CHAPTER 17. TRANSITIVE CATEGORICALS.
Consider the wide form of actual proposition ‘This thing, which was/is/will be S at some time, was/is/will be P at some (same or other) time’.
Note that the primary subject is ‘this thing’, an existent which is being mentally or physically being pointed to; however, that designation is further specified by the thing’s characterization as S at some (explicitly specified or tacitly intended) time, say t1. Next, note that the time at which P applies may be the same or a different time, call it t2. This form is uncommitted as to whether, at time t1, this thing is P or nonP, as well as S; nor is it specific as to whether, at time t2, this thing is S or nonS, as well as P.
a. We normally understand the form ‘this S is P’ to mean ‘this thing, which is now S, is now P’, implying S and P to be simultaneous. It is a static event, not implying process. The past and future tenses of this likewise only give a still snapshot of the situation at the given time. Such propositions may be called attributive, since their original function is to record the attributes of things.
The copula of ‘being’ is the most fundamental copula, but it is not the only kind of relation Logic needs to consider. The way categorical propositions of this kind are structured limits their applicability to certain phenomena (the majority, no doubt); certain other phenomena seem best dealt with using categorical propositions with other relational features, defined using attributives.
One such alternative copula is, let us say, ‘S changes to P’ (intended in the most generic sense of ‘something, which initially is S, later is P’). It is close to fundamental, though derived from ‘is’ (and less used). Such propositions might be called transitive, since they suggest a process across time. We have to be careful to distinguish between the subject-definition time, and the predicate-activation time.
b. However, this verb ‘to change to’ is colloquially used in many senses, and we should first isolate the ones which concern us most here. When we say ‘S has or will, change to, or is changing to, P’, what do we mean? A convention is needed, so we can develop two somewhat more specific forms describing change. Let us agree that:
‘This S will get to be P’ signifies a process starting at S (with or without P) at some earlier time, and ending at P with S at some later time.
‘This S will become P’ signifies a process starting at S (with or without P) at some earlier time, and ending at P without S at some later time.
We have dichotomized change into two kinds, each covering a segment of the phenomena we come across. These forms represent two species of transitive propositions. Sub-classes of these, or also classifications otherwise defined, can be developed as needed, of course. For example, ‘S ceases to be P’, for change from SP to nonP.
Note that in both cases, we start as S, without specifying whether P or nonP applies at the same initial time; and we end at P. The copula ‘will get to be’ is reserved for ‘S and P’ finales, and the copula ‘will become’ is reserved for ‘nonS and P’ finales. If we are not sure whether the finale contains S or nonS, we simply say ‘S will get to be or become P’, to specify our knowledge of P at least; this is equivalent to the generic ‘change to’ copula.
These two copula are enough to cover all processes. With nonP as predicate, we say ‘will get to be nonP’ or ‘will become nonP’; with nonS as subject, we say ‘this nonS will get to be’ or ‘this nonS will become’. To deny processes, we may negate the copula, using ‘won’t’ (or will-not) get to be or become. If we wish to specify whether the initial state includes or excludes P, we may do so separately, by saying ‘this S is at first P (or not P)’.
This distinction allows us to express whether the change in question is: an alteration, with the starting point S remaining or reappearing (= ‘eventual being’) at the end, when P arises; or a mutation, with the original subject S disappearing (= ‘becoming’) at the end, when P arises. Thus, we may speak of alterative and mutative propositions.
c. Note in passing that there are other ‘types of change’. The above types are based on natural or temporal modality; they concern real changes in objects, transitions through which individual things go, over time. However, there is also ‘extensional change’, as when we say that a genus ‘changes’ differentiae from one species to another; this refers to relatively static differences, and treats a universal as if it was an individual going through changes (because the mind works serially). More broadly, we have ‘logical change’, which traces the mental realization of a previously unknown yet existing truth, as if its to us novel appearance was equivalent to its coming into being.
a. In the interim, between the beginning at S and the result at P, any state or combination of states may take place (S and P, S and nonP, nonS and P, nonS and nonP), or the change may be abrupt and without intervening state. The forms adopted are left open in that respect. If indeed complications arise in the interim, they can be specified separately, in additional statements.
The transition itself may be gradual or sudden; also, there may be a slow build up of surrounding forces before the process in question is put in motion, or it may take off immediately.
b. In the past tense, ‘got to be’ and ‘became’ may be used. They report the culmination of processes. In the present continuous tense, ‘is getting to be’ (implying ‘will get to be’) and ‘is becoming’ (implying ‘will become’) may be used. Note that these ‘presents’ do not in fact describe the present stage (i.e. whether S or nonS, P or nonP, are at this time active), but merely define the beginning and end of a process and tell us that it is in progress.
In the cases of ‘will change’ or ‘is changing’, if ‘must change’ is not meant, there is not a real actuality, like for ‘has changed’, but a certain probability of success, assuming no extraordinary interference, without totally excluding failure. A specific example of such future is of course the expression of intention, of resolve of will-power, including free-will.
