1. Propositional Logic

2. Propositions Any statement that is either True (T) or False (F) is a proposition Propositional variables: a variable that can assume a value of T or F Propositional constants: T or F Atomic proposition: A proposition consisting of only a single propositional variable or constant Logical connectives: logical operators

3. Truth Table Gives the values of a proposition under all possible assignments of its variables Used to define connectives P PP TF FT

4. FF TF FT TT QP T F F T P  Q Biconditional “equivalent” T T F T P  Q Conditional “implies” F T T T P  Q Disjunction “or” F F F T P  Q Conjunction “and” Connectives P implies Q if P then Q Q if P (e.g., Q :- P) P only if Q P is sufficient for Q Q is necessary for P P is equivalent to Q P if and only if Q P is necessary and sufficient for Q P iff Q

5. Compound Propositions Also called logical expressions, formulas, and well-formed formulas (wffs) Well-formed formulas are defined inductively: – Basis: T and F are wffs (these are the constants) P, Q, … are wffs (these are the variables) – Induction: if A and B are wffs, then so are: (  A) (A  B), (A  B), (A  B), (A  B)

6. Parentheses Well-formed formulas are fully parenthesized: (((  (P  Q))  ((  P)  Q))  R) We can remove some parentheses: – Outside parentheses can be removed – Use precedence:      – Use associativity  always left associative

7. Evaluating Logical Expressions F T T T T T F F FF TF FT TT ((  P)  Q) QP  P  Q

8. T T T T T T F T 6 F F T T F F F F 4 T T T T F F F F 2 T T T T T T F F 5 F F F F F F T T 1 T T T T T T F F 3 FFF TFF FTF TTF FFT TFT FTT TTT  R  Q PP)  PQPQ (( RQP  (P  Q)   P  Q  R Evaluating Logical Expressions

9. In general… – To evaluate expressions using truth tables with k variables and n operations is O(2 k n). – If we have one operator (n = 1) and if we can substitute in T or F and evaluate in 1  sec, then ktime 3020 minutes 4014 days 5040 years 6040,000 years! Not practical to use truth tables for “large” k! Evaluating with Truth Tables

10. Logical Expressions Tautology – a logical expression that is true for all variable assignments – The symbol |= (read “ entails ” ) denotes that what follows holds or is true, so long as what precedes it is true – Since tautologies are always true, we sometimes write “ |= B ” to denote that B is a tautology independent of what precedes it Contradiction – a logical expression that is false for all variable assignments Contingent – a logical expression that is neither a tautology nor a contradiction

11. F F T T T F F T  (P   P)P   P PP P T T T T T T T F F F F T FF TF FT TT  (P  Q)  Q  (P  Q)P  Q QP Sample Tautologies

12. Sound Reasoning A logical argument has the form: A 1  A 2  …  A n  B and is sound if when A i = T for all i, B = T (i.e., if the premises are all true, then the conclusion is also true) This happens when A 1  A 2  …  A n  B is a tautology

13. Logical Arguments Consider the following statements 1.if you study then you succeed 2.you study 3.you succeed These three statements create a logical argument – Lines 1 and 2 are the premises – Line 3 is the conclusion This logical argument is sound… 1. If P then Q 2. P ------------------ 3. Q ( ( P  Q )  P )  Q

14. Modus Ponens T T T T F F F T T T F T FF TF FT TT ((A  B)  A)  B(A  B)  A(A  B) BA AB A  B T?T Hence, modus ponens is sound! A  B A B

15. Important! We are dealing with the validity of an argument, NOT with the validity of the result! In logic, it doesn’t matter if a logical statement makes sense or not What does matter is that: – IF the premises are correct – THEN so is the conclusion 1. If P then Q 2. P ------------------ 3. Q

16. P:I study hard Q:I do well on my exam P:cows give milk Q:doors open P:I obey Q:I am happy P:I sail past the end of the world Q:I will fall off Makes sense Doesn’t make sense Makes sense Doesn’t make sense Valid Logical Arguments

17. Disjunctive Syllogism T T F F T T T T F T F F F T T T AA FF TF FT TT (A  B)   A  B(A  B)   AA  B BA Hence, disjunctive syllogism is sound! A  B  A B

