Phase Two: Microanalysis
Chapter 11 – Piecemeal Microanalysis
1. Binary Coding and Unraveling
4. Intersection, Nullification and Merger
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:
In the items columns: | 1 = present | 0 = absent | |
In the modus column(s): | 1 = possible | 0 = impossible | ∙ = open |
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 2^{2} = 4 digits; if with three items (say, P, Q, R), it will have 2^{3} = 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”.
Items | S | 2 alternative moduses | ||
P | R | m | 10 | 12 |
1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | ∙ | 0 | 1 |
0 | 0 | 1 | 1 | 1 |
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”.
Items | S | 2 alternative moduses | ||
P | R | n | 10 | 14 |
1 | 1 | 1 | 1 | 1 |
1 | 0 | ∙ | 0 | 1 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 1 | 1 | 1 |
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”.
Items | S | 16 alternative moduses | |||||||||||||||||
P | (Q) | R | p | 149 | 150 | 151 | 152 | 157 | 158 | 159 | 160 | 181 | 182 | 183 | 184 | 189 | 190 | 191 | 192 |
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 1 | ∙ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | ∙ | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | ∙ | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | ∙ | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
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”.
Items | S | 16 alternative moduses | |||||||||||||||||
P | (Q) | R | q | 42 | 46 | 58 | 62 | 106 | 110 | 122 | 126 | 170 | 174 | 186 | 190 | 234 | 238 | 250 | 254 |
1 | 1 | 1 | ∙ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | ∙ | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | 0 | ∙ | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 0 | ∙ | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
As above stated, we shall presently look into the summary moduses of absolute partial and contingent causation. As we shall see, they are much wider than those relative to a given complement, dealt with above. This is natural, since absolute weak causations are vaguer forms than relative weak causations.
Comparing the summary moduses of complete and necessary causation with two identical items, 10.1 and 1.01, we see more clearly in what sense they are ‘mirror images’ of each other: the strings are identical, viewing one from left to right and the other from right to left. Similarly for the summary moduses of partial and contingent causation with three identical items, 10.1.1.. and ..1.1.01.
It should also be noted that a weak cause and its complement have the same summary modus. That is, partial causation of forms P(Q)R and Q(P)R have the same 10.1.1.. summary; and contingent causation of forms P(Q)R and Q(P)R have the same ..1.1.01 summary. This was obvious from the original definitions of these determinations, in which P and Q had the same relations to each other and to R; the distinction of one or the other of P, Q as complement was purely one of convenience or focus.[7]
Another observation we can make at this stage is that a generic determination and its (appropriate) converse would have one and the same summary modus.
That is true for the strong and absolute weak[8] determinations, which all concern two items. For instance, “P is a complete cause of R” and “R is a necessary cause of P” (note the change of determination, as well as that of item positions) are both here described by the string “10.1”. This can be ascertained by reading Table 11.2 in a different order, starting with the first row, then the third, then the second, then the fourth.
That is also true for the weak determinations, which involve three items. For instance, “P (complemented by Q) is a partial cause of R” is convertible to “R (complemented by notQ) is a contingent cause of P” (note well the change of complement polarity, as well as of determination and item positions) are both here described by the string “10.1.1..”. Again, we can prove this by rereading Table 11.4 in a different order, starting with the third row, then the seventh, then the first, then the fifth, then the fourth, then the eighth, then the second, then the sixth.
Indeed, we can say that convertibility is to be explained by such identity of moduses. Clearly, it follows that we cannot express direction of causation by reference to summary moduses. The orientations “from P to R” and “from R to P” must have some meaning – they are not empty verbal distinctions – but that meaning is not apparent in the way of a difference between moduses. It has to be sought in other properties, as already argued.
Now, the above account does not allow us to compare the moduses of the strong determinations with those of the weak ones, nor tell us how to distinguish absolute from relative weak determinations. To enable such comparisons, we need to develop two processes: (a) contraction of a three-item modus into a two-item modus, and (b) expansion of a two-item modus into a three-item modus….
