The completeness of Copi's system of natural deduction.
John A. Winnie · Notre Dame Journal of Formal Logic · 1970
I.Introduction.This note will outline a proof of the completeness of the system of sentential logic developed by Copi in [2] which also provides an effective proof-method for this system.Although the completeness of the Copi system is well known, the method to be used here does not involve a detour through an auxiliary axiomatic system (as in [l], where the completeness of the system presented in [3] is established).Since the method is of some interest in itself, the general procedure is sketched first.Let Pi, P 2 , > Pn, Q be any sequence of sentential schemata.Then a sentential system of natural deduction is here said to be complete if and only if there is a derivation in the system of Q from P l9 P 2 , . . ., P n whenever the schema (P λ P 2 , . . ., P n ) 3 Q is a (standard) truth-table tautology.The notion of a derivation used here will, of course, depend on the particular rules of inference or rules of replacement which are peculiar to the system under study.In the method used below, completeness is demonstrated as follows.First, we show that any tautology is derivable in the system from any non-empty sequence of sentences whatsoever.It now follows as a corollary that (P x P 2 , . . ., P n ) ^ Q is derivable from P l9 P 2 , . . ., P w whenever {P x P 2 , . . ., P n ) D Q is a tautology.Repeated use of the rule of conjunction (or an equivalent device) will now yield (P x P 2 , . . ., P«).A single application of modus ponens (i.e., the rule of detachment) then gives us Q, the desired result.In what follows, it is assumed that the reader is familiar with the inference and replacement rules of [2], here called CND (Copi's system of natural deduction), along with their abbreviations. 1