Logical and Rule-Based Reasoning Part I

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

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

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

Logical Reasoning The goal is find a way to Logic 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

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:

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

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

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.

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 Þ  q   (~p) \/  q "Switcheroo" law

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

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

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

Using Equivalences: Example

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

More Tautologies and Equivalences

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

More Tautologies and Equivalences

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

More Tautologies and Equivalences

More Tautologies and Equivalences

More Tautologies and Equivalences

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:

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?

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?

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.

Examples of Inference Rules

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.

Applying Inference Rules Example: Modus Tollens (MT)

Applying Inference Rules Some general rules to remember:

Proof Example Prove that the following argument is valid

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

Proof Example Prove that the following argument is valid

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

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!

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).

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

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