Thursday, June 14, 2012

Jottings on Logical Tense

Suppose we take a fairly simple argument of more than single step, e.g.,

p) All black cats are ninjas.
q) Whatever is a ninja is well-trained.
r) Therefore all black cats are well-trained.
s) Some black cats are pets who never get out of bed.
t) Therefore some pets who never get out of bed are well-trained.

This is a series of propositions, but they have a logical order. Some propositions are premises for others, while others are conclusions. If we take a proposition as a reference-point, like (r), we find that (p) and (q) are in the premiseward direction, and that (t) is in the conclusionward direction. (s) is, so to speak, simultaneous with (r). On the basis of this we could build up a modal logic to describe the relations among propositions.

Let us use the following to indicate "p is a premise everywhere for r" (I'll get to the reason for using 'everywhere' in a bit):

Hpr

and let us use the following to indicate "t is a conclusion somewhere for r".

Ftr

From these we can develop two new descriptions. The following says "t is a conclusion everywhere for r":

Gtr

And the following says "p is a premise somewere for r"

Ppr

The reason for the 'everywhere' and 'somewhere' is that there is an asymmetry, not logically necessary but practically sensible, in how we talk casually about premises and conclusions. When we say something is a premise, we usually mean that it is a premise everywhere; once a premise, always a premise, it can be used whenever you please as a premise. When we call something a conclusion, however, we do not usually mean that it is always a conclusion; many conclusions are path-dependent: you can only get them in the conclusionward direction if other things intervene. To say that a proposition is a conclusion everywhere with respect to p requires that it follow immediately from p. In other words. As I said, there is no particular logical reason why we should talk this way; we could just as easily talk in the reverse way. So it makes sense to allow two ways in which a proposition can be premiseward and two ways in which a proposition can be conclusionward.

We can then talk about conditions on arguments in these terms. For instance, this pair tells us there are no contradictory premises for any given proposition x:

Ppx -> ~H~p,x ("If something is a premise anywhere, its contradictory is not a premise everywhere")
Hpx -> ~P~p,x ("If something is a premise everywhere, its contradictory is not a premise anywhere")

And these are the corresponding formulae for cases where there are no contradictory conclusions:

Fpx -> ~G~p,x ("If something is a conclusion anywhere, its contradictory is not a conclusion everywhere")
Gpx -> ~F~p,x ("If something is a conclusion everywhere, its contradictory is not a conclusion anywhere")

Likewise, we can say "if p is an always available conclusion, it is a conclusion available at some point":

Gpx -> Fpx

And we can say that if (p->q) is a premise then, if p is a premise, q is also a premise:

G(p->q),x -> (Gpx -> Gqx)

Nested operators end up being meta-operators. For instance, FFpx says that Fp is a conclusion at some point with respect to x. Thus for most contexts, this will be true:

FFp,x -> Fpx

But this will not:

Fpx -> FFp,x

This is all a tense logic, of course, although it simply indicates direction rather than time (premises are the past, conclusions are the future), but we've picked out a particular proposition to serve as a reference point rather than just assuming a 'now'. Another difference is that this is not a standard tense logic, that is, it is not based on K or Kt modal systems.

We can also quantify if we please, e.g.,

∀x Hpx ("p is an everywhere-available premise for every proposition")
∃x Hpx ("p is an everywhere-available premise for some proposition")
∀x ~Fpx ("p is the conclusion of no proposition")
∃x ~Fpx ("there is a proposition for which p is not a conclusion")

Of course, one would also want to indicate sections of argument, using multiple reference points, so one would ideally have equivalents for Since and Until. And if one added truth-values as modal operators, one could discuss things like validity of arguments.