## Selected papers on Substructural Logics## John Slaney
This is the abstract of the paper: John Slaney. This paper presents F, a substructural logic designed to treat vagueness. Weaker than Łukasiewicz's infinitely valued logic, it is presented first in a natural deduction system, then given a Kripke semantics in the manner of Routley and Meyer's ternary relational semantics for R and related systems, but in this case, the points are motivated as degrees to which the truth could be stretched. Soundness and completeness are proved, not only for the propositional system, but also for its extension with first-order quantifiers. The first-order models allow not only objects with vague properties, but also objects whose very existence is a matter of degree.
This is the abstract of the paper: Tomasz Kowalski and John Slaney. It is shown that the pure (strict) implication fragment of the modal logic S3 has finitely many non-equivalent formulae in one variable. The exact number of such formulae is not known. We show that this finiteness result is the best possible, since the analogous fragment of S4, and therefore of S3, in two variables has infinitely many non-equivalent formulae.
More Proofs of an Axiom of Lukasiewicz This is the abstract of the paper: John Slaney.
This paper reports results and some new problems in one of the domains to
which automatic first-order theorem provers have been most successfully
applied: axiomatics of non-classical propositional logics. It is well known
that one of the standard axioms of the denumerable-valued pure implication
logic of Lukasiewicz becomes derivable from the remainder in the presence of
negation. Here it is shown that the same axiom is similarly derivable using
conjunction and disjunction instead of negation. This closes a problem left
open by Harris and Fitelson
This is the abstract of the paper: John Slaney and Robert Meyer. This is an account of the semantics of a family of logics whose paradigm member is the relevant logic R of Anderson and Belnap. The formal semantic theory is well worn, having been discussed in the literature of such logics for over a quarter of a century. What is new here is the explication of that formal machinery in a way intended to make sense of it for those who have claimed it to be esoteric, `merely formal' or downright impenetrable. Our further goal is to put these logics in the service of practical reasoning systems, since the basic concept of our treatment is that of an agent a reasoning to conclusions using as assumptions the theory of agent b, where a and b may or may not be the same. This concept requires true multi-agent reasoning, as opposed to what is merely reasoning by multiple agents.
Finite Models for some Substructural Logics This is the abstract of the paper: J K Slaney. This paper is a report of computer-driven research into the modelling of propositional systems related both to linear logic and to relevant logic. One result is an exponential lower bound on the number of models of given size validating such systems. Another theorem explores dualities within models for such systems in order to explain striking regularities in the distribution of models of the system C. In these researches the computer figures not as a prover of theorems, nor even as a proof assistant, but as a source of quasi-empirical data.
Classical Versions of BCI, BCK and BCIW Logics This is an abstract of the paper: John K Slaney and Martin W. Bunder.
We give an implicational formula
This is the introduction to the paper: John Slaney.
This is an essay in logic in the traditional sense: the formal theory of
inference. In it I attempt a unified account of a fairly wide range of logical
systems, some very well known and others less so. These systems include
classical logic, relevant logics such as Anderson and Belnap's R, close
relatives of fuzzy logic, some modal logics and many weaker, but still
interesting, nonstandard systems. These diverse logics, quite different in
their philosophical underpinnings, are displayed as variations on a single and
simple theme. Although its concern is with formal logic this paper is designed
to be accessible to non-specialists. It only assumes familiarity with a
natural-deduction formulation of elementary logic such as that of Lemmon's
Little of the formal material of this paper is really new. In particular,
following the work of Dunn, Meyer, Sylvan etc, relevant logicians have been
using the most important technical device of my presentation at least since
1973 (see J.M. Dunn's `Gentzen system for Positive Relevant Implication' in
Anderson and Belnap's
Solution to a Problem of Ono and Komori This is the introduction to the paper: John Slaney.
The logics DBCK and DBCC result from BCK and BCC respectively by the addition
of a postulate for the distribution of conjunction over disjunction. Ono and
Komori (
Reduced Models for Relevant Logics Without WI This is an abstract of the paper: John Slaney.
The object of this paper is to prove a reduced modelling theorem for a wide
range of logics which are weaker than the usual relevant logics in that they
lack the rule of contraction. Reduced modelling means that in the
Routley-Meyer semantics for these systems the base world
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 |