A Note on Nicod’s Postulate is a short technical paper in formal logic rather than a full-length book. Its subject is the reduction of the elementary propositional calculus to a single axiom and a single rule of inference by means of the Sheffer stroke, the connective of alternative denial. Nicod had established that the whole of Russell’s elementary propositional calculus can be derived from one axiom, expressed entirely with the Sheffer stroke, together with a single rule of inference. Russell had merely mentioned that such a derivation was possible, and Nicod himself gave only brief indications of how it was to be carried out. The paper takes as its starting point J. Jörgensen’s A Treatise of Formal Logic, which, according to Quine, was the first work to set out the derivation in detail.
Quine’s note is primarily a close examination of Jörgensen’s execution of that derivation. It does not propose a new logical system or a new foundation for logic; instead, it scrutinizes an existing formal deduction and corrects it. Quine shows that Jörgensen’s derivation contains two errors. Both errors occur in the derivation of the so-called principle of identity, \(t/(t/t)\), which is the formula that serves as the basis for the further steps of the deduction. Quine argues that these mistakes arose because Nicod’s somewhat obscure indications were misunderstood. Having identified the two faulty steps, Quine demonstrates that the resulting gap can be filled in a rigorous and sound way, so that the deduction as a whole retains its validity. The note thus functions as a correction and repair of a published derivation rather than as an independent exposition of Nicod’s postulate.
The logical background is the Russellian elementary propositional calculus and the project of deriving it from a single axiom. Nicod’s axiom, written with the Sheffer stroke, is a single formula from which, with the accompanying rule, the standard truths of propositional logic are meant to follow. The interest of the note lies in the fact that the derivation is not merely stated but actually carried through in detail, and in Quine’s insistence that the details be formally correct. Because the principle of identity \(t/(t/t)\) is used as a foundation for later derivations, an error at that point would threaten the entire deduction; Quine’s repair is therefore essential to the claim that the deduction is valid. The paper belongs to the literature on the Sheffer stroke, on the single-axiom basis for the propositional calculus, and on the formal correctness of derivations in symbolic logic. It is a focused piece of logical criticism and reconstruction, concerned with the validity of a specific formal proof rather than with broader philosophical questions about logic.