Let us first consider contraction of the three-item moduses of p or q (in their relative forms). Take first the case of partial causation by P of R, with reference to Table 11.3, above.
· The conjunction (P + Q + R) is possible, since the first row is always coded 1, whereas (P + notQ + R) is open, since the third row is sometimes coded 0 and sometimes 1. Nevertheless, it follows that the conjunction (P + R) is possible (i.e. to be coded 1), since “(P + Q + R) is possible” implies that “(P + R) is possible”. Note that if we regarded (P + R) as merely open, we would fail to record that there is no column with 0s in both the first and third cells.
Note this well: it is a finding we altogether missed in macroanalysis, and which may therefore affect some of our results.
· The same reasoning applies for the conjunctions (P + notR), comprising the second and fourth rows, and (notP + notR), comprising the sixth and eighth rows. They are both possible conjunctions, and not merely open.
· On the other hand, the conjunction (notP + R), comprising the fifth and seventh rows, must be declared open (i.e. be coded ∙), since it is conceivable for both (notP + Q + R) and (notP + notQ + R) to be found impossible (as in the columns numbered 149, 150, 181, 182).
In this way the three-item modus for relative partial causation “10.1.1..” becomes the two-item modus “11.1”. Similarly with contingent causation: its three-item modus “..1.1.01” becomes the two-item modus “1.11”.
Thus, in case of need, we can contract a three-item modus into a two-item one, by changing a combination of 1 and 0, or 1 and ∙, in corresponding locations, into a 1. Also, a combination of two dots yields one dot. Note well the rule of contraction:
1. Where there is a 1 in the 3-item modus, there must be a 1 in the 2-item modus.
Additionally note, though we have not yet encountered cases:
2. The only way we could obtain a 0 in a two-item modus, from a three-item modus, would be to find only 0s along both rows of the latter.
3. If we find cases of ‘11’,’10’ and/or ‘01’ mixed with cases of ‘00’ in the three-item modus, we must conclude a dot (∙) in the two-item modus.
Now, what have we found here? We started with weak causations by P of R, relative to some complement Q specifically, and ended with weak causations by P or R, without specification of Q, i.e. absolutely. The three-item modus for p or q relative to Q could not be equated to the same relative to some other complement, say Q_{1}; their matrices are superficially similar, but the items involved (namely PQR and PQ_{1}R) are quite different. But if we contract both kinds to two-item moduses, they would be indistinguishable, since the items involved (namely PR) are exactly identical.
Thus, the two-item modus of absolute (which includes relative) partial causation of form PR (symbolized, according to context, by p or p_{abs}, or most precisely p_{PR}) is by contraction found to be “11.1”. This is, through the following table, worked out to have two conceivable realizations, namely “1101” or “1111” (labeled respectively Nos. 14, 16).
Table 11.5. Matrix of “P is a partial cause of R”.
Items | S | 2 alternative moduses | ||
P | R | p | 14 | 16 |
1 | 1 | 1 | 1 | 1 |
1 | 0 | 1 | 1 | 1 |
0 | 1 | ∙ | 0 | 1 |
0 | 0 | 1 | 1 | 1 |
In contrast, the two-item modus, summarily put, of absolute (which includes relative) contingent causation of form PR (symbolized, according to context, by q or q_{abs}, or most precisely q_{PR}) is by contraction found to be “1.11”. This is, through the following table, worked out to have two conceivable realizations, namely “1011” or “1111” (labeled respectively Nos. 12, 16).
Table 11.6. Matrix of “P is a contingent cause of R”.
Items | S | 2 alternative moduses | ||
P | R | q | 12 | 16 |
1 | 1 | 1 | 1 | 1 |
1 | 0 | ∙ | 0 | 1 |
0 | 1 | 1 | 1 | 1 |
0 | 0 | 1 | 1 | 1 |
Notice the similarity between the summary moduses of m (10.1) and p_{abs} (11.1), or those of n (1.01) and q_{abs} (1.11). Where for the strong determination we have a ‘0’ code, in the corresponding absolute weak determination we have a ‘1’; the remaining codes being identical. This, as we shall see in a later chapter[9], allows us to define the absolute weak determinations in formal terms.
