CHAPTER 15. MAIN MODAL SYLLOGISMS.
We called a mood of syllogism, a combination of formally fully specified premises and conclusion in a given figure (e.g. 1/AAA). We will call mode, any combination of symbols which does not by itself fully specify a syllogistic form, but which abstracts a specific aspect of such, in a given figure (e.g. 1/uuu). It was shown, in Aristotelean logic, that the primary valid modes of polarity and quantity are as in the following table.
Table 15.1 Valid modes of Polarity and Quantity.
Figure  First  Second  Third  Fourth 
Polarities  +++  +–  +++  + 
 +  +  + 

Quantities  uuu  uuu  upp  upp 
 upp  upp  pup 

 uss  uss  ssp 

We can at the outset, prior to systematic validation, predict that the valid modes for natural and temporal modality will be the following, by analogy to the results obtained for extensional modality.
Table 15.2 Valid Modes of Natural and Temporal Modalities.
Figure  First  Second  Third  Fourth 
Natural Modality  aaa  aaa  aaa  aaa 
 nnn  nnn  npp  npp 
 npp  npp  pnp 

Temporal Modality  mmm  mmm  mmm  mmm 
 ccc  ccc  ctt  ctt 
 ctt  ctt  tct 

Note the slight difference between quantity modes and modality modes. The modes aaa and mmm are valid in all figures, whereas sss is not (3/ssp is exceptional, and anyway does not yield an s conclusion). This is due to modality standing outside the relationship between the terms, whereas quantity concerns the subject more directly.
Natural and temporal modality being essentially analogous, we can concentrate on developing the theory of syllogism for the former, and then generalize the results to the latter. Apart from the above we will need to investigate the valid modes of mixed, natural and temporal syllogism.
In the broadest sense, of course, all syllogism is modal. But for the sake of convenience we will often find it useful to call nonmodal, syllogism both of whose premises are actual or momentary (aaa or mmm); so that syllogism with one or both premises necessary or possible, can be called modal. Aristotelean logic can then be said to have concerned nonmodal syllogism, while this thesis concerns modal syllogism.
If we combine together the valid modes of polarity and quantity for a given valid mode of modality, in each of the figures, we should obtain the valid moods of syllogism. Let us now do so, using the valid natural modality modes, to develop a full list of natural syllogism, including both the nonmodal (Aristotle’s achievement) and the modal (the new contribution). This is the principal goal of our whole formal research. The notation system used for this, consists in applying modality subscripts (n, p, a) to the six standard symbols, A, E, I, O, R, G.
We see in the list below that only 56 primary moods emerge as logically valid, not counting derivative syllogism. There are 18 valid moods in each of the first three figures, and 2 in the fourth. Since 19 of the above moods are actual, only 37 are original forms.
