THE LOGIC OF CAUSATION

Phase Two: Microanalysis

Chapter 12Systematic Microanalysis

4.Enumeration of Moduses, for Three Items

5.Comparing Frameworks

1.Grand Matrices.

Our study of causative propositions, in a first phase, consisted in conception of positive forms, their dissection into defining clauses, and their matricial analysis, or more precisely their macroanalysis. That provided us with the means to solve various problems, including many syllogistic issues; but it left us without practical means to answer questions concerning negative forms. We consequently, in a second phase, opted for a more detailed and deep method of study, microanalysis. We thus somewhat improved our predictive abilities; but serious difficulties remained, due to our approach being piecemeal.

To resolve outstanding issues, we must approach microanalysis in a more systematic manner. Instead of constructing matrices for each propositional form, we shall proceed in the opposite direction and conceive agrand matrixfor the items concerned in which each and every propositional form can be located. A grand matrix tabulatesall conceivablemoduses for a given number of items, and assigns a numerical label (an address, as it were) to each such logical possibility. Once this is developed, we can identify the places of the various determinations within such a broad framework, and easily predict all their interactions.

Through grand matrices, we have an overview of all possible relations between the items concerned. We can then focus on particular segments of the matrix as signifying this or that specific relation.

Two items (P, R) give rise to a table with 22= 4 rows (with PR sequences 11, 10, 01, 00, conventionally so ordered), and 24= 16 modus columns (conventionally ordered with the maximum number of zeros on the left and the maximum number of ones on the right, then numbered 1-16). Such a table defines thegeneralrelation of any pair of items, and is the same whatever they happen to be.

Aspecificrelation proposed for two particular items is then expressed by highlighting the modus column(s) corresponding to that specific relation (or by stating their numerical labels). The degree of determination involved is visually represented by the pattern of zeros and ones which stand out against the background of the grand matrix in which they are imbedded.

The grand matrix prefigures all ‘potential’ configurations for the number of items involved; while the highlighted alternative(s) depict the apparent or supposed ‘actual’ configuration for the particular items under scrutiny, which constitutes the distinctive determination relating them with each other.

In the case of three items (P, Q, R), the table has 23= 8 rows and 28= 256 modus columns, conventionally ordered in a similar manner. For four items (P, Q, R, S) we can expect a table with 24= 16 rows and 216= 65,536 modus columns. And so forth. Note well that the concrete content of the items is irrelevant to the structure of the grand matrix; it looks the same for any given number of items.

From an epistemological and ontological point of view, a grand matrix depictsthe universe of imaginable relations between any two (or more) items in the world or in knowledge taken at random. In reality, i.e. in the experienced world or at a given stage of knowledge development, only some of these relations (alternative moduses, i.e. conjunctions of presences and absences) will be found applicable to the items under scrutiny.

Thus, we can visualize the ‘distance’ (their separation in space-time, or their conceptual difference) between any two or more items in the world or in knowledge as inhabited by a belt[1]with strips of zeros and ones (a grand matrix with alternative moduses), of which some are highlighted or potent in the case concerned, and the rest are neutralized or inactive. We thus propose a very binary structure for the world and for knowledge, appealing by its universality and simplicity.

Indeed, in this perspective, we can even conceive of a ‘universal matrix’, comprising the umpteen items in the world or in knowledge, and an enormous tapestry of logically possible relations with zillions of zeros and ones in their every combination and permutation. For x items, this matrix would have y = 2xrows and z = 2ycolumns.

With this image in mind, the pursuit of knowledge can be considered as an attempt to pinpoint – on the basis of sensory and other experience, as well as of mental speculation and logical insight – the applicable moduses within such broad ranges, for the items concerned. A specific relation like ‘causation’ or ‘complete causation’ is thus a selection of moduses proposed as applicable to the concrete items concerned. The applicable alternative moduses constitute the ‘bond’ (of some degree) between the items in a given case.

Identification of applicable moduses proceeds gradually,inductively(with deduction as but a tool of induction). They are not known immediately, without residual doubts. Intellectual work is required.

We start with a mass of phenomena in flux. Appearances are presented to consciousness, perceptually (concretes) or conceptually (abstracts). We stratify some as ‘given’ (pure) and others as ‘speculative’ (mental projections about the pure), and try through logical insight to judge the hypotheses most fitting for the overall context of currently available data.

Much of our ‘thinking’ in relation to causation consists simply in trying to encapsulate the data available in the different forms of causation. This is a trial and error process, which may be characterized as successive formulation and (if need arise) elimination of hypotheses. Our approach may be passive, unconscious; or proactive, purposeful.

