CHAPTER39.EXTENSIONALS: FEATURES, OPPOSITIONS, EDUCTIONS.
Very different from naturals and temporals, are the conditionals built on
extensional modality. These are quite important, because they broaden the theory
of classification, providing us with the formal means for more complex thinking
processes.
4.
Translations and
Eductions.
a.
Actual
forms. The following are
prototypical forms of extensional conditional, those with three terms. The
antecedent and consequent might in this context be called ‘occurrences’. We will
first consider forms with actual occurrences, and thereafter deal with those
with modal ones.
(The forms, if need be, could be symbolized like their categorical
analogues, except for, say, an ampersand ‘&‘
as prefix, to distinguish them also from natural or temporal conditionals.)
&A: Any S which is P, is Q
&E: No S which is P, is Q
&R: This S is P and Q
&G: This S is P and not Q
&I: Some S which are P, are Q
&O: Some S which are P are not Q
b.
Basis
and Connection.
The basis of all these forms is a particular proposition of the form
‘Some S are P and Q (or nonQ)’, which incidentally implies that ‘some S are P’
and ‘Some S are Q (or nonQ)’. The basis is a particular conjunction of the same
modality as the occurrences.
Note well, the difference between such extensional basis, and the basis
‘All/this/some S can be, or sometimes is/are, P and Q (or nonQ)’ of natural or
temporal conditionals, which is a potential or temporary conjunction of the same
quantity as the events. Contrast also to the basis of hypotheticals.
The connection implicit in ‘Any S which is P, is Q’ is the general
proposition ‘No S are both P and not Q’; and that in ‘No S which is P, is Q’
(meaning, ‘Any S which is P, is not Q’) is ‘No S is P and Q’. Note that ‘Any
S…’ can be expressed in many ways, like ‘In any case that S…’, or ‘Whatever
S…’, or ‘Where S….’ In the forms ‘Some S which are P, are Q (or nonQ)’, the
connection is identical with the basis.
Thus, to define the general forms of extensional conditional, we must
mention both the connection and basis; the connection alone provides us only
with a sort of logical conditional — an adequate basis is additionally
required to form an extensional conditional. For the particular forms, the basis
is all we need to define them. For singulars, we must present a specific case
which fits the description; the basis follows incidentally.
The modal qualification of the
relation as a whole, here, is the quantity. Note that in practise we often say
‘In such case as S is P, it must or may be Q (or nonQ)’, with the intent to mean
an extensional conditional; here, ‘must’ signifies generality, and ‘may’
particularity. What matters, is that we mean the relationship here discussed,
however we choose to verbalize it.
In extensional conditionals, it is the (general, singular or particular)
quantity which expresses the (extensional) necessity, existence or possibility
of the relationship, so that it is essential to the relation. In contrast, in
natural or temporal conditionals, the quantity is merely incidental, allowing us
to summarize many individual events in one statement.
The forms ‘This S is P and Q (or nonQ)’ signify that we have found an
instance of the subject-concept which displays the said conjunction. An
‘extensional possibility’ concerning the universal S, has been found ‘realized’,
in this pointed-to instance of S. We could have written ‘In this case, S is P
and Q (or nonQ)’.
The singular versions are also often expressed as ‘There is (or this is)
an S which is P, which is Q (or nonQ)’, or ‘This S is a P, which is Q (or
nonQ)’, to emphasize the mediative role played (which is more evident in
plurals). These forms inform us, with reference to the sample of S, of the
factual relationship between P and Q.
The expression ‘which‘ is interesting.
It strings together two extreme terms, through the medium of a merely particular
middle term. Because extensional conditionals have three terms, we do not need
the distributive middle term of categorical syllogism to express the passage
from minor to major term. The syllogism ‘This S is P and some P are Q, so this S
is Q’ is invalid — unless we have inside information assuring us that the
middle term is known to overlap in this case. That assurance is given us by the
‘which’.
Note lastly that the consequent may be positive or negative. Needless to
say, the antecedent in the above forms may equally be negative: ‘In such case as
S is not P,….’
c.
Function.
Extensional conditionals describe ‘cases’ of correspondences between the
manifestations of distinct universals. Though their quantity is dispensive, as
in categoricals, their focus is not so much the behavior of cases as that of
universals.
