Formal reason why proof by arbitrary element works.

Question

When we prove $\forall x \in D, P(x),$ we often do it so by taking an arbitrary element of $D,$ and proving for it $P(x).$

I was thinking for more of a formal reason why the arbitrary method works. I know the verbal reason why arbitrary works: arbitrary element is a generic element of the set and nothing extra is assumed for it; it represents the elements of the set. But I wanted a more concrete reason.

I think that when we prove by the arbitrary method, what we are indeed doing is actually proving the implication $(x \in D \implies P(x))$ true. To prove this implication, we suppose an $x$ of $D,$ and just from that fact, we deduce $P(x).$ The main highlight is that our the hypothesis is quite simple $-$ that is to say, the hypothesis demands that $x$ just needs to be an element of $D,$ and no special conditions are required for $x$ other than that it has to be an element of $D.$

So, we have $(x \in D \implies P(x))$ true. Now, we come to set $D.$ It is true that every element of $D$ belongs to $D.$ This means that every element of $D$ satisfies the hypothesis of $(x \in D \implies P(x)),$ and because the implication is true, we can therefore deduce $P(x)$ for every element of $D.$ I think that's the reason why the arbitrary generalizes to whole set.

It leads me to think that when we say let $x$ be an arbitrary element of $D,$ we are actually supposing the simple hypothesis $x \in D$ of the implication $(x \in D \implies P(x)).$

I agree with Bram28 that my take above is "non-formal."

I'd greatly appreciate rectifications !

Answer 1

I am not sure how what you do is helping matters much:

It seems to me that what you really want to show is $\forall x (x \in D \to P(x))$. And I also note that you just provided a non-formal 'proof' for exactly this in your post. That is, you effectively said: Let $x$ be a completely arbitrary element, and let's show that for that element $x$ it is true that if $x \in D$, then $P(x)$. So ... you end up making a non-formal justification for the validity of the universal proof technique ... maybe you should try to formalize that?

Maybe you can start to see the futility in keep asking for a formal proof, as that will just ;lead to an infinite regress. At some point we're just going to have to refer to our logical intuitions and say that it is exactly those that are made hard by some formalization ... so that formalization has no further formal justification.

Answer 2

In predicate logic "syntax", the statement will only be $\forall x P(x)$

The "semantics" of predicate logic, require you to define a structure, which has a "universe" in which all possible $x$ reside. An example universe is $\mathbb{R}$ - the real numbers.

If $D$ is your universe, then $\forall x P(x)$ suffices.
If not, then it will be something like $\forall x ( \in D (x) \implies P(x) )$, where "$\in D $" is the unary predicate, true only for $x$ that belong to $D$.