Converting statements with term 'only' and 'any' to predicate logic

Question

How to convert following statement into predicate logic?

1)"Only dogs are mammals"
2)"Any dog is a mammal"

Is there a difference between "Any dog is a mammal" and "All dogs are mammals" fro the view point of first order logic?

"Only dogs are mammals" becomes "every mammal is a dog." – goblin GONE Apr 14 '14 at 14:13 — goblin GONE, Apr 14 '14 at 14:13

score 2 · Accepted Answer · answered Apr 14 '14 at 14:12

2

The first is claiming that of all possible things or creatures, only dogs are mammals (I.e., anything other than a dog is not a mammal. I.e. all mammals are dogs.)

The second is indeed equivalent to "All dogs are mammals": i.e. "Considering all creatures, if any of them is a dog, then it is a mammal."

The second (unlike the first) does NOT rule out the fact that there may be non-dog creatures that are also mammals.

answered Apr 14 '14 at 14:12

amWhy

209,954

So here are my answers. Am I correct? 1)∀x (mammmal(x) -> dog(x) ) 2)∀x (dog(x) -> mammal(x) ) – Samitha Chathuranga Apr 14 '14 at 14:20
Yes, that's correct! – amWhy Apr 14 '14 at 14:22
2

One has to be careful with "any". In ordinary mathematical English, it can sometimes mean "some" and sometimes mean "all". See http://math.stackexchange.com/a/509828/630 and the comments below it for some examples. – Carl Mummert Apr 14 '14 at 19:13
E.g. "My plan will handle animals in the zoo that are not striped. So if no animal in the zoo is striped, I am fine. But if any animal in the zoo is striped, my plan will fail." - in this sentence, "any animal in the zoo" is an existential quantifier, not a universal quantifier. – Carl Mummert Apr 14 '14 at 19:15
Good points, @CarlMummert. Thanks for pointing out the caution (and knowledge of context) needed when handling "any". – amWhy Apr 14 '14 at 19:48
@CarlMummert U re correct. Thanks for ur comments. – Samitha Chathuranga Apr 16 '14 at 04:23

Converting statements with term 'only' and 'any' to predicate logic

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