CHAPTER 16. OTHER MODAL SYLLOGISMS.
Concerning subaltern valid syllogism. Any combination of premises not included in the above list of primary valid moods, but implying one which is included, can obviously be listed as a derivatively valid mood. Likewise, propositions implied by one of the conclusions to the valid moods form subaltern moods with the same premises. There are also fourth figure moods to take into account, which though valid are insignificant.
We have two stages to consider. To begin with, applying the primary valid modal modes to the secondary valid actual (or momentary) moods. And then, listing the secondary valid modal modes, which can be applied to both primary and secondary valid actual (or momentary) moods. These two lists together make up the full list of secondary valid modal moods. We shall for a start deal with the first three figures, before turning our attention to the less regular fourth figure.
Table 16.1 Secondary Modes of the Regular Figures.
Figure  First  Second  Third 
Quantity  uup  uup  uup 
 usp  usp  usp 


 sup 
Natural Modality  nna  nna  nna 
 naa  naa  naa 
 ana  ana  ana 
 nnp  nnp  nnp 
 nap  nap  nap 
 anp  anp  anp 
 aap  aap  aap 
Similarly for Temporal Modality.
The subaltern quantity modes were implicit in the list of secondary moods established previously, when considering Aristotelean syllogism. Note that the subaltern modality modes are the same in the three figures. They could be mostly predicted by analogy; but let us derive them quickly from the primary modes, at least in the case of natural modality. In these three figures, nna, naa or ana are derived from aaa, whose premises theirs imply; and nnp, nap, anp, and aap follow from these by virtue of their subaltern conclusions. Temporal modality modes can similarly be dealt with.
Now let us count the number of subaltern moods which we can expect to encounter in these three figures. The full list will not be drawn up, being too large and relatively unimportant; the numbers are interesting, however, as will be seen. Each of these figures has 2 valid polarity modes. These are combinable with 3 valid primary quantity modes in each figure; plus 2 valid subaltern quantity modes each, in the first and second figure, and 3 of them, in the third. These are in turn combinable, in any of these figures, with 3 valid primary natural modality modes, plus 7 valid subaltern natural modality modes.
We thus obtain, in the first figure, a total of 2X(3+2)X(3+7) = 100 valid moods. In the second figure, we have the same results. In the third figure, our total is 2X(3+3)X(3+7) = 120. In each of the three figures, there are 2X3X3 = 18 primary moods, and the rest are secondary. Identical results are of course obtainable for temporal modality.
Now, whereas in the first three figures any valid polarity mode can be correctly combined with any valid quantity or modality modes, in the fourth figure only specific combinations are permissible. The fourth figure, as earlier indicated, lacks uniformity, and seems to in effect contain three different sets of valid moods, which we may call 4a, 4b, 4c, for the sake of convenience.
Figure 4a, whose polarity mode is +, is the significant one; and we saw that it contained two valid primary moods, EIO and EnIpOp. Figures 4b and 4c are insignificant, being mere derivatives of the first figure by transposition of premises and conversion of conclusion. The prototype of 4b is AEE, with polarity +–; and that of 4c is IAI with polarity +++. The corresponding modal forms are AnEnEn and IpAnIp. We shall now list the implicit quantity and modality modes of these forms and their subalterns.
Table 16.2 Secondary modes of the Fourth Figure.
Subfigure  4a.  4b.  4c. 
Polarity  (+)  +–  +++ 
Quantity  (upp)  uuu  pup 
 uup  uup  uup 
 usp 
 sup 
Natural Modality  (npp)  nnn  pnp 
 (aaa)  aaa  aaa 
 nna  nna  nna 
 naa  naa  naa 
 ana  ana  ana 
 nnp  nnp  nnp 
 nap  nap  nap 
 anp  anp  anp 
 aap  aap  aap 
Similarly for Temporal Modality.
The primary valid modes of the fourth figure are included in the above table in brackets to facilitate reading of the sources of their subaltern modes; these are in 4a. With regard to 4b and 4c, the first quantity and the first two modalities listed for each are the sources of the others, but all are viewed as secondary, as well as the corresponding polarities.
Let us now count the number of moods implied valid. For 4a, 1X(1+2)X(2+7) = 27, of which only two are primary. For 4b, 1X2X9 = 18, and for 4c, 1X3X9 = 27; all these being secondary in one way or another. Thus, figure four consists of two primary valid moods, and another 70 secondary valid moods. This is said for natural modality, and can be repeated for temporal modality, as usual.
Thus, to conclude this section, there are a total of 100+100+120+72 = 392 valid moods in each type of modality, of which 56 are primary, and 336 are secondary. The full list of secondary moods is easily developed given the above lists of modes.