Normally, we first try out the strongest form of causation (mn), then lesser forms (mqornp), and finally the weakest (pq); if none of these work, we conclude with non-causation. Alternatively, we may proceed on a deeper level, with reference to if-then statements or, more cautiously, to moduses, before we build up comprehensive causative propositions.

As the empirical context changes, growing and becoming more focused, our opinion may vary. We may also discover, through deductive reasoning, inconsistencies between different conclusions. What seemed previously a successful summary of information then has to be reviewed. But eventually things seem to settle down and solidify, and we may presume that our opinion at last corresponds to (or more closely than ever approaches) the ‘real’ state of affairs, and may be regarded as knowledge.

Logic, after working out matricial configurations, immediately imposesone universal restriction: the alternative modus in any grand matrix consisting only of zeros, with no ones, cannot be true. Whatever the grand matrix, i.e. for any number and content of items, only alternative moduses involving at least one ‘1’ code are at all credible; in every such matrix, the first modus, composed entirely of ‘0’ codes, has no credibility.

This is just a restatement with regard to matricial analysis of the Laws of Non-Contradiction and of the Excluded Middle. Since the rows of our matrixalreadypredict every conceivable combination of the items in their positive and negative forms, at least one of these rows has to possibly exist; if a column means that none of these combinations may occur, it contradicts that setup and lays claim toyet anothercombination of items. Such a claim would be absurd, and may be rejected at the outset.

All other moduses are logically soundper se, though they might well be excludedwithin a given context. Indeed, the knowledge enterprise may be viewed as a search for good reasons for the elimination of as many moduses as we can, so as to be left with a limited number of moduses which signify an interesting specific relation like causation. We thus move from the vaguely conceivable, to a more focused and pondered evaluation.

We cannot say at the outset which relation (expressed by one or more moduses) applies in a given case. There is bound to besomerelation, but as we shall soon see logic does not insist on a specificallycausativerelation, it allows for a non-causative relation.Ab initio, all logic stipulates is that the modus consisting only of zeros can never apply.

This is the nearest thing to a ‘law of causation’ we can foresee at this stage; which by itself implies that there is no law of causation in the traditional senses, or that if there is one it must be sought for in other ways. We shall, of course, return to this topic in more detail, in a later chapter.

2.Moduses in a Two-Item Framework.

We shall first consider a two-item framework, and catalogue all its conceivable moduses, then enumerate those applicable to each category of proposition. In the following table, P is looked upon as a putative cause, while R is looked upon as a putative effect. Their conceivable combinations define rows, and columns refer to all initially conceivable alternative moduses for them.

In a two-item grand matrix, there are 4 rows and 16 columns, as we have seen, and therefore 64 cells. Each cell may equally be coded 0 (impossible) or 1 (possible), so that each code will occur a total of 32 times. The matrix is constructed by coding: in the first row, 0 in the first 8 cells then 1 in the last 8 cells; for the second row, 0 in the first and third set of 4 cells then 1 in the second and fourth set of 4 cells; in the third row, we have a succession of pairs, 00, 11, 00, 11, and so forth; finally, in the fourth row, we coded 0, 1, 0, 1, in succession. We are thus sure to have foreseen every possible interplay of 0 and 1 codes.

Take the time to notice that we have ordered the alternative moduses in a progressive manner, starting with a maximum number of 0s in a column (no cell coded 1) and ending with a maximum number of 1s in a column (no cell coded 0). We then conventionally number (or label) the columns so ordered, 1-16. The rows, note well, are also in a conventional arrangement, with four PR sequences 11, 10, 01, 00, respectively (labeled a-d, if need be).

Now, the column labeled No. 1 is animpossiblemodus, since at least one row has to have a ‘1’, by the Laws of Non-Contradiction and of the Excluded Middle. Significantly, this is the only combination excluded universally by those logical laws, as already explained. Concerning the remaining 15possiblemoduses, they areexhaustive(one of them must be true) andmutually exclusive(no more than one may be true at once).

Here, then, is the grand matrix for two items, a catalogue of all conceivable alternative moduses for any two items, like P, R:

Table 12.1.Catalogue of moduses for the four conjunctions of two items (P, R).

 Row Items ** Possible moduses, labeled 2-15 label P R 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 a 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 b 1 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 c 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 d 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

**Column labeled No. 1 is an impossible modus.

The following table interprets the preceding, by enumeration of the alternative moduses of the main causative forms. It is based on the known characteristics of positive strong and absolute weak generics, i.e. the moduses given in Tables 1, 2, 5 and 6 of the previous chapter. From this initial information, we can, using the processes of negation, intersection and merger, infer the alternative moduses of derivative forms, i.e. negatives, as well as joints and vaguer forms (s,w,c), and their negations.

