THE LOGIC OF CAUSATION

Phase One: Macroanalysis

Chapter 2 – The Generic Determinations.

The strongest determination of causation, which we identified as the paradigm of causation, may be called complete and necessary causation. We shall now repeat the three constituent propositions of this form and their implications, all of which must be true to qualify:

(i) If C, then E;

(ii) if notC, then notE;

(iii) where: C is contingent and E is contingent.

As we saw, these propositions together imply the following:

The conjunction (C + E) is possible;

the conjunction (notC + notE) is possible.

Clauses (i) and (iii) signify complete causation. With reference to this positive component, we may call C a complete cause of E and E a necessary effect of C. Where there is complete causation, the cause is said to make necessary (or necessitate) the effect[1]. This signifies that the presence of C is sufficient (or enough) for the presence E.

Clause (ii) and (iii) signify necessary causation. With reference to this negative component, we may call C a necessary cause of E and E a dependent effect of C. Where there is necessary causation, the cause is said to make possible (or be necessitated by) the effect. This signifies that the presence of C is requisite (or indispensable) for the presence E[2].

Clause (iii) is commonly left tacit, though as we saw it is essential to ensure that the first two clauses do not lead to paradox. Strictly speaking, it would suffice, given (i), to stipulate that C is possible (in which case so is E) and E is unnecessary (in which case so is C). Or equally well, given (ii), that C is unnecessary (in which case so is E) and E is possible (in which case so is C). The possibilities of the conjunctions (C + E) and (notC + notE), logically follow, and so need not be included in the definition.

Looking at the paradigm, we can identify two distinct lesser determinations of causation, which as it were split the paradigm in two components, each of which by itself conforms to the paradigm through an ingenuous nuance, as shown below.

Also below, I list the various clauses of each definition, renumbering them for purposes of reference. Then a table is built up, including all the causal and effectual items involved (positive and negative) and all their conceivable combinations[3]. The modus of each item or combination, i.e. whether it is defined or implied as possible or impossible, or left open, is then identified. In each case, the source of such modus is noted, i.e. whether it is given or derivable from given(s).

Complete causation:

(i) If C, then E;

(ii) if notC, not-then E;

(iii) where: C is possible.

 Table 2.1. Complete causation. No. Element/compound Modus Source/relationship 1 C possible (iii) 2 notC possible implied by (ii) 3 E possible implied by (i) + (iii) 4 notE possible implied by (ii) 5 C E possible implied by (i) + (iii) 6 C notE impossible (i) 7 notC E open 8 notC notE possible (ii)

Complete causation conforms to the paradigm of causation by means of the same main clause (i); whereas its clause (ii), note well, concerning what happens in the absence of C, substitutes for the invariable absence of E (i.e. “then notE”), the not-invariable presence of E (i.e. “not-then E”). However, remember, contraposition of (i) implies that “If notE, then notC”, meaning that in the absence of E we can be sure that C is also absent[4].

Clause (ii) means that (notC + notE) is possible, so we are sure from it that C is unnecessary and E is unnecessary; also it teaches us that C and E cannot be exhaustive. Technically, it would suffice for us to know that notE is possible, for we could then infer clause (ii) from (i); but it is best to specify clause (ii) to fit the paradigm of causation. As for clause (iii), we need only specify that C is possible; it follows from this and clause (i) that (C + E) is possible and so that E is also possible.

Note well the nuance that, to establish such causation, the effect has to be found invariably present in the presence of the cause, otherwise we would commit the fallacy of post hoc ergo propter hoc; but the effect need not be invariably absent in the absence of the cause: it suffices for the effect not to be invariably present.

The segment of the above table numbered 5-8 (shaded) may be referred to as the matrix of complete causation. It considers the possibility or impossibility of all conceivable conjunctions of all the items involved in the defining clauses or the negations of these items.

Necessary causation:

(i) If notC, then notE;

(ii) if C, not-then notE;

(iii) where: C is unnecessary.

 Table 2.2. Necessary causation. No. Element/compound Modus Source/relationship 1 C possible implied by (ii) 2 notC possible (iii) 3 E possible implied by (ii) 4 notE possible implied by (i) + (iii) 5 C E possible (ii) 6 C notE open 7 notC E impossible (i) 8 notC notE possible implied by (i) + (iii)

Necessary causation conforms to the paradigm of causation by means of the same main clause (i)[5]; whereas its clause (ii), note well, concerning what happens in the presence of C, substitutes for the invariable presence of E (i.e. “then E”), the not-invariable absence of E (i.e. “not-then notE”). However, remember, contraposition of (i) implies that “If E, then C”, meaning that in the presence of E we can be sure that C is also present[6].

Clause (ii) means that (C + E) is possible, so we are sure from it that C is possible and E is possible; also it teaches us that C and E cannot be incompatible. Technically, it would suffice for us to know that E is possible, for we could then infer clause (ii) from (i); but it is best to specify clause (ii) to fit the paradigm of causation. As for clause (iii), we need only specify that C is unnecessary; it follows from this and clause (i) that (notC + notE) is possible and so that E is also unnecessary.

Note well the nuance that, to establish such causation, the effect has to be found invariably absent in the absence of the cause, otherwise we would commit the fallacy of post hoc ergo propter hoc; but the effect need not be invariably present in the presence of the cause: it suffices for the effect not to be invariably absent.

Note the matrix of necessary causation, i.e. the segment of the above table numbered 5-8 (shaded).

