Dubislav and Classical Monadic Quantificational Logic

Abstract

Among the members of the Berlin group, the scholar most devoted to the philosophy of logic and mathematics was Walter Dubislav (1891–1937). In a paper written in 1929, titled “Elementarer Nachweis der Widerspruchslosigkeit des Logik-Kalküls” (“Elementary proof of the consistency of the calculus of logic”), Dubislav made use of a procedure of “quasi truth valuation” (suggested by Emil L. Post in 1921) that was meant to prove the consistency of quantificational logic. Dubislav then employed this procedure as a “purely calculatory criterion for definitions” in the third edition (1931) of his monograph Die Definition. In a book written in 1932, entitled Die Philosophie der Mathematik in der Gegenwart (Contemporary Philosophy of Mathematics), Dubislav claimed that his approach provides a decision procedure for quantificational logic, explaining its application via examples from classical monadic quantificational logic. Whereas the decidability of the latter had been known since 1915, the undecidability of the full range of quantificational logic could not have been known to Dubislav in 1932. The fact that Dubislav’s procedure does not qualify as a decision procedure for even the monadic case was, after some cautious criticism raised in the 1930s, clearly stated for the first time in Reichenbach’s Elements of Symbolic Logic in 1947. This paper offers an elementary and detailed elucidation of the history surrounding these developments, and of their theoretical background.