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FUTURE LOGIC

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

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CHAPTER 41. MODALITIES OF SUBSUMPTION.

We need to analyze our presuppositions regarding the modalities of subsumption by the terms of categoricals, as distinct from the copulative modalities.

1. Formal Review.

2. Impact.

3. Primitives.

4. Transformations.

5. Imaginary Terms.

1. Formal Review.

In formulating the logic of modal categoricals so far, we have taken for granted certain ideal assumptions, which will now be reviewed.

a. Singular subsumption. We granted that ‘All S are P’ implies ‘This S is P’. However, closer inspection suggests the truth of such subalternation, only on the proviso that we have directed our attention to something, which we designate by ‘this’, and have discerned that ‘this is S’.

For, whereas ‘all S’ can be talked about without needing to be attentive any one S, the indicative ‘this’ requires a definite act of focusing on one thing, and judging whether or not it is S. This psychological requirement also means for logic that ‘this S is P’ and ‘this S is not P’ are both deniable at once, by saying ‘but this is not an S’.

Thus, A and R, or E and O, are only relatively subalternative, since this relation only works conditionally; absolutely speaking they are neutral to each other. Likewise, although R and G are relatively contradictory, they are absolutely only contrary. When the preliminary judgements regarding subsumption are settled, the relative opposition comes into effect; otherwise, the absolute opposition is operative. Similarly for modal singulars.

b. Actual subsumption. We granted that ‘All S must be P’ implies ‘All S are P’. However, closer inspection suggests the truth of such subalternation, only on the proviso that there be Ss in the present actuality. We have to consider the two modalities of ‘all S’.

Normally, we understand An to refer to ‘all S, ever’ (i.e. past, present, or future); although it could refer more restrictively to ‘all now S’. In the timeless (i.e. across time) case, there is no guarantee that any S exist in the present actuality, taken at random. In contrast, A is normally understood to refer to ‘all S now’, since any absent S are out of the present picture; although, if we view all the scattered actualities as one actuality, then we could say that the implication holds in the timeless case.

Thus, An and A may have distinct extensions. If they both mean the same ‘all S’, the subalternation holds. But if An means ‘all S at all times’ and A only means ‘all S at this time’, then An ceases to imply A, unless we have already established that ‘some S are actual’.

We can argue in the same way that ‘All S are P’ implies ‘All S can be P’, provided they have the same extension; if A means ‘all S now’ and Ap is understood to mean ‘all S ever’, the inference is illicit, and we can only accept that ‘Some S can be P’.

Thus, An implies A, and likewise for En and E, only conditionally. Also, A implies Ap, and likewise for E and Ep, only conditionally, though they still respectively imply Ip and Op unconditionally.

For the same reason, A and O, or E and I, may both be false, if it happens that ‘No S are actual’. Their contradictions apply at such times, but there are times when both can be denied. And likewise, the subcontrariety of I and O is only relative to there being actual Ss.

With regard to the interrelationships of modal propositions, since normally the subsumption of ‘all S’ has the same modality for all of them, such problems do not arise. They all imply, and presuppose, that Ss are potential. (It is true that if Ss do not exist even potentially, then modals behave like actuals without actual subjects; but this is another issue, dealt with later.)

Just as singulars like ‘This S is P’ presuppose ‘this is S’, so with any actual propositions we have to assume that ‘there are Ss at this time’: these are separate, preliminary judgements, which affect the logical properties of the propositions that conceal them.

c. Subsumption by the predicate. The above concerns subsumption by the subject. With regard to the predicate, it seems obvious that, if the subject of an affirmative, actual or necessary, proposition is actual, then so is the predicate, for the same extension. On this basis, we can convert ‘all or some S are P’ to ‘some P are S’. Also, since necessity implies actuality when the subject is actual, An and In can be converted to I, under those conditions; otherwise, only to Ip.

In the case of the corresponding negatives, it would at first sight not be thought that the predicate needs be actual. However, if ‘No S is P’ is to be converted, there has to be actual Ps to support the actuality of the inference; if this precondition is not met the eduction is invalid. If only some P are actual, then E is convertible, but only to ‘Some P are not S’; if all P are actual, E is convertible fully to ‘No P are S’; if no P are actual, nothing actual about Ps may be denied or affirmed.

Also, since En implies E, conversion of ‘No S can be P’ to ‘No P is S’ is only conditionally feasible, even though that to ‘No P can be S’ is independent of actuality. (Note however, in passing, that the conversion of En to En does presuppose the potentiality of the predicate.) For On and O, such problem does not arise, since they are inconvertible in any case.

d. All the above can be repeated with reference to temporal modality.

