What is wrong with this syllogism?
All candy is yummy.
No vegetables are candy.
Therefore, no vegetables are yummy.
This is not homework. Promise. This was presented in class as an example and it just doesn’t sound right.
Was this presented as a correct example, or as an example of an incorrect argument?
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It relies on the assumption that the only yummy food is candy -which is not established as fact.
^^^^^Same** invalid form **as:
All sheep are mammals.
No pigs are sheep.
Therefore, no pigs are mammals.
Also known as (or at least could be reformulated as) the Fallacy of the Inverse:
If candy, then yummy.
Vegetables are not candy.
Therefore, vegetables are not yummy.
Suppose, for example, that I find all candy yummy, and I find broccoli yummy. And, to be clear, I consider broccoli a vegetable, but don’t consider anything to be both a vegetable and a candy. Then both premises (all candy is yummy, and no vegetable is candy) are true, yet the conclusion (no vegetable is yummy) is false. That’s what’s wrong with it.
A correct formulation would be:
All candy is yummy
No vegetables are yummy
Therefore, no vegetables are candy
As presented, candy is a subset of “yummy things”, and while vegetables are not a subset of “candy” they could still be another subset of “yummy things”, so the conclusion is not demonstrably true.
There is no common middle term. To get a valid syllogism from the first proposition, the second one would have to be either about candy or about yummy things, but it isn’t, it is about vegetables.
This, though, is just another way of saying what the other replies say. The syllogism is clearly invalid. There are various ways to characterize its invalidity.
There is obviously something wrong with it.
What, then, are you seeking clarification on? If it’s already clear to you that this is not a valid argument form, what are you looking to hear in response to “What is wrong with this syllogism?”. What’s wrong with it is just that it’s not a valid argument form, in that it can take you from true premises to a false conclusion. There’s no reason to have thought it was a valid argument form, unless you advance one for us to pick apart.
Note also that a syllogism doesn’t need to be true to be valid. For instance,
All candy is yummy.
All vegetables are candy.
Therefore, all vegetables are yummy
is a valid syllogism, despite the second premise and the conclusion both being false.
This is logic. You can’t just say there is something wrong with it without giving a reason.
It has two premises and a conclusion.
It is in the form All A are B, C is A, Therefore C is B.
It has one universal premise A.
No, it quite manifestly isn’t.
And if it were an argument in that form, it would be valid. But what you actually have is: All A are B. C is ~A. Therefore C is ~B. That’s the crucial difference.
With negation, the argument needs to be: All A are B. C is ~B. Therefore C is ~A. And as you can clearly see, “All candy is yummy. All vegetables are not-yummy. Therefore vegetables are not-candy” is perfectly cromulent.
You can say “All members of this Class have this Attribute. X is a member of this Class, therefore X has this Attribute” and that’s fine. Or you can say “…X does not have this Attribute, therefore X is not a member of this Class.” That’s fine too. The other way around, not so much.
I think this is a good example of a syllogism that can be clearly illustrated with a Venn diagram, in this case showing how it’s faulty. I drew up a very rough one and posted it here: http://kelworthfiles.files.wordpress.com/2013/10/venn_syll.png
The first premise, ‘All Candy is Yummy’ blacks out two sectors of the diagram, the two that fall within ‘Candy’ and not within ‘Yummy’ (One is within ‘Veggies’ and one without.)
The second premise, ‘No veggies are Candy’ would black out both sectors where veggies and Candy overlap. One of these is blacked out already, so we also black out the triple overlap sector.
Now, if we’d proved the conclusion that ‘No veggies are yummy’, both sectors where Veggies and Yummy overlap would be blacked out. Instead, we see that there is still a theoretical possibility of a yummy veggie that is not candy. 
What’s throwing you is that you are taking A to be “candy”, B to be “yummy”, and C to be “no vegetables”.
But when people speak of the form “All A are B, C is A, Therefore C is B”, they don’t mean for “no vegetables” to be the sort of thing you can substitute for C [at least, not in the way this will end up being interpreted in conventional English, as a statement with the “no” serving as a top-level quantifier]. They only mean for you to place an actual object (or, if you like, collection of objects) in the role of C.
The form in which these syllogisms have been presented to you are not meant to allow for arbitrary textual substitution, with the results then interpreted as capriciously as ordinary language would lead them to be. The substitutions are to be at a semantic level, of objects or collections of objects.