14 Modal Oppositions & Eductions

We have already encountered the oppositions of actuals or momentaries in classical logic. There is subalternation from A to R, to I; and from E to G, to O. A and E, A and G, R and E, are pairs of contraries; A and O, I and E, R and G, are pairs of contradictories, R and O, I and G, I and O, are pairs of subcontraries. These relationships were shown to proceed from analysis of the forms' meanings and application of the laws of thought. In the wider context of modal logic, we are concerned with the oppositions of, not only these six forms, but 24 more.

Remember that subalternation is one-way implication, contradictories can neither be both true nor both false, contraries cannot be both true but may be both false, subcontraries may be both true but cannot be both false, and unconnecteds do not affect each others' truth or falsehood.

At this point, I would like show how, given a certain oppositional relation to exist between two singular propositions (s1, s2), referring to the same instance of the same subject-concept, we can systematically predict the oppositions involving one or two of the corresponding universal (u1, u2) and particular (p1, p2) forms. This doctrine may be called quantification of oppositions, meaning more precisely opposition of quantified forms. It allows us to introduce quantity into basic figures of opposition, such as that between the categories or types of modality which will presented in the next sections. Consider the following general-model figure of opposition.

Grant that we already know the subalternations, labeled (1), to be true, since universality includes singularity, which includes particularity. For any given opposition between singulars, labeled (2) horizontal, we need to discover the remaining lines of oppositions, namely (2) diagonal, (3), and (4). The following results are obtained.

If the singulars are implicants, then all horizontal lines signify implicance, and all diagonals signify subalternation, downward. Proof for the horizontals: since it is given any pair of singular forms s1, s2 mutually imply each other, then any full or partial enumeration of such pairs, as in u1, u2, or p1, p2, will likewise mutually imply each other, provided the extensions involved are the same. For the diagonals: since u1 implies s1, and s1 implies s2, then u1 implies s2. Since u1, s1, imply s2, and s2 implies p2, then they also imply p2. Likewise, u2, s2 can be shown to imply s1, p1.

If the singulars are subalternative, left implying right, then all horizontal or left down to right diagonals signify subalternation in that direction, and all right down to left diagonals signify unconnectedness. Proof: similar to previous case, though the relations involved here are unidirectional. Unconnectedness, of course, applies when no more finite opposition can be established.

If the singulars are contradictory, then all lines labeled (2) signify contradiction, all lines labeled (3) contrariety, all lines labeled (4) subcontrariety. Proof for the upper square: given that s1 and s2 cannot both be true, then any enumerations which include them both, such as u1 + s2, s1 + u2, or u1 + u2, cannot be both true (so, for instance, if u1=T, then u2=F; i.e. if u1, then not-u2). Proof for the lower square: given that s1 and s2 cannot both be false, then any enumerations which exclude them both such as not-p1 + not-s2, not-s1 + not-p2, or not-p1 + not-p2, cannot both be true (so, for instance, if not-p1 = true, then not-p2 = false; i.e. if not-p1, then p2). So far, we have proven the claimed contrarieties and subcontrarieties. But what of the contradictions of u1 + p2, or p1 + u2? If we affirm such a pair, we do not necessarily thereby affirm a specific s1 + s2 pair true, but we do imply that some unspecified pair(s) of s1 and s2, referring to one and the same individual, would be posited together; this shows the incompatibility of u1 + p2, or p1 + u2. Likewise, for the incompatibility of not-u1 + not-p2, or not-p1 + not-u2, there is bound to be some unspecified case(s) of not-s1 + not-s2 subsumed, against our given information.

If the singulars are contrary, then all lines labeled (2) or (3) signify contrariety, and all lines labeled (4) unconnectedness. Proof: see the relevant ('not both true') parts of the arguments above for contradiction.

If the singulars are subcontrary, then all lines labeled (2) or (4) signify subcontrariety, and all lines labeled (3) unconnectedness. Proof: see the relevant ('not both false') parts of the arguments above for contradiction.

These general rules of opposition can now be used in any context, saving us from having to deal with each case of quantification anew.

The following diagram concerns singular propositions only, and is designed to illustrate the relationships of the different categories of modality, whether of the natural type or of the temporal type (each type separately).

The above is equivalent to the figure of oppositions of the six quantities of Aristotelean propositions, and may be established by similar argument. The vertical, downward subalternations proceed from the definitions of the concepts involved; 'all' the circumstances or times includes any 'this one' we pick, and any specific 'this one' implies 'some' unspecified number.

The horizontal contradiction is simply the axiomatic presence and absence incompatibility. The diagonal contradictions between necessity and unnecessity, or impossibility and possibility, follow, on the basis that there would otherwise be individual circumstance(s) or time(s) which contained both presence and absence, or neither.

For the rest, the proofs are very mechanical consequences of the above. For example, using the symbols n, a, p, with subscripts + and -, we can say: n+ implies a+ implies not{a-}, whereas not{n+} does not imply not{a+}, nor therefore a-, so that n+ and a- are contrary; or again, not{p+} implies not{a+} implies a-, whereas p+ does not imply a+, nor therefore not{a-}, so that p+ and a- are subcontraries.

With regard to contingency; being defined as the sum of possibility and unnecessity, it subalternates p+ and p-, and is contrary to n+ and n-. Incontingency, its negation, therefore means either necessity or impossibility, and is subalternated by n+ and n-, and subcontrary to p+ and p-. Contingency and incontingency are both oppositionally unconnected to presence and absence. These relationships could be represented in a wedge-shaped diagram.

As for the oppositions of probability forms. Probability (most cases) subalternates improbability-not (few cases), and probability-not subalternates improbability; which is why we speak of degrees or levels of probability. By definition, probability (covering over half the times or circumstances) and improbability (half or less of them, let us say) are contradictory; likewise probability of negation and improbability of negation. Therefore, probability and probability-not are contrary, and their negations are subcontrary. These relations could be illustrated by a square.

Furthermore, necessity implies probability, and impossibility implies probability-not. Improbability implies unnecessity, and improbability-not implies possibility. It follows that high or low probability are contrary to necessity of opposite polarity, and subcontrary to possibility of opposite polarity. These wider relations are easily established.

We can view necessity as the highest form of probability. Also, probability, whether high or low, is merely a more defined form of possibility. If we express a more specific proportion of cases (e.g. 75% or 33%), we obtain sub-categories of probability. Lastly, of course, none of the probability forms are connected oppositionally to the presence/absence forms. Nevertheless, the whole idea of probability thinking is to try and predict the chances of realization of presence or absence.

If we take each of the oppositional relations between singulars of natural modality and quantify them with the general rules, we obtain the following table of opposition for all the forms of natural modality.

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