thing (in the universe of discourse) must be a member of one class or the other (law of excluded middle). Post (1921): The propositional calculus is consistent and complete As part of his PhD thesis "Introduction to a general theory of elementary propositions" Emil Post proved "the system of elementary propositions of Principia [PM]" i.e. its "propositional calculus" described by PM's first 8 "primitive propositions" to be consistent. The definition of "consistent" is this: that by means of the deductive "system" at hand (its stated axioms, laws, rules) it is impossible to derive (display) both a formula S and its contradictory ~S (i.e. its logical negation) (Nagel and Newman 1958:50). To demonstrate this formally, Post had to add a primitive proposition to the 8 primitive propositions of PM, a "rule" that specified the notion of "substitution" that was missing in the original PM of 1910. Given PM's tiny set of "primitive propositions" and the proof of their consistency, Post then proves that this system ("propositional calculus" of PM) is complete, meaning every possible truth table can be generated in the "system": "...every truth system has a representation in the system of Principia while every complete system, that is one having all possible truth tables, is equivalent to it. ... We thus see that complete systems are equivalent to the system of Principia not only in the truth table development but also postulationally. As other systems are in a sense degenerate forms of complete systems we can conclude that no new logical systems are introduced." A minimum set of axioms? The matter of their independence Then there is the matter of "independence" of the axioms. In his commentary before Post 1921, van Heijenoort states that Paul Bernays solved the matter in 1918 (but published in 1926) – the formula ❋1.5 Associative Principle: p ⋁ (q ⋁ r) ⊃ q ⋁ (p ⋁ r) can be proved with the other four. As to what system of "primitive-propositions" is the minimum, van Heijenoort states that the matter was "investigated by Zylinski (1925), Post himself (1941), and Wernick (1942)" but van Heijenoort does not answer the question. Model theory versus proof theory: Post's proof Kleene (1967:33) observes that "logic" can be "founded" in two ways, first as a "model theory", or second by a formal "proof" or "axiomatic theory"; "the two formulations, that of model theory and that of proof theory, give equivalent results"(Kleene