Note that the antecedent and consequent occurrences may coincide in time,
or be unequal or separate, like any two events. They may be transient, or
permanent; they may be qualitative or concern action. But the message of such
forms is not primarily these dynamic details, but the extensional relations
between them.
It is as if the universal involved is regarded as an individual,
something in itself, which changes over time. In fact, no actual, objective
change needs be taking place. The time lapse involved may be subjective,
relating merely to the observer, as he or she focuses on one instance after
another of the unchanging universal. In extensional modality, opposites may
happen simultaneously in objective time, because they happen in different
instances.
Extensional conditional propositions differ from naturals and temporals,
in that they study (record, report) the behaviors of universals, instead of
individuals, as if the various manifestations of a universal are like the various
states of an individual. Extensional contingency is diversity; incontingency
is positive or negative universality.
Extensional modality is concerned with instances of the subject-concept;
instances are its ‘modal units’, instead of surrounding circumstances or times.
The effective subject of such a proposition is S-ness as such. The varying cases
of S, signal varying hidden (extensional) conditions, and thus serve a function
analogous to the various circumstances or times in the existence of an
individual thing, which are natural or temporal conditions. This explains why
all these modal types have many similar characteristics.
We see here an important underlying assumption concerning universals,
that they are ruled by a kind of static and plural causality, similar to and yet
distinct from the mobile causality relating individual events. For natural or
temporal conditioning, real change is implied; for extensional conditioning,
only real difference is implied.
Here, we are still concerned with real-world causality, but it is of a
clearly different type. Natural and temporal causality essentially concern the
changes within individual things stretching across time and the links between
them (this is true for quantified forms as well as singulars, by subsumption).
Whereas, extensional causality refers to the differences and ties affecting
universals as such.
The logic of conditioning for this type of modality, investigates more
intricate relationships, than those dealt with by Aristotle’s categorical
propositions. These relationships have analogies to those found in natural and
temporal conditioning, and even in logical conditioning, but they also have
their own peculiar attributes and properties. We must therefore study them
separately.
This research results in a better understanding of quantity and
universals, and a powerful verbal and conceptual tool. The clarity of language
it offers, will become apparent when we look into class-logic.
The main function of extensional conditionals is classification,
ordering of data. These forms record the impacts of universals on each
other, with reference to some or all of their instances. Extensionals are thus
useful in explaining differences in structure or behavior patterns by reference
to certain characteristics of the species.
For example, in biology. Suppose the species S1, S2, S3, display the attributes
or properties {P1, Q1, R1}, {P2, Q2, R2}, {P3, Q3, R3}, respectively; we might
infer that they stand in a hierarchy, proportional to the differences of degree
between P1, P2, P3, or Q1, Q2, Q3, or R1, R2, R3. In this way, we conclude that,
say, birds are related to reptiles, or men to monkeys. Although we have no film
footage of natural and temporal transitions, we presume common ancestries (theory of evolution) with reference to character continuities.
But of course, strictly speaking, as our analysis of the definitional
features of the various types of conditioning show, extensional comparisons are not proof of natural or temporal causation.
Awareness of the type of modality involved is therefore very important.
a.
Modal
Forms.
The antecedent and consequent of an extensional need not be both actual
propositions (as above), but may involve any combination of natural and temporal
modalities. I use the actuals as standard forms, because they suffice to analyze
the main logical properties of extensional conditioning, but any natural or
temporal category is a fitting occurrence.
To begin with, consider an extensional conditional of the form ‘This S
can be P and can be Q’. Its intent is only to record that these two
potentialities are each consistent with the subject-concept in the given case.
The form does not insist that this S can be both P and Q at once. If we wanted
to specify the latter, we would have to elaborate with a natural conditional of
the form ‘When this S is P, it can be Q’. Note well the difference.
Thus, the said extensional is a wider, vaguer conjunction of two
categoricals: ‘This S can be P, and this S can be Q’, whereas the corresponding
natural presents the special case: ‘This S can be {P and Q}’. The natural form
therefore subalternates the extensional form.
The purpose of the extensional is to specifically inform us of the
identity of the indication ‘this S’ in the two potential occurrences, leaving
open the issue as to whether or not their potentials can actualize in tandem.
