Jon Michael Dunn

(1941 - 2021)

We present a Kripke model for Girard's Linear Logic (without exponentials) in a conservative fashion where the logical functors beyond the basic lattice operations may be added one by one without recourse to such things as negation. You can either have some logical functors or not as you choose. Commutatively and associatively are isolated in such a way that the base Kripke model is a model for noncommutative, nonassociative Linear Logic. We also extend the logic by adding a coimplication operator, similar to Curry's subtraction operator, which is resituated with Linear Logic's contensor product. And we can add contraction to get nondistributive Relevance Logic. The model rests heavily on Urquhart's representation of nondistributive lattices and also on Dunn's Gaggle Theory. Indeed, the paper may be viewed as an investigation into nondistributive Gaggle Theory restricted to binary operations. The valuations on the Kripke model are three valued: true, false, and indifferent. The lattice representation theorem of Urquhart has the nice feature of yielding Priestley's representation theorem for distributive lattices if the original lattice happens to be distributive. Hence the representation is consistent with Stone's representation of distributive and Boolean lattices, and our semantics is consistent with the Lemmon-Scott representation of modal algebras and the Routley-Meyer semantics for Relevance Logic.

In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) in relevant arithmetic.

We study an application of gaggle theory to unary negative modal operators. First we treat negation as impossibility and get a minimal logic system Ki that has a perp semantics. Dunn 's kite of different negations can be dealt with in the extensions of this basic logic Ki. Next we treat negation as “unnecessity” and use a characteristic semantics for different negations in a kite which is dual to Dunn 's original one. Ku is the minimal logic that has a characteristic semantics. We also show that Shramko's falsification logic FL can be incorporated into some extension of this basic logic Ku. Finally, we unite the two basic logics Ki and Ku together to get a negative modal logic K-, which is dual to the positive modal logic K+ in [7]. Shramko has suggested an extension of Dunn 's kite and also a dual version in [12]. He also suggested combining them into a “united” kite. We give a united semantics for this united kite of negations.

The implicational fragment of the logic of relevant implication, $R_{\to}$ is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on $LR_{\to}$, a Gentzen-style calculus. In this paper, we add the truth constant $\mathbf{t}$ to $LR_{\to}$, but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve $\mathbf{t}$. This calculus, $LT_\to^{\text{\textcircled{$\mathbf{t}$}}}$, extends the consecution calculus $LT_{\to}^{\mathbf{t}}$ formalizing the implicational fragment of ticket entailment. We introduce two other new calculi as alternative formulations of $R_{\to}^{\mathbf{t}}$. For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of $R_{\to}^{\mathbf{t}}$. These results serve as a basis for our positive solution to the long open problem of the decidability of $T_{\to}$, which we present in another paper.

This paper explores the development of mathematics on a quantum logical base when mathematical postulates are taken as necessary truths. First it is shown that first-order Peano arithmetic formulated with quantum logic has the same theorems as classical first-order Peano arithmetic. Distribution for first-order arithmetical formulas is a theorem not of quantum logic but rather of arithmetic. Second, it is shown that distribution fails for second-order Peano arithmetic without extensionality. Third, it is shown that distribution holds for second-order Peano arithmetic (second-order quantum logic) with extensionality. Some remarks about extensions to quantum set theory are made.

Symmetric generalized Galois logics (i.e., symmetric gGl s) are distributive gGl s that include weak distributivity laws between some operations such as fusion and fission. Motivations for considering distribution between such operations include the provability of cut for binary consequence relations, abstract algebraic considerations and modeling linguistic phenomena in categorial grammars. We represent symmetric gGl s by models on topological relational structures. On the other hand, topological relational structures are realized by structures of symmetric gGl s. We generalize the weak distributivity laws between fusion and fission to interactions of certain monotone operations within distributive super gGl s. We are able to prove appropriate generalizations of the previously obtained theorems—including a functorial duality result connecting classes of gGl s and classes of structures for them.

This paper explores allowing truth value assignments to be undetermined or "partial" and overdetermined or "inconsistent", thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson 's "constructible falsity"

The implicational fragment of the logic of relevant implication, $R_\to$ is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, $T_\to$ is decidable. Our proof is based on the consecution calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of $T_\to$ to the decidability problem of $R_\to$. The decidability of $T_\to$ is equivalent to the decidability of the inhabitation problem of implicational types by combinators over the base $\{\textsf{B},\textsf{B}',\textsf{I},\textsf{W}\}$.