103 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] |
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] |
16383 | Puzzled Pierre has two mental files about the same object [Recanati on Kripke] |
19906 | All countries are in a mutual state of nature [Locke] |
19882 | We are not created for solitude, but are driven into society by our needs [Locke] |
19864 | In nature men can dispose of possessions and their persons in any way that is possible [Locke] |
19865 | There is no subjection in nature, and all creatures of the same species are equal [Locke] |
19866 | The rational law of nature says we are all equal and independent, and should show mutual respect [Locke] |
19872 | The animals and fruits of the earth belong to mankind [Locke] |
19907 | There is a natural right to inheritance within a family [Locke] |
19863 | Politics is the right to make enforceable laws to protect property and the state, for the common good [Locke] |
5654 | The Second Treatise explores the consequences of the contractual view of the state [Locke, by Scruton] |
19888 | A society only begins if there is consent of all the individuals to join it [Locke] |
6702 | If anyone enjoys the benefits of government (even using a road) they give tacit assent to its laws [Locke] |
19909 | A politic society is created from a state of nature by a unanimous agreement [Locke] |
19910 | A single will creates the legislature, which is duty-bound to preserve that will [Locke] |
19893 | Anyone who enjoys the benefits of a state has given tacit consent to be part of it [Locke] |
19894 | You can only become an actual member of a commonwealth by an express promise [Locke] |
19892 | Children are not born into citizenship of a state [Locke] |
19885 | Absolute monarchy is inconsistent with civil society [Locke] |
19886 | The idea that absolute power improves mankind is confuted by history [Locke] |
19903 | Despotism is arbitrary power to kill, based neither on natural equality, nor any social contract [Locke] |
19905 | People stripped of their property are legitimately subject to despotism [Locke] |
19904 | Legitimate prisoners of war are subject to despotism, because that continues the state of war [Locke] |
19895 | Even the legislature must be preceded by a law which gives it power to make laws [Locke] |
19900 | The executive must not be the legislature, or they may exempt themselves from laws [Locke] |
19902 | Any obstruction to the operation of the legislature can be removed forcibly by the people [Locke] |
19908 | Rebelling against an illegitimate power is no sin [Locke] |
19911 | If legislators confiscate property, or enslave people, they are no longer owed obedience [Locke] |
19901 | The people have supreme power, to depose a legislature which has breached their trust [Locke] |
19887 | Unanimous consent makes a united community, which is then ruled by the majority [Locke] |
19913 | A master forfeits ownership of slaves he abandons [Locke] |
19883 | Slaves captured in a just war have no right to property, so are not part of civil society [Locke] |
19870 | If you try to enslave me, you have declared war on me [Locke] |
19871 | Freedom is not absence of laws, but living under laws arrived at by consent [Locke] |
19880 | All value depends on the labour involved [Locke] |
19873 | We all own our bodies, and the work we do is our own [Locke] |
19884 | There is only a civil society if the members give up all of their natural executive rights [Locke] |
19879 | A man owns land if he cultivates it, to the limits of what he needs [Locke] |
6580 | Locke (and Marx) held that ownership of objects is a natural relation, based on the labour put into it [Locke, by Fogelin] |
20520 | Locke says 'mixing of labour' entitles you to land, as well as nuts and berries [Wolff,J on Locke] |
19875 | A man's labour gives ownership rights - as long as there are fair shares for all [Locke] |
19874 | If a man mixes his labour with something in Nature, he thereby comes to own it [Locke] |
19877 | Fountain water is everyone's, but a drawn pitcher of water has an owner [Locke] |
19876 | Gathering natural fruits gives ownership; the consent of other people is irrelevant [Locke] |
19878 | Mixing labour with a thing bestows ownership - as long as the thing is not wasted [Locke] |
19898 | Soldiers can be commanded to die, but not to hand over their money [Locke] |
19881 | The aim of law is not restraint, but to make freedom possible [Locke] |
19868 | It is only by a law of Nature that we can justify punishing foreigners [Locke] |
19867 | Reparation and restraint are the only justifications for punishment [Locke] |
19912 | Self-defence is natural, but not the punishment of superiors by inferiors [Locke] |
19869 | Punishment should make crime a bad bargain, leading to repentance and deterrence [Locke] |
19899 | The consent of the people is essential for any tax [Locke] |