95 ideas
| 5728 | The concept of truth was originated by the senses [Lucretius] |
| 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] |
| 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] |
| 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] |
| 9520 | The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q [Lemmon] |
| 5702 | The senses are much the best way to distinguish true from false [Lucretius] |
| 5729 | If the senses are deceptive, reason, which rests on them, is even worse [Lucretius] |
| 5697 | The only possible standard for settling doubts is the foundation of the senses [Lucretius] |
| 7413 | Without confidence in our beliefs, how should we actually live? [Tuck] |
| 5727 | Most supposed delusions of the senses are really misinterpretations by the mind [Lucretius] |
| 5714 | Even simple facts are hard to believe at first hearing [Lucretius] |
| 5718 | The mind is in the middle of the breast, because there we experience fear and joy [Lucretius] |
| 5717 | The mind is a part of a man, just like a hand or an eye [Lucretius] |
| 21387 | The separate elements and capacities of a mind cannot be distinguished [Lucretius] |
| 5709 | The actions of the mind are not determinate and passive, because atoms can swerve [Lucretius] |
| 5695 | Only bodies can touch one another [Lucretius] |
| 5711 | The earth is and always has been an insentient being [Lucretius] |
| 5712 | Particles may have sensation, but eggs turning into chicks suggests otherwise [Lucretius] |
| 5719 | The mind moves limbs, wakes the body up, changes facial expressions, which involve touch [Lucretius] |
| 5724 | Lions, foxes and deer have distinct characters because their minds share in their bodies [Lucretius] |
| 5713 | You needn't be made of laughing particles to laugh, so why not sensation from senseless seeds? [Lucretius] |
| 6611 | One man's meat is another man's poison [Lucretius] |
| 5730 | Our bodies weren't created to be used; on the contrary, their creation makes a use possible [Lucretius] |
| 5726 | The dead are no different from those who were never born [Lucretius] |
| 5705 | Nature only wants two things: freedom from pain, and pleasure [Lucretius] |
| 5716 | Nature runs the universe by herself without the aid of gods [Lucretius] |
| 5704 | There can be no centre in infinity [Lucretius] |
| 5703 | The universe must be limitless, since there could be nothing outside to limit it [Lucretius] |
| 5693 | Everything is created and fed by nature from atoms, and they return to atoms in death [Lucretius] |
| 5701 | If an object is infinitely subdivisible, it will be the same as the whole universe [Lucretius] |
| 5708 | In downward motion, atoms occasionally swerve slightly for no reason [Lucretius] |
| 17004 | Nothing can break the binding laws of eternity [Lucretius] |
| 5706 | Atoms move themselves [Lucretius] |
| 5696 | If there were no space there could be no movement, or even creation [Lucretius] |
| 5700 | It is quicker to break things up than to assemble them [Lucretius] |
| 5698 | We can only sense time by means of movement, or its absence [Lucretius] |
| 5715 | This earth is very unlikely to be the only one created [Lucretius] |
| 5694 | Nothing can be created by divine power out of nothing [Lucretius] |
| 5699 | If matter wasn't everlasting, everything would have disappeared by now [Lucretius] |
| 5707 | The universe can't have been created by gods, because it is too imperfect [Lucretius] |
| 5710 | Gods are tranquil and aloof, and have no need of or interest in us [Lucretius] |
| 5731 | Why does Jupiter never hurl lightning from a blue sky? [Lucretius] |
| 5720 | Spirit is mortal [Lucretius] |
| 5722 | For a separated spirit to remain sentient it would need sense organs attached to it [Lucretius] |
| 5725 | An immortal mind couldn't work harmoniously with a mortal body [Lucretius] |
| 5721 | The mind is very small smooth particles, which evaporate at death [Lucretius] |
| 5723 | If spirit is immortal and enters us at birth, why don't we remember a previous existence? [Lucretius] |