Also note that the absolute weak determinations are convertible, just like the strong ones (as we pointed out in the previous section). For instance, “P is a partial cause of R” converts to “R is a contingent cause of P” (note the change in determination, as well as that of item positions), since these two forms have the same summary modus “11.1”.
The next question to ask is: what are the three-item moduses of m or n, or of p or q in their absolute forms? We can answer this question by means of expansion, as follows.
Consider, to begin with, the strong determinations. In the case of complete causation by P of R, the following can be said:
· Knowing the conjunction (P + notR) is impossible, it follows that both (P + Q + notR) and (P + notQ + notR) are impossible conjunctions (whence the initial modus 0 becomes two moduses 0).
· Whereas, since the conjunction (P + R) is possible, it follows only that at least one of (P + Q + R) and (P + notQ + R) is a possible conjunction (i.e. they cannot both be impossible) – but we cannot predict which one is possible, so both conjunctions must be declared open (whence, the initial modus 1 becomes two moduses ∙). Similarly, mutadis mutandis, for (notP + notR).
· Lastly, since the conjunction (notP + R) is open, so will a fortiori its two derivatives be (i.e. the initial modus ∙ becomes two moduses ∙).
In this way the two-item modus “10.1” for complete causation becomes the three-item modus “.0.0….”. Similarly with necessary causation: its two-item modus “1.01” becomes the three-item modus “….0.0.”. Notice the loss of information occasioned by the change in each case, due to the fact that ones become dots; the results of such expansions are vaguer than their sources.
Thus, in case of need, we can expand a two-item modus into a three-item one, by changing a zero into two zeros in the appropriate locations, and a one or dot into two dots as appropriate. Here, note well the ‘appropriate locations’ are not adjacent rows: they are the first and third, the second and fourth, the fifth and sixth, etc., reflecting a correspondence in combination of items – such as (P + notR) becoming (P + Q + notR) or (P + notQ + notR), which means moving from a PR sequence 10 to the PQR sequences 110 and 100, which signify the second and fourth rows of the matrix.
However, note well the restrictions implied in the following rules of expansion:
- Where there is a 0 in the 2-item modus, there must be two 0s in the 3-item modus.
- Where there is a 1 in the 2-item modus, there cannot be two 0s in the 3-item modus.
- Where there is a dot in the 2-item modus, there might be any combinations of 0s and/or 1s in the 3-item modus.
With regard to ‘zero’ moduses (impossibility), they are universalized as it were from the initial row to the corresponding expanded rows. With regard to ‘ones’ (possibility), what is universalized from the single initial row to the two subsumed rows is the interdiction of zeros: just as in the two-item modus 0 is excluded by 1, so the three-item expansion cannot include columns (moduses) having 0s in both the corresponding rows. A fortiori, in the case of ‘dots’ (which might include zeros or ones), we cannot predict combinations in the two cells concerned, since all pairs are allowed, i.e. 0 and 0, 0 and 1, 1 and 0, 1 and 1.
Consider now the weak determinations, in accord with the rules of expansion just ascertained. If we similarly expand the two-item modus of absolute (or relative) partial causation, namely 11.1, into a three-item modus, we obtain “……..”, since all ones or zeros become dots. Likewise, by expansion of the two-item modus of absolute (or relative) contingent causation, namely 1.11, into a three-item modus, we obtain “……..”.
Note well that the result in both these cases is a string of dots, signifying complete uncertainty, the least possible amount of information. Each initial one or dot was expanded into two dots, so that all remaining specificity in the initial string was dissolved in its derivatives.
Note also the marked difference between the three-item strings of the absolute weak determinations “……..”, and those of the corresponding relative forms, namely “10.1.1..” and “..1.1.01” respectively.
