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Introduction to Symbolic Logic. (part 2) San Diego Math Circle David W. Brown. Theorems, Tautologies, Proofs. Oh my!. We’ve just encountered some statements that are, in fact, “theorems” of propositional logic Laws of contraposition ( p → q ) ↔ ( ~q → ~p ) ( q → p ) ↔ ( ~p → ~q )
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IntroductiontoSymbolic Logic (part 2) San Diego Math Circle David W. Brown
We’ve just encountered some statements that are, in fact, “theorems” of propositional logic Laws of contraposition ( p → q ) ↔ ( ~q → ~p ) ( q → p ) ↔ ( ~p → ~q ) Biconditional introduction/elimination (p → q) (p ← q) ↔ (p ↔ q) We demonstrated the truth of these equivalences by comparing the truth tables for each side of the equivalence symbol. Since the truth values of each side (either T or F) were the same in every row, we “proved” the equivalence to be true for all possible truth values of propositions p and q.
Tautology, Contradiction, Contingency A tautology is a statement that is true for all possible truth values of its atomic substatements; i.e., is equivalent to the constant-true function T and thus expresses a universal truth. A contradiction is a statement that is false for all possible truth values of its atomic substatements; i.e., is equivalent to the constant-false function F and thus expresses a universal falsehood. A contingency is a statement that is true for some possible truth values of its atomic substatements, and false for others; i.e., is not equivalent to either constant-true function T, nor the constant-false function F. Most propositions are contingencies.
Definitions: b = “The boy did it.” d = “The dog did it.” c = “The cat did it.” Givens: The boy says: (~b d) The dog says: (~d c) The cat says: (~c b) Proof: (~b d) (~d c) (~c b) (~b b) (~d d) (~c c) F F F F Contradiction All givens in a proof are presumed true. Here, this means that the boy, the dog, and the cat are given to be speaking the truth. Conjunction of givens Commutative & associative laws Law of noncontradiction Idempotency → Somebody’s lying. Contradiction Example
Tautology The equivalences we just proved are tautologies: ( p → q ) ↔ ( ~q → ~p ) ( q → p ) ↔ ( ~p → ~q ) (p → q) (p ← q) ↔ (p ↔ q) Tautologies of equivalence allow us to “substitute” one side of the equivalence for the other in any logical expression. Other tautologies (e.g., of implication) allow us to form inferences. This is essential for developing a “calculus” for proving theorems.
Laws of negation: p ~p ↔ F p ~p ↔ T Laws of identity p T ↔ p p F ↔ p Laws of domination p F ↔ F p T ↔ T Laws of idempotency p p ↔ p p p ↔ p Laws of reflexivity p ↔ p p → p Vacuous truth / Double negation F → p ~~p ↔ p “Unary” Tautologies
Spotlight on Laws of negation • p ~p ↔ F Law of noncontradiction No wff can be both T and F • p ~p ↔ T Law of the excluded middle Every wff must be T or F – no “maybe”
Some Basic “Binary” Tautologies • Laws of simplification (conjunction elimination) • (p q ) → p • (p q ) → q • Laws of addition (disjunction introduction) • p → (p q) • q → (p q) • Laws of absorption • p (p q) ↔ p • p (p q) ↔ p
Algebra-like Tautologies Commutative Laws: “like:” • p q ↔ q p“commutative law of x” • p q ↔ q p“commutative law of +” Associative Laws: • (p q) r ↔ p (q r)“associative law of x” • (p q) r ↔ p (q r)“associative law of +” Distributive Laws: • p (q r) ↔ (p q) (p r ) “distributive law of x over +” • p (q r) ↔ (p q) (p r )“distributive law of + over x” X Transitive Laws: • (p ↔ q) (q ↔ r) → (p ↔ r)“transitive property of =” • (p → q) (q → r) → (p → r)“transitive property of >”
Note: • In ordinary algebra, commutative laws, associative laws, distributive laws, etc. are axioms – primitive statements that stand without proof. • In logic, the similar statements are proven from more primitive truths; this is a reflection of the fact that logic is more fundamental than the higher mathematics built upon it.
De Morgan’s Laws De Morgan’s Laws tell us how to negate conjunctions and disjunctions; something frequently necessary in practice ~(p q) ↔ ~p ~q ~(p q) ↔ ~p ~q Note that this is quite different from the relation between negation (-) and the binary operations of algebra, (+) and (x).
Truth Tables for De Morgan’s Laws The equivalence of the rightmost columns proves the tautologies
Truth Tables for De Morgan’s Laws Negate p Negate p
Truth Tables for De Morgan’s Laws Negate q Negate q
Truth Tables for De Morgan’s Laws Form disjunction of p and q Form conjunction of p and q
Truth Tables for De Morgan’s Laws Negate disjunction of p and q Negate conjunction of p and q
Truth Tables for De Morgan’s Laws Form conjunction of ~p and ~q Form disjunction of ~p and ~q
Truth Tables for De Morgan’s Laws The equivalence of the rightmost columns proves the tautologies
Rules of Inference “The proof of proof”
Classical vs. Propositional Notations for Inference • Modus Ponens • (p → q ) p → q • Modus Tollens • (p → q ) ~q → ~p
Modus Ponens Modus Tollens Syllogisms: Disjunctive Hypothetical Reductio ad Absurdam Proof by Contradiction Indirect Proof Dilemmas: Constructive Destructive Proof by Cases Some Common Inference rules
Modus Ponens Latin: “modus ponendo ponens” – the mode that affirms by affirming. That is, the consequent is affirmed (proven true) by affirming (asserting the truth of) the antecedant. If p and p → q, then q. -or- (p (p → q )) → q (truth flows forward)
Modus Tollens Latin: “modus tollendo tollens” - mode that denies by denying That is, the antecedant is denied (proven false) by denying (asserting the falseness of) the consequent. If p → q and ~q, then ~p. -or- ((p → q ) ~q) → ~p (falsehood flows backward)
Disjunctive Syllogism The principle of the disjunctive syllogism is that since at least one of the two propositions comprising a true disjunction must be true, if one of them is known to be false, the other can be concluded to be true: If ~p and p q, then q. -or- (~p (p q )) → q
Hypothetical Syllogism The hypothetical syllogism expresses the transitive property of logical implication: If p → q and q → r, then p → r. -or- ((p → q ) (q → r )) → (p → r ) Note that this relates three propositions, which involves a domain of ordered triples (p,q,r) requiring a truth table of 23 rows.
