Prior's metric temporal logic comes with two sentential operators, Pn and Fn, which respectively mean something like "n ago in the past" and "in n in the future", where n is a duration. On Prior's metric logic, omnitemporal universal quantification becomes a triple conjunction. Thus,
- Always, all ravens are black
Suppose now that time is in fact discrete. Then one could use a temporal logic with only one sentential operator @n, where n is an integer, which means "in n moments". Since n could be negative, zero or positive, this can be used to talk about the past, present and future. And (1) becomes:
But logic shouldn't be tied to how things actually are. So this temporal logic is only plausible if time not only is (which is controversial enough) but must be discrete. And that would be very controversial.
Could one get this temporal logic working if time weren't discrete? One would need some way of measuring temporal distance that would allow distances to be negative (past), zero (present) and positive (future). Fixing a unit system, say seconds, makes it easy to do that. But there are three problems with such an approach. First, logic should not be dependent on a unit system. Second, temporal logic should apply in all worlds with time, while all the units of time (Planck times, seconds, etc.) that we have in our world are dependent on the particular laws of nature (just look at how a Planck time or a second is defined). Finally, a third problem is that if times ranged over, say, the real numbers, then the system wouldn't work in worlds with a non-Archimedean timeline.
One could, however, try the following move. The basic tense logic operator is @(n,u) where n is a real number and u is a duration, and we can read it as: in n us. Then (1) becomes something like:
So if presentism is true, our best bet for an induction-friendly metric temporal logic is if time is discrete.
What if eternalism is true? Then our most plausible temporal logic is a non-metric one with an operator like #t, which says that something is true at t, where t is a possible time. Our omnitemporal quantification (1) becomes:
But perhaps the presentist could explain Time(t) tenselessly. Maybe: Time(t) if and only if #t(0=0). (This won't work on Crisp-style ersatz times. But it might work for times that are durations from a beginning--assuming there has to be a beginning.) I.e., t is a time of the actual world if and only if something (and what better candidate than a tautology?) is true at t. Without the triple disjunction in the definition of an actual time, we now have some hope.
All that said, I doubt that a non-metric #t operator with a tenseless and changeless definition of Time(t) will be attractive to the presentist. For I think the typical presentist wants truths about what happens at different times to be grounded in tensed truths, something like in Prior's metric temporal logic.
Note: The metric logic renderings also need something that says that a given n isn't out-of-range (before a beginning of time or after an end of time).