87 ideas
2056 | Philosophers are always switching direction to something more interesting [Plato] |
2083 | Either a syllable is its letters (making parts as knowable as whole) or it isn't (meaning it has no parts) [Plato] |
2086 | Understanding mainly involves knowing the elements, not their combinations [Plato] |
2082 | A rational account is essentially a weaving together of things with names [Plato] |
2052 | Eristic discussion is aggressive, but dialectic aims to help one's companions in discussion [Plato] |
15854 | A primary element has only a name, and no logos, but complexes have an account, by weaving the names [Plato] |
9535 | 'Contradictory' propositions always differ in truth-value [Lemmon] |
9514 | If A and B are 'interderivable' from one another we may write A -||- B [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] |
9512 | We write the 'negation' of P (not-P) as ¬ [Lemmon] |
9511 | We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q [Lemmon] |
9508 | The sign |- may be read as 'therefore' [Lemmon] |
9509 | That proposition that both P and Q is their 'conjunction', written P∧Q [Lemmon] |
9510 | That proposition that either P or Q is their 'disjunction', written P∨Q [Lemmon] |
9533 | A 'implies' B if B is true whenever A is true (so that A→B is tautologous) [Lemmon] |
9534 | Two propositions are 'equivalent' if they mirror one another's truth-value [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] |
9518 | A 'theorem' is the conclusion of a provable sequent with zero assumptions [Lemmon] |
9529 | A wff is 'inconsistent' if all assignments to variables result in the value F [Lemmon] |
9532 | 'Subcontrary' propositions are never both false, so that A∨B is a tautology [Lemmon] |
9528 | A wff is a 'tautology' if all assignments to variables result in the value T [Lemmon] |
9519 | A 'substitution-instance' is a wff formed by consistent replacing variables with wffs [Lemmon] |
9530 | A wff is 'contingent' if produces at least one T and at least one F [Lemmon] |
9531 | 'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology [Lemmon] |
9393 | A: we may assume any proposition at any stage [Lemmon] |
9396 | DN: Given A, we may derive ¬¬A [Lemmon] |
9399 | ∧E: Given A∧B, we may derive either A or B separately [Lemmon] |
9401 | ∨E: Derive C from A∨B, if C can be derived both from A and from B [Lemmon] |
9394 | MPP: Given A and A→B, we may derive B [Lemmon] |
9400 | ∨I: Given either A or B separately, we may derive A∨B [Lemmon] |
9395 | MTT: Given ¬B and A→B, we derive ¬A [Lemmon] |
9402 | RAA: If assuming A will prove B∧¬B, then derive ¬A [Lemmon] |
9397 | CP: Given a proof of B from A as assumption, we may derive A→B [Lemmon] |
9398 | ∧I: Given A and B, we may derive A∧B [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] |
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] |
9522 | 'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q [Lemmon] |
9521 | 'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q [Lemmon] |
9523 | De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions [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] |
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] |
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] |
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] |
10216 | We master arithmetic by knowing all the numbers in our soul [Plato] |
2060 | There seem to be two sorts of change: alteration and motion [Plato] |
2084 | If a word has no parts and has a single identity, it turns out to be the same kind of thing as a letter [Plato] |
15843 | The whole can't be the parts, because it would be all of the parts, which is the whole [Plato] |
15844 | A sum is that from which nothing is lacking, which is a whole [Plato] |
2080 | Things are only knowable if a rational account (logos) is possible [Plato] |
16126 | Expertise is knowledge of the whole by means of the parts [Plato] |
2050 | It is impossible to believe something which is held to be false [Plato] |
2076 | How can a belief exist if its object doesn't exist? [Plato] |
2045 | Perception is infallible, suggesting that it is knowledge [Plato] |
2067 | Our senses could have been separate, but they converge on one mind [Plato] |
2069 | Thought must grasp being itself before truth becomes possible [Plato] |
2068 | With what physical faculty do we perceive pairs of opposed abstract qualities? [Plato] |
2078 | You might mistake eleven for twelve in your senses, but not in your mind [Plato] |
2089 | An inadequate rational account would still not justify knowledge [Plato] |
2085 | Parts and wholes are either equally knowable or equally unknowable [Plato] |
2091 | Without distinguishing marks, how do I know what my beliefs are about? [Plato] |
2087 | A rational account might be seeing an image of one's belief, like a reflection in a mirror [Plato] |
2090 | A rational account involves giving an image, or analysis, or giving a differentiating mark [Plato] |
2081 | Maybe primary elements can be named, but not receive a rational account [Plato] |
2088 | A rational account of a wagon would mean knowledge of its hundred parts [Plato] |
2047 | What evidence can be brought to show whether we are dreaming or not? [Plato] |
2053 | If you claim that all beliefs are true, that includes beliefs opposed to your own [Plato] |
2054 | Clearly some people are superior to others when it comes to medicine [Plato] |
2059 | How can a relativist form opinions about what will happen in the future? [Plato] |
1389 | If memory is the sole criterion of identity, we ought to use it for other people too [Shoemaker] |
1390 | Bodily identity is one criterion and memory another, for personal identity [Shoemaker, by PG] |
2058 | God must be the epitome of goodness, and we can only approach a divine state by being as good as possible [Plato] |
2057 | There must always be some force of evil ranged against good [Plato] |