[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 G Heck Jr
rgheck at brown.edu

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