Consider the statement "I'm moving". It seems to me that this statement can be both true and false. That is, because motion is relative, I may not be moving relative to the Earth (i.e. standing still), but I'm always moving relative to the Sun, for example (or virtually any heavenly body for that matter). It looks as though the whole thing is a kind of a cheap trick, but what is it exactly?
1 Let's start with a simpler case first. While it is true that the statement
(a) "I am moving"
is true at some points in time and false at others, it doesn't mean that given any point t in time we have:
(b) "I am moving at time t " ∧ "I am not moving at time t ".
Given any time t, either you are moving at time t or you are not. The fact that there are times at which you are moving and that there are times at which you are not moving, does not contradict that fact.
2 Similarly, "I am moving" can be true with respect to some objects in space and false with respect to others. For example, if you're on a train, you're moving with respect to the external environment of the train, but are stationary with respect to the person sitting next to you. It doesn't follow from this that:
(c) "I am moving with respect to an object o " ∧ "I am not moving with respect to that o ".
Given any object o in space, either you are moving with respect to o or you are not. The fact that there are objects (e.g. the Sun) with respect to which you are moving and that there are objects (e.g. the Earth) with respect to which you are not, does not contradict that fact.
3 The impression that you have that there is some trick being played here might be caused by the fact that the context of utterance is often implicit. The context determines things such as the time and place of the utterance, the person uttering the sentence, and so on. Once the context is made explicit, it's possible to make formally precise the idea that the same sentence can have different truth-values depending on different context-sensitive factors, such as the time of utterance and so on.
Ending a sentence with "moving" is ending it on a preposition, i.e. the sentence is incomplete and what you are moving in relation to is implied or (hopefully) obvious from context. So yes, it is a question of the language being ambiguous.
To me, it seems that we are talking of language being subjective.
"It looks as though the whole thing is a kind of a cheap trick, but what is it exactly?" I think it is the dynamic of language and communication. Language is a tool for communication which implies a certain degree of common understanding between the speaker(s)/transmitter(s) and the listener(s)/receiver(s).
In the case of "I'm moving", the listeners are assumed to use the Earth as the reference.
Isn't every single word an artificial construct? And, they could even mean different things in different languages. If what we call a "leaf" were to be renamed to "fael" tomorrow, would a leaf cease to be what it is?