THE LOGIC OF CAUSATION

© Avi Sion, 1999. All rights reserved.

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 as m, 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, namely mp and nq to be inconsistent. This was due to incompatibilities between clauses of their definitions, or in other words, certain rows of their matrices. Thus, row 6 of m (C + notE is impossible) is in conflict with row 22 (C1 + notE is possible) of p; similarly, row 7 of n (notC + E is impossible) is in conflict with row 23 (notC1 + E is possible) of q.

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 treat negations of generic determinations. To define notm, notn, notp, notq in verbal terms would be extremely arduous and confusing. I will therefore for now merely affirm to you that combinations of any one positive generic determination with the negations of the three other generic determinations, for the very same terms, are inconsistent.

By elimination, we are left with only four consistent 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, since m implies the negation of p and n implies the negation of q, ‘mn’ (meaning both complete and necessary causation) tacitly implies ‘notp and notq’ (neither partial nor contingent causation, with whatever complement); the latter negations need not therefore be mentioned. Similarly, an expression like m-alone signifies the affirmation of one generic determination (here, m) and the denial of all three others (i.e. notn and 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, namely mn, mq, np, pq, are formed by conjunction of positive causative propositions; these we shall call (following J. S. Mill’s nomenclature) joint determinations. It follows from the above table that each generic determination has only two species. Each generic determination may therefore be interpreted as a disjunction of its two possible embodiments; thus, m means mn or mq; n means mn or np; p means np or pq; and q means mq or pq. Also note, we could refer to mn as ‘only-strong causation’ and to pq ‘only-weak causation’, while mq and np are ‘mixtures of strong and weak’.

The four specific determinations formed by composing positive causative propositions with negative ones, namely m-alone, n-alone, p-alone, q-alone, will be called lone 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 as m with the corresponding specific symbol m-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 concern absolute partial or contingent causation, i.e. they are irrespective of complement and mean m-alone_abs etc. But as we shall later see, when they involve relative partial or contingent causation, i.e. when some complement is specified (in p_rel or q_rel or their negations), so that they mean m-alone_rel etc., 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 “m or n or p or q”, or the more specific “mn or mq or np or pq”. Such positive propositions merely imply causation, 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 of all four generic or joint determinations, note well. That is, it means “neither m nor n nor p nor q” or equally “neither mn nor mq nor np nor pq”.

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 causation and one of non-causation). For instance “not(mn)” means mq, np, pq, or non-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 (m in the case of p, or n in the case of q), 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 of p or q, as already explained in the previous chapter. 

2.   The Joint Determinations. 

We shall now examine in detail the four joint determinations, symbolized by mn, mq, np, and pq, 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, mn is a specific case of m and a specific case of n; 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 Necessary causation 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 Contingent causation 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.