Questions: All, Some, and None

Question 1

A student says 'All rectangles are squares.' Which of the following disproves this claim?

• A. Finding a rectangle that IS a square
• B. Finding a square that is NOT a rectangle
• C. Finding a rectangle that is NOT a square — like a 3x5 rectangle
• D. Finding a triangle

Answer: (C)

To disprove an 'all' claim, you need one counterexample — one rectangle that is not a square. A 3x5 rectangle is a rectangle (four right angles, opposite sides equal) but not a square (sides are not all equal). One counterexample is enough to make 'all rectangles are squares' false. Finding a rectangle that IS a square does not help — it supports the claim, it does not disprove it.

Question 2

In logic, 'some cats are black' is true even if all cats are black.

• T. True
• F. False

Answer: True

In logic, 'some' means 'at least one.' If all cats are black, then certainly at least one is — so 'some cats are black' is true. This is different from everyday speech, where 'some' often implies 'some but not all.' In logic, 'some' is the minimum claim: at least one exists. It does not exclude the possibility that all do.

Question 3

What is the fewest number of counterexamples needed to disprove the claim 'No dogs can swim'?

• A. You need to show that most dogs can swim
• B. You need to show that all dogs can swim
• C. Just one dog that can swim is enough
• D. You need to find at least ten swimming dogs

Answer: (C)

A 'none' claim says zero members satisfy the condition. Finding just one dog that can swim shows that the number is not zero, disproving the claim. You do not need to show that most or all dogs can swim — one counterexample is sufficient. 'None' claims are fragile in this way: a single exception breaks them.

Question 4

Explain why 'all' claims are easy to disprove but hard to prove, while 'some' claims are easy to prove but hard to disprove.

Model answer: An 'all' claim says every single member satisfies the condition. To disprove it, you only need one counterexample. But to prove it, you must check every member — which might be impractical or impossible. A 'some' claim says at least one member satisfies the condition. To prove it, you only need to find one example. But to disprove it, you must check every member and show that none satisfy it. The strength of the claim determines the difficulty: strong claims ('all,' 'none') are easy to attack but hard to defend; weak claims ('some') are easy to defend but hard to attack.

This asymmetry is fundamental to logic, science, and law. Scientific theories make 'all' claims ('all objects fall at the same rate in a vacuum') that can be disproved by one experiment. Legal defenses often use 'some' strategies ('there exists reasonable doubt'). Understanding quantifier strength is understanding the logic of evidence.