# [FOM] relevant logic and paraconsistent logic

Katalin Bimbo kbimbo at indiana.edu
Fri Mar 17 13:37:20 EST 2006

```I'd like to add four references -- a recent book and 3
forthcoming handbook chapters to the discussion --
plus some remarks.

Relevance logics, following, for example, Church and Ackermann's
work, are logics that do not contain A -> (B -> A) as a
theorem (where `->' is relevant implication, i.e., A -> B is
not equivalent to -A v B).

There are interesting relevance logics that do not contain
a negation connective.  When negation occurs in a relevance
logic, it is typically not ``classical negation,'' rather
De Morgan negation.  This negation is characterized by
being order inverting and of period two; and A & -A
need not be less than an arbitrary B.  However, it is
easy to construct De Morgan lattices of larger that
any preset finite size with the property that negation
has a fixed point (i.e., A = -A for some A).  (Both
prime factorization lattices and Sugihara lattices are
suitable examples.)

Thus, one of the interesting results about T, E, and R
(the logics of "ticket entailment," of "entailment" and
of "relevant implication") is that the so-called gamma
rule is admissible.  That is, if A and -A v B are theorems
then so is B.  (The first proof of this theorem is by
Meyer and Dunn, JSL, 1969, and other proofs are in
Anderson and Belnap, Entailment, vol 1, and in Dunn,
Relevance logic and entailment, in Handbook of Philosophical
Logic, vol. 3, 1986.)  A consequence of the admissibility
of gamma is that the rule A, -A implies B is also admissible
(though inapplicable).

The references are:

R. Brady (ed.), Relevant Logics and their Rivals, vol. II,
(A continuation of the work of R. Sylvan, R. Meyer, V.
Plumwood and R. Brady), Burlington (VT), Ashgate, 2003.

G. Restall, Relevant and substructural logics, in Handbook
of the History of Logic, vol. 6, D. M. Gabbay and J. Woods
(eds.), Amsterdam, Elsevier, 2006.

N. C. A. da Costa, D. Krause and O. Bueno, Paraconsistent
logics and paraconsistency, in Handbook of the Philosophy
of Logic, D. Jacquette (ed.), Amsterdam, Elsevier
(North-Holland), 2006.

K. Bimbo, Relevance logics, in Handbook of the Philosophy
of Logic, D. Jacquette (ed.), Amsterdam, Elsevier
(North-Holland), 2006, pp. 723--789.

Kata Bimbo
-----------------------------
Indiana University
School of Informatics
Bloomington, IN 47408
E-mail:  <kbimbo at indiana.edu>

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