143 ideas
| 2136 | Philosophers become as divine and orderly as possible, by studying divinity and order [Plato] |
| 23767 | The winds of the discussion should decide its destination [Plato] |
| 23682 | It would be absurd to be precise about the small things, but only vague about the big things [Plato] |
| 2151 | Dialectic is the only method of inquiry which uproots the things which it takes for granted [Plato] |
| 2154 | The ability to take an overview is the distinguishing mark of a dialectician [Plato] |
| 4011 | For Plato, rationality is a vision of and love of a cosmic rational order [Plato, by Taylor,C] |
| 2093 | You must never go against what you actually believe [Plato] |
| 2130 | People often merely practice eristic instead of dialectic, because they don't analyse the subject-matter [Plato] |
| 9535 | 'Contradictory' propositions always differ in truth-value [Lemmon] |
| 9511 | We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q [Lemmon] |
| 9510 | That proposition that either P or Q is their 'disjunction', written P∨Q [Lemmon] |
| 9512 | We write the 'negation' of P (not-P) as ¬ [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] |
| 9514 | If A and B are 'interderivable' from one another we may write A -||- B [Lemmon] |
| 9509 | That proposition that both P and Q is their 'conjunction', written P∧Q [Lemmon] |
| 9508 | The sign |- may be read as 'therefore' [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] |
| 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] |
| 9531 | 'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology [Lemmon] |
| 9534 | Two propositions are 'equivalent' if they mirror one another's truth-value [Lemmon] |
| 9530 | A wff is 'contingent' if produces at least one T and at least one F [Lemmon] |
| 9532 | 'Subcontrary' propositions are never both false, so that A∨B is a tautology [Lemmon] |
| 9533 | A 'implies' B if B is true whenever A is true (so that A→B is tautologous) [Lemmon] |
| 9528 | A wff is a 'tautology' if all assignments to variables result in the value T [Lemmon] |
| 9518 | A 'theorem' is the conclusion of a provable sequent with zero assumptions [Lemmon] |
| 9398 | ∧I: Given A and B, we may derive A∧B [Lemmon] |
| 9397 | CP: Given a proof of B from A as assumption, we may derive A→B [Lemmon] |
| 9394 | MPP: Given A and A→B, we may derive B [Lemmon] |
| 9402 | RAA: If assuming A will prove B∧¬B, then derive ¬A [Lemmon] |
| 9395 | MTT: Given ¬B and A→B, we derive ¬A [Lemmon] |
| 9400 | ∨I: Given either A or B separately, we may derive A∨B [Lemmon] |
| 9401 | ∨E: Derive C from A∨B, if C can be derived both from A and from B [Lemmon] |
| 9396 | DN: Given A, we may derive ¬¬A [Lemmon] |
| 9393 | A: we may assume any proposition at any stage [Lemmon] |
| 9399 | ∧E: Given A∧B, we may derive either A or B separately [Lemmon] |
| 9521 | 'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q [Lemmon] |
| 9522 | 'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q [Lemmon] |
| 9525 | We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q) [Lemmon] |
| 9524 | We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q [Lemmon] |
| 9523 | De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions [Lemmon] |
| 9527 | The Distributive Laws can rearrange a pair of conjunctions or disjunctions [Lemmon] |
| 9526 | We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q) [Lemmon] |
| 2145 | In mathematics certain things have to be accepted without further explanation [Plato] |
| 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] |
| 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] |
| 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] |
| 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] |
| 9520 | The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q [Lemmon] |
| 8726 | Geometry can lead the mind upwards to truth and philosophy [Plato] |
| 9863 | We aim for elevated discussion of pure numbers, not attaching them to physical objects [Plato] |
| 9864 | In pure numbers, all ones are equal, with no internal parts [Plato] |
| 8727 | Geometry is not an activity, but the study of unchanging knowledge [Plato] |
| 9861 | The same thing is both one and an unlimited number at the same time [Plato] |
| 9862 | To become rational, philosophers must rise from becoming into being [Plato] |
| 21818 | Being depends on the Good, which is not itself being, but superior to being [Plato] |
| 2061 | The best things (gods, healthy bodies, good souls) are least liable to change [Plato] |
| 6562 | Plato's reality has unchanging Parmenidean forms, and Heraclitean flux [Plato, by Fogelin] |
| 2142 | The plurality of beautiful things must belong to a single class, because they have a single particular character [Plato] |
| 5094 | Plato's Forms are said to have no location in space [Plato, by Aristotle] |
| 12043 | Forms are not universals, as they don't cover every general term [Plato, by Annas] |
| 2159 | Craftsmen making furniture refer to the form, but no one manufactures the form of furniture [Plato] |
| 17 | A Form applies to a set of particular things with the same name [Plato] |
| 12122 | Plato mistakenly thought forms were totally abstracted away from matter [Bacon on Plato] |
| 5574 | Plato's Forms not only do not come from the senses, but they are beyond possibility of sensing [Plato, by Kant] |
| 2133 | Knowledge must be of the permanent unchanging nature of things [Plato] |
| 2162 | If theory and practice conflict, the best part of the mind accepts theory, so the other part is of lower grade [Plato] |
| 2140 | True belief without knowledge is like blind people on the right road [Plato] |