Lastly, notice that complete and necessary causation are ‘mirror images’ of each other. All their characteristics are identical, except that the polarities of their respective cause and effect opposite: C is replaced by notC, and E by notE, or vice-versa. The one represents the positive aspect of strong causation; the other, the negative aspect. Accordingly, their logical properties correspond, mutadis mutandis (i.e. if we make all the appropriate changes).

Following the preceding analysis of necessary and complete causation into two distinct components each of which independently conforms to the paradigm, we can conceive of complete causation without necessary causation and necessary causation without complete causation. These two additional determinations of causation are conceivable, note well, only because they do not infringe logical laws; that is, we already know that the various propositions that define them are individually and collectively logically compatible.

Before looking into weaker determinations of causation, we must deal with the phenomenon of parallelism.

The definition of complete causation does not exclude that there be some cause(s) other than C – such as say C1 – having the same relation to E. In such case, C and C1 may be called parallel complete causes of E. The minimal relation between such causes is given by the following normally valid 2nd figure syllogism (see FL, ch. 29.1):

If C, then E (and if notC, not-then E / and C is possible);

and if C1, then E (and if notC1, not-then E / and C1 is possible);

therefore, if notC1 not-then C (= if notC, not-then C1 – by contraposition).

The possibility of parallel complete causes is clear from the logical compatibility of these premises, which together merely imply that in the absence of E both C and C1 are absent. The main clauses of the premises can be merged in a compound proposition of the form “If notE, then neither C nor C1”, which by contraposition yields “If C or C1, then E”. Thus, such parallel causes may be referred to as ‘alternative’ complete causes (in a large sense of the term ‘alternative’).

Since the conclusion of the above syllogism is subaltern to each of the propositions “if notC1, then notC” and “if notC, then notC1”, it may happen that C implies C1 and/or C1 implies C – but they need not do so. Likewise, since the conclusion is compatible with the proposition “if C1, then notC” or “if C, then notC1”, it may happen that C and C1 are incompatible with each other – but they do not have to be. The conclusion merely specifies that C and C1 not be exhaustive (i.e. be neither contradictory nor subcontrary; this is the sole formal specification of the disjunction in “If C or C1, then E”).

Similarly, still in complete causation, E need not be the exclusive necessary effect of C; there may be some other thing(s) – such as say E1 – which invariably follow C, too. In such case, E and E1 may be called parallel necessary effects of C. The minimal relation between such effects is given by the following normally valid 3rd figure syllogism (see FL, ch. 29.1):

If C, then E (and if notC, not-then E1 / and C is possible);

and if C, then E1 (and if notC, not-then E1 / and C is possible);

therefore, if E1, not-then notE (= if E, not-then notE1 – by contraposition).

The possibility of parallel necessary effects is clear from the logical compatibility of these premises, which together merely imply that in the presence of C both E and E1 are present. The main clauses of the premises can be merged in a compound proposition of the form “If C, then both E and E1”. Thus, such parallel effects may be said to be ‘composite’ necessary effects.

Since the conclusion of the above syllogism is subaltern to each of the propositions “if E1, then E” and “if E, then E1”, it may happen that E1 implies E and/or E implies E1 – but they need not do so. Likewise, since the conclusion is compatible with the proposition “if notE1, then E” or “if notE, then E1”, it may happen that E and E1 are exhaustive – but they do not have to be. The conclusion merely specifies that E and E1 not be incompatible (i.e. be neither contradictory nor contrary).

Again, mutadis mutandis, the definition of necessary causation does not exclude that there be some cause(s) other than C – such as say C1 – having the same relation to E. In such case, C and C1 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 FL, ch. 29.1):

If notC, then notE (and if C, not-then notE / and notC is possible);

and if notC1, then notE (and if C1, not-then notE / and notC1 is possible);

therefore, if C1, not-then notC (= if C, not-then notC1 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 C1 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 C1”, which by contraposition yields “If notC or notC1, 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 C1, then C” and “if C, then C1”, it may happen that C1 implies C and/or C implies C1 – but they need not do so. Likewise, since the conclusion is compatible with the proposition “if notC1, then C” or “if notC, then C1”, it may happen that C and C1 are exhaustive – but they do not have to be. The conclusion merely specifies that C and C1 not be incompatible (i.e. be neither contradictory nor contrary; this is the sole formal specification of the disjunction in “If notC or notC1, 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 E1 – which are invariably preceded by C, too. In such case, E and E1 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 FL, ch. 29.1):

If notC, then notE (and if C, not-then notE / and notC is possible);

and if notC, then notE1 (and if C, not-then notE1 / and notC is possible);

therefore, if notE1, not-then E (= if notE, not-then E1 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 E1 are absent. The main clauses of the premises can be merged in a compound proposition of the form “If notC, then neither E nor E1”. 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 notE1, then notE” and “if notE, then notE1”, it may happen that E implies E1 and/or E1 implies E – but they need not do so. Likewise, since the conclusion is compatible with the proposition “if E1, then notE” or “if E, then notE1”, it may happen that E and E1 are incompatible with each other – but they do not have to be. The conclusion merely specifies that E and E1 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:

· parallel complete causes cannot be exhaustive (since “if notC, not-then C1” is true for them); and parallel necessary effects cannot be incompatible (since “if E, not-then notE1” is true for them);

· parallel necessary causes cannot be incompatible (since “if C, not-then notC1” is true for them); and parallel dependent effects cannot be exhaustive (since “if notE, not-then E1” is true for them);

… it follows that either one of parallel causes C and C1 may be a complete or necessary cause of the other; and likewise, either one of parallel effects E and E1 may be a complete or necessary cause of the other.

In certain situations, as we shall see in a later chapter, it is possible to infer such causal relations between parallels. But, it must be stressed, the mere fact of parallelism does not in itself imply such causal relations.

In sum, complete and/or necessary causation should not be taken to imply exclusiveness (i.e. that a unique cause and a unique effect are involved); such relation(s) allow for plurality of causes or effects in the sense of parallelism as just elucidated.

Indeed, it is very improbable that we come across exclusive relations in practice, since every existent has many facets, each of which might be selected as cause or effect. Our focusing on this or that aspect as most significant or essential, is often arbitrary, a matter of convenience; though often, too, it is guided by broader considerations, which may be based on intuition of priorities or complicated reasoning.

In any case, it is important to distinguish plurality arising in strong causation, which signifies alternation of causes or composition of effects, as above, from plurality arising in weak causation, which signifies composition of causes or alternation of effects, which we shall consider in the next section.

Having clarified the complete and necessary forms of causation, as well as parallelism, we are now in a position to deal with lesser determinations of causation. Let us first examine partial causation; contingent causation will be dealt with further on.

Partial causation:

(i) If (C1 + C2), then E;

(ii) if (notC1 + C2), not-then E;

(iii) if (C1 + notC2), not-then E;

(iv) where: (C1 + C2) is possible.

 Table 2.3. Partial causation. No. Element/compound Modus Source/relationship 1 C1 possible implied by (iii) or (iv) 2 notC1 possible implied by (ii) 3 C2 possible implied by (ii) or (iv) 4 notC2 possible implied by (iii) 5 E possible implied by (i) + (iv) 6 notE possible implied by (ii) or (iii) 7 C1 E possible implied by (i) + (iv) 8 C1 notE possible implied by (iii) 9 notC1 E open 10 notC1 notE possible implied by (ii) 11 C2 E possible implied by (i) + (iv) 12 C2 notE possible implied by (ii) 13 notC2 E open 14 notC2 notE possible implied by (iii) 15 C1 C2 possible (iv) 16 C1 notC2 possible implied by (iii) 17 notC1 C2 possible implied by (ii) 18 notC1 notC2 open 19 C1 C2 E possible implied by (i) + (iv) 20 C1 C2 notE impossible (i) 21 C1 notC2 E open 22 C1 notC2 notE possible (iii) 23 notC1 C2 E open 24 notC1 C2 notE possible (ii) 25 notC1 notC2 E open 26 notC1 notC2 notE open

Two phenomena C1, C2 may be called partial causes of some other phenomenon E, only if all the above conditions (i.e. the four defining clauses) are satisfied. In such case, we may call E a contingent effect of each of C1, C2. Of course, the compound (C1 + C2) is a complete cause of E, since in its presence, E follows (as given in clause (i)); and in its absence, i.e. if not(C1 + C2), E does not invariably follow (as evidenced by clauses (ii) and (iii)). Rows 19-26 of the above table (shaded) constitute the matrix of partial causation.

We may thus speak of this phenomenon as a composition of partial causes; and stress that C1 and C2 belong in that particular causation of E by calling them complementary partial causes of it. Indeed, instead of saying “C1 and C2 are complementary partial causes of E”, we may equally well formulate our sentence as “C1 (complemented by C2) is a partial cause of E” or as “C2 (complemented by C1) is a partial cause of E”. These three forms are identical, except for that the first treats C1 and C2 with equal attention, whereas the latter two lay stress on one or the other cause. Such reformatting, as will be seen, is useful in some contexts.

We may make a distinction between absolute and relative partial causation, as follows. The ‘absolute’ form specifies one partial cause without mentioning the complement(s) concerned; it just says: “C1 is a partial cause of E”, meaning “C1 (with some unspecified complement) is a partial cause of E”. This is in contrast to the ‘relative’ form, which does specify a complement, as in the above example of “C1 (complemented by C2) is a partial cause of E”. This distinction reflects common discourse. Its importance will become evident when we consider negations of such forms.

One way to see the appropriateness of our definition of partial causation, its conformity to the paradigm of causation, is by resorting to nesting (see FL, ch. 27.3). We may rewrite it as follows:

 From (i) if C2, then (if C1, then E); from (ii) if C2, then (if notC1, not-then E); from (iii) if notC2, not-then (if C1, then E).[7]

Clause (i) tells us that given C2, C1 implies E. Clause (ii) tells us that given C2, notC1 does not imply E. Thus, under condition C2, C1 behaves like a complete cause of E. Moreover, clause (iii) shows that under condition notC2, C1 ceases to so behave. Similarly, mutadis mutandis, C2 behaves conditionally like a complete cause of E.[8]

Let us now examine the definition of partial causation more closely. The terminology adopted for it is obviously intended to contrast with that for complete causation.

Clause (i) informs us that in the presence of the two elements C1 and C2 together, the effect is invariably also present. However, that clause alone would not ensure that both C1 and C2 are relevant to E, participants in its causation. We need clause (ii) to establish that without C1, C2 would not by itself have the same result. And, likewise, we need clause (ii) to establish that without C2, C1 would not by itself have the same result.

Suppose, for instance, clause (ii) were false; then, combining it with (i), we would obtain the following simple dilemma:

If (C1 + C2), then E – and – if (notC1 + C2), then E;

therefore, if C2, then E.

That is, C2 would be a complete cause of E, without need of C1, which would in such case be an accident in the relation “If (C1 + C2), then E”, note well. Similarly, if clause (iii) were false, it would follow that C1 is sufficient by itself for E, irrespective of C2. In the special case where both (ii) and (iii) are denied, C1 and C2 would be parallel complete causes of E (compatible ones, since they are conjoined in the antecedent of clause (i)). Therefore, as well as clause (i), clauses (ii) and (iii) have to specified for partial causation.

Furthermore, our definition of partial causation thus mentions three combinations of C1, C2 and their respective negations, namely:

· C1 + C2

· notC1 + C2

· C1 + not C2

And it tells us what happens in relation to E in each of these situations: in the first, E follows; in the next two, it does not. One might reasonably ask, what about the fourth combination, namely:

· notC1 + notC2?[9]

Well, for that, there are only two possibilities: either E follows or it does not. Note first that both these possibilities are logically compatible with clauses (i), (ii) and (iii).

Suppose that “If (notC1 + notC2), then E” is true. In that case, notC1 and notC2 would each have the same relation to E that C1 and C2 have by virtue of clauses (i), (ii), (iii). For if we combine this supposed additional clause with clauses (ii) and (iii), we see that, whereas E follows the conjunction of notC1 and notC2, E does not follow the conjunction of not(notC1) with notC2 or that of notC1 with not(notC2). In that case, we would simply have two, instead of just one, compound causes of E, namely (C1 + C2) and (notC1 + notC2), sharing the same clauses (ii) and (iii) which establish the relevance of each of the elements. Though at first sight surprising, such a state of affairs is quite conceivable, being but a special case of parallel causation! Thus, the proposition “If (notC1 + notC2), then E” may well be true. But may it be false? Suppose that its contradictory “If (notC1 + notC2), not-then E” is true, instead. Here again, the causal significance of the first three clauses remains unaffected. We can thus conclude that what happens in the situation “notC1 + notC2”, i.e. whether E follows or not, is irrelevant to the roles played by C1 and C2. Our definition of partial causation through the said three clauses is thus satisfactory.

Lastly the following should be noted. If we replaced clauses (ii) and (iii) by “If not(C1 + C2), not-then E”, to conform with clause (i) to the definition of complete causation, we would only be sure that the compound (C1 + C2) causes E. It does not suffice to establish that both its elements are involved in that causation, since it could be adequately realized by the eventuality that “If (notC1 + notC2), not-then E”. For this reason, too, clauses (ii) and (iii) are unavoidable.

Regarding clause (iv), which serves to ensure that the first three clauses do not lead to paradox, it is easy to show that the possibility of the conjunction (C1 + C2) is the minimal requirement. For this through clause (i) implies that E is possible and (C1 + C2 + E) is possible. Additionally, clause (ii) means that (notC1 + C2 + notE) is possible, and therefore implies that (notC1 + C2) is possible and each of notC1, C2, notE is possible. Similarly, clause (iii) means that (C1 + notC2 + notE) is possible, and therefore implies that (C1 + notC2) is possible and each of C1, notC2, notE is possible. It is thus redundant to specify these various contingencies.

The methodological principle underlying the definition of partial causation is well known to scientists and oft-used. It is that to establish the causal role of any element such as C1, of a compound (C1 + C2…) in whose presence a phenomenon E is invariably present, we must find out what happens to E when the element C1 is absent while all other elements like C2 remain present. That is, we observe how the putative effect is affected by removal of the putative cause while keeping all other things equal[10]. Only if a change in status occurs (minimally from “then E” to “not-then E”), may the element be considered as participating in the causation, i.e. as a relevant factor.

Once this is understood, it is easy to generalize our definition of partial causation from two factors (C1, C2) to any number of them (C1, C2, C3…), as follows:

(i) If (C1 + C2 + C3…), then E;

(ii) if (notC1 + C2 + C3…), not-then E;

(iii) if (C1 + notC2 + C3…), not-then E;

(iv) if (C1 + C2 + notC3…), not-then E;

…etc. (if more than three factors);

and (C1 + C2 + C3…) is possible.

Clause (i) establishes the complete causation of the effect E by the compound (C1 + C2 + C3…). But additionally there has to be for each element proof that its absence would be felt: this is the role of clauses (ii), (iii), (iv)…, each of which negates one and only one of the elements concerned. Thus, the number of additional clauses is equal to the number of factors involved.

Whatever the relation to E of other possible combinations of the elements and their negations, the partial causation of E by elements C1, C2, C3… is settled by the minimum number of clauses specified in our definition. As we saw, with two factors the combination “notC1 + notC2” is not significant. Similarly, we can show that with three factors the following combinations are not significant:

· notC1 + notC2 +C3

· notC1 + C2 + notC3

· C1 + notC2 + notC3

· notC1 + notC2 + notC3

And so forth. Generally put, if the number of elements is n, the number of insignificant combinations will be is 2n – (1 + n). Whether any of these further combinations implies or does not imply E does not affect the role of partial causation signified by the defining clauses for the factors C1, C2, C3… per se. Other causations may be involved in certain cases, but they do not disqualify or diminish those so established.

The very last clause, that (C1 + C2 + C3…) is possible, is required and sufficient, for reasons already seen.

Clearly, we can say that the more factors are involved, the weaker the causal bond. If C is a complete cause of E, it plays a big role in the causation of E. If C1 is a partial cause of E, with one complement C2, it obviously plays a lesser role than C. Similarly, the more complements C1 has, like C2, C3…, the less part it plays in the whole causation of E. We may thus view the degree of determination involved as inversely proportional to the number of causes involved, though we may (note well) be able to assign different weights to the various partial causes[11].

Note finally that we can facilitate mental assimilation of multiple (i.e. more than two) partial causes through successive reductions to pairs of partial causes, one of which is compound. Thus, (C1 + C2 + C3 + …) may be viewed as (C1 + (C2 + C3 +…)), provided all the above mentioned conditions are entirely satisfied.

Let us now turn our attention to contingent causation.

Contingent causation:

(i) If (notC1 + notC2), then notE;

(ii) if (C1 + notC2), not-then notE;

(iii) if (notC1 + C2), not-then notE;

(iv) where: (notC1 + notC2) is possible.

 Table 2.4. Contingent causation. No. Element/compound Modus Source/relationship 1 C1 possible implied by (ii) 2 notC1 possible implied by (iii) or (iv) 3 C2 possible implied by (iii) 4 notC2 possible implied by (ii) or (iv) 5 E possible implied by (ii) or (iii) 6 notE possible implied by (i) + (iv) 7 C1 E possible implied by (ii) 8 C1 notE open 9 notC1 E possible implied by (iii) 10 notC1 notE possible implied by (i) + (iv) 11 C2 E possible implied by (iii) 12 C2 notE open 13 notC2 E possible implied by (ii) 14 notC2 notE possible implied by (i) + (iv) 15 C1 C2 open 16 C1 notC2 possible implied by (ii) 17 notC1 C2 possible implied by (iii) 18 notC1 notC2 possible (iv) 19 C1 C2 E open 20 C1 C2 notE open 21 C1 notC2 E possible (ii) 22 C1 notC2 notE open 23 notC1 C2 E possible (iii) 24 notC1 C2 notE open 25 notC1 notC2 E impossible (i) 26 notC1 notC2 notE possible implied by (i) + (iv)

Two phenomena C1, C2 may be called contingent causes of some other phenomenon E, only if all the above conditions (i.e. the four defining clauses) are satisfied. In such case, we may call E a tenuous effect[12] of each of C1, C2. Of course, the compound (notC1 + notC2) is a necessary cause of E, since in its presence, notE follows (as given in clause (i)); and in its absence, i.e. if not(notC1 + notC2), notE does not invariably follow (as evidenced by clauses (ii) and (iii)). Rows 19-26 of the above table (shaded) constitute the matrix of contingent causation.

We may thus speak of this phenomenon as a composition of contingent causes; and stress that that C1 and C2 belong in that particular causation of E by calling them complementary contingent causes of it. Indeed, instead of saying “C1 and C2 are complementary contingent causes of E”, we may equally well formulate our sentence as “C1 (complemented by C2) is a contingent cause of E” or as “C2 (complemented by C1) is a contingent cause of E”. These three forms are identical, except for that the first treats C1 and C2 with equal attention, whereas the latter two lay stress on one or the other cause. Such reformatting, as will be seen, is useful in some contexts.

We may make a distinction between absolute and relative contingent causation, as follows. The ‘absolute’ form specifies one contingent cause without mentioning the complement(s) concerned; it just says: “C1 is a contingent cause of E”, meaning “C1 (with some unspecified complement) is a contingent cause of E”. This is in contrast to the ‘relative’ form, which does specify a complement, as in the above example of “C1 (complemented by C2) is a contingent cause of E”. This distinction reflects common discourse. Its importance will become evident when we consider negations of such forms.

Here again, we can demonstrate that our definition of contingent causation conforms to the paradigm of causation through nesting. We may rewrite it as follows:

 From (i) if notC2, then (if notC1, then notE); from (ii) if notC2, then (if C1, not-then notE); from (iii) if C2, not-then (if notC1, then notE).

Clause (i) tells us that given notC2, notC1 implies notE. Clause (ii) tells us that given notC2, C1 does not imply notE. Thus, under condition notC2, C1 behaves like a necessary cause of E. Moreover, clause (iii) shows that under condition C2, C1 ceases to so behave. Similarly, mutadis mutandis, C2 behaves conditionally like a necessary cause of E.

Note well that the main clause of contingent causation is not “If not(C1 + C2), then notE”[13], but more specifically “If (notC1 + notC2), then notE”. Considering that in partial causation the antecedent is (C1 + C2) and that this compound behaves as a complete cause, one might think that in contingent causation the antecedent would be a negation of the same compound, i.e. not(C1 + C2), which would symmetrically behave as a necessary cause. But the above demonstration of conformity to paradigm shows us that this is not the case. The explanation is simply that two of the alternative expressions of “If not(C1 + C2), then notE”, namely “If (C1 + notC2), then notE” and “If (notC1 + C2), then notE” are contradictory to clauses (ii) and (iii), respectively. Therefore, only “If (notC1 + notC2), then notE” is a formally appropriate expression in this context. Our definition of contingent causation is thus correct.

We need not repeat our further analysis of partial causation for contingent causation; all that has been said for the former can be restated, mutadis mutandis, for the latter. For partial and contingent causation are ‘mirror images’ of each other. The one represents the positive aspect of weak causation; the other, the negative aspect. All their characteristics are identical, except that the polarities of their respective causes and effect are opposite: C1 is replaced by notC1, C2 by notC2, and E by notE, or vice-versa.

Note that partial and contingent causation each involves a plurality of causes, though in a different sense from that found in parallelism.

We should also mention that partial causation often underlies alternation or plurality of effects.

Consider the form “If C, then (E or E1)”, which may be interpreted as “the conjunction (C + notE + notE1) is impossible”, and therefore implies “If (C + notE), then E1” and “If (C + notE1), then E”. Take the latter, for instance, and you have a type (i) clause. If additionally it is true that (notC + notE1 + notE), (C + E1 + notE), (C + notE1) are possible conjunctions, you have clauses of types (ii), (iii) and (iv), respectively. In such case[14], C is a partial cause of E (the other partial cause being notE1 or, more precisely, some complete and necessary cause of notE1).

Just as we may have plurality of effects in partial causation, so we may have it in contingent causation.

Note, concerning the term ‘occasional’. When parallel complete causes may occur separately (i.e. neither implies the other), they are often called occasional causes; however, note well, the same term is often used to refer to partial causes, in the sense that each of them is effective only when the other(s) is/are present. The term occasional effect is used with reference to alternation of effects; i.e. when a cause has alternative effects, each of the latter is occasional; but the term is also applicable more generally, to any effect of a partial cause as such, i.e. to contingent effects.

Partial and contingent causation may conceivably occur in tandem or separately; i.e. no formal inconsistency arises in such cases.

Before going further let us here deal with parallelism in relation to the weaker determinations of causation.

In partial causation, this would mean, that there are two (or more) sets of two (or more) partial causes, viz. C1, C2… and C3, C4… (and so forth), with the same effect E:

If (C1 + C2…), then E; etc.

If (C3 + C4…), then E; etc.

Clearly, we have ‘plurality of causes’ in both senses of the term at once, here. By “etc.”, I refer to the further clauses involved in partial causation, such as “if (C1 + notC2), not-then E” and so on, here left unsaid to avoid repetitions. Such statements may be merged; thus, the above two become a single statement in which each bracketed conjunction constitutes an alternative complete cause:

If (C1 + C2…) or (C3 + C4…) or…, then E; etc.

The bracketed conjunctions, as we have seen when dealing with parallel complete causes, may be interrelated in various ways except be exhaustive. These interrelations would be expressed in additional statements. The resulting information, including the above statement where all the conjunctions are disjoined in a single antecedent and all statements not explicitated[15] here, can then be analyzed in great detail by tabulating all the items and their negations, and considering the modus of each combination. We can, in this way, have a clear picture of all eventualities, and avoid all ambiguity.

Similarly, mutadis mutandis, for contingent causation:

If (notC1 + notC2…), then notE; etc.

If (notC3 + notC4…), then notE; etc.

We may merge these complex causal statements, consider additional specifications regarding the opposition of alternatives, and analyze the mass of information through a table.

Note the following special cases of the above parallelisms.

A partial cause may be found common to two (or more) such causations with the same effect; if say C3 is identical with C1, C1 would have C2… as complement(s) in the first relation and C4… as complement(s) in the second, without problem. But may something (say C1) be a partial cause in one relation and its negation (say, notC1 = C3) a partial cause in the other? Yes, since the negation of E would imply both not(C1 + C2…) and not(notC1 + C4…), which is consistent; except that in such case the two compounds could not occur together.

Similarly, a contingent cause may be found common to two (or more) such causations with the same effect; if say C3 is identical with C1, notC1 would have notC2… as complement(s) in the first relation and notC4… as complement(s) in the second, without problem. But may something (say C1) be a contingent cause in one relation and its negation (say, notC1 = C3) a contingent cause in the other? Yes, since the negation of notE would imply both not(notC1 + notC2…) and not(C1 + notC4…), which is consistent; except that in such case the two compounds could not occur together.

We have found the minimal formal definitions of, respectively, complete, necessary, partial and contingent causation. We are now in a position to begin synthesizing our accumulated findings concerning these determinations of causation. Remember how we developed these four concepts….

We started with the paradigm of causation (later named complete and necessary causation). From this we abstracted two constituent forms, or (strong) determinations, which we called complete causation and necessary causation. Then we derived by means of an analogy two additional forms, or (weak) determinations, which we called by way of contrast partial causation and contingent causation.

These four constructs apparently exhaust what we mean by causation, in view of their respective conceptual derivations from the paradigm of causation, and of their symmetry in relation to each other and the whole. No further expressions of the concept of causation, direct or indirect, seem conceivable.

The four forms thus identified can thus be referred to as the genera of causation, or as its generic determinations. And we can safely postulate that:

Nothing can be said to be a cause or effect of something else (in the causative sense), if it is not related to it in the way of at least one of these four genera of causation.

We shall need symbols for these four genera, to facilitate their discussion. I propose (remember them well) the following letters, simply:

n for Necessary causation,

m for coMplete causation (to rhyme with n),

p for Partial causation, and

q for ‘Qontingent’ causation (to rhyme with p)[16].

This notation will be found particularly useful when we deal with causative syllogism. We will also occasionally distinguish between absolute and relative partial or contingent causation, by means of the symbols: pabs and qabs for absolutes (i.e. those not mentioning any complement) and prel and qrel for relatives (i.e. those specifying some complement). Unless specified as relative, p and q may always be considered absolute.

It follows from what we have just said that we may interpret the causative proposition “P is a cause of Q” as “P is a complete or necessary or partial or contingent cause of Q (or a consistent combination of these alternatives)”.

It is easy to demonstrate that any compounds of the four genera involving both m and p, and/or both n and q, are inconsistent, i.e. formally excluded. That is, one and the same thing cannot be both a complete and partial cause of the same effect; for if clause (i) of m, namely “if C1, then E”, is true, then clause (iii) of p, namely “if (C1 + notC2), not-then E”, cannot be true, and vice-versa. Similarly, necessary and contingent causation, i.e. n and q, are incompatible. We shall see at a later stage that certain other combinations are also formally impossible.

We shall consider the remaining, consistent compounds involving the four generic determinations, which we shall call the specific determinations, in the next chapter.

We may, as already suggested, refer to something as a strong cause, if it is a complete and/or necessary cause; and to something as a weak cause, if it is a partial and/or contingent cause. Conversely, a necessary and/or dependent effect may be said to be a strong effect; and a contingent and/or tenuous effect, it may be said to be a weak effect. Mixtures of these characters are conceivable, as we shall see.

Another classification based on common characters: if something is known to be a complete or partial cause, it may be called a ‘contributing cause[17]; and if something is known to be a necessary or contingent cause, it may be called a ‘possible cause’. Likewise, if something is known to be a necessary or contingent effect, it may be called a ‘possible effect’; and if something is known to be a dependent or tenuous effect, it may be called (say) a ‘subject effect’.

Moreover: we have characterized complete and partial causation as positive aspects of causation; and necessary and contingent causation as its negative aspects, comparatively. We may in this sense, relative to a given set of items, speak of ‘positive’ or ‘negative’ causation. The latter, of course, should not be confused with negations of causation. Accordingly, we may refer to positive or negative causes or effects.

The reader is referred to the Appendix on J. S. Mill’s Methods, for comparison of our treatment of causation in this chapter (and the next).

So far, we have only considered in detail positive causative propositions, i.e. statements affirming causation of some determination. We must now look at negative causative propositions, i.e. statements denying causation of some determination or any causation whatever. For this purpose, to avoid the causal connotations implied by use of symbols like C and E for the items involved, we shall rather use neutral symbols like P and Q.

Statements denying causation may be better understood by studying the negations of conditional propositions.

A ‘positive hypothetical’ proposition has the form “If X, then Y” (which may be read as X implies Y, or X is logically followed by Y); it means by definition “the conjunction (X + notY) is impossible”. Its contradictory is a ‘negative hypothetical’ proposition of the form “If X, not-then Y”[18] (which may be read as X does not imply Y, or X is not logically followed by Y); it means by definition “the conjunction (X + notY) is possible”.

In the positive form, though X and notY are together impossible, they are not implied (or denied) to be individually impossible. In the negative form, since X and notY are possible together, each of X, notY is also formally implied as possible. In either form, there is no formal implication that notX be possible or impossible, or that Y be possible or impossible. As for the remaining conjunctions (X + Y), (notX + Y), (notX + notY) – nothing can be inferred concerning them, either.

However, as we have seen, when such statements appear as implicit clauses of causation, the interactions between clauses will inevitably further specify the situation for many of the items concerned.

The negation of complete causation or necessary causation, through statements like “P is not a complete cause of Q” or “P is not a necessary cause of Q”, is feasible if any one or more of the three constituent clauses of such causation is deniable. That is, such negation consists of a disjunctive proposition saying “not(i) and/or not(ii) and/or not(iii)”, which may signify non-causation or another determination of causation (necessary instead of complete, or vice-versa, or a weaker form of causation).

To give an example: the denial of “P is a complete cause of Q” means “if P, not-then Q” and/or “if notP, then Q” and/or “P is impossible”. These alternatives may give rise to different outcomes; in particular note that if “P is impossible” is true, then P cannot be a cause at all, and if “if P, then Q” and “if notP, then Q” are both true, then Q is necessary, in which case Q cannot be an effect at all.

The negation of strong causation as such means the negation of both complete and necessary causation.

With regard to negation of partial or contingent causation, we must distinguish two degrees, according as a given complement is intended or any complement whatever.

The more restricted form of negation of partial causation or contingent causation mentions a complement, as in statements like “P1 (complemented by P2) is not a partial cause of Q” or “P1 (complemented by P2) is not a contingent cause of Q”. Such negation is feasible if any one or more of the four constituent clauses of such causation is deniable. That is, such negation consists of a disjunctive proposition saying “not(i) and/or not(ii) and/or not(iii) and/or not(iv)”.

In contrast, note well, the negation of partial causation or contingent causation through statements like “P1 is not a partial cause of Q” or “P1 is not a contingent cause of Q”, is more radical. “P1 is not a partial cause of Q” means “P1 (with whatever complement) is not a partial cause of Q” – it may thus be viewed as a conjunction of an infinite number of more restricted statements, viz. “P1 (complemented by P2) is not a partial cause of Q, and P1 (complemented by P3) is not a partial cause of Q, and… etc.”, where P2, P3, etc. are all conceivable complements. Similarly with regard to “P1 is not a contingent cause of Q”.

A restricted negative statement is very broad in its possible outcomes: it may signify that P1 is not a cause of Q at all, or that P1 is instead a complete or necessary cause of Q, or that P1 is a weak cause of Q but a contingent rather than partial one or a partial rather than contingent one, or that P1 is a partial or contingent cause (as the case may be) of Q but with some complement other than P2.

A radical negative statement comprises many restricted ones, and is therefore less broad in its possible outcomes, specifically excluding that P1 be involved in a partial or contingent causation (as the case may be) with any complement(s) whatsoever. A restricted negation is relative to a complement (say, P2); a radical negation is a generality comprising all similar restricted negations for the items concerned (P1, Q), and is therefore relative to no complement (neither P2, nor P3, etc.).

The negation of weak causation as such means the negation of both partial and contingent causation, either in a restricted sense (i.e. relative to some complement) or in a radical sense (i.e. irrespective of complement).

This brings us to the relation of non-causation, which is also very complex.

As we saw, the positive causative proposition “P is a cause of Q” may be interpreted as “P is a complete or necessary or partial or contingent cause of Q”. Accordingly, we may interpret the negative causative proposition “P is not a cause of Q” as “P is not a complete and not a necessary and not a partial and not a contingent cause of Q”, i.e. as a denial of all four genera of causation in relation to P and Q (with whatever complement).

