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
'Contradictory' propositions always differ in truth-value
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / a. Symbols of PL
We write the conditional 'if P (antecedent) then Q (consequent)' as P→Q
The sign |- may be read as 'therefore'
That proposition that either P or Q is their 'disjunction', written P∨Q
We write the 'negation' of P (not-P) as
That proposition that both P and Q is their 'conjunction', written P∧Q
If A and B are 'interderivable' from one another we may write A -||- B
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
A 'theorem' is the conclusion of a provable sequent with zero assumptions
A 'substitution-instance' is a wff formed by consistent replacing variables with wffs
A wff is 'inconsistent' if all assignments to variables result in the value F
A wff is a 'tautology' if all assignments to variables result in the value T
A wff is 'contingent' if produces at least one T and at least one F
'Contrary' propositions are never both true, so that (A∧B) is a tautology
'Subcontrary' propositions are never both false, so that A∨B is a tautology
Two propositions are 'equivalent' if they mirror one another's truth-value
A 'implies' B if B is true whenever A is true (so that A→B is tautologous)
A 'well-formed formula' follows the rules for variables, , →, ∧, ∨, and ↔
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
∨E: Derive C from A∨B, if C can be derived both from A and from B
RAA: If assuming A will prove B∧B, then derive A
∧E: Given A∧B, we may derive either A or B separately
MTT: Given B and A→B, we derive A
DN: Given A, we may derive A
CP: Given a proof of B from A as assumption, we may derive A→B
∧I: Given A and B, we may derive A∧B
∨I: Given either A or B separately, we may derive A∨B
A: we may assume any proposition at any stage
MPP: Given A and A→B, we may derive B
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
We can change conditionals into negated conjunctions with P→Q -||- (P ∧ Q)
'Modus tollendo ponens' (MTP) says P, P ∨ Q |- Q
The Distributive Laws can rearrange a pair of conjunctions or disjunctions
We can change conjunctions into negated conditionals with P→Q -||- (P → Q)
We can change conditionals into disjunctions with P→Q -||- P ∨ Q
De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions
'Modus ponendo tollens' (MPT) says P, (P ∧ Q) |- Q
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
Truth-tables are good for showing invalidity
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
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
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
Write '(∀x)(...)' to mean 'take any x: then...', and '(∃x)(...)' to mean 'there is an x such that....'
'Gm' says m has property G, and 'Pmn' says m has relation P to n
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
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
Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules
Universal Elimination (UE) lets us infer that an object has F, from all things having F
Universal elimination if you start with the universal, introduction if you want to end with it
With finite named objects, we can generalise with &-Intro, but otherwise we need ∀-Intro
UE all-to-one; UI one-to-all; EI arbitrary-to-one; EE proof-to-one
If you pick an arbitrary triangle, things proved of it are true of all triangles
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / d. Universal quantifier ∀
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 ∃
'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
The paradoxes of material implication are P |- Q → P, and P |- P → Q