The Hexagon of Opposition

From John Sanders:

First, we should note that while a determinate future can be expressed on the Square by the single propositions “S will obtain” and “S will not obtain,” there is no single proposition expressing future indeterminacy. To express this third possibility, we must conjoin the two subcontraries “might” and “might not.” In other words, determinacy (“will” and “will not”) is given primitive status on the Square, while indeterminacy must be inferred. This asymmetry between determinacy and indeterminacy perhaps explains why “might” and “might not” have tended to be understood exclusively in terms of their individual subaltern relations to “will” and “will not.” That is, while “will” and “will not” have been allowed to express states of affairs, “might” and “might not” have tended to be limited to expressing merely the epistemological preconditions of those two determinate states. If it is true that “S will obtain,” it must also be true that “S might obtain,” viz. it must be possible for S to obtain. So too, for it to be true that “S will not obtain,” it must also be true that “S might not obtain,” viz. it must be possible for S not to obtain. But what has not been adequately appreciated in the western tradition is that the subcontraries “might” and “might not” may be conjointly true and the contraries “will” and “will not” conjointly false. In this case, “might” and “might not” are no longer related as subalterns to “will” and “will not.” Rather, when they are conjointly true, they have the same relation to “will” and “will not” that “will” and “will not” have to each other. In other words, they express a third distinct possibility –future indeterminacy – that stands in a contrary relationship to both the positive future determinacy expressed by “will” and the negative future determinacy expressed by “will not.” For any possible future state of affairs, one of the three – “will,” “will not” and “might and might not” – must be true and the other two false. But, because “might” and “might not” must be conjoined to play this third, indeterminate, contrary role, the possibility of their playing this role has been largely overlooked. Consequently, the possibility that the future is in some respects indeterminate and known by God as such has been largely overlooked.

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