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
That proposition that either P or Q is their 'disjunction', written P∨Q
That proposition that both P and Q is their 'conjunction', written P∧Q
We write 'P if and only if Q' as P↔Q; it is also P iff Q, or (P→Q)∧(Q→P)
If A and B are 'interderivable' from one another we may write A -||- B
We write the 'negation' of P (not-P) as
The sign |- may be read as 'therefore'
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / b. Terminology of PL
A 'implies' B if B is true whenever A is true (so that A→B is tautologous)
A 'theorem' is the conclusion of a provable sequent with zero assumptions
A wff is a 'tautology' if all assignments to variables result in the value T
A wff is 'inconsistent' if all assignments to variables result in the value F
A 'well-formed formula' follows the rules for variables, , →, ∧, ∨, and ↔
A 'substitution-instance' is a wff formed by consistent replacing variables with wffs
A wff is 'contingent' if produces at least one T and at least one F
The 'scope' of a connective is the connective, the linked formulae, and the brackets
'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
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
MPP: Given A and A→B, we may derive B
A: we may assume any proposition at any stage
DN: Given A, we may derive A
MTT: Given B and A→B, we derive A
∨I: Given either A or B separately, we may derive A∨B
∧I: Given A and B, we may derive A∧B
∧E: Given A∧B, we may derive either A or B separately
RAA: If assuming A will prove B∧B, then derive A
∨E: Derive C from A∨B, if C can be derived both from A and from B
CP: Given a proof of B from A as assumption, we may derive A→B
4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / d. Basic theorems of PL
'Modus tollendo ponens' (MTP) says P, P ∨ Q |- Q
'Modus ponendo tollens' (MPT) says P, (P ∧ Q) |- Q
We can change conjunctions into negated conditionals with P→Q -||- (P → Q)
We can change conditionals into disjunctions with P→Q -||- P ∨ Q
We can change conditionals into negated conjunctions with P→Q -||- (P ∧ Q)
The Distributive Laws can rearrange a pair of conjunctions or disjunctions
De Morgan's Laws make negated conjunctions/disjunctions into non-negated disjunctions/conjunctions
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....'
The 'symbols' are bracket, connective, term, variable, predicate letter, reverse-E
'Gm' says m has property G, and 'Pmn' says m has relation P to n
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
Universal Elimination (UE) lets us infer that an object has F, from all things having F
Predicate logic uses propositional connectives and variables, plus new introduction and elimination rules
Universal elimination if you start with the universal, introduction if you want to end with it
If you pick an arbitrary triangle, things proved of it are true of all triangles
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
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