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FUTURE LOGIC

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

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CHAPTER 39. 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.

1. Main Features.

2. Modal and Other Forms.

3. Oppositions.

4. Translations and Eductions.

1. Main Features.

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.

2. Modal and Other Forms.

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.

3. Oppositions.

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.

2016-08-20T07:18:38+00:00