Selected papers on Paraconsistent Logics

John Slaney

Relevant Logic and Paraconsistency

This is the abstract of the paper:

John Slaney.
Relevant Logic and Paraconsistency.
Bertossi, Hunter and Schaub (eds), Inconsistency Tolerance Springer, 2004.

This is an account of the approach to paraconsistency associated with relevant logic. The logic fde of first degree entailments is shown to arise naturally out of the deeper concerns of relevant logic. The relationship between relevant logic and resolution, and especially the disjunctive syllogism, is then examined. The relevant refusal to validate these inferences is defended, and finally it is suggested that more needs to be done towards a satisfactory theory of when they may nonetheless safely be used.

A, Still Adorable

This is the abstract of the paper:

Robert Meyer and John Slaney.
A, Still Adorable.
Paraconsistency: The Logical Way to the Inconsistent (Proceedings of the world congress on paraconsistency held in Saõ Paulo) (2002): 241-260.

This paper expands on an earlier paper, "Abelian Logic (from A to Z)", by the same authors. The main result is that the Abelian logic A is rejection-complete; i.e., that each formula B is either a theorem of A or else leads in the style of Lukasiewicz to a proof of the variable p. Meredith's single axiom formulation is introduced for the implicational fragment of A and the finite model property proved. An interesting normal form is shown for A, using distribution laws not normally available to relevant logics.

Realistic Belief Revision

This is the abstract of the paper:

Greg Restall and John Slaney.
Realistic Belief Revision.
Proceedings of the second World Conference on the Fundamentals of Artificial Intelligence (1995): 367-378.

In this paper we consider the implications for belief revision of weakening the logic under which belief sets are taken to be closed. A widely held view is that the usual belief revision functions are highly classical, especially in being driven by consistency. We show that, on the contrary, the standard representation theorems still hold for paraconsistent belief revision. Then we give conditions under which consistency is preserved by revisions, and we show that this modelling allows for the gradual revision of inconsistency.

The Implications of Paraconsistency

This is the abstract of the paper:

John Slaney.
The Implications of Paraconsistency.
Proceedings of the twelfth International Joint Conference on Artificial Intelligence (IJCAI-12) (1991): 1052-1057.

This is a connected series of arguments concerning paraconsistent logic. It is argued first that paraconsistency is an option worth pursuing in automated reasoning, then that the most popular paraconsistent logic, fde, is inadequate for the reconstruction of essential first order arguments. After a case is made for regarding quantifiers as dyadic rather than monadic operators, it is shown that the addition of such quantifiers to fde allows an implication connective to be defined yielding the known logic BN4. Refining the treatment of implication in a manner similar to that found in intuitionist logic leads to the more interesting system BN.

RWX is not Curry Paraconsistent

This is an abstract of the paper:

John Slaney.
RWX is not Curry Paraconsistent.
Paraconsistent Logic: Essays on the Inconsistent Priest, Routley and Norman (eds), Munich, Philosophia-Verlag, 1989: 472-480.

It is shown that the naive comprehension axiom leads to triviality in the context of the logic RWX (relevant logic without contraction, plus the law of the excluded middle) given that an absurd constant can be defined. The version of the semantic and set-theoretic antinomies noted here is more damaging than the usual ones, in that it afflicts a contraction-free logic.

Abelian Logic

This is an abstract of the paper:

Robert Meyer and John Slaney.
Abelien Logic (from A to Z).
Paraconsistent Logic: Essays on the Inconsistent Priest, Routley and Norman (eds), Munich, Philosophia-Verlag, 1989: 245-288.

We axiomatise and discuss Abelian logic: the logic which has as its models the lattice-ordered abelian groups. Although published in 1989, this work was presented to a meeting of the Australasian Association for Logic in 1979, and has existed in a technical report version since then.

Consistency of Propositional Comprehension

This is an abstract of the paper:

John Slaney.
A Simple Proof of Consistency for a Naive Axiom of Propositional Comprehension.
Journal of Non-classical Logic 4 (1987): 23-31.

This is a proof of the consistency of the continuum-valued logic of Lukasiewicz with all the standard connectives plus propositional quantifiers and a naive axiom scheme of propositional comprehension: for every formula A there is a proposition p such that p is equivalent to A. That includes cases in which the variable p occurs free in A - for example, where A is "not-p"! The fact that this theory is consistent is already known; what is new is the very simple model-theoretic proof which establishes a connection between paraconsistent logic and real analysis.

	      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