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
- Decide whether a sentence makes a claim that can be true or false
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Statements vs. Non-Statements): Worked example: Which rule or habit best matches Statements vs. Non-Statements?
- Statements vs. Non-Statements targets a specific reasoning habit.
- A statement must make a claim that can be true or false.
- The other choices either overclaim or change the logical relationship.
Answer: A statement must make a claim that can be true or false.
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
- 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?
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- It makes a mathematical claim.
- A logical statement needs a possible truth value.
- This sentence qualifies.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
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
- Warm-up: First identify exactly what the question is asking: Practice case B (Statements vs. Non-Statements): Which is a compound statement?
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- A compound statement joins simpler claims.
- The word and connects two claims.
- So the first choice is compound.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
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
- 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."
- Read the statement in plain English first, then match the symbol, connective, quantifier, or argument form to that meaning.
- The symbol ¬ means not.
- Place ¬ before the statement letter.
- The symbolic form is ¬p.
- Check the response against the original logical form: it should match ¬p without changing the meaning.
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
- 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?
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- The sentence with x makes a claim.
- Its truth depends on the value of x.
- So it is unknown until x is specified.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: x is greater than 10.
5. Practice case E (Statements vs. Non-Statements): Lesson focus: Which transfer task would show readiness for Statements vs. Non-Statements?
Choices: Classify a new sentence as a statement, question, command, or fragment. · Repeat the exact worked example without changing the context. · Pick an answer before identifying the claim parts. · Use a rule from another lesson because the words sound close.
Show solution
- A transfer task keeps the same objective but changes the surface context.
- Classify a new sentence as a statement, question, command, or fragment.
- That is a better mastery signal than memorizing a practice prompt.
Answer: Classify a new sentence as a statement, question, command, or fragment.
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