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 ) 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.
Contradiction Example 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.
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.
“Unary” Tautologies 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
Spotlight on Laws of negation ~p ↔ F T p ~p ↔ T F 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 p q ~p ~q (p q) ~(p q) ~p ~q T F The equivalence of the rightmost columns proves the tautologies p q ~p ~q (p q) ~(p q) ~p ~q T F
Truth Tables for De Morgan’s Laws p q ~p ~q (p q) ~(p q) ~p ~q T F Negate p p q ~p ~q (p q) ~(p q) ~p ~q T F
Truth Tables for De Morgan’s Laws p q ~p ~q (p q) ~(p q) ~p ~q T F Negate q p q ~p ~q (p q) ~(p q) ~p ~q T F
Truth Tables for De Morgan’s Laws p q ~p ~q (p q) ~(p q) ~p ~q T F Form disjunction of p and q conjunction of p q ~p ~q (p q) ~(p q) ~p ~q T F
Truth Tables for De Morgan’s Laws p q ~p ~q (p q) ~(p q) ~p ~q T F Negate disjunction of p and q conjunction of p q ~p ~q (p q) ~(p q) ~p ~q T F
Truth Tables for De Morgan’s Laws p q ~p ~q (p q) ~(p q) ~p ~q T F Form conjunction of ~p and ~q disjunction of p q ~p ~q (p q) ~(p q) ~p ~q T F
Truth Tables for De Morgan’s Laws p q ~p ~q (p q) ~(p q) ~p ~q T F The equivalence of the rightmost columns proves the tautologies p q ~p ~q (p q) ~(p q) ~p ~q T F
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 p→q Major Premise p Minor Premise q Conclusion p→q Major Premise ~q Minor Premise ~p Conclusion
Some Common Inference rules Modus Ponens Modus Tollens Syllogisms: Disjunctive Hypothetical Reductio ad Absurdam Proof by Contradiction Indirect Proof Dilemmas: Constructive Destructive Proof by Cases
Modus Ponens If p and p → q, then q. -or- (p (p → q )) → q 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)
Truth Table Proof of Modus Ponens q p→q p(p→q) [p(p→q)]→q T F
Truth Table Proof of Modus Ponens q p→q p(p→q) [p(p→q)]→q T F
Truth Table Proof of Modus Ponens q p→q p(p→q) [p(p→q)]→q T F
Truth Table Proof of Modus Ponens q p→q p(p→q) [p(p→q)]→q T F
Condensed Truth Table Proof of Modus Ponens q [(p → q) p] T F
(falsehood flows backward) 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)
Truth Table Proof of Modus Tollens q ~p ~q p→q (p→q)~q [(p→q)~q]→~p T F
Truth Table Proof of Modus Tollens q ~p ~q p→q (p→q)~q [(p→q)~q]→~p T F
Truth Table Proof of Modus Tollens q ~p ~q p→q (p→q)~q [(p→q)~q]→~p T F
Truth Table Proof of Modus Tollens q ~p ~q p→q (p→q)~q [(p→q)~q]→~p T F
Truth Table Proof of Modus Tollens q ~p ~q p→q (p→q)~q [(p→q)~q]→~p T F
Truth Table Proof of Modus Tollens q ~p ~q p→q (p→q)~q [(p→q)~q]→~p T F
Condensed Truth Table Proof of Modus Tollens q [(p → q) ~ q] T F
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
Truth Table Proof of Disjunctive Syllogism q ~p p q ~p(p q) [~p(p q)]→q T F
Truth Table Proof of Disjunctive Syllogism q ~p p q ~p(p q) [~p(p q)]→q T F
Truth Table Proof of Disjunctive Syllogism q ~p p q ~p(p q) [~p(p q)]→q T F
Truth Table Proof of Disjunctive Syllogism q ~p p q ~p(p q) [~p(p q)]→q T F
Truth Table Proof of Disjunctive Syllogism q ~p p q ~p(p q) [~p(p q)]→q T F
Condensed Truth Table Proof of Disjunctive Syllogism q ~p (p q) → T F
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.
Truth Table Proof of Hypothetical Syllogism q r p→q q→r (p→q)(q→r) p→r [(p→q)(q→r)]→(p→r) T F
Truth Table Proof of Hypothetical Syllogism q r p→q q→r (p→q)(q→r) p→r [(p→q)(q→r)]→(p→r) T F
Truth Table Proof of Hypothetical Syllogism q r p→q q→r (p→q)(q→r) p→r [(p→q)(q→r)]→(p→r) T F
Truth Table Proof of Hypothetical Syllogism q r p→q q→r (p→q)(q→r) p→r [(p→q)(q→r)]→(p→r) T F
Truth Table Proof of Hypothetical Syllogism q r p→q q→r (p→q)(q→r) p→r [(p→q)(q→r)]→(p→r) T F
Truth Table Proof of Hypothetical Syllogism q r p→q q→r (p→q)(q→r) p→r [(p→q)(q→r)]→(p→r) T F
Condensed Truth Table Proof of Hypothetical Syllogism q r [(p → q) (q r)] (p r) T F
ἡ εἰς ἄτοπον ἀπαγωγή Reductio ad Absurdam p q ~p ~q (p → q) ~q) T F p → q p → ~q ~p Reductio ad Absurdam involves concluding both a consequent and its negation from the same antecedent, thus proving the negation of the antecedent. This is only technically distinct from proof by contradiction, since by the law of noncontradiction we have that q~q↔F.
