## Michael Lynch’s _Truth as One and Many_ on Domain Logical Pluralism

~1500 words, ~1200 w/o footnotes

Recently I was looking through Lynch’s book on truth, Truth as One and Many. I was interested in its chapter-long discussion of domain-specific logical pluralism. There he writes about the natural idea that different logics should apply in different domains: say, intuitionism in moral or mathematical domains, but classical logic when talking about physical objects.

I wanted to see what he thought about what we ought to do in the case of sentences mixed across domains. For example, a disjunction might have its first disjunct from one domain and one logic but its second from another. (The logics might disagree about how to handle $\lor$, after all.)

Here’s his idea (and, what this whole post is about):

Modest
where a compound proposition or inference contains propositions from distinct domains, the default governing logic is that of the compound or inference’s weakest member. (100)

Lynch says that “the weakest logic in play is that which has the fewest logical truths or which sanctions the fewest valid inferences” (100).1

The proposal is promising, but I am worried it stumbles with a case Lynch discusses. Consider someone who hews to classical logic in one domain—say, the domain of physical objects—but, because of commitment to constructivism, prefers intuitionistic logic in another domain—say, for mathematics. And let us assume that they accept Modest.

Such a person will accept the law of excluded middle for the first domain but not the second. When it comes to physical objects, they will think that it is right that $\vDash_c p \lor \neg p$, even where we have for $p$ something that they will never be able to decide for or against. (Say it is some statement about a tiny detail of the distant past.) But $\nvDash_i q \lor \neg q$, for $q$ may be something that our mathematical constructions have yet to decide for or against.

Now what should our modest constructivist say about

Problem Inference (PI)
$\vDash_? (p \lor \neg p) \lor q$

According to Modest they should use $\vDash_i$ and so reject that inference. There are intuitionist models where it just so happens that none of those literals get decided. (Another route to the same end: for any disjunctive theorem of intuitionistic logic, one of the disjuncts must be a theorem—which is not the case here.)

There are two problems with this. First, it seems bizarre to deny the inference. Our modest constructivist should be happy to infer $\vDash_c p \lor \neg p$, at which point they know of whatever detail of the distant past that it either was or wasn’t so. Subsequently weakening by adding the disjunct $q$ is acceptable in both logics; for both $\vDash_c$ and $\vDash_i$, it is right that $p \lor \neg p \vDash (p \lor \neg p) \lor q$. So the constructivist would be applying a rule which is acceptable on all the logics they like to what they already know, which seems like a fine way to reason.

Modest predicts that our constructivist should not count the sentence in PI as a theorem, as something they can infer from no premises. But it seems like our constructivist has a reasonable informal justification for inferring it from no premises. So the first problem is that Modest simply appears to get the wrong result.

Not only is there an informal justification, we can match it with a formal derivation, which brings us to the second problem.

It appears that the constructivist can get to the same conclusion from no premises if they construct an actual derivation in steps, first getting $\vdash_c p \lor \neg p$, then noting that $p \lor \neg p \vdash_i (p \lor \neg p) \lor q$, and then using a straightforward discharge rule.

Now they’ve got a formal derivation from no premises for an inference whose conclusion they regard as not a theorem; something must give.

In order to prevent this, it seems like our modest intuitionist has got to deny the discharge rule and say we don’t get to simply conclude things and then squirrel them away into the premise set without further comment. We have to remember what kind of reasoning we used to get them, if we’re going to use them in the future, and use them to discharge across a $\vdash$ only if the premise is from the appropriate place. I, anyway, find that unintuitive. It ought to be that once I deduce that something is true, I can use it in modus ponens (in any logic that supports modus ponens).2 It should be immaterial what logic I used to discover it.

So the second problem is that the constructivist doesn’t get to conclude that things are true or false, simpliciter, and remember them that way; but they have to keep track of the logical origins of their inferences, so that they are sure not to use a logic which is stronger than those origins. I worry that this is close to the kind of fragmentation of the truth concept that Lynch wanted to avoid—as if what is being remembered is the sort of truth we found that claim to have. Perhaps some other account of what is going on is available, but even thenthis only allows them to cleanly deny PI, which (remember the first problem) isn’t really the intuitive judgement about it, anyway.

* * *

Of course I have my own proposal. (And of course it has its own problems.) Our constructivist should think to themselves that the propositions about physical objects are all decided. Semantically, they think that every node in the construction decides for or against such a proposition (using the Kripke frame semantics); proof theoretically, they think that for those propositions, it is always right that $p \lor \neg p$. So they can produce a logic for both domains at once by typing propositions and endorsing the law of excluded middle for one type but not the other.

This hopefully gets the right result that, if $p$ and $q$ are typed appropriately, the above inference works, both semantically and proof-theoretically. (If everything works out like I’m picturing, anyway; it’s not like I’ve hashed out all the details.) If you buy this way of thinking, then it’s nice for this particular logical pluralist.

But the resulting logic is stronger than bare intuitionist logic. So Modest has a problem. And it is weaker than classical logic. So the most obvious alternative to Modest doesn’t look so good, either.

Even worse: it is not so easy to see how to extend this way of dealing with the problem to a general account of statements mixed across domains. It seems to have drawn on the details of the specific case. Even extending this idea to mixing classical and intuitionistic first-order logic seems non-trivial. The general formula seems to be: figure out what the models of the two logics are like—find a way to splice together the models from the two logics—figure out what the proof theory is for what you just created. This doesn’t seem like the sort of thing that can be mechanically applied whenever you want to see how two logics should interact. Whatever else might be said about Modest, it is a clean, simple, and clear rule.

* * *

Was I unfair to Modest? Did I make some terrible mistake? Comments welcome!

## Footnotes:

1

Lynch clearly means an order by inclusion, not by cardinality; the latter would nearly trivialize the principle. Lynch also suggests the more sophisticated Modest*: take the intersection of the logics rather than the weaker one. This deals with the case where neither logic is weaker. But it shouldn’t matter for our example.

Modest’s assumption seems to be that the two logics are in the same language, that they contain the same symbols. I don’t think Modest makes it obvious how to deal with cases where this isn’t true, and it’s too drastic to deny mixed inferences in all such cases. So probably Modest needs to be augmented somehow for this.

It seems to me that the most natural way to approach combining the languages in such cases requires us to know about the intended models, not just the syntax and logical truths of each language. Say I am trying to consider a conjunction with each conjunct in a different modal logic. Say each language uses a box in the usual way. Should the conjunction act like the same symbol appeared twice? Or does it distinguish between $\square_1$ and $\square_2$? Surely the second if one box was supposed to mean ‘always’ and the other ‘necessarily’; probably the first if the thought was that both mean ‘necessarily’, but came from languages treating different objects. So not only does Modest need augmentation to deal with such cases: it needs to include considerations of an entirely different sort, to pay attention to the intended models.

2

I think that for Lynch ‘true’ here should mean true simpliciter, not true relative to some domain: it is truth simpliciter that is preserved across inferences (87).

If you’re thinking of classical logic and intuitionistic logic as two equally valid ways of treating the single domain of mathematical logic, but two ways that judge things differently, that discharge rule looks much worse. But we’re supposed to consider two different logics that are simultaneously correct in making judgments using the same truth concept about different domains of objects.