## Saturday, July 21, 2012

### Tense Logic and Order Theory

I've talked before about using tense logic to describe arguments, and computer programmers can and do use tense logic to describe computer programs. What time has in common with logical arguments and computer programs is primarily order, so this naturally suggests that you could have a tense logic to talk about order theory.

In essence, you can translate talk of orderings to tense logic talk in the following way:

x < y :: Pxy
x > y :: Fxy

Given these we could define an incomparability operator:

x ~ y :: Sxy :: neither Pxy nor Fxy

Then we'd have a semiorder when the following obtained (assuming I haven't slipped up anywhere):

∀x∀y¬(Pxy & Fxy)
∀x∀y∀w((Pxy & Syx & Pzw) → Pxw)
∀x∀y∀w((Pxy & Pyz & Syw) → ¬(Sxw & Szw))

To get preorders and weak partial orders we'd need a notation for x = y, and so forth. As with logical arguments, we aren't assuming that the tense logic system is standard (Kt).

To put it in other words: tense logics seem actually to be logics of direction in an order. Nothing about them requires that they be used only for time. It's just that time as we usually think of it is an obvious total order. (Thought of relativistically, of course, time gets much more complicated, but contrary to what once was thought, it's become increasingly clear that a relativistic tense logic is possible. And we see here why: special relativity doesn't imply that time has no order, it just implies that time can only be treated as a total order when certain conditions obtain.)