Note that relative weak determinations are not dealt with here, because, in a two-item framework, they have the same moduses as absolutes. They can only be distinguished as of a three-item framework, so we cannot analyze them and their derivatives till we get there.

Table 12.2.Enumeration of two-item moduses for the strong or absolute weak determinations and their derivatives (form PR).

 Determination Column number(s) Comment Strongs and their negations: M 10, 12 2 alternatives, by macroanalysis. N 10, 14 2 alternatives, by macroanalysis. not-m 2-9, 11, 13-16 All alternatives but those ofm;i.e. 13 cases. not-n 2-9, 11-13, 15-16 All alternatives but those ofn;i.e. 13 cases. Absolute weaks and their negations: pabs 14, 16 2 alternatives, by macroanalysis ofpreland contraction. qabs 12, 16 2 alternatives, by macroanalysis ofqreland contraction. not-pabs 2-13, 15 All alternatives but those ofpabs;i.e. 13 cases. not-qabs 2-11, 13-15 All alternatives but those ofqabs;i.e. 13 cases. Joints (absolute) and their negations: Mn 10 Their one common alternative, by intersection. mqabs 12 Their one common alternative, by intersection. npabs 14 Their one common alternative, by intersection. pabsqabs 16 Their one common alternative, by intersection.
 not(mn) 2-9, 11-16 All alternatives but that ofmn;i.e. 14 cases. not(mqabs) 2-11, 13-16 All alternatives but that ofmqabs;i.e. 14 cases. not(npabs) 2-14, 15-16 All alternatives but that ofnpabs;i.e. 14 cases. not(pabsqabs) 2-15 All alternatives but that ofpabsqabs; i.e. 14 cases. Strong causation and its negation: s = m or n 10, 12, 14 All their 3 alternatives, by merger. not-s = not-m+not-n 2-9, 11, 13, 15-16 All alternatives but the preceding; i.e. 12 cases. Absolute weak causation and its negation: wabs= pabsor qabs 12, 14, 16 All their 3 alternatives, by merger. not- wabs=not-pabs+not-qabs 2-11, 13, 15 All alternatives but the preceding; i.e. 12 cases. Causation (absolute) and its negation: cabs=m or n or pabsor qabs 10, 12, 14, 16 All their four alternatives, by merger. not-cabs=not-m+not-n+ not-pabs+not-qabs 2-9, 11, 13, 15 All alternatives but the preceding; i.e. 11 cases.

Let us highlight some of the information in the above table. First, take note of the ease with which we are now able to define any negative form, given the moduses of the corresponding positive form, by simply listing the leftover moduses. We can also readily define vaguer positive forms, likes,w,c, by merging the modus lists of their components. These forms were until here very difficult to define, remember.

Second, we can see at a glance that compatible forms are those which have a common modus (or more); for instance,mandn,mandqabs,nandpabs,pabsandqabscan be joined, because they share a modus (respectively, 10, 12, 14 and 16). Incompatibilities are also made evident by such a table; thus,mandpabshave no common modus, nor donandqabs; so these are incompatible pairs and give rise to no form.

Third, certain compounds of positives and negatives have not been listed in the above table, because they are equivalent to already listed forms, i.e. all their moduses are the same.Implicationsignifies thateverymodus of the implying form is a modus of the implied form; this is not mere overlap, note, but full inclusion of one form in the other.

Twoone-wayimplications (and their contraposites) must be noted:

·thatpabsimpliesnot-m(ormimpliesnot-pabs), and

·thatqabsimpliesnot-n(ornimpliesnot-qabs).

This is because the moduses Nos. 14, 16 ofpabsare both also moduses ofnot-m, and the moduses Nos. 12, 16 ofqabsare both also moduses ofnot-n. Given thatmimpliesnot-pabs, it follows that (m + not-pabs) is identical tom. Similarly, (n + not-qabs) =n; (not-m + pabs) =pabs; and (not-n + qabs) =qabs. There is therefore no need to list these four conjunctions separately.

Mutualimplication or equivalence occurs when the forms compared have the very same alternative modus list. Thus,

·(m + not-qabs) = (n + not-pabs) =mn(modus 10);

·(m + not-n) = (not-pabs+ qabs) =mqabs(modus 12);

·(not-m + n) = (pabs+ not-qabs) =npabs(modus 14); and

·(not-m + qabs) = (not-n + pabs) =pabsqabs(modus 16).

There is therefore no need to list these various conjunctions separately. In contrast, for instance,mandndo not imply each other, though they have one modus in common (No. 10), because each has a modus the other lacks. Likewise forpabsandqabs, they overlap only in one of their moduses (No. 16) and both have a distinct additional modus.

