Permutation: (p ⋁ q) ⊃ (q ⋁ p) ❋1.5 Associative Principle: p ⋁ (q ⋁ r) ⊃ q ⋁ (p ⋁ r) [redundant] ❋1.6 Principle of [logical] Summation: (q ⊃ r) ⊃ ((p ⋁ q) ⊃ (p ⋁ r)) ❋1.7 [logical NOT]: If p is an elementary proposition, ~p is an elementary proposition. ❋1.71 [logical inclusive OR]: If p and q are elementary propositions, (p ⋁ q) is an elementary proposition. Russell sums up these principles with "This completes the list of primitive propositions required for the theory of deduction as applied to elementary propositions" (PM:97). Starting from these eight tautologies and a tacit use of the "rule" of substitution, PM then derives over a hundred different formulas, among which are the Law of Excluded Middle ❋1.71, and the Law of Contradiction ❋3.24 (this latter requiring a definition of logical AND symbolized by the modern ⋀: (p ⋀ q) =def ~(~p ⋁ ~q). (PM uses the "dot" symbol ▪ for logical AND)). Ladd-Franklin (1914): "principle of exclusion" and the "principle of exhaustion" At about the same time (1912) that Russell and Whitehead were finishing the last volume of their Principia Mathematica, and the publishing of Russell's "The Problems of Philosophy" at least two logicians (Louis Couturat, Christine Ladd-Franklin) were asserting that two "laws" (principles) of contradiction" and "excluded middle" are necessary to specify "contradictories"; Ladd-Franklin renamed these the principles of exclusion and exhaustion. The following appears as a footnote on page 23 of Couturat 1914: "As Mrs. LADD·FRANKLlN has truly remarked (BALDWIN, Dictionary of Philosophy and Psychology, article "Laws of Thought"), the principle of contradiction is not sufficient to define contradictories; the principle of excluded middle must be added which equally deserves the name of principle of contradiction. This is why Mrs. LADD-FRANKLIN proposes to call them respectively the principle of exclusion and the principle of exhaustion, inasmuch as, according to the first, two contradictory terms are exclusive (the one of the other); and, according to the second, they are exhaustive (of the universe of discourse)." In other words, the creation of "contradictories" represents a dichotomy, i.e. the "splitting" of a universe of discourse into two classes (collections) that have the following two properties: they are (i) mutually exclusive and (ii) (collectively) exhaustive. In other words, no one thing (drawn from the universe of discourse) can simultaneously be a member of both classes (law of non-contradiction), but [and] every single