### Ideas of E.J. Lemmon, by Theme

#### [British, fl. 1960, Claremont College]

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###### 4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
 9535 'Contradictory' propositions always differ in truth-value
###### 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / a. Symbols of PL
 9511 We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q
 9512 We write the 'negation' of P (not-P) as ¬
 9508 The sign |- may be read as 'therefore'
 9509 That proposition that both P and Q is their 'conjunction', written P∧Q
 9510 That proposition that either P or Q is their 'disjunction', written P∨Q
 9514 If A and B are 'interderivable' from one another we may write A -||- B
 9513 We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P)
###### 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
 9529 A wff is 'inconsistent' if all assignments to variables result in the value F
 9533 A 'implies' B if B is true whenever A is true (so that A→B is tautologous)
 9532 'Subcontrary' propositions are never both false, so that A∨B is a tautology
 9531 'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology
 9534 Two propositions are 'equivalent' if they mirror one another's truth-value
 9516 A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔
 9518 A 'theorem' is the conclusion of a provable sequent with zero assumptions
 9530 A wff is 'contingent' if produces at least one T and at least one F
 9528 A wff is a 'tautology' if all assignments to variables result in the value T
 9519 A 'substitution-instance' is a wff formed by consistent replacing variables with wffs
 9517 The 'scope' of a connective is the connective, the linked formulae, and the brackets
###### 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
 9394 MPP: Given A and A→B, we may derive B
 9393 A: we may assume any proposition at any stage
 9398 ∧I: Given A and B, we may derive A∧B
 9402 RAA: If assuming A will prove B∧¬B, then derive ¬A
 9401 ∨E: Derive C from A∨B, if C can be derived both from A and from B
 9399 ∧E: Given A∧B, we may derive either A or B separately
 9397 CP: Given a proof of B from A as assumption, we may derive A→B
 9396 DN: Given A, we may derive ¬¬A
 9395 MTT: Given ¬B and A→B, we derive ¬A
 9400 ∨I: Given either A or B separately, we may derive A∨B
###### 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
 9521 'Modus tollendo ponens' (MTP) says ¬P, P ∨ Q |- Q
 9523 De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions
 9526 We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q)
 9524 We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q
 9527 The Distributive Laws can rearrange a pair of conjunctions or disjunctions
 9525 We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q)
 9522 'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q
###### 4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
 9538 A truth-table test is entirely mechanical, but this won't work for more complex logic
 9537 Truth-tables are good for showing invalidity
###### 4. Formal Logic / B. Propositional Logic PL / 4. Soundness of PL
 9536 If any of the nine rules of propositional logic are applied to tautologies, the result is a tautology
###### 4. Formal Logic / B. Propositional Logic PL / 5. Completeness of PL
 9539 Propositional logic is complete, since all of its tautologous sequents are derivable
###### 4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / a. Symbols of PC
 13902 'Gm' says m has property G, and 'Pmn' says m has relation P to n
 13911 The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E
 13909 Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....'
###### 4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / b. Terminology of PC
 13910 Our notation uses 'predicate-letters' (for 'properties'), 'variables', 'proper names', 'connectives' and 'quantifiers'
###### 4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / c. Derivations rules of PC
 13904 Universal Elimination (UE) lets us infer that an object has F, from all things having F
 13901 Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules
 13903 Universal elimination if you start with the universal, introduction if you want to end with it
 13906 With finite named objects, we can generalise with &-Intro, but otherwise we need ∀-Intro
 13908 UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one
###### 4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
 13905 If there is a finite domain and all objects have names, complex conjunctions can replace universal quantifiers
###### 4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
 13900 'Some Frenchmen are generous' is rendered by (∃x)(Fx→Gx), and not with the conditional →
###### 5. Theory of Logic / B. Logical Consequence / 8. Material Implication
 9520 The paradoxes of material implication are P |- Q → P, and ¬P |- P → Q