IN MEMORIAM: J. MICHAEL DUNN, 1941–2021

Abstract

The history of relevance logic cannot be written without mentioning J. Michael Dunn who played a prominent role in shaping this area of logic. In the late twentieth century, he was a doyen with a world-class reputation in the field of philosophical logic. Dunn’s research also encompassed logics outside of relevance logic, from 2-valued first-order logic to quantum logic and substructural logics such as the Lambek calculi, intuitionistic logic, linear logic, etc. Each of the disciplines of philosophy, mathematics, and computer science has been impacted in intrinsic ways by some of the theorems proved by Dunn. Jon Michael Dunn was born in Fort Wayne, Indiana, on June 19, 1941, and he passed away on April 5, 2021, in Bloomington, Indiana. After attending high schools in Fort Wayne and Lafayette, he obtained an AB degree in philosophy from Oberlin College, Ohio, in 1963. Dunn completed his Ph.D. Thesis entitled The Algebra of Intensional Logics at Pittsburgh University in 1966; his thesis supervisor was Nuel D. Belnap. In 1969, Dunn was appointed an associate professor in the Department of Philosophy at Indiana University in Bloomington, Indiana, and he stayed on the faculty at IU until 2007, when he retired as University Dean of the School of Informatics, Oscar R. Ewing Professor of Philosophy, Professor of Informatics, Professor of Computer Science, and Core Faculty in Cognitive Science. Dunn supervised 14 Ph.D. students in logic and 3 Ph.D. students in other areas of philosophy. He taught advanced graduate courses in logic, including courses on 2-valued logic (metalogic), modal logic, algebraic logics, and substructural logics. Dunn was the founding dean of the School of Informatics, and he served in other administrative positions such as Chair of the Department of Philosophy and Associate Dean of the College of Arts and Sciences in previous years. Dunn held multiple research grants and visiting positions at universities in the US, Europe, and Australia. Dunn, for his contributions to Indiana University, was honored by the IU Provost Medallion in 2007. The state of Indiana bestowed on Dunn the rank and title of Sagamore of the Wabash the same year. Dunn was elected a Fellow of the American Academy of Arts and Sciences in 2010. The logic of relevant implication R combines Church’s “weak implication” with lattice connectives and De Morgan negation. Alternatively, R results from Ackermann’s system Π′ by omitting a rule (the so-called rule) and adding permutation. Dunn started to investigate R from an algebraic point of view in his Ph.D. thesis [14]. This research continued the study of distributive lattices with De Morgan negation already underway in [1, 4, 5, 35, 43]. Dunn showed that 4, the four-element lattice with two incomparable elements on which negation has fixed points, plays a fundamental role among De Morgan lattices, and hence, for first-degree entailments fde; (the implication-free fragment of R and of the logic of entailment E). Dunn proved—using methods similar to those Stone utilized in his representation of Boolean algebras—that every De Morgan lattice is embeddable into a product of 4, that is, into ∏ i<κ 4i , where κ ≤ 2 and is the cardinality of the De Morgan lattice. Whenever possible, Dunn generalized theorems to complete lattices, complete homomorphisms, complete embeddings, and similar notions, which, strictly speaking, go beyond the purely algebraic approach. He also defined a new interpretation for fde that relies on pairs of sets of situations. Later, Dunn defined the four values (true, false, both, and neither) that emerge in the interpretation of fde as subsets of {T, F }, and Belnap provided motivations for these values by appeal to databases. Nowadays, this logic is often referred to as Dunn–Belnap (or Belnap–Dunn) logic (cf. [48]). The algebraization of R revolves around two concepts: residuation and the intensional truth constant (denoted by t). Relevant