107 ideas
20910 | Everything happens necessarily, and for a reason [Democritus] |
9535 | 'Contradictory' propositions always differ in truth-value [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] |
9514 | If A and B are 'interderivable' from one another we may write A -||- B [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] |
9516 | A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔ [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] |
9517 | The 'scope' of a connective is the connective, the linked formulae, and the brackets [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] |
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] |
9520 | The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q [Lemmon] |
16146 | Two can't be a self-contained unit, because it would need to be one to do that [Democritus, by Aristotle] |
20901 | True Being only occurs when it is completely full, with atoms and no void [Democritus, by Aristotle] |
20902 | Being does not exist more than non-being [Democritus, by Aristotle] |
20904 | The non-existent exists as much as the existent, because it has causal powers [Democritus] |
20903 | The only distinctions are Configuration (shape), Disposition (order) and Turning (position) [Democritus, by Aristotle] |
20893 | Nothing comes from non-existence, or passes into it [Democritus, by Diog. Laertius] |
20896 | It is not possible to know what sort each thing is [Democritus] |
3357 | Democritus denies reality to large objects, because atomic entities can't combine to produce new ones [Benardete,JA on Democritus] |
598 | Democritus said that substances could never be mixed, so atoms are the substances [Democritus, by Aristotle] |
490 | Everything happens by reason and necessity [Leucippus] |
1532 | Sensible qualities can't be real if they appear different to different creatures [Democritus, by Theophrastus] |
20894 | Man is separated from reality [Democritus] |
517 | All evidence comes from senses, so they are indispensable to the mind [Democritus] |
20897 | Obscure knowledge belongs to the five senses, and genuine knowledge is the other type [Democritus] |
577 | Democritus says there is either no truth, or it is concealed from us [Democritus, by Aristotle] |
20895 | We actually know nothing, and opinions are mere flux [Democritus] |
1528 | We in fact know nothing, but we each restructure our reality with beliefs [Democritus] |
492 | It is obviously impossible to understand the reality of each thing [Democritus] |
515 | We know nothing in reality; for truth lies in an abyss [Democritus] |
20892 | Democritus was devoted to discovering causal explanations [Democritus, by Eusebius] |
5882 | Democritus says soul consists of smooth round bodies brought together in accidental collision [Democritus, by Cicero] |
6034 | Atomists say soul has a rational part in the chest, and a diffused non-rational part [Democritus, by Aetius] |
20912 | The soul is the same as the mind [Democritus, by Aristotle] |
20916 | Animals have a share of reason [Democritus, by Porphyry] |
20914 | The directive centre is located in the whole head [Democritus, by Ps-Plutarch] |
6033 | Democritus said everything happens of necessity, by natural motion of atoms [Democritus, by Cicero] |
5088 | Some say there is a determinate cause for every apparently spontaneous event [Democritus, by Aristotle] |
21670 | Democritus said atoms only move by their natural motions, which are therefore necessary [Democritus, by Cicero] |
24041 | Democritus says spherical atoms are fire, and constitute the soul [psuche] [Democritus, by Aristotle] |
20913 | Democritus says the soul is the body, and thinking is thus the mixture of the body [Democritus, by Theophrastus] |
1540 | Pleasure and pain guide our choices of good and bad [Democritus] |
495 | Wisdom creates a healthy passion-free soul [Democritus] |
1537 | Happiness is identifying and separating the pleasures [Democritus, by Stobaeus] |
20917 | Contentment comes from moderation and proportion in life [Democritus, by Stobaeus] |
13551 | Democritus says wealth is a burden to the virtuous mind [Democritus, by Seneca] |
20899 | Atoms cling together, until a stronger necessity disperses them [Democritus, by Aristotle] |
20898 | Atoms are irregular, hooked, concave, convex, and many other shapes [Democritus, by Aristotle] |
17542 | 'Full' and 'Void' secularised Parmenides's Being and Not-being [Democritus, by Heisenberg] |
1525 | Atomists say there are only three differences - in shape, arrangement and position [Democritus, by Aristotle] |
493 | Experiences are merely convention; only atoms and the void are real [Democritus] |
5947 | If only atoms are real and the rest is convention, we wouldn't bother to avoid pain [Democritus, by Diogenes of Oen.] |
13219 | When atoms touch, why don't they coalesce, like water drops? [Aristotle on Democritus] |
1533 | Because appearance is infinitely varied, atomists assume infinitely many shapes of atom [Democritus, by Aristotle] |
20908 | There could be an atom the size of the world [Democritus, by Ps-Plutarch] |
1527 | There must be atoms, to avoid the absurdity of infinite division down to nothing [Democritus, by Aristotle] |
20909 | The basic atoms are without qualities - which only arise from encounters between atoms [Democritus, by Galen] |
1536 | If a cone is horizontally sliced the surfaces can't be equal, so it goes up in steps [Democritus] |
23314 | Greeks explained regularity by intellectual design, not by laws [Democritus, by Frede,M] |
20905 | Growth and movement would not exist if there were no void to receive them [Democritus] |
24059 | Democritus is wrong: in a void we wouldn't see a distant ant in exact detail [Aristotle on Democritus] |
5101 | Movement is impossible in a void, because nothing can decide the direction of movement [Aristotle on Democritus] |
20911 | There are unlimited worlds of varying sizes, some without life or water [Democritus, by Hippolytus] |
1535 | Democritus said people imagined gods as the source of what awed or frightened them [Democritus, by Sext.Empiricus] |
20915 | The soul is destroyed with the body [Democritus, by Ps-Plutarch] |