Theory of generalized tautology in revised Kleene system

This paper is a complement and extension of the theory of generalized tautology which was first proposed by Wang Guojun in revised Kleene system, Some interesting results are obtained: (i) accessible α+-tautology and generalized contradiction which are dual theory to generalized tautology have been introduced; (ii) congruence partition about  has been given in logic system , W, Wk; (iii) in logic system Wk, tautologies can be obtained by employing the upgrade algorithm at most (k+1)/(2) times to an arbitrary formula; (iv) in logic system (W), tautologies cannot be obtained by employing upgrade algorithm to non-tautologies within finitely many times; (v) the deduction rule (1)/(2)+-MP holds in logic system (W).

This paper is a complement and extension of the theory of generalized tautology which was first proposed by Wang Guojun in revised Kleene system, Some interesting results are obtained: (i) accessibleα +-tautology and generalized contradiction which are dual theory to generalized tautology have been introduced; (ii) congruence partition about—has been given in logic system $\bar W$ ,W k ; (iii) in logic systemW k , tautologies can be obtained by employing the upgrade algorithm at most $\left[ {\frac{{k + 1}}{2}} \right]$ times to an arbitrary formula; (iv) in logic system $\bar W(W)$ , tautologies cannot be obtained by employing upgrade algorithm to non-tautologies within finitely many times; (v) the deduction rule $\left( {\left[ {\left( {\frac{1}{2}} \right)^ + } \right] - MP} \right)$ holds in logic system $\bar W(W)$ .