true of all (members of the domain) is true of some (members of the domain)", and "What is not true of all (members of the domain) is true of none (of the members of the domain)". The "dictum" appears in Boole 1854 a couple places: "It may be a question whether that formula of reasoning, which is called the dictum of Aristotle, de Omni et nullo, expresses a primary law of human reasoning or not; but it is no question that it expresses a general truth in Logic" (1854:4) But later he seems to argue against it: "[Some principles of] general principle of an axiomatic nature, such as the "dictum of Aristotle:" Whatsoever is affirmed or denied of the genus may in the same sense be affirmed or denied of any species included under that genus. ... either state directly, but in an abstract form, the argument which they are supposed to elucidate, and, so stating that argument, affirm its validity; or involve in their expression technical terms which, after definition, conduct us again to the same point, viz. the abstract statement of the supposed allowable forms of inference." But the first half of this "dictum" (dictum de omni) is taken up by Russell and Whitehead in PM, and by Hilbert in his version (1927) of the "first order predicate logic"; his (system) includes a principle that Hilbert calls "Aristotle's dictum" (x)f(x) → f(y) This axiom also appears in the modern axiom set offered by Kleene (Kleene 1967:387), as his "∀-schema", one of two axioms (he calls them "postulates") required for the predicate calculus; the other being the "∃-schema" f(y) ⊃ ∃xf(x) that reasons from the particular f(y) to the existence of at least one subject x that satisfies the predicate f(x); both of these requires adherence to a defined domain (universe) of discourse. Gödel's restricted predicate calculus To supplement the four (down from five; see Post) axioms of the propositional calculus, Gödel 1930 adds the dictum de omni as the first of two additional axioms. Both this "dictum" and the second axiom, he claims in a footnote, derive from Principia Mathematica. Indeed, PM includes both as ❋10.1 ⊦ ∀xf(x) ⊃ f(y) ["I.e. what is true in all cases is true in any one case" ("Aristotle's dictum", rewritten in more-modern symbols)] ❋10.2 ⊦∀x(p ⋁ f(x)) ⊃ (p ⋁ ∀xf(x)) [rewritten in more-modern symbols] The latter asserts that the logical sum