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FUTURE LOGIC

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

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CHAPTER 13. MODAL PROPOSITIONS.

1. Categories and Types.

2. List and Notation.

3. Distributions.

1. Categories and Types.

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.

2. List and Notation.

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).

An

All S must be P

En

No S can be P

Rn

This S must be P

Gn

This S cannot be P

In

Some S must be P

On

Some S cannot be P

A

All S are P

E

No S is P

R

This S is P

G

This S is not P

I

Some S are P

O

Some S are not P

Ap

All S can be P

Ep

All S can not-be P

Rp

This S can be P

Gp

This S can not-be P

Ip

Some S can be P

Op

Some S can not-be P



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.

Ac

All S are always P

Ec

No S is ever P

Rc

This S is always P

Gc

This S is never P

Ic

Some S are always P

Oc

Some S are never P

A

All S are now P

E

No S is now P

R

This S is now P

G

This S is not now P

I

Some S are now P

O

Some S are not now P

At

All S are sometimes P

Et

All S are sometimes not P

Rt

This S is sometimes P

Gt

This S is sometimes not P

It

Some S are sometimes P

Ot

Some S are sometimes not P



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.

3. Distributions.

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.

2016-08-17T04:35:22+00:00