More logics without tautologies.
Leo Simons · Notre Dame Journal of Formal Logic · 1978
Logic without tautologies"* 1 describes a system of sentential natural deduction (there called C) as "next-strongest'': the addition to C of any general rule of inference yields a system which is either inconsistent or is a system of natural deduction for classical sentential calculus.C is not uniquely next-strongest.Another (non-equivalent) system, call it Ci, is also next-strongest.C and Ci and other systems related to them are all of them logics without tautologies, and are all completable by the addition to them of the law of excluded middle in the form S x h S 2 v ~ S 2 . Replacement rulesTo discuss these matters, the results and conventions of notation and abridgement of proofs and the like of "Logic without tautologies" are herewith assumed, but with some simplifications.Below are repeated the primitive replacement rules of C, which are common to all systems under discussion.CIO.