THE LOGIC OF CAUSATION

Phase One: Macroanalysis

Chapter 4 – Immediate Inferences.

The logical interrelations between the truths and falsehoods of propositions involving the same items are referred to as their ‘oppositions’. This expression is unfortunate, because in everyday speech (and often in logical discourse) it connotes more specifically a ‘conflict’ between propositions; whereas in the science of logic, the term is intended more broadly as a ‘face-off’. Thus, the possible oppositions between two propositions are:

· contradiction (provided they cannot be both true and they cannot be both false);

· contrariety (provided they cannot be both true);

· subcontrariety (provided they cannot be both false);

· subalternation (provided one, called the subalternant, cannot be true if the other, called the subaltern, is false – though the latter may be true if the former is false);

· equivalence (provided neither can be true if the other is false, and neither can be false if the other is true);

· neutrality (provided either can be true or false without the other being true or false).

Contradictory or contrary propositions are incompatible or mutually exclusive; propositions otherwise related are compatible or conjoinable. Equivalents mutually imply each other; among subalternatives, the subalternant implies but is not implied by the subaltern. Propositions are neutral to each other if they are not related in any of the other ways above listed.

Let us now consider the oppositions between the four generic determinations. We can show, with reference to the definitions in the preceding chapter, that:

· If “P is a complete cause of Q”, then “(whatever P is complemented by) P is not a partial cause of Q”;

· if “P (with whatever complement) is a partial cause of Q”, then “P is not a complete cause of Q”.

· If “P is a necessary cause of Q”, then (whatever P is complemented by) “P is not a contingent cause of Q”;

· if “P (with whatever complement) is a contingent cause of Q”, then “P is not a necessary cause of Q”.

For clause (i) of complete causation, viz. “if P, then Q”, implies both “if (P + R), then Q” and “if (P + notR), then Q”, for any item R whatsoever; whereas clause (iii) of partial causation implies that there is an item R such that “if (P + R), then Q”. Similarly, necessary and contingent causation have conflicting implications, and therefore cannot both be true.

More briefly put, m and p are incompatible and n and q are incompatible. Other than that, no incompatibilities or implications exist between the four generic determinations. P may have no causative relation to Q at all, without any inconsistency ensuing. Thus, m and p are contrary (but not contradictory) and n and q are contrary (but not contradictory). As for the pairs m and n, m and q, n and p, p and q – they are all neutral to each other.

With regard to the negations of generic determinations, it follows that they are all neutral to each other. Of course, by definition, not-m is the contradictory of m, not-n is the contradictory of n, not-p is the contradictory of p, and not-q is the contradictory of q. Also, as above seen, m implies not-p, and n implies not-q. But between the four negations themselves, no incompatibilities or implications exist.

It should be stressed that partial causation is not to be considered as identical with the negation of complete causation, but only as one of the possible outcomes of such negation. That is, it would be illogical to infer from “P is not a complete cause of Q” that “P is a partial cause of Q”, or from “P is not a partial cause of Q” that “P is complete a cause of Q”. The labels ‘complete’ and ‘partial’ could be misleading, connoting a relation of inclusion between whole and part; here, note well, ‘complete’ excludes ‘partial’, and vice-versa. Similarly, of course, contingent causation is not equivalent to the negation of necessary causation.

Most importantly, keep in mind the inferences already mentioned in the last section of the preceding chapter, namely:

• If m is true, then n or q must be true.
• If n is true, then m or p must be true.
• If p is true, then n or q must be true.
• If q is true, then m or p must be true.
• If neither m nor p is true, then neither n nor q can be true.
• If neither n nor q is true, then neither m nor p can be true.

With regard to the oppositions between the four joint determinations.

Each of the four joint determinations obviously implies but is not implied by its constituent generic determinations. That is, m and n are subalterns of mn, m and q are subalterns of mq, and so forth. It follows that each joint determination is contrary to the negations of its constituent generic determinations. That is, mn is contrary to not-m and to not-n; and so forth. Or in other words, if either or both of its constituent generic determinations is denied, the joint determination as a whole must be denied.

