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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 13.
MODAL PROPOSITIONS.
Let us review some of the modal concepts introduced thus far, before
examining them in more detail.
Modality in its widest sense is an attribute of relationships. The
paradigm of modality is the quantity attribute of (the terms of) propositions.
When phenomena are observed to be alike in some way, they may be grouped into a
class, and be regarded as instances of that class.
We may refer to such units in various ways. The units intended by a
reference are said to be included in it; those not so, excluded. When a unit is
focused on individually and specifically (if only through a pointing to it), the
reference is singular; otherwise, our focus is plural.
When we refer to a fraction of the class, it is particular; when to its
totality, it is general. The greater division of a class is a majority; the
smaller, a minority. Singular and particular frequencies concern mere incidence;
the other plurals — generality, or majority or minority — are relative
frequencies, and describe prevalence.
Quantity is one type of modality, namely the extensional. Other types of
concern to us here are temporal modality and natural Modality. These have in
common with quantity the mode of analysis defined above. However, the classes
under consideration are not the terms of propositions, but respectively the
temporal existence or the causal conditions of the connection between the terms.
Just as quantity concerns the application of a term to one, some, all,
most or few of its instances; so temporal modality analyses the application of
the predicate to one, some, all, most or few of the moments of its given
subject's existence; and natural modality concerns the application of the
subject and predicate relation to one, some, all, most or few, of the
circumstances surrounding such happening.
These common factors may be called the categories of modality. They are:
presence (unitary event), possibility (partial reference to the events-class),
necessity (complete reference to it), high or low probability (inclusive of more
or less than half the units). Derivative concepts are: absence (presence of
negation, or negation of presence), possibility-not (possibility of negation, or
negation of necessity), contingency (sum of possibility and possibility-not),
impossibility (negation of possibility, or necessity of negation), and
incontingency (either necessity or impossibility). These general categories may
be given specialized names when applied to each type of modality.
In extensional modality, the main ones are, as we have seen, singularity,
particularity, generality (or universality). In temporal modality, we will use
the words momentariness, temporariness, constancy, for the corresponding
concepts. In natural modality, actuality, potentiality, necessity.
Note that the sub-categories of possibility should not be taken to imply
contingency, as often the case in everyday discourse; they are compatible with
necessity. Also note our double use of words such as necessity for both abstract
categories and especially natural modality sub-categories.
Additionally, let us point out that presence may be usefully viewed
either as stemming from necessity or as an occasion of contingency. This way of
viewing presence, as the realization of a deeper phenomenon of necessity or
contingency, follows from the oppositional relations between these concepts,
which will be analyzed below. Accordingly a singular instance may be viewed as
the concretization, of either a generality or a distinction. A momentary event
may be viewed as the eventualization, of either a constancy or a variability. An
actual occurrence may be viewed as the actualization of either a (natural)
necessity or a (natural) contingency. Similarly, on the negative side.
We reserve the following terminologies in formal treatment of these three
types. This, some, all, most, few, will express quantity. Now, sometimes,
always, usually, rarely, will be used to express temporal modality. Is, can be,
must be, is likely to be, is unlikely to be, will express natural modality. In
ordinary discourse, these various expressions of frequency, quantifiers and
modifiers, are of course often interchanged.
It is stressed that all plural such expressions are intended to include
the units they subsume on a one by one basis. That is, 'in some or all cases'
means 'in each and every one of the cases in the part or whole of the group
under consideration'. It is not a collective reference to the units considered
together. This quality applies equally to all three types of modality, each in
its own domain (extension, time, circumstances).
Every proposition has quantity (implicitly if not explicitly); and every
proposition has either temporal or natural modality. The unitary forms of these
latter two modalities coincide; but their plural forms cannot be combined, being
factors in one and the same continuum. That is, when we colloquially say 'X can
always be Y', for instance, we may mean formally-speaking 'All X can be Y', but
it is not possible to combine 'can' or 'must' with 'sometimes' or 'always' in
the reserved senses of words, because, strictly, must implies always implies
sometimes implies can, i.e. these concepts are related in specific ways, as will
be seen.
Aristotelean logic recognized six main propositional forms, as we have
seen, labeled A, E, I,
O and R,
G. Actually, classical logic is usually developed in terms of the
first four of these, i.e. the universal and particular. I added on the last two,
i.e. the singular, to complete the picture systematically; they were not unknown
to Aristotle, anyway. The labeling above mentioned is of course mere convention.
Another notation could have been devised, using the letters u,
p, s
for quantity specification, and +, -
for polarity. In that case, A=u+,
E=u-, I=p+,
O=p-, R=s+,
G=s-. Generally, I have
found it practical to continue using the letters A, E, R,
G, I,
O, in most work, though the separate labeling of quantity and
polarity are sometimes valuable.
The value of this alternative notation becomes more evident once modality
is introduced, because the laws of inference in Aristotelean logic can thereby
be brought out more clearly. (Note how I often use the term modality in a
restrictive sense excluding quantity.) By analogy to u,
p, s,
we may introduce the symbols c, t,
m, for constant, temporary and
momentary propositions, respectively; and n,
p, a
for naturally necessary, potential and actual propositions, respectively. (The
equivocal use of 'p' for
particularity and potentiality is perhaps unfortunate, but context will always
make clear which of the two is meant, so it is not serious). The modality
symbols may be used as subscripts to the standard six letters. The following is
a list of all the categorical forms under consideration in this study.
a.