Fourth, some compositions have not been listed in the above table, because they do not constitute an interesting concept. Falling in this category arem orqabs(moduses 10, 12, 16) and its negation (not-m + not-qabs), or againn orpabs(moduses 10, 14, 16) and its negation (not-n + not-pabs).

Fifth, certain conjunctions of positives and negatives have not been listed in the above table, because they give rise to no forms. Note especially that(absolute) lone determinations are excludedfrom consideration (or nullified) by this technique. That is, we cannot form the following conjunctions of positive and negatives, because they do not share a single common alternative modus:

·m-aloneabs=m+not-n+not-pabs+not-qabs= null-class;

·n-aloneabs=n+not-m+not-pabs+not-qabs= null-class;

·p-aloneabs=pabs+not-m+not-n+not-qabs= null-class;

·q-aloneabs=qabs+not-m+not-n+not-pabs= null-class.

Thus, for instance,mshares modus 12 withnot-nand (needless to say, since it implies it) withnot-pabs, but this modus is absent innot-qabs. And so forth, for the other absolute lones. These symbolically contrived conjunctions are therefore impossible in fact:by reference to the moduses we can definitively establish this fact and understand it.

This is an important formal principle, which may be looked upon as a ‘law of causation’ (among others)[2]. Had (absolute) lone determinations been possible, our view of the causative relation would have been much less deterministic. Before microanalysis, we could not ascertain whether or not the generic determinationsm,n,pabsorqabsmay logically exist without intersection; now we know for sure that they can only exist within joint determinations.

The following equations follow from the nullification of lones:

·m= (mn or mqabs), andn= (mn or npabs);

·pabs= (npabsor pabsqabs), andqabs= (mqabsor pabsqabs).

Again,s= (mn or mqabsor npabs), andwabs= (mqabsor npabsor pabsqabs). Consequently,cabs= (mn or mqabsor npabsor pabsqabs); and it is equivalent to (m or pabs) and to (n or qabs). Also, by negation,not-cabsis equivalent to (not-m + not-pabs) and to (not-n + not-qabs).

These various compounds are therefore implicit in the above table, and need not be listed.

Lastly, we should notice thegenus-speciesrelations between forms. Thus,mnis a species ofmand a species ofn, because it shares a modus (No. 10) with each of them, and has none they lack; the latter forms are more generic or less definite, since they involve additional alternatives. Similarly,sis vaguer or broader in possibilities thanmorn, and therefore a genus of theirs; likewise,pabsandqabsare species ofwabs. Causation (c) is clearly the summum genus for all the positive forms. Negatives can be examined in the same perspective.

It is also worth noticing what underlies therelative strengths of determinations. Note that the alternative moduses of the strong determinations (10, 12, 14) involve more zeros than those of the weaks (12, 14, 16). In particular, ignoring the common moduses (12, 14), compare modus 10 (two 0s) with modus 16 (no 0s). Clearly,mandnare stronger thanpandq, because they involve more impossibility (two extra zeros); zeros more firmly delimit a relation. Similarly, comparing joints with each other; the more zeros in the modus, the stronger the determination.

3.Catalogue of Moduses, for Three Items.

Let us now consider a three-item framework. We shall here catalogue all its conceivable moduses; and in the next section, we shall enumerate those applicable to each category of proposition. In the following table, P and Q are looked upon as putative causes, while R is looked upon as a putative effect. Their conceivable combinations define rows, and columns refer to all initially conceivable alternative moduses for them.

In a three-item grand matrix, there are 8 rows and 256 columns, as we have seen, and therefore 2048 cells. Each cell may equally be coded 0 (impossible) or 1 (possible), so that each code will occur a total of 1024 times. This matrix is constructed in the same manner as the preceding one, by coding 0s and 1s progressively throughout it, so symmetrically that we can be sure it is exhaustive.

The columns (representing the alternative moduses), so ordered, are then numbered (or labeled) 1-256. Since the order of the rows is also fixed conventionally, with eight PQR sequences 111, 110, 101, 100, 011, 010, 001, 000 (which can, if need be, be labeled a-h, respectively), the modus number suffices to symbolize the modus concerned.[3]

Now, the column labeled No. 1 is animpossiblemodus, since at least one row has to have a ‘1’, by the Laws of Non-Contradiction and of the Excluded Middle. Significantly, this is the only combination excluded universally by those logical laws, as already explained. Concerning the remaining 255possiblemoduses, they areexhaustive(one of them must be true) andmutually exclusive(no more than one may be true at once).

