# help with answering the set of statements : All cats are lions, some lions are mice, all mice are giraffes

Consider the three statements A,B and C to be true even if they are different from the commonly known facts

I drew two euler diagrams for this

and concluded that conlusion II and IV follows and hence option e.) is correct but in the book option d.) is given correct.

Which solution is the correct one? e) (mine) or d) (the book's)? And why?

• You are correct. Also conclusion III entails conclusion II, so option (d) doesn't make sense. Here's a counter-example to conclusion III: think of a world consisting only of 1 cat, 1 cat-lion and 1 mouse-giraffe-lion. This satisfies all three premises but conclusion III is false.
– E...
Commented Mar 18, 2016 at 22:27
• @Eliran H - Your example as stated doesn't work, because "1 cat" contradicts "all cats are lions". But if you just got rid of that one, and said the world consisted of only 1 cat-lion and 1 mouse-giraffe-lion, this would satisfy the premises but III would be false. Commented Mar 18, 2016 at 22:31
• Yeah I meant that (I edited the example a few times and missed it) :)
– E...
Commented Mar 18, 2016 at 22:33
• I would hazard a guess that the book contains a typo, and (d) was intended to read "II and IV". Commented Jun 22, 2016 at 14:22

I. doesn't follow. All cats are lions, but not necessarily all lions are cats. So there may be lions that are not cats. Some lions are mice, but they could be those lions that are not cats.

II. follows. If all mice are giraffes, and some lions are mice, then those lions that are mice must necessarily be giraffes, because all mice are giraffes.

III. doesn't follow. While, from II, some giraffes must be lions, they could perfectly be those lions that are not cats.

IV. follows. If all mice are giraffes, then some giraffes must be mice.

(all this supposes that there are any lions, cats, mice, and giraffes; if some or all of these sets are empty, then we would have a problem with equally empty referents, which would make the truth value of these statements more complicated.)

So, only II and IV follow. As this doesn't match any of a/b/c/d options, then the correct option is (e), none of the above.

III, "Some giraffes are cats" doesn't follow.

Assume there are just two animals:

Animal 1 is at the same time a cat, not a dog, a lion, not a mouse, not a giraffe. Animal 2 is at the same time a dog, not a cat, a lion, a mouse, and a giraffe.

The statements about cats, lions, mice and giraffes are all true, but no giraffe is a cat.

PS. Looks like the question has changed since this answer. With the question as it is now, "Some lions are mice" implies that there is at least one lion which is a mouse and therefore a giraffe, and as a consequence there is at least one mouse which is therefore a giraffe, so II and IV are true.

There is no reason why any mice should be cats, and even less reason why any giraffes should be cats, so I and III are false. "All cats are lions" doesn't even imply the existence of a single cat!

• The body of the question from the image remain unaltered from the first version... I've only changed the title of the question away from some thing completely meaningless. Commented Jun 22, 2016 at 8:42