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THE LOGIC OF CAUSATION

Phase Two: Microanalysis

Chapter 14 Main Three-Item Syllogisms

3.    The Moduses of Conclusions.

4.    Dealing with Vaguer Propositions.

We shall now begin to use microanalytic methods for the understanding and solution of syllogistic problems. In this way, we shall develop an advanced and general theory of the syllogism, using a tool more powerful and universal than any previously used. Ciao Aristotle, hello future logic. We shall proceed in stages, from the simpler cases to the more complex.

As we are already aware, syllogism basically involves three items (terms or theses) in three propositions. The propositions (in this context, of causative form) are the major premise, the minor premise and the conclusion. The items involved are the minor item (which occurs somewhere in the minor premise, and as the cause in the conclusion), the major item (which occurs somewhere in the major premise, and as the effect in the conclusion), and the middle item (present in both premises, but ignored in the conclusion).

Syllogism, then, is argument aimed at discovering or establishing the relation (the conclusion) between two items (the minor and major), by way of their given relations (the premises) to a third item (the middle). If the premises are compatible, they will either imply a certain conclusion, in which case the syllogism is ‘valid’; or fail to imply any conclusion, in which case the syllogism is ‘invalid’.

And as we have seen, there are three main ‘figures’ for such argument, traditionally labeled Nos. 1, 2 and 3. In the first figure, the middle item has the roles of effect in the minor premise and cause in the major premise; in the second figure, it has the role of effect in both premises; and in the third figure, it has the role of cause in both figures.

Ordinarily, the three items occur only in the positions just said. But we may conceive of special cases where the items recur in other positions, as complements. This gives rise to arrangements we have called ‘subfigures’. We need not concern ourselves with these complications at this stage, but will only focus our attention on the essential, Aristotelian arrangements.

Figures are, of course, abstractions regarding the positions of the items. Syllogism is concretized in individual formal ‘moods’, when the exact relations between these items are specified - i.e., in the present context, the determination and polarity of the causative relation in each premise. The latter are likewise useless without specification of the figure involved.

Thus, to end this brief review, we will here to begin with reexamine, using the microanalytic method, three-item causative syllogism in the three main figures (without consideration of the fourth figure or of subfigures). We shall start with positive moods, i.e. moods with both premises affirmative; after that, we shall deal with negative moods, i.e. those involving at least one negative premise.

Still later, we shall gradually treat all other conceivable situations, and thus demonstrate the universality of this technique.

Now, putting aside all the above mentioned details, syllogism is nothing more than a logical conjunction of two propositions (the premises), and all we seek through it is an evaluation or inference of their intersection (the conclusion). That is:

The major premise + the minor premise = the conclusion (if any).

It follows that, knowing the moduses of the premises, we can ascertain those of the conclusion. The conclusion may thus be viewed as a summary the information of the premises taken together; i.e. as a statement of their combined value in knowledge. Each premise, taken separately, has a certain value - expressed in its modus list; in the conclusion, we find out their mutual impact - which is merely their common moduses, if any. We can refer to an image in mechanics, we have two vectors and we wish to calculate their resultant force.

The following simple example will illustrate how a valid conclusion arises. Mood 1/m/m (which, you will recall, we labeled 155) consists of the premises “Q is a complete cause of R” (the major, which we may label mQR) and “P is a complete cause of Q” (the minor, which we may label mPQ). We know by macroanalysis that its conclusion is “P is a complete cause of R” (which we may label mPR); as shown in the table below, microanalysis yields the same result.

Table 14.1.       Microanalysis of a mood - the example of mood 1/m/m (No. 155).

 Proposition Moduses Comments The major premise, mQR 10, 12, 25-28, 42, 44, 57-60, 130, 132, 138, 140, 145-148, 153-156, 162, 164, 170, 172, 177-180, 185-188 36 alternatives, with reference to items Q and R. The minor premise, mPQ 66-68, 70-72, 74-76, 78-80, 130-132, 134-136, 138-140, 142-144, 194-196, 198-200, 202-204, 206-208 36 alternatives, with reference to items P and Q. The conclusion, mPR 130, 132, 138, 140 4 moduses in common to the above premises.

Note that the premises, being in the first figure, have respectively forms QR (major) and PQ (minor), while the conclusion must be of form PR. Once the modus or moduses of the premises are ascertained (as explained in the next section), we need only find our which modus or moduses, if any, they have in common. The form or forms which include all such common modus(es) is/are implied by them, and may be considered as constituting the conclusion. This intersection may be illustrated as follows (Diagram 14.1):

In our example, four moduses are found in common. With reference to Table 12.4 of the chapter on systematic microanalysis, which concerns causative propositions of form PR, we see that these four moduses are all among the 36 alternatives of m. It follows that mPR is our conclusion or part thereof. Continuing our search, we see that not all of these four moduses are included in mn (only No. 132 is so) or mqabs (only Nos. 132, 138, 140 are so), therefore we cannot obtain a more precise positive conclusion. Thus, mQR + mPQ is mPR.

