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I am taking a course on logical equations and I found this exercise while reading about proofs and how to prove a given sentence and what kind of mistakes usually occur when you are trying to prove something specific. Here is a problem:

You are asked to find the mistake in the following proofs. Each proof is given for the following statement: Prove that the sum of any two rational numbers is a rational number itself.

On the same exercise there is the following proof for the statement:

Two rational numbers sum up into a rational number when added. So if R and S are rational numbers then R+S is a rational number too. This completes the proof.

I know this proof is not correct because it is based on a fallacy of presumption.

However with this one:

Proof: We assume we have the rational numbers 1/4 and 1/2. The sum of 1/2+1/4 is 3/4 which is a rational number. This completes the proof.

I also know it's not a legit proof, I just can't figure out why.

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1  
I tend to jokingly call this "proof by example" –  AlexR Nov 15 '13 at 14:24
6  
“Every man wears glasses”. Proof: ”I wear glasses”. –  egreg Nov 15 '13 at 14:25
    
The second proof tries to prove a general statement by taking a single example, which is false. Like @egreg said, if you say "I wear glasses" that does not mean that every man wears glasses. You could use mathematical induction. –  Nickolas Nov 15 '13 at 14:49
    
Idea-wise, I find the second proof very close to a correct proof. –  Adam Nov 22 '13 at 1:09

2 Answers 2

up vote 2 down vote accepted

It uses a hasty generalization, a fallacy that looks like this: $$\exists x \exists yPxy\to\forall x\forall yPxy,$$ which is not valid.

Let $Pxy$ be "$x+y$ is a rational number".

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We cannot prove a generalized statement with a single case. So ${1\over 4}+{1\over 2}={3\over 4}$ is not a proof that the sum of two rational numbers is a rational number. We have to consider all rational numbers and show that the statement is true no matter which two rational numbers we sum.

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