Logic

Statements, Connectives, and Quantifiers In symbolic logic, we only care whether statements are true or false – not their content. In logic, a statement or a proposition is a declarative sentence that is either true or false. We often represent statements by lowercase letters such as p, q, r or s.

Examples of statements: Today is Wednesday Today is Wednesday = = 4 Kobe Bryant will play in the Super Bowl Kobe Bryant will play in the Super Bowl Chris Bosh plays for the Toronto Raptors Chris Bosh plays for the Toronto Raptors

Examples of non-statements: What day is today? - this is not a declarative statement – it is a question. Come here – this is a command. This statement is false – this is a paradox. It cannot be either true or false. Why? That was a good movie – this is an ambiguous sentence – there may be no agreement on what makes a movie good.

Try Classify each of the following sentences as a statement or not a statement When is your next class? That was a hard test! Four plus three is eight Gordon Campbell is a great Premier Vancouver is the capital of BC Is Vancouver the capital of BC?

Truth Value Because we deal only with statements that can be classified as “” or “”, we can assign a to a statement. Because we deal only with statements that can be classified as “true” or “false”, we can assign a truth value to a statement p. We use T to represent the value “true” and F to represent the value “false”.

Compound Statements A compound statement may be formed by combining two or more statements or by negating a single statement. Example: Today is not Tuesday. Nanaimo has no mayor but Victoria has one..

Connectives The words or phrases used to form compound statements are called connectives. Some of the connectives used in English are: Or; either…or; and; but; if…then.

Compound Statements Decide whether each statement is compound. 1. If Jim wrote the test, then he passed The car was fixed by Jack and Jill. 3. He either brought it to your house or he sent it to school Craig loves Math and Psychology.

Connectives The connectives used in logic generally fall into five categories: Negation Conjunction Disjunction Conditional Biconditional

Negations The negation of a given statement p is a statement that is true when p is false and is false when p is true. We denote the negation of p by ~p. Example:- p: Victoria is the capital of BC ~p: Victoria is not the capital of BC.

Connectives Negation Example:

Quantifiers A quantifier tells us “how many” and fall into two categories. Universal quantifiers Existential quantifiers

Quantifiers

Negating Quantifiers Suppose we want to negate the statement “All professional athletes are wealthy.” (universal) Correct: “Some athletes are not wealthy” or “Not all athletes are wealthy.” (both existential) Incorrect: “All athletes are not wealthy.” (universal)

Negating Quantifiers Negate the statement “Some students will get a scholarship.” (existential) Correct: “No students will get a scholarship.” (universal) Incorrect: “Some students will not get a scholarship.” (existential)

Negating Quantifiers Write a negation of each statement. 1.The flowers are not watered. 2.Some people have all the luck. 3.Everyone loves a winner. 4.The Olympics will start on 12 th February. 5.Everybody loves somebody sometime. 6.All the balls are red. 7.Some students did not write the test.

Negating Inequalities

Connectives Example:

Connectives Example:

Connectives DEFINITION Conditional A conditional or an implication expresses the notion of if … then. We use an arrow, , to represent a conditional. Example: Suppose that p represents “The Raptors win the NBA Playoffs” and q represents “Bosh will win the MVP”. We would read the statement p  q as “If The Raptors win the Play- offs, then Bosh will win the MVP. Write the statement “If The Raptors do not win the Playoffs, then Bosh will not win the MVP symbolically. ~p  ~q

Connectives - Conditional We read p  q as “p implies q” or “if p, then q”. In the conditional p  q, the statement p is the antecedent or hypothesis, while q is the consequent or conclusion.

Connectives - Conditional Examples: If it rains, then I carry my umbrella. If US President Obama comes to the Olympics, then security will be tight. If the package does not arrive today, I will call to find out why.

Connectives - Conditional Examples: Statement: All equilateral triangles have acute angles. If-then form: If a triangle is equilateral, then it has acute angles.

Connectives - Conditional Try: Write each statement in if-then form. 1. “Winners never quit” 2. It is difficult to study when you are distracted. If you are a winner, then you never quit. If you are distracted, then it is difficult to study.

Connectives - Conditional Givenp: “I have a Math test” q: “I do not have time for breakfast” Write the following statements in symbolic form: 1. If I have time for breakfast, then I have a Math test. 2. If I have a Math test, then I do not have time for breakfast. 3. If I don’t have a Math test, then I have time for breakfast.

Truth Tables Truth tables are often used to show all possible true-false patterns for statements. For example, when statement p is true, ~p is false and when p is false, ~p is true. Thus the truth table for negation is given by Statement p Negation ~p TF FT

Truth Table for the Conjunction p and q Let us decide on the truth values of the conjunction. Recall that a conjunction represents the idea of “both”. The compound statement = 4 and Victoria is the capital of BC is true because each component statement is true. However, the compound statement = 4 and Vancouver is the capital of BC is false, even though part of the statement is false.

Truth Table for the Conjunction p and q For the conjunction to be true, both p and q must be true. The truth table for the conjunction is given below. p and q Pq TTT TFF FTF FFF

Truth Table for the Disjunction p or q Let us decide on the truth values of the disjunction. Recall that a disjunction represents the idea of “either”. The compound statement: I have a quarter or I have a dime is true whenever I have either a quarter, a dime, or both. The only way this disjunction could be false would be if I had neither a quarter nor a dime.

Truth Table for the Disjunction p or q For the disjunction to be false, both p and q must be false. The truth table for the disjunction is given below. p or q Pq TTT TFT FTT FFF

Truth Tables Example: Let p: “5 > 3” and q: “ - 3 > 0”. Find the truth value of:

Truth Tables Example: Suppose that p is false, q is true and r is false. What is the truth value of the compound statement (Parentheses first)? 1. 2.

Truth Tables Example: Let p: “3 > 2”, q: “ 5 < 4” and r: 3 < 8. Determine whether each statement is true or false ~ p is false so the and is false. p is true and q is false so is false and hence the statement is TRUE.

Constructing Truth Tables Consider the statement. a) Construct a truth table. b) Suppose both p and q are true. Find the truth value of the statement. We begin by listing all possible combinations of truth values for p and q. We then list the truth values of ~p pq ~p TTF TFF FTT FFT

Constructing Truth Tables pq ~p TTF TFF FTT FFT Now we use the columns for ~p and q along with the and truth table to find the truth values of. pq~p TTFF TFFF FTTT FFTF

Constructing Truth Tables Finally, we include a column for ~q and use the or truth table to combine with ~q. pq~p TTFF TFFF FTTT FFTF pq ~q TTFFFF TFFFTT FTTTFT FFTFTT b) When both p and q are true, the statement is FALSE.

Constructing Truth Tables TRY: Construct truth tables for a) b) What do you observe? They have the same truth values. pq TTF TFF FTF FFT

The Lady or the Tiger

The Island of Questioners

Equivalent Statements Logically equivalent statements express the same meaning.

DeMorgan’s Laws