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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 35.
NATURALS CONDITIONALS: OPPOSITIONS AND EDUCTIONS.
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.
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.
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. |