Furthermore, the four joint determinations are all mutually exclusive. That is, if any one of them is true, the three others have to be false. For if mn is true, mq cannot be true (since n and q are incompatible), and np cannot be true (since m and p are incompatible), and pq cannot be true (for both reasons). Similarly, if we affirm mq, we must deny the combinations mn, np, pq; and so forth. On the other hand, the negation of any joint determination has no consequence on the others; they may all be false without resulting inconsistency.

Immediate inference is inference of a conclusion from one premise, in contrast to syllogistic (or mediate) inference. ‘Opposition’ is one form of it, in which the items concerned retain the same position and polarity. ‘Eduction’ is another form of it, involving some change in position and/or polarity of the items occurs.

Let us now look into the feasibility of eductions from causative propositions, with reference to their definitions. We shall for now ignore the issue of direction of causation, dealt with further on. All the usual eductive processes[1], namely inversion, conversion, and contraposition, obversion, obverted-inversion, obverted-conversion, and obverted-contraposition, can be used in the ways shown below. First however, we must consider eduction by negation of the complement, a process applicable to the weak determinations.

a. Negations of the complement (from R to notR) for the same items (P-Q). This concerns the weak determinations, and results in a negative conclusion.

· “P (complemented by R) is a partial cause of Q” implies “P (complemented by notR) is not a partial cause of Q”.

Proof: Clause (i) of “P (complemented by R) is a partial cause of Q” and clause (iii) of “P (complemented by notR) is a partial cause of Q” contradict each other; therefore they are incompatible propositions.

In contrast, note, “P (complemented by R) is a partial cause of Q” is compatible with “P (complemented by notR) is a contingent cause of Q”.

· “P (complemented by R) is a contingent cause of Q” implies “P (complemented by notR) is not a contingent cause of Q”.

Proof: In a similar manner, mutadis mutandis.

In contrast, note, “P (complemented by R) is a contingent cause of Q” is compatible with “P (complemented by notR) is a partial cause of Q”.

Negation of the complement for the joint determination pq follows by conjunction:

· If P (complemented by R) is a partial and contingent cause of Q, then P (complemented by notR) is neither a partial nor a contingent cause of Q.

b. Inversions (changes from P-Q to notP-notQ); the conclusion is called the inverse of the premise.

All four generic determinations are invertible to a positive causative proposition, simply by substituting not{notP} for P, not{notQ} for Q. In the case of weak determinations, additionally, not{notR} replaces R; and moreover, eduction by negation of the complement of the positive conclusion yields a further negative conclusion. Thus,

· “P is a complete cause of Q” implies “notP is a necessary cause of notQ”.

And vice-versa. In contrast, note, “P is a complete cause of Q” and “notP is a complete cause of notQ” are merely compatible.

· “P is a necessary cause of Q” implies “notP is a complete cause of notQ”.

And vice-versa. In contrast, note, “P is a necessary cause of Q” and “notP is a necessary cause of notQ” are merely compatible.

· “P (complemented by R) is a partial cause of Q” implies “notP (complemented by notR) is a contingent cause of notQ”.

And vice-versa. It follows by negation of the complement that:

· “P (complemented by R) is a partial cause of Q” implies “notP (complemented by R) is not a contingent cause of notQ”.

In contrast, note, “P (complemented by R) is a partial cause of Q” is compatible with “notP (complemented by R) is a partial cause of notQ” and with “notP (complemented by notR) is a partial cause of notQ”.

· “P (complemented by R) is a contingent cause of Q” implies “notP (complemented by notR) is a partial cause of notQ”.

And vice-versa. It follows by negation of the complement that:

· “P (complemented by R) is a contingent cause of Q” implies “notP (complemented by R) is not a partial cause of notQ”.

In contrast, note, “P (complemented by R) is a contingent cause of Q” is compatible with “notP (complemented by R) is a contingent cause of notQ” and with “notP (complemented by notR) is a contingent cause of notQ”.

Notice, with regard to the positive implications of the weak determinations, that P, Q, and R all change polarity. Evidently, inversions involve a change of determination from positive (complete or partial) to negative (necessary or contingent, respectively), or vice-versa.

With regard to the joint determinations, their inversions follow from those relative to the generic determinations.

Inversion of mn or pq is possible, without change of determination (i.e. to mn or pq, respectively), since the changes for each constituent determination balance each other out; and all items change polarity. Thus:

· If P is a complete and necessary cause of Q, then notP is a complete and necessary cause of notQ.

