I suppose that the definition of state as "a way that a system can be" isn't very clear to me. According to the wikipedia article you linked, it seems like a state space is defined to be the image of some function. But as no function or even codomai…
I'm having difficulty with the idea of regarding a set X as a set of states for some system and PX propositions of that system. Perhaps I'm pushing the analogy too far, but here goes.
Let X be the states of the Pacific ocean. How do we deal with s…
Here are two more examples of preorders. Take your favorite set X.
1) For all x in X, setting \(x \leq x\) gives you a preorder.
2) For all x,y in X, setting \(x \leq y \) gives you a preorder.