Copi's method of deduction.
Frederick A. Johnson · Notre Dame Journal of Formal Logic · 1979
In [1] Bradley pointed out that it was superfluous for Copi to refer to the completeness and analyticity of RS to show that the method of deduction set forth in Chapter 3 of Symbolic Logic, 3rd ed.[3], is complete.Since in the 4th edition [4] Copi continues to make his proof of completeness depend upon the completeness and analyticity of RS, it seems worthwhile to give a proof which clearly stands on its own.To do this, it is necessary to formalize Copi's method of deduction.We will call the formalization CMD.*We will let the capital letters, with or without subscripts, from the earlier part of the alphabe't be the simple well-formed formulas in CMD and the capital letters, with or without subscripts, from the later part of the alphabet be variables in our meta-language which range over the well-formed formulas of CMD.The well-formed formulas of CMD are defined inductively in the classical way.Well-formed arguments have the form x -» Q, where x is the empty symbol or a well-formed formula of CMD.The intended reading of 'P-+ Q' is Q follows from P; the intended reading of ζ x -* Q\ where x is the empty symbol, is Q follows from the empty premise, or Q follows from any premise.It will become evident that all of the theorems (and axioms) of CMD are well-formed arguments.The axiom schema for CMD areas follows, where 'hP Q' abbreviates Q)'P-*Q (M.P.) Ax2.h \-P'Q-+P (Simp.) Ax8.\-P Q-> P'Q (Conj.)Ax9.\-P-*PvQ (Add.)