· If P (complemented by R) is a partial and contingent cause of Q, then notP (complemented by notR) is a partial and contingent cause of notQ; also, notP (complemented by R) is not a partial or contingent cause of notQ.

Inversion of mq or np is possible, though with changes of determination (i.e. to np or mq, respectively); and all items change polarity. Thus:

· If P is a complete and (complemented by R) a contingent cause of Q, then notP is a necessary and (complemented by notR) a partial cause of notQ; also, notP (complemented by R) is not a partial cause of notQ.

· if P is a necessary and (complemented by R) a partial cause of Q, then notP is a complete and (complemented by notR) a contingent cause of notQ; also, notP (complemented by R) is not a contingent cause of notQ.

With regard to negative causative propositions, we can easily derive analogous inversions on the basis of[2] the above findings:

· “P is not a complete cause of Q” implies “notP is not a necessary cause of notQ”.

· “P is not a necessary cause of Q” implies “notP is not a complete cause of notQ”.

· “P (complemented by R) is not a partial cause of Q” implies “notP (complemented by notR) is not a contingent cause of notQ”.

· “P (complemented by R) is not a contingent cause of Q” implies “notP (complemented by notR) is not a partial cause of notQ”.

Similarly for the negations of joint determinations.

c. Conversions (changes from P-Q to Q-P); the conclusion is called the converse of the premise.

The strong generic determinations are convertible, as follows:

· “P is a complete cause of Q” implies “Q is a necessary cause of P”.

Proof: Clause (i) of the given proposition may be contraposed to “if notQ, then notP”; clauses (i) and (iii) together imply that (P + Q) is possible, which means that “if Q, not-then notP”; and clause (ii) implies “notQ is possible”. Thus, the conditions for the said conclusion are satisfied, and conversion is feasible.

And vice-versa. In contrast, note, “P is a complete cause of Q” and “Q is a complete cause of P” are merely compatible.

· “P is a necessary cause of Q” implies “Q is a complete cause of P”.

Proof: In a similar manner, mutadis mutandis.

And vice-versa. In contrast, note, “P is a necessary cause of Q” and “Q is a necessary cause of P” are merely compatible.

The weak generic determinations are also convertible, as follows:

· “P (complemented by R) is a partial cause of Q” implies “Q (complemented by notR) is a contingent cause of P”.

Proof: Clause (i) of the given proposition means that (P + R + notQ) is impossible, which may be restated as “if (notQ + R), then notP”; clauses (i) and (iv) together imply that (P + R + Q) is possible, which means that “if (Q + R), not-then notP”; clause (iii) means that (P + notR + notQ) is possible, which may be restated as “if (notQ + notR), not-then notP”; and clause (ii) implies “(notQ + R) is possible”. Thus, the conditions for the said conclusion are satisfied (reading not{notR} instead of R), and conversion is feasible.

And vice-versa. It follows by negation of the complement that:

· “P (complemented by R) is a partial cause of Q” implies “Q (complemented by R) is not a contingent cause of P”.

In contrast, note, “P (complemented by R) is a partial cause of Q” is compatible with “Q (complemented by R) is a partial cause of P” and with “Q (complemented by notR) is a partial cause of P”.

· “P (complemented by R) is a contingent cause of Q” implies “Q (complemented by notR) is a partial cause of P”.

Proof: In a similar manner, mutadis mutandis.

And vice-versa. It follows by negation of the complement that:

· “P (complemented by R) is a contingent cause of Q” implies “Q (complemented by R) is not a partial cause of P”.

In contrast, note, “P (complemented by R) is a contingent cause of Q” is compatible with “Q (complemented by R) is a contingent cause of P” and with “Q (complemented by notR) is a contingent cause of P”.

Note well, with regard to the positive implications of the weak determinations, that R changes polarity, while P, Q do not; in this sense, their conversion may be qualified as imperfect. Evidently, conversions involve a change of determination from positive (complete or partial) to negative (necessary or contingent, respectively), or vice-versa.

With regard to the joint determinations, their conversions follow from those relative to the generic determinations.

Conversion of mn is possible, without change of determination (i.e. to mn), since the changes for each constituent determination balance each other out. Thus:

· If P is a complete and necessary cause of Q, then Q is a complete and necessary cause of P.