Clearly, for all four generic determinations, expansion of a two-item modus “1” (possible) into two three-item moduses “∙“ (open) results in a loss of data; i.e. the information that ‘at least one of the two conjunctions concerned must be possible: i.e. they cannot both be impossible’ is no longer coded in our table. A calculus of causation should be so designed as to avoid all loss of information due to mere linguistic inadequacies[10]. Thus, we have to find a way to express, through a special code in the modus, say µ (Gk. letter mu), that at least one of the two (or more) conjunctions so coded is implicitly a “1”.[11]
Complete causation | = µ0µ0.µ.µ |
Necessary causation | = µ.µ.0µ0µ |
Partial causation (absolute) | = µµµµ.µ.µ |
Contingent causation (absolute) | = µ.µ.µµµµ |
This measure by itself is not enough; to save all available information, we would have to specify the rows concerned, say by labeling them a-h. For instance, if at least one of rows ‘a’ and ‘c’ has to have modus 1, each would have to be coded more specifically as µ_{ac}. Such coding means that the PQR sequences signified by the labels a and c (namely, 111 and 101) may have moduses with 0 and 1, 1 and 0, 1 and 1 – but they may not have the pair 0 and 0. It follows that:
· if a = µ_{ac} and c = 0, then a = 1 (since 00 is inconceivable), and
· if a = µ_{ac} and c = 1, then a = ∙ (since 01 and 11 are both conceivable);
and likewise, of course, if c = µ_{ac} and a = 0 then c = 1, and if c = µ_{ac} and a = 1 then c = ∙.
All this information can be considered implicit in a table like the following (the relative weak determinations of form PQR are included for comparison):
Table 11.7. Summary moduses for the six generic determinations of form PR or PQR.
Row | Items | m | n | p_{abs} | q_{abs} | p_{rel} | q_{rel} | ||
label | P | Q | R | PR | PR | PR | PR | PQR | PQR |
a | 1 | 1 | 1 | µ_{ac} | µ_{ac} | µ_{ac} | µ_{ac} | 1 | ∙ |
b | 1 | 1 | 0 | 0 | ∙ | µ_{bd} | ∙ | 0 | ∙ |
c | 1 | 0 | 1 | µ_{ac} | µ_{ac} | µ_{ac} | µ_{ac} | ∙ | 1 |
d | 1 | 0 | 0 | 0 | ∙ | µ_{bd} | ∙ | 1 | ∙ |
e | 0 | 1 | 1 | ∙ | 0 | ∙ | µ_{eg} | ∙ | 1 |
f | 0 | 1 | 0 | µ_{fh} | µ_{fh} | µ_{fh} | µ_{fh} | 1 | ∙ |
g | 0 | 0 | 1 | ∙ | 0 | ∙ | µ_{eg} | ∙ | 0 |
h | 0 | 0 | 0 | µ_{fh} | µ_{fh} | µ_{fh} | µ_{fh} | ∙ | 1 |
All this may seem pretty complicated, but as we shall see it simplifies a lot of things. Through the summary moduses in the above table, we can identify precisely the alternative moduses in a three-item framework implied by each of the four determinations (for the items PR or PQR).
As we shall see in the next chapter, the strong determinations m and n turn out to have 36 such alternative moduses each, while the weak determinations p and q in absolute form (as here), have 108 alternative moduses each (to compare to the 16 moduses of relative weaks). We shall list these moduses in the next chapter, so no need to do so here; they are easy to unravel by substituting zeros and ones for dots as previously explained.
4. Intersection, Nullification and Merger.
We shall now consider certain inferences from the above data.
The joining of generic determinations can be considered as the intersection of their respective summary moduses. By such conjunction of two propositions (or more), two classes (generic determinations) are used to express a more restrictive class (a joint determination), with whatever they have in common.