2. Impact.

Thus, the singular ‘this’ and the plural ‘all’ or ‘some’ are more weakly related than previously intimated. Also, actual copulae require at least actual subsumption by the terms, whereas modal copulae (whether necessary or possible) need only possible subsumption by the terms. The type of modality subsuming the terms corresponds to the type affecting the copula. In natural modal propositions, the subsumptions are potential; in temporals, they are temporary.

Thus, we have seen that many processes adopted as standard by both actual and modal logic, are only conditionally true. Some other logical processes, which depend on those considered above for their validity, may be expected to in turn become equally conditional. For example, if E is only conditionally convertible, then obverted conversion or inversion of E is likewise restricted.

Even syllogism may be affected. We have to look at the results of arguments, to make sure they are unconditional with reference to modal subsumption. For example, the mood 4/EIO does convert both its premises unconditionally, because the middle term in the minor premise, allows conversion of the middle term in the major premise. In contrast, the mood 3/RRR was rejected, essentially because the degree of specificity of the middle term could not be transferred to the minor term; but we could equally view this mood as conditionally valid, if we can indicate the subject.

We had made some ideal assumptions, to better emphasize the essential natures of the forms under consideration. These assumptions are reasonable — one would not normally formulate a proposition unless its subsumptive conditions seemed fulfilled; it is only in further ratiocination that an illicit process may occur, which yields a presumptive subsumption. However, we must be made aware of the exceptions and provisos, so that the system as a whole remain unassailable.

Thus, an avenue for further logical analysis is to check the unconditionality or conditionality of all our validations or rejections of logical processes. That investigation is left to the reader.

Note well that these theoretical requirements are not necessarily fulfilled in practise. There is a difference between ‘common parlance’, which is more flexible and approximate, and the ideal language of formal logic, which must needs have fixed and precise meanings.

For example, when in practise we say ‘All S are P’, we often mean A, but may also mean An or Ac or Ap or At, or even sometimes just I or Ip or It. Also, we may mean ‘all now’ or ‘all ever’. We may even misrepresent the terms. This is all harmless, if our thought is clear enough to oneself and successfully conveyed to others. One can reason logically with the rough sentences of everyday language, but there is less likelihood of error using formal language.

So long as the normative system is capable of verbalizing all situations encountered in practise, it is successful and sufficient. Thus, the science of Logic must extend its tentacles as far as necessary, enough to make possible the verbalization of any intention we may encounter in practise.

In that case, all casual statements must be carefully reformulated, to fit the standard forms provided by Logic, before they can be subjected to its rigid analysis. It is impossible to develop a system of Logic which parallels common practise exactly, because the variations in it are too arbitrary and too subjective.

Obviously, if the standard forms are not properly used, if the translation picks the wrong forms to express our pre-verbal intention, the results are likely to go awry. The process of forming a clarified thought is by no means automatic and guaranteed.

3. Primitives.

Completely categorical propositions may be called primitives. They vary in degree of specificity, but conceal no conditions.

Indication is the instrument of full specification. Only something which is precisely indicated — extensionally, naturally and temporally — is fully specified.

The indicative, singular and actual: ‘this thing, at this time, and in these circumstances, is so and so’, refers to an unnamed, pointed-to thing, existing in a pointed at time and set of circumstances. This form is specific extensionally, and temporally and naturally.

‘This’ (or that or these or those) is a sui generis term, which is meaningless without the presence in front of one of what is being referred to. One can say that ‘this is not so and so’ (to deny a statement starting with ‘this so and so is…’), but one cannot say “this is not a ‘this'”.

The next level of specificity is the indefinite, particular, actual: ‘there are, at this time and in these circumstances, some things which are so and so’, which informs us that, out-there somewhere in the world, ‘some things are so and so’. This form is unspecific extensionally, though still specific with regard to time or circumstance.

Further down the scale, the indicative singular modal ‘this thing is possibly so and so’, and the particular modal ‘some things are possibly so and so’, are indeed categorical, but unspecific. Note well that ‘this thing’ in singular modals is less specific than ‘this thing’ in singular actuals; because the latter concerns an actual relation, whereas the former concerns a modal one. The indicative is less demanding, here. Likewise for ‘some things’, the modality of subsumption depends on the modality of the copula.

The above mentioned primitive forms are the only absolutely categorical propositions. All other ‘categorical’ forms used by formal logic are more complex, and thus implicitly conditional. Their categorical format is somewhat conventional, artificial — hiding their compositeness.