Conversion of pq is possible, without change of determination (i.e. to pq), for the same reason; but the subsidiary item (R) changes polarity in the positive implication. Thus:

· If P (complemented by R) is a partial and contingent cause of Q, then Q (complemented by notR) is a partial and contingent cause of P; also, Q (complemented by R) is not a partial or contingent cause of P.

Conversion of mq or np is possible, though with changes of determination (i.e. to np or mq, respectively); also, the subsidiary item (R) changes polarity in the positive implication. Thus:

· If P is a complete and (complemented by R) a contingent cause of Q, then Q is a necessary and (complemented by notR) a partial cause of P; also, Q (complemented by R) is not a partial cause of P.

· If P is a necessary and (complemented by R) a partial cause of Q, then Q is a complete and (complemented by notR) a contingent cause of P; also, Q (complemented by R) is not a contingent cause of P.

With regard to negative causative propositions, we can easily derive analogous conversions on the basis of the above findings:

· “P is not a complete cause of Q” implies “Q is not a necessary cause of P”.

· “P is not a necessary cause of Q” implies “Q is not a complete cause of P”.

· “P (complemented by R) is not a partial cause of Q” implies “Q (complemented by notR) is not a contingent cause of P”.

· “P (complemented by R) is not a contingent cause of Q” implies “Q (complemented by notR) is not a partial cause of P”.

Similarly for the negations of joint determinations.

d. Contrapositions (changes from P-Q to notQ-notP); the conclusion is called the contraposite of the premise.

All four generic determinations are contraposable, simply by conversion of their inverses:

· “P is a complete cause of Q” implies “notQ is a complete cause of notP”.

And vice-versa. In contrast, note, “P is a complete cause of Q” and “notQ is a necessary cause of notP” are merely compatible.

· “P is a necessary cause of Q” implies “notQ is a necessary cause of notP”.

And vice-versa. In contrast, note, “P is a necessary cause of Q” and “notQ is a complete cause of notP” are merely compatible.

· “P (complemented by R) is a partial cause of Q” implies “notQ (complemented by R) is a partial cause of notP”.

And vice-versa. It follows by negation of the complement that:

· “P (complemented by R) is a partial cause of Q” implies “notQ (complemented by notR) is not a partial cause of notP”.

In contrast, note, “P (complemented by R) is a partial cause of Q” is compatible with “notQ (complemented by R) is a contingent cause of notP” and with “notQ (complemented by notR) is a contingent cause of notP”.

· “P (complemented by R) is a contingent cause of Q” implies “notQ (complemented by R) is a contingent cause of notP”.

And vice-versa. It follows by negation of the complement that:

· “P (complemented by R) is a contingent cause of Q” implies “notQ (complemented by notR) is not a contingent cause of notP”.

In contrast, note, “P (complemented by R) is a contingent cause of Q” is compatible with “notQ (complemented by R) is a partial cause of notP” and with “notQ (complemented by notR) is a partial cause of notP”.

Notice, with regard to the positive implications of the weak determinations, that while P, Q change polarity, R does not; in this sense, their contraposition may be qualified as imperfect. Evidently, contrapositions distinctively do not involve changes of determination.

With regard to the joint determinations, their contrapositions follow from those relative to the generic determinations.

Contraposition of mn is possible, without change of determination (i.e. to mn). Thus:

· If P is a complete and necessary cause of Q, then notQ is a complete and necessary cause of notP.

Contraposition of pq, mq or np is possible, without change of determination (i.e. to pq, mq or np, respectively); and the subsidiary item (R) does not change polarity in the positive implication. Thus:

· If P (complemented by R) is a partial and contingent cause of Q, then notQ (complemented by R) is a partial and contingent cause of notP; also, notQ (complemented by notR) is not a partial or contingent cause of notP.

· If P is a complete and (complemented by R) a contingent cause of Q, then notQ is a necessary and (complemented by R) a partial cause of notP; also, notQ (complemented by notR) is not a partial cause of notP.

· If P is a necessary and (complemented by R) a partial cause of Q, then notQ is a complete and (complemented by R) a contingent cause of notP; also, notQ (complemented by notR) is not a contingent cause of notP.

