These doubts led Russell (1912) to propose a multiple relation theory of judgment, to replace the standard two-place relational theory (which is discussed at length in section ). To use Russell's example, in judging that Desdemona loves Cassio, Othello stands, not in a binary relation to a proposition, but rather in a multiple or many-placed relation to Desdemona, loving, and Cassio. Othello's judgment is true when there is a fact of Desdemona loving Cassio and otherwise false. This theory, and its contemporary incarnations, is discussed in a supplementary document.

Suppose we have two propositions, p and q . Let P be the subset of S corresponding to p , and let Q be the subset of S corresponding to q . The subset corresponding to !p is the complement of the subset corresponding to p , P c . The subset corresponding to the proposition (p | q) is the union of the set corresponding to p and the set corresponding to q , (P ∪ Q) . The subset of S corresponding to the proposition (p & q) is the intersection of the set corresponding to p and the set corresponding to q , PQ . The set corresponding to the proposition (p → q) is (P c ∪ Q) . If P is a subset of Q , then