The negatives of those actual forms are interpreted accordingly. ‘Hasn’t changed’ denies completion of process so far; ‘won’t change’ denies it for the future, or some part thereof; and ‘is not changing’ denies a process is taking place.
However, remember that ‘will’ and ‘won’t’ may strictly-speaking be both wrong as predictions, even though it is, by the law of contradiction, ex-post-facto true in general that the event in question must ultimately either happen or not-happen. Modalities are also useable here, to specify more exact statistics of past culminations of process, giving the degree of ‘will’ or ‘won’t’.
c. These forms may, of course, all be quantified and naturally or temporally modalized. Thus, quantity and modality, like polarity, are characteristics of relations which transcend the specific copula involved. The plurals ‘all’ and ‘some’ used for these purposes are, as usual, intended subsumptively, addressing each unit singly; the meaning is more rarely collectional or collective.
Note that in plural forms, the times at which the implied individual events happen, and the relative times of start and finish of each event, fade in importance, in comparison to singular propositions; the various events span across time, indefinitely, ‘whenever they happen’.
Potentiality, as in ‘S can get to be or become P’, means that in some circumstances, S does get to be or become P’. Natural necessity here signifies inevitability of the alteration or mutation, in contrast to the suggestion of invariability in the static ‘S must be P’. If ‘can’ is combined with ‘can not-‘ (get to be or become), we have contingency. ‘Cannot’ of course means ‘in no circumstances’ does the event occur.
Note that potential alteration and potential mutation are compatible, although they cannot actually happen together, since the one finally keeps its subject while the other loses it. It is also conceivable that ‘S cannot get to be P’ and ‘S cannot become P’ are both true: together they mean that P is entirely closed to S, by whatever process.
Likewise, with temporal modalities.
d. The oppositions between alterative or mutative propositions, and also those of these two groups with each other, and with attributives, need to all be worked out in detail, of course. But this job will not be carried out here, to avoid expanding this paper more than necessary.
Simply, beginning at the singular level, examine each form’s definition for implications, and compare pairs of forms for points of agreement or disagreement; once singular oppositions are established, they can be quantified following the rules developed for attributives.
Similarly for eductive arguments.
a. Consider the following examples:
· Attributive: This egg is soft (or hard).
· Alterative: This egg has hardened (gotten to be hard).
· Mutative: This soft egg has become hard (or a hard egg).
A soft egg can’t be or get to be a hard egg, but only become one; however, soft and hard eggs are both still eggs. The transition from soft to hard is gradual, depending on heat supply; and the terms soft and hard may be viewed as extremes separated by unnamed in-between states, or we could say that all states before hard are degrees of soft, depending on whether we define soft as raw or uncooked.
In terms of class-thinking, each of these copulae plays a distinct role. The first merely classifies. The second indicates a process leading to a classification. The third declassifies on one side and reclassifies on the other. The logic of classification would be incomplete if we did not consider the more dynamic relations of change.
Note that, whereas ‘This S is or gets to be nonS’ are self-contradictory, ‘This S becomes nonS’ is not so, but rather implied whenever an S becomes anything (P or nonP); almost every S eventually becomes a nonS in this world. On the other hand, whereas ‘This S is or gets to be S’ are formally acceptable, ‘This S becomes S’ is indeed self-contradictory, unless we add the word ‘again’, to mean: ‘This S becomes nonS, which in turn becomes S’.
b. The similarities and differences between attributives and alteratives should be noticed. ‘This S will be P’ and ‘This S will get to be P’ are obviously very close in meaning; in fact, if you look at the definitions, the former is a special case of the latter, and is implied by it. The difference is merely in the time-definition of the subject; when quantity or modality are introduced, the difference blurs, because that time is less definite.
Still, the one is static and the other dynamic. This is more noticeable in the past tense; compare ‘This S was P’ to ‘This S got to be P’. Similarly, in the present continuous sense, ‘is’ and ‘is getting to be’ are obviously different. For this reason, two distinct copulae are needed.
c. We should henceforth not confuse ‘getting to be’ and ‘becoming’, with each other or with the more general ‘changing to’. We make this convention, although these expressions, and others like them (end up as, turn out to be), are colloquially interchangeable, because logic needs fixed terminologies to proceed.
In alteration, the initial term, defining the subject, effectively remains in force at the end, underlying the predication.
Note that, normally, ‘S gets to be P’ implies ‘S is not initially P’, suggesting a switch of predicates. But this issue is best left open, since the form is then more widely useable for cases starting and ending at SP, yet having intermediate stages, of nonS and/or nonP, or even for cases involving no change. Often, the terms S and P are two extremes in a range, and the motion from the one to the other is a gradual transition whose intermediate stages are irrelevant, and often unnamed. If the process starts and ends at SP, yet passes through other states, we would say ‘S gets to be again P ‘.