With regard to negative causative propositions, we can easily derive analogous contrapositions on the basis of the above findings:

· “P is not a complete cause of Q” implies “notQ is not a complete cause of notP”.

· “P is not a necessary cause of Q” implies “notQ is not a necessary cause of notP”.

· “P (complemented by R) is not a partial cause of Q” implies “notQ (complemented by R) is not a partial cause of notP”.

· “P (complemented by R) is not a contingent cause of Q” implies “notQ (complemented by R) is not a contingent cause of notP”.

Similarly for the negations of joint determinations.

e. Obversions (changes from P-Q to P-notQ); the conclusions are called obverses of the premise.[3]

All four generic determinations are obvertible in various ways, though the obverses are negative causative propositions.

· “P is a complete cause of Q” implies “P is not a complete cause of notQ” and “P is not a necessary cause of notQ”.

Proof: Clauses (i) and (iii) of “P is a complete cause of Q” together imply that (P + Q) is possible, whereas clause (i) of “P is a complete cause of notQ” implies that conjunction impossible; therefore they are incompatible propositions. Also, clause (i) of “P is a complete cause of Q” and clause (ii) of “P is a necessary cause of notQ” contradict each other; therefore they are incompatible.

· “P is a necessary cause of Q” implies “P is not a necessary cause of notQ” and “P is not a complete cause of notQ”.

Proof: In a similar manner, mutadis mutandis.

· “P (complemented by R) is a partial cause of Q” implies “P (complemented by R) is not a partial cause of notQ” and “P (complemented by notR) is not a contingent cause of notQ”.

Proof: Clauses (i) and (iv) of “P (complemented by R) is a partial cause of Q” together imply that (P + R + Q) is possible, whereas clause (i) of “P (complemented by R) is a partial cause of notQ” implies that conjunction impossible; therefore they are incompatible propositions. Also, clause (ii) of “P (complemented by R) is a partial cause of Q” and clause (i) of “P (complemented by notR) is a contingent cause of notQ” contradict each other; therefore they are incompatible. Notice in the latter case, the change in polarity of the complement (from R to notR), as well as the change in determination (from p to q).

In contrast, note well, “P (complemented by R) is a partial cause of Q” is compatible with “P (complemented by R) is a contingent cause of notQ”, and with “P (complemented by notR) is a partial cause of notQ”.

· “P (complemented by R) is a contingent cause of Q” implies “P (complemented by R) is not a contingent cause of notQ” and “P (complemented by notR) is not a partial cause of notQ”.

Proof: In a similar manner, mutadis mutandis. Notice in the latter case, the change in polarity of the complement (from R to notR), as well as the change in determination (from q to p).

In contrast, note well, “P (complemented by R) is a contingent cause of Q” is compatible with “P (complemented by R) is a partial cause of notQ”, and with “P (complemented by notR) is a contingent cause of notQ”.

With regard to the joint determinations, their obversions follow from those relative to the generic determinations.

· If P is a complete and necessary cause of Q, then P is neither a complete nor a necessary cause of notQ.

· If P is a complete and contingent cause of Q, then P is neither a complete, nor (complemented by R) a contingent, cause of notQ, and P is neither a necessary, nor (complemented by notR) a partial, cause of notQ.

· If P is a necessary and partial cause of Q, then P is neither a necessary, nor (complemented by R) a partial, cause of notQ, and P is neither a complete, nor (complemented by notR) a contingent, cause of notQ.

· If P (complemented by R) is a partial and contingent cause of Q, then P (whether complemented by R or notR) is neither a partial nor a contingent cause of notQ.

f. Obverted inversions (changes from P-Q to notP-Q); the conclusions are called obverted-inverses of the premise.

All four generic determinations may be subjected to obverted-inversion, by successive inversion then obversion. The conclusions are therefore negative causative propositions.

· “P is a complete cause of Q” implies “notP is not a complete cause of Q” and “notP is not a necessary cause of Q”.

· “P is a necessary cause of Q” implies “notP is not a necessary cause of Q” and “notP is not a complete cause of Q”.

· “P (complemented by R) is a partial cause of Q” implies “notP (complemented by R) is not a partial cause of Q” and “notP (complemented by notR) is not a contingent cause of Q”.