Here, then, is the grand matrix for three items, a catalogue of all conceivable alternative moduses for any three items, such as P, Q, R:

Table 12.3.Catalogue of moduses for the eight conjunctions of three items (P, Q, R).

 Items ** Possible moduses, labeled 2-16 P (Q) R 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 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 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

**Column labeled No. 1 is an impossible modus.

Same table continued.

 Items Moduses, labeled 17-32 P (Q) R 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 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 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Same table continued.

 Items Moduses, labeled 33-48 P (Q) R 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 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 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Same table continued.

 Items Moduses, labeled 49-64 P (Q) R 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 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 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Table 3 continued.

 Items Moduses, labeled 65-80 P (Q) R 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Same table continued.

 Items Moduses, labeled 81-96 P (Q) R 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Same table continued.

 Items Moduses, labeled 97-112 P (Q) R 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 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 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Same table continued.

 Items Moduses, labeled 113-128 P (Q) R 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 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 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Table 3 continued.

 Items Moduses, labeled 129-144 P (Q) R 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 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 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Same table continued.

 Items Moduses, labeled 145-160 P (Q) R 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 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 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Same table continued.

 Items Moduses, labeled 161-176 P (Q) R 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 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 1 0 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 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Same table continued.

 Items Moduses, labeled 177-192 P (Q) R 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 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 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Table 3 continued.

 Items Moduses, labeled 193-208 P (Q) R 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Same table continued.

 Items Moduses, labeled 209-224 P (Q) R 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Same table continued.

 Items Moduses, labeled 225-240 P (Q) R 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 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 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

Same table continued.

 Items Moduses, labeled 241-256 P (Q) R 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 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 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 1 1 1 0 0 0 0 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

4.Enumeration of Moduses, for Three Items.

The following table interprets the preceding, by enumeration of the alternative moduses of the main causative forms. It is based on the known characteristics of positive strong and weak generics, i.e. the moduses given in Tables 1-6 of the previous chapter. From this initial information, we can, using the processes of negation, intersection and merger, infer the alternative moduses of derivative forms, i.e. negatives, as well as joints and vaguer forms (s,w,c), and their negations.

We shall deal here only with the absolute weak determinations and their derivatives; relative weaks and their derivatives will be considered in the next chapter.

Table 12.4.Enumeration of three-item moduses for the generic determinations and their derivatives (form PR).

