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 Example: