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THE LOGIC OF CAUSATION

© Avi Sion, 1999. All rights reserved.

Phase One: Macroanalysis

Chapter 2 -  The Generic Determinations.

1.    Strong Determinations.

2.    Parallelism of Strongs. 

3.    Weak Determinations.

4.    Parallelism of Weaks.  

5.    The Four Genera of Causation.  

6.    Negations of Causation.  

1.    Strong 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. 

 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. 

 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. 

2.    Parallelism of Strongs. 

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, p.151): 

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, p. 151-153): 

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, p.151): 

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, p. 151-153): 

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. 

3.    Weak Determinations. 

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.