It is noteworthy that we cannot theoretically define non-causation except through negation of all the concepts of causation, which have to be defined first[19]. In contrast, on a practical level, we proceed in the opposite direction: in accord with general rules of induction, we presume any two items P and Q to be without causative relation, until if ever we can establish inductively or deductively that a causative relation obtains between them.[20]

Nevertheless, ‘non-causation’ refers to denial of causation, and is not to be confused with ignorance of causation; it is an ontological, not an epistemological concept.

Note well that non-causation is not defined by the propositions “if P, not-then Q, and if notP, not-then notQ”. Such a statement, though suggestive of non-causation, is equally compatible with partial and/or contingent causation; so it cannot suffice to distinguish non-causation. To specify a relation of non-causation, we have to deny every determination of causation.

Furthermore, “P is not a cause of Q” refers to relative non-causation – it is relative to the items P and Q specifically, and does not exclude that Q may have some other cause P1, or that P may have some other effect Q1. Two items, say P and Q, taken at random, need not be causatively related at all (even in cases where they happen to be respectively causatively related to some third item, as will be seen when we study syllogism in later chapters). In such case, P and Q are called accidents of each other; their eventual conjunction is called a coincidence.

Relative non-causation is an integral part of the formal system of deterministic causality. We have to acknowledge the possibility, indeed inevitability, of such a relation. If I say “the position of stars does not affect[21] people’s destinies”, I mean that there is no causal relation specifically between stars and people; yet I may go on to say that stars affect other things or that people are affected by other things, without contradicting myself.

Relative non-causation should not be confused with absolute non-causation. The causelessness of some item A would be expressed as “nothing causes A”, a proposition summarizing innumerable statements of the form “B does not cause A; C does not cause A;…etc.”, where B, C,… are all existents other than A. Similarly, the effectlessness of some item A would be expressed as “nothing is caused by A”, a proposition summarizing innumerable statements of the form “B is not caused by A; C is not caused by A;… etc.”, where B, C,… are all existents other than A.

We thus see that whereas positive causative propositions are defined by conjunctions of clauses, negative ones are far more complex in view of their involving disjunctions.

The negations of determinations, or the negation altogether of causation, should not themselves be regarded as further determinations, since they by their breadth allow for non-causation (between the items concerned), note.

[1] The expression “X makes Y impossible” means that X makes notY necessary, incidentally.

[2] We commonly say, in such case, that C is a sine qua non (Latin for ‘without which not’) or proviso of E.

[3] I use the word ‘item’ to refer to a cause or effect (or the negation of a cause or effect), indifferently. An item is, thus, for the logician, primarily a thesis (in the largest sense), i.e. a categorical or other form of proposition. But an item may also signify a term, since theses are ultimately predications. An item, then, is a thesis, or term within a thesis, involved in a causal proposition.

[4] In some but not all cases, notE not only implies but causes notC, note.

[5] Notice that clause (i), here, in necessary causation, was labeled as clause (ii) in complete and necessary causation. The numbering is independent.

[6] In some but not all cases, E not only implies but causes C, note.

[7] These three forms are implied, respectively, by our first givens; but they do not imply them unconditionally.

[8] We can also, incidentally, view the matter as follows, by focusing on the nested clauses. Clauses (i) and (iii) mean that the partial cause C2 of E may be regarded as a complete cause of the new effect “if C1, then E”. Similarly, mutadis mutandis, clauses (i) and (ii) can be taken to mean that C1 is a complete cause of “if C2, then E”.

[9] Note that the combination “notC1 + notC2” may occasionally be impossible. In such case, notC1 implies C2 and notC2 implies C1. But according to syllogistic theory (see FL, ch. 29.1), this would not allow us to abbreviate clauses (ii) and (iii) of the definition to “If notC1, not-then E” and “If notC2, not-then E”. Thus, even in such case, the definition remains unaffected.

[10] This phrase “keeping all other things equal” is not mine – but a consecrated phrase often found in textbooks. I do not know who coined it first.

[11] For instance, with reference to concomitant variations (see Appendix on J. S. Mill’s Methods); if the C1 and C2 enter in a mathematical formula like, say, E = C12 + C2, C1 has less weight than C2.

[12] I use the name “tenuous effect” for lack of a better one, to signify a lesser degree of non-independence than a “dependent effect”. Alternatively, broadening the connotation of dependence, we might say that the effect of a necessary cause is strongly dependent (it depends on that one cause) and the effect of a contingent cause is weakly dependent (it depends on that cause, if no other is available).

[13] This form, note well, does not specify which of the three alternative combinations (C1 + notC2), (notC1 + C2) or (notC1 + notC2) implies notE; it means only that at least one of them does.

[14] As can be seen by renaming C as “C1” and notE1 as “C2”.

[15] i.e. made explicit.

[16] I have previously used, in my work Future Logic, the letters n and p for the modalities of necessity and possibility (or more specifically, particularity or potentiality). These should not be confused, note well. In any case, their relations are very different. In modality, n implies p (i.e. if something is necessary, it is possible). But here, in causation, as we shall soon see, n and p are merely compatible (i.e. a necessary cause need not be a partial cause, though something may be both a necessary and partial cause).

[17] In the sense that it is a cause to some extent, sufficient or not.

[18] The proposition “If X, not-then Y” is not to be confused with “If X, then notY”, note well. The latter implies but is not implied by the former.

[19] We shall later see that this truism is ignored by some philosophers.

[20] The philosophical problems of defining causation (its forms) and identifying specific cases of causation (its contents), are distinct, as we shall see.

[21] To ‘affect’ some thing is to cause a change in it.

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