Table 15.3 Primary Valid Moods of Natural Syllogism.
Mode/Figure  First  Second  Third  Fourth 
aaa  AAA  AEE  AII  EIO 
EAE  EAE  EIO  
AII  AOO  IAI  
EIO  EIO  OAO  
ARR  AGG  RRI  
ERG  ERG  GRO  
nnn  AnAnAn  AnEnEn  
EnAnEn  EnAnEn  
AnInIn  AnOnOn  
EnInOn  EnInOn  
AnRnRn  AnGnGn  
EnRnGn  EnRnGn  
npp  AnApAp  AnEpEp  AnIpIp  EnIpOp 
EnApEp  EnApEp  EnIpOp  
AnIpIp  AnOpOp  InApIp  
EnIpOp  EnIpOp  OnApOp  
AnRpRp  AnGpGp  RnRpIp  
EnRpGp  EnRpGp  GnRpOp  
pnp  ApInIp  
EpInOp  
IpAnIp  
OpAnOp  
RpRnIp  
GpRnOp 
We will now present these 37 valuable new forms in full, for the record.
a. First Figure. Form: MP, SM, SP.
AnAnAn  EnAnEn 
All M must be P  No M can be P 
All S must be M  All S must be M 
All S must be P  No S can be P 
AnInIn  EnInOn 
All M must be P  No M can be P 
Some S must be M  Some S must be M 
Some S must be P  Some S cannot be P 
AnRnRn  EnRnGn 
All M must be P  No M can be P 
This S must be M  This S must be M 
This S must be P  This S cannot be P 
AnApAp  EnApEp 
All M must be P  No M can be P 
All S can be M  All S can be M 
All S can be P  All S can notbe P 
AnIpIp  EnIpOp 
All M must be P  No M can be P 
Some S can be M  Some S can be M 
Some S can be P  Some S can notbe P 
AnRpRp  EnRpGp 
All M must be P  No M can be P 
This S can be M  This S can be M 
This S can be P  This S can notbe P 
b. Second Figure. Form: PM, SM, SP.
AnEnEn  EnAnEn 
All P must be M  No P can be M 
No S can be M  All S must be M 
No S can be P  No S can be P 
AnOnOn  EnInOn 
All P must be M  No P can be M 
Some S cannot be M  Some S must be M 
Some S cannot be P  Some S cannot be P 
AnGnGn  EnRnGn 
All P must be M  No P can be M 
This S cannot be M  This S must be M 
This S cannot be P  This S cannot be P 
AnEpEp  EnApEp 
All P must be M  No P can be M 
All S can notbe M  All S can be M 
All S can notbe P  All S can notbe P 
AnOpOp  EnIpOp 
All P must be M  No P can be M 
Some S can notbe M  Some S can be M 
Some S can notbe P  Some S can notbe P 
AnGpGp  EnRpGp 
All P must be M  No P can be M 
This S can notbe M  This S can be M 
This S can notbe P  This S can notbe P 
c. Third Figure. Form: MP, MS, SP.
AnIpIp  EnIpOp 
All M must be P  No M can be P 
Some M can be S  Some M can be S 
Some S can be P  Some S can notbe P 
InApIp  OnApOp 
Some M must be P  Some M cannot be P 
All M can be S  All M can be S 
Some S can be P  Some S can notbe P 
RnRpIp  GnRpOp 
This M must be P  This M cannot be P 
This M can be S  This M can be S 
Some S can be P  Some S can notbe P 
ApInIp  EpInOp 
All M can be P  All M can notbe P 
Some M must be S  Some M must be S 
Some S can be P  Some S can notbe P 
IpAnIp  OpAnOp 
Some M can be P  Some M can notbe P 
All M must be S  All M must be S 
Some S can be P  Some S can notbe P 
RpRnIp  GpRnOp 
This M can be P  This M can notbe P 
This M must be S  This M must be S 
Some S can be P  Some S can notbe P 
d. Fourth Figure. Form: PM, MS, SP.
EnIpOp 

