FUTURE LOGIC

CHAPTER 25.HYPOTHETICALS: OPPOSITIONS AND EDUCTIONS.

1.Connection and Basis.

2.Oppositions.

3.Hierarchy.

4.Eductions.

1.Connection and Basis.

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 logicalconnectionbetween 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 logicalbase(s)of the theses, or thebasisof 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.

2.Oppositions.

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 conjunctivesH1n,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, not-then 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, not-then Q’ and ‘If P, not-then 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, not-then 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 not-then) 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 not-then) Q’ and ‘If nonP, then (or not-then) 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 not-then) nonQ’ and ‘If nonP, then (or not-then) 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 truth-table, in which given the truth of a form under headingT, or the falsehood of one under headingF, the status of the others along the same row is revealed.

Table 25.2 Truth-Table for Opposing Hypotheticals.

(key: T = true, F = false, . = undetermined.)

3.Hierarchy.

The square of opposition shown in the previous section, you will notice, is the familiar one encountered for the categorical propositionsA,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, not-then notQ’ (note the double negation) is the lowercase form corresponding to the uppercase ‘If P, then Q’; likewise, ‘If P, not-then Q’ is the subaltern form of ‘If P, then notQ’, ‘If notP, not-then notQ’ is the subaltern form of ‘If notP, then Q’, and ‘If notP, not-then 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, not-then 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, not-then 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 inter-relationships 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, contingency-based hypotheticals, rather than hypotheticals with one or both theses incontingent. It applies to normal logic, rather than abnormal or general-case forms.

4.Eductions.

Here again, we will first consider normal hypotheticals, and then mention merely-connective hypotheticals and abnormals.

We need only, to begin with, deal with the primary hypothetical forms, ‘If P, then Q’ and ‘If P, not-then 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 (‘not-then’ 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 P-Q to P-nonQ.

b. Conversion.From P-Q to Q-P.

Not applicable.

c.Obverted Conversion.From P-Q to Q-nonP.

d. Conversion by Negation.From P-Q to nonQ-P. 

e.Contraposition.From P-Q to nonQ-nonP.

f. Inversion.From P-Q to nonP-nonQ.

Not applicable.

g.Obverted Inversion.From P-Q to nonP-Q.

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 not-nonP’ 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 contingency-based 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 not-nonQ’, rather than merely ‘If P, not-then 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 ‘then-not’ (and its negation, ‘not-then-not’), to complement ‘then’ (and ‘not-then’). But I think such multiplication of ‘nots’ is without value.