A Relevant Framework for Barriers to Entailment

In her recent book, Russell (2023) examines various so-called “barriers to entailment, ” including Hume’s law, roughly the thesis that an ‘ought’ cannot be derived from an ‘is. ’ Hume’s law bears an obvious resemblance to the pro- scription on fallacies of modality in relevance logic, which has traditionally formally been captured by the so-called Ackermann property. In the context of relevant modal logic, this property might be articulated thus: No conditional whose antecedent is box-free and whose consequent is box-prefixed is valid (for the connection, interpret box deontically). While the deontic significance of Ackermann-like properties has been observed before, Russell’s new book sug- gests a more broad-scoped formal investigation of the relationship between bar- rier theses of various kinds and corresponding Ackermann-like properties. In this paper, I undertake such an investigation by elaborating a general relevant bimodal logical framework in which several of the barriers Russell examines can be given formal expression. I then consider various Ackermann-like properties corresponding to these barriers and prove that certain systems satisfy them. Finally, I respond to some objections Russell makes against the use of relevance logic to formulate Hume’s law and related barriers. 2020 Mathematics Subject Classification. Primary: 03B47, Secondary: 03B44, 03B45. An early version of this paper was presented on April 13, 2024 at the Impromptu CUNY Logic Workshop held at the CUNY Graduate Center in New York, USA. Mature versions were presented on May 13, 2024 at the 2nd Third Workshop held at the University of Alberta in Edmonton, Canada, and on September 27, 2024 at the Logic Colloquium of the University of Connecticut in Storrs, USA. I am grateful to the attendees of these talks for helpful feedback. Further thanks are due to two anonymous referees for the journal, Katalin Bimbó, Lloyd Humberstone, and Shawn Standefer.

Authors

• Yale Weiss

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