93 ideas
| 19250 | Everything interesting should be recorded, with records that can be rearranged [Peirce] |
| 19228 | Sciences concern existence, but philosophy also concerns potential existence [Peirce] |
| 19241 | An idea on its own isn't an idea, because they are continuous systems [Peirce] |
| 19227 | Philosophy is a search for real truth [Peirce] |
| 19218 | Metaphysics is pointless without exact modern logic [Peirce] |
| 19229 | Metaphysics is the science of both experience, and its general laws and types [Peirce] |
| 19219 | Metaphysical reasoning is simple enough, but the concepts are very hard [Peirce] |
| 19231 | Metaphysics is turning into logic, and logic is becoming mathematics [Peirce] |
| 19247 | The one unpardonable offence in reasoning is to block the route to further truth [Peirce] |
| 19246 | 'Holding for true' is either practical commitment, or provisional theory [Peirce] |
| 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] |
| 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] |
| 9401 | ∨E: Derive C from A∨B, if C can be derived both from A and from 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] |
| 19237 | Deduction is true when the premises facts necessarily make the conclusion fact true [Peirce] |
| 9520 | The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q [Lemmon] |
| 19256 | Our research always hopes that reality embodies the logic we are employing [Peirce] |
| 19238 | The logic of relatives relies on objects built of any relations (rather than on classes) [Peirce] |
| 19226 | We now know that mathematics only studies hypotheses, not facts [Peirce] |
| 19240 | Realism is the belief that there is something in the being of things corresponding to our reasoning [Peirce] |
| 19239 | There may be no reality; it's just our one desperate hope of knowing anything [Peirce] |
| 19252 | Objective chance is the property of a distribution [Peirce] |
| 19232 | In ordinary language a conditional statement assumes that the antecedent is true [Peirce] |
| 19223 | We act on 'full belief' in a crisis, but 'opinion' only operates for trivial actions [Peirce] |
| 19253 | We talk of 'association by resemblance' but that is wrong: the association constitutes the resemblance [Peirce] |
| 19224 | Scientists will give up any conclusion, if experience opposes it [Peirce] |
| 19243 | If each inference slightly reduced our certainty, science would soon be in trouble [Peirce] |
| 19225 | I classify science by level of abstraction; principles derive from above, and data from below [Peirce] |
| 19234 | 'Induction' doesn't capture Greek 'epagoge', which is singulars in a mass producing the general [Peirce] |
| 19235 | How does induction get started? [Peirce] |
| 19236 | Induction can never prove that laws have no exceptions [Peirce] |
| 19251 | The worst fallacy in induction is generalising one recondite property from a sample [Peirce] |
| 19222 | Men often answer inner 'whys' by treating unconscious instincts as if they were reasons [Peirce] |
| 19220 | We may think animals reason very little, but they hardly ever make mistakes! [Peirce] |
| 19255 | Generalisation is the great law of mind [Peirce] |
| 19242 | Generalization is the true end of life [Peirce] |
| 19249 | 'Know yourself' is not introspection; it is grasping how others see you [Peirce] |
| 19257 | Whatever is First must be sentient [Peirce] |
| 19248 | Reasoning involves observation, experiment, and habituation [Peirce] |
| 19221 | Everybody overrates their own reasoning, so it is clearly superficial [Peirce] |
| 19233 | Indexicals are unusual words, because they stimulate the hearer to look around [Peirce] |
| 19230 | People should follow what lies before them, and is within their power [Peirce] |
| 19245 | We are not inspired by other people's knowledge; a sense of our ignorance motivates study [Peirce] |
| 19244 | Chemists rely on a single experiment to establish a fact; repetition is pointless [Peirce] |
| 19254 | Our laws of nature may be the result of evolution [Peirce] |
| 9425 | Lewis later proposed the axioms at the intersection of the best theories (which may be few) [Mumford on Lewis] |