Note well that, to conclude mPR, we do not need to find all its moduses to be in common to the premises - just one would suffice; what is important is that all the moduses which are in common to the premises be included in the list of 36 for that putative conclusion.

Also note that the above mentioned table[1], which enumerates three-item moduses for propositions of form PR, is appropriate for evaluating the conclusion, which has that same form in all figures.

But the moduses of the premises cannot be identified with reference to that table, since they have forms other than PR, namely QR or PQ (in the first figure). In their case, we must revert to the grand matrix for three items, i.e. the preceding table in the same chapter, Table 12.3. How this is done is explained in detail in the next section, where the results for all conceivable strong or absolute weak premises in whatever figure are also tabulated.

We can, at the outset, formulate the following general rules of inference, based on our knowledge of the intersection process:

1.    If the premises have no modus in common, they are incompatible. This signifies that the premises are already incoherent and cannot credibly occur together in knowledge. If we encounter such premises (without common modus, inconsistent) in practice, as often happens, not only can we draw no conclusion from them, but one or both of them must be reviewed and rejected or corrected.

2.    If the premises have one or more modus(es) in common and such modus(es) is/are all included by some causative propositional form(s), the latter constitute(s) our conclusion. Note well again that, though all common moduses of the premises must be moduses of the putative conclusion, not all the moduses of the latter need be common to the former. For a form, remember, is implied by any one of its alternative moduses. Also note, a conclusion may consist of one causative form or a conjunction of such forms, of whatever polarities.

3.    But if the premises have two or more moduses in common and these moduses are not all included by some causative propositional form(s), though some of them are included by a form and others by its contradictory, we have no conclusion. In this case, the premises are indeed compatible, since they have common moduses; but those common moduses do not give rise to a causative proposition, however vague, being too scattered[2]. The common moduses may of course still give rise to a proposition of form other than causative; but this is not of interest in the present context.

In order to systematically apply the microanalytic method to the solution of syllogistic problems, we need first to identify the moduses of all conceivable premises. For we already have the moduses of eventual conclusions, which always have form PR: these are given in Table 12.4 of the chapter on systematic microanalysis.

It is marvelous that we can refer to the same grand matrix for three items, with 255 possible moduses, which we used to interpret three-item propositions, to interpret three-item syllogisms. For syllogisms with four items, we shall need to consider 65,535 possible moduses![3]

Regarding the premises in the three figures, we see that they involve four possible pairs and arrangements of items, namely QR or RQ for the major premise and PQ or QP for the minor premise. The summary moduses of the various generic forms of interest to us here are as follows. Note well that in the following table, the weak determinations p, q are intended as absolute, though not so specified.

Table 14.2.  Three-item summary moduses for strong or absolute weak generic positive premises.

 row labl Items m m m m n n n n p p p p q q q q P Q R QR RQ PQ QP QR RQ PQ QP QR RQ PQ QP QR RQ PQ QP a 1 1 1 ae ae ab ab ae ae ab ab ae ae ab ab ae ae ab ab b 1 1 0 0 ∙ ab ab ∙ 0 ab ab bf ∙ ab ab ∙ bf ab ab c 1 0 1 ∙ 0 0 ∙ 0 ∙ ∙ 0 ∙ cg cd ∙ cg ∙ ∙ cd d 1 0 0 dh dh 0 ∙ dh dh ∙ 0 dh dh cd ∙ dh dh ∙ cd e 0 1 1 ae ae ∙ 0 ae ae 0 ∙ ae ae ∙ ef ae ae ef ∙ f 0 1 0 0 ∙ ∙ 0 ∙ 0 0 ∙ bf ∙ ∙ ef ∙ bf ef ∙ g 0 0 1 ∙ 0 gh gh 0 ∙ gh gh ∙ cg gh gh cg ∙ gh gh h 0 0 0 dh dh gh gh dh dh gh gh dh dh gh gh dh dh gh gh

Where two letters (among a-h) appear in two cells of a column, the intent is to indicate that this pair of cells can never both be zero, i.e. a ‘µ‘ is intended[4]. Thus, for instance, instead of writing µae, I write ‘ae’ in both rows a and e. This is done just to improve visibility of the information.

Notice the following details in the above table. First, note the similarity of sequence of modalities in two columns read in opposite directions (for instance the modus of mQR read from top to bottom and that of nQR read from bottom to top); this signifies mirroring. Second, note the identity of moduses of various pairs of propositions, like mQR and nRQ, or pQR and qRQ; this signifies convertibility from one to the other.