The purpose of the natural is to inform us of the concurrence of actual events,
and not merely their potentialities, in the indicated instance, in some
circumstances.
If the form ‘There is an S which can be P, which can be Q’ was taken to
imply that that S, as a P, can be Q, then in cases where a P cannot be Q we would have
to say ‘There is an S which can be P, which can become Q’. It follows that in cases of uncertainty about the
compatibility of P and Q, we would say: ‘There is an S which can be P, which can
be or become Q’.
It is therefore better to admit the extensional form in its widest sense,
only implying that S can be Q, without determining whether SP can be or become
Q. An extensional is concerned specifically with the extensional aspects of the
relation (the coincidence of modal occurrences), and leaves the issue of
circumstantial compatibility of the actual events to a natural proposition.
Their functions are distinct.
The basis and connection of the corresponding general form ‘Any S which
can be P, can be Q’ are: ‘Some S can be P, and (at least) these S can be Q’ and
‘No S both can be P and cannot be Q’, respectively. In every case, the implied
basis is a positive conjunction of particular
propositions (of equal extension), each of which has the
same natural or temporal modality as the occurrence it underlies, note well.
The connection, for general conditionals, is a general denial of the conjunction
of the antecedent modality with the negation of the consequent modality. The
basis and connection of the corresponding particular form ‘Some S which can be
P, can be Q’, are one and the same proposition ‘Some S can be P, and these S can
be Q’
It may be mentioned here, that the colloquialism ‘S can or can not be P’,
does not disjoin ‘can’ and ‘can not’, but rather (redundantly) disjoins ‘P’ and
‘nonP’; it should more strictly be expressed as ‘S can and can not be P’ (the
antinomy between P and nonP being given by the law of contradiction, anyway).
Modal extensionals, one or both of whose occurrences is/are of natural
necessity, have different basis and connection. Thus, ‘an S which must be P, can
be Q’ is based on ‘Some S must be P, and these can be Q’, whereas ‘an S which
can be P, must be Q’ is based on ‘Some S can be P, and these must be Q’; and
similarly with two natural necessities. Although such forms happen to imply that
‘these S can (or even must) be {P and Q}’ (and therefore that ‘some P can be
Q’), that is not the primary message, and they are still very different from the
natural conditionals with the same implications.
The reader is encouraged to always mentally compare, as we proceed with
our study, the logical behavior of extensionals, with that of natural and
temporal conditionals and hypotheticals of similar appearance. The evident
differences in attributes and properties, serve to justify our making a
distinction between these various forms.
We can similarly analyze other combinations of natural and/or temporal
modalities, of whatever polarities and quantities. In all cases, the natural or
temporal modality is effectively a part of the occurrence it appears in, and
does not qualify the relation as a whole; it is the quantity which performs the
task of modalizing the relation. (In that large sense, all plurals are ‘modal’,
be their internal components actual or modal — in contrast to singulars which
are ‘nonmodal’ with respect to extensional modality.)
Some random examples of occurrences of mixed modality are: ‘Any S which
must be P, is Q’, or ‘Some S are sometimes P and always Q’, or ‘There are S
which can be P, yet are never Q’.
In this text, we shall of course try to use a uniform terminology, at
least in strictly formal presentations. But in practise, people are not always
consistent in their choice of words to express the modal type of a conditional
proposition. We may for example say ‘If or When S are P, they must be or are
always Q’ and yet mean ‘All S which are P, are Q’.
To complicate matters further, we sometimes intend conditioning of mixed
modal type — in structure, not just content. We may say ‘when any S is P, it
must be Q’, and mean both that ‘All S can be both P and Q’ and that ‘Some S are
both P and Q’; here, the extensional ‘Any S which is P, is Q’ is tacitly
understood. Effectively, we are constructing a distinct type of conditioning,
using a compound type of modality, which expresses a two-edged probability
argument.
(Note, concerning symbolization: the seeming actuality of the symbols
&A, &E,
&R, &G, &I,
&O, is irrelevant, what matters is that they specify the polarity and
extensional modality concisely. If we insist on a symbolic notation to indicate
the natural or temporal modalities in antecedent and consequent, we could insert
two suffixes of modality, as in &Anp for example. But it is better to avoid complications; if we need
to, we can always write a proposition in full.)
b.
Other
Forms.
