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 states that are identically true or false? For instance, an element of X (i.e. a state of the ocean) is C:="the ocean contains more water than can fit in my coffee mug". Passing to PX, we get a singleton {C}. But because the state of C is always occupied, we have true implies {C}. In this model, that means {C} contains PX which seems absurd. What am I missing here?

Let X be the states of the Pacific ocean. How do we deal with states that are identically true or false? For instance, an element of X (i.e. a state of the ocean) is C:="the ocean contains more water than can fit in my coffee mug". Passing to PX, we get a singleton {C}. But because the state of C is always occupied, we have true implies {C}. In this model, that means {C} contains PX which seems absurd. What am I missing here?