84 ideas
16241 | The metaphysics of nature should focus on physics [Maudlin] |
16257 | Kant survives in seeing metaphysics as analysing our conceptual system, which is a priori [Maudlin] |
16276 | Wide metaphysical possibility may reduce metaphysics to analysis of fantasies [Maudlin] |
16244 | If the universe is profligate, the Razor leads us astray [Maudlin] |
16255 | The Razor rightly prefers one cause of multiple events to coincidences of causes [Maudlin] |
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] |
9509 | That proposition that both P and Q is their 'conjunction', 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] |
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] |
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] |
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] |
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] |
3299 | In logic identity involves reflexivity (x=x), symmetry (if x=y, then y=x) and transitivity (if x=y and y=z, then x=z) [Baillie] |
16243 | The Humean view is wrong; laws and direction of time are primitive, and atoms are decided by physics [Maudlin] |
16271 | Lewis says it supervenes on the Mosaic, but actually thinks the Mosaic is all there is [Maudlin] |
16273 | If the Humean Mosaic is ontological bedrock, there can be no explanation of its structure [Maudlin] |
16275 | The 'spinning disc' is just impossible, because there cannot be 'homogeneous matter' [Maudlin] |
16258 | To get an ontology from ontological commitment, just add that some theory is actually true [Maudlin] |
16259 | Naïve translation from natural to formal language can hide or multiply the ontology [Maudlin] |
16253 | A property is fundamental if two objects can differ in only that respect [Maudlin] |
16263 | Fundamental physics seems to suggest there are no such things as properties [Maudlin] |
16260 | Existence of universals may just be decided by acceptance, or not, of second-order logic [Maudlin] |
16277 | Logically impossible is metaphysically impossible, but logically possible is not metaphysically possible [Maudlin] |
16249 | A counterfactual antecedent commands the redescription of a selected moment [Maudlin] |
16254 | Induction leaps into the unknown, but usually lands safely [Maudlin] |
16245 | Laws should help explain the things they govern, or that manifest them [Maudlin] |
16248 | Evaluating counterfactuals involves context and interests [Maudlin] |
16250 | We don't pick a similar world from many - we construct one possibility from the description [Maudlin] |
16268 | The counterfactual is ruined if some other cause steps in when the antecedent fails [Maudlin] |
16267 | If we know the cause of an event, we seem to assent to the counterfactual [Maudlin] |
16269 | If the effect hadn't occurred the cause wouldn't have happened, so counterfactuals are two-way [Maudlin] |
16247 | Laws are primitive, so two indiscernible worlds could have the same laws [Maudlin] |
16272 | Fundamental laws say how nature will, or might, evolve from some initial state [Maudlin] |
16242 | Laws of nature are ontological bedrock, and beyond analysis [Maudlin] |
16251 | 'Humans with prime house numbers are mortal' is not a law, because not a natural kind [Maudlin] |
16270 | If laws are just regularities, then there have to be laws [Maudlin] |
16264 | I believe the passing of time is a fundamental fact about the world [Maudlin] |
16265 | If time passes, presumably it passes at one second per second [Maudlin] |
16266 | There is one ordered B series, but an infinitude of A series, depending on when the present is [Maudlin] |