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Implicational tonoid logics and their relational semantics have been introduced by Yang and Dunn. This paper extends this investigation to implicational partial Galois logics. For this, we first define some implicational partial gaggle logics as special kinds of implicational tonoid logics called “implicational partial Galois logics.” Next, we provide Routley–Meyer-style relational semantics for finitary those logics.

This paper combines two classes of generalized logics, one of which is the class of weakly implicative logics introduced by Cintula and the other of which is the class of gaggle logics introduced by Dunn. For this purpose we introduce implicational tonoid logics. More precisely, we first define implicational tonoid logics in general and examine their relation to weakly implicative logics. We then provide algebraic semantics for implicational tonoid logics. Finally, we consider relational semantics, called Routley–Meyer–style semantics, for finitary those logics.

The decidability of the logic of pure ticket entailment means that the problem of inhabitation of simple types by combinators over the base { B, B′, I, W } is decidable too. Type-assignment systems are often formulated as natural deduction systems. However, our decision procedure for this logic, which we presented in earlier papers, relies on two sequent calculi and it does not yield directly a combinator for a theorem of $${T_\to}$$ . Here we describe an algorithm to extract an inhabitant from a sequent calculus proof—without translating the proof into another proof system.

We begin by discussing the history of quantum logic, dividing it into three eras or "lives." The first life has to do with Birkhoff and von Neumann's algebraic approach in the 1930's. The second life has to do with attempt to understand quan- tum logic as logic that began in the late 1950's and blossomed in the 1970's. And the third life has to do with recent developments in quantum logic coming from its connections to quantum computation. We discuss our own work connecting quantum logic to quantum computation (viewing quantum logic as the logic of quantum reg- isters storing qubits), make some speculations about mathematics based on quantum principles, and end by quickly describing the papers by the other authors contribut- ing to this special issue on "Quantum Logic Inspired by Quantum Computing." The overall aim of this issue is to show how work on this topic is leading to interesting new areas of logic.

One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley–Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing a general conception of conditionality that may unify the three given conceptions.

Both I and Belnap, motivated the “Belnap-Dunn 4-valued Logic” by talk of the reasoner being simply “told true” (T), and simply “told false” (F), which leaves the options of being neither “told true” nor “told false” (N), and being both “told true” and “told false” (B). Belnap motivated these notions by consideration of unstructured databases that allow for negative information as well as positive information (even when they conflict). We now experience this on a daily basis with the Web. But the 4-valued logic is deductive in nature, and its matrix is discrete: there are just four values. In this paper I investigate embedding the 4-valued logic into a context of probability. Josang’s Subjective Logic introduced uncertainty to allow for degrees of belief, disbelief, and uncertainty. We extend this so as to allow for two kinds of uncertainty—that in which the reasoner has too little information (ignorance) and that in which the reasoner has too much information (conflicted). Josang’s “Opinion Triangle” becomes an “Opinion Tetrahedron” and the 4-values can be seen as its vertices. I make/prove various observations concerning the relation of non-classical “probability” to non-classical logic.

Symmetric generalized Galois logics (i.e., symmetric gGls) are distributive gGls 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 gGls by models on topological relational structures. On the other hand, topological relational structures are realized by structures of symmetric gGls. We generalize the weak distributivity laws between fusion and fission to interactions of certain monotone operations within distributive supergGls. We are able to prove appropriate generalizations of the previously obtained theorems—including a functorial duality result connecting classes of gGls and classes of structures for them.

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.

This paper explores allowing truth value assignments to be undetermined or "partial" (no truth values) and overdetermined or "inconsistent" (both truth values), 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 Łukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's (first-degree) 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".

