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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 'negation' of P (not-P) as
1.2 p.7 We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q
1.2 p.11 The sign |- may be read as 'therefore'
1.3 p.19 That proposition that both P and Q is their 'conjunction', written P∧Q
1.3 p.19 That proposition that either P or Q is their 'disjunction', 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 MPP: Given A and A→B, we may derive B
1.5 p.39 A: we may assume any proposition at any stage
1.5 p.40 ∧I: Given A and B, we may derive A∧B
1.5 p.40 ∧E: Given A∧B, we may derive either A or B separately
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 either A or B separately, 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 DN: Given A, we may derive A
1.5 p.40 MTT: Given B and A→B, we derive A
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 tollendo ponens' (MTP) says P, P ∨ Q |- Q
2.2 p.61 'Modus ponendo tollens' (MPT) says P, (P ∧ Q) |- Q
2.2 p.62 We can change conditionals into disjunctions with P→Q -||- P ∨ Q
2.2 p.62 We can change conditionals into negated conjunctions with P→Q -||- (P ∧ Q)
2.2 p.62 We can change conjunctions into negated conditionals with P→Q -||- (P → Q)
2.2 p.62 The Distributive Laws can rearrange a pair of conjunctions or disjunctions
2.2 p.62 De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions
2.3 p.68 A wff is 'contingent' if produces at least one T and at least one F
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.69 'Subcontrary' propositions are never both false, so that A∨B is a tautology
2.3 p.69 'Contrary' propositions are never both true, so that (A∧B) is a tautology
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 Two propositions are 'equivalent' if they mirror one another's truth-value
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 If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers
3.2 p.105 Universal Elimination (UE) lets us infer that an object has F, from all things having F
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