Today’s Topics

1. Today’s Topics • Review the Idea of a Proof • Rules of Inference • Rules of Equivalence

2. A proof is a finite set of formulae, beginning with the premises of an argument and ending with its conclusion, in which each formula following the premises is derived from the preceding formulae according to established rules of inference and equivalence.

3. NOTE: Constructing a proof for an argument definitively establishes that the argument is valid. • HOWEVER, failure to construct a proof proves nothing. It may be because the argument is non-valid, or it may be because YOU can’t construct the proof.

4. Valid Argument Forms Justify Two Types of RULES • Rules of Inference • Operate on whole lines only • Generate new lines with unique truth values • Rules of Equivalence • Operate on whole lines or parts of lines • Generate lines equivalent to the original lines

5. Rules of Inference • Generate new lines whose truth value follows from, but is not identical to, the truth of the source lines. • Operate on lines whose statement forms match the statement forms of the lines in the argument form of the rule. • Can be applied ONLY to entire lines, not parts of lines.

6. Modus Ponens (MP) From a conditional and a line identical to its antecedent, you may derive a line identical to its consequent Modus Tollens (MT) From a conditional and the negations of its consequent, you may derive the negation of its antecedent Disjunctive Syllogism (DS) From a disjunction and the negation of one disjunct, you may derive the other disjunct Hypothetical Syllogism (HS) From 2 conditionals, if the consequent of the first is identical to the antecedent of the second, you may derive a new conditonal whose antecedent is identical to the antecedent of the first and whose consequent is identical to the consequent of the second. Eight Basic Inference Rules

7. Modus Ponens Modus Tollens p  q p  q p ~q q~p Disjunctive Syllogism Hypothetical Syllogism p  q p  q ~p q  r qp r

8. Simplification From a conjunction you may derive either conjunct. Conjunction From any 2 lines you may derive a conjunction which has those lines as conjuncts Addition From any line you may derive a disjunction with that line as a disjunct Constructive Dilemma From a disjunction and 2 conditionals, if the antecedents match the disjuncts, you may derive a disjunction of the consequents

9. Simplification p  q p q Conjunction p q p  q Addition p p v q Constructive Dilemma p v q p  r q  s r v s

10. Rules of Equivalence • Equivalent expressions are true and false under exactly the same circumstances. So, one expression, or part of an expression, can be replaced with an equivalent expression (or part) without any change in meaning. • Rules of Equivalence allow replacement of “equals for equals” • Rules of equivalence are bi-directional

11. Use rules of equivalence to manipulate the shape of a line to make it fit the strict pattern of an inference rule.

12. Double Negation (DN) p :: ~ ~ p ~ ~ p :: p DeMorgan (DM) ~(p v q) :: (~p  ~q) ~(p  q) :: (~p v ~q) Association (Assn) (p v q) v r :: p v (q v r) (p  q)  r :: p  (q  r) Commutation (Comm) p v q :: q v p p  q :: q  p Rules of Equivalence

13. Distribution (Dist) [p  (q v r)] :: [(p  q) v (p  r)] [p v (q  r)] :: [(p v q)  (p v r)] Implication (Impl) (p  q) :: (~p v q) Exportation (Exp) ((p q)  r) :: (p  (q  r))

14. Contraposition (Contra) (p  q) :: (~q  ~p) Tautology (Taut) p :: (p  p) p :: (p v p) Equivalence (Equiv) (p  q) :: [(p  q) v (~p ~q)] (p  q) :: [(p  q)  (q  p)]

15. That’s It! We now have all 18 of the rules of inference and equivalence • MP, MT, DS, HS, Conj, Simp, Add, CD • DN, DeM, Assoc, Comm, Dist, Imp, Exp, Cont, Taut, Equiv

16. Applying Rules of Equivalence to Formulas

17. Have a good weekend!