Unit 9 Review and Checkpoint
A free Logic lesson from the “Quantifiers and Predicates” unit, with a worked example and practice problems including step-by-step solutions.
This checkpoint checks whether learners can reason with all and some without overclaiming. Learning objective: Review predicates, domains, all/some claims, and quantified negation. Prerequisite: Review the lessons in this unit before starting.. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: A truth-table question asks for cases; a counterexample question asks for one case that breaks a claim. Example 2: A validity question asks whether the conclusion must follow, not whether the sentences sound realistic. 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
- Review predicates, domains, all/some claims, and quantified negation
- Choose the reasoning tool that matches the statement
- Explain why an answer is valid, invalid, true, false, or unsupported
Worked example
Problem. Example case A (Unit 9 Review and Checkpoint): Worked example: Which rule or habit best matches Unit 9 Review and Checkpoint?
- Unit 9 Review and Checkpoint targets a specific reasoning habit.
- Unit 9 Review and Checkpoint focuses on review predicates, domains, all/some claims, and quantified negation.
- The other choices either overclaim or change the logical relationship.
Answer: Unit 9 Review and Checkpoint focuses on review predicates, domains, all/some claims, and quantified negation.
Practice problems
1. Practice case A (Unit 9 Review and Checkpoint): Which phrase signals a universal statement?
Choices: all · some · at least one · there exists
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case A (Unit 9 Review and Checkpoint): Which phrase signals a universal statement?
- For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
- Universal statements talk about every object in the domain.
- The word all signals that.
- Some and exists are existential.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: all
2. Practice case B (Unit 9 Review and Checkpoint): Which mistake is common when negating "All dogs bark"?
Choices: Writing 'No dogs bark' instead of 'At least one dog does not bark' · Changing all to every · Keeping the same topic · Looking for a counterexample
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case B (Unit 9 Review and Checkpoint): Which mistake is common when negating "All dogs bark"?
- For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
- The opposite of all is not none.
- To make all false, one counterexample is enough.
- No dogs bark is stronger than needed.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: Writing 'No dogs bark' instead of 'At least one dog does not bark'
3. Practice case C (Unit 9 Review and Checkpoint): "Some A are B" means:
Choices: At least one object is in both A and B · Every A is B · No A is B · Every B is A
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case C (Unit 9 Review and Checkpoint): "Some A are B" means:
- For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
- Some means at least one.
- For set diagrams, that object lies in the overlap.
- It does not mean all.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: At least one object is in both A and B
4. Practice case D (Unit 9 Review and Checkpoint): What is the negation of "Some triangles are equilateral"?
Choices: No triangles are equilateral. · All triangles are equilateral. · Some triangles are not equilateral. · At least one triangle is equilateral.
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case D (Unit 9 Review and Checkpoint): What is the negation of "Some triangles are equilateral"?
- For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
- Some means at least one.
- The exact opposite is that none exist.
- So no triangles are equilateral.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: No triangles are equilateral.
5. Practice case E (Unit 9 Review and Checkpoint): The sentence "It is not false that p" is equivalent to:
Choices: p · ¬p · p ∧ ¬p · unknown
Show solution
- Checkpoint Practice: First identify exactly what the question is asking: Practice case E (Unit 9 Review and Checkpoint): The sentence "It is not false that p" is equivalent to:
- Compare each choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- Not false means true in two-valued logic.
- The double negative cancels.
- So the statement is equivalent to p.
- Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.
Answer: p
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