THE LOGIC OF CAUSATION

© Avi Sion, 2003. All rights reserved.

Phase Two: Microanalysis

Chapter 11 -   Piecemeal Microanalysis

1.    Binary Coding and Unraveling

2.    The Generic Determinations

3.    Contraction and Expansion

4.    Intersection, Nullification and Merger

5.    Negation

1.    Binary Coding and Unraveling.

We have developed a theory of causative propositions and arguments (eductions and syllogisms) by means of an analysis of the possibilities and impossibilities implied for the various combinations of the items concerned. This was characterized as ‘matricial analysis’, because of our recourse to tables for assessing and recording results.

But thus far we have only really engaged in elementary matricial analysis, which may be called macroanalysis. We shall now introduce a more advanced approach, which may be called microanalysis. They are not different methods. Microanalysis is based on macroanalysis; it is merely a more detailed examination, digging deeper into the issues concerned, in an attempt to solve outstanding problems.

As we have seen, the determinations of causation are best expressed through a matrix, a table composed of ‘items’ and ‘moduses’. The items are the terms or theses related by the causative proposition concerned. Each conceivable conjunction of these items, in positive or negative form, defines a row of the matrix. The modus for each such conjunction is a statement regarding its logical possibility or impossibility, or ‘openness’ (the latter in cases where the conjunction is in some unspecified contexts possible and in others impossible, so that an uncertainty remains). The moduses for the various conjunctions of items together constitute an additional column of the matrix.[1]

If we array the items of a matrix in a conventional arrangement (presenting the same row always in the same place), then the modus columns of all matrices will be comparable. By such standardization, we can express a determination of causation by merely writing down a string of moduses (i.e. its modus column), which we may call the modus of the determination concerned as a whole, or (for reasons we shall see presently) its summary modus.

To simplify things, we may revert to binary codes. We may express the presence or absence of each item in the matrix by a 1 or 0 notation. Similarly, we may code the modus for each row by a 1, 0 or  ∙ (dot - meaning blank). The zeros or ones have different meanings in the items and modus cells of the matrix, note well:

Binary codes:

Such notation is merely convenient abbreviation, allowing us to express the summary modus of any determination as a relatively short string of digits and see the whole matrix in one sweep of the eyes. It is also, obviously, useful for computer programming purposes. Of course, if we are dealing with two items (say, P, R), the modus string will have 2² = 4 digits; if with three items (say, P, Q, R), it will have 2³ = 8 digits; and so forth. Whether the string of digits is distinctive for each determination, we shall look into further on[2].

As we said, the rows of a matrix are defined and (conventionally) located by combinations of items. Thus, for two items, P and R, the four possible PR sequences are 11, 10, 01, 00, which may be labeled a, b, c, d if need be. We may choose this order of combinations as our standard arrangement (any other permutation is equally conceivable, but we conventionally settle on this one[3]). Similarly, for three items, P, Q and R, there are eight possible PQR sequences, which may be labeled a-h if need be. And so forth, for more items.

We may thus, to begin with, present the matrices of the generic determinations of causation as in the following tables. These include (in the first two or three columns) the items in positive (1) or negative (0) forms, arrayed in standard combinations; followed by the summary modus for each propositional form (shaded column, symbol S), which you will recall we developed at the beginning of our research (in Phase 1, chapter 2) by analyzing the meaning of each of its constituent clauses and assessing the result of their interactions.

New columns are then introduced, which present all the conceivable realizations of the summary modus. These realizations, called alternative moduses, are obtained simply by substituting, successively, a 0 (for ‘impossible’) or 1 (for ‘possible’) for each dot (‘open’ position) encountered in the summary modus, so that no dots are leftover. This process can be called unraveling. The alternative moduses thus make explicit all cases inherent in the summary modus; and conversely, the latter is a summary of all the information contained in the former.

Note that the alternative moduses are themselves, ultimately, summaries, too. For while a zero (for impossibility) signifies that the combination of items concerned is in every context or always absent, a one (for possibility) signifies that it is in some contexts or sometimes present[4]. Thus, to remove all implicit modality, and consider only actualities, we would have to dissect each such modus into an unspecifiable number of actualizations, where ‘0’ means absent and ‘1’ means present, simply. However, such further analysis is not needed for our purposes; the moduses as above defined are sufficiently informative.[5]

Consideration of a summary modus constitutes macroanalysis; that of alternative moduses, microanalysis. That is all the difference between these two methods of matricial analysis: one of degree of detail. In the former, we have a rough idea of the relations involved; in the latter, it is as if we scrutinize them under a microscope.

The similar strings of zeros and ones used by computer programmers to code letters of the alphabet and symbols (I am thinking of ASCII codes) were arbitrary, pure conventions. But here, note well, once the meanings of zeros and ones, and the order of their presentation, are decided, there is nothing conventional about the string for each determination; it is a logical property of it, objectively given information.

2.    The Generic Determinations.

In the four tables below, the precise significance of the numbers heading the columns of alternative moduses will be made clear in the next chapter; for now, just consider them as arbitrary labels. It should be stressed at the outset that these modus numbers are not to be confused with the determination numbers or mood numbers used in earlier chapters. Note also that Tables 11.1 and 11.2 concern two items (P, R), whereas Tables 11.3 and 11.4 concern three items (P, Q, R)[6]; the summary moduses of these two sets are therefore not directly comparable, the former being within a ‘two-item framework’, the latter within a ‘three-item framework’.

The two-item modus of complete causation of form PR (symbolized by m, or more precisely m_PR) was previously established to be “10.1”. This is, through the following table, worked out to have two conceivable realizations, namely “1001” or “1011” (labeled respectively Nos. 10, 12).

Table 11.1.            Matrix of “P is a complete cause of R”.

In contrast, the two-item modus, summarily put, of necessary causation of form PR (symbolized by n, or most precisely n_PR) was previously established to be “1.01”. This is, through the following table, worked out to have two conceivable realizations, namely “1001” or “1101” (labeled respectively Nos. 10, 14).

Table 11.2.            Matrix of “P is a necessary cause of R”.

The three-item summary modus of relative partial causation of form P(Q)R (symbolized, according to context, by p or p_rel, or p_Q or most precisely p_PQR) was previously established to be “10.1.1..” . This is, through the following table, worked out to have sixteen conceivable realizations, as shown below (labeled respectively Nos. 149-152, 157-160, 181-184, 189-192). Note well that this is true relative to complement Q; we shall consider absolute partial causation further on.

Table 11.3.     Matrix of “P (complemented by Q) is a partial cause of R”.

The three-item summary modus (column S) of contingent causation of form P(Q)R (symbolized, according to context, by q or q_rel, or q_Q or most precisely q_PQR) was previously established to be “..1.1.01”. This is, through the following table, worked out to have sixteen conceivable realizations, as shown below (labeled respectively Nos. 42, 46, 58, 62, 106, 110, 122, 126, 170, 174, 186, 190, 234, 238, 250, 254). Note well that this is true relative to complement Q; we shall consider absolute contingent causation further on.

Table 11.4.            Matrix of “P (complemented by Q) is a contingent cause of R”.