# Does this fit as a counterexample?

Still learning about conditional statements, I'm now learning about Proofs in the same subject from here: How to make this hypothesis in a negative form?

It's given that I should rewrite the statements these ways, where P is the first half of the statement and Q is the other half:

• A. P -> Q
• B. Q -> P
• C. P <-> Q

And the given statement is:

It is an eagle, it's a bird.

So after some rewriting, it is given that I should make a counterexample of the statements if they are false. Both B and C are false, so I have to rewrite it with some counterexample.

So I rewrote B as:

If it's a bird, then it could be an eagle.

As to the previous:

If it's a bird, then it's an eagle.

I'm not sure if my counterexample fits well with the example statement. What other ways can I write my counterexample?

If it's a bird, then it's an eagle.

is a way writing of B in English. It's not the only way. Other possible answers could be

It is a bird implies that it is an eagle

or

All birds are eagles.

Note that "If it is a bird then it could be an eagle" is incorrect. It doesn't express Q -> P since P is "it is an eagle" not "It could be an eagle".

A counterexample is a specific bird that proves that this is not true • I'm trying to figure what could fit for "If it's a bird, then it ______." Any suggestions? – Water Water Apr 9 at 4:34
• What do you mean? A counterexample is an example that proves a general fact wrong. It needs to be specific. "'A duck' is a counterexample to the claim that all birds are eagles." – James K Apr 9 at 4:43
• Of course you still need to prove that this duck is a bird, and that this duck isn't an eagle. That is left as an exercise for the reader. – James K Apr 9 at 4:46

I'm not entirely sure I follow what you're saying, but here's how I would go about it:

A. If it is an eagle, then it is a bird. (E implies B)
B. If it is a bird, then it is an eagle. (B implies E)
C. A bird is the same thing as an eagle. (E implies B and B implies E)

(I'm not sure I have C right; it's been a while since I took a logic course. I understand the ⇔ operator to mean that the two statements are logically equivalent, is that right?)

A counterexample is not a rewriting of the statement; it is a real-world example that disproves the statement. So I would say:

Proposition A is correct.
A counterexample to B is: A pigeon. A pigeon is a bird, but is not an eagle. Therefore to say "If it is a bird, it is an eagle" must not be correct.
I believe the same counterexample could apply to C.

• My problem is what I could use to make a counterexample for B and C, because they seem to be false. My answer in C would be "It's an eagle if and only if it's a bird." – Water Water Apr 9 at 4:37
• @Water, a thing must be false for there to be a counterexample of it. What exactly are your instructions? A counterexample is not a rewriting of a statement, it is a thing (an example) that fits the structure of the statement but not the logic. – randomhead Apr 9 at 4:40
• Rewrite the following statements given the illustrations (which are already listed in the question). If it's false, give a counterexample. – Water Water Apr 9 at 4:42
• @Water, you must first rewrite each statement, and then — if the statement is not actually true — provide an example that disproves the statement. – randomhead Apr 9 at 4:43