FUTURE LOGIC

14.MODAL OPPOSITIONS AND EDUCTIONS.

We have already encountered the oppositions of actuals or momentaries in classical logic. There is subalternation fromAtoR, toI; and fromEtoG, toO.AandE,AandG,RandE, are pairs of contraries;AandO,IandE,RandG, are pairs of contradictories,RandO,IandG,IandO, 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.

1.Quantification of Oppositions.

2.Basic Intramodal Oppositions.

3.Quantified Intramodal Oppositions.

4.Intermodal Oppositions.

5.Eductions.

1.Quantification of Oppositions.

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.

Diagram 14.1Quantification of Oppositions.

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 areimplicants, then all horizontal lines signify implicance, and all diagonals signify subalternation, downward. Proof for the horizontals: since it is given any pair of singular formss1,s2mutually imply each other, then any full or partial enumeration of such pairs, as inu1,u2, orp1,p2, will likewise mutually imply each other, provided the extensions involved are the same. For the diagonals: sinceu1impliess1, ands1impliess2, thenu1impliess2. Sinceu1,s1, implys2, ands2impliesp2, then they also implyp2. Likewise,u2,s2can be shown to implys1,p1.

If the singulars aresubalternative, 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 arecontradictory, then all lines labeled (2) signify contradiction, all lines labeled (3) contrariety, all lines labeled (4) subcontrariety. Proof for the upper square: given thats1ands2cannot both be true, then any enumerations which include them both, such asu1+s2,s1+u2, oru1+u2, cannot be both true (so, for instance, ifu1=T, thenu2=F; i.e. ifu1, then not-u2). Proof for the lower square: given thats1ands2cannot 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, thenp2). So far, we have proven the claimed contrarieties and subcontrarieties. But what of the contradictions ofu1+p2, orp1+u2? If we affirm such a pair, we do not necessarily thereby affirm a specifics1+s2pair true, but we do imply that some unspecified pair(s) ofs1ands2, referring to one and the same individual, would be posited together; this shows the incompatibility ofu1+p2, orp1+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-s2subsumed, against our given information.

If the singulars arecontrary, 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 aresubcontrary, 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.

2.Basic Intramodal Oppositions.

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).

Diagram 14.2Oppositions of Main Categories of Modality.

The above is equivalent to the figure of oppositions of the six quantities of Aristotelian 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 symbolsn,a,p, with subscripts+and–, we can say:n+impliesa+implies not{a-}, whereas not{n+} does not imply not{a+}, nor thereforea-, so thatn+anda-are contrary; or again, not{p+} implies not{a+} impliesa-, whereasp+does not implya+, nor therefore not{a-}, so thatp+anda-are subcontraries.

With regard to contingency; being defined as the sum of possibility and unnecessity, it subalternatesp+andp-, and is contrary ton+andn-. Incontingency, its negation, therefore means either necessity or impossibility, and is subalternated byn+andn-, and subcontrary top+andp-. 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, see remarks inAppendix 2.

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.

3.Quantified Intramodal Oppositions.

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.

Table 14.1Table of Oppositions in Natural Modality.

These relationships may be clarified by use of a truth-table. In the table below, given the truth of a proposition listed under column headingT, or the falsehood of one underF, we see the reactions, along the same row, of all other propositions, listed as column headings. This data follows from the preceding table.

Table 14.2Truth-Table for Natural Modality.

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

Needless to say, the easiest way to visualize and transmit all the above information is by means of a figure of opposition. However, since in this context the required diagram is three-dimensional, it is rather difficult to present on paper. Below is a sketch of it, but without the various lines of opposition. Note that the shaded planes have already been presented earlier, with all their lines of opposition shown. The reader can work out the remaining planes, with reference to the above two tables, or more radically to the principles of ‘quantification of oppositions’ developed earlier.

Diagram 14.3Figure of Oppositions of Natural Propositions.

Identical results are obtainable for temporal modality, substitutingcforn, andtforp, throughout.

4.Intermodal Oppositions.

Lastly, needing elucidation, is the inter-opposition of natural and temporal modalities. The following diagram shows the continuum of modality, including both natural and temporal types together. Parts of this diagram have already been presented, when we dealt with each modal type separately. But it is interesting to have an overview, anyway.

Diagram 14.4Oppositions between Modality Types.

This diagram concerns singulars. We know from our analysis of modality thatnimpliesc, which impliesaorm, which impliest, which impliesp, for either polarity; that is, the illustrated subalternations proceed from the meanings of the concepts involved. From these, and the already established intramodal oppositions, it is easy to infer the contrariety betweennandc, ornandt, forms of opposite polarity (upper diagonals), and the subcontrariety betweencandp, ortandp, forms of opposite polarity (lower diagonals).

These relationships between singulars can now be quantified by reference to the general rules of opposition, and the results tabulated as follows.

Table 14.3Table of Oppositions between Natural and Temporal Modalities.

In the table above, given the truth of a proposition listed under column heading T, or the falsehood of one under F, we see the reactions, along the same row, of all other propositions, listed as column headings.

5.Eductions.

The following is a list of the eductions possible from propositions with natural modality. The methods of validation used for these are similar to those developed for Aristotelean forms. That is, conceptual analyses and appeal to the laws of thought, in the cases of obversion and conversion; and reduction to these first two process, in the other cases.

a.Obversion(S-P to S-nonP).

S must be P implies S cannot be nonP; S cannot be P implies S must be nonP; S can be P implies S can not-be nonP; S can not-be P implies S can be nonP. These are true irrespective of the quantity (all/this/ some) involved; and the obverse has in all cases the same quantity as the obvertend. These results follow from the definitions of the concepts involved and the law of contradiction.

b.Conversion(S-P to P-S).

Affirmatives, be they necessary or potential, general or particular, all convert to a particular potential, Some P can be S, but no better. In the case of negatives, only No S can be P (En) is convertible, and that fully to No P can be S;Ep,Gn,Gp,On,Opare not convertible. These results can be established by consideration of the subsumptions of circumstance involved.

c.Obverted Conversion(S-P to P-nonS).

This process is applicable only to convertibles, which are then all obvertible. Thus, affirmatives all yield Some P can not-be nonS; andEnyields All P must be nonS.

d.Conversion by Negation(S-P to nonP-S).

This is obversion, followed by conversion. Thus, all originally negative propositions can be converted by negation, to yield Some nonP can be S. But of originally affirmative propositions, only All S must be P (An) can be so processed, to yield No nonP can be S;Ap,Rn,Rp,In,Ipare not convertible by negation.

e.Contraposition(S-P to nonP-nonS).

This requires conversion by negation, followed by obversion. Therefore, all negatives are contraposable, and that to Some nonP can not-be nonS. Whereas, in the case of affirmatives, onlyAncan be so processed, yielding All nonP must be nonS.

f.Inversion(S-P to nonS-nonP).

Of affirmatives, onlyAncan be so treated, by contraposing then converting it, to obtain Some nonS can be nonP. Of negatives, onlyEnis invertible, by converting then contraposing it, with the result Some nonS can not-be nonP.

g.Obverted Inversion(S-P to nonS-P).

This being inversion followed by obversion is applicable only to universal necessaries,Anyielding Some nonS can not-be P, andEnyielding Some nonS can be P.

We note, in conclusion, that onlyAnandEn(as well asAandE) can be subjected to all six of these processes.

Similar results can easily be established regarding propositions with temporal modality.