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© Avi Sion
All rights reserved
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
MODALITIES OF SUBSUMPTION.
We need to analyze our presuppositions regarding the modalities of
subsumption by the terms of categoricals, as distinct from the copulative
In formulating the logic of modal categoricals so far, we have taken for
granted certain ideal assumptions, which will now be reviewed.
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.
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
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.
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.
All the above can be repeated with reference to temporal modality.
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
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
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.
Completely categorical propositions may be called primitives. They vary
in degree of specificity, but conceal no conditions.
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
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
I: 'Some things are now S and
O: 'Some things are now S and
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
Gp: 'This thing can be S and
Ip: 'Some things can be S and
Op: 'Some things can be S and
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.)
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
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
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
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
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.