Metric temporal logics and induction

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,

1. Always, all ravens are black

becomes:

2. (x)(Rx→Bx)&(n)Pn(x)(Rx→Bx)&(n)Fn(x)(Rx→Bx),

i.e., all ravens are black, all ravens have always been black and all ravens will always be black. But this is unsatisfactory. Suppose my data is that all ravens have always been observed to be black. This data does not significantly support the first and third conjunct of (2), but only the second. To conclude to (2) is like trying to draw inferences about jade, which is in fact a disjunction of jadeite and nephrite, on the basis of observations of jadeite alone. Consequently, if Prior's metric temporal logic captures fundament temporal structure, induction from past to future is dubious indeed.

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:

3. (n)@n(x)(Rx→Bx),

an elegant and simple hypothesis that it is reasonable to take to be supported by the fact that all pastly observed ravens were black.

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:

4. (n)(u)@(n,u)(x)(Rx→Bx).

This even works in non-Archimedean timelines—u could be an infinitesimal or infinite duration. But I think there is a difficulty. All our observations of past black ravens can be explained by:

5. (u)@(−1,u)(x)(Rx→Bx),

i.e., the claim that all ravens have been black, which is just as simple as (4), but less committive. Why not go for (5) instead of (4)? Induction is still in danger. Besides, all of this metric time stuff is going to be a failure in worlds where there is no metric for time.

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:

6. (t)(Time(t)→#t(x)(Rx→Bx)),

where Time(t) says that t is a time of the actual world. Could a presentist endorse (6)? I think so. But I also think there would be a pressure on the presentist to explain Time(t) as something like a triple disjunction: t is a past time or t is the present time or t is a future time. Certainly, that's how it will have to look for Crisp-style ersatz times. And now we have the problem of induction again, because plausibly our observations should be taken instead to justify belief that:

7. (t)(Past(t)→#t(x)(Rx→Bx)).

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).