# Selected papers on Relevant Logic

## John Slaney

This is the abstract of the paper:

John Slaney and Edward Walker.
The One-Variable Fragment of T.
Journal of Philosophical Logic 43 (2014): 867-878.

We show that there are infinitely many pairwise non-equivalent formulae in one propositional variable p in the pure implication fragment of the logic T of "ticket entailment" proposed by Anderson and Belnap. This answers a question posed by R. K. Meyer.

This is the abstract of the paper:

JC Beall, Ross Brady, Mike Dunn, Allen Hazen, Ed Mares, Roberty Meyer, Graham Priest, Greg Restall, David Ripley, John Slaney and Richard Sylvan.
On the Ternary Relation and Conditionality.
Journal of Philosophical Logic 41 (2012): 595-612.

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.

This is the abstract of the paper:

John K. Slaney, Robert K. Meyer and Greg Restall.
Linear Arithmetic Desecsed.
Logique et Analyse 155-156 (1996): 379-387.

In classical and intuitionistic arithmetics, any formula implies a true equation, and a false equation implies anything. In weaker logics fewer implications hold. In this paper we rehearse known results about the relevant arithmetic R#, and we show that in linear arithmetic LL# by contrast false equations never imply true ones. As a result, linear arithmetic is desecsed. A formula A which entails 0=0 is a secondary equation; one entailed by ~(0=0) is a secondary unequation. A system of formal arithmetic is secsed if every extensional formula is either a secondary equation or a secondary unequation. We are indebted to the program MaGIC for the simple countermodel SZ7, on which 0=1 is not a secondary formula. This is a small but significant success for automated reasoning.

This is the abstract of the paper:

John Slaney.
Sentential Constants in Systems Near R.
Studia Logica 52 (1993): 443-455.

An Ackermann constant is a formula of sentential logic built up from the sentential constant t by closing under connectives. It is known that there are only finitely many non-equivalent Ackermann constants in the relevant logic R. In this paper it is shown that the most natural systems close to R but weaker than it - in particular the non-distributive system LR and the modalised system NR - allow infinitely many Ackermann constants to be distinguished. The argument in each case proceeds by construction of an algebraic model, infinite in the case of LR and of arbitrary finite size in the case of NR. The search for these models was aided by the computer program MaGIC (Matrix Generator for Implication Connectives) developed by the author at the Australian National University.

This is the introduction to the paper:

John K Slaney and Robert K Meyer.
A Structurally Complete Fragment of Relevant Logic.
Notre Dame Journal of Formal Logic 33 (1992): 561-566

This note contains a proof that the implication-conjunction fragment of the relevant logic R is structurally complete: in it, every admissible rule is derivable. The analogous result for the same fragment of intuitionist logic has been known at least since the early 1970s. The extension to relevant logic, however, is nontrivial and so far has been secured only for the one fragment.

This is the abstract of the paper:

John K Slaney.
The Ackermann Constant Problem: A Computer-Assisted Investigation.
Journal of Automated Reasoning 7 (1991): 453-474.

This is a report on researches carried out at the Australian National University some ten years ago which led to proofs of the results reported in [Slaney, Journal of Symbolic Logic 1985] and [Slaney, Notre Dame Journal of Formal Logic 1989]. While not automated reasoning per se this project made heavy and essential use of a computer as a proof assistant in ways to be detailed below. Some of the algorithms are described very briefly in a technical report (ANU 1988), but they have never been published. Nor has the story been told of how a solution to a non-trivial programming problem was used to obtain interesting and important theorems in pure logic. The present paper, then, is long overdue. I am grateful to the Australian National University for providing the opportunity to write it up.

This is the abstract of the paper:

John K Slaney.
On the Structure of De Morgan Monoids, with Corollaries on Relevant Logic and Theories.
Notre Dame Journal of Formal Logic 30 (1989): 117-129.

A De Morgan monoid is constant iff it is generated by its identity alone. It is shown that the only nontrivial proper homomorphic image of a prime De Morgan monoid in a constant one is the 4-element algebra C4. Moreover, the only element mapped by such a homomorphism to the lattice 0 of C4 is the lattice 0 of the original. These facts are used to obtain results on De Morgan monoids with idempotent generators. The paper concludes with some applications to the relevant logic R and particularly to the arithmetic R#.

This is the abstract of the paper:

John K Slaney.
3088 Varieties: A Solution to the Ackermann Constant Problem.
Journal of Symbolic Logic 50 (1985): 487-501.

It is shown that there are exactly six normal De Morgan monoids generated by the identity element alone. The free De Morgan monoid with no generators but the identity is characterised and shown to have exactly three thousand and eighty-eight elements. This result solves the "Ackermann constant problem" of describing the structure of sentential constants in the logic R.

```	      Dr J K Slaney                       Phone (Aus.):  (026) 125 8607
Theory Group                        Phone (Int.): +61 26 125 8607
Research School of Computer Science Fax (Aus.):    (026) 125 8651
Australian National University      Fax (Int.):   +61 26 125 8651
Canberra, ACT, 0200, AUSTRALIA
```