Propositions involving natural modality. These, for the purposes of
definition, could equally be expressed in the form 'In all/this/some
circumstance(s), all/this/some S is/is-not P' (Or, 'Under any/the given/certain
conditions, all/this/some S is/is-not P'.) Note well the difference between
'cannot be' (which should have been written 'not-can be', to signify negation of
potentiality) and 'can not-be' (signifying potentiality of negation).
b.
Propositions characterized by temporal modality. These can be defined by
the overall form 'At all/this/some time(s), all/this/some S is/is-not P'. Note
that we here use the word 'now' equivalently to 'at this time', to avoid getting
involved with issues of tense in this context.
It will be observed that, in the above listing, we left out subscription
of actual propositions with an 'a',
and momentary propositions with an 'm'.
This was an intentional ambiguity, which will now be explained. If we analyze
common usage of the form 'S is P', we find that it is really very vague and
capable of many interpretations. This is not said as a criticism of Aristotle's
logic; in a way it has been one of its strengths, the reason why he seemed to
have succeeded in describing human thought processes fully. But logic requires
that ambiguities be brought out in the open, to ensure that nothing is left to
chance. That is precisely why I have taken the trouble to develop a theory of
modal logic, and researched it in such detail.
In its broadest sense, 'S is P' could be understood to mean any of the
following: 'S must be P' (an absolute sense, often though not exclusively
encountered in theoretical sciences), or 'S is always P' (a timeless sense,
often found in empirical sciences), or 'S is in the present circumstances P' or
'S is at the present time P' (such meanings are usually intended in everyday
descriptions of social events), or even no more than 'S can be P' or 'S is
sometimes P' (with the qualification left tacit for purposes of stress). We are
sometimes not aware of just how high or low on this scale our thoughts or
statements fall; sometimes, though aware, we allow our meaning to be suggested
by the context, or regard the distinction as not important enough to call for
explicit expression. Sometimes, of course, our intention is not left tacit, and
we say exactly what we mean.
To further complicate matters, the 'S is P' form is sometimes used in a
likewise indefinite, but more restricted sense; that is, one not including
natural necessity or potentiality, but broad enough to include any temporal
modality. In this sense, 'S is P' signifies a generic actuality, capable of
embracing either constancy or momentariness or temporariness.
As far as formal logic is concerned, the 'broadest sense' described
above, means no more than 'S can be P', which is its least assuming
interpretation. Likewise, the 'more restricted sense' next described, must be
taken by formal logic at its minimal power, meaning 'S is sometimes P'. Thus,
paradoxically, the broader the possible meaning, the lower is its logical value;
that is, given a more or less indefinite 'S is P' statement, without further
specification, we are forced to adopt its most all-inclusive interpretation.
Logical science therefore ignores such vague references, and prefers to deal in
fully specified forms.
This leaves us with one more ambiguity. If an 'S is P' statement is not
intended in the above vague senses, is it intended in the sense of actuality (in
this circumstance) or in that of momentariness (at the present time)? Are these
parallel but different, or are they essentially one and the same? I suggest that
the latter answer is ultimately to be preferred. The concepts of 'present
circumstance' and 'present time' indeed have somewhat different conceptual
roots, namely causality and time; but they represent the point of intersection
of these two frameworks.
Just as a singular proposition points to 'this' instance and not merely
'an' unspecified instance of the subject-concept, so in natural and temporal
modality, there is an mentally understood environment to the event under
scrutiny (i.e. S being P). In a natural modality perspective, we view this vague
environment as the surrounding disposition or layout of other objects,
constituting an undefined set of causal conditions, which may have given rise to
our event. In a temporal modality perspective, we merely locate the event in
time, but it is taken for granted that the underlying circumstances, however
unclear precisely which, may be involved somehow in our event.
Thus, the difference between a-forms and m-forms, in their most definite
senses, is merely one of perspective, but they both point to the same factual
material. We may therefore regard them as identical, when the interactions of
natural and temporal modal propositions are analyzed.
We thus have 18 natural modality forms and 18 frequency forms, or a total
of only 30 forms, according to our perspective. We may deal with the two
modalities as separate phenomena, or as part of the same continuum of modality.
The interrelationships between these various forms will be much clarified by
oppositional analysis.
The concept of distribution of terms, which was developed in the context
of Aristotelean logic, can be broadened to apply to modality. It has been found
a useful doctrine, often aided by pictorial representations, for understanding
the workings of arguments, and its utility would be increased. We defined a term
as being distributive if, as a result of the structure of the proposition, it
was found to be referring to all the instances of the class concerned;
otherwise, the term was being used undistributively. Now, this concerns
quantity, the extensional type of modality, and could be called extensional
distribution.
We could then by analogy consider a term as naturally distributive if it
was being referred to under all conditions, and naturally undistributive if the
reference was dependent on circumstance. Likewise, temporal distribution would
indicate reference to all or some of the times concerning a term. The following
properties can then be formulated.
a.
Whatever the polarity, concerning the subject: universals are
extensionally distributive; but particulars are not; necessaries are naturally
distributive, but not potentials; constants are temporally distributive, but not
so temporaries.
b.
The predicates of negatives are distributive in all three senses, whereas
those of affirmatives are in all senses undistributive.
Thus, a given proposition may be distributive of this or that term in one
sense, but not in another. In this way, we can explain why a certain inference
is possible, or why another is not. This is not a very important doctrine, but,
as already stated, a useful tool. |
