In mathematics often (always) one proves that a property is true for the general element of a set. From that, one can say that that property is true for all the elements of that set.
Is that a principle of logic or set theory? What is it called? When I make this kind of reasoning, what should I think of?
Thanks, Luca
The Universal Generalization basically shows that if a particular property holds for an arbitrary element of a set, then it holds generally for any element of it. The reason this rule is valid is because we assume absolutely nothing about the element $x$ we pick up.
From Velleman's How to Prove It (2006), p.108:
To prove a goal of the form ∀x P(x):
Let $x$ stand for an arbitrary object and prove $P(x)$. The letter $x$ must be a new variable in the proof. If $x$ is already being used in the proof to stand for something, then you must choose an unused variable, say $y$, to stand for the arbitrary object, and prove $P(y)$.
Hence the chosen element must be arbitrary, that is, we express this by stating that the letter chosen to denote it must have no previous ocurrences in the proof (a new variable).
If you want to prove $\forall a\in S:P(a)$, you might proceed as follows:
Let $x$ be any element of set $S$, i.e. you assume that $x\in S$, where $x$ was not previously introduced as a free variable in your proof.
If you can subsequently prove that $P(x)$ is true, then you can generalize, $\forall a\in S: P(a) $ provided:
$a$ is does not occur in $P(x)$
No free variables occur in $P(x)$ that were introduced after the above assumption
No other assumptions were made after the above assumption that were not already discharged (i.e not still active).
For related methods of mathematical proof, you might have a look at the tutorial that is included with my proof software available free at my website.
