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.
Answer
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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