By this process, in a two-item framework (where the weak determinations are absolute), the joint determinations are found to have the following summary moduses:
· complete-necessary causation, mn = 10.1 + 1.01 = 1001 (modus No. 10);
· complete-contingent causation, mq = 10.1 + 1.11 = 1011 (modus No. 12);
· necessary-partial causation, np = 1.01 + 11.1 = 1101 (modus No. 14);
· partial-contingent causation, pq = 11.1 + 1.11 = 1111 (modus No. 16).
As can be seen, the result in each case is a single alternative modus (mentioned in brackets), which represents what the joined generics have in common. Thus, for instance, m has moduses 10 and 12, and n has moduses 10 and 14; therefore mn (meaning m + n) will have modus 10. The resulting summary modus is more defined than its sources, i.e. there are less dots, there are less uncertainties in the relation between the items.
This operation is merely an application of the well-known rule of class logic, that the logical product of two classes (such as m and n, each of which subsumes two subclasses, namely 10 and 12 for m and 10 and 14 for n) is the elements they have in common (namely, modus 10, in the case of mn). This can be seen for example in an Euler diagram, comprising two circles which overlap: their common area is the outcome of their product, and usually smaller than the circles (in our example, modus 10).
Note that by ‘logical product’ logicians mean that the two (or more) classes are conjoined together (i.e. mn means m + n)[12]. It must be stressed that modus lists are disjunctive not conjunctive, so that underlying this formula is another one (namely mn = ‘modus 10 or modus 12’ and ‘modus 10 or modus 14’, which means ‘in any event, modus 10’, i.e. it refers to the leftover after removing from consideration the elements ‘modus 12’ and ‘modus 14’, which are exclusive in either disjunct.
Similarly, in a three-item framework (where the weak determinations may be absolute or relative), intersection of the generic determinations yields the joint determinations, with the following summary moduses:
· complete-necessary causation:
mn | = µ0µ0.µ.µ + µ.µ.0µ0µ | = µ0µ00µ0µ (9 alternative moduses) |
· complete-contingent causation,
mq_{abs} | = µ0µ0.µ.µ + µ.µ.µµµµ | = µ0µ0µµµµ (27 alternative moduses) |
mq_{rel} | = µ0µ0.µ.µ + ..1.1.01 | = .0101.01 (4 alternative moduses) |
· necessary-partial causation:
np_{abs} | = µ.µ.0µ0µ + µµµµ.µ.µ | = µµµµ0µ0µ (27 alternative moduses) |
np_{rel} | = µ.µ.0µ0µ + 10.1.1.. | = 10.1010. (4 alternative moduses) |
· partial-contingent causation:
p_{abs}q_{abs} | = µµµµ.µ.µ + µ.µ.µµµµ | = µµµµµµµµ (81 alternative moduses) |
p_{rel}q_{rel} | = 10.1.1.. + ..1.1.01 | = 10111101 (1 alternative modus) |
Here again, the result signifies the alternative moduses that the joined generics have in common; we shall not list them at this stage: the list will be given in the next chapter. In the case of p_{rel}q_{rel}, exceptionally, the result is a fully specifying summary modus, i.e. a single alternative modus (that labeled #190, as we shall see later). The resulting summary modus fuses together the most definite elements of the initial summary moduses; some dots become µ’s, and some dots or µ’s become more specifically a 0 or a 1. The µ’s concerned are in pairs like µ_{ac} remember; the subscripts are not mentioned here for brevity.
Some of these results correspond to those obtained by macroanalysis, note. To grasp the rules of intersection, let us review the examples shown above:
- The summary moduses are never conflicting in a given position (1 in one case and 0 in the other); this simply means that the determinations joined are compatible.
- For each position identical in both generic summary moduses, or more definite (µ or 1 or 0) in one and indefinite (∙ or µ) in the other, the resulting corresponding position in the joint summary modus has that equal or more definite value.
We cannot join two determinations whose summary moduses have conflicting elements in the same position (a 0 in one and a 1 in the other): they are incompatible propositions, it is an impossible conjunction. In alternative modus terms, it means that these determinations do not have even one modus in common; in class logic terms, it means that the given classes (generic determinations) do not overlap: they have no intersection. Such logically empty concepts are known as null classes; we might therefore refer to the act of judging a class to be null as nullification.
