Abstract

Ushenko presented his philosophy of logic in vehement opposition to "the postulationist theory." In the endeavor to amputate logic from philosophy and absorb it within mathematics, the postulationists viewed logic as an isolated object-logic to be discussed in meta-logic and construed its symbolic formulas as a game played according to arbitrarily established rules. The objections Ushenko raised are no longer novel, but twenty years ago the entire controversy was new. Above all, he stressed the numerous difficulties entangling the meta-logic. He scored the menace of an infinite regress of meta-logics, and insisted that the consequences of Gödel's work necessarily frustrate the initial great expectations of the postulationists. No purely formal system can be internally proved to be self-consistent and certainly no formal language can ever become as comprehensive as English, though the price of such comprehensiveness is the inevitable occurrence of contradictory sentences. Moreover, he argued that, despite the postulationists' pretense that rules like the principle of non-contradiction were mere conventions for playing the game of logic, these rules proved ubiquitous by trespassing from the object-logic and intruding into the ultimate reaches of the meta-logic.