| 2096 | Is the function of the mind management, authority and planning - or is it one's whole way of life? [Plato] |
| 6009 | Psychic conflict is clear if appetite is close to the body and reason fairly separate [Plato, by Modrak] |
| 6041 | There is a third element to the mind - spirit - lying between reason and appetite [Plato] |
| 2127 | The mind has parts, because we have inner conflicts [Plato] |
| 1737 | The soul seems to have an infinity of parts [Aristotle on Plato] |
| 5945 | The 'Republic' is a great work of rhetorical theory [Lawson-Tancred on Plato] |
| 23316 | For Plato and Aristotle there is no will; there is only rational desire for what is seen as good [Plato, by Frede,M] |
| 16 | We avoid evil either through a natural aversion, or because we have acquired knowledge [Plato] |
| 16565 | Without the surface decoration, poetry shows only appearances and nothing of what is real [Plato] |
| 2160 | Representation is two steps removed from the truth [Plato] |
| 2163 | Artists should be excluded from a law-abiding community, because they destroy the rational mind [Plato] |
| 2135 | Truth is closely related to proportion [Plato] |
| 2141 | I suggest that we forget about trying to define goodness itself for the time being [Plato] |
| 1869 | The good cannot be expressed in words, but imprints itself upon the soul [Plato, by Celsus] |
| 4115 | Plato found that he could only enforce rational moral justification by creating an authoritarian society [Williams,B on Plato] |
| 4547 | Plato measured the degree of reality by the degree of value [Nietzsche on Plato] |
| 2094 | A thing's function is what it alone can do, or what it does better than other things [Plato] |
| 2095 | If something has a function then it has a state of being good [Plato] |
| 2129 | Goodness is mental health, badness is mental sickness [Plato] |
| 12 | If we were invisible, would the just man become like the unjust? [Plato] |
| 2168 | Clever criminals do well at first, but not in the long run [Plato] |
| 2137 | The main aim is to understand goodness, which gives everything its value and advantage [Plato] |
| 2139 | Every person, and every activity, aims at the good [Plato] |
| 2143 | Good has the same role in the world of knowledge as the sun has in the physical world [Plato] |
| 2147 | The sight of goodness leads to all that is fine and true and right [Plato] |
| 4007 | For Plato we abandon honour and pleasure once we see the Good [Plato, by Taylor,C] |
| 2144 | Goodness makes truth and knowledge possible [Plato] |
| 2164 | Bad is always destructive, where good preserves and benefits [Plato] |
| 2138 | Pleasure is commonly thought to be the good, though the more ingenious prefer knowledge [Plato] |
| 2070 | Even people who think pleasure is the good admit that there are bad pleasures [Plato] |
| 2157 | Nice smells are intensive, have no preceding pain, and no bad after-effect [Plato] |
| 2134 | Philosophers are concerned with totally non-physical pleasures [Plato] |
| 2156 | There are three types of pleasure, for reason, for spirit and for appetite [Plato] |
| 2158 | Pleasure-seekers desperately seek illusory satisfaction, like filling a leaky vessel [Plato] |
| 2123 | Excessive pleasure deranges people, making the other virtues impossible [Plato] |
| 2166 | We should behave well even if invisible, for the health of the mind [Plato] |
| 2097 | Isn't it better to have a reputation for goodness than to actually be good? [Plato] |
| 19946 | Morality is a compromise, showing restraint, to avoid suffering wrong without compensation [Plato] |
| 5 | Justice is merely the interests of the stronger party [Plato] |
| 7 | Surely you don't return a borrowed weapon to a mad friend? [Plato] |
| 8 | Is right just the interests of the powerful? [Plato] |
| 15 | Sin first, then sacrifice to the gods from the proceeds [Plato] |
| 5944 | For Plato, virtue is its own reward [Lawson-Tancred on Plato] |
| 2155 | True goodness requires mental unity and harmony [Plato] |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| 2126 | A good community necessarily has wisdom, courage, self-discipline and morality [Plato] |
| 23562 | If the parts of our soul do their correct work, we will be just people, and will act justly [Plato] |
| 2092 | Simonides said morality is helping one's friends and harming one's enemies [Plato] |
| 19889 | People need society because the individual has too many needs [Plato] |
| 19890 | All exchanges in a community are for mutual benefit [Plato] |
| 10 | After a taste of mutual harm, men make a legal contract to avoid it [Plato] |
| 23561 | People doing their jobs properly is the fourth cardinal virtue for a city [Plato] |
| 2149 | Reluctant rulers make a better and more unified administration [Plato] |
| 2132 | Only rule by philosophers of integrity can keep a community healthy [Plato] |
| 2131 | Is there anything better for a community than to produce excellent people? [Plato] |
| 2152 | Dialectic is the highest and most important part of the curriculum [Plato] |
| 2148 | To gain knowledge, turn away from the world of change, and focus on true goodness [Plato] |
| 2153 | Compulsory intellectual work never remains in the mind [Plato] |
| 2630 | If Plato's God is immaterial, he will lack consciousness, wisdom, pleasure and movement, which are essential to him [Cicero on Plato] |
| 14 | If the gods are non-existent or indifferent, why bother to deceive them? [Plato] |
| 2165 | Something is unlikely to be immortal if it is imperfectly made from diverse parts [Plato] |
| 13 | Is the supreme reward for virtue to be drunk for eternity? [Plato] |
| 2120 | God is responsible for the good things, but we must look elsewhere for the cause of the bad things [Plato] |