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© Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.

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1. Basis and Connection.

2. Quantification.

3. Other Features.

4. Natural Disjunction.

1. Basis and Connection.

There are six singular forms of natural conditionals with three terms, as follows. These forms are so structured that we can analyze the behavior of individual subjects, the relationships between their predicates, independently of other individuals. Note the three categories of modality and two polarities they feature.

(We could if need be use the same symbolic conventions as we did for categoricals, only perhaps prefix them with, say, a paragraph (§), to remind us of the differences.)

§Rn: When this S is P, it must be Q

§Gn: When this S is P, it cannot be Q

§R: This S is P and Q

§G: This S is P and not Q

§Rp: When this S is P, it can be Q

§Gp: When this S is P, it can not-be Q

Let us examine the structure of these forms in more detail:

a. The expression ‘when’ used here signifies a conditionality of the type ‘in such circumstances as’; and it is intended to imply that the condition ‘this S is P’ is potential. Note well that the reference here is to natural circumstances; we are dealing with a real, objective type of causality.

‘When’ suggests that the underlying ‘this S can be P’ is an established fact, and not merely something logically conceivable. Thus, it is not equivalent to the ‘if’ of hypothetical propositions, which only signifies that the condition might turn out to be true, not being so far inconsistent with the context of knowledge.

Needless to say, by now, we are not always careful, in everyday discourse, to use ‘when’ (instead of ‘if’ or similar expressions) wherever natural conditioning is intended, or ‘must’ (instead of ‘is’) wherever necessity is intended. There is no harm in confusing words in practise, provided we know what we mean.

The S being P condition is called the antecedent; it is only operative when actual, and needs be at least potential to fit in this formal position. The S being or not-being Q conjunction is called the consequent; here too, the relevant modality is actuality, and potentiality is formally implied. These two actualities may be called ‘events’.

The implied potential of the events and their conjunction is called the ‘basis’, and the natural modality qualifying the conjunction as a whole specifies the ‘connection’ involved.

b. Basis. Every natural conditional proposition may be said to be ‘based on’ the natural possibility, the potentiality, of the antecedent’s eventual actualization. Each of the six forms introduced above logically implies the categorical proposition ‘this S can be P’.

Likewise, since when the condition is actualized, the consequence will also be actualized, whether unconditionally or under certain unspecified additional conditions, it follows that the consequent is also logically implied to be potential. That is, ‘this S can be (or can not-be) Q’ may be educed from these same forms (with the appropriate polarity).

More precisely, the full basis of these forms is the conjunctive categorical ‘this S can be both P and Q (or can be both P and nonQ, in negative cases)’, which incidentally implies the two above-mentioned separate potentialities. The conditional proposition implicitly guarantees that the said base potentiality exists. This joint potentiality underlying every natural conditional is the foundation on which the subjunction is built.

Potentiality signifies that certain unspecified surrounding circumstances, may underlie the specified event. This refers to the various postures of the real world, the situation of the rest of the material, mental, and even spiritual world. Since potentiality is compatible with both necessity and contingency, items in the wider environment may or not be responsible for triggering the reaction or inhibiting it.

Although the original function of the form is to capture actualization of naturally contingent phenomena, it is so engineered that one or both of the events could in fact be naturally necessary. The formal basis of any natural conditional is the potentiality of the events, not their natural contingency.

The precise function of a natural conditional is thus only to point out to us the intersections, inclusions or exclusions, between the circumstances surrounding the two events. This may be compared to the doctrine of ‘distribution of terms’, in categorical propositions.

That is, though each form is based on the potentiality of antecedent and consequent and their conjunction, this does not logically necessitate that both the events be conditional, but admits as logically possible that one or both of the events exist(s) under all natural circumstances.

Thus, though for instance the necessary form ‘When this S is P, it must be Q’ implies ‘This S can be P and Q’, it is still logically compatible with any of the conjunctions ‘This S can be nonP and nonQ’ (double contingency), or ‘This S must be P and must be Q’ (double necessity), or ‘This S can not-be P, and must be Q’ (contingency with necessity).

However, that necessary form is logically incompatible with the conjunction ‘This S must be P, and can not-be Q’ (necessity with contingency), because of the connection, as we shall see. Similarly, with a negative consequent (substitute nonQ for Q throughout).

In contrast, the corresponding actual and potential forms, allow for all those eventual modal conjunctions, though only the said basic joint potentiality is formally implied.

If one or both of the events is necessary, the conditioning is admittedly effectively redundant, since a necessary event exists independently, it is ‘already there’; but the relationship is still formally true.

c. Connection. Although the modal qualification (the ‘must’, ‘cannot’, ‘can’, or ‘can not’ modifier) is placed on the side of the consequent — it is not part of the consequent, but properly concerns the relation between it and the preliminary condition, that is, the subjunction as a whole. This should be grasped clearly: the antecedent and consequent of natural conditional propositions can only be actualities or actualizations.

That is, ‘When this S is P, it must be Q’ does not say that the phenomenon ‘this S is P’ will be followed by the phenomenon of natural necessity ‘this S must be Q’, for it admits that ‘this S can not-be Q’ might be true. Rather ‘this S is P’ will, whatever the surrounding circumstances, be followed by the phenomenon of actuality ‘this S is Q’. Likewise for a negative consequence.

Similarly, ‘When this S is P, it can be Q’ does not say that the phenomenon ‘this S is P’ will be followed by the phenomenon of potentiality ‘this S can be Q’, for that is already given as part of the basis. Rather ‘this S is P’ will, in some unspecified surrounding circumstances, be followed by the phenomenon of actuality ‘this S is Q’. Likewise for a negative consequence.

It is thus very appropriate to regard the antecedent actuality and the consequent actuality, as the two ‘events’ referred to by the proposition. The modality merely acts as a bridge between them.

Note well that, even in the case of necessary conditioning, the natural circumstances in which the antecedent is actualized are not specified. What is specified, is that the conditions which suffice to actualize the antecedent, whatever they be, will also be sufficient to actualize the consequent.

That directional link between the events is formally expressed by saying that ‘When this S is P, it must be Q’ implies ‘This S cannot be {P and nonQ}’; and ‘When this S is P, it cannot be Q’ implies ‘This S cannot be {P and Q}’. These implications, in the form of naturally impossible conjunctions, are the connections between the events.

Thus, to define a necessary conditional, we must specify two categorical conjunctions (with appropriate polarities): the basis ‘this S can be both P and Q (or nonQ)’, and the connection ‘this S cannot be both P and nonQ (or Q)’. We cannot, with such natural conditioning (unlike with logical conditioning), ignore one or the other of these specifications; both must be kept in mind.

In the case of potential conditioning, the link between the events is formally expressed by contradicting the above necessary connections, and saying that ‘When this S is P, it can not-be Q’ implies ‘This S can be {P and nonQ}’; and ‘When this S is P, it can be Q’ implies ‘This S can be {P and Q}’. We see that, here, the implied basis and connection are one and the same naturally possible conjunction.

In merely potential conditionals, the (unspecified) conditions for actualization of the antecedent will not be enough to bring about the consequent; some additional (also unspecified) conditions are required for that. Clearly, these propositions enable us to express cases of partial, instead of complete, causality of natural phenomena; their subjunctive form is not artificial.

It is understood that there are some sets of circumstances, like say R, which in conjunction with P will suffice to cause Q (or nonQ, as the case may be) in this S. That is, for instance, ‘When this S is P, it can be Q, and when it is not P, it can not-be Q’ minimally implies ‘When this S is P and R, it must be Q’, for at least one (known or unknown) ‘R’.

However, that specifically concerns fully deterministic systems, and does not take free will into account. Indeed, denying such implication altogether is the way we can begin to formally develop the topic of spontaneous events. For this reason, I will not go into these issues in greater detail in the present study.

d. Definitions. In summary, we can define modal natural conditionals entirely through categorical conjunctions, but all the implied categoricals must be specified.

Thus, ‘When this S is P, it must be Q’ means ‘This S can be both P and Q, but cannot be P without being Q’; similarly, ‘When this S is P, it cannot be Q’ means ‘This S can be P without being Q, but cannot be both P and Q’. In contrast, ‘When this S is P, it can be Q’ means no more than ‘This S can be both P and Q’; and ‘When this S is P, it can not-be Q’ only means ‘This S can be both P and nonQ’.

It follows from these understandings that each of the necessary forms subalternates the potential form of like polarity (identical with their basis). Natural conditionals thus constitute a modal continuum, as did categoricals.

The actual forms, ‘This S is P and Q’ and ‘This S is P and not Q’, refer to conjunctions of events existing ‘in the present natural circumstances’. They obviously imply, as their bases, the propositions ‘This S can be both P and Q’ and ‘This S can be P and nonQ’, respectively, since what is true of ‘one specified circumstance’ is equally true of ‘some unspecified circumstance(s)’.

The position of these actual conjunctions in the modal hierarchy of conditionals, to some extent parallels the position of single actuals among modal categoricals, since they are the way the potential conjunctions, which are the basis of all modal conditionals, are actualized. However, the analogy is limited, because in the field of conditionals, natural necessity does not imply actuality, though both necessity and actuality do imply potentiality.

This is obvious from the greater complexity of the necessary forms. A connective like ‘This S cannot be both P and Q’ remains problematic with respect to which of the alternative positive conjunctions ‘P and nonQ’, ‘nonP and Q’, ‘nonP and nonQ’ will actually take the place of the excluded ‘P and Q’. We cannot even be sure that all these conjunctions are even potential; the only one formally given as potential is the one serving as basis, namely ‘P and nonQ’, the others may or not be so. Similarly, with appropriate polarity changes, for ‘This S cannot be both P and nonQ’.

Thus, a naturally impossible conjunction involves a certain amount of leeway, like a logically impossible conjunction. It does not by itself formally fully determine any actuality or even all potentialities. However, to repeat, a natural connective is not by itself ground enough to form a conditional proposition; an adequate basis is also required for that (whereas in the case of hypotheticals, logical basis is varied and optional).

Note, however, in exceptional cases, our use of the expression ‘when and if’ to suggest that we know the natural connection to apply (as suggested by the ‘when’), but we do not know the natural basis to be applicable (whence the ‘if’ proviso). But this expression may have other meanings (see section 3b further on).

e. Note well that actual ‘conditionals’ are in fact conjunctions, and cannot meaningfully be written in conditional form, with a ‘when’. With regard to the seemingly nonmodal conditional form ‘When this S is P, it is Q’, which we commonly use to describe habitual, voluntary actions or events, the following may be said:

A proposition such as ‘When she is happy, she sings’, should not be regarded as an actual conditional, but rather as a form vaguely expressing a degree of natural probability below necessity. It means, in ordinary circumstances, so and so is very likely, but in extraordinary circumstances, it is less to be expected. Alternatively, the intention may be to express a temporal modality, as in ‘When this S is P, it is always (or usually or sometimes) Q’; in which case the form properly belongs under the heading of temporal conditionals.

Ultimately, volitional conditioning involves a type of modality different from natural conditioning. Note that the antecedent of a natural conditional proposition may be voluntary, since even something freely willed may have naturally necessary consequences. What distinguishes volitional conditioning is that, whether the antecedent is emerges naturally or voluntarily, the consequence is voluntarily chosen and brought about. For example, ‘If you do this, I will do that’ involves two voluntary actions.

Volitional conditionals are thus statements of conditional intention. The ‘will’ involved, concerns another type of causality, than the ‘must’ of naturals. Volition is a special domain within Nature (in the broadest sense), where otherwise common relations (those of natural modality) do not all apply. Volition denotes a greater than usual degree of agency.

However, this type of modality will not be dealt with in this treatise, but belongs in a work on aetiology.

2. Quantification.

Quantification of the six prototypes expands the list of such natural conditional propositions to 18.

