The basics of propositional logic Section 1.3 The basics of propositional logic

Liars and Truthtellers Raymond Smullyan has written a number of wonderful books filled with challenging logic puzzles. A recurring theme in these books is an Island where each inhabitant is either a Liar or a Truthteller. Here is an example: You come across two inhabitants of the island, let’s call them A and B. A says, “We are both telling the truth,” and B says, “A is lying.” Can you tell who is a liar and who is a truthteller?

Liars and Truthtellers To solve the puzzle, we consider in table form the four possibilities for the types of people A and B are: Scenario #1 A is a TruthTeller B is a TruthTeller Scenario #2 B is a Liar Scenario #3 A is a Liar Scenario #4

Liars and Truthtellers In each of these situations, we consider the two statements that were made. To keep from getting dizzy, we will initially make our assessment purely with regard to the actual statements and with no regard to who makes the statements. Both A and B are TruthTellers A is a Liar Scenario #1 A is a TruthTeller B is a TruthTeller Scenario #2 B is a Liar Scenario #3 Scenario #4

Propositional logic Propositional logic is essentially a shorthand for the expression of sentences along with some rules that govern when compound expressions are true or false. In the previous example, we might use the propositional variable p for the statement, “A is a truthteller,” and the propositional variable q for the statement, “B is a truthteller.”

Propositional logic Propositional logic is essentially a shorthand for the expression of sentences along with some rules that govern when compound expressions are true or false. The logical connectives Ù, Ú, and Ø stand for the English words “and,” “or,” and “not,” respectively. So the statement “A and B are both Truthtellers” can be expressed p Ù q, and the statement, “A is lying” can be expressed as Ø p.

Truth tables for formal propositions Using these shorthand conventions, we can write the table that allowed us to solve the previous problem in the following concise form: p q p Ù q Ø p T F

Who is the truthteller? Who is the liar? Now consider which person, A or B, made each statement. Consider each row in the table and determine if each outcome could represent the real situation.

Problem 2: Suppose that you meet three inhabitants, A, B, and C and that A says, “B or C is lying,” B says, “C is lying,” and C says, “A and I are both telling the truth.” Who if anyone is telling the truth. Set up a truth table to solve this problem.

Propositional logic Sasha is a soccer player who also plays chess. Practice Problem. Write each of the following statements in propositional logic using the propositional variable c for “Sasha is a chess player” and the variable s for “Sasha plays soccer.” Sasha is a soccer player who also plays chess. Sasha is a soccer player but has not learned chess yet. Sasha has time to participate in soccer or chess but not both.

Truth tables for compound statements The key to making a truth table is to analyze the truth of statements from simple to complex in each of the possible scenarios for the values of the propositional variables. Example. Give the truth table for the statement, “Ø (p Ú q)” p q p Ú q Ø (p Ú q) T F

Truth tables for compound statements Practice. Give the truth table for the statement, “(Ø p) Ù (Ø q)” p q Ø p Ø q (Ø p) Ù (Ø q) T F

Negations of formal propositions When a propositional statement is false, its negation is a true propositional statement. For example, if the statement “Sasha plays soccer and chess” is false, then the statement “Sasha doesn’t play chess or Sasha doesn’t play soccer” is true. An advantage of formal logic is that we can present formal rules for negating compound statements like this.

Negations of formal propositions The truth table examples we saw earlier give us one such rule for negation. The truth tables for the statements Ø(p Ú q) and (Ø p) Ù (Ø q) were identical.

Logical equivalence When two statements have exactly the same truth table, we say that the two statements are logically equivalent. The pair Ø(p Ú q) and (Ø p) Ù (Ø q) of logically equivalent statements is one of DeMorgan’s Laws, and it gives us a handy way to rewrite potentially clumsy negated statements.

Which of the following statements are true?

Before next time you should… Read Section 1.3 of the text completing all Practice Problems as you go. Practice truth tables by going to the Flash section of the Discrete Math website and completing all problems from Section 1.3. See if you have questions on any of the assigned questions from Section 1.3.