### 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
 9508 The sign |- may be read as 'therefore'
 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)
 9512 We write the 'negation' of P (not-P) as ¬
 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
###### 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
 9517 The 'scope' of a connective is the connective, the linked formulae, and the brackets
 9533 A 'implies' B if B is true whenever A is true (so that A→B is tautologous)
 9534 Two propositions are 'equivalent' if they mirror one another's truth-value
 9530 A wff is 'contingent' if produces at least one T and at least one F
 9518 A 'theorem' is the conclusion of a provable sequent with zero assumptions
 9519 A 'substitution-instance' is a wff formed by consistent replacing variables with wffs
 9528 A wff is a 'tautology' if all assignments to variables result in the value T
 9529 A wff is 'inconsistent' if all assignments to variables result in the value F
 9532 'Subcontrary' propositions are never both false, so that A∨B is a tautology
 9516 A 'well-formed formula' follows the rules for variables, ¬, →, ∧, ∨, and ↔
 9531 'Contrary' propositions are never both true, so that ¬(A∧B) is a tautology
###### 4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
 9393 A: we may assume any proposition at any stage
 9394 MPP: Given A and A→B, we may derive B
 9395 MTT: Given ¬B and A→B, we derive ¬A
 9396 DN: Given A, we may derive ¬¬A
 9397 CP: Given a proof of B from A as assumption, we may derive A→B
 9398 ∧I: Given A and B, we may derive A∧B
 9399 ∧E: Given A∧B, we may derive either A or B separately
 9401 ∨E: Derive C from A∨B, if C can be derived both from A and from B
 9400 ∨I: Given either A or B separately, we may derive A∨B
 9402 RAA: If assuming A will prove B∧¬B, then derive ¬A
###### 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
 9522 'Modus ponendo tollens' (MPT) says P, ¬(P ∧ Q) |- ¬Q
 9526 We can change conjunctions into negated conditionals with P→Q -||- ¬(P → ¬Q)
 9525 We can change conditionals into negated conjunctions with P→Q -||- ¬(P ∧ ¬Q)
 9524 We can change conditionals into disjunctions with P→Q -||- ¬P ∨ Q
 9523 De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions
 9527 The Distributive Laws can rearrange a pair of conjunctions or disjunctions
###### 4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
 9537 Truth-tables are good for showing invalidity
 9538 A truth-table test is entirely mechanical, but this won't work for more complex logic
###### 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
 13909 Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....'
 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
###### 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
 13901 Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules
 13904 Universal Elimination (UE) lets us infer that an object has F, from all things having F
 13903 Universal elimination if you start with the universal, introduction if you want to end with it
 13907 If you pick an arbitrary triangle, things proved of it are true of all triangles
 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