In contrast, note well, “P (complemented by R) is a partial cause of Q” is compatible with “notP (complemented by notR) is a partial cause of Q”, and with “notP (complemented by R) is a contingent cause of Q”.

· “P (complemented by R) is a contingent cause of Q” implies “notP (complemented by R) is not a contingent cause of Q” and “notP (complemented by notR) is not a partial cause of Q”.

In contrast, note well, “P (complemented by R) is a contingent cause of Q” is compatible with “notP (complemented by notR) is a contingent cause of Q”, and with “notP (complemented by R) is a partial cause of Q”.

With regard to the joint determinations, their obverted-inversions follow from those relative to the generic determinations, as usual.

g. Obverted conversions (changes from P-Q to Q-notP); the conclusions are called obverted-converses of the premise.

All four generic determinations may be subjected to obverted-conversion, by successive conversion then obversion. The conclusions are therefore negative causative propositions.

· “P is a complete cause of Q” implies “Q is not a complete cause of notP” and “Q is not a necessary cause of notP”.

· “P is a necessary cause of Q” implies “Q is not a necessary cause of notP” and “Q is not a complete cause of notP”.

· “P (complemented by R) is a partial cause of Q” implies “Q (complemented by R) is not a partial cause of notP” and “Q (complemented by notR) is not a contingent cause of notP”.

In contrast, note well, “P (complemented by R) is a partial cause of Q” is compatible with “Q (complemented by notR) is a partial cause of notP”, and with “Q (complemented by R) is a contingent cause of notP”.

· “P (complemented by R) is a contingent cause of Q” implies “Q (complemented by R) is not a contingent cause of notP” and “Q (complemented by notR) is not a partial cause of notP”.

In contrast, note well, “P (complemented by R) is a contingent cause of Q” is compatible with “Q (complemented by notR) is a contingent cause of notP”, and with “Q (complemented by R) is a partial cause of notP”.

With regard to the joint determinations, their obverted-conversions follow from those relative to the generic determinations, as usual.

h. Obverted contrapositions, also known as conversions by negation (changes from P-Q to notQ-P); the conclusions are called obverted-contraposites of the premise.

All four generic determinations may be subjected to obverted-contraposition, by successive contraposition then obversion. The conclusions are therefore negative causative propositions.

· “P is a complete cause of Q” implies “notQ is not a complete cause of P” and “notQ is not a necessary cause of P”.

· “P is a necessary cause of Q” implies “notQ is not a necessary cause of P” and “notQ is not a complete cause of P”.

· “P (complemented by R) is a partial cause of Q” implies “notQ (complemented by R) is not a partial cause of P” and “notQ (complemented by notR) is not a contingent cause of P”.

In contrast, note well, “P (complemented by R) is a partial cause of Q” is compatible with “notQ (complemented by notR) is a partial cause of P”, and with “notQ (complemented by R) is a contingent cause of P”.

· “P (complemented by R) is a contingent cause of Q” implies “notQ (complemented by R) is not a contingent cause of P” and “notQ (complemented by notR) is not a partial cause of P”.

In contrast, note well, “P (complemented by R) is a contingent cause of Q” is compatible with “notQ (complemented by notR) is a contingent cause of P”, and with “notQ (complemented by R) is a partial cause of P”.

With regard to the joint determinations, their obverted-contrapositions follow from those relative to the generic determinations, as usual.

We may finally note the following derivative eductions, though they are virtually useless except that they partly summarize the preceding findings:

· If P is a strong cause of Q, then notP is a strong cause of notQ (inversion), and Q is a strong cause of P (conversion), and notQ is a strong cause of notP (contraposition).

· If P (complemented by R) is a weak cause of Q, then notP (complemented by notR) is a weak cause of notQ (inversion), and Q (complemented by notR) is a weak cause of P (conversion), and notQ (complemented by R) is a weak cause of notP (contraposition).

· If P is a cause of Q, then notP is a cause of notQ (inversion), and Q is a cause of P (conversion), and notQ is a cause of notP (contraposition).

Moreover, we can say:

· If P is a strong cause of Q, then P is not a strong cause of notQ (obversion), and notP is not a strong cause of Q (obverted inversion), and Q is not a strong cause of notP (obverted conversion), and notQ is not a strong cause of P (obverted contraposition).

