Maybe it would help if we give concrete examples of empirical proofs and non-empirical proofs.
First example, dictionary definitions. Dictionary definitions are empirical evidence of how words are used and of what they mean.
So, to know what the word “empirical” means, we can look at the empirical evidence provided by a dictionary definition of what the word “empirical” means:
Second example of empirical evidence. Absence of evidence is evidence of absence. If you look into your bathroom and there is no evidence of any man-eating dragon, you take this as evidence there is no man-eating dragon. The few people who don’t are usually regarded as being insane, suggesting this is something most of us do and do even without thinking about it. It’s obvious, although again you’ll find people who don’t understand the idea.empirical
a. Relying on or derived from observation or experiment: empirical results that supported the hypothesis.
b. Verifiable or provable by means of observation or experiment: empirical laws.
2. Guided by practical experience and not theory, especially in medicine.
Third example of empirical evidence. My own subjective experience is empirical evidence. If I feel pain, I will infer there’s something wrong somewhere in my body. And again, I assume that’s something nearly all of us do. We all take the feeling of pain as evidence there’s a real something. And science helps us understand why. The feeling of pain is scientifically accepted as being in normal conditions the result of the perception of some actual state or condition present in your own body, so we can talk of pain as the feeling resulting from the perception of harm or injury, which is called “nociception” (and not “perception of pain” as some dictionaries put it).
Fourth example of empirical evidence. I can use my own logical intuitions as empirical evidence to work out a method of logical calculus. I could do it, but it seems clear, although I accept this is debatable, that most methods of logic ever published have been developed on the basis of the logical intuitions people have had, starting with Aristotle and the few philosophers who, prior to Aristotle, discussed specific logical relations (without necessarily calling them as such). An apparent counterexample would be the definition of material implication, at least to the extent that you take material implication as a logical relation (I don’t), by mathematicians at the beginning the 20th century (essentially Frege and Russell initially).
And we have empirical evidence that people routinely use their own logical intuitions as empirical evidence, for example in the descriptions mathematicians make of the way they rely on their own intuitions to prove mathematical conjectures true. As I understand it, every proof, by mathematician or by theorem prover, is ultimately based on the intuition of the specialists, even when they are ostensibly based on a set of logical truths, since logical truths have accepted as such since the Antiquity on the basis of the intuitions the specialists had and on that reported by other people. Such intuitions include for example the logical truth “If it is true that it rains and it is true that I am hungry, then it is true that it rains”. Tell me if you think this isn’t obviously true.
And then, non-empirical evidence. Here, I’ll give just two examples.
One, that you don’t see a dragon is empirical evidence, but it is not empirical evidence that there is an invisible dragon. And it may well be evident to you that there is an invisible dragon. So, this in itself would be empirical evidence. But, that you don’t see a dragon is not evidence that there is an invisible dragon.
Second, that there are people who believe in God is empirical evidence but not empirical evidence that God exists.