93 ideas
1597 | Thales was the first western thinker to believe the arché was intelligible [Roochnik on Thales] |
23000 | Vicious regresses force you to another level; non-vicious imply another level [Baron/Miller] |
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] |
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] |
23024 | A traveller takes a copy of a picture into the past, gives it the artist, who then creates the original! [Baron/Miller] |
23008 | Grounding is intended as a relation that fits dependences between things [Baron/Miller] |
23018 | How does a changing object retain identity or have incompatible properties over time? [Baron/Miller] |
3013 | Nothing is stronger than necessity, which rules everything [Thales, by Diog. Laertius] |
1494 | Thales said water is the first principle, perhaps from observing that food is moist [Thales, by Aristotle] |
23011 | Modern accounts of causation involve either processes or counterfactuals [Baron/Miller] |
23013 | The main process theory of causation says it is transference of mass, energy, momentum or charge [Baron/Miller] |
23014 | If causes are processes, what is causation by omission? (Distinguish legal from scientific causes?) [Baron/Miller] |
23015 | The counterfactual theory of causation handles the problem no matter what causes actually are [Baron/Miller] |
23016 | Counterfactual theories struggle with pre-emption by a causal back-up system [Baron/Miller] |
1713 | Thales must have thought soul causes movement, since he thought magnets have soul [Thales, by Aristotle] |
23009 | There is no second 'law' of thermodynamics; it just reflects probabilities of certain microstates [Baron/Miller] |
23002 | In relativity space and time depend on one's motion, but spacetime gives an invariant metric [Baron/Miller] |
22988 | The block universe theory says entities of all times exist, and time is the B-series [Baron/Miller] |
22991 | How can we know this is the present moment, if other times are real? [Baron/Miller] |
22992 | If we are actually in the past then we shouldn't experience time passing [Baron/Miller] |
22994 | Erzatz Presentism allows the existence of other times, with only the present 'actualised' [Baron/Miller] |
22998 | How do presentists explain relations between things existing at different times? [Baron/Miller] |
23017 | Presentism needs endurantism, because other theories imply most of the object doesn't exist [Baron/Miller] |
23023 | How can presentists move to the next future moment, if that doesn't exist? [Baron/Miller] |
22995 | Most of the sciences depend on the concept of time [Baron/Miller] |
22993 | For abstractionists past times might still exist, althought their objects don't [Baron/Miller] |
23001 | The error theory of time's passage says it is either a misdescription or a false inference [Baron/Miller] |
22999 | It is meaningless to measure the rate of time using time itself, and without a rate there is no flow [Baron/Miller] |
22986 | The C-series rejects A and B, and just sees times as order by betweenness, without direction [Baron/Miller] |
22996 | The A-series has to treat being past, present or future as properties [Baron/Miller] |
23007 | The B-series can have a direction, as long as it does not arise from temporal flow [Baron/Miller] |
23003 | Static theories cannot account for time's obvious asymmetry, so time must be dynamic [Baron/Miller] |
23004 | The direction of time is either primitive, or reducible to something else [Baron/Miller] |
23005 | The kaon does not seem to be time-reversal invariant, unlike the rest of nature [Baron/Miller] |
23006 | Maybe the past is just the direction of decreasing entropy [Baron/Miller] |
23010 | We could explain time's direction by causation: past is the direction of causes, future of effects [Baron/Miller] |
22989 | Static time theory presents change as one property at t1, and a different property at t2 [Baron/Miller] |
23020 | If a time traveller kills his youthful grandfather, he both exists and fails to exist [Baron/Miller] |
23022 | Presentism means there no existing past for a time traveller to visit [Baron/Miller] |
22987 | The past (unlike the future) is fixed, along with truths about it, by the existence of past objects [Baron/Miller] |
22990 | The moving spotlight says entities can have properties of being present, past or future [Baron/Miller] |
22997 | The present moment is a matter of existence, not of acquiring a property [Baron/Miller] |
1742 | Thales said the gods know our wrong thoughts as well as our evil actions [Thales, by Diog. Laertius] |