1965 | Beginning Logic |
p.104 | 13901 | Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules |
1.2 | p.7 | 9511 | We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q |
1.2 | p.7 | 9512 | We write the 'negation' of P (not-P) as ¬ |
1.2 | p.11 | 9508 | The sign |- may be read as 'therefore' |
1.3 | p.19 | 9510 | That proposition that either P or Q is their 'disjunction', written P∨Q |
1.3 | p.19 | 9509 | That proposition that both P and Q is their 'conjunction', written P∧Q |
1.4 | p.29 | 9513 | We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P) |
1.5 | p.34 | 9514 | If A and B are 'interderivable' from one another we may write A -||- B |
1.5 | p.39 | 9394 | MPP: Given A and A→B, we may derive B |
1.5 | p.39 | 9393 | A: we may assume any proposition at any stage |
1.5 | p.40 | 9399 | ∧E: Given A∧B, we may derive either A or B separately |
1.5 | p.40 | 9402 | RAA: If assuming A will prove B∧¬B, then derive ¬A |
1.5 | p.40 | 9401 | ∨E: Derive C from A∨B, if C can be derived both from A and from B |
1.5 | p.40 | 9400 | ∨I: Given either A or B separately, we may derive A∨B |
1.5 | p.40 | 9398 | ∧I: Given A and B, we may derive A∧B |
1.5 | p.40 | 9395 | MTT: Given ¬B and A→B, we derive ¬A |
1.5 | p.40 | 9396 | DN: Given A, we may derive ¬¬A |
1.5 | p.40 | 9397 | CP: Given a proof of B from A as assumption, we may derive A→B |
2.1 | p.44 | 9516 | A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔ |
2.1 | p.47 | 9517 | The 'scope' of a connective is the connective, the linked formulae, and the brackets |
2.2 | p.50 | 9518 | A 'theorem' is the conclusion of a provable sequent with zero assumptions |
2.2 | p.53 | 9519 | A 'substitution-instance' is a wff formed by consistent replacing variables with wffs |
2.2 | p.60 | 9520 | The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q |
2.2 | p.61 | 9521 | 'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q |
2.2 | p.61 | 9522 | 'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q |
2.2 | p.62 | 9527 | The Distributive Laws can rearrange a pair of conjunctions or disjunctions |
2.2 | p.62 | 9523 | De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions |
2.2 | p.62 | 9524 | We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q |
2.2 | p.62 | 9526 | We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q) |
2.2 | p.62 | 9525 | We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q) |
2.3 | p.68 | 9530 | A wff is 'contingent' if produces at least one T and at least one F |
2.3 | p.68 | 9528 | A wff is a 'tautology' if all assignments to variables result in the value T |
2.3 | p.68 | 9529 | A wff is 'inconsistent' if all assignments to variables result in the value F |
2.3 | p.69 | 9532 | 'Subcontrary' propositions are never both false, so that A∨B is a tautology |
2.3 | p.69 | 9531 | 'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology |
2.3 | p.70 | 9533 | A 'implies' B if B is true whenever A is true (so that A→B is tautologous) |
2.3 | p.70 | 9534 | Two propositions are 'equivalent' if they mirror one another's truth-value |
2.3 | p.70 | 9535 | 'Contradictory' propositions always differ in truth-value |
2.4 | p.80 | 9536 | If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology |
2.4 | p.81 | 9537 | Truth-tables are good for showing invalidity |
2.5 | p.90 | 9539 | Propositional logic is complete, since all of its tautologous sequents are derivable |
2.5 | p.91 | 9538 | A truth-table test is entirely mechanical, but this won't work for more complex logic |
3.1 | p.96 | 13909 | Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....' |
3.1 | p.97 | 13910 | Our notation uses 'predicate-letters' (for 'properties'), 'variables', 'proper names', 'connectives' and 'quantifiers' |
3.1 | p.97 | 13900 | 'Some Frenchmen are generous' is rendered by (∃x)(Fx→Gx), and not with the conditional → |
3.1 | p.98 | 13902 | 'Gm' says m has property G, and 'Pmn' says m has relation P to n |
3.2 | p.104 | 13903 | Universal elimination if you start with the universal, introduction if you want to end with it |
3.2 | p.105 | 13905 | If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers |
3.2 | p.105 | 13904 | Universal Elimination (UE) lets us infer that an object has F, from all things having F |
3.2 | p.106 | 13906 | With finite named objects, we can generalise with &-Intro, but otherwise we need ∀-Intro |
3.2 | p.106 | 13907 | If you pick an arbitrary triangle, things proved of it are true of all triangles |
3.3 | p.111 | 13908 | UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one |
4.1 | p.139 | 13911 | The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E |