112 ideas
| 19073 | True philosophy aims at absolute unity, while our understanding sees only separation [Hegel] |
| 15624 | Free thinking has no presuppositions [Hegel] |
| 15631 | The ideal of reason is the unification of abstract identity (or 'concept') and being [Hegel] |
| 15612 | Older metaphysics naively assumed that thought grasped things in themselves [Hegel] |
| 21768 | Logic is metaphysics, the science of things grasped in thoughts [Hegel] |
| 21984 | We must break up the rigidity that our understanding has imposed [Hegel] |
| 22081 | Let thought follow its own course, and don't interfere [Hegel] |
| 15626 | Categories create objective experience, but are too conditioned by things to actually grasp them [Hegel] |
| 6675 | The heart has its reasons of which reason knows nothing [Pascal] |
| 15616 | If truth is just non-contradiction, we must take care that our basic concepts aren't contradictory [Hegel] |
| 15638 | Dialectic is the moving soul of scientific progression, the principle which binds science together [Hegel] |
| 21767 | Dialectic is seen in popular proverbs like 'pride comes before a fall' [Hegel] |
| 15639 | Socratic dialectic is subjective, but Plato made it freely scientific and objective [Hegel] |
| 15615 | Older metaphysics became dogmatic, by assuming opposed assertions must be true and false [Hegel] |
| 19070 | Superficial truth is knowing how something is, which is consciousness of bare correctness [Hegel] |
| 5644 | In Hegel's logic it is concepts (rather than judgements or propositions) which are true or false [Hegel, by Scruton] |
| 19072 | In the deeper sense of truth, to be untrue resembles being bad; badness is untrue to a thing's nature [Hegel] |
| 19071 | The deeper sense of truth is a thing matching the idea of what it ought to be [Hegel] |
| 9535 | 'Contradictory' propositions always differ in truth-value [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] |
| 9514 | If A and B are 'interderivable' from one another we may write A -||- B [Lemmon] |
| 9511 | We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q [Lemmon] |
| 9512 | We write the 'negation' of P (not-P) as ¬ [Lemmon] |
| 9510 | That proposition that either P or Q is their 'disjunction', written P∨Q [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] |
| 9516 | A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔ [Lemmon] |
| 9528 | A wff is a 'tautology' if all assignments to variables result in the value T [Lemmon] |
| 9530 | A wff is 'contingent' if produces at least one T and at least one F [Lemmon] |
| 9518 | A 'theorem' is the conclusion of a provable sequent with zero assumptions [Lemmon] |
| 9533 | A 'implies' B if B is true whenever A is true (so that A→B is tautologous) [Lemmon] |
| 9532 | 'Subcontrary' propositions are never both false, so that A∨B is a tautology [Lemmon] |
| 9531 | 'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology [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] |
| 9534 | Two propositions are 'equivalent' if they mirror one another's truth-value [Lemmon] |
| 9517 | The 'scope' of a connective is the connective, the linked formulae, and the brackets [Lemmon] |
| 9393 | A: we may assume any proposition at any stage [Lemmon] |
| 9400 | ∨I: Given either A or B separately, we may derive A∨B [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] |
| 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] |
| 9396 | DN: Given A, we may derive ¬¬A [Lemmon] |
| 9398 | ∧I: Given A and B, we may derive A∧B [Lemmon] |
| 9394 | MPP: Given A and A→B, we may derive B [Lemmon] |
| 9524 | We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q [Lemmon] |
| 9525 | We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q) [Lemmon] |
| 9521 | 'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q [Lemmon] |
| 9522 | 'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q [Lemmon] |
| 9526 | We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q) [Lemmon] |
| 9527 | The Distributive Laws can rearrange a pair of conjunctions or disjunctions [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] |
| 21595 | Excluded middle is the maxim of definite understanding, but just produces contradictions [Hegel] |
| 15628 | The idea that contradiction is essential to rational understanding is a key modern idea [Hegel] |
| 15629 | Tenderness for the world solves the antinomies; contradiction is in our reason, not in the essence of the world [Hegel] |
| 15630 | Antinomies are not just in four objects, but in all objects, all representations, all objects and all ideas [Hegel] |
| 22078 | Even simple propositions about sensations are filled with categories [Hegel] |
| 15634 | Thought about particulars is done entirely through categories [Hegel] |
| 21981 | The one substance is formless without the mediation of dialectical concepts [Hegel] |
| 15637 | Essence is the essential self-positing unity of immediacy and mediation [Hegel] |
| 15613 | Real cognition grasps a thing from within itself, and is not satisfied with mere predicates [Hegel] |
| 15636 | The Cogito is at the very centre of the entire concern of modern philosophy [Hegel] |
| 22300 | Existence is just a set of relationships [Hegel] |
| 15609 | The sensible is distinguished from thought by being about singular things [Hegel] |
| 15625 | Sense perception is secondary and dependent, while thought is independent and primitive [Hegel] |
| 22011 | The first principles of truth are not rational, but are known by the heart [Pascal] |
| 15619 | Empiricism made particular knowledge possible, and blocked wild claims [Hegel] |
| 15620 | Empiricism contains the important idea that we should see knowledge for ourselves, and be part of it [Hegel] |
| 15622 | Empiricism unknowingly contains and uses a metaphysic, which underlies its categories [Hegel] |
| 15621 | Empiricism of the finite denies the supersensible, and can only think with formal abstraction [Hegel] |
| 15632 | The Humean view stops us thinking about perception, and finding universals and necessities in it [Hegel] |
| 15623 | Humean scepticism, unlike ancient Greek scepticism, accepts the truth of experience as basic [Hegel] |
| 15617 | In abstraction, beyond finitude, freedom and necessity must exist together [Hegel] |
| 15608 | The act of thinking is the bringing forth of universals [Hegel] |
| 21986 | Hegel's system has a vast number of basic concepts [Hegel, by Moore,AW] |
| 15607 | We don't think with concepts - we think the concepts [Hegel] |
| 15610 | Active thought about objects produces the universal, which is what is true and essential of it [Hegel] |
| 6681 | We only want to know things so that we can talk about them [Pascal] |
| 6676 | Painting makes us admire things of which we do not admire the originals [Pascal] |
| 6680 | It is a funny sort of justice whose limits are marked by a river [Pascal] |
| 6677 | Imagination creates beauty, justice and happiness, which is the supreme good [Pascal] |
| 6678 | We live for the past or future, and so are never happy in the present [Pascal] |
| 20732 | If man considers himself as lost and imprisoned in the universe, he will be terrified [Pascal] |
| 6682 | Majority opinion is visible and authoritative, although not very clever [Pascal] |
| 6679 | It is not good to be too free [Pascal] |
| 15614 | Old metaphysics tried to grasp eternal truths through causal events, which is impossible [Hegel] |
| 15618 | If God is the abstract of Supremely Real Essence, then God is a mere Beyond, and unknowable [Hegel] |
| 15635 | The older conception of God was emptied of human features, to make it worthy of the Infinite [Hegel] |
| 21980 | God is the absolute thing, and also the absolute person [Hegel] |
| 15633 | We establish unification of the Ideal by the ontological proof, deriving being from abstraction of thinking [Hegel] |
| 7457 | Pascal is right, but relies on the unsupported claim of a half as the chance of God's existence [Hacking on Pascal] |
| 7455 | Pascal knows you can't force belief, but you can make it much more probable [Pascal, by Hacking] |
| 7456 | The libertine would lose a life of enjoyable sin if he chose the cloisters [Hacking on Pascal] |
| 6684 | If you win the wager on God's existence you win everything, if you lose you lose nothing [Pascal] |