We give a set of postulates for the minimal normal modal logicK + without negation or any kind of implication. The connectives are simply ∧, ∨, □, ◊. The postulates (and theorems) are all deducibility statements ϕ ⊢ ψ. The only postulates that might not be obvious are $$\diamondsuit \varphi \wedge \square \psi \vdash \diamondsuit (\varphi \wedge \psi )\square (\varphi \vee \psi ) \vdash \square \varphi \vee \diamondsuit \psi $$ . It is shown thatK + is complete with respect to the usual Kripke-style semantics. The proof is by way of a Henkin-style construction, with “possible worlds” being taken to be prime theories. The construction has the somewhat unusual feature of using at an intermediate stage disjoint pairs consisting of a theory and a “counter-theory”, the counter-theory replacing the role of negation in the standard construction. Extension to other modal logics is discussed, as well as a representation theorem for the corresponding modal algebras. We also discuss proof-theoretic arguments.

Applications philosophiques de la logique de la predication pertinente, mise au point par l'A., aux proprietes intrinseques et aux relations internes. Les proprietes intrinseques sont des predicats monadiques pertinents et les relations internes, des predicats polyadiques pertinents. L'A. compare sa theorie de la predication a celle de D. Lewis

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.

The validity of an entailment has nothing to do with whether or not the components are true, false, necessary, or impossible; it has to do solely with whether or not there is a necessary connection between antecedent and consequent. Hence it is a mistake (we feel) to try to build a sieve which will “strain out” entailments from the set of material or strict “implications” present in some system of truth-functions, or of truth-functions with modality. Anderson and Belnap (1962, p. 47).

The sentential calculus E-Mingle (EM) is obtained by adding the axiom scheme A->(A-*A) to the Belnap-Anderson system E of Eelevant Implication. We shall here presume that EM has been explicitly axioma tized as in [2], and shall cite various theorems of EM listed there by name, e.g., Transitivity. EM has a model theory in terms of Sugihara matrices, as was discovered by Meyer [9]1. Here, however, is presented a model theory using Kripke's device of model structures with a binary accessibility relation. The prin? cipal point of departure from Kripke is the consideration of models which allow sentences to be simultaneously both "true" and "false". When the basic results here were first obtained back in 19692 there was no need to explicitly mention that the semantics presented used a binary accessibility relation. That was surely a part of the ordinary meaning of the phrase "Kripke-style semantics". But since then the work of Eoutley and Meyer (cf. [13]) and others has shown how to extend Kripke-style semantics to allow for ternary accessibility relations. This has been particularly fruitful in the case of the relevance logics, and [13] its successors contains completeness results for various of these logics, including E and EM (cf. also [4]). It must be frankly confessed that the ingenuity and the power of the ternary semantics were so overwhelming to this author they caused delay until now of full publication of the binary semantics for EM. There had been at first the vain hope that the binary semantics could be extended to the other relevance logics, and then with the success of the ternary approach the binary semantics became to appear old-fashioned and special.

Given classical (2 valued) structures $$\mathfrak{A}$$ and $$\mathfrak{A}'$$ and a homomorphism h of $$\mathfrak{A}$$ onto $$\mathfrak{A}'$$ , it is shown how to construct a (non-degenerate) “3-valued counterpart” $$3\mathfrak{A}'$$ of $$\mathfrak{A}'$$ . Classical sentences that are true in $$\mathfrak{A}$$ are non-false in $$3\mathfrak{A}'$$ . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof for number theory was obtained earlier by R. K. Meyer and suggested the present abstract development).

This paper reformulates and decides a certain conjecture in Dunn's ‘Relevant Predication 1: The Formal Theory’ (Journal of Philosophical Logic16, 347–381, 1987). This conjecture of Dunn's relates his object-language characterisation of a property's being relevant in a variable x to certain grammatical characterisations of relevance, analogous to some given by Helman, in ‘Relevant Implication and Relevant Functions’ (to appear in Entailment: The Logic of Relevance and Necessity, vol. 2, by Alan Ross Anderson, Nuel Belnap, and J. Michael Dunn et al.) In the course of the investigation this paper also investigates Kit Fine's semantics for quantified relevance logics, which appears in his appropriately titled ‘Semantics for Quantified Relevance Logics’