Help with this logic problem please!

[cintinative]

According to the publisher, this is invalid. 

 

I can't wrap my mind around why. Please help. TIA!

 

"Since all mothers are parents, and all parents have children, then all children have mothers."

[happypamama]

I am thinking that it's not invalid because it's not true, but because it's not a direct consequence of the first two statements.  All mothers being parents and all parents having children doesn't, according to the way it's written, lead to all children having mothers.  All mothers can be parents and have children, and all parents can have children, but those don't mean that ALL children have mothers.  SOME children could have mothers, but the sentence doesn't allow for the fact that SOME children might not have mothers.  We know that all children have mothers because we know basic biology, but that statement doesn't make that true.

 

If it were written, "Since all beeps are dots, and all dots have squirts, then all squirts have beeps," you would say, "Well, no.  There could be some squirts that don't have beeps, even if the first two statements are true."  At least I think that's what it means.

[cintinative]

That sort of makes sense. I think it is the biology bit that is tripping me up.  The beeps and dots argument makes sense.

 

For example, this statement was valid (and is the question before the one I asked about):

 

"Since all mothers are parents, and all parents have children, therefore, all mothers have children."

[Pegs]

It's checking your understanding of the law of syllogism.

 

m => p,

 

and

 

p => c

 

Therefore,

 

m => c.

 

But see how there's no arrow leading from c to anything else? This means we can't conclude "Children have...".

[cintinative]

pegs, yes, that helped me a lot.

[happypamama]

Pegs has a much better way of putting it than I do. :)

[Caroline]

It says all mothers are parents and all parents have children, but it doesn't say that all parents are mothers or that all children have parents.

[EKS]

"Since all mothers are parents, and all parents have children, then all children have mothers."

 

Not all parents are mothers.  Not all children have parents.

[Bluegoat]

It's tricky because we know that in fact, all children do have mothers, and we tend to forget to separate the idea of a parent and a mother in the second  statement - we intuitively think of "parents" as "a mother and a father".  But that is not part of the statement.

 

Although - it's interesting - I wonder if we could say that in fact, "parents" does necessarily mean, at least with humans, a mother and a father?

 

But that is rather outside of what they are trying to demonstrate!

[regentrude]

"Since all mothers are parents, and all parents have children, then all children have mothers."

 

I think the problem is that we know all children have mothers as a true biological fact, and it is hard to separate the truth of that from the faulty logic of the statement.

Substitute something else:

 

"Since all poodles are dogs, and all dogs have four legs, then all four legged creatures are poodles." is obviously not correct.

[Farrar]

I'm curious what people think of the purpose of the problem. Obviously, it's to test the student's knowledge of syllogisms, which is clearer when the problem can be revealed through common sense or when it uses nonsense concepts like the example given above. But I wonder if it's also to push students to understand how logic will be incomplete without more complete information. If you start with limited information, your conclusions will also be limited in their scope, maybe even to the point of being misleading, which I think is the case here. Again, logically, it's not misleading because the conclusion is limited. We cannot say from this information that all children have mothers. But in an argument, it makes it sound to the casual listener like maybe some children don't have mothers or like the science on that is still out and can't be concluded, which would be a false assumption.