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Truth Values: True, False, and Unknown

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 has a truth value even when the learner does not know it yet. Unknown means not enough information has been provided; it does not mean false. Learning objective: Distinguish true, false, and unknown claims. 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: 'All squares are rectangles' is a claim; 'Draw a square' is a command. Example 2: In 'All squares are rectangles, and this is a square, so this is a rectangle,' the first two sentences are premises and the last sentence is the conclusion. 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: Clear reasoning helps students explain algebra steps, evaluate claims, and ask better questions when something does not follow.

Worked example

Problem. Example case A (Truth Values: True, False, and Unknown): Worked example: Which rule or habit best matches Truth Values: True, False, and Unknown?

  1. Truth Values: True, False, and Unknown targets a specific reasoning habit.
  2. Unknown means not enough information has been given; it does not mean false.
  3. The other choices either overclaim or change the logical relationship.

Answer: Unknown means not enough information has been given; it does not mean false.

Practice problems

1. Practice case A (Truth Values: True, False, and Unknown): Classify the sentence "The number 18 is divisible by 3."

Choices: Statement · Question · Command · Fragment

Show solution
  1. It makes a claim that can be checked.
  2. Logic starts by deciding whether a sentence makes a true-or-false claim.
  3. The best classification is Statement.

Answer: Statement

2. Practice case B (Truth Values: True, False, and Unknown): In the argument "All multiples of 6 are even. 24 is a multiple of 6. Therefore, 24 is even," which sentence is the conclusion?

Choices: 24 is even. · All multiples of 6 are even. · 24 is a multiple of 6. · 6 is even.

Show solution
  1. The conclusion is what the premises support.
  2. The word therefore signals the conclusion.
  3. So the conclusion is 24 is even.

Answer: 24 is even.

3. Practice case C (Truth Values: True, False, and Unknown): Which wording is most precise for a math claim?

Choices: Every integer greater than 2 is positive. · Numbers are usually big. · It works most of the time. · That answer feels right.

Show solution
  1. Precise claims say exactly which objects are included.
  2. Every integer greater than 2 is positive can be checked.
  3. The other choices use vague language.

Answer: Every integer greater than 2 is positive.

4. Practice case D (Truth Values: True, False, and Unknown): A valid argument is best described as one where:

Choices: the conclusion must follow if the premises are true · the premises are always short · the conclusion is probably popular · the words sound mathematical

Show solution
  1. Validity is about structure.
  2. Assume the premises are true and ask whether the conclusion is forced.
  3. That is the definition of validity.

Answer: the conclusion must follow if the premises are true

5. Practice case E (Truth Values: True, False, and Unknown): Lesson focus: Which transfer task would show readiness for Truth Values: True, False, and Unknown?

Choices: Decide whether a new claim is true, false, or still unknown from the information given. · 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
  1. A transfer task keeps the same objective but changes the surface context.
  2. Decide whether a new claim is true, false, or still unknown from the information given.
  3. That is a better mastery signal than memorizing a practice prompt.

Answer: Decide whether a new claim is true, false, or still unknown from the information given.

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