Ivan Efimovič Orlov’s paper “The calculus of the compatibility of propositions”, published in Russian in 1928, is fascinating for anyone interested in the early history of relevance, modal or intuitionistic logic. This is a translation of that paper.

Ivan Efimovič Orlov’s paper “The calculus of the compatibility of propositions”, published in Russian in 1928, is fascinating for anyone interested in the early history of relevance, modal or intuitionistic logic. This is a guide that outlines Orlov's life and work, analyses the content of the paper, and relates it to work of his contemporaries and successors.

The present paper is inspired by Sylvan and Plumwood’s logicBM defined in “Non-normal relevant logics” and by their treatmentof negation with the ∗-operator in “The semantics of first-degree en-tailment”. Given a positive logic L including Routley and Meyer’sbasic positive logic and included in either the positive fragment of Eor in that of RW, we investigate the essential De Morgan negation ex-pansions of L and determine all the deductive relations they maintainto each other. A Routley-Meyer semantics is provided for each logicdefined (...) in the paper. (shrink)

This note extends earlier results on geometrical interpretations of the logic KR to prove some additional results, including a simple undecidability proof for the four-variable fragment of KR.

This paper examines ten possible topological interpretations of connection and for each interpretation, identifies sufficient conditions under which a significant class of topological spaces provides models of General Extensional Mereotopology with Closure Conditions (GEMTC) in which some key mereotopological ideas align with their topological analogues. In particular, there is an interpretation under which the non-empty sets of any symmetric topology are a model of GEMTC with alignment between the mereotopological and topological definitions of (self-)connection, open and closed entities, interior, exterior, (...) closure, and boundary. (shrink)

In this article, we perform a detailed proof theoretic investigation of a wide number of relevant logics by employing the well-established methodology of labelled sequent calculi to build our intended systems. At the semantic level, we will characterise relevant logics by employing reduced Routley-Meyer models, namely, relational structures with a ternary relation between worlds along with a unique distinct element considered as the real (or actual) world. This paper realizes the idea of building a variety of modular labelled calculi by (...) reflecting, at the syntactic level, semantic informations taken from reduced Routley-Meyer models. Central results include proofs of soundness and completeness, as well as a proof of cut- admissibility. (shrink)

On some views, we can be sure that parties to a dispute over the logic of ‘exists’ are not talking past each other if they can characterise ‘exists’ as the only monadic predicate up to logical equivalence obeying a certain set of rules of inference. Otherwise, we ought to be suspicious about the reality of their disagreement. This is what we call a proof- theoretic argument. Pace some critics, who have tried to use proof-theoretic arguments to cast doubts about the (...) reality of disagreements about the logic of ‘exists’, we argue that proof-theoretic arguments can be deployed to establish the reality of several such disagreements. Along the way, we will also utilise this technique to establish similar results about some disagreements over the logic of identity. (shrink)

We show how one can give a clear formal account of Boole’s notorious “indefinite" (or “auxiliary”) symbols by treating them as variables that range over functions from classes to classes rather than just over classes while, at the same time, following Hailperin’s proposal of binding them existentially.

In this article, we explore the semantic characterization of the (right) pure refined variable inclusion companion of all logics, which is a further refinement of the nowadays well-studied pure right variable inclusion logics. In particular, we will focus on giving a characterization of these fragments via a single logical matrix, when possible, and via a class of finite matrices, otherwise. In order to achieve this, we will rely on extending the semantics of the logics whose companions we will be discussing (...) with infectious values in direct and in more subtle ways. This further establishes the connection between infectious logics and variable inclusion logics. (shrink)

In this paper, we provide a recipe that not only captures the common structure of semantic paradoxes but also captures our intuitions regarding the relations between these paradoxes. Before we unveil our recipe, we first talk about a well-known schema introduced by Graham Priest, namely, the Inclosure Schema. Without rehashing previous arguments against the Inclosure Schema, we contribute different arguments for the same concern that the Inclosure Schema bundles together the wrong paradoxes. That is, we will provide further arguments on (...) why the Inclosure Schema is both too narrow and too broad. We then spell out our recipe. The recipe shows that all of the following paradoxes share the same structure: The Liar, Curry's paradox, Validity Curry, Provability Liar, Provability Curry, Knower's paradox, Knower's Curry, Grelling-Nelson's paradox, Russell's paradox in terms of extensions, alternative Liar and alternative Curry, and hitherto unexplored paradoxes. We conclude the paper by stating the lessons that we can learn from the recipe, and what kind of solutions the recipe suggests if we want to adhere to the Principle of Uniform Solution. (shrink)

This paper introduces a new framework, based on the notion of compatibility space, obtained by adding a primitive incompatibility relation to a state space in the sense of Fine. The key idea inspiring the framework is to modify Fine's truthmaker semantics by taking the notion of incompatibility as primitive, and use it to define other notions. We discuss some interesting features of the framework and explore its advantages over the standard framework of state spaces. We review some applications of the (...) framework, including proofs of soundness and completeness theorems for a number of logics, one way to use compatibility states to mirror Kripke frames, and an exploration of a way to use the new framework to provide truthmaking clauses for modal formulas. (shrink)

In, Kosterec attempts to provide ``model-theoretic proofs'' of certain theses involving the normal modal operators $\Diamond$ and $\square$ and the truth-in-fiction operator $F$ which he then goes on to show have counterexamples in Kripke models. He concludes from this that the embedding of modal logic under the truth-in-fiction operator is unsound. We show instead that it is the ``model-theoretic proofs'' that are themselves unsound, involving illicit substitution, a subtle error that nevertheless allows us to draw an important conclusion about intensional (...) contexts and semantic equivalences. (shrink)

