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Subsection 2.1.4 Using Operators to Build Complex Statements

In arithmetic, operators, such as \(+ \) and \(* \) have meanings. If we know the value of \(y\text{,}\) then we know the value of \(y + 2 \text{.}\) Similarly, the logical operators that we are about to define also have meanings. In particular, we will define them so that they correspond to natural ways of reasoning. For now, assume that they mean approximately what you think they mean. For example, “\(p\) and \(q\)” is true just in the case where both \(p\) and \(q\) are true. Similarly, “not \(p\)” is true only in the case where \(p\) is false. We will give precise meanings to all of our operators shortly (using a clever device called a truth table).

But, before we do that, we can notice that the job of all of the operators is to construct truth values from the truth value(s) of their operand(s). This means that every logical expression, no matter how many operators it contains, has a truth value. So it meets our definition of a statement. Thus:

  • Any logical expression that is formed by combining operands with operators is a statement. It may be either true or false. Using the definitions of the logical operators it contains, we can compute its truth value from the truth values of its operands

For example:

  • “Jim is tall” is a statement. It may be true or false.

  • “Joe is short” is a statement. It may be true or false.

So

  • “Jim is tall or Joe is short” is a statement. It is true if at least one of the two simple claims is true.

  • “Jim is tall and Joe is short” is a statement. It is true only if both of the two simple claims are true.

Big Idea

We can build logical expressions from operands and meaningful operators in much the same way that we build arithmetic expressions.

Exercises Exercises

1.

Which of the following expressions is not a logical statement?

  1. Rome is the capital of France.

  2. The sidewalks are wet and Paris is the capital of France.

  3. The sidewalks are wet and Rome is the capital of France.

  4. What is the capital of France?

Answer.
Correct answer: D
Solution.
Explanation: All but, “What is the capital of France?” have truth values (including F). It doesn’t. Questions don’t.

2.

Which of the following expressions is not a logical statement?

  1. The moon is made of lemon juice..

  2. The sidewalks are wet Paris is the capital of France.

  3. Dragons breathe fire or thunder.

  4. Not not Rome is the capital of Italy.

Answer.
Correct answer: B
Solution.
Explanation: “The sidewalks are wet Paris is the capital of France” is a list of two statements, but there is no operator connecting them so we don’t know what it means. To form one statement, we’d need to combine those two with an operator like and or or. Notice that the last one is a fine statement. It applies the operator not once to create a statement. Then it applies it a second time to that statement, producing yet another statement.

3.

Assume the world in which we live. Which of the following is not a true statement:

  1. Rome is the capital of France or Rome is not the capital of France.

  2. Paris is the capital of France.

  3. Paris is the capital of France and Paris is not the capital of France.

  4. Paris is the capital of France and Rome is the capital of Italy.

Answer.
Correct answer: C
Solution.
Explanation: We’ll see later that, for any statement p, the compound statement “p and not p” is never true.