To systematically cover all possibilities of combination, we now need to investigate mixed syllogism, that is, syllogism involving a mixture of natural modalities (n, p, a) and temporal modalities (c, t, m), in their premises and/or conclusions. It will be seen that valid such combinations are entirely derivable from syllogisms previously encountered under this or that type of modality separately, so that we can say that mixed modes are in fact all secondary.
Analysis shows that the mixed modes we seek are all derivable from the primary modes of temporal modality of each figure. We have seen that these are: in figures 1 and 2, mmm, ccc, ctt; in figure 3, mmm, ctt, tct; in figure 4a, mmm, ctt; in figure 4b, mmm, ccc; in figure 4c, mmm, tct. If we analyze the subalternations (through premises and/or conclusion) possible for each of these four modes, we get the following results. From mmm: nnt, ncm, nct, ncp, cnm, cnt, cnp, nmt, mnt, ccp, cmp, mcp (12 modes, in common to all figures). From ccc: nnc, ncc, cnc (3 modes, applicable to figures 1, 2, and 4b). From ctt: ntt, ntp, ctp (3 modes, applicable to figures 1, 2, 3, and 4a). From tct: tnt, tnp, tcp (3 modes, applicable to figures 3 and 4c).
We thus obtain, for the first three figures, 18 valid mixed modes each; for each subset of figure four, 15 valid modes. Other mixed modes are found not to follow from any valid nonmixed modes, or to be only apparently mixed because of the different symbolization of actual and momentary propositions.
The valid mixed modes may each be combined with the polarities and quantities existing in their respective figures. It follows that the number of valid mixed moods are as follows: 2X5X18 = 180, for each of figures 1 and 2; 2X6X18 = 216 for figure 3; and 3X15 = 45 for figure 4a, 2X15 = 30 for figure 4b, 3X15 = 45 for figure 4c. The total number of valid mixed moods is therefore 696. These valid moods, to repeat, count as secondary. Most may be practically useless, but they had indicated for completeness.
The above listed mixed modes, you will observe, do not include combinations involving the aform and certain combinations involving the mform. The reason for this is simply that the actual and momentary forms are essentially identical, although they appear different. Their distinction is one of perspective, and a verbal one sometimes, but their logical value is the same. It follows, not only that aaa and mmm are equivalent, but also that other combinations of a and m, namely amm, ama, mam, maa, are all identical. Furthermore, combinations involving modal propositions together with one or both these, are also redundant; this includes groups such as nnm, cca, ncm, nca, nmm, nma, nam, and so on.
We must of course avoid the duplicate listing as mixed modes, of syllogism which have already been presented as nonmixed modes, merely because they superficially appear different through the use of different notation, so combinations such as those just mentioned must be left out of our accounts. On the other hand, some combinations involving nonmodal proposition(s) mixed with modal(s), are noteworthy, even though subaltern, because they provide additional logical information. We need only to select either of the symbols a or m, to represent nonmodal forms, and work with that exclusively. I selected m as more appropriate, after finding that all valid mixed modes could be derived from solely temporal syllogism. But this is strictlyspeaking mere convention; modes such as ncm or nmt could equally have been written nca or nat. The underlying meaning is the same.
If we examine the principal 37 new syllogism introduced in this paper, it is clear that they are not at first sight obviously valid. An effort of thought is needed to see their truth. This shows that our enterprise, the development of a modal logic, was a worthwhile endeavor, a valuable addition to human knowledge. The justification is still greater, if we analyze our work in this chapter statistically, and sumup the number of new syllogistic forms introduced.
We saw earlier that there are 2X3X5 = 30 possible categorical forms of the kind under study, a proposition may have one of two polarities (+ or –), one of three quantities (s, u, or p), and one of five modalities (a or m, or n, c, t, or p). A syllogism contains three propositions, in any of four possible figures; therefore the total number of imaginable combinations is (30 cubed)X4 = 108,000 moods, whether valid or invalid. Of these, ((2X3)cubed)X4 = 864 would be wholly nonmodal moods; (((2X3X3)cubed)X4)864 = 22,464 would be natural modal moods; and again 22,464 would be temporal modal moods; the remainder 62,208 moods would be of mixed modal type.
Now, let us calculate how many out of this theoretical total of possibilities, are in fact valid. We saw that in nonmodal logic, there are 44 valid moods, of which 19 are primary and 25 are secondary. Next, in modal logic, we established 37 primary moods for natural modality and 37 for temporal modality; and we found these to have 336 and 336 secondary moods, respectively. Lastly, we identified 696 mixed moods as valid, and pronounced them all secondary. Thus, the total number of valid moods obtained is 1486, of which only 93 are primary, and the remaining 1393 are secondary.