No P can be M 

Some M can be S 

Some S can notbe P 

A similar listing would be obtained for temporal syllogism. Secondary modes, valid derivatively and of lesser significance, will discussed later. Mixed syllogism will also be dealt with separately.
We have seen that each figure has a method of validation most appropriate to it. Aristotelean syllogism being identical with our nonmodal (actual or momentary) forms, the task of validation of modal syllogisms is much facilitated. Similar approaches can be used with regard to modal syllogism; and moreover we can appeal, if we need to, to correct nonmodal argument, in the process. The following description of validation and rejection processes for natural modal syllogism, can all be repeated for temporal modes.
a. First figure.
We previously defended Aristotle’s valid moods in the first figure, on the basis of the principle that one must mean what one says. Some phenomena have been observed, perceptually and/or conceptually; within a complex of appearances, certain aspects have been distinguished; names have been assigned to their various components; thereby, a framework is established which we are logically required to adhere to; such recognition guarantees the accord between thought and reality (that is, longterm, overall, appearance.)
Now, granting the six valid actual moods of this figure, the corresponding moods in the modes nnn and npp, are to be demonstrated valid.
A necessary proposition ‘X must be Y’ may be viewed as merely a collection of actual propositions ‘In circumstance 1, X is Y’, ‘in circumstance 2, X is Y’, ‘in circumstance 3, X is Y’, and so on; it says ‘Whatever the surrounding circumstances, X is Y’. Likewise, a potential proposition may be viewed as a partial enumeration of circumstances in which the stated relationship of X and Y is actualized. An actual proposition indicates a specific single circumstance in which the event occurs.
Now, let us consider a group of three propositions which, in the actual mode aaa, constitute a valid syllogism, e.g. AAA. In the case of AnAnAn, the nnn equivalent, we can predict that in each and any circumstance we may select, we will find the two premises AA actual, and yielding the conclusion A. It follows that, given the premises AnAn, we can say, ‘Whatever the circumstances, the conclusion A occurs’; which means that the An conclusion is valid. Thus, any mood valid in aaa mode is equally valid in nnn mode. With similar reasoning, we can demonstrate the validity of npp, since the ‘all circumstances’ in the major premise includes the ‘some circumstances’ in the minor premise, which are in turn posited as framing the conclusion, too.
With regard to invalidation of invalid modal modes. Although the onus of proof is on anyone who wants to defend them, as it were, it is important to give special attention to the mode pnp, which might at first sight seem reasonable. We might think that the ‘in all circumstances’ of the minor premise, includes the ‘some circumstances’ of the major premise, so that a potential conclusion can be drawn. However, in any modal proposition, the circumstances under consideration apply primarily to the subject of the proposition. When we refer to all the circumstances surrounding the subject’s existence, we do not claim these to be the only circumstances which can coincide with the predicate’s existence, or any other subject’s existence.
In the case of npp, the ‘some circumstances’ under consideration, are implied for the middle term since the minor premise is affirmative, and concern the same subject in minor premise and conclusion. But in the case of pnp, the specific conditions under which the middle term is addressed in the major premise do not necessarily coincide with any condition concerning the minor term in the minor premise, and so cannot be transferred to it in the conclusion. The change of subject being qualified makes the process illicit. The invalidity of ppp is all the more obvious, since it has no misleading unconditional premise. Thus the analogy of valid modal modes to valid quantitative modes is complete.
What of aaa, which is posited as valid, although we reject sss? Here too, one could argue that the unitary circumstance referred to by each of the two premises may not coincide, since their subjects differ. In truth, this argument against aaa is justified, and serves to warn us that the aaa mode is valid only on the condition that we know the unitary circumstance involved in the two premises to be one and the same. However, actual propositions by definition concern an ostensible circumstance (which though left tacit is understood). So aaa is a valid mode, when we know the sousentendu circumstance to be common.
Although we might attach a similar proviso for the validity of sss, we in fact cannot, because of a structural difference between actuality and singularity. The ‘this’ in a singular proposition is more firmly attached to the subject; it identifies the subject itself, and not a circumstance surrounding it. Comparing one subject’s ‘this’ to another’s is nonsensical, as is the idea of moving our mental finger from one to the other; because all they have in common is ‘thisness’, not this one ‘thisness’. In actuals, on the other hand, the focus of the ‘this’ is a circumstance standing outside the subject of the proposition, though bounded by its existence; it is not the subject as such which is focused on by that ‘this’. It follows that, here, the two ‘this’ occurrences in the premises may be compared, and the specification transferred to their conclusion.
b. Other figures.
The valid moods of the second figure are established by reduction ad absurdum through the first figure. We attach the denial of the conclusion to the major premise, and see that the result would be denial of our original minor premise. Thus, 2/nnn is reduced through 1/npp, and 2/npp follows from 1/nnn; always of course provided the underlying actual mood has valid polarity and quantity properties. Invalid modes in the second figure are dealt with similarly, by showing that the combination of the major premise with the suggested conclusion results in a contradiction or a nonsequitur, through the first figure.
The third figure modes could be reduced ad absurdum to the first figure for systematic validation; the denial of the conclusion would be combined with the minor premise and result in denial of the original major premise. Rejection of invalid modes could be achieved similarly. However, exposition reflects more accurately the way we deal with this figure in practise. We can reproduce our arguments for the first figure, showing that the circumstances in the premises intersect, and are passed on to the conclusion. This is facilitated by the fact that, in this figure, the two premises have the same subject (the middle term).
For the fourth figure, direct reduction is the appropriate approach. There is, furthermore, only one primary valid mood to consider. The premises EnIp are both converted, allowing us to process them in the first figure, and obtain the desired Op conclusion. Other validations, and invalidations, are likewise easy to deal with.