But similar negative implications are not possible for “P (complemented by R) is a weak cause of Q”, in view of variations in the complement in such cases. It follows that similar negative implications are not possible for “P is a cause of Q”.

Finally, concerning the weak determinations, it should be noted that wherever the inference results in no change of complement, i.e. wherever the premise and conclusion concern the same complement, the complement need not be mentioned at all. That is, we can in some cases simply say: if “P is a partial (or contingent) cause of Q”, then “(the new cause) is (or is not) a partial (or contingent) cause of (the new effect)” (as the case may be), on the tacit understanding that the complement, whatever it happens to be, has not been altered.

More broadly, whether or not the complement changes polarity, it is clear that we do not need to specify or even remember its precise content, in order to perform the inference. When the complement is unchanged, we need not mention it at all (or, to be sure, we can say in the conclusion “with the same complement, whatever it be”); and when it is changed, we can add in the conclusion “with the negation of the initial complement, whatever it be, as complement”. It is good to know this, because it allows us to proceed with inferences without immediately having to or being able to pin-point the complement involved.

Note lastly, that all immediate inferences could also be validated or invalidated, as the case may be, by means of matricial analysis (see later). I have here preferred the less systematic, but also less voluminous, method of reduction to conditional arguments.

All these inferences add to our knowledge and understanding of causative propositions, of course. Some of them will prove useful for validations or invalidations of causative syllogisms by direct reduction to others.

Now, the implications between different forms of causative propositions identified above, such as that “P is a complete cause of Q” implies “Q is a necessary cause of P”, demonstrate that our definitions of causation were incomplete. For we well know that causation has a direction! However, bear with me – we deal with this issue.

Strictly-speaking, when we utter a statement of the form “P is a (complete, necessary, partial, contingent) cause of Q”, we imply a tacit clause specifying the direction (or ordering of items), in addition to the various clauses (treated in the preceding chapter) concerning determination. This means that denial of the tacit clause on direction would suffice to deny the causation concerned, even if all the other clauses are affirmed. However, there are good reasons why in our formal treatment we are wise to keep the issue of direction separate.

First of these is the epistemological fact that the direction of causation is not always known. We may by inductive or deductive means arrive at knowledge of all the other clauses, and yet be hard put to immediately specify the direction. If we wished to summarize our position in such case, and were not permitted to use the language of causation, we would have to introduce a relational expression other than “is a … cause of” (say, “is a … determinant of”) to allow us to verbalize the situation. Causation would then be defined as the combination of this relation (“determination”) with a directional clause. This is feasible, but in my view redundant; we can manage without such an artifice.

Secondly, we have to consider the ontological fact that causation does not always occur in only one direction: it may occur in both. Sometimes, the direction is exclusively from P to Q, or from Q to P; but sometimes, the causal relation is two-way or reversible. Moreover, reversible causation is not always reciprocal: there may be one determination in one direction, and another in the opposite sense; or there may, in the case of weak causations, be different complements in each direction. For this reason, too, we are wise to handle the issue of direction flexibly, considering it expressed in an additional clause, but left ‘hidden’ or ignored until specifically dealt with. This is the course adopted in the present work.

The directional clause for a causative proposition can be a phrase qualifying the sentence “P is a … cause of Q”, a phrase of the form “in the direction from P to Q” (which is identical with ‘notP to notQ’) and/orin the direction from Q to P” (which is identical with ‘notQ to notP’). We must additionally allow for (hopefully temporary) ignorance with the phrase “direction unknown“.

We allow for only two directions, not four, note well. “P to Q” and its inverse “notP to notQ” are one and the same direction; likewise “Q to P” and “notQ to notP” are identical in direction. In this manner, causative statements remain always or formally invertible – but strictly-speaking only sometimes or conditionally convertible or contraposable, specifically when the causation is known to be reversible. That is, whereas inversion is ontologically universal, conversion and contraposition have the status of formal artifices until and unless their ontological applicability is established in a given case. The latter two eductions, of course, go together; if either is applicable, so is the other (since the contraposite is the inverse of the converse).

As we shall see, consideration of direction of causation affects other deductive processes in a similar manner, i.e. making them conditional instead of universal. Thus, in causative syllogism, arguments in the first figure guarantee the direction implicit in the conclusion (given the directions implied by the premises), whereas arguments in the other two figures cannot do so.