§An: When any S is P, it must be Q

§En: When any S is P, it cannot be Q

§In: When certain S are P, they must be Q

§On: When certain S are P, they cannot be Q

§A: All S are P and Q

§E: No S is P and not Q

§I: Some S are P and Q

§O: Some S are P and not Q

§Ap: When any S is P, it can be Q

§Ep: When any S is P, it can not-be Q

§Ip: When certain S are P, they can be Q

§Op: When certain S are P, they can not-be Q

We have already analyzed the expression ‘when’, signifying the natural conditionality, and the features of polarity of consequent (is or is not Q), and modality (in all, the given, or some circumstances). Here, we introduce plural quantity (any, certain), in place of the singular indicative (this).

The first thing to note is that the quantifiers are here intended as dispensive, and not collective or collectional. They refer to the instances of the subject severally, each one singly, so that the plural forms are merely a shorthand rendition of a number of singular propositions. The all or some units of the subject-concept do not have to simultaneously fulfill the condition for the consequence to follow, and the two predicates apply to the individual units, and not to a group of such units as a whole.

For this reason, the words ‘any’ and ‘certain’ are preferably used in this context, less misleading (however, the word ‘certain’ should understood as meaning ‘at least some’, and not ‘only some’).

Secondly, the basis of the general conditional propositions should be a categorical generality. For instance, ‘When any S is P, it must be Q’ implies that ‘all S can be P and Q’. In practise, we tend to confuse or mix the methodology of natural and extensional modality (see the discussion of the latter, in a later chapter), and often intend only a particular basis for a seemingly general natural conditional; however, here, the stated generality will be regarded as genuine. The corresponding particular conditionals only have particular bases, obviously.

Note that, although we have dealt with forms with a negative consequent, we did not so far mention forms with a negative antecedent, like ‘When this S is not P,….’ Obviously, we could construct another 18 forms (6 singulars and 12 plurals; or 6 actuals and 12 modals), with this added feature in mind.

I will not devote much attention to these extra forms, because their logic is easily derived. With reference to the eductions feasible from forms with positive antecedents, we can infer their oppositions to those with negative antecedents. And all the inferences feasible with the former can be duplicated with the latter, by simply substituting ‘nonP’ for ‘P’ throughout.

I do not here mean to underrate negative antecedents. Taking the antecedent as a whole, its polarity is of course logically irrelevant. Undeniably, forms with antithetical antecedents are important, because they complement each other.

For instance, a form like ‘When this S is P, it must be Q’ does not by itself communicate change, but combined with ‘When this S is not P, it cannot be Q’, we get a sense of the dynamics involved. That is, not merely is the static actuality of P accompanied by that of Q (or nonP by nonQ), but the actualization of P brings about that of Q (or nonP, nonQ).

We may view this as a formal implication, by certain combinations of conditionals involving actualities and inactualities, of similar conditionals concerning the triggering or prevention of actualizations (using the transitive copula, ‘gets to be’).

Besides natural conditionals with three terms, there are other varieties: those with four terms, such as ‘When this S1 is P, that S2 is Q’. Quite often used and important, is the case: ‘When any S1 is P, the corresponding (or some unspecified) S2 is Q’. Here, the mediation provided by an explicit common subject is lacking, though some hidden thread links the two events. For examples, ‘When a car runs out of fuel, its motor stops’ or ‘When evil is let loose, somebody somewhere suffers’.

Often, of course, we use still more complex versions, involving composite antecedent and/or consequent, such as ‘When {S is P1 and P2} and {S2 is P3}, {S3 is Q1 and Q2}’, say.

The logical mechanisms applicable to these more complex varieties should be similar to those for the standard three-term forms we are focusing on. So long as we clearly understand which individual subjects are denoted, so that we know precisely which one affects which, there should be no logical confusion.

3. Other Features.

The forms mentioned thus far deal with most natural conditioning situations. In this section, we will mention various notable departures from these norms.

a. The order of sequence, or chronology, of the antecedent and consequent events must be kept in mind to avoid errors of judgment with regard to natural conditionals. Here, I assume that a consequence takes place as soon as and so long as its antecedent. More broadly:

The antecedent may accompany the consequent immediately (and thus be simultaneous), or later in time, or earlier in time; and the time lapse between them may be mentioned explicitly, or tacitly understood (as we do here).

