The Australasian Journal of Logic

Introduction to the Issue

Shay Allen Logan — 2025-09-10

This paper is an introduction to the special issue in honour of Ross T. Brady.

The Content of Content Semantics

Nicholas Ferenz — 2025-09-10

In this paper, we investigate Brady's content semantics, which was an early attempt to get around the infamous incompleteness of the standard axiom systems for quantified relevant logics with regard to constant domain expansions of ternary relation semantics. We investigate this semantic framework by showing equivalence to a variation on a recently proposed algebraic version of semantics due to Mares and Goldblatt.

Content and Depth Revisited

Shay Allen Logan — 2025-09-10

In this paper I will compare depth relevance to what has come to be called depth hyperformalism. Ordinary formality, is typically taken to require closure under uniform substitutions. Depth hyperformality requires closure under depth substitutions—not-necessarily-uniform substitutions that are allowed to vary with depth. As it turns out, all depth hyperformal logics are depth relevant but not vice-versa.

So we’re left to ask the following question: for the particular projects Ross Brady is engaged in, should it be depth relevance or depth hyperformalism that should set the pace? More to the point, I’ll be interested in the following two claims:

Logics of meaning containment are necessarily depth hyperformal.
Logics of meaning containment are necessarily depth relevant.

The first entails the second. Brady seems to endorse the second. I’ll argue in this paper that he should also endorse the first.

On the Irrationality of the Square Root of 2

John Slaney — 2025-09-10

This paper presents a proof of the irrationality of √2 which not only avoids use of paradoxes of implication but also eschews the principle of contraction. The actual theorem proved, in the relevant arithmetic B-sharp, is
¬∃ x ∃ y(x'.x' = 2.y.y)
which is the theorem of natural number theory standardly expressing the irrationality of √2. The key move in the argument is to use provable cases of the law of the excluded middle, P ∨ ¬P, or its close relative P ∨ (P → (0 < 0)) to mimic the effect of contraction.

A 2 Set-up Ternary Relational Semantics for the Companions to Brady’s 4-valued logic BN4

Sandra M. López — 2025-09-10

The present paper is inspired by Ross T. Brady’s work on semantics for many-valued logics which also belong to the family of relevance. In particular, we aim to enhance his methodology for (meta)completeness results with respect to a 2 set-up ternary relational semantics. In 1982, Brady developed the 4-valued logic BN4 and endowed it with such semantics, providing both strong soundness and completeness theorems. In the recent literature, six new 4-valued logics have been defined as companions to the system BN4 and endowed with a bivalent Belnap-Dunn type semantics. The aim of this paper is to deepen the knowledge of these new companions to BN4 by providing a 2 set-up ternary relational semantics, thus following the same strategies Brady applied to BN4 in (Brady, 1982).

A note on the logic of Turing's halting paradox

Zach Weber — 2025-09-10

Projects directed at a universal logic (notably, Brady's [Brady 2006]) have long struggled with paradoxes. In just the domain of computability, Hilbert's call for a general decision procedure was scuttled by Turing, using a diagonal argument. Indeed, Turing's halting problem can be straightforwardly viewed as a paradox, of the same type as others like the Liar and Curry's. This is confirmed here by reconstructing a halting predicate in a sequent calculus setting, thereby fitting a ``recipe' for paradox in [Ahmad, AJL 2022]. The usual range of possible solutions then apply, including especially substructural approaches. Bringing this work as a response to Brady's ``"Starting the Dismantling of Classical Mathematics" [Brady, AJL 2018], then, we assess his proposals to drop the law of excluded middle and other logical principles---asking whether this strategy avoids all versions of the halting paradox. Brady has well begun, in his idiom, the re-construction of mathematics; there is more yet to do.

A quasi-relevant, connexive 4-valued implicative expansion of Belnap-Dunn logic defining material connexive logic MC

José Méndez — 2025-09-10

In this paper, it is defined a connexive 4-valued implicative expansion of Belnap-Dunn logic we have dubbed LMI4C. It is a quasi relevant logic in the sense that it enjoys the "quasi relevance property". Also, LMI4C defines "material connexive logic" MC. The fact that LMI4C defines classical positive logic C+ is used to provide it with a Hilbert-style formulation presenting LMI4C as an expansion of C+. Said formulation is obtained by using a Belnap-Dunn "two-valued semantics".

