Formal Logic 2. This lecture: Standard Procedure of Inferencing Normal forms Standard Deductive Proofs in Logic using Inference Rules

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1 ormal Logic 2 HW2 Due Now & ickup HW3 handout! Last lecture ropositional Logic ropositions, Statements, Connectives, ruth table, ormula W roperties: autology, Contradiction, Validity, Satisfiability Logical Equivalence autology Equivalence Laws his lecture: Standard rocedure of Inferencing Normal forms Standard Deductive roofs in Logic using Inference Rules 1 Normal orm wo special forms for formulas logically equivalent to a given formula : Disjunctive normal form (DN) and Conjunctive normal form (CN). DN: a formula G of m variables being a disjunction x 1 x 2 x k of k >= 0 terms/clauses, where each x i is a conjunction of m literals, i.e. x i = (y 1 y 2 y m ) CN: a formula G of m variables being a conjunction x 1 x 2 x k of k >= 0 terms/clauses, where each x i is a disjunction of m literals, i.e. x i = (y 1 y 2 y m ) heorem: Every formula is logically equivalent to a corresponding formula in DN (and a formula in CN) (in another word, every formula can be written in DN and CN) 2 1

2 Systematic way to find DN and CN Given S = (( ( R)) ( )) R S DN is ( R) ( R) ( R) i.e. take each row where S is rue. Check this! CN for S starts with DN of S, i.e. take each row where S is alse and form DN. hen, use DeMorganlaw to convert ( S) in DN to CN 3 DN for S is: ( R) ( R) ( R) ( R) ( R) Now S = ( S) : (( R) ( R) ( R) ( R) ( R) ) Use DeMorgan slaw: (push in 1 st ) ( R) ( R) ( R) ( R) ( R) ) More DeMorgan slaw to get CN for S: ( R) ( R) ( R) ( R) ( R) 4 2

3 Deductive roof Method erminology : axiomis a basic assumption about mathematical structure that needs no proof, i.e. things known to be true (facts or proven theorems or tautologies) A theoremis a statement that can be shown to be valid. A lemmais a simple theorem used as an intermediate result in the proof of another theorem. A corollaryis a proposition that follows directly from a theorem that has been proved. A conjectureis a statement whose truth value is unknown. Once it is proven, it becomes a theorem. 5 A theorem often has two parts Conditions(or hypotheses/premises) and a Conclusion A correct (deductive) proof is to establish that If all conditions are true, then the conclusion is true i.e., (Conditions Conclusion) is valid (a tautology) Deduction: a method which usestautology lawsand inference rulesto prove a theorem, i.e. Often there are missing pieces (not easily understandable) between conditions and conclusion. ill it by a sequence of applications of tautologies and inference rules - Starting from conditions and axioms - Generate a sequence of valid statements connected by proper rules of inference (new statements are generated from existing ones by these rules) 6 3

4 Logically valid inferences by using inference rules he general form of a rule of inference is: p 1 p 2. p n q if p 1 and p 2 and and p n are all true, then q is true as well. Each rule is an established tautology of p 1 p 2 p n q hese rules of inference can be used in any mathematical argument and do not require any proof. 7 Modus ponens Latin name, means: method of assertion An argumentconsists of one or more hypotheses and a conclusion. An argument isvalid(theorem), if whenever all its hypotheses are true, its conclusion is also true Example : Use deduction to prove that [[ ( )] ( )] is a valid argument (or theorem). Note : Using Conditional proof law, this is equivalent to [[ ( )] ] 1. ( ) (hypothesis) 2. (hypothesis) 3. (1& 2 modus ponens) 4. (2& 3 modus ponens) 8 4

5 Example : rove that ( ) ( S) ( S) is theorem. Note : Using Conditional proof law, this is equivalent to ( ) ( S) S 1. (hypothesis) 2. S (hypothesis) 3. (hypothesis) 4. (contraposition, 1, 4 equivalent) 5. (4, double negations) 6. (3, 5 modus ponens) 7. S (2, 6 modus ponens) 9 ew Inferences rules (more in the text book) conjunction disjunctive syllogism Modus ollens addition simplification 10 5

6 Example : rove that (A ) (A L) [A ( L)] is theorem Note : his is equivalent to (A ) (A L) A ( L) 1. A (hypothesis) 2. A L (hypothesis) 3. A (hypothesis) 4. (1, 3 modus ponens) 5. L (2, 3 modus ponens) 6. L (4, 5 conjunction inference rule) 11 Example : rove ( R) (R S W) ( S W) is a theorem Note : his is equivalent( R) (R S W) ( S) W 1. R (hypothesis) 2. R S W (hypothesis) 3. S (hypothesis) ** arget: Show W is true: ** 4. (3, simplification) 5. (4, addition) 6. R (1, 5 mp) 7. S W (2, 6 mp) 8. S (3, simplification) 9. W (7,8 disjunctive syllogism) 12 6