Incomplete translations of complete logics.
Stewart B. Shapiro · Notre Dame Journal of Formal Logic · 1977
Let J and K be sets of (interpreted) logical primitives and let LJ and LK be languages based on J and if respectively, but having a common set of variables and non-logical constants.Let £Jbe a logic on LJ.Suppose t is a function which carries formulas of LJ into logically equivalent formulas of LK.It has been known since at least 1958 [6] that the completeness of the logic on LK ( -> v ί Crossley thought that ^[Ί, &, 3] is complete, but it is not. 2 Similar examples may have motivated William Frank's recent article [3] in this Journal concerning the reasons why some translations do not preserve completeness.Unfortunately, there are two errors in the latter; it is the purpose of this article to set them straight.Frank's main theorem reads as follows: