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FUTURE LOGIC

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

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CHAPTER 42. CONDENSED PROPOSITIONS.

Conditional propositions provide us with a powerful formal language, enabling us to elucidate a large variety of derivative forms we commonly use.

1. Forms with Complex Terms.

2. Making Possible or Necessary.

1. Forms with Complex Terms.

We are now in a position to consider ‘condensed’ propositions, which have a conjunctive or disjunctive subject and/or predicate. These propositions are made to appear like single categoricals, through this device of complex terms, but in fact conceal two or more standard propositions.

a. The ‘subject’ may be a conjunction of subjects. Thus, ‘Every S1 and S2 is P’ normally means that whatever is {both S1 and S2}, is P, without implying (nor denying) that something S1 but not S2, or something S2 but not S1, satisfies the condition for being P.

Note in passing that complex propositions like ‘all ABC are XYZ’ are often used in practise, because we try to abbreviate a multiplicity of relations in a minimum of words. Such a form implies smaller statements like ‘some A are B’ or ‘some A are X’ or ‘some X are Y’. Such implicits may be the precise premises of categorical syllogisms, rather than the more complex form which is presented as a premise. This explains why theoretical logic may seem so much more bare than practical examples. For instance, ‘my computer sounds like a duck’ contains many smaller statements like: ‘I have a computer’ or ‘my computer makes sounds’.

b. The ‘subject’ may be a disjunction of subjects. Thus, ‘Every S1 or S2 is P’ normally means that every S1 is P and every S2 is P, without telling us whether anything may or may not be S1 but not S2, or S2 but not S1.

c. The ‘predicate’ may be a conjunction of predicates. Thus, ‘Every S is P1 and P2’ means that every S is P1 and every S is P2.

d. The ‘predicate’ may be a disjunction of predicates. Thus, ‘Every S is P1 or P2’ normally means that some S are P1 and some S are P2, without telling us whether or not any S is both P1 and P2.

However, other interpretations of some of these forms are feasible. Firstly, our interpretation depends on whether the ‘or’ is understood as an ‘either-or’ or as an ‘and/or’ or as an ‘or-else’, and on how definite these disjunctions are. Secondly, our interpretation depends on the type of modality intended: is the ‘or’ intended to convey an extensional disjunction, or a natural or temporal one, or even a problemacy?

Form (a) admits of singular or particular versions ‘this/some {S1 and S2} is/are P’. Note that we often in practise intend the conjunction more loosely, so that we really mean the same as form (b).

Form (b) was above understood extensionally, so that the predicate was dispensed to both subjects generally. In that case, there are no corresponding singular or particular versions. However, we can say ‘this/some S is/are {P1 or P2}’, if we regard the disjunctive clause as a whole, rather than the disjuncts, as the subject. In this case, how the ‘or’ is understood, and the type of modality involved, becomes more variable.

Form (c) is the least ambiguous of them all, and readily admits of singular or particular versions ‘this/some S is/are P1 and P2’.

Form (d) was above understood extensionally, so that both predicates were dispensed to the subject particularly. In that case, there are no corresponding singular or particular versions. However, we can say ‘this/some S is/are {P1 or P2}’, if we regard the disjunctive clause as a whole, rather than the disjuncts, as the predicate. In this case, how the ‘or’ is understood, and the type of modality involved, becomes more variable.

To illustrate alternative interpretation, consider the form ‘Every S is P1 or P2’ again. If we wanted our ‘or’ to suggest that S may be split into two groups S1, S2, such that no S1 are S2, and all S1 are P1 and all S2 are P2, it would not suffice for us to say that ‘some S are P1, and all other S are P2’. We would have to make use of extensional conditionals, as follows: ‘any S which is P1, is not P2; and any S which is P2, is not P1’.

This last form is important because it introduces fractionating of a subject, which topic will be dealt with in more detail later.

Similarly, natural conditionals may be used to express other interpretations, such as ‘when any S is not P1, it is P2; and when any S is not P2, it is P1’. And likewise with temporal modality. We can even understand the ‘or’ in a logical sense, even as a mere problemacy; for instance, ‘if all S are P1, no S is P2; and if all S are P2, no S is P1’.

Thus, we see that the ambiguities of such condensed forms are dealt with through the instrument of our more precise conditional forms in each type of modality, and we do not need to develop a new logic for them (except as an exercise).

The condensed forms presented above were all affirmative and actual. We may similarly analyze negative forms, like ‘No S1 and S2 is P’, or modal forms, like ‘this S must be P1 or P2’. Note in passing that exceptive propositions, like ‘all S but S1 are P’, can be similarly analyzed. Note also that conditional propositions may also involve complex terms, and are similarly analyzable in a multitude of ways.

I will not go into such detail, however: I must move on; the job is left as an exercise for the reader, and for other logicians. It is clear, in any case, that the same principles apply.

