CHAPTER 15. MAIN MODAL SYLLOGISMS.

1. Valid Modes.

2. Valid Moods.

3. Validations.

 

1. Valid Modes.

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.

 

2. Valid Moods.

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: M-P, S-M, S-P.

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 not-be 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 not-be 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 not-be P



b. Second Figure. Form: P-M, S-M, S-P.

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 not-be M

All S can be M

All S can not-be P

All S can not-be P



AnOpOp

EnIpOp

All P must be M

No P can be M

Some S can not-be M

Some S can be M

Some S can not-be P

Some S can not-be P



AnGpGp

EnRpGp

All P must be M

No P can be M

This S can not-be M

This S can be M

This S can not-be P

This S can not-be P



c. Third Figure. Form: M-P, M-S, S-P.

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 not-be 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 not-be 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 not-be P



ApInIp

EpInOp

All M can be P

All M can not-be P

Some M must be S

Some M must be S

Some S can be P

Some S can not-be P



IpAnIp

OpAnOp

Some M can be P

Some M can not-be P

All M must be S

All M must be S

Some S can be P

Some S can not-be P



RpRnIp

GpRnOp

This M can be P

This M can not-be P

This M must be S

This M must be S

Some S can be P

Some S can not-be P



d. Fourth Figure. Form: P-M, M-S, S-P.

EnIpOp

No P can be M

Some M can be S

Some S can not-be 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.

 

3. Validations.

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, long-term, 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 sous-entendu 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 ‘this-ness’, not this one ‘this-ness’. 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 non-sequitur, 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.