Tuesday, October 20, 2009

Logical Pluralism, Monism and Logically Impossible Worlds

Logics are either:

(a) Overall theories of reality, relative to the high level of generality and abstractness at which formal languages operate, or
(b) Codifications of what we take to be good reasoning, or
(c) Something else

For the moment, let’s put aside (c) and focus on the debate between (a) and (b). On initial consideration, (b) may sound more natural and obvious. We do, after all, often use “logical” and “reasonable” as if they were synonymous, and the use of terms like “inductive logic” and “informal logic” certainly lends itself most easily to (b). On the other hand, normal talk of “logically possible worlds,” “logical impossibility” and so on most naturally lends itself to (a), and (a) has a long and glorious philosophical lineage, it or something along very much the same lines having been advocated (in one form or another) by figures as diverse in their other views as, for example, Frege, Quine, Penelope Maddy and Graham Priest.*

That list includes two people who believe that logical truths are “analytic” in some sense and two who reject the very notion of analyticity, two who accept the web-of-belief model of logical theory change and two who do not, two people who accept classical logic and two who reject it, one who believes in truth-value gaps and three who do not and one who believes in true contradictions and three who do not. Moreover, those categories and battle lines crisscross in dizzingly complex enough ways just among the overall views of those four thinkers to pretty much demand philosophical baseball cards to keep it all straight....the point is just that, far from being an eccentric consequence of any one perspective, it can fit with a variety of different overall packages of beliefs.

Further, note that (a) is even compatible with a wide variety of views about how logic and reasoning do relate to each other. That said, we might be able to say some general things about that relationship without bringing in very many extra assumptions. The theory of what it means to say that one thing follows from another that most naturally fits with (a) it the traditional view that validity is truth-preservation. (Roughly: Q follows from P iff P has never been and will never be true without Q being true as well. This gives us an obvious way to connect the claim that logic is all about giving us rules of entailment with the claim that logic is an overall theory about how the world is. For example, a philosopher who takes classical logic to give us the right overall theory of reality will claim that any time the world is configured in such a way as to make a disjunction true and one of its disjuncts false, it will be configured in a way that makes the other disjunct true. The dialetheist, who takes it that sometimes the world makes one disjunct both true and false and the other disjuct false only, a possibility the classical theorist rules out, will disagree with that, and the two sides proceed to engage in an argument about which way the world actually is.) This, in turn, generates at least one obvious point of connection between logic and good reasoning: logically valid arguments will never lead you from true premises to false conclusions, so if it’s reasonable to believe a bunch of premises, it’s also reasonable to (on the basis of the logical connection) believe the conclusions that can be validly inferred from those premises, hence Frege’s observation that although logic is not about how men think, it is about how they must think “if they are not to miss the truth.” So even if logically valid inferences don’t exhaust the category of reasonable inferences (as, on this view, it’s pretty clear that they don’t), they still constitute an important subcategory of it.

With the conceptual distinction between (a) and (b) firmly in mind, let’s move on to a related distinction. Logical pluralists believe that various logics are suitable for various purposes, but that there is no such thing as the One True Logic. By contrast, logical monists believe that there is some sense in which one logic is uniquely correct.

(a) seems to pretty much demand logical monism, unless of course one takes reality to be spectacularly inconsistent in a way that it’s hard to make sense of even dialetheically. (Note, for example, that Priest is a monist.) On the other hand, (b) fits very naturally with pluralism. Think about the notion of underlying logics, where “underlying logic” means, very roughly, something like the following.

*Logic L is the underlying logic of some context C iff L formally captures the standards of reasoning that are relevant to C.

Of course, what it means to be “relevant to C” here is importantly non-obvious. Are we talking about the standards implicit in what most people treat as good reasoning about C, or if not most people, then some special class of people (e.g. subject matter experts)? Or are we, maybe, talking about something a bit more objective, like “the standards of reasoning that are the most likely to lead to the truth about C” or “the standards of reasoning that are most likely to lead to progress towards some goal (other than truth) that’s important in C” or something like that?