In conclusion, it should have become obvious by now that the issue of complex terms, involving a conjunction or disjunction of subjects or predicates, is ultimately an enlargement of the issue of modalities of subsumption. Also, it shows clearly that the distinction between categorical and conditional forms, is ultimately somewhat arbitrary; there is a continuum of forms running into each other.

2. Making Possible or Necessary.

Another, unrelated, family of forms which condense conditionals, can be mentioned at this juncture: making possible or making necessary. These relate to causality. Here again, the concepts involved can be applied to any type of modality.

‘P makes Q possible’ signifies, in logical modality, that if nonP, then nonQ (nonP and nonQ are possible, and ‘nonP and Q’ is impossible), whereas if P, not-then nonQ (P and Q are possible). This commutes to ‘Q is made possible by P’. Clearly, only on the condition of P being true, does the possibility of Q being true have an effective chance of arising; P is thus said to be an exclusive condition for Q, a sine-qua-non.

‘P makes Q necessary’ signifies, in logical modality, that if P, then Q (P and Q are possible, and ‘P and nonQ’ is impossible), whereas if nonP, not-then Q (nonP and nonQ are possible). This commutes to ‘Q is made necessary by P’. Clearly, the truth of P alone would raise the mere possibility (within a contingency) of Q’s truth to an effective necessity (more precisely, it is Q’s realization, rather than Q itself, which becomes necessary); P is thus said to be a sufficient condition for Q.

These concepts ‘making possible’ and ‘making necessary’ are obviously correlative. If P makes Q possible, then nonP makes nonQ necessary; and if P makes Q necessary, then nonP makes nonQ possible. Also, if P makes Q possible, then Q makes P necessary; likewise, if P makes Q necessary, then Q makes P possible.

We have other concepts of a similar nature. Thus, ‘P is necessary for (or to) Q’ means that without P, Q would not be true; which is equivalent to ‘P makes Q possible’. Again ‘P necessitates Q’ means that in order for P to be true, Q would be required to be true; which is equivalent to ‘P makes Q necessary’.

We can similarly analyze the concepts of ‘making impossible’ (implying prevention or inhibition) and ‘making unnecessary’.

All this can be duplicated in other types of modality. Thus, in natural modality, ‘When this S is not P, it cannot be Q, and when this S is P, it can be Q’ implies that P makes Q potential in this S; also, ‘When this S is P, it must be Q, and when this S is not P, it can not-be Q’ implies that P makes Q (that is, Q’s actualization) a natural necessity in this S. Similarly with regard to temporal modality. In extensional modality, ‘Any S which is not P, is not Q, and some S which are P, are Q’ implies that P makes Q possible for S’s; also, ‘Any S which is P, is Q, and some S which are not P, are not Q’ implies that P makes Q necessary for S’s.

When issues of sequence arise, the possibility or necessity involved may be, if not simultaneous, precedent or subsequent in time.

These concepts obviously refer us to the various types and categories of aetiology. They allow us to begin a classification of causes. Within each modality, some causes are both exclusive (or necessary) and sufficient (or necessitating); some are only the one or the other; some are neither of these, but rather occasional (or contingent), meaning that they depend on additional partial conditions (a conjunctive antecedent) to effect the consequence. Many more subdivisions of causality are of course possible.

Clearly, a formal logic of causality can be derived from the logic of conditioning. Forms like ‘A being B causes C to be D’ are commonly used, and capable of precise analyses, by specifying the category and type of modality involved. I will not here develop this field of logic, since it is derivable; but it is important, and should be eventually done.

Many statements conceal this sort of form. In some cases, syllables like ‘en-‘ or ‘-fy’, meaning ‘to make’, are used to signal causality, as in ‘he verified the statement’, meaning ‘he (did something which) caused the statement to be (accepted as) true’. In some cases, the verb entirely hides the causal aspect, as for instance in ‘water dissolves these crystals’, the verb ‘dissolves’ means ’causes to dissolve’, and we can rephrase the whole more precisely as ‘when some water is mixed with these crystals, a solution is obtained’.

We enter here, also, into the subordinate realm of teleology, the study of needs in the context of given goals or purposes. For examples, with making possible. In logical pursuits: what must I prove first, before I can prove so and so? In natural or temporal causation: what should I do, in order to achieve so and so? In extensional choices: which of these things should I choose, to obtain so and so?

These forms play an important role in the formal logic of ethical modality. Granting certain standards of value, all the ways and means follow, with reference to objective aetiological (and teleological) relations. Something is permissible if compatible with all our ends, impermissible if incompatible with some of them; something is imperative if a sine-qua-non of some of our final causes, unimperative if not a sine-qua-non of any of them. A full study of ethical modality would have to analyze volition, and discuss the source of our ultimate norms. These issues are of course beyond the scope of the present treatise.

2016-08-20T07:21:21+00:00