 Determination Modus numbers Comment Strongs and their negations: m 34, 36-40, 42, 44-48, 130, 132-136, 138, 140-144, 162, 164-168, 170, 172-176 36 alternatives,by macroanalysis. n 34, 37-38, 50, 53-54, 98, 101-102, 114, 117-118, 130, 133-134, 146, 149-150, 162, 165-166, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 36 alternatives,by macroanalysis. not-m 2-33, 35, 41, 43, 49-129, 131, 137, 139, 145-161, 163, 169, 171, 177-256 All alternatives but those ofm, i.e. 219 cases. not-n 2-33, 35-36, 39-49, 51-52, 55-97, 99-100, 103-113, 115-116, 119-129, 131-132, 135-145, 147-148, 151-161, 163-164, 167-177, 179-180, 183-193, 195-196, 199-209, 211-212, 215-225, 227-228, 231-241, 243-244, 247-256 All alternatives but those ofn, i.e. 219 cases. Absolute weaks and their negations: pabs 50, 52-56, 58, 60-64, 98, 100-104, 106, 108-112, 114, 116-120, 122, 124-128, 146, 148-152, 154, 156-160, 178, 180-184, 186, 188-192, 194, 196-200, 202, 204-208, 210, 212-216, 218, 220-224, 226, 228-232, 234, 236-240, 242, 244-248, 250, 252-256 108 alternatives,by macroanalysis ofprelthen contraction and expansion. qabs 36, 39-40, 42, 44-48, 52, 55-56, 58, 60-64, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 132, 135-136, 138, 140-144, 148, 151-152, 154, 156-160, 164, 167-168, 170, 172-176, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 108 alternatives,by macroanalysis ofqrelthen contraction and expansion. not-pabs 2-49, 51, 57, 59, 65-97, 99, 105, 107, 113, 115, 121, 123, 129-145, 147, 153, 155, 161-177, 179, 185, 187, 193, 195, 201, 203, 209, 211, 217, 219, 225, 227, 233, 235, 241, 243, 249, 251 All alternatives but those ofpabs, i.e. 147 cases. not-qabs 2-35, 37-38, 41, 43, 49-51, 53-54, 57, 59, 65-99, 101-102, 105, 107, 113-115, 117-118, 121, 123, 129-131, 133-134, 137, 139, 145-147, 149-150, 153, 155, 161-163, 165-166, 169, 171, 177-179, 181-182, 185, 187, 193-195, 197-198, 201, 203, 209-211, 213-214, 217, 219, 225-227, 229-230, 233, 235, 241-243, 245-246, 249, 251 All alternatives but those ofqabs, i.e. 147 cases.
 Joints (absolute) and their negations: mn 34, 37-38, 130, 133-134, 162, 165-166 Their 9 common alternatives, by intersection. mqabs 36, 39-40, 42, 44-48, 132, 135-136, 138, 140-144, 164, 167-168, 170, 172-176 Their 27 common alternatives, by intersection. npabs 50, 53-54, 98, 101-102, 114, 117-118, 146, 149-150, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 Their 27 common alternatives, by intersection. pabsqabs 52, 55-56, 58, 60-64, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 148, 151-152, 154, 156-160, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 Their 81 common alternatives, by intersection. not(mn) 2-33, 35-36, 39-129, 131-132, 135-161, 163-164, 167-256 All alternatives but those ofmn; i.e. 246 cases. not(mqabs) 2-35, 37-38, 41, 43, 49-131, 133-134, 137, 139, 145-163, 165-166, 169, 171, 177-256 all alternatives but those ofmqabs; i.e. 228 cases. not(npabs) 2-49, 51-52, 55-97, 99-100, 103-113, 115-116, 119-145, 147-148, 151-177, 179-180, 183-193, 195-196, 199-209, 211-212, 215-225, 227-228, 231-241, 243-244, 247-256 All alternatives but those ofnpabs; i.e. 228 cases. not(pabsqabs) 2-51, 53-54, 57, 59, 65-99, 101-102, 105, 107, 113-115, 117-118, 121, 123, 129-147, 149-150, 153, 155, 161-179, 181-182, 185, 187, 193-195, 197-198, 201, 203, 209-211, 213-214, 217, 219, 225-227, 229-230, 233, 235, 241-243, 245-246, 249, 251 All alternatives but those ofpabsqabs; i.e. 174 cases. Strong causation and its negation: s = m or n 34, 36-40, 42, 44-48, 50, 53-54, 98, 101-102, 114, 117-118, 130, 132-136, 138, 140-144, 146, 149-150, 162, 164-168, 170, 172-176, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 Their 63 separate and common alternatives (including overlap, i.e.mn), by merger. not-s = not-m+not-n 2-33, 35, 41, 43, 49, 51-52, 55-97, 99-100, 103-113, 115-116, 119-129, 131, 137, 139, 145, 147-148, 151-161, 163, 169, 171, 177, 179-180, 183-193, 195-196, 199-209, 211-212, 215-225, 227-228, 231-241, 243-244, 247-256 All alternatives but the preceding; i.e. 192 cases. Absolute weak causation and its negation: wabs= pabsor qabs 36, 39-40, 42, 44-48, 50, 52-56, 58, 60-64, 98, 100-104, 106, 108-112, 114, 116-120, 122, 124-128, 132, 135-136, 138, 140-144, 146, 148-152, 154, 156-160, 164, 167-168, 170, 172-176, 178, 180-184, 186, 188-192, 194, 196-200, 202, 204-208, 210, 212-216, 218, 220-224, 226, 228-232, 234, 236-240, 242, 244-248, 250, 252-256 Their 135 separate and common alternatives (including overlap, i.e.pabsqabs), by merger. not- wabs=not-pabs+ not-qabs 2-35, 37-38, 41, 43, 49, 51, 57, 59, 65-97, 99, 105, 107, 113, 115, 121, 123, 129-131, 133-134, 137, 139, 145, 147, 153, 155, 161-163, 165-166, 169, 171, 177, 179, 185, 187, 193, 195, 201, 203, 209, 211, 217, 219, 225, 227, 233, 235, 241, 243, 249, 251 All alternatives but the preceding; i.e. 120 cases.
 Causation (absolute) and its negation: cabs=m or n or pabsor qabs 34, 36-40, 42, 44-48, 50, 52-56, 58, 60-64, 98, 100-104, 106, 108-112, 114, 116-120, 122, 124-128, 130, 132-136, 138, 140-144, 146, 148-152, 154, 156-160, 162, 164-168, 170, 172-176, 178, 180-184, 186, 188-192, 194, 196-200, 202, 204-208, 210, 212-216, 218, 220-224, 226, 228-232, 234, 236-240, 242, 244-248, 250, 252-256 Their 144 separate and common alternatives (including overlap). not-cabs=not-m+not-n+ not-pabs+not-qabs 2-33, 35, 41, 43, 49, 51, 57, 59, 65-97, 99, 105, 107, 113, 115, 121, 123, 129, 131, 137, 139, 145, 147, 153, 155, 161, 163, 169, 171, 177, 179, 185, 187, 193, 195, 201, 203, 209, 211, 217, 219, 225, 227, 233, 235, 241, 243, 249, 251 All alternatives but the preceding; i.e. 111 cases.

