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.)
Best Answer
I like to think that mathematical truth is a mathematical model of "real world truth", similar in my mind to the way in which the mathematical real number line $\mathbb{R}$ is a mathematical model of a "real world line", and similarly for many other mathematical models.
In order to achieve the level of rigor needed to do mathematics, sometimes the description of the mathematical model has formal details that perhaps do not reflect anything in particular that one sees in the real world. Oh well! That's just the way things go.
So yes, the empty function is injective. It's a formal consequence of how we axiomatize mathematical truth.
And, by the way, yes, there are infinitely many primes. The classical proof by contradiction that you feel is a natural language proof and not really a "formal proof" is actually not very hard to formalize at all. Part of the training of a mathematician is (a) to use our natural intuition, experience, or whatever, in order to come up with natural language proofs and then (b) to turn those natural language proofs into formal proofs.