[FOM] relevant logic and paraconsitent logic

Richard Heck rgheck at brown.edu
Wed Mar 1 12:16:50 EST 2006

```Joseph Vidal-Rosset wrote:

>...[I]f I accept relevant logic am I involved to accept paraconsitent 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?
>
>
There are ways out. In standard relevant logics, you can't have both A
and ~A being true. You may wish to have a look at Priest's book
/Introduction to Non-Classical Logic/.

I have this dim sense that there's an emerging consensus that, at least
in certain kinds of cases, paraconsistent logics aren't nearly as crazy
as they seem. Consider, for example, vagueness and supervaluational
semantics. Such a semantics assigns to each predicate, in each model, a
range of possible interpretations. We say that a formula is true in a
supervaluational model (or "supertrue") if it is true in all of those
interpretations. Such a logic fails bivalence---it may be that neither A
nor ~A is supertrue---but all classical validities are nonetheless valid.

It was noted some time ago by Achille Varzi (I don't know if he was the
first) that, given the same formal structure, we can consider
"subvaluations": A formula is subtrue if it is true in /some/
interpretation. Then both A and ~A can be subtrue (though you don't ever
have "A & ~A" being subtrue), so you don't have: A, ~A |= B, for
arbitrary B.

I'm not sure if Achille makes this point, but it was (at least
subsequently) noticed that you get the same logic if you stick with
supervaluations but define an inference \Gamma |= B to be valid if,
whenever no statement in \Gamma is superfalse, B is not superfalse,
either: That is, if you characterize validity in terms of non-falsity
preservation rather than in terms of truth-preservation. So, if you
accepted supervaluational semantics but thought, for whatever reason,
that the right norm of assertion was "Just don't say anything false",
then you'd sound just like someone who accepted subvaluational
semantics. A point in this vicinity is made in a paper by Vann McGee:
There are trade-offs to be made between how one thinks of the semantics
and how one thinks of the relevant norms.

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.

Richard

--
Richard G Heck Jr
rgheck at brown.edu
http://bobjweil.com/heck/

```