Book review: Overfitting and heuristics in philosophy

Timothy Williamson’s Overfitting and Heuristics in Philosophy is both a work of philosophical methodology and a series of case studies in which that methodology is applied to various debates in metaphysics, epistemology, and the philosophy of language. Williamson is a paradigmatic bullet biter—he’s known for defending the views that there is a single hair that marks the boundary between being bald and not (1994) and that he (along with the rest of us and everything else) could not have failed to exist (2013). While other philosophers might bend over backwards to avoid commitment to such consequences, Overfitting and Heuristics in Philosophy borrows some methodological tools from contemporary science to argue that philosophers as a group are too quick to modify powerful, explanatory theories in the face of putative counterexamples. While I have my quibbles with various of the applications—see below—in my view, the book makes a powerful, persuasive case for adding the concepts of overfitting and heuristics to the philosophical toolkit, thereby nudging us towards more Williamsonian willingness to buy simplicity at the price of counterintuitiveness.

Heuristics first. Philosophers will likely be familiar with the basic idea: when exact calculation is costly, we rely on quick-and-dirty rules that are reliable enough in most environments. Williamson argues that this fact can undercut putative counterexamples in philosophy. For instance, take the above-mentioned judgment that one hair can’t make the difference between being bald and not. This and other “tolerance” judgments famously give rise to paradox. Williamson rejects them. He invokes heuristics as part of an error theory to make that rejection easier to swallow. Williamson argues that tolerance judgments are exactly what we’d expect from a generally reliable—but fallible—“persistence heuristic”. We default to assuming small changes make no difference because, in everyday life, that saves time and is usually safe. Williamson marshals a variety of considerations in support of the idea that some heuristic along these lines plays an essential role in making cognition tractable, and that it sometimes—even in cases that don’t involve vagueness—can go very wrong.

Next, overfitting. In machine learning a model with lots of free parameters can be tuned to match every training datum, yet that very flexibility makes it fragile on new cases. Williamson claims that a similar pathology afflicts many philosophical theories. Here’s an illustrative example. Influential, explanatorily powerful formal models of belief in philosophy and economics have the result that agents who believe P and P ⊃ Q thereby believe Q. This seems like a bad consequence, as it’s possible—isn’t it?—to believe P and P ⊃ Q without believing Q. Some philosophers respond by enriching their formal models of belief—characterizing the contents of belief in terms of impossible worlds, or some other hyperintensional machinery, until any pattern of assent and dissent can be represented. But, Williamson argues, this amounts to overfitting on intuitive data about the possibility of logical error; theoretical frameworks flexible enough to represent logical ignorance tend to be so flexible that they sacrifice all their explanatory power. Williamson’s counsel is to do the opposite: keep the streamlined, logically omniscient model of belief as the best formal framework we’ve got, acknowledge that ordinary thinkers fall short of that model in ways that it’s not particularly fruitful to try to capture in systematic theory, and bite whatever bullets that entails. Better to tolerate some apparent counterexamples than to torture your theory until it becomes a mirror to the data it pretends to explain.

A recurring theme in the book is Williamson’s defense of intensional over hyperintensional frameworks for doing metaphysics, semantics, and philosophy of mind and language. Very roughly, intensional frameworks cannot distinguish between necessary equivalents, while hyperintensional frameworks can. If it is necessary that an action maximizes happiness if and only if it’s right, then happiness-maximization and rightness are the very same property; an intensional metaphysician will not distinguish them. Similarly, if “Superman flies” and “Clark Kent flies” are necessarily equivalent, then they express the very same proposition. So, if propositions are the objects of belief, anyone who believes that Superman flies believes that Clark Kent flies. These consequences can seem absurd. Isn’t it part of utilitarian doctrine that happiness-maximization grounds rightness but not vice versa? This sort of asymmetric relationship is only possible if happiness-maximization and rightness are not the very same property. And isn’t it possible to believe that Superman flies without believing that Clark Kent flies? Intensional metaphysics seems to leave us unable to even express the positions metaphysicians want to debate.

Williamson argues, however, that hyperintensional frameworks fail to explain these distinctions. Rather, they merely provide a language in which those distinctions can be stated. To get the desired results, one typically has to put them in by hand—stipulating that “Clark Kent” and “Superman” receive distinct semantic values, or that rightness and utility-maximization are distinct properties—without deriving those distinctions from independent, predictively fruitful principles. To see why this is a problem, note that we don’t want to be able to capture a difference between believing that a god created the universe and believing that a deity created the universe. But the flexibility afforded by hyperintensional metaphysics can just as easily distinguish “god” and “deity” as it can distinguish “Superman” and “Clark Kent”. Better, Williamson argues, to have leaner, intensional models that fail to fit some data, for the sake of being able to genuinely explain the data that they do capture.

I referred in the previous paragraph to “data”. Williamson is explicit in Overfitting and Heuristics in Philosophy in adopting a non-factive conception of data, and distinguishing data from evidence (56). Evidence, according to Williamson, is knowledge, and knowledge must be true (2000). But data can be false. He seems to conceive of the debates between himself and his opponents as concerning how best to explain a shared body of data, consisting in the contents of various philosophical judgments. Williamson will grant that it’s a datum that one grain of sand can’t make the difference between a heap and a non-heap, or that someone can believe that Superman flies without believing that Clark Kent flies. He just holds that when one considers these data together with lots of other relevant data and aims to adopt the models that best explain the totality of the data, one will end up favoring models in which the data hyperintensionalists emphasize end up being understood as something like outliers.

