God is blessed, imperishable, happy and good [Sext.Empiricus] |
| 22739 | God must suffer to understand suffering [Sext.Empiricus] |
| 1891 | How can we agree on the concept of God, unless we agree on his substance or form or place? [Sext.Empiricus] |
| 22738 | The Divine must lack the virtues of continence and fortitude, because they are not needed [Sext.Empiricus] |
| 23348 | Both god and the good bring benefits, so their true nature seems to be the same [Epictetus] |
| 22734 | God is defended by agreement, order, absurdity of denying God, and refutations [Sext.Empiricus] |
| 1892 | The existence of God can't be self-evident or everyone would have agreed on it, so it needs demonstration [Sext.Empiricus] |
| 22736 | God's sensations imply change, and hence perishing, which is absurd, so there is no such God [Sext.Empiricus] |
| 22740 | God without virtue is absurd, but God's virtues will be better than God [Sext.Empiricus] |
| 22735 | The original substance lacked motion or shape, and was given these by a cause [Sext.Empiricus] |
| 22732 | The perfections of God were extrapolations from mankind [Sext.Empiricus] |
| 22728 | Gods were invented as watchers of people's secret actions [Sext.Empiricus] |
| 22737 | An incorporeal God could do nothing, and a bodily god would perish, so there is no God [Sext.Empiricus] |
| 22731 | It is mad to think that what is useful to us, like lakes and rivers, are gods [Sext.Empiricus] |
| 23360 | Each of the four elements in you is entirely scattered after death [Epictetus] |
| 1893 | If God foresaw evil he would presumably prevent it, and if he only foresees some things, why those things? [Sext.Empiricus] |