THE LOGIC OF CAUSATION

THE LOGIC OF CAUSATION

Phase One: Macroanalysis

Chapter 3 –The Specific Determinations.

1.The Species of Causation.

2.The Joint determinations.

3.The Significance of Certain Findings.

1.The Species of Causation.

We shall now look into the consistent combinations of the four genera of causation, symbolized asm,n,p,q, with each other or their negations. Implicit in our gradual development of these concepts of causation from a common paradigm, was the idea that they are abstractions, indefinite concepts that are eventually concretized in the more specific and definite compounds.

We have already found some of their combinations, namelympandnqto beinconsistent. This was due to incompatibilities between clauses of their definitions, or in other words, certain rows of their matrices. Thus, row 6 ofm(C + notE is impossible) is in conflict with row 22 (C1 + notE is possible) ofp; similarly, row 7 ofn(notC + E is impossible) is in conflict with row 23 (notC1 + E is possible) ofq.

It is also possible to prove certain other combinations to be logically impossible. This can be done formally, but not at the present stage of development, because we do not yet have the technical means at this stage to treatnegationsof generic determinations. To definenotm,notn,notp,notqin verbal terms would be extremely arduous and confusing. I will therefore for now merely affirm to you thatcombinations ofany onepositivegeneric determination withthe negations of the three othergeneric determinations, for the very same terms, are inconsistent.

By elimination, we are left with onlyfourconsistent compounds, i.e. remaining combinations give rise to no inconsistency, i.e. whose respective clauses do not contradict each other. This means that, from the logical point of view, they are conceivable, and therefore worthy of further formal treatment. We may refer to them as the specific determinations, or species of causation.

The following table (where + and – signify, respectively, affirmation and denial of a determination) lists all combinations of the generics and identifies the logically possible specifics among them:

The formulae given in the above table for each specific determination is as brief as possible. For instance, sincemimplies the negation ofpandnimplies the negation ofq, ‘mn’ (meaning both complete and necessary causation) tacitly implies ‘notpand notq’ (neither partial nor contingent causation,with whatever complement); the latter negations need not therefore be mentioned. Similarly, an expression likem-alonesignifies the affirmation of one generic determination (here,m)andthe denial of all three others (i.e. notnand notq, as well as notp). This notation is far from ideal, but suffices for our current needs, since many combinations are eliminated at the outset.

We see that four specific determinations, namelymn,mq,np,pq, are formed by conjunction of positive causative propositions; these we shall call (following J. S. Mill’s nomenclature)jointdeterminations. It follows from the above table that each generic determination has onlytwospecies. Each generic determination may therefore be interpreted as a disjunction of its two possible embodiments; thus,mmeansmnormq;nmeansmnornp;pmeansnporpq; andqmeansmqorpq. Also note, we could refer tomnas ‘only-strong causation’ and topq‘only-weak causation’, whilemqandnpare ‘mixtures of strong and weak’.

The four specific determinations formed by composing positive causative propositions with negative ones, namelym-alone,n-alone,p-alone,q-alone, will be calledlone determinations. This expression is introduced at this stage to contrast it with generic and joint determinations. Clearly, one should not confuse an isolated generic symbol such asmwith the corresponding specific symbolm-alone; I use this heavy notation to ensure no confusion arises. Moreover,nota bene: In the above table, these forms are eliminated at the outset, because they concernabsolutepartial or contingent causation, i.e. they are irrespective of complement and meanm-alone_absetc. But as we shall later see, when they involverelativepartial or contingent causation, i.e. when some complement is specified (inp_relorq_relor their negations), so that they meanm-alone_reletc., they remain possible forms. This need not concern us at the moment, but is said to explain why these forms need to be named.

We would label as, simply,causation(or ‘any causation’), the disjunctive proposition “mornorporq”, or the more specific “mnormqornporpq”. Such positive propositions merelyimplycausation, if they involve less disjuncts or an isolated generic or joint determination. The contradictory of causation,non-causation, is the only remaining allowable combination, our table being exhaustive. This last possible combination involves negation ofall fourgeneric or joint determinations, note well. That is, it means “neithermnornnorpnorq” or equally “neithermnnormqnornpnorpq”.

The above table also allows us to somewhat interpret complex negations. The negation of any compound is equivalent to the disjunction of all remaining four compounds (three of causationandone of non-causation). For instance “not(mn)” meansmq,np,pq, ornon-causation. Similarly with any other formula.

Note that where one of the weak determinations is denied by reason of the affirmation of the contrary strong determination (min the case ofp, ornin the case ofq), any and all proposed complements are denied. Where one of the weaks is affirmed (even if the other is radically denied), at least one complement is implied; and of course, the contrary strong determination is denied. In all other cases, we must remember to be careful and distinguish between restricted and radical negations ofporq, as already explained in the previous chapter.

2.The Joint Determinations.

We shall now examine in detail the four joint determinations, symbolized bymn,mq,np, andpq, each of which is obtained by consistent conjunction of two generic determinations. Each is thus a species shared by the two genera constituting it. Thus,mnis a specific case ofmand a specific case ofn; and so forth.

We have already encountered one of these joint determinations, viz. complete and necessary causation, the paradigm of causation. We shall now examine it in further detail, and also treat the other three joint determinations.

Complete and Necessarycausation by C of E:

(i)If C, then E;

(ii)if notC, not-then E (may be left tacit);

(iii)where:C is possible.

And:

(iv)if notC, then notE;

(v)if C, not-then notE (may be left tacit);

(vi)where:C is unnecessary.

Notice how the merger of clauses (i), (ii) and (iii) with (iv), (v) and (vi) renders clauses (ii) and (v) redundant (though still implicit). Rows 5-8 of the above table (shaded) constitute the matrix of complete-necessary causation.

Complete but Contingentcausation by C1 of E:

(i)If C1, then E;

(ii)if notC1, not-then E (may be left tacit);

(iii)where: C1 is possible (may be left tacit).

And:

(iv)if (notC1 + notC2), then notE;

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

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

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

Notice how the merger of clauses (i), (ii) and (iii) with (iv), (v), (vi) and (vii) renders clauses (ii) and (iii) redundant (though still implicit). Rows 19-26 of the above table constitute the matrix of complete-contingent causation.

Concerning the four positions labeledopenin the above table, note that the moduses of Nos. 12 and 24 are tied and likewise those of Nos. 15 and 19. Proof for the first two: if #12 (C2 + notE) is impossible, #24 (notC1 + C2 + notE) must also be impossible; if #24 (notC1 + C2 + notE) is impossible, then knowing #20 (C1 + C2 + notE) to be impossible, #12 (C2 + notE) must also be impossible; the rest follows by contraposition. Proof for the other two: if #15 (C1 + C2) is impossible, #19 (C1 + C2 + E) must also be impossible; if #19 (C1 + C2 + E) is impossible, then knowing from (i) that #20 (C1 + C2 + notE) is impossible, #15 (C1 + C2) must also be impossible; the rest follows by contraposition. The interpretation of these open cases is as follows.

(a) Suppose #12 is impossible; this means that “If C2, then E”. We know from #14 that “If notC2, not-then E”; and from #3 that “C2 is possible”. Whence, C2 satisfies the definition for being acompletecause of E, just like C1. Thus, in such case, C1 and C2 are simplyparallelcomplete (and contingent) causes of E. This is quite conceivable, and as we have seen in an earlier section such causes may be compatible or incompatible. If #15 is possible, they are compatible; and if#15 is impossible, they are incompatible.

(b) Suppose #12 is possible; this means that “If C2, not-then E”, in which case C2 is not a complete cause of E. This is quite conceivable, covering situations where one of the contingent causes (namely, C1) is also complete, while the other (C2) is not complete. Additionally, we can say: if #15 is possible, they are compatible; and if#15 is impossible, they are incompatible; there is no problem of consistency either way.

However, a very interesting question arises in such case: is a contingent but not complete cause (like C2, here) bound to be a partial cause? C2 is certainly not a partial cause of E in conjunction with C1, since C1 is a complete cause of E. Therefore,ifC2 is a partial cause of E, it will be so in conjunction withsome otherpartial cause of E, say C3. But since C3 is unmentioned in our original givens, its existence is not formally demonstrable. We thus have no certainty that anincompletecontingent cause is implicitly apartialcontingent cause! We will return to this issue later.