We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R and compare this with arithmetic based on a suitable logic of meaning containment, which was developed in Brady [7]. We argue in favour of the latter as it better captures the key logical concepts of meaning and truth in arithmetic. We also contrast the two approaches to classical recapture, again favouring our approach in [7]. We then consider our previous development of Peano arithmetic including (...) primitive recursive functions, finally extending this work to that of general recursion. (shrink)

Meyer and Mortensen’s Alien Intruder Theorem includes the extraor- dinary observation that the rationals can be extended to a model of the relevant arithmetic R♯, thereby serving as integers themselves. Al- though the mysteriousness of this observation is acknowledged, little is done to explain why such rationals-as-integers exist or how they operate. In this paper, we show that Meyer and Mortensen’s models can be identified with a class of ultraproducts of finite models of R♯, providing insights into some of the (...) more mysterious phenomena of the rational models. (shrink)

This is a brief note about the history of the analysis of the collection of theories, RM3modn, in Meyer and Mortensen "Inconsistent Models for Relevant Arithmetics" Journal of Symbolic Logic 49 (1984).

This paper explores the model theory of relevant arithmetic, emphasizing the structure of nonstandard natural numbers in the relevant arithmetic R#. In particular, the authors prove the “Alien Intruder Theorem” guaranteeing the existence of a model of R# including the rational numbers in which each rational acts as a nonstandard natural number. The authors conclude by considering some consequences of and open questions about the construction used in the theorem.

The purpose of this paper is to formulate first-order Peano arithmetic within the resources of relevant logic, and to demonstrate certain properties of the system thus formulated. Striking among these properties are the facts that it is trivial that relevant arithmetic is absolutely consistent, but classical first-order Peano arithmetic is straightforwardly contained in relevant arithmetic. Under, I shall show in particular that 0 = 1 is a non-theorem of relevant arithmetic; this, of course, is exactly the formula whose unprovability was (...) sought in the Hilbert program for proving arithmetic consistent. Under, I shall exhibit the requisite translation, drawing some Goedelian conclusions therefrom. Left open, however, is the critical problem whether Ackermann’s rule γ is admissible for theories of relevant arithmetic. The particular system of relevant Peano arithmetic featured in this paper shall be called R♯. Its logical base shall be the system R of relevant implication, taken in its first-order form RQ. Among other Peano arithmetics we shall consider here in particular the systems C♯, J♯, and RM3♯; these are based respectively on the classical logic C, the intuitionistic logic J, and the Sobocinski-Dunn semi-relevant logic RM3. And another feature of the paper will be the presentation of a system of natural deduction for R♯, along lines valid for first-order relevant theories in general. This formulation of R♯ makes it possible to construct relevantly valid arithmetical deductions in an easy and natural way; it is based on, but is in some respects more convenient than, the natural deduction formulations for relevant logics developed by Anderson and Belnap in Entailment. (shrink)

This paper offers an elementary proof that formal arithmetic is consistent. The system that will be proved consistent is a first-order theory R♯, based as usual on the Peano postulates and the recursion equations for + and ×. However, the reasoning will apply to any axiomatizable extension of R♯ got by adding classical arithmetical truths. Moreover, it will continue to apply through a large range of variation of the un- derlying logic of R♯, while on a simple and straightforward translation, (...) the classical first-order theory P♯ of Peano arithmetic turns out to be an exact subsystem of R♯. Since the reasoning is elementary, it is formalizable within R♯ itself; i.e., we can actually demonstrate within R♯ (or within P♯, if we care) a statement that, in a natural fashion, asserts the consistency of R♯ itself. The reader is unlikely to have missed the significance of the remarks just made. In plain English, this paper repeals Goedel’s famous second theorem. (That’s the one that asserts that sufficiently strong systems are inadequate to demonstrate their own consistency.) That theorem (or at least the significance usually claimed for it) was a mis- take—a subtle and understandable mistake, perhaps, but a mistake nonetheless. Accordingly, this paper reinstates the formal program which is often taken to have been blasted away by Goedel’s theorems— namely, the Hilbert program of demonstrating, by methods that everybody can recognize as effective and finitary, that intuitive mathematics is reliable. Indeed, the present consistency proof for arithmetic will be recognized as correct by anyone who can count to 3. (So much, indeed, for the claim that the reliability of arithmetic rests on transfinite induction up to ε0, and for the incredible mythology that underlies it.). (shrink)

This is the introduction to the special issue on Robert K. Meyer and the philosophy of arithmetic.

This is a bibliography of R.K. Meyer's published articles on relevant arithmetic.

In this paper, we explore the possibility of constructing algebra-valued models of set theory based on Priest's Logic of Paradox. We show that we can build a non-classical model of ZFC which has as internal logic Priest's Logic of Paradox and validates Leibniz's law of indiscernibility of identicals. This is achieved by modifying the interpretation map for $\in$ and $=$ in our algebra-valued model. We end by comparing our model constructions to Priest's model-theoretic strategy and point out that we have (...) a tradeoff between desirable model-theoretic properties and the validity of ZFC and its theorems. (shrink)

In 1978, the authors began a paper, “S,” henceforth [S4S], intended as a philosophical companion piece to the technical solution [SPW] of the Anderson-Belnap P–W problem. [S4S] has gone through a number of drafts, which have been circulated among close friends. Meanwhile other authors have failed to see the point of the semantics which we introduced in [SPW]. It will accordingly be our purpose here to revisit that semantics, while giving our present views on syllogistic matters past, present and future, (...) especially as they relate to not begging the question via such dubious theses as A →’ A. We shall investigate in particular a paraconsistent attitude toward such theses. (shrink)