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The Logician © Avi Sion All rights reserved

FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 25.
HYPOTHETICALS: OPPOSITIONS AND EDUCTIONS.
We defined positive and negative hypothetical propositions in terms of
the logical impossibility or possibility, respectively of a certain conjunction.
This phenomenon refers to the logical connection
between the theses concerned. Taken by itself, such a relation does not require
that the theses be more than problematic; we need not know whether each of them
is contingent, necessary or impossible.
However, in everyday discourse, we commonly regard the logical modality
of the theses as tacitly, mutually understood. That is, we take for granted that
the respondent has the same idea as the speaker with regard to the contingency,
necessity or impossibility of each of the theses. This phenomenon refers to the
logical base(s) of the theses, or the basis
of a hypothetical proposition.
Normally,
in most cases we ordinarily encounter, this underlying modality is logical
contingency, for both the theses. Abnormally, in rare cases of a usually philosophical nature, the
modality of one or both of the theses is found to be logical necessity or
impossibility. For this reason, we may refer to two broad classes of
hypotheticals, the normal or the abnormal.
As we shall see, hypotheticals behave according to different logics.
'Baseless' hypotheticals, those with a problematic basis, representing only
various connections, without specifying the logical modality of the theses —
display what may be called the general or absolute or unconditional behavior
patterns. Normal hypotheticals, which have contingent bases, and abnormal
hypotheticals, which have one or both theses incontingent, each display slightly
different patterns, their own particular or relative or conditional patterns.
Thus, we could develop considerably different logics for each variety of
hypothetical. In this volume, we will try to highlight the main features of
hypothetical logic, sometimes for unspecified basis, sometimes for specified
bases, normal (fully contingent) or abnormal (partly or fully incontingent), as
appropriate.
Note that we could similarly regard conjunctions as having a variety of
bases. The logics would parallel those of hypotheticals of specified bases.
a.
The absolute oppositions, between the forms of hypothetical proposition
whose bases are unspecified, proceed from the definitions of connections as
modal conjunctions. They are identical to the oppositions between the
conjunctives H1n, H2n, H3n, H4n,
K1p, K2p, K3p,
K4p, which we discussed in a previous chapter.
Here, our purpose is to identify the oppositions between hypotheticals,
especially in cases where the logical modality of the theses is more
specifically known. We will first deal with merely connective and/or normal
hypotheticals, for which the theses may be assumed both contingent, and
thereafter consider some of the differences in oppositional properties for
abnormal hypotheticals.
b.
Normal hypotheticals are opposed as follows. Note well the unstated
condition that the theses are logically contingent. Let us consider, to begin
with, the four forms with a common antecedent P. Diagram
25.1 Square of Opposition for
Hypotheticals with Common Antecedent.
Since 'If P, then Q' and 'If P, notthen Q' inform that the conjunction
'P and nonQ' is, in the former case, impossible, and, in the latter case,
possible, they are contradictory. Likewise for the other diagonal.
The contrariety of 'If P, then Q' and 'If P, then nonQ' is obtained by
supposing them both true; in that case, if P was true, Q and nonQ would be both
true; therefore, these hypotheticals are incompatible; on the other hand,
supposing them both false yields no impossible result.
The subcontrariety of 'If P, notthen Q' and 'If P, notthen nonQ'
follows, since if they were both false, their contradictories would be both
true, though incompatible; on the other hand, supposing them both true yields no
impossible result.
Finally, if 'If P, then Q' is true, then 'If P, then nonQ' is false, by
contrariety; then 'If P, notthen nonQ' is true, by contradiction; whereas
nothing can be shown concerning the latter if 'If P, then Q' is false; so their
subalternative relation (downward) holds. The other subalternation can be
likewise shown.
