Notes on the Seven Fold Relations between Propositions

There are seven types of possible relation between propositions. Any two propositions will be related in any of the seven ways. These relations are (i) Independence, (ii) Equivalence,

iii) Contradictory, (iv) Contrary, (v) Sub-contrary, (vi) Sub-altern and (vii) Super-altern. Let us discuss each of these.

(i) Independence

Let P and Q be any two propositions such that neither the truth or falsity of P implies the truth or falsity of Q nor the truth nor falsity of Q implies the truth or falsity of P. Then P and Q are called independent. The relation in this case is called independence. For example, “All men are mortal’ and “All horses are animal” are clearly independent propositions as any one of them does not imply the other.

From the truth or falsity of one, the truth or falsity of the other cannot be inferred. We may note that in a valid deductive argument the conclusion cannot be independent of the premise or premises.

(ii) Equivalence

Two propositions P and Q are equivalent when they are so related that if P is true then Q is true and if Q is true then P is true; and if P is false, Q is false and if Q is false the P is false. If P and Q are equivalent propositions then it is impossible for P to be true and Q to be false or Q to be true and P to be false. For example, “No men are perfect” and “No perfect beings are men” are clearly equivalent to each other. The relation is called the relation of equivalence.

(iii) Contradictory

Two propositions are contradictorily related if the truth of one implies the falsity of the other and the falsity of one implies the truth of the other. In other words, P and Q are contradictorily related if and only if (i) the truth of P implies the falsity of Q and conversely and (ii) the falsity of P implies the truth of Q and conversely. Thus two contradictorily related propositions can neither be true together nor they can be false together. For example, “All men are mortal” and “Some men are not mortal” are contradictorily related. A proposition and its negation are contradictorily related.

(iv) Contrary relation (or contrariety)

Two propositions are contrarily related if the truth of one implies the falsity of the other but the falsity of one does not imply the truth of the other. Here both the propositions can be simultaneously false but they cannot be simultaneously true. ‘All S is P’ and ‘No S is P’ are contraries. Thus A and E-propositions with the same subject and same predicate are contrarily related. Here from the falsity of one we cannot derive the truth or falsity of the other.

(v) Sub-contrary relation (or Sub-Contrariety)

Two propositions are sub-contrarily related if and only if the falsity of any one of them implies the truth of the other but not conversely. Here both the propositions can be true together but they cannot be simultaneously false. For example, “Some men are rich” and “Some men are not rich” can be simultaneously true but they cannot be simultaneously false.

The falsity of one implies the truth of the other. Note that in case of contrary relation the truth of any one of the propositions implies the falsity of the other where as in case of sub-contrary relation the falsity of any one of the propositions implies the truth of the other. Further, for any proposition P and Q if P is contrarily related to Q then from the falsity of P or Q nothing can be inferred with respect to the truth or falsehood of the other.

(vi) Super alternation

Two propositions P and Q may be related in such a manner, that if P is true then Q is true but P is false then Q remains undecided. Then this relation is called super-alternation and P is superaltern to Q. This relation holds between A and I proposition and E and O proposition with the same subject and same predicate,

(Vii) Sub alternative

Two propositions P and Q may be related in such a manner that if Q is false then P is raise, but if Q is true then P remains undecided. This relation is called sub alternation. Here Q is filed subaltern to P. This relation holds between I and A proposition and also between O and E preposition with the same subject and same predicate. Here I-proposition is called subaltern to proposition and similarly, O-proposition in the subaltern to E-proposition.