For instances, the conjunctions mp, nq are null classes. Since m has moduses 10, 12 and p has moduses 14, 16, they have no common ground, no modus in which to coexist. Similarly for n and q, mutadis mutandis. This we know already from macroanalysis. More interesting, is the capacity nullification gives us to judge the feasibility of lone determinations, as we shall see in the next chapter.
Let us now consider another logical composition, that of merger, which disjoins two (or more) propositions, to obtain a single, vaguer proposition. In alternative modus terms, this process puts together all the alternative moduses listed for the given propositions in a larger list for the merged proposition. In class logic terms, this means that the two (or more) classes together become a single class covering all the areas they have exclusively as well as those they have in common.
This corresponds to the ‘logical sum’ of classes, where the two initial classes merge into a larger class by inclusive disjunction (expressed by operator or, which here means ‘and/or’, i.e. ‘not both not’; this is often symbolized by a ‘v’, or in some computer languages by a ‘½‘)[13]. Inclusive disjunction means that all the elements subsumed by the given classes are to be included in the larger class; if a subclass subsumes x elements and another involves y elements, then the larger class covers (x + y) elements. In contrast, in conjunction, only the elements subsumed by all the given classes are selected, forming a narrower class.[14]
We can, for instance, merge joint determinations into generics; thus, “mn v mq” is equivalent to just “m”, “mn v np” becomes “n”, “np v pq” becomes “p”, and “mq v pq” results in “q”. We can likewise merge generic determinations into broader concepts, such as strong or weak causation or causation, as shown below. Merger is easy if we work directly with alternative moduses; but it becomes very complicated if we refer to summary moduses, due to the inadequacies of the ad hoc notation system we have used so far.
In a two-item framework, it is feasible if we introduce an additional symbol, say l (Gk. letter lambda), signifying that the two positions in the formula where it occurs cannot both be coded ‘1’ (in contrast to µ, which signifies that they cannot both be ‘0’, remember). In such case, we can predict the summary moduses of the following vague propositions (s, w, c) on the basis of the generics merged in them:
· strong causation, symbol s = m or n = 10.1 v 1.01 = 1ll1 (moduses Nos. 10, 12, 14)[15];
· absolute weak causation, symbol w_{abs} = p_{abs} or q_{abs} = 11.1 v 1.11 = 1µµ1 (moduses Nos. 12, 14, 16);
· relative weak causation, symbol w_{rel} = p_{rel} or q_{rel} = same two-item summary modus as for absolute weak causation;
· causation, symbol c = m or n or p_{abs} or q_{abs} = 10.1 v 1.01 v 11.1 v 1.11 = 1..1 (moduses Nos. 10, 12, 14, 16).
Note that ‘causation’ here means some causation, causation of any determination whatever, whether m, n, p_{abs} or q_{abs}. As we will show in the next chapter, ‘contributory causation’ (m or p) and ‘possible causation’ (n or q) are different from it only with reference to relatives; in absolute terms, they are identical to each other and to causation (because m implies not-p_{abs}, and n implies not-q_{abs}).
The same four operations in a three-item system all apparently yield one and the same conclusion, namely “µ.µ..µ.µ” (try and see) – which is the summary modus of causation, covering 144 alternative moduses, as we shall see. This is of course an absurd result, because, as we shall see in the next chapter, strong causation in fact covers 63 moduses; absolute weak causation, 135 moduses; and relative weak causation, 31 moduses! It follows that our notation system is inadequate for merger operations other than:
c = µ0µ0.µ.µ v µ.µ.0µ0µ v µµµµ.µ.µ v µ.µ.µµµµ = µ.µ..µ.µ (144 moduses).