Thus, only 1486 out of 108,000 = 1.4% of possible combinations are logically valid; versus 98.6% chances of erroneous reasoning. This shows the importance of our thesis, that modality needed to be considered and systematically analyzed by logical science. The number 1486 is of course quite large in itself; this shows the value of the notation system I invented, which made it possible for me to analyze so many combinations with a certainty of exhaustiveness and in a minimum of space.
Of the valid moods, only 6.3% are primary and 93.7% are secondary. Primary moods are the most significant and independent forms of reasoning; secondary moods are relatively less significant and more derivative. This does not mean, however, that secondary moods are necessarily less commonly used in practise; although many of them occur rather rarely, many may nonetheless be as important as primary moods. For example, naa moods in figure 1 or 2 are quite valuable, although technically subaltern to aaa.
It must be stressed, also, that the recognition of invalid modes of thought is as important as the knowledge of valid modes. We indicated, for example, how at first sight one might suppose moods such as 1/ApAnAp valid; an analytical effort is required to understand the error involved. Some invalid moods are of course instantly seen to be wrong; but some contain pitfalls for the logically untrained mind.
Further research, which I will not develop in detail in this paper, shows that some invalid moods may be made to yield imperfect conclusions, of the types ‘Some nonS can be nonP’ or ‘Some nonS are sometimes nonP’. Similar cases arose in nonmodal syllogism, with ‘Some nonS are nonP’ conclusions, the reader will recall.
Another kind of atypical conclusion is drawable in some cases; for example, 1/ApAp (All M can be P and All S can be M) does not conclude Ap (All S can be P), but does allow us to infer that ‘All S either can be or can become P’. Indeed, this may be viewed as an explanation why the simple ‘All S can be P’ cannot be inferred.
Syllogism which mix copula in this way might be characterized as mixedform. They involve another kind of copula (becoming, instead of being). The logic of becoming is a large field on its own, which will be touched upon further in the next chapter.
To conclude, general rules of the syllogism, which summarize, or explain, the valid moods, may be presented. They reveal the underlying principles of such reasoning, or the common attributes.
a. Rules of Polarity. If the extreme terms are positively connected to the middle term in the premises, they will be connected in the conclusion, if any. But if either extreme term is unconnected to the middle term in its premise, it will be unconnected to the other extreme term in the conclusion, if any. If neither extreme term is positively connected to the middle term in the premises, no conclusion can be drawn.
b. Rules of Quantity. At least some instance(s) of the middle term must be found in common to both premises, for a conclusion to be conceivable. And no instance of either extreme term may be found referred to in a suggested conclusion which was not covered in the premise. However, instances found in a premise may notreappear in the conclusion.
c. Rules of Modality. There must be found some circumstance(s) and time(s) in common to both premises, for a conclusion to be capable of being drawn from them. And no circumstance or time may make its appearance in the conclusion, which was not already mentioned in a premise. Though, of course, the conclusion may be circumstantially or temporally more narrow than the premises.
These are common sense conditions; we can see at a glance that they are reasonable. They tell us that the information in our proposed conclusion must have been implicit in our premises, if we are to claim that it was drawn from them. These rules can be expressed graphically or in the language of ‘distribution’.
a. The middle term must occur in both premises, and be distributive once at least, extensionally, naturally and temporally, for any conclusion to be possible. Breach of this rule is labeled fallacy of the undistributive middle term.
b. One of the terms of each premise must be found in the conclusion, and if such an extreme term was undistributive in its premise, extensionally, naturally, or temporally, it must remain likewise undistributive in the conclusion, if any. Breach of this rule is labeled fallacy of illicit process of the major or minor term, as the case may be.
Such breaches of logic are essentially commissions of the fallacy of four terms. A syllogism has a valid structure only if these rules are obeyed; otherwise it is a paralogism. If the middle term is really different somehow in the two premises, it is as if there were no middle term, and therefore no basis for deduction. Likewise, if either term in the conclusion has a meaning or scope different from that given in the premises, it is as if a new term has been introduced in the equation, so that it cannot be called deduction.
Lastly, note the possibility of sorites with modal syllogisms, as with actual ones. A regular sorites, consisting entirely of first figure arguments, would look as follows:
All (or Some) A must (or can) be B,
All B must be C, All C must be D, All D must be E,
All E must be F (or No E can be F)
therefore, All (or Some) A must (or can) be F (or the negative equivalent).
The rules of quantity or polarity are the same as before: only the first premise (the most minor) may be particular, only the last one (the most major) may be negative, and the conclusion follows accordingly. Here, we may add the rule of modality, that only the first premise may be potential, and the conclusion follows accordingly.