We can now, with reference to the rules implied by the above table, easily read the grand matrix for three items[5] in the usual manner, to obtain the alternative moduses of all the premises of concern to us here. For now, we need only deal with positive premises - first, the generics, based on the above summary moduses; then, the joint determinations, by recourse to intersections between the generics. The results are as follows:

Table 14.3.  Enumeration of three-item alternative moduses for strong or absolute weak positive premises, generic or joint, for any figure of syllogism.

 Determ. Major QR Major RQ Minor PQ Minor QP m 10, 12, 25-28, 42, 44, 57-60, 130, 132, 138, 140, 145-148, 153-156, 162, 164, 170, 172, 177-180, 185-188 10, 14, 25-26, 29-30, 74, 78, 89-90, 93-94, 130, 134, 138, 142, 145-146, 149-150, 153-154, 157-158, 194, 198, 202, 206, 209-210, 213-214, 217-218, 221-222 66-68, 70-72, 74-76, 78-80, 130-132, 134-136, 138-140, 142-144, 194-196, 198-200, 202-204, 206-208 66-68, 82-84, 98-100, 114-116, 130-132, 146-148, 162-164, 178-180, 194-196, 210-212, 226-228, 242-244 n 10, 14, 25-26, 29-30, 74, 78, 89-90, 93-94, 130, 134, 138, 142, 145-146, 149-150, 153-154, 157-158, 194, 198, 202, 206, 209-210, 213-214, 217-218, 221-222 10, 12, 25-28, 42, 44, 57-60, 130, 132, 138, 140, 145-148, 153-156, 162, 164, 170, 172, 177-180, 185-188 66-68, 82-84, 98-100, 114-116, 130-132, 146-148, 162-164, 178-180, 194-196, 210-212, 226-228, 242-244 66-68, 70-72, 74-76, 78-80, 130-132, 134-136, 138-140, 142-144, 194-196, 198-200, 202-204, 206-208
 pabs 14, 16, 29-32, 46, 48, 61-64, 74, 76, 78, 80, 89-96, 106, 108, 110, 112, 121-128, 134, 136, 142, 144, 149-152, 157-160, 166, 168, 174, 176, 181-184, 189-192, 194, 196, 198, 200, 202, 204, 206, 208-224, 226, 228, 230, 232, 234, 236, 238, 240-256 12, 16, 27-28, 31-32, 42, 44, 46, 48, 57-64, 76, 80, 91-92, 95-96, 106, 108, 110, 112, 121-128, 132, 136, 140, 144, 147-148, 151-152, 155-156, 159-160, 162, 164, 166, 168, 170, 172, 174, 176-192, 196, 200, 204, 208, 211-212, 215-216, 219-220, 223-224, 226, 228, 230, 232, 234, 236, 238, 240-256 82-84, 86-88, 90-92, 94-96, 98-100, 102-104, 106-108, 110-112, 114-116, 118-120, 122-124, 126-128, 146-148, 150-152, 154-156, 158-160, 162-164, 166-168, 170-172, 174-176, 178-180, 182-184, 186-188, 190-192, 210-212, 214-216, 218-220, 222-224, 226-228, 230-232, 234-236, 238-240, 242-244, 246-248, 250-252, 254-256 70-72, 74-76, 78-80, 86-88, 90-92, 94-96, 102-104, 106-108, 110-112, 118-120, 122-124, 126-128, 134-136, 138-140, 142-144, 150-152, 154-156, 158-160, 166-168, 170-172, 174-176, 182-184, 186-188, 190-192, 198-200, 202-204, 206-208, 214-216, 218-220, 222-224, 230-232, 234-236, 238-240, 246-248, 250-252, 254-256 qabs 12, 16, 27-28, 31-32, 42, 44, 46, 48, 57-64, 76, 80, 91-92, 95-96, 106, 108, 110, 112, 121-128, 132, 136, 140, 144, 147-148, 151-152, 155-156, 159-160, 162, 164, 166, 168, 170, 172, 174, 176-192, 196, 200, 204, 208, 211-212, 215-216, 219-220, 223-224, 226, 228, 230, 232, 234, 236, 238, 240-256 14, 16, 29-32, 46, 48, 61-64, 74, 76, 78, 80, 89-96, 106, 108, 110, 112, 121-128, 134, 136, 142, 144, 149-152, 157-160, 166, 168, 174, 176, 181-184, 189-192, 194, 196, 198, 200, 202, 204, 206, 208, 209-224, 226, 228, 230, 232, 234, 236, 238, 240, 241-256 70-72, 74-76, 78-80, 86-88, 90-92, 94-96, 102-104, 106-108, 110-112, 118-120, 122-124, 126-128, 134-136, 138-140, 142-144, 150-152, 154-156, 158-160, 166-168, 170-172, 174-176, 182-184, 186-188, 190-192, 198-200, 202-204, 206-208, 214-216, 218-220, 222-224, 230-232, 234-236, 