Partial yet Necessarycausation by C1 of E:

(i)If notC1, then notE;

(ii)if C1, not-then notE (may be left tacit);

(iii)where: C1 is unnecessary (may be left tacit).

And:

(iv)if (C1 + C2), then E;

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

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

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

Notice here again how the merger of clauses (i), (ii) and (iii) with (iv), (v), (vi) and (vii) renders clauses (ii) and (iii) redundant (though still implicit). Rows 19-26 of the above table (shaded) constitute the matrix of partial-necessary causation.

Concerning the four positions labeledopenin the above table, note that the moduses of Nos. 13 and 21 are tied and likewise those of Nos. 18 and 21. These statements may be proved in the same manner as done for the preceding table; this is left to the reader as an exercise. We can also interpret these situations in similar ways. If #13 is impossible, C2 is a partial and necessary cause of E, parallel to C1; and notC2 is either compatible or incompatible with notC1 according to whether #18 is possible or impossible. If #13 is possible, C2 is a partial but not necessary cause of E, and notC2 is either compatible or not with notC1, according to whether #18 is possible or not.

However, it is not formally demonstrable that anunnecessarypartial cause is implicitly acontingentpartial cause; and the implications of this finding (or absence of finding) will have to be considered later.

Partial and Contingentcausation by C1 of E:

(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.

And:

(v)if (notC1 + notC2), then notE;

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

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

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

Rows 19-26 of the above table (shaded) constitute the matrix of partial-contingent causation. We note that here none of the original clauses are made redundant by the combination of partial and contingent causation. Furthermore, no position in the above table is left open, with regard to the possibility or impossibility of the item or combination concerned.

Additionally we can say that if C1 and C2 are, as here,complementarypartial contingent causes of E, then they have the same set of relations to each other and to E. But this does not mean that if C1 and C2 are complementary partial causes of E, they are bound to be complementary contingent causes of E, since as we have seen both or just one of them may be necessary cause(s) of E. Similarly, we cannot say that if C1 and C2 are complementary contingent causes of E, they are bound to be complementary partial causes of E, since as we have seen both or just one of them may be complete cause(s) of E.

There may, of course, be more than one complement to C1 (i.e. complements C3, C4…, in addition to C2) in the last three joint determinations,mq,nporpq. Such cases may be similarly treated, as we have explained when considering the weaker generic determinations separately.

It is with reference to the joint determinationsmqandnpthat the utility of reformatting sentences about partial or contingent causation becomes apparent. Anmqproposition is best stated as “C1 is a complete and (complemented by C2) a contingent cause of E”, and anpproposition is best stated as “C1 is a necessary and (complemented by C2) a partial cause of E”.

We must now consider thehierarchybetween the above four forms, since there are clearly differences in degree in the ‘bond’ between cause(s) and effect. Causation is obviously at itsstrongestwhen both complete and necessary (mn). It is difficult to say which of the next two forms (mqornp) is the stronger and which the weaker, they are not really comparable to each other; all we can say is that they are both less determining than the first and more determining than the last; let us call themmiddlingdeterminations. Causation isweakestfor each factor involved in partial and contingent causation (pq).

With regard toparallelism, we can infer that it is conditionally possible with reference to our previous findings in the matter.

Two complete-necessary causes, C, C₁, of the same effect E, may be parallel, provided they are neither exhaustive nor incompatible with each other, i.e. provided “if C, not-then notC₁and if notC, not-then C₁” is true.

For complete-contingent causation, it is conceivable that C1, C2 have this relation to E and C3, C4 have this same relation to E, provided the complete causes C1 and C3 are not exhaustive and the compounds (notC1 + notC2) and (notC3 + notC4) are not exhaustive. An interesting special case is when C2 = C4, i.e. when the two complete causes have the same complement in the contingent causation of E.

