Temporal Logic

Introduction

Each time slice of the actual world is a possible world - one that can be indexed by the time at which it exists. It naturally suggests that a modal logic that deals with truths about possible worlds could be adapted to deal with truths about temporal relations in the actual world. In that case we'd need some way to think of the accessibility relations: the obvious choice is to think that n accesses k means either that n follows k or that n precedes k.

If that's how we look at the accessibility relation, then we know that it has the property of transitivity but not the properties of symmetry or reflexivity. A normal modal logic that satisfies this requirement is K4 (K + Sym.)

For example, suppose we have a set of worlds T = {n, k, l, j} and a transitive access relation ® for which the set forms a chain  n ® k ® l ® j. If the elements of T are time slices of the actual world, and the access is interpreted as  a ® b iff a is before b, then we have a K4 Temporal Logic. It gives the following interpretations of the modal operators:

<>p =: at some time in the Future, p (also written 'Fp')

[]p =: at all times in the future, p is Going to be (also written 'Gp')

Note that these only allow us to talk about things in the future.

Suppose that on T we defined the relation ¬ to be interpreted as a ¬ b iff a is before b, then we have another K4 Temporal Logic by which we can talk about things in the past. It gives the following interpretations of the modal operators:

<>p =: at some time in the Past, p (also written 'Pp')

[]p =: at all times in the past, p Has been (also written 'Hp')

Obviously, if we can combine these logics - using the letter operators to avoid ambiguity - we can have a logic that can treat both past and future. We could also define other operators, like:

Lp =: Hp & p & Gp (meaning 'p always was, is, and always will be.')

To relate the past and future we appeal to these two principles:

PGp É p. Meaning that if at some past time it was always going to be that p, then p right now.

FHp É p. Meaning that if at some future time p was always going to have been, then p right now.

The tree rules for a combined logic of this sort are modelled on the tree rules for K4 (orthodox version) given above. We start by picking one of the logics that we want to combine. Suppose we choose the first one. Rename the accessibility relation '®' as B (for Before), and replacing A in the original definitions for the K4 tree rules by B. So:

FR                      FX                w/v      

                            ...                              

                          wBv

                            X                  v

                         where v is new to this path

GR                     GX                w\v      

                          wBv

                           ...                              

                            X                  v

Trans                wBv

                           vBt

                           ...                              

                           wBt

Now for the second logic we use the same K4 rules but we reverse the order of the worlds in them. So:

PR                     PX                w/v      

                            ...                              

                          vBw

                            X                  v

                         where v is new to this path

GR                     HX                w\v      

                          vBw

                           ...                              

                            X                  v

And for the operator L we can define

LR                     LX                w\v      

                           ...                              

                            X                  v

                         for any v.

By these rules, the two principles we wanted to include are valid in this logic. This particular logic is thus defined as:

CRTr = Ptr + MN + {PR, HR, FR, GR, LR, Trans}

The CR stands for Cocchiarella's Relativistic causal time. Why relativistic? because nothing in the definition of CR makes every element accessible to every other element, so some events are neither before nor after other events, which is suggestive of the way that some events in the real world can be causally separated (outside each others' light cones) and thus neither event can cause or be caused by the other.

According to C4, possible topologies of time include a single Now with multiple branching pasts and futures, or with a single linear past and branching futures, or with single linear pasts and futures. All very odd; but it also allows circular time (where for T we have n ® k ® l ® j ® n) (Indian philosophers have always been keen on this) or time with a tail (... ® -2 ® -1 ® n ® k ® l ® j ® n) (which is something that threatens the USS Enterprise on a regular basis.) It also allows the Parmenidean universe of the permanent Now (n ® n.)

Note that the branching pasts and futures shouldn't be confused with the possible ways for the world to be (even in the past or future.) And just because Fp or Pp it doesn't immediately follow that []p. In fact it doesn't even follow from Lp: consider G, the Constant of Gravitation for example. Nor do Fp or Pp immediately follow from <>p. 

Suppose we don't want such temporal liberality; how do we insist on sensible topologies? How do we insist upon linear time, for example?

We add Connectedness: if a and b are before c then either a is before b or b is before a. And similarly for 'After.' Thus:

Connec             vBw                                 wBv

                           tBw                                 wBt

                            ...                                      ...                              

                          /     \                                 /     \

                   vBt          tBv                   vBt          tBv

Try this argument:

                         F(Gq & ~p)

                         ----------------------

                         G(p É (Gp É q)

The resultant logic is Cocchiarella's Linear time:

CLTr = CRTr + Connec

CL still allows circularity. There are two ways we can get rid of that. Either we can make the accessibility relation irreflexive by simply declaring that in no case will nBn be allowed, or we can make the relation asymmetric, so that if aBb then it cannot be that bBa. Since B is transitive, in any circle we will find that each element in the circle is accessible to any other and B thus allows symmetry. (Note, nevertheless, it doesn't have to be a symmetric relation.) Antisymmetry is imposed with the rule:

Asym                vBw

                             ...

                           wBv

                             X  

This gives us the system CLA.

CLATr = CLTr + Asym 

Suppose we decide that time has no beginning and no end: then we have to believe that at any time there is a time before and a time after. In terms of our accessibility relation, it would  need to be the case that for any world n there is both a world k for which kBn and a world j for which nBj. This you'll recognise as being very similar to the serial property by which we made K4 into the system D. If we go back to CR - which you'll recall is the temporal version of K4 - we can add analogous versions of seriality to get a temporal version of D4, called D4_t. We can either add seriality directly to the accessibility relation in the Orthodox fashion or we can adopt a Hintikka- style rule thus:

GD                     GX                w      

                             ...                              

                            FX                w

HD                     HX                w      

                             ...                              

                            PX                w

This gives us 

D4_tTr = CR + {HD, GD}

The effect of this is to make Gp É Fp and Hp É Pp (analogues to []p É <>p) valid formulas in the new logic, and to make time infinite in both directions. 

As a minimal extension of CR, D4_t still allows both branching and circles, but exactly the same modifications as worked for CR will work for D4_t yielding this time:

CSTr = D4_tTr + Connec

CSATr = CSTr + Asym 

Suppose we decide that time does have a beginning or an end: then we have to believe that there is a time that is first amongst all times and a time that is last amongst all times. In terms of our accessibility relation, it would  need to be the case that there is a time a such that for any world n aBn and a world w for which nBw. Obviously this means the consistency analogues of GD and HD that occur in the CS systems can't be admitted here, but we could include it in, say, CLA.

Given that system, try the formulas: 

                         Pp É P~Pp

                         Fp É F~Fp

Other obvious conditions will allow us to treat times with beginnings but no ends or vice versa. 

Note that the tree method doesn't seem to be effective in these logics. You have to be able to judge when a tree is going to become inconsistent at infinite length and must therefore be closed. For example the tree for the first of those formulas could go on forever, but that would contradict the condition that there is a first time (even if we don't know when that was) so it has to be closed.

'Time for Philosophers' from the ABC's The Philosophers' Zone, 19^th April 2008.