[FOM] relevant logic and paraconsistent logic

Joao Marcos botocudo at gmail.com
Sat Mar 4 07:25:57 EST 2006

Quoting Joseph Vidal-Rosset:
> (1) I feel uneasy with logical systems accepting "true contradictions"

Me too.  At any rate, it is worth calling attention again to the following fact:

Quoting Arnon Avron:
>  Only in case this relation is defined semantically using the notion
> of "model" in the standard way (T |- A iff every "model" of T is a model
> of A), then it requires models in which both P and its negation are "true".

So, paraconsistent logics "by themselves" (that is, in terms of their
underlying consequence relations) do not require anyone to accept any
formula to be "both true and false".  These logics might require, at
most, if one abides by the standard notion of model, that there should
be (preferably _non-trivial_) models that satisfy both some specific
formula A and its negation ~A.  Of course, you might still call this
conjunctive event a "true contradiction", if you wish (unless it still
makes you feel uneasy, of course).  The point is that it is not at all
_necessary_ to assign more than one truth-value to a given formula for
the semantics of a paraconsistent logic to start making sense.

Quoting Panu Raatikainen:
> In a way, this is an instance of a very old point: The simplest
> example of a paraconsistent logic is the one you get by taking
> some ordinary three-valued truth-tables (say, the Kleene tables)
> and treating "neither true nor false" as a designated value.
> If you spoke Kleene but thought that the relevant norm was
> "Just don't say anything false", then you'd sound just like
> someone who thought there were true contradictions.

This 3-valued "dual-Kleene" example is indeed a very nice one.
Probably the first time it ever appeared was in Florencio Asenjo's "A
calculus of antinomies" (NDJFL, 7:103–105, 1966).  Later on it was of
course reused in Graham Priest's "The logic of paradox" (JPL,
8(2):219–241, 1979).

If only to dispute the "utmost simplicity" of the above example, there
is another variety of paraconsistent logics that I repute to be at
least equally simple, if not even simpler, namely the ones obtained
from any normal Kripke structure in which negation is interpreted as

M, w |= ~A iff there is some v such that wRv and M, v |/= A

The basic intuition is clearly DUAL to the one that sustains the
interpretation of negation (as "necessarily-not") in intuitionistic
logic.  Some standard references to the study of (paraconsistent)
negation as "possibly-not" can be found in my paper "Nearly every
normal modal logic is paranormal" (Logique et Analyse
48(189–192):279–300, 2005).

Quoting Panu Raatikainen:
> Quoting Edwin Mares:
> > A paraconsistent logic is any logic that rejects as valid the rule
> > scheme: A, ~A => B.
> So, minimal logic (which is a weakening of intuitionistic logic) is a
> paraconsistent logic.

Indeed, *this is so*.  In my paper "A taxonomy of C-systems" (in
Paraconsistency: The logical way to the inconsistent, vol.228 of
Lecture Notes in Pure and
Applied Mathematics, pages 1–94, Marcel Dekker, 2002) I called a logic
L _partially explosive_ if there is some schema \sigma(q) such that

(1)  => \sigma(B) is not a valid rule,
(2) A, ~A => \sigma(B) is valid.

Obviously, the above mentioned _explosion rule_ rejected by
paraconsistent logics is but a very particular case of partial
explosion.  Now, a logic L may be called _boldly paraconsistent_ in
case it is not partially explosive.  As far as I know, the first paper
to call attention to bold paraconsistency might have been Igor Urbas's
"Paraconsistency" (Studies in Soviet Thought, 39:343–354, 1990).

A nice survey of partially explosive paraconsistent logics can be
found in Sergei Odintsov's "On the structure of paraconsistent
extensions of Johansson's logic" (Journal of Applied Logic, 3(1):
43–65, 2005).  The immense majority of paraconsistent logics from the
literature, however, are *boldly paraconsistent*.  On that issue, be
sure to check also my chapter "Logics of Formal Inconsistency",
forthcoming at v.14 of the 2nd edition of the Handbook of
Philosophical Logic.

Joao Marcos

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