The Algebra of Intensional Logics

<p>J. Michael Dunn's PhD dissertation occupies a unique place in the development of the algebraic approach to logic.  In <em>The Algebra of Intensional Logics</em> Dunn introduced De Morgan monoids a class of algebras in which the algebra of <strong>R</strong> (the logic of relevant implication) is free.  This is an example where a logic's algebra is neither a Boolean algebra with further operations nor a residuated distributive lattice.  De Morgan monoids served as a paradigm example for the algebraization of other relevance logics including <strong>E</strong> the logic  of entailment and <strong>R-M</strong>ingle (<strong>RM</strong>) the extension of  <strong>R</strong> with the mingle axiom.</p><p><br />De Morgan monoids extend De Morgan lattices which algebraize the  logic of first-degree entailments that is a common fragment of <strong>R</strong> and <strong>E</strong>.  Dunn studied the role of the four-element De Morgan algebra <em>D</em> in the representation of De Morgan lattices and from this he derived a completeness theorem for first-degree entailments.  He also showed that every De Morgan lattice can be embedded into a 2-product of Boolean algebras and proved related results about De Morgan lattices in which negation has no fixed point.  Dunn also developed an informal interpretation for first-degree entailments utilizing the notion of aboutness which was motivated by the representation of De Morgan lattices by sets.</p><p><br />Dunn made preeminent contributions to several areas of relevance logic in his career spanning more than half a century.  In proof theory he developed sequent calculuses for positive relevance logics and a tableaux system for first-degree entailments; in semantics he developed a binary relational semantics for the logic <strong>RM</strong>.  The use of algebras remained a central theme in Dunn's work from the proof of the admissibility of the rule called γ to his theory of generalized Galois logics (or ``gaggles'') in which the residuals of arbitrary operations are considered.  The representation of gaggles---utilizing relational structures---gave a new framework for relational semantics for relevance and for so-called substructural logics and led to an information-based interpretation of them. </p>