The results obtained in Table 12.4 can be made to conveniently stand out by color coding each form’s moduses in Table 12.3. This is left to the reader to do.

We need not repeat here what was said before, with reference to the similar table for a two-item framework (Table 12.2); the same comments apply, because the relationships there established are true irrespective of framework. We will, however, highlight something which was less visible before, namely the consistency between various results.

There are never overlaps between contradictory propositions, and their alternatives sum up to 255; also, each generic sums up to two joints (since absolute lones do not exist). For instance,mcomprises 36 alternative moduses, the 9 ofmnplus the 27 ofmqabs; whilenot-mhas the 219 remaining alternatives. Similarly, with regard ton. Likewise,pabscomprises 108 alternatives, the 27 ofnpabsplus the 81 ofpabsqabs; whilenot-pabshas the 147 remaining alternatives. Similarly, with regard toqabs.

Moreover, the number of moduses corresponding to the vaguer forms are predictable. Thus,s(=m or n) comprises the 36 moduses ofmplus the 36 ofn, less the 9 ofmn[4], a total of 63 alternatives; and its negation has 255 – 63 = 192 alternatives. We can similarly predict the moduses ofwabs(=pabsorqabs) to be 108 + 108 – 81 = 135; and a residue of 120 alternatives for its negation. Forc(=s or wabs) we have 63 + 135 – 2*27 = 144 (the 54 subtracted being those ofmqabsandnpabs– i.e. ofswabs); for its negation, 111.

Thus, incidentally, causation in all its forms covers more than half the matrix, but still leaves a large space to non-causation.

5.Comparing Frameworks.

Now let us compare the results in Tables 2 and 4. They are essentially the same tables, except that each modus of the first is, as it were, further subdivided into a number of moduses in the second. However, the subdivision is evidentlynotproportional, say in the ratio 16:256; you cannot just say that to each two-item modus there corresponds 16 three-item ones. The following table makes this disproportionality clear:

Table 12.5.Numbers of Moduses for Positive Forms, in Different Frameworks.

 Framework m,n pabs,qabs mn mqabs,npabs pabsqabs s wabs c Two-Item 2 2 1 1 1 3 3 4 Three-Item 36 108 9 27 81 63 135 144

The explanation is easy.Expansion of a two-item alternative modus into a number of three-item moduses depends on how many zero or one codes it involves. For, as we saw in the previous chapter (with the proviso of appropriate locations), each ‘0’ in a two-item framework has a single expression (‘0 0’) in the three-item framework; whereas each ‘1’ in the former has three expressions in the latter (‘0 1’, ‘1 0’ or ‘1 1’ – i.e.any but‘0 0’).

Thus, if a two-item modus involves four ‘zeros’ and no ‘one’, its three-item equivalent will consist of 1*1*1*1 = 1 (equally impossible) modus; if the former involves three zeros and a one, the latter will consist of 1*1*1*3 = 3 moduses; if the former involves two zeros and two ones, the latter will consist of 1*1*3*3 = 9 moduses; if the former involves one zero and three ones, the latter will consist of 1*3*3*3 = 27 moduses; and if the former involves no zero and four ones, the latter will consist of 3*3*3*3 = 81 moduses.

Whence, the strongsm,n, which each involves two two-item moduses, one with two zeros (No. 10) and one with a single zero (no. 12 or 14), will have 9 + 27 = 36 three-item moduses; whereas, the weakspabs,qabs, which each involves two two-item moduses, one with a single zero (no. 12 or 14) and one with no zero (No. 16) and will have 27 + 81 = 108 three-item moduses.

The numbers of three-item moduses for the conjunctions and disjunctions of these forms follow. The jointmn(two-item modus No. 10) will have 9 of them;mqabs(modus No. 12) andnpabs(modus No. 14) will each have 27; andpabsqabs(modus 16) will have 81. The vague forms(moduses 10, 12, 14) will have 9 + 2*27 = 63;wabs(moduses 12, 14, 16) will have 2*27 + 81 = 135; andc(moduses 10, 12, 14, 16) will have 9 + 2*27 + 81 = 144.

We can proceed in a like manner to predict expansions of negative forms. Furthermore, given the two-item modus(es) of a form, we can predict not onlyhow manymoduses it will have in a three-item framework, but preciselywhichmoduses it will have. Thus, a table of equivalencies between the two frameworks can be constructed without difficulty. In short, we have here a functioning calculus.

