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Ideas of E.J. Lemmon, by Text

[British, fl. 1960, Claremont College]

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