For partial-necessary causation, it is conceivable that C1, C2 have this relation to E and C3, C4 have this same relation to E, provided the necessary causes C1 and C3 are not incompatible and the compounds (C1 + C2) and (C3 + C4) are not exhaustive. An interesting special case is when C2 = C4, i.e. when the two necessary causes have the same complement in the partial causation of E.

For partial-contingent causation, the same condition of non-exhaustiveness between the parallel compounds involved applies. And here, too, note the special case when C2 = C4 as interesting.

Tables involving all the items concerned and their negations in all combinations may be constructed to analyze the implications of such parallelisms in detail.

Thenegationsof the four joint determinations may be reduced to the denial of one or both of their constituent generic determinations. That is,not(mn)means ‘not-mand/or not-n’;not(mq)means ‘not-mand/or not-q’;not(np)means ‘not-pand/or not-n’; andnot(pq)means ‘not-pand/or not-q’. Each of these alternative denials in turn implies denial of one or more of the constituent clauses, obviously.

3.The Significance of Certain Findings.

Let us review how we have proceeded so far. We started with the paradigm of causation, namely, complete necessary causation. We then abstracted its constituent “determinations”, the complete and the necessary aspects of it, and by negation formulated another two generic determinations, namely partial and contingent causation. We then recombined these abstractions, to obtain all initially conceivable formulas. Some of these formulas (mp,nq) could be eliminated as logically impossible by inspecting their definitions and finding contradictory elements in them. Others (thelonedeterminations, obtained by conjunction of only one generic determination and the negations of all three others) were eliminated on the basis of later findings not yet presented here. This left us with only five logically tenable specific causative relations between any two items, namely the fourjointdeterminations (the consistent conjunctions generic determinations) and non-causation (the negation of all four generic determinations).

When I personally first engaged in the present research, I was not sure whether or not the (absolute) lone determinations were consistent or not. Because each lone determination involves three negative causative propositions in conjunction, and each of these is defined by disjunction of the negations of the defining clauses of the corresponding positive form, it seemed very difficult to reliably develop matrixes for them. I therefore, as a logician[1], had to assume as a working hypothesis that they were logically possible. It is only in a later phase, when I developed “matricial microanalysis” that I discovered that they can be formally eliminated. Take my word on this for now. This discovery was very instructive and important, because it signified thatcausation is more “deterministic” than would otherwise have been the case.

If lone determinations had been logically possible, causation would have been moderately deterministic. For two items might be causatively related on the positive side, but not on the negative side, or vice-versa. Something could beonlya complete cause (oronlya partial cause) of another without having to also be a necessary or contingent one; or it could beonlya necessary cause (or only a contingent cause) of another without having to also be a complete or partial one. But as it turned out there is logically no such degree of freedom in the causative realm.

If two things are causatively related at all, theyhave to beultimately related in one (and indeed only one) of the four ways described as the joint determinations[2], i.e. in the way ofmn,mq,np, orpq. The conceptsm,n,p,qare common aspects of these four relations and no others. There is no “softer” causative relation. Causation is “full” or it is not at all; no “holes” are allowed in it.

We can formulate the following “laws of causation” in consequence:

• If something is a complete or partial cause of something, it must also be either a necessary or (with some complement or other) a contingent cause of it.
• If something is a necessary or contingent cause of something, it must also be either a complete or (with some complement or other) a partial cause of it.
• In short, since a lone determination is impossible, if something is at all a causative of anything, it must be related in the way of a joint determination with it.

These laws have the following corollaries:

• If something is neither a necessary nor contingent cause of something, it must also be neither a complete nor (with whatever complement) a partial cause of it.
• If something is neither a complete nor partial cause of something, it must also be either neither a necessary nor (with whatever complement) a contingent cause of it.
• In short, since a lone determination is impossible, if two things are known not to be related in the way of either pair of contrary generic determinations (i.e.mandp, ornandq), they can be inferred to be not causatively related at all.

Also:

• Thecomplementof a partial cause of something, being also itself a partial cause of that thing, must either be a necessary or (with some complement or other) a contingent cause of that thing.
• Thecomplementof a contingent cause of something, being also itself a contingent cause of that thing, must either be a complete or (with some complement or other) a partial cause of that thing.

