5.1: Logic Statements

Learning Objectives

Logic is the study of the methods and principles of reasoning. In logic, statement is a declarative sentence that is either true or false, but not both. The key to constructing a good logical statement is that there must be no ambiguity. To be a statement, a sentence must be true or false. It cannot be both. In logic, the truth of a statement is established beyond ANY doubt by a well-reasoned argument.

So, a sentence such as "The house is beautiful" is not a statement, since whether the sentence is true or not is a matter of opinion.

A question such as "Is it snowing?" is not a statement, because it is a question and is not declaring that something is true.

Some sentences that are mathematical in nature often are not statements because we may not know precisely what a variable represents. For example, the equation \( 3x + 5 = 10\) is not a statement, since we do not know what \(x \) represents. If we substitute a specific value for \( x\) (such as \(x = 4\)), then the resulting equation, \( 3x + 5 = 10\) is a statement (which is a false statement).

Statement

A statement is a sentence that is either true or false.

In logic, lower case letters are often used to represent statements such as \(p\), \(q\) or \(r\).

Example \(\PageIndex\)

The following are statements:

  1. Zero times any real number is zero.
  2. \(1+1 = 2.\)
  3. All birds can fly. (This is a false statement. How can you establish that?)

The following are not statements:

  1. Come here.
  2. Who are you?
  3. I am lying right now. (This is a paradox. If I'm lying I'm telling the truth and if I'm telling the truth I'm lying.)
Try It \(\PageIndex\)

Which of the following are statements?

  1. I like sports cars.
  2. \( 2+3=6\).
  3. Where are you?

Only b is correct since you cannot determine if a is true, and c is a question that is neither true nor false.

Negation

The negation of a statement is a statement that has the opposite truth value of the original statement.

Notation: \(\sim p\) (read: "the negation of \(p\)" or "not \(p\)")

The negation of a true statement must be a false statement and vice-versa. A simple statement can often be negated by adding or removing the word "not." A statement can also be negated by adding "It is not true that . (statement)" or "It is not the case that . (statement)."

Example \(\PageIndex\)

Find the negation of the following statements:

  1. \(p\): The car is red.
  2. \(q\): Homework is not due today.
Solution
  1. \(\sim p\): The car is not red. Note that \(p\) is true and \(\sim p\) is false.
  2. \(\sim q\): Homework is due today. Note that \(q\) and \(\sim q\) have opposite truth values. If \(q\) is true, then \(\sim q\) is false, or if \(q\) is false, then \(\sim q\) is true.

Logical statements are related to sets and set operations. Words that describe an entire set, such as “all”, “every”, or “none”, are called universal quantifiers because that set could be considered a universal set. In contrast, words or phrases such as “some”, “one”, or “at least one” are called existential quantifiers because they describe the existence of at least one element in a set.

Quantifiers

A universal quantifier states that an entire set of things share a characteristic.

An existential quantifier states that a set contains at least one element.

Something interesting happens when we negate a quantified statement. When we negate a statement with a universal quantifier, we get a statement with an existential quantifier, and vice-versa.

Negating a quantified statment

At least one A is not B.