Bolzano, (the Appropriate) Relevant Logic, and Grounding Rules for Implication

Abstract

AbstractThe Bohemian philosopher Bernard Bolzano introduced the notion of logical deducibility, anticipating by almost one hundred years the much more famous Tarskian result. Besides the notion of logical deducibility, he also introduced the notion of grounding, which, on the one hand, he distinguished from logical deducibility, but he also regarded as importantly connected to it. In particular, he saw a close connection between grounding and exact deducibility (a special case of deducibility). This paper examines this latter notion and compares it to notions of relevant entailment. Although the literature on this subject compares the notion of exact deducibility with that of relevant entailment as defined by Anderson and Belnap, here it is argued that Neil Tennant’s system CR is the best model for Bolzano’s ideas: it is shown that the two approaches share many important features, the most notable of which is the lack of full transitivity. It is also argued that Tennant’s system CR can serve as a framework for developing grounding rules for conditionals.