These are important distinctions, and it’s usually hard to tell, when reading bits of underlying-logic-talk, exactly which position is implicitly being taken on these issues. Fortunately, at the moment, we won't have to clarify them for the purposes this discussion, so let’s put all of those issues to one side, and consider one last distinction. Given that we decide to get into the business of using underlying-logic-talk, either we take it to be the case that:

(d) Different contexts of reasoning involve different underlying logics, or
(e) There’s just one logic which is the unique underlying logic of every context of reasoning, or perhaps just of all the ones that are worth engaging with.

If we make the background assumption that (b) is the correct view about what logics are, then (d) just is logical-pluralism-expressed-in-underlying-logic-talk and (e) is logical-monism-expressed-in-underlying-logic-talk. Now, again relative to the assumption of (b), it’s pretty easy to argue for (d) and against (e).

After all, any given context of actual human reasoning is going to be finite. (Note that this is true even in the total context of "all reasoning that has ever, or will ever occur in any particular context whatsoever.”) Only so many inferences will be made, and thus only so many inferences will be classified as reasonable or unreasonable relative to that context. Formal logics, on the other hand, are total systems whose rules classify every inference as valid or invalid. As such, a variety of logics—in fact, an infiite number of possible systems of formal logic—will license all of the inferences regarded as good reasoning in that context. Thus, far from (e) making sense, it doesn’t look like we can specify a singular underlying logic even within a single context.

If you start out with a strong committment to (b), the obvious move is to take this as a reductio against logical monism. On the other hand, if you don’t start by making that crucial background assumption, it might make more sense to take it as a reductio against the usefulness of underlying-logic-talk.

With all that kept in mind, let’s backtrack from the special difficulties faced by (b)-plus-(e)-monism (which, obviously, don't touch (a)-monism) to a more general issue.

One might plausibly think that counterpossible reasoning is a problem for any sort of logical monism. Here's why:

The process of debate between deeply entrenched philosophical views seems to necessarily involve reasoning about cases in which each side takes it not just that the other side’s views are wrong, but that they *couldn’t* be right. (Although this sort of thing goes on all the time in other areas of philosophy, it’s particularly unavoidable in phil of logic, for obvious reasons.) That said, if we’re going to actually reason with each other about which of the disputing views is correct, rather than simply retreating behind our respective barricades to glare at each other, than we’ve got to make the case for our view (in part) by drawing out unappealing consequences of the views we reject. “X is absurd. After all, if X were true, then Y would be true.”

Now, on the most orthodox theory of counterfactual conditionals, “if X were true, Y would be true” just means that the closest possible worlds at which X is true are also worlds at which Y is true. This has the unfortunate consequence that all counterpossible conditionals come out as (vacuously) true. This is both wildly counterintuitive and (for the reasons just discussed) would, if taken seriously, make philosophical debate about basic issues impossible. As such, an attractive move to make here is to tweak the Lewissian rules for evaluating counterfactuals, such that a statement of the form “if X were true, Y would be true” is true iff the closest worlds, possible or impossible, at which X is true are also worlds at which Y is true. (A natural assumption to make is that possible worlds are just definitionally always closer to the actual world than even the closest impossible worlds.) So, while arguing against quantum logician’s claim that the distributive law (the conjunction of “A” and “B or C” entails the disjunction of “A and B” or “A and C”) fails, a partisan of distribution can say that “if the distributive law failed, then counter-intuitive consequence C would hold,” and we take it that that conditional is true if and only if the closest (impossible) worlds where the conjunctions and disjunctions of the properties of subatomic particles really do fail to distribute in the way the quantum logician claims that they fail to in the actual world are also worlds at which C holds.

So far, so good.

But wait.

