The herbrand manifesto: Thinking inside the box

Abstract

The traditional semantics for relational logic (sometimes called Tarskian semantics) is based on the notion of interpretations of constants in terms of objects external to the logic. Herbrand semantics is an alternative that is based on truth assignments for ground sentences without reference to external objects. Herbrand semantics is simpler and more intuitive than Tarskian semantics; and, consequently, it is easier to teach and learn. Moreover, it is stronger than Tarskian semantics. For example, while it is not possible to finitely axiomatize integer arithmetic with Tarskian semantics, this can be done easily with Herbrand semantics. The downside is a loss of some common logical properties, such as compactness and inferential completeness. However, there is no loss of inferential power—anything that can be deduced according to Tarskian semantics can also be deduced according to Herbrand semantics. Based on these results, we argue that there is value in using Herbrand semantics for relational logic in place of Tarskian semantics. It alleviates many of the current problems with relational logic and ultimately may foster a wider use of relational logic in human reasoning and computer applications.