The Consistency and Completeness of Formal Systems

Summary

Of every formal system it is desirable to show that the system is (1) consistent, and (2) complete.² This is shown for the system P_LT’ in this chapter. A decision procedure is first developed for P_LT’ and by means of this, the desired results are demonstrated. It will be shown that (a) every theorem of P_LT’ is a tautology and that every tautology of propositionallogic is a theorem of P_LT’. The results thus derived will be extended to the system P_LT. Of P + it will be shown that while the system is consistent (in an appropriate sense) it is not complete.

In the Introduction it was shown that the primitive rules of the system P_LT’, substitution and modus ponens, were L-truth-preserving, i.e., that if the premises of a proof are L-true then the conclusion is L-true. (In this chapter the term “L-true” is used interchangeably with the term “tautology”.) This being the case, it is possible to establish the following metatheorem concerning P_LT’.

(The present chapter is more difficult than its predecessors. Consequently it may be passed over by those using the work as an elementary text book in symbolic logic.)

The metalogical proofs here presented employ methods due to Post, Kalmár and Church.

In S-systems substantially more comprehensive than the propositional calculus (e.g., LF’_LT of Ch. 7), these desiderata cannot both be obtained.