Logical and Rule-Based Reasoning Part I

1. Logical and Rule-Based Reasoning Part I

2. Logical Models and Reasoning Big Question: Do people think logically?

3. Exercise • You are given 4 cards each with a letter on one side, and a number on the other. You can see one side of each card only: • Rule: “if a card has a vowel on one side, then it has an odd number on the other” • In order to check whether the rule is true of these cards, what is the minimal number of cards cards do you need to turn over and which ones? 1 2 3 4 E 7 K 2

4. Exercise • Now assume each card has a beverage on one side, and the drinker's age on the other : • Rule: “if someone drinks beer, then she is 21 years or older” • In order to check whether the rule is true of these cards, what is the minimal number of cards cards do you need to turn over and which ones? 1 2 3 4 Beer Coke 23 years 19 years

5. Logical Reasoning • The goal is find a way to • state knowledge explicitly • draw conclusions from the stated knowledge • Logic • A "logic" is a mathematical notation (a language) for stating knowledge • The main alternative to logic is "natural language" i.e. English, Swahili, etc. • As in natural language the fundamental unit is a “sentence” (or a statement) • Syntax and Semantics • Logical inference

6. Propositional Logic: Syntax • Sentences • represented by propositional symbols (e.g., P, Q, R, S, etc.) • logical constants: True, False • Connectives: ~, Ù, Ú, Þ, Û • Þ is also shown as  and Û as  • Examples:

7. Interpretations and Validity • A logical sentence S is satisfiable if it is true at least in one situation • (under at least one “interpretation”) • S is valid if it is true under all interpretations (S is a tautology) • S is unsatisfiable if it is false for all interpretations (S is inconsistent) • A sentence T follows (is entailed by) S, if any time S is true, T is also true

8. Propositional Logic: Semantics • In propositional logic, the semantics of connectives are specified by truth tables: • Each assignment of truth values to individual propositions (e.g., P, Q, R) in the sentence represents one interpretation  a row in the truth table • Truth tables can also be used to determine the validity of sentences

9. Notes on Implication • If p and q are both true, then p Þ q is true. • If 1+1 = 2 then the sun rises in the east. • Here p: "1+1 = 2" and q: "the sun rises in the east." • If p is true and q is false, then p Þ q is false. • When it rains, I carry an umbrella. • p: "It is raining," and q: "I am carrying an umbrella." • we can rephrase as: "If it is raining then I am carrying an umbrella." • On days when it rains (p is true) and I forget to bring my umbrella (q is false), the statement p Þ q is false • If p is false, then p Þ q is true, no matter whether q is true or not. For instance: • If the moon is made of green cheese, then I am the King of England.

10. Notes on Implication • Using truth tables we notice that the only way the implication p Þ q can be false is for p to be true and q to be false. • In other words, p Þ q is logically equivalent to (~p) \/ q. p Þq (~p) \/ q "Switcheroo" law

11. Propositional Inference • Let S be (A \/ C) /\ (B \/ ~C) and let R be A \/ B. Does R follow from S? • check all possible interpretations (involving A, B, and C); R must be true whenever S is true

12. Checking Validity and Equivalences • Suppose we want to determine if a sentence: • is valid: • Construct the truth table for the sentence using all possible combinations of true and false assigned to P and Q • As intermediate steps, can create columns for different components of the compound sentence (PÞQ)Û(~PÚQ) This sentence is a tautology because it is true under all interpretations

13. Conversion between => and \/ DeMorgan’s Laws Distributivity Some Useful Tautologies (equivalences)

14. Using Equivalences: Example

15. Some Online Practice Exercises • http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/logic/logic3.html

16. More Tautologies and Equivalences

17. More Tautologies and Equivalences • Can also check it with truth tables:

18. More Tautologies and Equivalences

19. More Online Practice Exercises • http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/logic/logic4.html

20. More Tautologies and Equivalences

21. More Tautologies and Equivalences

22. More Tautologies and Equivalences

23. Summary of Tautological Implications and Equivalences • http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/logic/logic4.html See tables A and B at the following page:

24. Exercise: The Island of Knights & Knaves • We are in an island all of whose inhabitants are either knights or knaves • knights always tell the truth • knaves always lie • Problem: • you meet inhabitants A and B, and A tells you “at least one of us is a knave” • can you determine who is a knave and who is a knight?

25. Exercise: The Island of Knights & Knaves • Problem 1 • you meet inhabitants A and B. A says: “We are both knaves.” What are A and B? • Problem 2 • you meet inhabitants A, B, and C. You walk up to A and ask: "are you a knight or a knave?" A gives an answer but you don't hear what she said. B says: "A said she was a knave." C says: "don't believe B; he is lying.” • What are B and C? How about A?

26. Logical Inference • Given a set of assumptions (premises), logically inferring a new statement (conclusion) is done by a step-by-step derivation using “rules of inference” • Rules of inference are the Tautological Implications and Tautological Equivalences we saw before (e.g., Modus Ponens) • The derivation starting from the premises and leading to the conclusion is called a “proof” or and “argument” • See the middle column of Tables A and B in Section 4 of the Logic Web site.

27. Examples of Inference Rules

28. Applying Inference Rules • Example: Modus Ponens (MP) • Suppose we have 3 statements we know to be true: • Applying MP to statements 1 and 3, we conclude: (r /\ ~s) as the conclusion. • Note that MP has the form: • Here A stands for (p \/ q) and B stands for (r /\ ~s). Premise 2 in this case was not used.

29. Applying Inference Rules • Example: Modus Tollens (MT)

30. Applying Inference Rules • Some general rules to remember:

31. Proof Example • Prove that the following argument is valid

32. Proof Example • Prove that the following argument is valid Do Exercise 2P on Section 6 of the Logic Web site

33. Proof Example • Prove that the following argument is valid

34. More Examples & Exercises • In Section 6 of the Logic Web Site: • Proof Strategies: Examples 4 and 5, and exercise 5P • Forward and Backward: Examples 6 and 7, and exercise 7P • Different types of arguments: Examples 8-10 • Logical Reasoning: Example 11

35. Extra Credit Contest • You are to write down and submit a statement • Rules of the contest: (Note: I can’t violate the rules) • There are two prizes: • Prize 1: you get a couple of m&m’s • Prize 2: you get 10 extra credit points on your next assignment • If your statement is true, then I have to give you one of the prizes • If it is false, you get nothing • The challenge: come up with a statement that guarantees you get prize 2!

36. Predicate Logic • Consider: • p: All men are mortal. q: Socrates is a man. r: Socrates is mortal. • We know that from p and q we should be able to prove r. • But, there is nothing in propositional logic that allows us to do this. • Need to represent the relationship between all men and one man in particular (Socrates).

37. Predicate Logic • Instead we need to use quantifiers and predicates: For all x, if x is a man, then x is mortal  x [ man(x)  mortal(x) ] Universal quantifier predicates

38. Predicate Logic • Second quantifier is the existential quantifier (“there exists”): “Everybody loves somebody” “for every person x, there is a person y so that x loves y” x [ person(x)  y [ person(y) /\ loves(x,y) ] ] Existensial quantifier predicates