Page 14 - The Logic of Causation

14 THE LOGIC OF CAUSATION

that the phenomena are bound together, and either can be accessed through the other; the
labels applied to them become a matter of convenience for purposes of discourse.
Finally, the term “invariably” has to be stressed. How such constancy is established is not
the issue here; we shall consider that elsewhere. In the paradigm of causation given above,
it would not do for the conjunction of the cause and effect, or the conjunction of their
negations, to be merely occasional. We would not regard such varying conjunctions as
signifying genuine causation, but quite the opposite as signs of mere coincidence,
happenstance of togetherness. Post hoc ergo propter hoc. The problem is complicated in
lesser determinations of causation; but as we shall see it can be overcome, a constancy of
conjunction or of non-conjunction is always ultimately involved.
In this context, a warning is in order. When something is invariably accompanied by
another, we say that the first (the presence or absence of the cause) “is followed by” the
second (the presence or absence of the effect). This refers to causal sequence and should
not be confused with temporal sequence; the term “followed” is ambivalent (indeed, it is
also used in relation to spatial or numerical series). Even though causal and temporal
sequence are often both involved (which is why the term “to follow” is equivocal), causal
sequence may occur without temporal sequence (even in natural causation) or in a direction
opposite to temporal sequence (though supposedly not in natural causation, certainly in
logical causation, and by abstraction of the time factor also in extensional causation). The
context usually makes the intent clear, of course.
Now, for some formal analysis:
In our present treatment of causation, we shall focus principally on the logical ‘mode’ of
causation, note well. There are (as we shall later discuss) other modes, notably the natural,
the temporal, the spatial and the extensional, whose definitions differ with respect to the
type of modality considered. Having investigated modality and conditioning in detail in a
previous treatise (Future Logic, 1990), I can predict that most of the behavior patterns of
logical causation are likely to be found again in the other modes of causation; but also, that
some significant differences are bound to arise.
Returning now to the paradigm of causation, it may be expressed more symbolically as
follows, using the language of logical conditioning (as developed in my Future Logic, Part
III):

If C, then E; and
if notC, then notE.

A sentence of the form “If P, then Q” means “the conjunction of P and the negation of Q is
impossible”, i.e. there are no knowledge-contexts where this conjunction (P + notQ)
credibly occurs. Such a proposition can be recast in the contraposite form “If notQ, then
notP”, which means “the conjunction of notQ and the negation of notP is impossible” – the
same thing in other words.
Such a proposition, note, does not formally imply that P is possible or that notQ is possible.
Normally, we do take it for granted that such a proposition may be realized, i.e. that P is
possible, and therefore (by apodosis) Q is possible and the conjunction “P and Q” is
possible; and likewise that notQ is possible, and therefore (by apodosis) notP is possible
and the conjunction “notQ and notP” is possible.
However, in some cases such assumption is unjustified. It may happen that, though “If P,
then Q” is true, P is impossible, in which case “If P, then notQ” must also be true; or it may
happen that, though “if P, then Q” is true, notQ is impossible, in which case “If notP, then
Q” must also be true. These results are paradoxical, yet quite logical. I will not go into this
matter in detail here, having dealt with it elsewhere (see Future Logic, ch. 31). It is not