97 ideas
| 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] |
| 9358 | There are several logics, none of which will ever derive falsehoods from truth [Lewis,CI] |
| 9520 | The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q [Lemmon] |
| 9357 | Excluded middle is just our preference for a simplified dichotomy in experience [Lewis,CI] |
| 9364 | Names represent a uniformity in experience, or they name nothing [Lewis,CI] |
| 9362 | Necessary truths are those we will maintain no matter what [Lewis,CI] |
| 9365 | We can maintain a priori principles come what may, but we can also change them [Lewis,CI] |
| 23101 | Intuitions don't prove things; they just receptivity to interpretations [Kekes] |
| 9361 | We have to separate the mathematical from physical phenomena by abstraction [Lewis,CI] |
| 23086 | Liberals say we are only responsible for fully autonomous actions [Kekes] |
| 23100 | Collective responsibility conflicts with responsibility's requirement of authonomy [Kekes] |
| 23093 | Moral and causal responsibility are not clearly distinct [Kekes] |
| 23096 | Morality should aim to prevent all evil actions, not just autonomous ones [Kekes] |
| 23087 | Much human evil is not autonomous, so moral responsibility need not be autonomous [Kekes] |
| 23098 | Effects show the existence of moral responsibility, and mental states show the degree [Kekes] |
| 23089 | Evil people may not be autonomously aware, if they misjudge the situation [Kekes] |
| 23094 | Ought implies can means moral responsibility needs autonomy [Kekes] |
| 23095 | Why should moral responsibility depend on autonomy, rather than social role or experience? [Kekes] |
| 23090 | Liberals assume people are naturally free, equal, rational, and morally good [Kekes] |
| 23117 | Love should be partial, and discriminate in favour of its object [Kekes] |
| 23119 | Sentimental love distorts its object [Kekes] |
| 23088 | Evil is not deviation from the good, any more than good is a deviation from evil [Kekes] |
| 23097 | What matters for morality is the effects of action, not the psychological causes [Kekes] |
| 23099 | It is said that if an agent is not autonomous then their evil actions don't reflect on their character [Kekes] |
| 23118 | Awareness of others' suffering doesn't create an obligation to help [Kekes] |
| 23109 | The veil of ignorance is only needed because people have bad motivations [Kekes] |
| 23114 | The chief function of the state is to arbitrate between contending visions of the good life [Kekes] |
| 23116 | Citizenship is easier than parenthood [Kekes] |
| 23103 | Power is meant to be confined to representatives, and subsequent delegation [Kekes] |
| 23107 | Prosperity is a higher social virtue than justice [Kekes] |
| 23081 | Liberal basics are pluralism, freedom, rights, equality, and distributive justice - for autonomy [Kekes] |
| 23085 | The key liberal values are explained by the one core value, which is autonomy [Kekes] |
| 23092 | Agents have little control over the capacities needed for liberal autonomy [Kekes] |
| 23102 | Liberals are egalitarians, but in varying degrees [Kekes] |
| 23084 | Are egalitarians too coercive, or not egalitarian enough, or lax over morality? [Kekes] |
| 23079 | Liberal justice ignores desert, which is the essence of justice [Kekes] |
| 23091 | Why do liberals not see a much wider range of values as basic? [Kekes] |
| 23112 | Liberals ignore contingency, and think people are good and equal, and institutions cause evil [Kekes] |
| 23082 | Liberal distribution cares more about recipients than donors [Kekes] |
| 23106 | To rectify the undeserved equality, we should give men longer and women shorter lives [Kekes] |
| 23121 | It is just a fact that some people are morally better than others [Kekes] |
| 23105 | It is not deplorable that billionaires have more than millionaires [Kekes] |
| 23120 | The problem is basic insufficiency of resources, not their inequality [Kekes] |
| 23108 | Justice combines consistency and desert; treat likes alike, judging likeness by desert [Kekes] |
| 23083 | Liberal welfare focuses on need rather than desert [Kekes] |
| 23113 | Sexual morality doesn't require monogamy, but it needs a group of sensible regulations [Kekes] |
| 9363 | Science seeks classification which will discover laws, essences, and predictions [Lewis,CI] |