Daniel wrote:

> Let \\(X\\) be the states of the Pacific ocean. How do we deal with states that are identically true or false?

That doesn't make sense: states don't have truth values! It's propositions that have truth values.

States are _ways a system can be_. For example in the classical mechanics of a particle on the line, the set of states is taken to be \\(\mathbb{R}^2\\). A point in here is a position-momentum pair. The idea is that the particle can have any real number as its position and any number as its momentum, and the position and momentum tell us everything there is to know about the particle. We say they tell us its "state".

The set of states of some system is usually called its **state space**. The concept of state space is important not only in physics but also computer science, game theory... and pretty much everything where you have a "system" that has a set of "ways it can be":

* Wikipedia, [State space](https://en.wikipedia.org/wiki/State_space).

Now, about this:

> "the ocean contains more water than can fit in my coffee mug."

That's not a state. That's a proposition: something that can be true or false! In many situations, a proposition can be seen a subset \\(S\\) of the state space \\(X\\): in this case, all the states of the Pacific Ocean in which that ocean contains more water than can fit in my coffee mug. If the proposition is identically true, then \\(S = X\\). If it's identically false, then \\(S = \emptyset\\).

Note that every state \\(x \in X\\) gives a proposition, namely the singleton \\(\\{x\\}\\). You can think of this as the proposition "the system is in state \\(x\\)". But there are lots of _other_ propositions.

Warning: when we get to quantum mechanics we should generalize these ideas. One way is to replace the category of sets by some other category. Then the "state space" becomes an object \\(X\\) in that category, and propositions become elements of the poset of subobjects of \\(X\\). There is much more to say, but this is a start.

> Let \\(X\\) be the states of the Pacific ocean. How do we deal with states that are identically true or false?

That doesn't make sense: states don't have truth values! It's propositions that have truth values.

States are _ways a system can be_. For example in the classical mechanics of a particle on the line, the set of states is taken to be \\(\mathbb{R}^2\\). A point in here is a position-momentum pair. The idea is that the particle can have any real number as its position and any number as its momentum, and the position and momentum tell us everything there is to know about the particle. We say they tell us its "state".

The set of states of some system is usually called its **state space**. The concept of state space is important not only in physics but also computer science, game theory... and pretty much everything where you have a "system" that has a set of "ways it can be":

* Wikipedia, [State space](https://en.wikipedia.org/wiki/State_space).

Now, about this:

> "the ocean contains more water than can fit in my coffee mug."

That's not a state. That's a proposition: something that can be true or false! In many situations, a proposition can be seen a subset \\(S\\) of the state space \\(X\\): in this case, all the states of the Pacific Ocean in which that ocean contains more water than can fit in my coffee mug. If the proposition is identically true, then \\(S = X\\). If it's identically false, then \\(S = \emptyset\\).

Note that every state \\(x \in X\\) gives a proposition, namely the singleton \\(\\{x\\}\\). You can think of this as the proposition "the system is in state \\(x\\)". But there are lots of _other_ propositions.

Warning: when we get to quantum mechanics we should generalize these ideas. One way is to replace the category of sets by some other category. Then the "state space" becomes an object \\(X\\) in that category, and propositions become elements of the poset of subobjects of \\(X\\). There is much more to say, but this is a start.