In the case of simultaneity, the events may happen at the same time, and yet not be contemporaneous, that is, not last for equal lengths of time. All the more, in cases of nonsimultaneity, the lasting power of the events may be very different; for instances, a flash of lightning may cause permanent damage, a long burning fuse may end in a sudden explosion. These issues should be taken into consideration in reasoning from natural conditionals.

In real causality, the cause is immediately or after some time followed by the effect; if we place the effect temporally before the cause, we are considering it as an ‘index’, ‘sign’ or ‘symptom’ of the cause’s presence or absence. Natural conditionals mirror reality if expressed in the right sequence, otherwise they are a logical artifice (wherein, instead of cause causing effect, the knowledge of the effect ’causes’ the knowledge of the cause).

In any case, the temporal qualification of the events is usually relative; the time of one event is defined as before or after the other’s time, by so much. In some cases, we refer to ‘absolute’ time — that is, date and o’clock; the relative time follows by inference.

Note also the different ways time may be specified: we can say that an event does or does not happen at some (stated or undefined) point or segment of time; or permanently, in past and/or future areas of time.

b. Modalities of Actualization. The ‘when’ should be taken in its weakest sense, as suggested in the expression ‘when and if’ or ‘if ever’, and not as implying the inevitability of actualization of the condition. This sense of ‘it can happen, but is not bound to happen’ is to be preferred as our standard because it is broader, more generally applicable.

We could work out a specialized logic for inevitable antecedents. A complex proposition like ‘When S gets to be P, and it is bound to be P eventually, it must be Q’, would have as its first base that ‘although this S can be and can not-be P, sooner or later it is must change from nonP to P’, and imply the inevitability of Q too (unless already actual through other causes).

Note well that inevitability of actualization signifies an underlying natural contingency of actuality, it is only the transition from nonP to P which is naturally necessary. Obviously, this should not be confused with the more static natural necessity of actuality, which we are usually concerned with, which is the antithesis of contingency.

All this brings to mind the wider field of natural conditionals for transitive events, incidentally. No one has researched it.

However, these are relatively narrow topics, and will not be discussed further here.

c. Within natural modality, we also need to recognize the phenomenon of acquisition or loss of ‘powers’. The concept of a power is rather difficult to define. By a power, we mean a potential close to actualization; something readily available, without too many preparatory measures. But this definition is too vague for formal work.

Anyway, something may remain outside the powers of a subject for a part of its existence and then eventually appear (e.g. through the maturing of an organism); or a subject may initially have a power and then lose it irrecoverably (e.g. the use of a hand which is cut off). Thus, we can talk of actualization of the presence or absence of powers.

Obviously, an ‘acquired power’ was always potential, even before it became more accessible; so the concept of a ‘acquired power’ is subsidiary to the concept of a potentiality, and included in it as a special case. However, a ‘lost power’ is something previously potential and henceforth naturally impossible; so this concept introduces a serious complication into modal logic, namely the logical possibility for changes in bases and connections.

Thus, in some cases, the modality given within a natural conditional, may be intended to be an intrinsic part of the antecedent or consequent. Such modal specifications are effectively actualities, as far as the conditional proposition as a whole is concerned, and should not be confused with the modality of the relation between them.

Powers may be indicated by use of modal expressions like ‘is able to be’ (which is less demanding than ‘is’, but more specific than ‘can be’) or ‘is unable to be’ (which lies between ‘cannot be’ and ‘is not’); or more dynamically, ‘is henceforth able to be’ or ‘is no longer able to be’ (more explicitly implying a change in powers). Likewise for ‘not-be’.

Thus, ‘When this S is able to be P, it is Q’ would mean ‘when this S has the (actual) power to be P, it is Q’. Likewise, ‘When this S is P, it is able to be Q’ would mean ‘When this S is P, it has the (actual) power to be Q’. More precisely, the latter statement should be modal, like all conditionals; that is, we mean ‘it must be able to be Q’ or ‘it can be able to be Q’, where ‘must’ or ‘can’ define the connection, while ‘is able to be’ signifies an actuality of power. Similarly for the interpretation of negatives.

