***

A well-known feature of standard paraconsistent logics, such as LP, is that they are weak in a certain sense. Thus, they do no validate the Disjunctive Syllogism, even though there are clearly cases where we would want to use it. I have argued that a way around this apparent problem is to employ a non-monotonic extension of LP, LPm, which is stronger.1LPm can be applied to any situation, and in that sense it is a universal logic: it gives classical reasoning in consistent situations and an inference engine at least as generous as LP in inconsistent situations.

There has to be more to matters than this, though. After all, reasoning classically in all situations would do that job as well. This is why what I called Reassurance is important. It guarantees that if a theory is non-trivial under LP consequence, it is non-trivial under LPm consequence as well – unlike classical consequence, which explodes any inconsistent information, producing triviality. In other words, taking triviality to be some kind of incoherence, LPm will never turn a coherent situation into an incoherent one.2

Jc Beall has recently objected to the strategy of using LPm.3 His complaint is that the use of LPm may deliver some other kind of incoherence. What we should require, he says (Section 3), is General Reassurance: if the consequences of some information under LP are all true, so are all the consequences under LPm. And this we do not have. For example, if p! is true and q is untrue, then p! ∨ q is true, and p! ∨ qm q.4

General Reassurance, however, is too much to ask. LPm is a nonmonotonic (aka inductive) logic. And it is precisely the definition of such logics that they may lead us from truth to untruth. The point is as old a Hume (‘The sun has risen every day so far. So the sun will rise tomorrow.’) and as new as that much over-worked member of the spheniscidae (‘Tweety is a bird. So Tweety flies.’) If they did not have this property, these logics would be deductive logics, which they are not. This is not a bug of such logics; it is a feature. Such logics do not preserve truth, by definition.5

The only way for Beall’s point to have force is, thus, to endorse the old Hume/Popper complaint about using non-deductive reasoning. This is not the place to review past debates on this claim. Let me just say that I take Hume and Popper to have lost that debate. Investigating most things using only deductive logic is like going into a fight with both hands (and a foot) tied behind one’s back.

Beall (in correspondence) tells me that he was not so much worried about inferring falsehoods, as inferring particularly absurd falsehoods, such as that you are a fried egg. This may not be triviality, but it is some lesser kind of incoherence. This objection misses the mark as well. The reason that such things are absurd is that we already have good reason to suppose them to be false: their negation is part of our total current information. LPm will not allow such things to be established if LP does not. Thus, though p! ∨ qm q, ¬q, p! ∨ qm q. There are mi models of the premises where p is both true and false, and q is just false. Generally, suppose that ∑ ⊭LPA. Then there is a model of ∑ where A is not true. Hence, there is an mi model, forumla, of ∑ where A is not true.6 Since ¬A is true in this interpretation, forumla is also an mi model of {¬A} ∪ ∑. (Any interpretation more consistent than forumla is not a model of ∑, and a fortiori of {¬A} ∪ ∑.) Hence, {¬A} ∪ ∑ ⊭m A.

There may or may not, of course, be better ways of going about the ‘recapture of classical logic in consistent situations’ than using LPm. But the considerations Beall has adduced do not show the approach using LPm to be flawed.7

1 See, e.g. Priest 2006: Ch. 16, and Ch. 19, §10.
2 Actually, Reassurance may be more than is required here. It might be quite sufficient if mostly, or normally, LPm did not turn a non-trivial inconsistent situation into a trivial one. If there are some exceptions, and LPm is otherwise robust, we might take this fact to speak against the coherence of inconsistent situations it shows to be trivial.
3 Beall (2012). He refers to LPm as MiLP.
4 I follow Beall in matters of notation. Note that p!, p! ∨ qm q.
5 Ironically, one place where LPm is guaranteed to preserve truth is with trivial information!
6 Priest, 2006: 226, Lemma 2. A similar result can be proved when the language is first-order, but the matter is more complex, and the details are still, to a certain extent, sub-judice. (See Crabbé 2011.) So I will not go into the details here, since Beall himself is concerned only with the propositional case.
7 Thanks go to JC Beall for helpful discussions of an earlier draft of this note.

References

Beall
J.
Why Priest’s reassurance is not so reassuring
Analysis
 , 
2012
, vol. 
72
 (pg. 
517
-
25
)
Crabbé
M.
Reassurance for the logic of paradox
Review of Symbolic Logic
 , 
2011
, vol. 
4
 (pg. 
479
-
85
)
Priest
G.
In Contradiction
 , 
2006
2nd edn.
Oxford
Oxford University Press