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FUTURE LOGIC

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

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CHAPTER 35. NATURALS CONDITIONALS: OPPOSITIONS AND EDUCTIONS.

1. Translations.

2. Oppositions.

3. Eductions.

1. Translations.

We may call ‘translation’ the reformulation of a proposition in other form, such as the change from conditional to categorical, or vice versa.

Natural conditionals are reducible to their categorical definitions, their implicit bases and connections, of course. Thus, for instance, ‘When any S is P, it must be Q’ implies ‘All S can be P’ and ‘No S can be P and not Q’, and therefore ‘All S can be P and Q’ and ‘All S can be Q’. These implications could be viewed as distinct immediate inferences, which are collectively though not individually reversible.

Another way to translate such natural conditionals into categoricals would be by joining the antecedent predicate to the subject, to form a new, narrower, subject. Thus, for instance, ‘When any S is P, it must be Q’ would become ‘All SP must be Q’. However, the new class ‘SP’ would have to be actual, or such a necessary categorical must be regarded as not implying actuality and so tacitly still conditional.

Modern logicians tend to regard all categoricals as involving a conditional subject, and so would regard such translation of conditionals into categoricals as formally true. However, I beg to differ with current opinion on this point. My contention is that, logically, there has got to be categoricals which are genuinely so, before we can build up conditional forms; categoricals are logically prior to conditionals, since the latter correlate the former.

Cases where the subject is not actual are only artificially categorical; they are made to seem so, but in fact are still conditional. (This argument also holds for imaginary subjects, where there is a hidden hypothesis ‘Though the subject is nonexistent, if it existed, so and so would follow’.)

Thus, the hidden conditionality in some categoricals is an exception, rather than the rule. The position taken by certain logicians to the contrary is not logically tenable, in my view. This issue is further discussed in the chapter on modalities of subsumption.

2. Oppositions.

The form ‘When this S is P, it must be Q’ means ‘this S can be P, but it cannot be P without being Q’, which implies that ‘this S can be P and Q’. It follows that the logical contradictory of this form is ‘This S cannot be P, or it can be P without being Q’, and not merely ‘This S can be P without being Q’. That is, ‘When this S is P, it can not-be Q’ is not formally contradictory, but only contrary; it is contradictory only if we take for granted that ‘This S can be P’.

Similarly, ‘When this S is P, it cannot be Q’ is on an absolute level merely contraried by ‘When this S is P, it can be Q’, and becomes contradicted only in such case as ‘This S can be P’ is already given.

On the other hand, the form ‘When this S is P, it must be Q’ implies that ‘When this S is P, it can be Q’, since the latter means no more than ‘This S can be P and Q’, which is the tacit basis of the former. Likewise, ‘When this S is P, it cannot be Q’ implies ‘When this S is P, it can not-be Q’.

It follows that ‘When this S is P, it must be Q’ and ‘When this S is P, it cannot be Q’ are invariably contrary to each other, since they imply each other’s contraries.

As for ‘When this S is P, it can be Q’ and ‘When this S is P, it can not-be Q’, they may be both be true, since ‘This S can be P, with or without Q’ occurs in some cases; and they may both be false, since it is conceivable that ‘this S neither can be P and Q, nor can be P and not Q’, as occurs in the case of ‘this S cannot be P’ being true. Thus, these two bases are normally neutral to each other, though if ‘This S can be P’ is granted, they become subcontrary.

With regard to actuality, ‘When this S is P, it must be Q’ does not imply, nor exclude, that ‘this S is P (and thereby Q)’, although ‘This S is P and Q’ does imply that ‘when this S is P, it can be Q’. Thus, the necessary form is ontologically a relationship which exists potentially, even when not actually operative. It is, of course, conceivable that ‘This S is P and Q’ in the actual circumstance but not in all circumstances, or in some circumstance(s) but not the actual one. The same can be said about the forms negating the consequent.

As for the parallel forms which negate the antecedent, their basis is different, namely ‘This S can not-be P and be (or not-be) Q at once’.

Therefore, ‘When this S is not P, it must be Q’ is compatible with ‘When this S is P, it must be Q’ (these together would imply that ‘this S must be Q’), and likewise with ‘When this S is P, it cannot be Q’ (in which case, we have a sine-qua-non situation every which way). All the more, the potential versions are all compatible. We need not, for our present purposes, go beyond this degree of detail.

These oppositions concern singular forms, note well; the corresponding oppositions for plural forms follow automatically, in accordance with the general rules of ‘quantification of oppositions’, which we dealt with in the chapter on opposition of modal categoricals. Thus, for example, ‘When any S is P, it must be Q’ is ordinarily contrary to ‘When certain S are P, they can not-be Q’; but if it is established that ‘All S can be P’, they become contradictory.

3. Eductions.

Eduction from conditionals consists in changing the position and/or polarity of antecedent and consequent.

a. With regard to actuals, suffices to say that ‘This S is P and Q’ and ‘This S is Q and P’ are, from our point of view, equivalent. For the rest:

Obversion obviously applies to all the forms, without loss of modality. Thus ‘When this S is P, it can be (or must be) Q’ imply ‘When this S is P, it can not-be (or cannot be) nonQ’; likewise, ‘When this S is P, it can not-be (or cannot be) P’ imply ‘When this S is P, it can be (or must be) nonQ’.

‘When this S is P, it can be or must be Q’ convert to ‘When this S is Q, it can be P’, since ‘This S can be P and Q’ is implicit basis of the source.

‘When this S is P, it can not-be or cannot be Q’ convert by negation to ‘When this S is not Q, it can be P’, since the latter target means ‘This S can be P and not Q’, which is given in the original proposition; note well, they are not convertible to ‘When this S is Q, it can not-be P’, since the source contains no basis for ‘This S can be Q and not P’.

These results are of course in turn obvertible.

We note that these simple eductions, other than obversion, yield a potential conclusion, even from a necessary premise.

b. However, a necessary conclusion may be drawn, if we are granted that the negation of the consequent is potential. This process may be called complex contraposition, and viewed either as a deduction from two premises, or as an eduction from a compound premise. The following is the primary valid mood:

When this S is P, it must be Q

and This S can not-be Q

hence, When this S is not Q, it cannot be P

The proof of this argument is by reduction ad absurdum. The denial of the conclusion implies either that ‘This S must be Q’ (base denied) or that ‘This S can be nonQ and P’ (connection denied); but either way this results in the denial of the minor or major premises; therefore, the conclusion is valid.

From this mood we may derive the following, by obversion:

When this S is P, it cannot be Q

and This S can be Q

hence, When this S is Q, it cannot be P

Thus, full contraposition is feasible, but only on the proviso that the basis of the conclusion is in advance given as true; without this additional information, it is not permissible. The reason for this is that the original conditional is in principle compatible with the categorical necessity of its consequent.

Note that the above arguments incidentally yield the conclusion that ‘This S can not-be P’. This may be viewed as modal apodosis from the given premises.

c. When quantity is introduced into all these equations, it is important to note that it is unaffected, unlike the modality. That is, a general natural conditional, is general for both the antecedent and consequent, implying that ‘all S can be Q’ as well as ‘all S can be P’.

So ‘When any S is P, it can be or must be Q’ converts to ‘When any S is Q, it can be P’, and ‘When any S is P, it can not-be or cannot be Q’ converts by negation to ‘When any S is not Q, it can be P’. Similarly, a particular premise is convertible, though to a particular conclusion.

Likewise, given that ‘All S can not-be Q’, the necessary ‘When any S is P, it must be Q’ contraposes to ‘When any S is not Q, it cannot be P’. Also, note well, when only one of the premises is general, whichever one — that is, given ‘When any S is P, it must be Q’ and ‘Some S can not-be Q’, or given that ‘When certain S are P, they must be Q’ and ‘All S can not-be Q’ — we can still infer that ‘When certain S are not Q, they cannot be P’ (and so that ‘Some S can not-be P’). However, if both premises are particular, contraposition is not permitted. Similarly, throughout, for propositions with negative predicates.

Derivative processes behave accordingly. For instance, inversion, being contraposition followed by conversion or vice-versa, requires two premises at least one of which is general, and always results in a potential conclusion.

Lastly note, these changes all essentially concern the predicates of natural conditionals. We might additionally have considered changes affecting the subject, such as conversions within the antecedent or consequent clause. But the idea seems somewhat artificial in this context, unlike in hypotheticals.

2016-08-05T10:52:11+00:00