After finding the quantized character of charge, in 1891 George Stoney proposed the unit 'electron' for this fundamental unit of electrical charge. This was before the discovery of the particle by J.J. Thomson in 1897.
Well positive and negative charge are convention. What we call positive charge we could've called negative charge, and what we call negative charge we could've called positive charge. There are devices that detect charges, they use charged objects. Imagine I have a negatively charged ball. Remember that we could've called that ball positively charged, but the whole of the scientific community decided to call it negatively charged. If I bring a charge ball next to it, if it repels that charged ball is what we call negative. If it attracts then that charged ball is what we call positive. This convention goes down all the way to particles. In the end, someone just decided to say that electrons are negatively charged, and everything followed after.
So, no theorems are used to determine the nature of a charge. It similar like saying up/down, attrat/repel etc.
Positive and negative are just a mathematical way to express these kind of things