With regard to the epistemological question, as to how these causative relations are to be established, we may say that they are ultimately based oninduction(including deduction from induced propositions): we have no other credible way to knowledge. Causative propositions may of course be built up gradually, clause by clause (see definitions in the previous chapter).

As I showed in my workFuture Logic, thepositivehypothetical (i.e. if/then) forms, from which causatives are constructed, result from generalizations from experience of conjunctions between the items concerned (which generalizations are of course revised by particularization, when and if they lead to inconsistency with new information). Thenegativehypothetical (i.e. if/not-then) forms are assumed true if no positive forms have been thus established, or are derived by the demands of consistency from positive forms thus established. In their case, an epistemological quandary may be translated into an ontologicalfait accompli(at least until if ever reason is found to prefer a positive conclusion).

We may first, by such induction (or deduction thereafter), propose one of the four generic determinations in isolation. The proposed generic determination is effectively treated as a joint determination “in-waiting”, a convenient abstraction that does not really occur separately, but only within conjunctions. We are of course encouraged by methodology to subsequentlyvigorously researchwhich of the four joint determinations can be affirmed between the items concerned. In cases where all such research efforts prove fruitless, we are simply left with aproblematicstatement, such as (to give an instance) “P is a complete cause, and either a necessary or a contingent cause, of Q”.

But, since lone determination does not exist, we can never opt for anegativeconclusion, like “P is a complete cause, but neither a necessary nor a contingent cause, of Q”. We maynotin this context effectivelygeneralizefrom “I did not find” to “there is not” (a further causative relation). We may not interpret a structural doubt as a negative structure, an uncertainty as an indeterminacy.

In the history of Western philosophy, until recent times, the dominant hypothesis concerning causation has been that it is applicableuniversally. Some philosophers mitigated this principle, reserving it for ‘purely physical’ objects,exceptingbeings with volition (humans, presumably G-d, and even perhaps higher animals). A few, notably David Hume, denied any such “law of causation” as it has been called.

But in the 20th Century, the idea that there might, even in Nature (i.e. among entities without volition), be ‘spontaneous’ events gained credence, due to unexpected developments in Physics. That idea tended to be supported by the Uncertainty Principle of Werner Heisenberg for quantum phenomena, interpreted by Niels Bohr as an ontological (and not merely epistemological) principle of indeterminacy, and the Big-Bang theory of the beginning of the universe, which Stephen Hawking considered as possibly implying anex nihiloand non-creationist beginning.

We shall not here try to debate the matter. All I want to do at this stage is stress the following nuances, which are now brought to the fore. The primary thesis of determinism is thatthere is causationin the world; i.e. that causal relations of the kind identified in the previous chapter (the four generic determinations)do occurin it. Our above-mentioned discovery that such causation has to fit in one of the four specific determinations may be viewed as a corollary of this thesis, or a logically consistent definition of it.

This is distinct from various universal causation theses, such as that nothing can occur except through causation (implying that causation is the only existing form of causality), or that at least nothing in Nature can do so (though for conscious beings other forms of causality may apply, notably volition), among others.

We shall analyze such so-calledlawsof causation in a later chapter; suffices for now to realize that they are extensions, attempted generalizations, of the apparentfactof causation, and not identical with it. Many philosophers seem to be unaware of this nuance, effectively regarding the issue as either ‘causation everywhere’ or ‘no causation anywhere’.

The idea that causation is presentsomewherein this world is logically quite compatible with the idea that there may bepocketsorborderswhere it is absent, a thesis we may call ‘particular (i.e. non-universal) causation’. We may even, more extremely, consider that causation is poorly scattered, in a world moved principally by spontaneity and/or volition.

The existence of causation thus doesnotin itself exclude the spontaneity envisaged by physicists (in the subatomic or astronomical domains); and it doesnotconflict with the psychological theory of volition or the creationist theory of matter[3].

Apparently, then, though determinism may be the major relation between things in this world, it leaves some room, however minor (in the midst or at the edges of the universe), for indeterminism.

We will give further consideration to these issues later, for we cannot deal with them adequately until we have clarified the different modes of causation.