18. Logical Implication If A and B are two logical expressions and if A  B is a tautology, we say that A logically implies B, and we write A  > B  > is a meta-symbol to say a logical argument is sound (P  Q)  P  > Q (P  Q)   P  > Q If A  B is a tautology, then A  > B Modus ponens Disjunctive syllogism

19. Logical Equivalence If A and B are two logical expressions and if A and B always have the same truth value, then A and B are said to be logically equivalent, and we write A  B  is a meta-symbol to say that A  B is a tautology. T T F T F T  AA  A A Thus, A  A  A Which means, you can replace A  A with A. A  B if and only if A  B is a tautology

20. Laws of , , and  Excluded middle law Contradiction law P   P  T P   P  F NameLaw Identity laws P  F  P P  T  P Domination laws P  T  T P  F  F Idempotent laws P  P  P P  P  P Double-negation law  (  P)  P

21. Commutative laws P  Q  Q  P P  Q  Q  P NameLaw Associative laws (P  Q)  R  P  (Q  R) (P  Q)  R  P  (Q  R) Distributive laws (P  Q)  (P  R)  P  (Q  R) (P  Q)  (P  R)  P  (Q  R) De Morgan’s laws  (P  Q)   P   Q  (P  Q)   P   Q Absorption laws P  (P  Q)  P P  (P  Q)  P

22. Can prove all laws by truth tables… T F T F T T T F T T F F T T T T F F F T T T T F FF TF FT TT QQ PP  (P  Q)  QP De Morgan’s law holds!

23. Two Other Useful Laws Law of implication – P  Q   P  Q Law of contrapositive – P  Q   Q   P Note that the converse of P  Q is Q  P, and the two are NOT equivalent

24. Duals To create the dual of a logical expression 1) swap propositional constants T and F 2) swap connective operators  and  P   P  TExcluded Middle   P   P  FContradiction The dual of a law is always a law! Thus, most laws come in pairs  pairs of duals

25. Normal Forms Normal forms are standard forms, sometimes called canonical or accepted forms A logical expression is said to be in disjunctive normal form (DNF) if it is written as a disjunction, in which all terms are conjunctions of literals Similarly, a logical expression is said to be in conjunctive normal form (CNF) if it is written as a conjunction of disjunctions of literals

26. Disjunctive Normal Form (DNF) (.. .. .. )  (.. .. .. )  …  (.. .. ) Term Literal, i.e. P or  P Conjunctive Normal Form (CNF) (.. .. .. )  (.. .. .. )  …  (.. .. ) Examples:(P  Q)  (P   Q) P  (Q  R) DNF and CNF Examples:(P  Q)  (P   Q) P  (Q  R)

27. Converting Expressions to DNF or CNF The following procedure converts an expression to DNF or CNF: 1.Remove all  and  2.Move  inside (use De Morgan’s law) 3.Use distributive laws to get proper form Simplify as you go (e.g., double-neg., idemp., comm., assoc.)

28. CNF Conversion Example (.. .. .. )  (.. .. .. )  …  (.. .. )  ((  P  Q)  R  (P  Q))   ((  P  Q)  R  (  P  Q)) impl.   (  P  Q)   R   (  P  Q) deM.  (  P   Q)   R  (  P   Q) deM.  (P   Q)   R  (P   Q) double neg.  (P   Q)  (P   Q)   R comm.  (P   Q)   R idemp.  (P   R)  (  Q   R) distr. (DNF) (CNF)

29. DNF Expression Generation F T F F F T T F FFF TFF FTF TTF FFT TFT FTT TTT  RQP (P  Q   R) (P   Q  R) (  P   Q  R)   (P  Q   R)  (P   Q  R)  (  P   Q  R) minterms The only definition of  is the truth table

30. CNF Expression Generation 1.Find . 2.Find the DNF of . 3.Then, use De Morgan’s law to get the CNF of  (i.e.  (  )   ) T F T F  FFF TTF FFT TTT  QP (P   Q) (  P  Q) (  P   Q) (P  Q)   (P   Q)  (  P   Q) DNF of    f   ((P   Q)  (  P   Q))   (P   Q)   (  P   Q) DeM.  (  P  Q)  (P  Q) DeM., double neg. max terms } Form a conjunction of max terms