This topic requires further study, but will not be pursued further here.

d. Note that in practise if one finds natural modality expressions, like ‘can’ or ‘must’ (or their negative equivalents), appearing in the antecedent or being intended as an intrinsic feature of the consequent — it does not follow that the conditioning is of the natural type.

On the contrary, this usually signifies that the conditional proposition is of the logical or extensional type. For examples, ‘If S must be P, then it can be Q’ is supposedly a hypothetical, and ‘In cases where S can be P, it must be Q’ is supposedly an extensional conditional, even though the antecedents and consequents are in natural modality.

As earlier pointed out, in practise the words we use are not always consistent with the intended modality of conditioning. One should therefore be careful to identify just what type of conditioning is intended, because their logics are considerably different.

4. Natural Disjunction.

Disjunction has traditionally been approached as an essentially logical relation. But our analysis of the types of modality shows clearly that disjunction also exists in nature. It can be understood with reference to natural conditioning.

a. There are various modalities and polarities of natural disjunction. Consider the simplest case of three terms, in the singular:

The necessary form ‘This S must be P or Q’, can be taken to mean that ‘When this S is not P, it must be Q, and when it is not Q, it must be P’, it follows that the implied connection is that ‘This S cannot be both nonP and nonQ’, and the implied basis is that ‘This S can be nonP and Q, and it can be P and nonQ’, which in turn imply that ‘This S can be and can not-be P, and can be and can not-be Q’. Note well the implied natural contingency of the individual events.

The corresponding potential form ‘This S can be P or Q’ accordingly means ‘When this S is not P, it can be Q, and when it is not Q, it can be P’ (same as the above basis).

As for the parallel negative forms: ‘This S can not-be P or Q’ has to mean ‘This S can be both nonP and nonQ’ (contradicting the above connection), and ‘This S cannot be P or Q’ may therefore be understood as ‘This S must be both nonP and nonQ’ (subalternating the preceding).

These various forms can of course be quantified.

b. Other manners of disjunction may also be used:

To describe a specifically ‘P and/or Q’ situation, we would have to add to the said ‘This S must be P or Q’ definitions, that ‘This S can be both P and Q’.

The natural disjunction ‘This S must be nonP or nonQ’ can be similarly interpreted, by substituting antitheses for theses throughout; briefly put, it means ‘When P, nonQ; when Q, nonP’. To describe a specifically ‘P or else Q’ situation, we would have to add to the said ‘nonP or nonQ’ definitions, that ‘This S can be both nonP and nonQ’.

An ‘either-or’ situation would be represented by a compound of the two disjunctions ‘P or Q’ and ‘nonP or nonQ’, meaning four natural conditional propositions.

c. Also, analogous forms involving more than three terms can be constructed, constituting multiple natural disjunctions. Their connections can be defined like multiple logical disjunctions, except with reference to numbers of actualities or inactualities, instead of truths or falsehoods.

However, here, note well, every one of the alternatives must be, taken individually, naturally contingent, as the two-alternative paradigm makes clear. Otherwise, the basis of disjunction is not properly, entirely natural, but closer to merely logical. Natural disjunction has a very different basis from logical disjunction; much more information is demanded of us, before we can formulate a natural one.

Note in any case that a logical ‘cannot’ implies, but is not implied, by a natural ‘cannot’; and therefore potentiality implies, but is not implied by, logical possibility.

After thus defining the various types of natural disjunction through naturally modal, categorical and conjunctive propositions, their logical interrelationships and processes can be worked out with little difficulty. The reader is invited to do this work.

In practise, it is not always clear whether we intend a disjunctive proposition looking like the above as natural or as logical. For instance, even though there is no such thing as actual natural disjunction, a proposition of the form ‘S is P or Q’ might be intended to mean ‘S must be P or Q’, rather than imply mere logical disjunction. But such ambiguities need not deter us from investigating the respective logical properties of these two types, and learning their differences. Some more comments will be made on this topic, in the chapter on condensed propositions.