Proof by Contradiction → F) T Proof by contradiction proceeds by negating the proposition representing the desired proof goal, and from this concluding any universal falsehood.
Indirect Proof p q ~p ~q (~q → ~p) (p q) Indirect proof also follows as a special case of the equivalence of any implication and its contrapositive. (p → q) ↔ (~q → ~p)
Dilemmas Constructive Dilemma Destructive Dilemma (p q) (p → r) (q → s)→ (r s) related to modus ponens truth feeds forward Destructive Dilemma (p → r) (q → s) (~r ~s) → (~p ~q) related to modus tollens falsehood flows backward
Constructive Dilemma (p → r) ⇗ ⇘ (p q) or (r s) (q → s) modus ponens ⇘ (p q) or (r s) (q → s) (p q) (p → r) (q → s) → (r s) Note that four propositions requires a truth table of 24 rows. “Dilemma” generalizes to “multi-lemma”, with more disjuncts.
Condensed Truth Table Proof of Constructive Dilemma q r s [(p → r) (q s)] (p q) (r s) T F
Condensed Truth Table Proof of Constructive Dilemma q r s [(p → r) (q s)] (p q) (r s) T F
Condensed Truth Table Proof of Constructive Dilemma q r s [(p → r) (q s)] (p q) (r s) T F
Condensed Truth Table Proof of Constructive Dilemma q r s [(p → r) (q s)] (p q) (r s) T F
Condensed Truth Table Proof of Constructive Dilemma q r s [(p → r) (q s)] (p q) (r s) T F
Condensed Truth Table Proof of Constructive Dilemma q r s [(p → r) (q s)] (p q) (r s) T F
Condensed Truth Table Proof of Constructive Dilemma q r s [(p → r) (q s)] (p q) (r s) T F
Condensed Truth Table Proof of Constructive Dilemma q r s [(p → r) (q s)] (p q) (r s) T F
Destructive Dilemma (p → r) ⇗ ⇘ (~r ~s) or (~p ~q) (q → s) modus tollens ⇘ (~r ~s) or (~p ~q) (q → s) (p → r) (q → s) (~r ~s) → (~p ~q) Note that four propositions requires a truth table of 24 rows. “Dilemma” generalizes to “multi-lemma”, with more disjuncts.
Condensed Truth Table Proof of Destructive Dilemma q r s (p → r) (q s) (~r ~s) (~p ~q) T F
(p q r) (p → s) (q → s) (r → s) → s Proof by Cases (p q r) (p → s) (q → s) (r → s) → s (p → s) ⇗ modus ponens ⇘ (p q r) or (q → s) s (r → s) For the purposes of illustration, we have chosen three cases; the inference is valid for any number of cases – the truth tables have many rows! Note that “proof by cases” is a special case of constructive “dilemma” or “multi-lemma” in which all the consequents are the same proposition.
Beyond Truth Tables Truth tables are powerful means of proving “simple” results involving small numbers of statements; however: The number of columns grows in rough proportion to the complexity of the statements to be proven, and The number of rows grows as 2n, where n is the number of atomic statements in the compound. For more than the most basic statements, truth tables can overwhelm manual calculation. Ultimately, the subject of symbolic logic is about using a relatively modest set of basic theorems to prove complex statements through algebra-like procedures.
Fallacies
Formal vs. Informal Fallacy a formal fallacy is a deductive argument that has an invalid form, whereas an informal fallacy is any other invalid mode of reasoning whose flaw is not in the form of the argument; e.g., in flawed premises. We are concerned only with formal fallacies. A formal fallacy is not necessarily a “contradiction” as earlier defined – a universal falsehood – it is only necessary that the argument admits an unexpected truth value in some case(s).
This is the fallacy of the undistributed middle: Penguin Logic Definitions: p = You are a penguin o = You are an old TV show b = You are black & white Premises: p → b = If you are a penguin, then you are black and white o → b = If you are an old TV show, then you are black and white Penguin conclusion: p → o = If you are a penguin, then you are an old TV show This is the fallacy of the undistributed middle: (p → b) (o → b) → (p → o) invalid which can be confused with hypothetical syllogism in which the “middle” term is properly “distributed”: (p → b) (b → o) → (p → o) valid
Affirming the Consequent Fallacy confused with Modus Ponens Tautology M.P. can be viewed as “affirming the antecedent” Affirming the Consequent Fallacy One final truth value is “F”, invalidating the desired tautology p q (p → q) T F p q (p → q) T F
Denying the Antecedent Fallacy confused with Modus Tollens Tautology M.T. can be viewed as “denying the consequent” Denying the Antecedent Fallacy One final truth value is “F”, invalidating the desired tautology p q (p → q) ~ T F p q (p → q) ~ T F
Affirming a Disjunct Fallacy confused with Disjunctive Syllogism Tautology D.S. can be viewed as “denying a disjunct” Affirming a Disjunct Fallacy A.D. can be viewed as confusing “” with “” p q (p q) ~ → T F p q (p q) → ~ T F
LogiCola LogiCola is a free downloadable logic tutorial program that permits students to challenge themselves with a variety of logical exercises at a variety of difficulty levels. (Note that Logicola uses 0 & 1 rather than T & F, and lays out its truth tables differently than we have done here.) The program is written by a college professor and intended as a companion to his own college-level logic course. It is thus fairly high in content and low on user frills, but with a little patience and practice, anyone can get the “knack” of using it. http://www.jcu.edu/philosophy/gensler/lc/index.htm