I'm trying to wrap my head around the relationship between truth in formal logic, as the value a formal expression can take on, as opposed to commonplace notions of truth.
Personal background: When I was taking a class in formal logic last semester, I found that the most efficient way to do my homework was to forget my typical notions of truth and falsehood, and simply treat truth and falsehood as formal, abstract values that formal expressions may be assigned. In other words, I treated everything formally, which from the name of the course is presumably what I was supposed to do while solving problems.
On the other hand, I came out of the course more confused than enlightened about what truth refers to in mathematics. For example, on what levels are each of the following two statements "true"?
- There are infinitely many prime numbers.
- The empty function $f\colon \emptyset \to \mathbb R $ is injective.
For the first one, I can obviously see that there would be a contradiction if there were only finitely many prime numbers. To me, the classic proof by contradiction is not a "formal proof" or anything; it is merely a natural language argument that proves (in the everyday sense of the word "proves") why the statement must be true (in the everyday sense of the word "true").
On the other hand, I run into trouble when I try to think about the second one. The very concept of the "empty function" doesn't even feel like it makes sense, but if I think about it as the relation between $\emptyset$ and $\mathbb R$ containing no elements, and then try to write out the statement formally, I get (if I did it correctly)
$ \forall x \forall y ( ((x\in \emptyset) \land (y\in \emptyset) \land (f (x) = f (y))) \implies (x=y)) $
which I think has to be true in a formal sense (since the antecedent is always false?). But to be honest, I don't really know how to think about "truth" here; the situation feels much more confusing than with the first statement.
So, in conclusion, my questions are:
In what sense is each of the above statements "true"? And, more generally,
Is the notion of truth in (mathematical) logic just a formal value assigned to expressions? Or should I think of it as encompassing, but also generalizing, the intuitive notion of a true statement?
(Any insightful comments or answers are appreciated, even if they don't address all of my questions directly.)