238-240, 246-248, 250-252, 254-256 82-84, 86-88, 90-92, 94-96, 98-100, 102-104, 106-108, 110-112, 114-116, 118-120, 122-124, 126-128, 146-148, 150-152, 154-156, 158-160, 162-164, 166-168, 170-172, 174-176, 178-180, 182-184, 186-188, 190-192, 210-212, 214-216, 218-220, 222-224, 226-228, 230-232, 234-236, 238-240, 242-244, 246-248, 250-252, 254-256 mn 10, 25-26, 130, 138, 145-146, 153-154 10, 25-26, 130, 138, 145-146, 153-154 66-68, 130-132, 194-196 66-68, 130-132, 194-196 mqabs 12, 27-28, 42, 44, 57-60, 132, 140, 147-148, 155-156, 162, 164, 170, 172, 177-180, 185-188 14, 29-30, 74, 78, 89-90, 93-94, 134, 142, 149-150, 157-158, 194, 198, 202, 206, 209-210, 213-214, 217-218, 221-222 70-72, 74-76, 78-80, 134-136, 138-140, 142-144, 198-200, 202-204, 206-208 82-84, 98-100, 114-116, 146-148, 162-164, 178-180, 210-212, 226-228, 242-244 npabs 14, 29-30, 74, 78, 89-90, 93-94, 134, 142, 149-150, 157-158, 194, 198, 202, 206, 209-210, 213-214, 217-218, 221-222 12, 27-28, 42, 44, 57-60, 132, 140, 147-148, 155-156, 162, 164, 170, 172, 177-180, 185-188 82-84, 98-100, 114-116, 146-148, 162-164, 178-180, 210-212, 226-228, 242-244 70-72, 74-76, 78-80, 134-136, 138-140, 142-144, 198-200, 202-204, 206-208 pabsqabs 16, 31-32, 46, 48, 61-64, 76, 80, 91-92, 95-96, 106, 108, 110, 112, 121-128, 136, 144, 151-152, 159-160, 166, 168, 174, 176, 181-184, 189-192, 196, 200, 204, 208, 211-212, 215-216, 219-220, 223-224, 226, 228, 230, 232, 234, 236, 238, 240-256 16, 31-32, 46, 48, 61-64, 76, 80, 91-92, 95-96, 106, 108, 110, 112, 121-128, 136, 144, 151-152, 159-160, 166, 168, 174, 176, 181-184, 189-192, 196, 200, 204, 208, 211-212, 215-216, 219-220, 223-224, 226, 228, 230, 232, 234, 236, 238, 240-256 86-88, 90-92, 94-96, 102-104, 106-108, 110-112, 118-120, 122-124, 126-128, 150-152, 154-156, 158-160, 166-168, 170-172, 174-176, 182-184, 186-188, 190-192, 214-216, 218-220, 222-224, 230-232, 234-236, 238-240, 246-248, 250-252, 254-256 86-88, 90-92, 94-96, 102-104, 106-108, 110-112, 118-120, 122-124, 126-128, 150-152, 154-156, 158-160, 166-168, 170-172, 174-176, 182-184, 186-188, 190-192, 214-216, 218-220, 222-224, 230-232, 234-236, 238-240, 246-248, 250-252, 254-256

To repeat, the same information for causative propositions of form PR has already been tabulated, in a previous chapter; and those results are applicable to reading conclusions. The above table concerns positive premises which may arise in three-item syllogism. Though the same propositional form has always the same number of moduses, the modus numbers differ according to the items involved and their positions. Note this well, and compare and contrast.

Next Section (continuation of same chapter)

[1]               i.e. Table 12.4 of the chapter on systematic microanalysis.

[2]               Suppose, for instance, the premises have common moduses 147 and 148. Modus 147 implies not-cabs, whereas modus 148 implies cabs; therefore, neither of these, and indeed no other, causative propositions can be concluded. It is always best to first test the common moduses with reference to causation or non-causation; if they all fit into the one or the other, we can then check out whether they also fit some more precise (positive or negative) form(s).

[3]               For that, at a rate of 64 moduses to a page, we would need to construct a table of over 1,000 pages.

[4]               As in Table 11.7 of the chapter on piecemeal microanalysis.

[5]               See Table 12.3 of the chapter on systematic microanalysis.

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