While I doubt there is any strict inconsistency between this approach and his earlier work, I do want to draw attention to some differences in emphasis. In both (2000) and (2007), evidence is a central category, and its factivity plays an important role in the dialectic of both books. The skeptic who tries to convince you that you don’t know you have hands, or that you don’t know that there are composite objects, can’t really be debated but should instead be dismissed. In particular, it’s a mug’s game to retreat to some dialectically uncontroversial body of evidence—consisting of how things seem to you or what you’re so inclined to judge—and to then try to argue for your views over skeptical alternatives appealing only to that impoverished set of considerations.

In this book, however, it’s Williamson in the position of the skeptic, arguing that intensional frameworks shouldn’t be dismissed on the basis of putative facts inconsistent with them. Now he’s the one “psychologizing the evidence” (or rather, the data), arguing that, e.g., he can explain why we make the judgment that one grain of sand can’t make the difference between a heap and a non-heap in terms of the operation of fallible heuristics. It’s a fruitful exercise, I think, to imagine how hyperintensional metaphysicians might appeal to Williamson’s earlier work to resist such a strategy. “If my only relevant evidence was that I judged that there’s a difference between believing that Superman flies and believing that Clark Kent flies, then your strategy might be apposite. But my evidence is that there is such a difference, and therefore I can know that your intensional metaphysics is false.” Of course, it’s also a running theme in Williamson’s work that we can be mistaken about what our evidence is, and I’m sure that’s how he’d diagnose my hypothetical hyperintensionalist.

A few small quibbles. In my judgment, Williamson makes a powerful case for thinking of the persistence heuristic as a general-purpose cognitive shortcut that may misfire when thinking about the semantics of vagueness. I think he’s on weaker ground in his discussion of what he calls the “suppositional heuristic” for the indicative conditional. Many writers have some version of the following view about indicative conditionals: “If A, B” expresses a conditional endorsement of B, on the supposition that A. This idea is associated with the “Ramsey test” for conditionals—to decide whether to endorse “If A, B”, first suppose A, and then within the scope of that supposition, consider whether B. Whatever your attitude towards B under the supposition that A, that should be your attitude towards “If A, B”. Williamson’s view, defended in (2020), is that the natural language indicative conditional is just the material conditional: “If A, B” is true just in case A ⊃ B. The Ramsey test, rather than playing a constitutive role in determining the meaning of the indicative conditional, is merely a heuristic we use to decide whether the material conditional is true.

If we wanted heuristics for fallibly deciding whether the material conditional is true, we could do a lot better than the suppositional heuristic. Consider a case where A is obviously overwhelmingly unlikely, while B is difficult to evaluate on the supposition that A. Executing the suppositional heuristic will be cognitively costly and may well lead to a false judgment. For instance: “If alien spaceships land on earth tomorrow, world leaders will respond by attacking them.” If you want a heuristic for evaluating that claim considered as a material conditional, it’s very easy: just affirm it. You’re almost certain to be right, since it’s almost certain that alien spaceships won’t land on earth tomorrow. By contrast, if you use the suppositional heuristic, you’ll have to think pretty hard—how would world leaders respond?—and if you judge, understandably enough, that they wouldn’t attack and so end up denying the conditional, you’re almost certainly going wrong. If we really were trying to guess at the truth of the material conditional, it would make much more sense to reserve the suppositional heuristic for conditionals in which the antecedent has a decent chance of being true.

My last nit to pick concerns guise-relative belief, to which Williamson appeals in attempting to make his treatment of Frege cases more palatable. It’s a familiar thought that the same proposition can be believed under different guises and that action varies with the guise. But Williamson offers no theory of guises, and no constraints on how beliefs-under-guises interact with values to rationalize choice. If, like me, you take the smooth interface between intensionalism and rational choice theory to be a chief virtue of the view—economists model belief along intentionalist lines—it is deflating to adopt an intensionalism that gives up on explaining data concerning rational choice. Intentionalists should work hard not to sideline this data. While Williamson mentions fragmentationist versions of intentionalism only to set them aside (248), in my view, they provide the most promising route for reconciling intentionalism with rational choice theory (Elga and Rayo, 2021).

Even where I disagree with his applications, I find it natural and fruitful to do so in his idiom—I agree that we can debunk counterexamples by identifying heuristics but just disagree about whether he’s plausibly identified a heuristic in this or that case. I agree that we can dismiss recalcitrant data as outliers if it’s necessary for retaining an overall simple and explanatory theory, but I suspect he’s too quick to dismiss too much data about rational choice and that it can be captured without overfitting. That I want to frame these disagreements in his own terms is, to my mind, the best proof that the book succeeds on its methodological promise.

REFERENCES

Elga, A. and Rayo, A. (2021). Fragmentation and logical omniscience. Noûs, 56(3):716–741.

Williamson, T. (1994). Vagueness. Routledge, London.

Williamson, T. (2000). Knowledge and its Limits. Oxford University Press.

Williamson, T. (2007). The Philosophy of Philosophy. Blackwell, Oxford.

Williamson, T. (2013). Modal Logic as Metaphysics. Oxford University Press, Oxford, England.

Williamson, T. (2020). Suppose and Tell: The Semantics and Heuristics of  Conditionals. Oxford, England.