Notes on type-free axiomatic theories of positive truth

This paper investigates type-free axiomatic theories of positive truth. We derive the TKF system by strengthening specific axioms of the positive disquotational system PUTB and develop LKF by weakening certain axioms of the Kripke–Feferman system KF. We establish that LKF—which resolves some limitations of both PUTB and KF—is equivalent to TKF. Furthermore, we provide specific standard models for LKF and examine its arithmetical and truth-theoretical strength. We further analyze variants of positive truth theories constructed by incorporating either the consistency axiom (CONS) or the completeness axiom (COMP), identifying a spectrum of positive truth systems spanning from PUTB to KF+CONS. Noteworthily, PUTB+CONS and PUTB+COMP suffice to prove the liar sentence and its negation, respectively. Our findings reveal that, with the exception of the weakest theory in this spectrum, the internal logics of positive truth theories cannot achieve both consistency and completeness. Finally, we argue that LKF+CONS serves as a more suitable system for deflationism than either PUTB or KF.