# [FOM] relevant logic and paraconsitent logic

Arnon Avron aa at tau.ac.il
Thu Mar 2 03:08:45 EST 2006

On Wed, Mar 01, 2006 at 11:25:56AM +0100, Joseph Vidal-Rosset wrote:

> I would be glad if experts in Logic of this list would be kind enough
> to develop on the list the links between paraconsistent logic and
> relevant logic.

Paraconsistent logics are (by definition) logics in which
the following fails:

P, ~P |- Q

where P and Q are atomic formulas and ~ is the official negation
connective of the logic. This definition for itself has nothing
to do with the question whether both P and ~P can be "true" in some
sense. It has to do only with the consequence relation (= logic).
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". However,
there are other methods to define logics (= consequence relations):
proof-theoretical, or even semantic. Thus one standard way
to define the consequence relation in relevance logics
is to let  A_1,...,A_n|-B iff the sentence A_1 & A_2 & ... & A_n -> B
is valid (and then validity of sentences may again be defined either
proof-theoretically or semantically). Alternatively, it is also
common there to define that A_1,...,A_n|-B iff the sentence
A_1 -> (A_2 -> ... ->(A_n -> B))...) is valid (these two methods
are *not* equivalent in relevance logics, and in general it is often
unclear what is the logic (= consequence relations)
which relevantists have in mind when they speak about relevance logic -
see my paper "Whither Relevance logic?", Journal of Philosophical Logic 21,
243-281, 1992). Now it is possible to provide simple semantics
in which {P,~P} has no model, but both ~P->(P->Q) and
~P & P -> Q are still semantically false. See e.g. my papers
"Relevant Entailment - Semantics and formal systems" (Journal of
Symbolic Logic 49, 334-342, 1984), "Relevance and Paraconsistency - A New
Approach (Journal of Symbolic Logic 55, 707-732, 1990) and
"Relevance and Paraconsistency - A New Approach,  Part II: the Formal
systems" (Notre Dame Journal of Formal Logic 31, 169-202, 1990).

In any case, relevance logics form a subclass of the class of
paraconsistent logics according to the official definition of
paraconsistent logics (and indeed relevance logicians regularly
took part in all the congresses on paraconsistent logics). However,
there are many families of paraconsistent logics which have nothing
to do with relevance logics.

> My request is mainly motivated by a logico-philosophical position:
>
> (1) I feel uneasy with logical systems accepting "true contradictions"
> (paraconsistent logics), but

As I wrote above, paraconsistent logics do not necessarily accept
"true contradictions". Some of then do. Others do not.

> Apart from the clarification I still need about paraconsistent logics, I
> would be thankful to f.o.m. subscribers if they could help me to reply
> technically to this logical question: if I accept relevant logic am I
> involved to accept paraconsistent logic? If (p & ~p) -> q is not valid,
> hence (p & ~p) must be true, or can I avoid this conclusion with the
> rejection of the universality of Bivalence principle or is there
> another way out?

Not to have (p & ~p) -> q as logically valid definitely does not
mean that "(p & ~p) must be true" - not even that there are
cases in which p & ~p is true. First, the invalidity of
(p & ~p) -> q does not imply the validity of ~((p & ~p) -> q).
What we can at most say (assuming some semantics which respects
LEM) is that from a semantic
point of view, if (p & ~p) -> q is not *valid* then there
should be cases in which ~((p & ~p) -> q) is *true*. However,
in relevance logic (and in most paraconsistent logics I know)
the classical tautology ~(A->B)->A is not valid, hence we cannot
infer p & ~p from ~((p & ~p) -> q)  (in relevance logics
~(A->B)->A is in fact equivalent to ~A->(A->B) - a "fallacy").
What's more: as I noted above one can give simple,
intuitive semantics in which it is
possible for both (p & ~p) -> q and (p & ~p) to be false.

> I need help. Of course I believe that logico-philosophical polemics on
> this topic will be welcome, because we are on the FOM list. :)

Personally I don't believe that paraconsistent logics in general
and relevance logics in particular are relevant to *mathematics*.
The only possible relevance might be in connection with the paradoxes
of naive set theory. Some people have suggested to accept that set theory
is inconsistent, and so to develop mathematics within
some paraconsistent logic. Without even discussing the plausibility
of these suggestions, they fail because naive set theory
remains trivial even if we do not use the negation connective at
all. This is due to Curry Paradox (given a proposition A
define the "set" {x| x\in x -> A}, and then derive A using this "set").
This paradox uses only the implication connective, and is based
on properties of this connective that relevance logics accept (in particular:
contraction). There were attempts to develop naive set theory
within logics that reject contraction (like Lukasiwixz logic
and Girard's Linear logic), but I don't find them very convincing.

Arnon Avron