One could make the following objection here. “On the old Lewissian picture where only logically possible worlds are allowed, the logical monist can evaluate whether the closest worlds at which X is true are worlds at which Y is true by using whatever they regard as the actually correct logic to reason about whether Y follows from X. Now, the (b)-plus-(d)-pluralist, who changes his logic due to variations of contexts of reasoning as easily as the rest of us change our shirts to due to variations in weather conditions, can handle this in an easy, straightforward way. When reasoning about whether C holds in the closest quantum-logical worlds, he can just reason about these worlds using quantum logic. But the logical monist faces a dilemma. Either she uses her favored (non-quantum) logic to reason about what’s true at the logically impossible world where distribution fails, and, given this inconsistency, she comes to many strange and irrelevant conclusions, or else she uses quantum logic when reasoning about the world where quantum logic is correct, and thus in effect becomes a logical pluralist.”**

Now, there are two things wrong with all of this. The first is that it’s not entirely true, even on the old only-logically-possible-worlds-need-apply model of counterfactual reasoning, that anyone reasoned about whether the closest worlds at which X was true were worlds at which Y was true by “using” the logic they used to determine the conditions for counting as a “logically possible world,” or, for that matter, by using any logic at all. In fact, it looks to me like interesting and informative counterfactual conditionals are very rarely ones in which the consequent is logically derivable from the antecedent. (After all, if Y is logically derivable from X, then Y is true at all of the logically possible worlds at which X is true, not juts the closest ones.) For example “if Hitler had won World War II, Germany would be the dominant power in post-War Europe” certainly seems to be true, but not because there’s a logically valid inference connecting the antecedent to the consequent.

Now, *if* we’re OK with underlying logic talk, we can re-phrase the point to get around this issue by saying that at least, there’s no reason to think that the underlying logic of reasoning about the actual world needs to be different from the underlying logic of reasoning about other logically possible worlds, whereas reasoning about what's true at some logically impossible world seems to pretty much demand a different underlying logic. Given this new phrasing, the first problem with the objection disappears. The problem posed for underlying logic talk by the failure of strictly logically valid inferences to exhaust the whole category of intuitively rational inferences doesn’t look like more of a problem for ordinary counterfactual reasoning than it does for factual reasoning, and assuming that there’s some sort of good solution to that problem when it comes to factual reasoning, that solution should carry over just as well to counterfactual reasoning.

To get a handle on the second and more important thing that’s wrong with the objection, we need to first answer the following question: Which sort of logical monism is this directed against? If the answer is that it’s a problem for any logical monist who wants to engage in counterpossible reasoning, we should still disambiguate monisms and try it out separately as an objection against each kind.

Taken as an objection to (b)-plus-(e) monists who are interested in counterpossible reasoning, it looks to me like it goes through. On the other hand, it’s a bit redundant. After all, as we saw above, (b)-plus-(e) monism fails on much more basic grounds than this, and the more basic objection works just as well even against (b)-plus-(e) monists who don’t think that it’s possible or desirably to (non-trivially) reason about counterpossible scenarios.

OK, how about (a)-monists? Well, an (a)-monist who accepts the notion of “underlying logics” can cheerfully admit that, when they reason about what’s the case in logically impossible worlds, the underlying logic of their reasoning is something other than the logic they take to be the uniquely correct overall theory of our reality. This semi-pluralism doesn’t, however, threaten, or indeed have much of anything to do with, the sense of logical monism that they endorse. (I say 'semi-pluralism' because to get to logical pluralism from these considerations, you need to combine your endorsement of (d) with an endorsement of (b).) On the other hand, I’ve already suggested that, if we don’t start out by assuming (b), the problem about uniquely specifying underlying logics sketched out above might be more naturally taken as a reductio against the very idea of underlying logics. An (a)-monist who took that route would not only be able to resist the push towards pluralism on the basis of counterpossible reasoning, but even the push towards the semi-pluralism involved in combining (a) with (d).