Reflections on Brady's Logic of Meaning Containment

Greg Restall — 2025-09-10

Abstract: This paper is a series of reflections on Ross Brady’s favourite substructural logic, the logic MC of meaning containment. In the first section, I describe some of the distinctive features of MC, including depth relevance, and its principled rejection of some concepts that have been found useful in many substructural logics, namely intensional or multiplicative conjunction (sometimes known as ‘fusion’), the Church constants (⊤ and ⊥), and the Ackermann constants (t and f). A further distinctive feature of the axiomatic formulation of MC is its meta-rule, which is a unique feature of MC Hilbert proofs. This meta-rule gives rise to one further special property of MC, in that the logic is distributive in one sense, and non-distributive in another. The distribution of additive conjunction over disjunction (the step from p∧(q∨r) to (p∧q)∨(p∧r)) holds in MC as a rule, but not as a provable conditional, and in this way, MC is distinctive among popular substructural logics. (Anderson and Belnap’s favourite logics R and E are distributive in both senses, while Girard’s linear logic is distributive in neither.)

In this paper, I aim to enhance our understanding of each of these distinctive features of MC, giving an account of what it might take for a propositional logic to meet these constraints. I will start with a presentation of Hilbert proofs for MC, and then showing how Brady Lattices (a natural class of algebraic models for MC) can help us understand each of these special features of Brady’s logic of meaning containment.

On three-valued logics with the variable sharing property

Gemma Robles — 2025-09-10

The class of the logics free from paradoxes of relevance determined by all natural implicative expansions of Kleene’s strong 3-valued matrix with two designated values is defined. These logics are free from paradoxes of relevance in the sense that they have the "variable sharing property". They have "natural conditionals" in the sense that the function defining them coincides
with the classical function when restricted to the "classical values", satisfies Modus Ponens and, finally, assigns a designated value to a conditional whenever its antecedent and its consequence are assigned the same value. These logics are defined by using a "two-valued" overdetermined Belnap-Dunn semantics. Thus, the interpretation of the three values is crystalline. The logics here introduced enjoy the properties customarily demanded of many-valued implicative logics except, of course, the satisfaction of the rule "Verum ex quodlibet".

A Normalized Natural Deduction System for the Logic MC

Ross Brady — 2025-09-10

Currently, there is no decidability proof for the sentential logic MC of meaning containment, due it not quite fitting standard semantic or Gentzen methods. This paper presents a normalized Fitch-style natural deduction system for MC, building on one previously published by the author for the neighbouring logic DW, but with modifications due to the presence in MC of Conjunctive Syllogism and the absence of Distribution. Decidability is achieved for MC through the use of the Subformula Property associated with normalization and the identity between the depth of a subformula in the final formula under test and the depth of subproof in which such a subformula can be located in the normalized natural deduction proof.

From depth relevance to connexivity

Fernando Cano-Jorge — 2025-09-10

Brady [4] used the matrix for Meyer’s crystal lattice CL to build a
hierachical model structure for his deep relevant logic DRd. In this paper
we modify the matrix for CL so as to define a connexive conditional. In
doing so, we arrive at a family of connexive logics satisfying the depth
relevance property. As a result, we show a way to satisfactorily combine
connexivity and relevance without trivializing the logic and without vali-
dating unappealing theorems

Non-Canonical Models of Relevant Logics

Joseph Zalewski — 2025-09-10

We construct, in several ways, Fine-style frame models of the relevant logic $\textbf{B}$ which verify exactly the theorems of $\textbf{B}$ but differ from the canonical model in some of their properties: they contain multiple points representing the same (formal) $\textbf{B}$ theory, or their points fail to represent some formal $\textbf{B}$ theories (in fact, fail to represent almost all of them). We briefly discuss the implications of this for the adequacy of frame semantics for relevant logics.

Proof Invariance

Blane Worley — 2025-09-10

We explore depth substitution invariance, or hyperformalism, and extend known results in this realm to justification logics extending weak relevant logics. We then examine the surprising invariance of justifications over formulas and restrict our attention to the substitution of proofs in the original relevant logic. The results of this paper indicate that depth invariance is a recalcitrant feature of the logic and that proof structures in hyperformal logics are quite inflexible.