However, this is not a great difficulty, because we know that wherever a causative conclusion is drawn, the direction of causation has to be either as implied by that conclusion or as implied by its formal converse or both. Thus, the issue of direction is relatively minor. It is without impact on the inferred ‘bond’, on the fact that there is a certain (strong or weak) causative relationship between the items concerned; the only problem it sets for us is in which form this relationship is expressed, as ‘P-Q’ or its converse ‘Q-P’.

The real problem with direction of causation is identifying how it is to be induced in the first place. We shall try to solve this problem later, in the chapter on induction of causation. For now, suffices to say the following. In de dicta (logical) causation, theses are hierarchized by their epistemological roles (an axiom causes but is not caused by a resulting theorem, even if the latter implies the former, for instance); in de re (natural, temporal, spatial or extensional) causation, the order of things is often dictated by temporal or spatial sequences, for instances (logical issues also come into play).

There are cases in practice where deciding which item is the cause and which is the effect is virtually a matter of convention. This may occur in reciprocal causation, as well as in causation with permanently unknown (i.e. practically unknowable) direction. In such cases, the expressions “the cause” and “the effect” merge into one, becoming mere verbal differentiations. This is often true in the logical mode, and in the spatial and extensional modes; it occurs more rarely in the temporal and natural modes. The reason being that the only really absolute rule of direction we know is temporal sequence; other rules, though credible, are open to debate.

It should be stressed that the concept of direction (or orientation) concerns not only causation, but more broadly space and time in a variety of guises. It is therefore an issue in a wider and deeper ontological and epistemological context, not one reserved to causation. It might be viewed as one of the fundamental building-blocks of knowledge, and therefore not entirely definable with reference to other concepts.

It may be exemplified concretely by drawing a line on paper (this expresses its spatial component), and running a finger along it first one way, then the other (this expresses its temporal component, since the movement takes time to cover space); and saying “though the path covered is the same in both instances, the first movement is to be distinguished from the second – and this difference will be called one of direction”. In this manner, the words ‘from’ and ‘to’, though very abstract[4], are shown to be meaningful, i.e. to symbolize a communicable distinction, which can by analogy be applied in other contexts.

Such visual and mechanical demonstration merely aids the intuition[5] in focusing upon the intention of verbal expressions of direction. It does not, of course, by itself suffice to clearly define directionality in the context of causation, or to establish the direction of causation in particular cases. We must search for more precise means to achieve these ends. But we at least have a sort of ostensive-procedural definition of directionality in general, which gives some meaning to clauses like “from P to Q” and “from Q to P”.

The propositions “P causes Q” and “Q causes P” are simply declared unequal. Causation in general is symbolized by a string of words, namely “P”-“causation”-“Q”, analogous to a line; this line of relation is, however, to be taken as two-fold, i.e. as occasionally different in the senses P-Q and Q-P. What this difference signifies more deeply in formal terms, we cannot yet say; but we do believe that it exists, and wish to prepare for its linguistic expression by such declaration.

[1] The terminology here used is the same as that traditionally used in other fields of logic, except for “negation of complement”.

[2] Specifically, by contraposition of the positive implications.

[3] The form “P causes notQ” is often reworded in everyday speech as “P prevents Q” (or other similar words like hinders, etc. – see your thesaurus). We could treat the latter expression as a form in its own right, and look into all the logic of its four genera: complete, necessary, partial and contingent prevention. But we do not need to do so, for all that logic is implicit in the work here in process.

[4] We cannot physically point to cases of ‘direction’, we can only point a finger in the direction of concrete objects (dogs, trees) – which is not the same act. A special mental capability and effort is required to transcend the object pointed to and shift attention to the act and significance of pointing, of which direction is an abstract aspect.

[5] Incidentally, even animals seem to intuitively (in the sense of wordlessly, effectively) understand direction. This is suggested, for instance, by their homing abilities, based on visual, auditory, and other sensory data (such as olfactory or gustative, in the case of ants, or magnetic, in the case of migrating birds). They can evidently even grasp direction of causation, knowing and remembering who or what hurt or pleased them. In my experience, however, dogs do not seem to understand finger-pointing; but some people claim that they do.

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