Temporal Logic in Relativity

Classical Temporal Logic
In classical theory any two events satisfy a time ordering. We shall call this classical temporal logic CTL.

The rules of classical temporal logic are very simple. Since they are just the rules of inequalities of real numbers:

A>B   →  B<A

!(A>B)  → A<B or A=B

A>B & B>C → A>C

Or equivalently given that x → y is equivalent to saying that (!x or y) is true, these rules can be given by saying that in CTL these statements are always true:

!(A>B) or B<A

A>B or A=B or A<B

!(A>B) or !(B>C) or A>C

With these rules we can dispense with the underlying geometric idea of events as points in space-time and we have all we need to know to compare the temporal ordering of events.

Relativity

In relativity there are four possibilities for events. Either event A is is in the future light-cone of event B in which case we write A>B or A is in the past light-cone of B so B<A. Maybe the events occur at the same point in space-time in which case A=B. The fourth possibility is that the events are space-like separated. One event is neither in the past nor future light-cone of the other. In which case we write A~B.

We could add to our set of symbols, those cases where two or more relationships work. For example x≤y for (x<y or x=y). For the moment we shall not choose to use these symbols.

The rules of temporal logic in relativity depend on the dimension of space. Therefore there are an infinite number of temporal logics which we can label TL(D).

0+1 Dimensions

The temporal logic in 0 dimension of space and 1 dimension of time is isomorphic to classical temporal logic. In other words TL(0)=CTL. This is due to there being no space-like separations.

1+1 Dimensions

In 1 spacial dimension and above these rules always hold:

!(A>B) or B<A

!(A>B) or A=B or A~B or A<B
!(A~B) or A=B or A<B or A>B

The following is true in TL(1) but not always true in TL(2):

!(A<B and B>C and C<D and D>E and A<F and F>E and B~E and A~D and C~F)

This is due to the geometric fact that in 1 dimensions, if each of three segments overlap the other two then there must be a point where all three overlap. This is not true in 2 dimensions where each of three circles can overlap the other two without all three overlapping.

The obvious question is whether we can, like in TL(0), dispense with the underlying geometry and derive all information about the possible ordering of events from a set of logical rules?

Geometric rules

A>B  <==>   a0>b0 and (b0-a0)^2 < (b1-a1)^2
A<B  <==>   a0<b0 and (b0-a0)^2 < (b1-a1)^2
A~B  <==>  (b0-a0)^2 > (b1-a1)^2


Distance Geometry
And interesting parallel is that of distance geometry. In distance geometry we dispense with the coordinate system and just deal with the distance between points. We can find the volume of a tetrahedron given the length of all it's sides. This volume then is identically zero in 2 dimensional space but not necessarily zero in higher dimensional space. Thus this gives a condition for the dimensionality of space without reference to coordinates.