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The Logician © Avi Sion All rights reserved |
FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 34.
NATURAL CONDITIONALS: FEATURES.
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).
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.
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.
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.
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. |