The precise three-item modus(es) corresponding to each two-item modus are given in the following table:

Table 12.6.Correspondences between two- and three item frameworks.

 Two-item modus No. of zeros in it Corresponding three-item modus numbers No. of moduses 1 4 1 1 2 3 2, 5, 6 3 3 3 3, 9, 11 3 4 2 4, 7, 8, 10, 12-16 9 5 3 17, 65, 81 3 6 2 18, 21-22, 66, 69-70, 82, 85-86 9 7 2 19, 25, 27, 67, 73, 75, 83, 89, 91 9 8 1 20, 23-24, 26, 28-32, 68, 71-72, 74, 76-80, 84, 87-88, 90, 92-96 27 9 3 33, 129, 161 3
 10 2 34, 37-38, 130, 133, 134, 162, 165-166 9 11 2 35, 41, 43, 131, 137, 139, 163, 169, 171 9 12 1 36, 39-40, 42, 44-48, 132, 135-136, 138, 140-144, 164, 167-168, 170, 172-176 27 13 2 49, 97, 113, 145, 177, 193, 209, 225, 241 9 14 1 50, 53-54, 98, 101-102, 114, 117-118, 146, 149-150, 178, 181-182, 194, 197-198, 210, 213-214, 226, 229-230, 242, 245-246 27 15 1 51, 57, 59, 99, 105, 107, 115, 121, 123, 147, 153, 155, 179, 185, 187, 195, 201, 203, 211, 217, 219, 227, 233, 235, 243, 249, 251 27 16 0 52, 55-56, 58, 60-64, 100, 103-104, 106, 108-112, 116, 119-120, 122, 124-128, 148, 151-152, 154, 156-160, 180, 183-184, 186, 188-192, 196, 199-200, 202, 204-208, 212, 215-216, 218, 220-224, 228, 231-232, 234, 236-240, 244, 247-248, 250, 252-256 81 16 Total number of moduses 256

Needless to say, each modus will occur only once in the above table, making a total of 16 or 256 moduses, according to the framework. Clearly, if we had developed this table earlier, we could have derived Table 12.4 from Table 12.2.[5]

Obviously, we can follow the same procedures to expand three-item alternative moduses (of which there are 256) into four-item alternative moduses (of which there are 65,536 – as seen earlier).

Thenumber and configurationof the latter will emerge from the each of the former, in accordance with the number of zero and one codes it contains and the way they are arrayed within it (i.e. the incidence, prevalence and locations of zero and one codes in it). Atable of correspondencescan thus be constructed, which details the results obtained in each case.

We have above identified the main lines of what might be called thetwo-three(2/3) table of correspondences, emerging from the operation ofexpansionof ‘0’ into ‘0 0’ and ‘1’ into ‘0 1’, ‘1 0’, ‘1 1’ (all pairs but ‘0 0’). We could thereafter, step by step, build similar tables of correspondence of size 3/4 or 4/5… and so forth on to infinity, if need arise to resolve eventual issues.

For instance, from a three-item matrix (which has 8 rows) to a four-item matrix, each combination of zeros and ones will result in a product of eight factors of 1 (for ‘0’ codes) or 3 (for ‘1’ codes) – e.g., a modus with 1 zero and 7 ones will become 1*3*3*3*3*3*3*3 = 2187 moduses, in various possible permutations. These are long-winded techniques, which may or may not be needed.

Next Section(continuation of same chapter)

[1]To stress this image, we could place the items at opposite ends of the matrix. For two items, the ‘belt’ would be flat; for three items voluminous in three dimensions; and so forth. Another idea is to imagine the matrix as somehow enveloping the items, with varying force of cohesion. Each alternative modus indeed signifies a centripetal or centrifugal force relating the items concerned.

[2]The expression ‘law of causation’ is traditionally used with reference to general statements such as “everything has a cause”, for which we have so far not found formal justification, though they might eventually be adopted as inductive principles. Here, the phrase is used in a more open sense, reflecting the usual usage of the term ‘law’. In this sense, as we saw earlier, the fact that alternative modus No. 1 (consisting only of zeros) is impossible is a law; and likewise the fact that absolute lone determinations do not exist. Indeed, in this sense, all formal processes about causation – including all oppositions, eductions, syllogisms – are laws.

[3]Needless to say, one should not confuse the modus numbers 1-16 in a two-item framework, with the first 16 of 256 modus numbers used for a three-item framework. These are mere homonyms. The framework concerned should always be specified, if not implicitly clear. (See Table 12.6 below for precise correspondences.)

You can purchase a paper copy of this bookat The Logician’s secure online Bookshop.