One way or another, anyone who accepts (a) has no reason to accept the notion that all reasoning involves “using a logic,” but, crucially, for the evaluation of what would and would not be the case in certain counterpossible scenarios, an (a)-monist might find it useful to consult the rules of a logic other than the one they take to the One True Logic. For example, when trying to figure out what’s the case in the failure-of-distribution-worlds that are the closest to the actual world (despite their logically impossibility), an important subcategory of good inferences will be those that are valid in quantum logic. Assuming that classical logic gives us the correct theory of the actual case, quantum logic will play the same role in relationship to our reasoning about what’s true at failure-of-distribution worlds that classical logic plays in relationship to our reasoning about what’s actually true. What of it? Once we reject (b), nothing in that picture gives us any more reason to abandon classical monism about logic than the uses of the details of Lemarckian biological theory for determining what’s true at the closest worlds at which acquired traits are genetically inherited gives us the slightest reason to abandon Darwinian monism about biology.

*This is clearest in the cases of Maddy, who explicitly claims in her book Second Philosophy that the laws of logic are like the laws of physics at a different level of generality, and Quine, who famously argues in Two Dogmas Of Empiricism that even logic is (rationally) revisable in the light of empirical evidence, and explicitly points to the possibility that classical logic will be falsified by quantum anomolies. Priest and Frege are more difficult cases, given that both men subscribe to some notion of "analytic truth," but at least in Priest's case, given his endorsement of the web-of-belief model in Two Dogmas Of Quineanism, and the explicit arguments he has made in various contexts for rejecting classical logic on empirical grounds, seems to more or less comfortably fit in the Quine/Maddy camp on this issue. As far as Frege goes, the evidence is weaker, but a very natural reading of claims like "logic is the science of the general laws of truth" and that logic gives us not the laws of "how men think, but of how they must think if they are not to miss the truth" is to see him as a type-(a) theorist, albeit a very un-Quine-like type-(a) theorist who takes logical laws to describe reality at a "third realm," non-empirical level. At the very least, without getting too deeply into thorny questions of Frege-interpretation, we can point out that the quoted statements are ones that a theorist who was solidly in the Quine/Maddy/Priest camp would enthusiastically endorse.

**This is certainly an objection that could be levelled against Graham Priest's overall package of views, given his explicit endorsement of logical monism in Doubt Truth To Be A Liar and of logically impossible worlds (at which, for example, there are no true contradictions) in Towards Non-Being.


Tuomas said...

That's interesting Ben, and I think quite accurate, although from what you said in the beginning I was hoping to see some more comparison between (a) and (b). I'm of course in the (a)-monism camp.

However, there's one thing that should be kept in mind here: (a) and (b) are not necessarily mutually exclusive: I there are at least some reasons to think that good reasoning just is the type of reasoning which the one true logic suggests.

Of course there will be cases where two mutually exclusive forms of reasoning will be 'good' in different situations, but an (a)-monist could for instance consider them to be local variations in the one true logic. There may be other ways to combine (a) and (b) as well.

Have you read Field's recent paper? (http://philosophy.fas.nyu.edu/docs/IO/1158/field_pluralism_in_logic.pdf)

Ben said...

Thanks, Tuomas. I haven't read Field's paper yet, but I will now.

I'm not sure I see quite how the "local variations in the One True Logic" move is supposed to work quite yet. Could you say a bit more about that?

Tuomas said...

I haven't got a particularly well-formulated idea to offer, but I had in mind something like this: it might not be feasible to impose one standard of 'good' reasoning for all contexts, but the One True Logic could perhaps accommodate a number of such standards. For instance, if quantum logic is considered to reflect 'good' reasoning when it comes to quantum phenomena, but in most other contexts reasoning conforming to classical logic is good, then why not just make the One True Logic context-dependent in this regard?

Depending on one's views, one might want to say that quantum logic is actually true in the context of quantum phenomena, while classical logic is true of some other phenomena; then we would have local variations in the One True Logic. But it may be possible to accommodate the idea even if one thinks that classical logic or something like it is the One True Logic in all contexts: it could still be considered 'good' or useful reasoning to use quantum logic in quantum contexts, if for no other reason then because we don't yet know how to model quantum phenomena in the One True Logic (i.e., we don't yet have that logic).

I'm not sure if this flies, but you see the idea?

I'd be interested to hear what you think about Field's paper...