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Statements vs. Non-Statements

A free Logic lesson from the “Foundations of Logical Thinking” unit, with a worked example and practice problems including step-by-step solutions.

A statement is a sentence with a truth value. Questions, commands, wishes, and fragments can be meaningful, but they are not statements in logic because they do not claim something true or false. Learning objective: Decide whether a sentence makes a claim that can be true or false. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: '7 is prime' is a statement because it can be true or false. Example 2: 'Is 7 prime?' is not a statement because it asks a question instead of making a claim. A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.

What you'll learn

Why it matters: Statements and non-statements appear in instructions, word problems, proofs, and computer conditions.

Worked example

Problem. Example case A (Statements vs. Non-Statements): Is "Every square has four equal sides." a logical statement?

  1. Worked Example: First identify exactly what the question is asking: Example case A (Statements vs. Non-Statements): Is "Every square has four equal sides." a logical statement?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. It makes a mathematical claim.
  4. A logical statement needs a possible truth value.

Answer: Yes, it makes a claim.

Practice problems

1. Practice case A (Statements vs. Non-Statements): Is "Every square has four equal sides." a logical statement?

Choices: Yes, it makes a claim. · No, it is only a command. · No, it is only a question. · No, statements cannot use numbers.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (Statements vs. Non-Statements): Is "Every square has four equal sides." a logical statement?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. It makes a mathematical claim.
  4. A logical statement needs a possible truth value.
  5. This sentence qualifies.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Yes, it makes a claim.

2. Practice case B (Statements vs. Non-Statements): Which is a compound statement?

Choices: x is positive and x is even. · x is positive. · What is x? · Solve for x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case B (Statements vs. Non-Statements): Which is a compound statement?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. A compound statement joins simpler claims.
  4. The word and connects two claims.
  5. So the first choice is compound.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x is positive and x is even.

3. Practice case C (Statements vs. Non-Statements): If p means "the number is even," write the symbolic form of "not p."

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case C (Statements vs. Non-Statements): If p means "the number is even," write the symbolic form of "not p."
  2. For data questions, identify what each statistic measures before calculating so the result matches the question.
  3. The symbol ¬ means not.
  4. Place ¬ before the statement letter.
  5. The symbolic form is ¬p.
  6. Check the result by substituting or estimating: the response should match ¬p and make sense in the original problem.

Answer: ¬p

4. Practice case D (Statements vs. Non-Statements): Which sentence has an unknown truth value because the variable has not been specified?

Choices: x is greater than 10. · 10 is greater than 5. · Close the book. · What is 10?

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case D (Statements vs. Non-Statements): Which sentence has an unknown truth value because the variable has not been specified?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The sentence with x makes a claim.
  4. Its truth depends on the value of x.
  5. So it is unknown until x is specified.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x is greater than 10.

5. Practice case E (Statements vs. Non-Statements): Which phrase usually signals a conclusion?

Choices: therefore · because · given that · assume

Show solution
  1. Therefore points to what follows.
  2. Because and given that often introduce reasons.
  3. So therefore is the conclusion signal.

Answer: therefore

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