Logic beyond completeness: The priority of expressive power

In the practice and teaching of logic, incomplete systems are typically not regarded as proper logical systems. Their study is marginal if not absent in most courses. There seems to be a tyranny of completeness. I find this problematic as many systems are being left behind. This article aims to challenge such tyranny, which I find to be a consequence of a traditional view of logic as a discipline that studies content-independent truths. I begin by showing how this view is inherited from Hilbert’s foundational program. As an alternative, I take a pragmatist view of logic from Frege, Brandom, and Frápolli. I then argue that this view implies a Principle of Expressive Power Priority (PEP) for picking out logical systems, and I show that the tyranny is simply a generalization of a singular case of application of PEP that is correct when we are interested in systems to be run exclusively by computers. But I argue that if the systems are to be employed by knowledgeable human beings, incomplete systems are still interesting. Thus, from this pragmatist view of logic we can overcome the tyranny and rekindle interest in incomplete systems without losing sight on the import of completeness.

Keywords:
Inferentialism; Logical Expressivism; Completeness; Hilbert; Frege