Two opposed propositions observe the principle of the exclusion of the co-truth of the contraries if they cannot be simultaneously true but they can be simultaneously false, thus satisfying the function of anticonjunction: D[T.sub.p][T.sub.q] = [F.sub.Dpq]; the principle of the exclusion of the subcontrary co-falsity if they cannot be false simultaneously and from the same point of view, but they can be simultaneously true, thus satisfying the disjunction matrix: A[F.sub.p] [F.sub.q] = [F.sub.Apq]; the principle of the exclusion of the co-valency of the contradictories if they can simultaneously be neither true, nor false, and therefore they meet the requirements of the strong disjunction: J[T.sub.p][T.sub.q] = J[F.sub.p][F.sub.q] = [F.sub.Jpq].
As prescriptions, logical principles determine the value or the result of an operation; the reunion of two contradictory notions is exhaustive (it coincides with the universe of discourse); the conjunction of contraries cannot be a true sentence, the intersection of two subcontrary notions cannot be an empty set; the conjunction of two contrarily-subcontrary propositions is (a) false (sentence); etc.
The contradiction between the statements "the soul is mortal" and "the soul outlives the body" will be read as a contrary opposition in the stated order, and as subcontrary opposition in the reversed order.
Two generic, or truth-fluctuating, propositions, now true, now false, seem to us compatible with one of the following relationships: interference (the weakest conditioning, neither necessary nor sufficient); subalternation (the sufficiently-necessary conditioning); superalternation (the necessarily-sufficient conditioning); equivalence (the necessary and sufficient conditioning); nonrelationality (the weakest opposition, neither contrary nor subcontrary); contrariety (the contrarily-contrary opposition); subcontrariety (the subcontrarily-subcontrary opposition); contradiction (the contrary and subcontrary opposition).
The terms contrary, subcontrary, necessary, sufficient qualify the alethic circuit between the correlated sentences, respectively the transition from true to false, from false to true, from false to false and from true to true.
Singling out the twofold "contrary" relation that figures so prominently in The Marriage leaves us to consider the subcontrary pair that remains.
But to affirm a pair of subcontrary propositions is to simultaneously deny or "contradict" both of a pair of contrary propositions.
Placing Urizen and his contrary Luvah/Orc at either end of the cross' long vertical axis would relegate the other two zoas to the shorter horizontal that Urizen obliquely and dismissively characterizes as "weakness." To imagine the relationship between these two, Tharmas and Urthona, as a "weaker" even subcontrary sort of "opposition" makes a certain sense.
For to a rigorous proponent of Aristotelian logic this subcontrary sort of relation that affirms both "life" and "death" necessarily denies the corresponding contraries, not just the possibility of Eternal death but of life Eternal as well.
Finally, just as "S must be P" cannot coexist with its contrary, "S cannot be P," but "S may be P" can coexist with its subcontrary
, "S may not be P," so "I must do X" cannot coexist with its contrary, "I cannot do X"; but "I may do X" can coexist with its subcontrary
, "I do not have to do X."
Formulas 4 and 5 are not contradictory either; they are subcontrary. Arminius, however, seems to have read the two complete sequences as mutually exclusive.
The two lowermost cornerpoints form subcontrary pairs; they can both be true, but not both be false.