Page 20 - The Logic of Causation

20 THE LOGIC OF CAUSATION

Again, mutatis mutandis, the definition of necessary causation does not exclude that there
be some cause(s) other than C – such as say C 1 – having the same relation to E. In such
case, C and C 1 may be called parallel necessary causes of E. The minimal relation between
such causes is given by the following normally valid 2nd figure syllogism (see Future
Logic, p. 162):

If notC, then notE (and if C, not-then notE / and notC is possible);
and if notC 1, then notE (and if C 1, not-then notE / and notC 1 is possible);
therefore, if C 1, not-then notC (= if C, not-then notC 1 by contraposition).

The possibility of parallel necessary causes is clear from the logical compatibility of these
premises, which together merely imply that in the presence of E both C and C 1 are present.
The main clauses of the two premises can be merged in a compound proposition of the
form “If E, then both C and C 1”, which by contraposition yields “If notC or notC 1, then
notE”. Thus, such parallel causes may be referred to as ‘alternative’ necessary causes (in a
large sense of the term ‘alternative’).
Since the conclusion of the above syllogism is subaltern to each of the propositions “if C 1,
then C” and “if C, then C 1”, it may happen that C 1 implies C and/or C implies C 1 – but they
need not do so. Likewise, since the conclusion is compatible with the proposition “if notC 1,
then C” or “if notC, then C 1”, it may happen that C and C 1 are exhaustive – but they do not
have to be. The conclusion merely specifies that C and C 1 not be incompatible (i.e. be
neither contradictory nor contrary; this is the sole formal specification of the disjunction in
“If notC or notC 1, then notE”).
Similarly, still in necessary causation, E need not be the exclusive dependent effect of C;
there may be some other thing(s) – such as say E 1 – which are invariably preceded by C,
too. In such case, E and E 1 may be called parallel dependent effects of C. The minimal
relation between such effects is given by the following normally valid 3rd figure syllogism
(see Future Logic, p. 162-164):

If notC, then notE (and if C, not-then notE / and notC is possible);
and if notC, then notE 1 (and if C, not-then notE 1 / and notC is possible);
therefore, if notE 1, not-then E (= if notE, not-then E 1 by contraposition).

The possibility of parallel dependent effects is clear from the logical compatibility of these
premises, which together merely imply that in the absence of C both E and E 1 are absent.
The main clauses of the premises can be merged in a compound proposition of the form “If
notC, then neither E nor E 1”. Thus, such parallel effects may be said to be ‘composite’
dependent effects.
Since the conclusion of the above syllogism is subaltern to each of the propositions “if
notE 1, then notE” and “if notE, then notE 1”, it may happen that E implies E 1 and/or E 1
implies E – but they need not do so. Likewise, since the conclusion is compatible with the
proposition “if E 1, then notE” or “if E, then notE 1”, it may happen that E and E 1 are
incompatible with each other – but they do not have to be. The conclusion merely specifies
that E and E 1 not be exhaustive (i.e. be neither contradictory nor subcontrary).

It happens that parallel causes or parallel effects are themselves causally related. That this
is possible, is implied by what we have seen above. Since each of the following pairs of
items may have any formal relation with one exception, namely: