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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

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    $\begingroup$ Ducks fly (unless actively prevented by a human action). However, ostriches and penguins do not. $\endgroup$
    – user228113
    Commented May 15, 2017 at 12:58
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    $\begingroup$ 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" $\endgroup$
    – Kaynex
    Commented May 15, 2017 at 12:58
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    $\begingroup$ 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! $\endgroup$ Commented May 15, 2017 at 12:58
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    $\begingroup$ 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. $\endgroup$ Commented May 15, 2017 at 13:04
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    $\begingroup$ @MauroALLEGRANZA I got the point. And ducks are making trouble here. Let's stick with the ostrich. $\endgroup$ Commented May 15, 2017 at 13:10

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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.

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