A similar square of opposition can be demonstrated for the forms with a
common antecedent nonP, namely, 'If nonP, then (or notthen) Q (or nonQ)'. We
can show that hypotheticals with a common consequent Q, but different
antecedents, P or nonP, fall into such a square of opposition, by contraposing
the forms (see next section on eduction). Likewise, if the common consequent is
nonQ, of course.
However, concerning propositions whose antecedents and consequents are
both different, namely, 'If P, then (or notthen) Q' and 'If nonP, then (or
notthen) nonQ', the same cannot be said. For their definitions as impossibility
(or possibility) of the conjunctions 'P and nonQ' and 'nonP and Q',
respectively, leave them quite compatible, and unconnected. Likewise, for
opposite pairs of the forms 'If P, then (or notthen) nonQ' and 'If nonP, then
(or notthen) Q'
The oppositions of the eight forms of hypothetical could be illustrated
by means of a cube. However, the following tables summarize all these results
for us, just as well. (The numbering of forms and symbols for oppositions used
in these tables is arbitrary.) Table
25.1
Table
of Oppositions between Hypotheticals.
These relationships may be clarified by means of a truthtable, in which
given the truth of a form under heading T,
or the falsehood of one under heading F,
the status of the others along the same row is revealed. Table
25.2 TruthTable
for Opposing Hypotheticals. (key: T = true, F = false, . = undetermined.)
The
square of opposition shown in the previous section, you will notice, is the
familiar one encountered for the categorical propositions A, E, I,
O. The analogy is not accidental. The
contrariety between 'If P, then Q' and 'If P, then nonQ' is obviously similar in
meaning to that between 'All S are P' and 'All S are nonP', and the diagonal
contradictions can also obviously be likened.
This analogy suggests that normal positive and negative hypotheticals
constitute a hierarchy, the former being 'uppercase'
forms similar to general propositions and the latter 'lowercase' forms similar
to particulars. Indeed, this is implicit in the definitions of hypotheticals.
Thus, 'If P, notthen notQ' (note the double negation) is the lowercase
form corresponding to the uppercase 'If P, then Q'; likewise, 'If P, notthen Q'
is the subaltern form of 'If P, then notQ', 'If notP, notthen notQ' is the
subaltern form of 'If notP, then Q', and 'If notP, notthen Q' is the subaltern
form of 'If notP, then notQ'. Each positive hypothetical includes the negative
hypothetical with like antecedent and unlike subsequent (i.e. consequent or
inconsequent).
This uppercase/lowercase classification will be found useful in
understanding of much hypothetical inference. By expressing the form 'If P, then
Q' as a generality 'All P occurrences are Q occurrences', and the form 'If P,
notthen notQ' as a particular 'Some P occurrences are Q occurrences', we will
be able to understand why, for instance, the major premise in first figure
hypothetical syllogism must be uppercase, and cannot be lowercase.
Now, what of the oppositions between the eight hypotheticals and the four
factual conjunctions referred to in their definitions? First, we note that any
pair of the four conjunctions are opposed to each other in the way of
contraries; that is, they cannot be both true, but may be both false.
Secondly, we know that each uppercase hypothetical form is contrary to
the conjunction which it denies as possible by definition; it is oppositionally
neutral to the remaining three conjunctions, since, taken as a pair with any one
of them, they may be both true or both false without problem. Thirdly, each
lowercase form is subaltern to (implied by) the conjunction which it affirms as
possible by definition; and unconnected oppositionally to the other
conjunctions.
From this we may conclude that while, for example, 'P and Q' implies 'If
P, notthen notQ', in the same way as a singular categorical implies a
particular, the analogy stops there. For 'P and Q' is not in turn implied by 'If
P, then Q', as analogy would require. That is, the conjunctions are not exactly
'middle case' forms, between the upper and lower cases.
This discussion of course serves to clarify the interrelationships of
the categories of logical modality. Uppercase is logical incontingency,
lowercase is logical possibility or unnecessity; and conjunction is plain fact,
lying in between. It concerns, of course, contingencybased hypotheticals,
rather than hypotheticals with one or both theses incontingent. It applies to
normal logic, rather than abnormal or generalcase forms.