What this means is that the symbolic language developed so far is too simple to express more complex relations than those intended by a 0, 1, ∙ or µ (or even l, just introduced to enable merger of the two-item summary moduses of strong determinations[16]). It does not generate a distinctive summary modus for each and every form. However, I will not bother to attempt improving on it, not wishing to get bogged down in inessential matters. For our primary goal here is not to develop a calculus of summary moduses, but to ascertain how generic propositions can be merged into vaguer forms. And this we can readily do with reference to the underlying alternative moduses, which is good enough.
Before moving on, let us review the ground covered thus far. We started with binary coding of the summary moduses of the generic determinations known to us thanks to macroanalyses performed at the very start of our research into causative propositions. We saw that these summary moduses involved uncertainties (coded ∙). To eliminate these information gaps, we had to unravel the summary moduses, that is, identify the underlying alternative moduses (involving 0 or 1 codes exclusively). We thus introduced microanalysis.
However, the strong determinations m, n were expressed in a two-item (PR) framework, while the relative weaks p_{rel}, q_{rel} were expressed in a three-item (PQR) framework – so these two sets of forms were not comparable. We therefore had to work out the means for contraction and expansion of their summary moduses (the latter process required that we introduce a fourth code, µ). This also allowed us to ascertain the two- and three- item summary moduses of absolute weak determinations p_{abs}, q_{abs} – first by contracting those of the relative weaks p_{rel}, q_{rel}, then by expanding these results.
Having thus obtained both the two- and three- item summary moduses of all six generic determinations, we had all the information we need to work out the matrices of all derivative propositions. Indeed, by means of intersection we can readily identify the alternative moduses of any conjunction of determinations: they are the alternative moduses the latter have in common. A special case of this is nullification: if the propositions we wish to conjoin have no alternative moduses in common, they are incompatible. And by means of merger we can readily identify the alternative moduses of any disjunction of determinations: they are the alternative moduses the latter have all taken together.
We thus dispose of the basic data and logical processes we need for microanalysis of all positive forms, be they generic, joint (i.e. narrower than the generics) or vague (i.e. broader than the generics). But we still lack the alternative moduses of negative forms of whatever breadth. We cannot obtain their summary moduses by macroanalysis, as we did for the generic positive forms, because of the underlying complexity of negative causative propositions. So we must look for more profound means.
Thus far, we have engaged in microanalysis that may be characterized as piecemeal. In the next chapter, we shall approach this topic with a more holistic perspective, which we may refer to as systematic microanalysis. That consists in considering all conceivable alternative moduses in a given framework (fixed by the number of items under consideration), and then locating the determination(s) under consideration within this full range of possibilities.
The alternative moduses of negative forms become easy to identify thereby. Having the list of all conceivable alternative moduses in a given framework, and the alternative moduses of a positive form, we can readily infer those of the corresponding negative form: they are all the remaining alternative moduses! This process, which we shall simply call negation[17], is akin to subtraction. If a class subsumes x elements and a subclass of it involves y elements, then the remaining area covers (x – y) elements.
Microanalysis thus ultimately enables us to distinctively define any and every causative proposition (and other, related forms, as we shall see), with little effort. Furthermore, such detailed matricial analysis turns out to be a panacea, providing us with resolutions to all deductive issues in causation.
In particular, note that once we identify the moduses of negative generics, we can ascertain those of lone determinations, which conjoin one positive generic with the negations of all others. As we shall see in the next chapter, absolute lones are nullified. However, as we shall see in a subsequent chapter, relative lones are not nullified. Let us here mention for the record their summary moduses, which may be constructed knowing their alternative moduses, there identified (check and see for yourself that these summaries give rise to the correct alternatives):
· m-alone_{rel} = µ0µ0µµµµ
· n-alone_{rel} = µµµµ0µ0µ
· p-alone_{rel} = 10.1µ1µ.
· q-alone_{rel} = .µ1µ1.01
If we compare these to the summary moduses of m, n, p_{rel} and q_{rel}, respectively (which are given in Table 11.7 above), as well as to those of joint determinations mn, mq_{rel}, np_{rel}, p_{rel}q_{rel} (given in the previous section), we may observe the following mutations.