The singular actual ‘This S is P (or not P)’ presupposes that ‘this thing is indeed S’, which may be said to specify the subject under discussion. As well, all actuals require and imply that the units subsumed by their terms be as actual as the copula between them (else, how would the relationship be viewed as actual?). Here, natural circumstances or times are being tacitly specified.

Plural actuals ‘All or Some S are P’ presuppose that ‘some things are indeed S’, which just means ‘there are actually unspecified Ss out there’. The specific actuality involved is supposed to be clearly understood.

Modals only require and imply that ‘this or some thing(s)’ — ‘are in some circumstances S’ (in the case of natural modality) or ‘are at some times S’ (in the case of temporal modality). Here, the circumstances or times for S remain unspecified, implying mere potentiality or temporariness of the subject, rather than a specified ‘this now’.

Similarly for the predicate, whatever the polarity of the copula, if conversion is accepted. We could alternatively, consistently, say that conversion of a universal negative is a valid process, only if the predicate is specific; in which case, the predicate of negative propositions does not need to be formally specific.

We cannot consistently say that all propositions are conditional, because then we would have no way to express categorically that the conditions have been met (as in apodosis). But it is logically permissible to regard the primitive statements ‘This thing is actually S’ and ‘There are actual Ss’ (= some things are S), as the only truly categorical forms, while all others as only relatively categorical.

4. Transformations.

Let us, therefore, reword the more complex categorical forms, in such a way that their implicit assumptions are brought out in the open, using primitives. We may call this ‘transformation’; it is done below, for actuals, then potentials, then naturally necessary propositions. A parallel listing can be made for temporal modality. We see that they all concern conjunctions involving the two terms, with varying degrees of specificity and complexity.

R: ‘This thing is now S and P’

G: ‘This thing is now S and not P’

I: ‘Some things are now S and P’

O: ‘Some things are now S and not P’

A: ‘Some things are now S and P, but nothing is now S and not P’

E: ‘Some things are now S and not P, but nothing is now S and P’

Rp: ‘This thing can be S and P’

Gp: ‘This thing can be S and not P’

Ip: ‘Some things can be S and P’

Op: ‘Some things can be S and not P’

Ap: ‘Some things can be S and P, and no other things can be S and P’.

Ep: ‘Some things can be S and not P, and no other things can be S and not P’.

Rn: ‘This thing can be S and P, but cannot be S and not P’

Gn: ‘This thing can be S and not P, but cannot be S and P’

In: ‘Some things can be S and P, but none of these things can be S and not P’

On: ‘Some things can be S and not P, but none of these things can be S and P’

An: ‘Some things can be S and P, but nothing can be S and not P’

En: ‘Some things can be S and not P, but nothing can be S and P’

Thus, the forms ‘All S are P’ or ‘Some S cannot be P’, and such, are really abbreviations, shorthand versions, of the above, more descriptive, forms. Their full definition shows many of them to be conjunctive, of two or more primitive categorical propositions.

Notice that the implicit conditionalities, may be a mix of extensional and natural, modal subjunctions. Plurals may be reworded in extensional conditional form, and modals in natural conditional form; so plural modals will involve both types of subjunction, one inside the other. Thus, we may have an extensional conditional, whose antecedent and consequent are two natural conditional propositions, involving different polarities.

For examples. Ap means: ‘For some things: in some circumstances, S and P coincide; but for other things: in no circumstances do S and P coincide’. An means ‘For some things: in some circumstances, S and P coincide; but for all things: in no circumstances do S and nonP coincide’.

Similarly with temporal modality, instead of natural, throughout.

I will not here analyze such forms further, although this is the obvious next step in the logical development of a complete system of modal logic. We would want to verify that the oppositions, eductions and syllogistic arguments, which were developed for complex categoricals, remain in force, when the later are transformed into their clearer, subjunctive versions. (If any inconsistencies in properties are uncovered, the transformations would have to be further perfected, until consistency is indeed achieved.)

5. Imaginary Terms.

Another issue relating to modality of subsumption is, how to view imaginary terms. This is a further complication, concerning logical modality.

An imaginary term may be built up out of certain suppositions and/or assumptions. ‘Supposition’ concerns what is already granted to be true in some cases, and/or in some circumstances or times, is singularized in an indicated instance and/or actualized in an indicated circumstance or time; whereas assumption concerns the granting of such particular, and/or potential or temporary subsumptions, to begin with.

Thus, supposition is based on given extensional, and/or natural or temporal possibility, and only presumes applicability to the specific instance or actuality; whereas assumption involves hypothetical constructs, it presumes the realization of what is merely logically conceivable. They differ in audacity, the former having more empirical grounds than the latter; but ultimately, they are both presumptive, bringing together certain events or characteristics in novel conjunctions, with more specificity than contextually justified.

