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Subsection 1.2.3 Statements: The Basic Building Blocks

A logical statements is an expression that has truth value. In other words, if we assume a particular world, it is either true or false. But not all English sentences correspond to logical statements.

Activity 1.2.6.

Here are some statements:

  1. The Earth revolves around the sun.

  2. Paris is the capital of France.

  3. The moon is made of green cheese.

Let’s assume the world we live in. (We can consider the logical possibility of other worlds later.) Then 1 and 2 are true. 3 is false.

But not all English sentences correspond to logical statements.

Activity 1.2.7.

Here are some fine English sentences that don’t correspond to logical statements:

  1. What does the Earth revolve around?

  2. Get off my grass!

  3. Why is grass green?

1 and 2 are questions. Particular answers to them will be statements, but the questions themselves are neither true nor false. 3 is a command and is neither true nor false.

Typically only declarative statements (not questions or commands) correspond to logical statements.

And recall (from our previous discussion of the bald king of France and John, the possible wife beater) that figuring out what logical statement actually corresponds to a particular English declarative sentence is not always obvious.

Problems 1.2.8.

(a)

Consider, “All purple unicorns live in Texas.” It:

  1. is a statement.

  2. is not a statement.

Answer.
i is correct.
Solution.
Explanation: This sentence is a statement because it is either true or false (depending on whether there are any purple unicorns living outside of Texas).

(b)

Consider, “Do you know any purple unicorns?” It:

  1. is a statement.

  2. is not a statement.

Answer.
ii is correct.
Solution.
Explanation: This sentence is not a statement because it asks a question. It is neither true nor false (although a proposed answer would be).

(c)

Consider, “Paint all the purple unicorns pink.” It:

  1. is a statement.

  2. is not a statement.

Answer.
ii is correct.
Solution.
Explanation: This sentence is not a statement. It is a command that is neither true nor false.