A code 0 or 1 for a generic is retained in a joint or lone including it. A µ found in m (or n, as the case may be) is retained in mn, and in m-alone_{rel} (or n-alone_{rel}), but not in mq_{rel} (or np_{rel}), because in the latter one µ is superseded by the 1 found in the corresponding position in p_{rel} (or q_{rel}), so that the remaining µ becomes a dot. There are no dots left in p_{rel}q_{rel} because all the dots in p_{rel} or q_{rel} have all been superseded by a 1 or 0. A dot in m or n becomes a µ in m-alone_{rel} or n-alone_{rel}, respectively. As for p-alone_{rel} or q-alone_{rel}, the dots in p_{rel} or q_{rel}not paired-off with a 1 become µ, whereas those paired-off with a 1 remain dots.
As already explained, a µ signifies that the pair of cells containing it (the first and third, the second and fourth, the fifth and seventh, or the sixth and eighth) may separately be 0 or 1, but cannot together be 0. No such restriction occurs where there are mere dots. Thus, what the above teaches us, especially, is that a relative lone determination has a slightly more restrictive modus than the corresponding generic determination, but is in all other respects identical.
[1] The word ‘modus’ was chosen to highlight the modal character of such statements. The plural form should perhaps be modera (just as genera is plural of genus); but we shall use moduses, anyway.
[2] The answer to that question is no.
[3] The labeling of columns (1-16) would change meaning in other permutations, but the meaning of the moduses would be unchanged. Our present study is in language ‘abcd’; 23 other languages could express the same information (since the rows might be ordered in 24 different ways). It might be interesting to compare these competing languages, in search for the most attractive; but we have adopted this one. (Note that the columns, also, could be ordered in umpteen different ways.)
[4] ‘Sometimes present’, remember, means ‘either always or only sometimes so’ – i.e. it allows for necessity as well as contingency.
[5] There are thus four senses of the word modus: the modus of a single conjunction of items (a cell in the grand matrix); the modus of all conceivable conjunctions of those items (a column, referring to the summary modus); the modus(es) which are the conceivable realizations of the summary modus (one or more columns, called the alternative modus(es)); and lastly, the actualizations underlying the possibilities inherent in the modal definition of the 0 and 1 codes (subsidiary columns, further subdividing each alternative modus), which we might call radical modus(es).
[6] We can, of course, symbolize the two or three items concerned by any letters we like. I have here chosen PR for two items to facilitate comparisons with P(Q)R for three items. The items could just as well have been labeled PQ and P(R)Q. These are mere matters of convention.
[7] The reader can ascertain this by taking the matrix of partial or contingent causation (Table 11.3 or 11.4) and reordering the rows: the columns are found to be in different order but have the same overall content.
[8] As we shall see further down, the summary modus of absolute partial causation is “11.1” and that of absolute contingent causation is “1.11”.
[9] See “Some More Microanalyses”, last section.
[10] I do not doubt that a better symbolic or mathematical logician than myself could develop neater approach. This is not my forte.
[11] When µ occurs, it occurs in pairs, note well. In contrast, within that notation, when a pair of dots occur, it means that both these positions may well be 0s.
[12] This process is therefore, despite its name, in some ways more akin to addition. See Future Logic, chapter 28, on logical compositions.
[13] This process turns out, despite its name, in some ways more akin to multiplication. See Future Logic, chapter 28, on logical compositions.
[14] Exclusive disjunction, note in passing, refers to the results of inclusive disjunction less those of conjunction, i.e. to the subsumptions of the given classes not common to them all.
[15] The two middle positions of the merged summary modus have to be l, because in the given summary moduses they may only be 00 or 01 (in the first) or 00 or 10 (in the second); i.e. the remaining possibility 11 is excluded.
[16] For three items, we would have to introduce, as well as the concepts ‘not 00’ (µ) and not ‘11’ (l), ‘not 01’ and ‘not 10’, among others (supposedly). All of which becomes more complicated than useful.
[17] Not to be confused with nullification, dealt with in the previous section.