Here again, we will first consider normal hypotheticals, and then mention
merelyconnective hypotheticals and abnormals.
We need only, to begin with, deal with the primary hypothetical forms,
'If P, then Q' and 'If P, notthen Q', as our source propositions, to elucidate
the processes. What is found valid for these, is mutadis mutandis applicable to
forms involving 'nonP' and/or 'nonQ' as the source theses. The educed
hypotheticals may have different polarity ('notthen' instead of 'then', the
reverse never occurs), may involve the antithesis of one or both of the original
theses as a new thesis, and may switch the positions of the theses. The valid
processes are: a.
Obversion.
From PQ to PnonQ.
b.
Conversion.
From PQ to QP. Not
applicable. c.
Obverted Conversion. From
PQ to QnonP.
d.
Conversion by Negation.
From PQ to nonQP.
e.
Contraposition. From PQ to
nonQnonP.
f.
Inversion.
From PQ to nonPnonQ. Not
applicable. g.
Obverted Inversion.
From PQ to nonPQ.
The primary process here is (e) contraposition. These eductions are
validated by reference to the forms' definitions. Since 'If P, then Q' means
that the conjunction 'P and nonQ' is impossible, and 'If nonQ, then nonP' that
'nonQ and notnonP' is impossible, and these two conjunctions are equivalent, it
follows that the two hypotheticals involved are also equivalent.
The same can be said with regard to the negative forms: they are defined
by the same possibility of conjunction, and therefore equal. Contraposition is
therefore a reversible process, and applicable as described to all hypotheticals
without loss of power.
This process applies to unspecific hypotheticals and abnormals, as well
as to contingencybased normals, because it only requires for its validity the
connection implied by the defining modal
conjunction.
The other processes, however, are only applicable to normal positive
hypotheticals, if at all, and always yield a weaker, negative result. These
processes are only applicable to normal hypotheticals, because they presume that
the theses are to be understood as both logically contingent.
They are proved by reductio ad absurdum, combining the source proposition
with the contradictory of the target proposition, to yield an inconsistency, in
some cases after some contraposition(s).
Thus, given 'If P, then Q' to be true, (a) if 'If P, then nonQ' was true,
it would follow that P implied both Q and nonQ, an absurdity, therefore the
stated obverse must be valid; (c) if 'If Q, then nonP' was true, we could
contrapose it and obtain the same absurdity, therefore the stated obverted
converse must be valid; (d) if 'If nonQ, then P' was true, it would follow,
after contraposing 'If P, then Q', that nonQ implied both P and nonP, an
absurdity, therefore the stated converse by negation must be valid; and (g) if
'If nonP, then Q' was true, we could contrapose both it and the source
proposition, and obtain the same absurdity, therefore the stated obverted
inverse must be valid.
All processes with theses P, Q in the source propositions, excluded from
the above list, cannot be likewise validated, and so are invalid.
By substituting the antitheses of P and/or Q in the above validated
processes, we get the following full list of possible eductions, which is useful
for reference purposes. a.
Obversions.
b.
Conversion. Not applicable. c.
Obverted Conversions.
d.
Conversion by Negations.
e.
Contrapositions.
f.
Inversion. Not applicable. g.
Obverted Inversions.
A final comment. We may observe in the above that obversion of uppercase
hypotheticals merely yields the corresponding lowercase form, so such eduction
yields no more than the oppositional inference of a subaltern.
We could have regarded the obverse of 'If P, then Q' to be 'If P, then
notnonQ', rather than merely 'If P, notthen nonQ'. This would obviously be
correct, and analogous to the obversion of 'All S are P' to 'No S are nonP'.
Effectively, we would be introducing a relational operator 'thennot' (and its
negation, 'notthennot'), to complement 'then' (and 'notthen'). But I think
such multiplication of 'nots' is without value.
