Considerations relevant to tense logic go back to Aristotle, but modern tense logic begins with Arthur Prior, who recognized that you could have with tenses an analogue of alethic modal logics (with necessities and possibilities). The Diamond or weak modality operators are:
P: at some point in the Past
F: at some point in the Future
The Box or strong modality operators are:
H: Has always been so in the past
G: Going to always be so in the future
Thus you can get a 'tense logic' in two different directions from a reference point, with pastward strong and weak modalities and futureward strong and weak modalities. In each direction, many standard things for modal logic hold true, including that the relevant strong and weak modalities are interdefinable, e.g.,
Pp = ~H~p
Fp = ~G~p
It follows from this that for every modal logic you can name, you can give a tense interpretation of it. Of course, many of these are weird tense logics -- but, by definition, they are logically possible ways a kind of tense could work. In terms of our ordinary grammatical tenses, we need to have a logic that has strings of modal operators. For instance, getting something like
It had been the case that John went to the store
requires that we be able to talk about what is to the past of something in the past:
PP(John goes to the store).
We also, of course, need to be able to talk about what is to the future of what is in the past, what is to the past of what is in the future, what is in the future of what is in the future, and so forth. One of the earliest important results in tense logic proved that, if we are dealing with a time that is completely linear (no branches), all possible ways of talking about it can be reduced to fifteen combinations of two tense operators. There are thus fifteen possible tenses, if we think of time as being something like a line. Most languages do not use all fifteen. (They also often combine tense with other things, so not all grammatical tenses can be captured by a tense logic alone.)
But of course, we can have tense logic interpretation for any modal logic we want, and we don't have to think of time as linear. It's common to think of time as branching in at least one direction (usually the future). You can also think of many times. For instance, instead of thinking of time as a line, you could think of it as a plane or volume. Thus, in addition to the past, we could have an eckwise and andwise direction (to borrow terms from the short story, "The Dark Tower"). Every point in time would have a pastward, a futureward, an eckwise, and an andwise direction. The eckwise and andwise would work exactly like past and future, but would just not be to the past or to the future. We couldn't guarantee that they were perpendicular, which requires not just tense but a precise way to measure time, but we could recognize them as not all on one line.
This might seem rather silly. But in fact this is not very different from how spacetime works in relativity theory; it gets more complicated when you bring in precise measurements, but if we are only looking at tenses, thinking of several dimensions of time is not any different from thinking of spacetime. Spatial directions too have 'tenses' (forward and backward).
And, of course, in science fiction, we often find people treating counterfactual possibilities as alternative tenses. This is actually older than you might expect; the late medieval scholastics in discussing the logical operation of ampliation identified five logical tenses: past, present, future, possible, imaginable. In medieval logic, it's generally taken to be the case that propositions (or 'enunciations') can be true or false depending on the present moment they are said, but it was also recognized that we sometimes 'ampliate' (make wider) what we are considering. 'John went to the store' is talking about John, who exists now, but is not confined to what John is doing now.
Because of this, we would expect to find analogues of counterfactual conditionals -- counterpresential conditionals, we might call them. And this is what we do find. They're not even very strange:
If John went to the store, he has already bought milk.
If John will go to the store, he will then get milk.
Can we just treat counterfactuals as just another kind of tense? This is almost built in -- tenses, at least in tense logic, are just interpretations of Box and Diamond, strong modality and weak modality, and counterfactuals, at least in alethic modal logics, are also such interpretations. But the kinds of modal logic that most people think make sense of temporal tenses are not obviously the ones you would propose for counterfactuals (and vice versa). The analogy is strongest when we think specifically of 'not being present'. But it maybe gets weaker if we really think about past, future, or alternate possibilities. But we also have to keep in mind that there is no one single view of how time works. Certainly counterfactuals seem more like alternate branches in branching time than parallel linear times. (If we try to understand counterfactuals in this way, we seem to need time to branch in both the pastward and the futureward direction. But perhaps counterfactuals also require an eckwise and an andwise direction.)