Just as, with regard to extensional, natural or temporal modality, the modalities of the copula and terms affect each other — so, with regard to logical modality, the modalities of the copula and terms, are proportional. If a proposition involves some term of less than established status, then its truth is correspondingly no more than conceivable.

A concept which is believed to involve no presumptions may be viewed as realistic, while a concept is imaginary to the degree that it involves suppositions and assumptions. If we are at a stage where the projected parameters are still conceivable, then our concept tends towards realism with varying success. If we know already that the projections are not realizable, then our concept will remain imaginary.

In science, we construct imaginary concepts in the hope of eventually establishing them as realistic. But literature allows for pure imagination, whether it is in the form of a novel built on the suppositions that certain particulars, potentials or temporaries are in effect, or in the form of science fiction or fantasy built on unrealistic assumptions. The latter kind of imagination has no pretensions of literal truth, it is mere entertainment or example setting.

Thus, we can say that, apart from deliberate fictions, the difference between imaginary and realistic concepts is one of degree of contextual credibility. The degree is greatest, if no presumption was involved; intermediate, if only supposition was involved (the less supposition, the more realistic); and least, if assumption was involved (the more assumption, the more imaginary).

Our belief of a proposition is a function of our belief in its terms. If a term is imaginary, then we do not in the fullest sense accept the proposition as true, even if the formula makes internal sense. As the chances that our term be realistic increase, so accordingly does the proposition as a whole become closer to ‘true’ in the ultimate sense.

Thus, the hypotheticality of a term influences the degree of truth in the proposition. But such conditioning must be the exception, rather than the rule. We cannot consider knowledge as hypothetical ad infinitum: there has to be some definite knowledge.

Some propositions must be admitted as categorically true; the proof is that, if we claim all knowledge hypothetical, we thereby posit that claim as unconditionally true, and thus contradict ourselves. Because some propositions are unconditionally true, then these at least must involve realistic terms: ergo, some concepts must be admitted as realistic.

In practise, we commonly call even propositions with fictional terms ‘true’ — this is in the sense of internal consistency within a narrow framework, without regard to the unrealizability of the terms. For example: ‘Dragons are lizard-like’ has a mythical subject and yet is in a sense ‘true’. Here, ‘truth’ merely signifies an accurate description of a mental image known to be fictional; effectively, there is a tacit bracket saying ‘all this is imaginary’.

Closer scrutiny reveals that our example really means (should be rephrased as) ‘We have formed a fantasy, to be called a dragon, with an arbitrary description including the shape of a lizard’: so formulated, the proposition is factual. The format ‘Dragons are lizard-like’ is merely an abbreviation of that true statement; but taken literally, it is false (since there are no dragons).

In practise, we often have a fictional predicate in a negative proposition, as in ‘Lizards are not dragons’. This is formally more justifiable, since we can regard the convertibility of universal negatives as conditional on the factuality of the predicate. We could then demand that all subjects be factual, since nothing can really be said about nonexistents, without insisting on the same requirement for predicates. If so, ‘X does not exist’ would have to be worded ‘No existents are X’.

In conclusion, just as, with regard to the extensional, natural and temporal subsumption, we said that, whatever the polarity the copula, the terms must be specified in primitive form (indicatively for singulars or actuals, through the corresponding possibility for plurals or modals) — so, with regard to logical modality, subsumption in fully true propositions must be factual (or necessary), whereas subsumption in logically modal propositions need only be logical possibility (of varying degree: from mere notion, through relevant and consistent, up to logically necessary).

The rules of subsumption are essentially the same for all types of modality. In logical modality, a proposition has to be conceivable at some level, however low. The minimum requirement is that the words involved all mean something. That something may be any kind of appearance: one either rightly believed in, or realistic but disbelieved or unsure, or wrongly believed in, or unrealistic and disbelieved or unsure. But there must in any case be some kind of appearance, whether empirical, conceptually arrived at, or imaginary, which serves as the intent of the word.

These restrictions concern any proposition presented as having some possibility of truth. False propositions are not subject to law; they can even be meaningless or self-contradictory. Likewise, the antithesis, ‘nonX’, of a meaningful and consistent term, need not itself be so conceivable.

The various types of modality should not be viewed as making up a hierarchy along one line. Rather, each is like a dimension, at right-angles to the others, with analogous categories of modality. They thus are capable of combining together, while remaining mutually independent continua.

2016-08-20T07:20:34+00:00