96 ideas
4171 | Philosophy considers only the universal, in nature as everywhere else [Schopenhauer] |
4186 | Everyone is conscious of all philosophical truths, but philosophers bring them to conceptual awareness [Schopenhauer] |
12171 | Absurdity is incongruity between correct and false points of view [Schopenhauer] |
21366 | Metaphysics must understand the world thoroughly, as a principal source of knowledge [Schopenhauer] |
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] |
21963 | It is possible that an omnipotent God might make one and two fail to equal three [Descartes] |
4168 | Matter and intellect are inseparable correlatives which only exist relatively, and for each other [Schopenhauer] |
21926 | Schopenhauer, unlike other idealists, says reality is irrational [Schopenhauer, by Lewis,PB] |
4167 | The knowing subject and the crude matter of the world are both in themselves unknowable [Schopenhauer] |
4165 | Descartes found the true beginning of philosophy with the Cogito, in the consciousness of the individual [Schopenhauer] |
21923 | Schopenhauer can't use force/energy instead of 'will', because he is not a materialist [Lewis,PB on Schopenhauer] |
4162 | The world only exists in relation to something else, as an idea of the one who conceives it [Schopenhauer] |
21922 | We know reality because we know our own bodies and actions [Schopenhauer] |
21913 | Kant rightly separates appearance and thing-in-itself [Schopenhauer] |
4164 | Direct feeling of the senses are merely data; perception of the world comes with understanding causes [Schopenhauer] |
4163 | All perception is intellectual [Schopenhauer] |
4166 | A consciousness without an object is no consciousness [Schopenhauer] |
21369 | We have hidden and unadmitted desires and fears, suppressed because of vanity [Schopenhauer] |
21367 | I know both aspects of my body, as representation, and as will [Schopenhauer] |
4175 | It is as perverse to resent our individuality being replaced by others, as to resent the body renewing itself [Schopenhauer] |
4176 | We all regard ourselves a priori as free, but see from experience that character and motive compel us [Schopenhauer] |
4170 | Man's actions are not free, because they follow strictly from impact of motive on character [Schopenhauer] |
4169 | Every true act of will is also at once and without exception a movement of the body [Schopenhauer] |
7187 | Schopenhauer was caught in Christian ideals, because he didn't deify his 'will' [Nietzsche on Schopenhauer] |
21365 | Only the will is thing-in-itself, seen both in blind nature and in human action [Schopenhauer] |
4173 | If we were essentially intellect rather than will, our moral worth would depend on imagined motives [Schopenhauer] |
21370 | Schopenhauer is a chief proponent of aesthetic experience as 'disinterested' [Schopenhauer, by Janaway] |
4182 | A principal pleasure of the beautiful is that it momentarily silences the will [Schopenhauer] |
21928 | The Sublime fights for will-less knowing, when faced with a beautiful threat to humanity [Schopenhauer, by Lewis,PB] |
21927 | Schopenhauer emphasises Ideas in art, unlike most romantics [Schopenhauer, by Lewis,PB] |
8116 | The will-less contemplation of art brings a liberation from selfhood [Schopenhauer, by Gardner] |
4174 | Man is more beautiful than anything else, and the loftiest purpose of art is to reveal his nature [Schopenhauer] |
21380 | The only aim of our existence is to grasp that non-existence would be better [Schopenhauer] |
21374 | We should no more expect ethical theory to produce good people than aesthetics to produce artists [Schopenhauer] |
4181 | Every good is essentially relative, for it has its essential nature only in its relation to a desiring will [Schopenhauer] |
5649 | Will casts aside each of its temporary fulfilments, so human life has no ultimate aim [Schopenhauer, by Scruton] |
4177 | Most people would probably choose non-existence at the end of their life, rather than relive the whole thing [Schopenhauer] |
4185 | Altruistic people make less distinction than usual between themselves and others [Schopenhauer] |
4183 | Only self-love can motivate morality, but that also makes it worthless [Schopenhauer] |
4172 | Happiness is the swift movement from desire to satisfaction, and then again on to desire [Schopenhauer] |
21371 | We can never attain happiness while our will is pursuing desires [Schopenhauer] |
4184 | Virtue must spring from an intuitive recognition that other people are essentially like us [Schopenhauer] |
4179 | The essence of nature is the will to life itself [Schopenhauer] |
4178 | Christianity is a pessimistic religion, in which the world is equated with evil [Schopenhauer] |
4180 | Religion is the mythical clothing of the truth which is inaccessible to the crude human intellect [Schopenhauer] |