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I took an interview last day. I remember one question:

If we know "mouse is a toy", "toys are funny", "some toys is harmful". Then which of the following propositions not necessarily true? (Some means at least one exists)

  1. Some of the mice are harmful.

  2. Some of the funny toys are harmful.

  3. All harmful mice are funny.

  4. Some mouse is funny.

Short answer solution is say (I) is true. could describe it for me, anyone?

wythagoras
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user3661613
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Suppose there are two toys: mice and rats.

Rats are harmful (since they can bite very badly (just kidding)). So there are some toys that are harmful. So it can be possible that there is no mouse that is harmful. Therefore, (I) is false.

Some of the toys are harmful

All toys are funny toys

Hence some of the funny toys are harmful. Hence, (II) is true.

All harmful mice are mice (implicit)

All mice are toys

All toys are funny

Hence all harmful mice are funny. Hence, (III) is true.

All mice are toys

All toys are funny

Hence all mice are funny, hence some mice are funny. Hence, (IV) is true.

Note: If it is possible that there are no mice, then this can be false. It depends on the conventions that you follow whether the statement "Every mouse is a toy" implies that there is at least one mouse.

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wythagoras
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    Are you sure you can conclude from the problem statement that there is at least one mouse? – awkward May 15 '16 at 12:11
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    @awkward Actually, this kind of logic originally is form Aristotle, and hence is more philosophical. One of the logic schools states that a statement "a mouse is a toy" implies that there is at least one mouse; the other states that it doesn't imply that. See https://en.wikipedia.org/wiki/Syllogism#Existential_import – wythagoras May 15 '16 at 17:37