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 asserts something that is true or false, even when its truth is unknown to us. Commands and questions do not assert claims. An unassigned free variable creates an open sentence; assign it a value or bind it with a quantifier before treating it as a closed statement.
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. Which sentence makes a claim with a truth value?
- A false declarative claim is still a statement.
- 51 is odd, so the selected statement is false rather than not a statement.
Answer: 51 is an even integer.
Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Which sentence makes a claim with a truth value?
Choices: Is 21 even? · What an interesting number! · 21 is an even integer. · Calculate 21 + 2.
Show solution
- A false declarative claim is still a statement.
- 21 is odd, so the selected statement is false rather than not a statement.
Answer: 21 is an even integer.
2. Classify “Please list the first 10 primes.”
Choices: Universal theorem · Command; it has no truth value · True statement · False statement
Show solution
- An instruction requests an action rather than asserting a claim.
Answer: Command; it has no truth value
3. The sentence “The sealed envelope contains a blue card” refers to one particular envelope. How should it be classified before opening it?
Choices: A statement whose truth is not yet known to us · Not a statement because it is unobserved · Automatically false · A command
Show solution
- Whether we know the truth is different from whether the sentence makes a definite claim.
Answer: A statement whose truth is not yet known to us
4. Over the integers, classify “x > 10” when no x is assigned or quantified.
Choices: A true statement · A false statement · A question · An open sentence (predicate)
Show solution
- Its truth depends on the value assigned to x.
- For example, x=11 satisfies it while x=9 does not.
Answer: An open sentence (predicate)
5. Is “20 is even and 21 is odd” a true statement?
Choices: True · False
Show solution
- 20=2(10); 21=2(10)+1.
- Both conjuncts hold, so the conjunction is true.
Answer: True
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