In mutation, there is a radical exchange; both terms are effectively subjects, and we swap one for the other; the initial subject is gone at the end, replaced by another. There may indeed be an implied substratum to the mutation, an underlying constancy; the changing thing remains always at least a ‘thing’, but often some narrower genus is understood. The paradigm of mutation is of course biological metamorphosis; but such change is found also in physics or other areas.
Note that, ‘This S can or must become P’ do not imply that ‘it cannot become nonP’, because the subject is not defined by P or nonP. There is no obversion here; the polarity of the copula (become) and of the predicate (P) are not interchangeable.
We will only concern ourselves here with the some of the more significant moods of syllogism. Our purpose is only to show how introducing transitive categoricals allows us to draw conclusions from arguments which would otherwise be sterile.
The various copula discussed above are in practise often used in tandem, to specify a situation with more precision. We might for instance say the conjunction ‘This S cannot be P, but can become it’. In syllogism, a conclusion in the form of a logical disjunction like ‘This S can (get to) be or become P’ is often possible where otherwise no other conclusion can be drawn.
Systematic treatment of syllogism involving transitives would proceed as follows. We are dealing with four families of proposition: attributives, transitives, alteratives and mutatives; the second of these being a genus for the last two. Each of these classes covers a number of forms varying in polarity, quantity, modality. We must consider, for each figure of the syllogism, every combination of premise, however mixed, and find out what conclusions, of whatever family, can be drawn from them.
However, for now, let us concentrate on a more limited task. We will look at the first figure only, and review only moods whose premises are attributive and/or mutative, and modal. The conclusions obtained have the generic transitive form ‘S can get to be or become P (or nonP)’.
a. When both premises are attributive, an attributive conclusion can only be drawn (if at all) from a necessary major premise; if the major is potential, we cannot draw an attributive conclusion. However, a transitive conclusion can be drawn, showing that the modality may have an impact on the copula. The following moods are valid:
All M can be P (or nonP),
All/This/Some S can (or must) be M,
ergo, All/This/Some S can get to be or become P (or nonP).
We know from the premises that what started as S, will in some circumstances be S and M; and that whatever is M, will in some further circumstances be M and P (or nonP); but we cannot predict whether the end result of this process includes or excludes S. It is conceivable that S stays on with P (or nonP), but it is also conceivable that S disappears prior to the arrival at P (or nonP). For this reason, our conclusion cannot be merely ‘S can be P (or nonP), but must be open to the ‘S can become P (or nonP)’ outcome.
b. The same can be argued with a mutative major premise, whatever the modalities involved. Thus, the following are valid:
All M can (or must) become P (or nonP),
All/This/Some S can (or must) be M,
ergo, All/This/Some S can get to be or become P (or nonP).
If one or both premises are potential, so is the conclusion, as above; but if both premises are necessary, a necessary conclusion can be drawn, as below:
All M must become P (or nonP),
All/This/Some S must be M,
ergo, All/This/Some S must get to be or become P (or nonP).
c. In cases where the minor premise is mutative, whether the major is attributive or mutative, similarly disjunctive conclusions may be drawn. We know from the minor premise that S will disappear to become M, but we cannot be sure whether, in the circumstances when M is or becomes P, S reappears or stays away.
Note that in all the cases so far considered, we could view the conclusion as a logical disjunction as we did, or we could say that a more specific conclusion can be drawn if we know one or the other alternative to be excluded. This would be equivalent to having a third premise, viz. ‘S cannot get to be P’ or ‘S cannot become P’. But formally speaking, this constitutes an additional argument (apodosis) after the syllogism as such.
d. All the above only concerns cases with both premises affirmative (whether the predicate be P or nonP). Now, the minor cannot be negative in the first figure, but what if the major premise is negative (in the sense of negating the copula, not merely the predicate)? In such case, we cannot draw a likewise negative conclusion, because we can construct a syllogism with compatible affirmative premises yielding a conflicting affirmative conclusion. Thus, for example, the following mood is invalid:
No M can become P,
This S must be M,
ergo, This S cannot get to be or become P.
This is invalid, because it is conceivable that, though no M can become P, all M can nonetheless be P, in which case the following syllogism could be constructed, as earlier established:
all M can be P,
This S must be M,
ergo, This S can get to be or become P.
We could interpret this to mean that, a (compound) negative conclusion is possible, if the negative major is compound, as in the following mood. Note that major premise and conclusion are conjunctions, not disjunctions, of negatives. The result is due to the attachment of S to M.
No M can be or become P,
This S must be M,
ergo, This S cannot get to be or become P.
If either or both of these premises were potential instead of necessary, a potential conclusion would be drawn.
I shall not, however, develop the matter further, but remain content with having shown some of the impact of transitives on syllogistic reasoning. A full theory should of course list all the valid moods, and do so in all figures. Also, all the above concerns natural modality, but similar arguments can be presented for temporal modality.