Extensional conditional propositions may also have more than three terms,
which may be related in noncategorical ways.
The subject may remain the same in antecedent and consequent, while its
predicates are more complex. For examples: ‘Any S which is P1 and P2, is Q’ has
a conjunction of categoricals as antecedent; ‘Any S among those which ‘when they
are P1, must be P2′, is Q’ has a natural conditional as antecedent. Likewise,
the consequent may be more complex.
Also, the antecedent and consequent may conceivably concern different
subjects. Since a ‘one for one’ correspondence is usually involved, though we
can expect some common substratum to underlie them, and make possible their
linkage somehow. For example, ‘For all S1 which are P, there is an S2 which is
Q’ would occur if S1 and S2 are both, say, aspects of the same entity S, or are
caused to occur together by some third thing S.
Extensional disjunction may be understood with reference to extensional
conditionals. It is quite distinct in its implications from other modal types of
disjunction.
With three terms and actual predications, the general form is ‘S are all
P or Q’, meaning ‘Any S which is not P, is Q, and any which is not Q, is P’.
This implies that ‘Some S are P and some not, and some S are Q, and some not’
(bases) and that ‘No S is {both nonP and nonQ}’ (connective). It does not imply
that all S can be P, nor that all S can be Q, note well.
Here again, the different senses of ‘or’ would need to be considered, as
well as the corresponding particular form, ‘S may be P or Q’, and the parallel
negative forms, ‘No S is P or Q’ and ‘Some S are not P or Q’. More broadly,
multiple disjunctions can be defined, with reference to the number of predicates
which are found to occur together or apart, in any instances of the subject.
Disjunction of modal predications is also feasible, of course. For
example, in ‘S all must be P or can not-be Q’, which means ‘Any S which can
not-be P, must be Q, and any S which must be Q, can not-be P, though some S must
be P, and some S can not-be Q’.
Note well that the natural modalities are parts of the occurrences, and
have nothing to do with the conditioning as such, which is itself extensional.
Also, do not confuse the above extensional interpretation, from that of a
similarly worded logical disjunction, meaning ‘{All S must be P} or {All S can
not-be Q}’.
Similarly, with any other internal polarities and modal categories and
types, in any combinations. We can also construct forms with more than three
terms, like ‘In all cases, an S1 is P or an S2 is Q’.
However, detailed analysis of these various forms will not be attempted
here. Our treatment of the analogous forms in other types of modality, should
serve as a model for further research in this area. The reader is invited to do
the job.
I shall only here sketch with a broad pen, the oppositions between
extensional conditionals, among each other and in relation to categoricals. The
reader should draw three-dimensional diagrams, to clarify all implications.
The singular form ‘This S is P and Q’ is contradicted by ‘This S is nonP
and/or nonQ’, in the sense of a logical disjunction.
The general form ‘Any S which is P, is Q’ means ‘Some S are P, and these
S are Q, and no S is both P and nonQ’; it may therefore be contradicted by
saying ‘No S is both P and Q, or some S are both P and nonQ’. But each of these
alternatives, whether denying the basis or denying the connection, taken by
itself, is only contrary to the form as a whole.
The particular form ‘Some S which are P, are Q’ is contradicted by saying
‘No S is both P and Q’. This may arise because ‘No S is P’ or ‘No S is Q’, but
it is also compatible with ‘Some S are P, and some (other) S are Q’. It follows
that general denial of the antecedent or of the consequent, only contraries the
basis.
Note that a proposition like ‘Any S which is P, is Q’, or its particular
version, does not exclude the logical possibility that ‘All S are P’ and/or that
‘All S are Q’.
In extensional conditioning, a general proposition subalternates a
particular one, since the latter is identical with the basis of the former, if
they are alike in polarities and modalities. But (here, unlike in natural or
temporal conditioning) a general
proposition does not subalternate a singular one; saying that ‘any S which
is P, is Q’ does not imply that this given S is among those which are P (and
therefore Q). However, a singular proposition subalternates a particular one;
saying that ‘this S is P and Q’ does imply that there are at least some cases of
S (if only this one) which are P and Q.
Comparing forms with consequents of opposite polarity, the singulars
‘This S is P and Q’ and ‘This S is P and nonQ’ are merely contrary, since they
may both be false, as in cases where ‘This S is not P’.
