Axiomatic Theories of Partial Ground II: Partial Ground and Hierarchies of Typed Truth

Abstract

This is part two of a two-part paper in which we develop an axiomatic theory of the relation of partial ground. The main novelty of the paper is the of use of a binary ground predicate rather than an operator to formalize ground. In this part of the paper, we extend the base theory of the first part of the paper with hierarchically typed truth-predicates and principles about the interaction of partial ground and truth. We show that our theory is a proof-theoretically conservative extension of the ramified theory of positive truth up to 0 and thus is consistent. We argue that this theory provides a natural solution to Fine’s “puzzle of ground” about the interaction of truth and ground. Finally, we show that if we apply the truth-predicate to sentences involving our ground-predicate, we run into paradoxes similar to the semantic paradoxes: we get ground-theoretical paradoxes of self-reference.