### Are truth-value gluts necessary to avoid explosion?

Before I get to the question in the title of the post, let me give a quick rehearsal of some uncontroversial material, for readers innocent of this particular topic. In classical logic, anything follows from a contradiction:

A & ~A, ∴ B

is a valid argument. This argument form is known as

*ex falso quodlibet*(EFQ); Graham Priest calls it 'explosion.' This clearly runs counter to our intuitions about what follows from what: it just doesn't seem like '2+2=200' follows from 'Grass is green and grass is not green.' Yet it does in classical logic.

Because this seems counterintuitive, people have devised logics in which EFQ is not a valid argument form. The most prominent is the family of relevance logics. So relevance logics score a point because they fit our intuitions about EFQ.

How do relevance logics avoid EFQ? Semantically/ Model-theoretically, they allow 'truth-value gluts', that is, a sentence can be both true and false. Now we can see why

*(A & ~A), ∴ B*is invalid in logics allowing gluts: assign A both true and false, and assign B false. Then all the premises are true and the conclusion is not true.

That was all set-up. Now the question: suppose it turns out that there are no truth-value gluts, i.e., no sentence (or proposition or whatever) is both true and false. Would (some) defenders of relevance logics then accept EFQ? Well, perhaps all the glut people need is that gluts are

*possible*, and not that

*actually*some sentence is both true and false. Then my question would be: would EFQ-deniers accept EFQ if gluts were impossible? From my limited exposure to the literature (Priest, C. Mortensen

*NDJFL*1983), it seemed like the answer might be

*yes*, because they say things like 'Disjunctive Syllogism (which is valid classically but not relevantly) is valid in all consistent reasoning-contexts.' I would've hoped that we could discard EFQ without taking on such a contentious idea as truth-value gluts...

And a further question just out of ignorance: does anyone characterize logical validity in such a way that it (i) avoids EFQ and (ii) does not require truth-preservation? I don't see any other way besides gluts to declare EFQ invalid, if we stick to the standard characterization of validity.

## 8 Comments:

"Well, perhaps all the glut people need is that gluts are possible, and not that actually some sentence is both true and false."

Whether this is a stable position or not is going to depend on your choice of logic. In LP you get the result that <>(p&~p) yields [<>(p&~p) & ~<>(p&~p)]. Aaron C has some discussion of this

here.

Sorry, I should say, the link opens a pdf.

There are a few ways to avoid explosion without allowing dialetheias. The "fuzzy logic" L (discussed in Graham Priest's "Introduction to Non-Classical Logic" in the chapter on "Fuzzy Logic") is paraconsistent (i.e. nonexplosive) but the highest truth-value (A^~A) can be assigned is .5, where entirely true=1 and entirely false=0. As a simpler example, if classical logic is altered so that the rule for disjunction introduction is changed to

A

B

->AvB

then you can retain disjunctive syllogism without contradictions exploding. Both of these still let you characterize logical validity as truth-preserving.

There is apparently a class of paraconsistent logics which are non-dialetheic, called "preservationist" logics. I only know about them from Brown's "Yes, Virginia, There Really Are Paraconsistent Logics", but they are out there.

Also, I suspect that the paraconsistentists would still reject

Quodlibeteven if dialetheias are impossible. For at least Priest is willing to entertain conditionals with impossible antecedents; he'd therefore want our logic to not lead us to infer erroneously from impossibilities, and soQuodlibethas to go, even if there are no true contradictions.If your notion of validity says that an argument is invalid only if it is possible that the premises are true and the conclusion false, then in that sense you'll have to accept the possibility of gluts in order to say that explosion is invalid. However, nothing here says that the notion of possibility has to be

metaphysicalpossibility - it can be the case that gluts are metaphysically impossible, even if they are logically possible, and this is what some relevantists are going to want to do. I suppose others might revise the notion of validity (but Beall and Restall are famous espousers of something of this form, and also both closely associated with some relevance logics).There is a question about what we mean by gluts in this context. Do we just mean any case where a sentence A and its negation ~A are both of them true? If so and if validity is truth-preservation, then the only way to avoid EFQ would be to allow gluts.

But: is validity truth-preservation? I guess that depends on what we mean by "validity". Are we talking about a technical consequence relation invented for the purposes of doing theoretical work in an impoverished language, or are we talking about some non-technical consequence relation that holds of natural language? If the latter, then it seems plausible that validity is

nottruth-preservation. I don't know what the right alternative account of validity would be, but just to throw out a toy theory, maybe validity is just some primitive notion. Then you can have failures of EFQ without gluts so long as EFQ is invalid according to the primitive, unanalyzable notion of validity which governs our language.Final thought. Suppose that validity is something like forward truth-preserving and backward falsity-preserving. Suppose also that some sentences of our language are indeterminate in truth value. Then you can potentially get failures of EFQ just on the grounds that you can have a false conclusion B and two premises A and ~A both of which are indeterminate, violating the falsity-preservation clause.

Thanks for the very helpful comments, everyone! Who needs to become acquainted with the literature when you've got a bunch of smart folks reading your blog?

Aidan: Thanks for the info and the pointer. Is that result limited to LP, or is it more widespread?

Daniel: Thanks for mentioning fuzzy logics; to me, assigning a sentence the semantic value 0.5 (like

A & ~A) is not assigning it the valuetrue. So that's a good example of a case where the logic avoids EFQ even though consequence is truth-preserving. (It's akin to Colin's 2nd point: by making consequence a stricter notion than just truth-preservation, we can cut off EFQ.)Kenny: Your response reminded me of something Priest said in the 1998

J.Phil."What's So Bad About Contradictions?" article -- though I prefer your way of putting it. Priest says: you might not believe that there are any true contradictions really out there, but you might still, in devising a logic, want to allow valuations to cover inconsistent scenarios, since we reason about impossible situations.Colin: Going primitive would certainly do the trick; so would simply specifying the syntactic rules for (e.g.) first degree entailment, and declaring them somehow more fundamental than any semantic icing you laid on top of them. And I like your 2nd suggestion too; though I haven't really thought about whether imposing a backwards-falsity preservation condition on consequence in a language with truth-value gaps would lead to unintuitive results.

Colin's point about backwards falsity preservation is of course from Belnap's "A Useful Four-Valued Logic", where validity is defined in terms of necessary truth-preservation and non-falsity-preservation.

Roy Cook makes an amusing use of the notion in his recent paper "Tonk Logic" - he uses the four Belnap values and then says that an inference is valid iff it is

eithertruth-preservingornon-falsity preserving, and then gives semantics for a connective that validates the tonk inferences. However, the entailment relation in this logic is non-transitive, which is of course strange. But in Belnap's system where both are required, it seems that you get exactly the behavior you might want.Thanks for the references, Kenny. I've been meaning to read Roy's paper on

tonkfor a while now. I think I first saw the idea of a backwards falsity preservation condition in a paper by Dana Scott written in the 70s; I don't know if that pre-dates "A Useful 4-Valued Logic." Now the historian in me is curious how far back into the history of many valued-logics we have to go to find the first expression of backwards falsehood preservation, and google is not being helpful...Post a Comment

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