The generals ‘Any S which is P, is Q’ and ‘No S which is P, is Q’
(meaning, ‘Any S which is P, is not Q’) share the same partial basis ‘Some S are
P’; but their connectives are respectively ‘No S is both P and nonQ’ and ‘No S
is both P and Q’; thus, they disagree on whether the S which are P, are or are
not Q, and are contrary.
Note well that ‘No S which is P, is Q’ means more than ‘No S is both P
and Q’ (its connective); the former has as basis ‘Some S are P and nonQ’,
whereas the latter does not have that implication, since it may be true because
‘No S is P’ and/or ‘No S is Q’.
As for ‘Some S which are P, are Q’ and ‘Some S which are P, are not Q’,
they are compatible, but neither implies the other, since they may be referring
to distinct cases of S. They are not subcontrary, since if ‘No S is P’ is true,
both are false; they are therefore neutral to each other.
The parallel forms negating the antecedent can similarly be dealt with.
Their antecedent is of course based on ‘Some S are not P’, instead of ‘Some S
are P’, so they are bound to be compatible, with forms which imply the latter
base. That is, for instance, ‘Any S which is P, is Q’ and ‘Any S which is not P,
is Q’ may both be true, implying that ‘Some S are P, some not, but all S are Q’.
Likewise for a negative consequent.
The four particular forms ‘Some S are P and Q’, ‘Some S are P and nonQ’,
‘Some S are nonP and Q’, ‘Some S are nonP and nonQ’, are together exhaustive:
one of them must be true, though up to four of them may be true.
We can also find the oppositions between extensional conditionals whose
occurrences have natural or temporal modalities other than actualities. The
oppositions between categoricals obviously affect this issue. For example, provided
‘some S must be P’ is given, ‘Any S which is P, is Q’ implies ‘Any S which must
be P, is Q’ (note well that the modality of Q is unaffected); but it may equally
be of course that ‘only those S which are P and can not-be P, are Q’, in which
case we must say so.
Still needing to be dealt with are the oppositions between extensional
conditionals, and natural and temporal conditionals. Samples of such
relationships have been hinted at throughout this chapter. A fuller picture is
left to the reader to try and work out.
We will not go into further detail here. Once the similarities between
extensional conditioning, and natural or temporal conditioning, are understood,
all their attributes and properties can be predicted by analogy, if only we
switch our focus to the appropriate modal type.
4.
Translations and Eductions.
Extensional conditionals may be translated into the form of conjunctions
of categoricals, by eliciting their defining basis and connection. One can also
abridge them without error, by forming a narrowed subject out of the original
subject and antecedent predication, as in ‘All/this/some SP is/are Q’, since it
is given that ‘some S are P’.
With regard to eduction. For singulars, note the following: ‘This S is
both P and Q’ is equivalent to ‘This S is both Q and P’, and implies ‘This S is
neither {P and nonQ}, nor {nonP and Q}, nor {nonP and nonQ}’.
For plurals, obversion is always possible, i.e. ‘Any SP is Q’ implies ‘No
SP is nonQ’, and ‘Some SP are Q’ implies ‘Some SP are not-nonQ’, obviously, and
vice versa.
‘Any S which is P, is Q’, like ‘Some S which are P, are Q’, converts only
to ‘Some S which are Q, are P’; ‘No S which is P, is Q’, like ‘Some S which are
P, are not Q’, only convert by negation, to ‘Some S which are not Q, are P’.
The polarities may not be changed, without their extensional possibility
being first given. Thus, only knowing that ‘Some S are not Q’ could we
contrapose ‘Any S which is P, is Q’ to ‘Any S which is not Q, is not P’;
likewise, only knowing that ‘Some S are Q’ could we contrapose ‘No S which is P,
is Q’ to ‘No S which is Q, is P’.
Similarly, with more complex forms involving natural or temporal
necessity or possibility. For example, ‘Any S which can be P, must be Q’
converts to ‘Some S which must be Q, can be P’ without proviso, but contraposes
to ‘Any S which can not-be Q, cannot be P’ only if we are additionally given
that ‘Some S can not-be Q’.
The subject S has remained the same throughout, note. Note well the
differences between all these immediate inferences, and those applicable to
similar looking natural or temporal conditionals.