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I am learning about proof construction. Please let me know what is wrong with this proof?

To be proved: There is a bird which can not fly.

Proof:

Every flying object has wings (Because flying is due to wings).

Every bird has wings. (It will not be a bird otherwise)

But wings don't imply flying (for example chicken or duck)

Every bird can not fly. QED

DivineCoder
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    Ducks fly (unless actively prevented by a human action). However, ostriches and penguins do not. –  May 15 '17 at 12:58
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    Just because wings don't imply flying doesn't mean there is a winged animal that doesn't fly. Are you not allowed to use a counter example? "Penguins don't fly, penguins are a bird, therefore not all birds fly. QED" – Kaynex May 15 '17 at 12:58
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    This is a bit of a strange question, imho, because the only valid proof here would be to simply provide an example of a bird that doesn't fly (e.g. an ostrich). Also, FYI, Ducks and Chickens can both fly! – Thomas Russell May 15 '17 at 12:58
  • helicopters don't really have wings. – Dando18 May 15 '17 at 12:59
  • The negation of "All birds can fly" is "Some bird cannot fly" (e.g. chickens) and not "Every bird can not fly" (clearly, eagles fly). – Mauro ALLEGRANZA May 15 '17 at 12:59
  • On behalf of ducks, I must protest. Ducks certainly fly. Some, like wood ducks, fly quite well. I have seen chickens fly, but I'll agree that it isn't their strong suit. – lulu May 15 '17 at 13:03
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    The argument: "Every flying animal has wings. Every bird has wings. Therefore: Every bird flies." is not valid exactly for the counter-example with chickens. Thus, "Every $A$ is $B$. Every $C$ is $B$. Therefore: Every $C$ is $A$." is not a valid syllogism. – Mauro ALLEGRANZA May 15 '17 at 13:04
  • @Dando18 Helicopters are often said to have rotary wings. But rockets fly without wings. In fact, according to one definition of "fly," any object that can be detached from the ground can fly, given a sufficient initial upward impulse, just not for very long. – David K May 15 '17 at 13:09
  • https://en.wikipedia.org/wiki/Mallard#/media/File:Flying_mallard_duck_-_female.jpg – skyking May 15 '17 at 13:09
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    @MauroALLEGRANZA I got the point.

    And ducks are making trouble here. Let's stick with the ostrich.

     – DivineCoder May 15 '17 at 13:10

1 Answers1

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In this proof attempt, your first two premises don't contribute anything to the conclusion. When you mention that there are birds that can't fly, you provide a counterexample, which is all you need to show that a general statement is not true. As mentioned in the comments, though, your counterexample should be some kind of bird that really is flightless.

Example: disprove the statement "All natural numbers are even".

Proof: The number $5$ is a natural number, and it is not even. Therefore, not all natural numbers are even.

G Tony Jacobs
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