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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.
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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 • 2 + 2 = 4 • Kobe Bryant will play in the Super Bowl • 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 “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. • If Jim wrote the test, then he passed. • The car was fixed by Jack and Jill. • 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
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. • The flowers are not watered. • Some people have all the luck. • Everyone loves a winner. • The Olympics will start on 12th February. • Everybody loves somebody sometime. • All the balls are red. • Some students did not write the test.
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 thatprepresents “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. “Winners never quit” 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 Given p: “I have a Math test” q: “I do not have time for breakfast” Write the following statements in symbolic form: If I have time for breakfast, then I have a Math test. If I have a Math test, then I do not have time for breakfast. 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
Truth Table for the Conjunctionp and q Let us decide on the truth values of the conjunction . Recall that a conjunction represents the idea of “both”. The compound statement 2 + 2 = 4 and Victoria is the capital of BC is true because each component statement is true. However, the compound statement 2 + 2 = 4 and Vancouver is the capital of BC is false, even though part of the statement is false.
Truth Table for the Conjunctionpandq For the conjunction to be true, both p and q must be true. The truth table for the conjunction is given below.
Truth Table for the Disjunctionporq 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 Disjunctionporq For the disjunction to be false, both p and q must be false. The truth table for the disjunction is given below.
Truth Tables Example: Let p: “5 > 3” and q: “ - 3 > 0”. Find the truth value of: 1. 2. 3.
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” andr: 3 < 8. Determine whether each statement is true or false. 1. 2. 3. ~ 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 . Construct a truth table. 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
Constructing Truth Tables Now we use the columns for ~p and q along with the and truth table to find the truth values of .
Constructing Truth Tables Finally, we include a column for ~q and use theor truth table to combine with ~q. 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.
Equivalent Statements • Logically equivalent statements express the same meaning.
