Today’s Topics

1. Today’s Topics Argument forms and rules (review) Rules of inference and equivalence Applying rules of inference Justifying steps in a proof Beginning to construct proofs

2. Argument Form • An argument form set of statement forms • Every substitution instance of an argument form is an argument • Argument forms are either valid or non-valid (truth tables verify the validity of valid argument forms)

3. 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

4. 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.

5. 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 negation 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

6. 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

7. 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

8. Simplification p  q q Conjunction p q p  q Addition p p ▼ q Constructive Dilemma p ▼ q p  r q  s r ▼ s

9. 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

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

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

12. Distribution (Dist) [p  (q ▼ r)] :: [(p  q) ▼ (p  r)] [p ▼ (q  r)] :: [(p ▼ q)  (p ▼ r)] Material Implication (MI) (p  q) :: (~p ▼ q) Exportation (Exp) ((p q)  r) :: (p  (q  r))

13. Contraposition (Cont) (p  q) :: (~q  ~p) Redundancy (Re) p :: (p  p) p :: (p ▼ p) Material Equivalence (ME) (p  q) :: [(p  q) ▼ (~p ~q)] (p  q) :: [(p  q)  (q  p)]

14. One Additional Rule of Equivalence: Absorption From a conditional you may derive a new conditional whose antecedent is that of the original and whose consequent is a conjunction of the original antecedent and the original consequent Absorption p  q : : p  (p ● q)

15. That’s It! We now have all 19 of the rules of inference and equivalence • MP, MT, DS, HS, Conj, Simp, Add, CD, Abs • NOTE: there are no inference rules for dealing with biconditionals • DN, DeM, Assoc, Comm, Dist, Imp, Exp, Cont, Taut, Equiv

16. Applying Rules of Equivalence to Formulas • Download the Handout on Equivalence Rules and apply the rules indicated to the formulas given. Discuss your results on the bulletin board.

17. Remember • 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.

18. Justifying Steps In a Proof • Each line in a proof must be justified. • Premises justify themselves, we assume them to be true. • Derived Lines (those lines after the premises) must be justified according to valid rules of inference or equivalence as following from previous lines.

19. Here’s how it works: • Consider the argument A  B, A B • First, list the premises and give them line numbers. • 1. A  B Premise • 2.A Premise • Now, justify lines leading to the conclusion; • 3.B 1,2 Modus Ponens (MP) • That is the proof.

20. Try a few • Download the Handout on Justification of Proofs and try to justify all the derived lines (those that follow the premises). • Now try the next homework set on justification of proofs.

21. Thinking About Proofs • Proofs in logic work just like proofs in geometry • The 18 rules we have allow us to manipulate a basic set of assumptions (the premises) so as to show that the conclusion is a logical consequence of them. • A proof is a set of instructions on how to get from the premises to the conclusion.

22. Constructing a proof is like giving instructions. The question is “How do I get there (the conclusion) from here (the premises)?” • The rules are the allowable moves or turns you can take. • Proceed stepwise. Suppose you want to get to D from A, B, and C. Well, if from A and B you can get to E, and from E and C you can get to D, you have your instructions. • That is all there is to constructing proofs

23. Key Ideas • Argument form • Rules of inference • Rules of equivalence • Proofs • Justifying steps in a proof