Counterexamples and Flexible Reasoning ACET Reviewer: Lesson and Practice

TEACHER ABI ACET LOGICAL REASONING

Counterexamples and Flexible Reasoning

Test sweeping claims, revise assumptions, and choose the conclusion that still works when the situation changes.

One focused competency27 original questionsProgressive difficultyACET-style transfer

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Counterexamples and Flexible Reasoning

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One possible exception can defeat a universal claim

Claim: Every student who studies with music remembers more.Possible case: Mara studies with music but remembers less because the lyrics distract her.

The case does not prove what usually happens. It does prove that the word every is too strong.

A counterexample must obey the conditions of the claim and make its conclusion false.

DO IT FAST

Use CLAIM → BUILD → CHECK → REVISE

1. CLAIM: Find the strong word: all, every, none, must, always, never.
2. BUILD: Imagine one case that satisfies the claim’s condition.
3. CHECK: Can the stated result fail in that case?
4. REVISE: If a rule changes, keep only conclusions supported by the new rule.

To challenge “All A are B,” find one A that is not B.
To challenge “No A are B,” find one thing that is both A and B.

WORKED EXAMPLES

Three ways ACET-style items test flexibility

1. Break an absolute claim

“Every rectangle has four equal sides.” A 3-by-5 rectangle is a rectangle without four equal sides.

That single valid case defeats the universal statement.

2. Protect the exact conditions

To test “Every metal conducts electricity,” a plastic ruler is useless because it is not metal.

A supposed counterexample fails if it does not meet the original condition.

3. Update after a rule changes

Old rule: Only members may enter. New rule: Members and invited guests may enter.

“Nonmembers cannot enter” is no longer safe; an invited guest may enter.

GUIDED TRY

Test the claim without arguing from habit

If a number is divisible by 4, it is even. Therefore, if a number is even, it is divisible by 4.

1. The conclusion reverses the original condition.
2. Find an even number.
3. Check whether it must also be divisible by 4.

Use 6. It is even but not divisible by 4, so it is a counterexample to the reversed claim.

COMMON TRAPS

What does not count as a counterexample?

Irrelevant case: It fails the condition before testing the result.

Another supporting example: It agrees with the claim instead of breaking it.

Unverified possibility: It merely says “maybe not” without constructing a coherent case.

Changed wording: It attacks a stronger or weaker claim than the one given.

FIVE-FORM SKILL CHECK

Do you need Foundations or Core Practice?

CHOOSE YOUR PRACTICE

Build the skill in stages

Foundations

Recognize valid counterexamples.

Core

Test rules and conditional claims.

ACET Transfer

Update conclusions under changing constraints.

FRESH MASTERY CHECK

Can you decide without the scaffolding?

Score 5/5 to verify this competency. If not, a fresh set loads.

QUICK ANSWERS

Counterexamples and Flexible Reasoning FAQ

Does one counterexample really disprove a claim?

It disproves a universal claim such as “all,” “none,” or “always.” It does not establish what happens most of the time.

Must a counterexample be real?

It must at least be logically possible under the stated rules. In mathematical items, an actual value is usually strongest.

Can an example support a claim without proving it?

Yes. Many confirming cases still cannot prove a universal statement, but one valid exception can refute it.

What is flexible reasoning?

It is the ability to update a conclusion when a premise, rule, category, or constraint changes.

What should I do when two options seem possible?

Check every word of the original condition. The correct option must satisfy the condition and produce the required logical effect.

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