Today s Topic: Propositional Logic
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1 Today s Topic: Propositional Logic What is a proposition? Logical connectives and truth tables Translating between English and propositional logic Logic is the basis of all mathematical and analytical reasoning Given a collection of known truths, logic allows us to deduce new truths Example Base facts: If it is raining, I will not go outside If I am inside, Stephanie will come over Stephanie and I always play Scrabble if we are together during the weekend Today is rainy Saturday Conclusion: Stephanie and I will play Scrabble today Logic allows us to advance mathematics through an iterative process of conjecture and proof
2 Propositional logic is a very simple logic Definition: A proposition is a precise statement that is either true or false, but not both. Examples: = 4 (true) All dogs have 3 legs (false) x 2 < 0 (false) Washington, D.C. is the capital of the USA (true) Not all statements are propositions Marcia is pretty Pretty is a subjective term. x 3 < 0 True if x < 0, false otherwise. Springfield is the capital True in Illinois, false in Massachusetts.
3 We can use logical connectives to build complex propositions We will discuss the following logical connectives: (not) (conjunction / and) (disjunction / or) (exclusive or) (implication) (biconditional) Negation The negation of a proposition is true iff the proposition is false What we know What we want to know One row for each possible value of what we know p p T F F T The truth table for negation
4 Negation Examples Negate the following propositions Today is Monday Today is not Monday 21 * 2 = * 2 42 What is the truth value of the following propositions (9 is a prime number) true (Pittsburgh is in Pennsylvania) false Conjunction The conjunction of two propositions is true iff both propositions are true p q p q T T T T F F F T F F F F The truth table for conjunction 2 2 = 4 rows since we know both p and q!
5 Disjunction The disjunction of two propositions is true if at least one proposition is true p q p q T T T T F T F T T F F F The truth table for disjunction Conjunction and disjunction examples Let: p x 2 > 0 q A lion weighs less than a mouse r 10 < 7 s Pittsburgh is located in Pennsylvania This symbol means is defined as or is equivalent to What are the truth values of these expressions: p q p s p q q r false true true false
6 Exclusive or (XOR) The exclusive or of two propositions is true if exactly one proposition is true p q p q T T F T F T F T T F F F The truth table for exclusive or Note: Exclusive or is typically used to natural language to identify choices. For example You may have a soup or salad with your entree. Implication The implication p q is false if p is true and q is false, and true otherwise Terminology p is called the hypothesis q is called the conclusion p q p q T T T T F F F T T F F T The truth table for implication
7 Implication (cont.) The implication p q can be read in a number of (equivalent) ways: If p then q p only if q p is sufficient for q q whenever p Implication examples Let: p Jane gets a 100% on her final q Jane gets an A What are the truth values of these implications: p q q p true false
8 Other conditional statements Given an implication p q: q p is its converse q p is its contrapositive p q is its inverse Why might this be useful? Note: An implication and its contrapositive always have the same truth value Biconditional The biconditional p q is true if and only if p and q assume the same truth value p q p q T T T T F F F T F F F T The truth table for the biconditional Note: The biconditional statement p q is often read as p if and only if q or p is a necessary and sufficient condition for q.
9 Truth tables can also be made for more complex expressions Example: What is the truth table for (p q) r? Subexpressions of what we want to know What we want to know 2 3 = 8 rows p q r p q r (p q) r T T T T F F T T F T T T T F T F F T T F F F T T F T T F F T F T F F T T F F T F F T F F F F T T Like mathematical operators, logical operators are assigned precedence levels 1. Negation q r means ( q) r, not (q r) 2. Conjunction 3. Disjunction q r s means (q r) s, not q (r s) 4. Implication q r s means (q r) s, not q (r s) 5. Biconditional In general, we will try to use parenthesis to disambiguate these types of expressions.
10 Group Exercises Problem 1: Show that an implication p q and its contrapositive q p always have the same value Hint: Construct two truth tables Problem 2: Construct the truth table for the compound proposition p ( q r) s English sentences can often be translated into propositional sentences But why would we do that?!? Philosophy and epistemology Reasoning about law Verifying complex system specifications
11 Example #1 Example: You can see an R-rated movie only if you are over 17 or you are accompanied by your legal guardian. Let: r You can see an R-rated movie o You are over 17 a You are accompanied by your legal guardian Translation: r (o a) Find logical connectives Translate fragments Create logical expression Example #2 Example: You can have free coffee if you are senior citizen and it is a Tuesday Let: c You can have free coffee s You are a senior citizen t It is a Tuesday Translation: (s t) c
12 Example #3 Example: If you are under 17 and are not accompanied by your legal guardian, then you cannot see the R-rated movie. Let: r You can see the R-rated movie u You are under 17 a You are accompanied by your legal guardian Translation: (u a) r Note: The above translation is the contrapositive of the translation from example 1! Logic also helps us understand bitwise operations Computers represent data as sequences of bits E.g., Bitwise logical operations are often used to manipulate these data If we treat 1 as true and 0 as false, our logic truth tables tell us how to carry out bitwise logical operations
13 Bitwise logic examples Group Exercises Problem 1: Translate the following sentences If it is raining then I will either play video games or watch a movie You get a free salad only if you order off of the extended menu and it is a Wednesday Problem 2: Solve the following bitwise problems
14 Final Thoughts Propositional logic is a simple logic that allows us to reason about a variety of concepts
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