No brute facts: The Principle of Sufficient Reason in ordinary thought

The Principle of Sufficient Reason (PSR) has been an influential thesis since the earliest stages of western philosophy. According to a simple version of the PSR, for every fact, there must be an explanation of that fact. In the present research, we investigate whether people presuppose a PSR-like principle in ordinary judgment. Across five studies ( N = 1121 in total, U.S., Prolific), we find that participants consistently make judgments that conform to the PSR. Such judgments predictably track the metaphysical aspects of explanation relevant to the PSR (Study 1) and diverge from related epistemic judgments about expected explanations (Study 2) and value judgments about desired explanations (Study 3). Moreover, we find participants ’ PSR-conforming judgments apply to a large set of facts that were sampled from random Wikipedia entries (Studies 4 – 5). Altogether, the present research suggests that a metaphysical presumption plays an important role in our explanatory inquiry, one that is distinct from the role of the epistemic and non-epistemic values that have been the focus of much recent work in cognitive psychology and philosophy of science.


Introduction
Explanation is essential to how we understand and act in the world around us. Often, we expect to find an explanation for what we experience, accept or dismiss evidence that bears on candidate explanations, and so on. We also typically want to find an explanation because we find it valuable, deem the search for explanations worth our effort, and so on. Much work in cognitive psychology and philosophy of science emphasizes the role of these epistemic and non-epistemic values during explanatory inquiry (see, e.g., Douglas, 2014;Lombrozo, 2016;Lipton, 2004). However, an intriguing possibility is that a distinctive metaphysical presumption also plays an important role. Perhaps, we presume that an explanation must exist for every fact-even if we judge we cannot come to know it, or it would not be valuable for us. If so, we presuppose a version of the Principle of Sufficient Reason (PSR), which states, roughly: for every fact, there has to be an explanation of that fact.
With this paper, we set out to investigate the extent to which people presuppose the PSR in ordinary judgment. Across five studies, we find that people indeed make PSR-conforming judgments that (i) predictably diverge from related epistemic and value judgments (Studies 1-3) and (ii) apply to a large set of facts sampled from random Wikipedia entries . Altogether, the present research suggests that a metaphysical presumption plays an important role in our explanatory lives, one that is distinct from the role of epistemic and non-epistemic values that have been the focus of much recent work in cognitive psychology and philosophy of science. To begin, we will briefly introduce the philosophical importance of PSR and then detail how a PSR-like presumption can be differentiated from other components of explanatory cognition.

The PSR in philosophy
The PSR has a prominent place in the history of western philosophy (cf. Amijee, 2020;Melamed & Lin, 2021). Most notably, it lies at the core of rationalist metaphysics, old (Descartes, 1641(Descartes, /1984Leibniz, 1714Leibniz, / 1989Spinoza, 1632Spinoza, /1985 and new (Amijee, 2021;Dasgupta, 2016;Della Rocca, 2010). Famously, some or other version of the PSR is a crucial premise in Leibniz's (1714Leibniz's ( /1989), Spinoza's (1631Spinoza's ( /1985, and Descartes's (1641Descartes's ( /1984 cosmological arguments in favor of the existence of God. 1 A very simple version of the argument says that the existence of the universe would be unexplained unless God exists. And, as the PSR states, there are no unexplained things. Hence, God must exist and explain the universe. 2 Beyond the cosmological argument, the PSR is crucial to a number of notable arguments in rationalist philosophy. For instance, it is central to Leibniz's argument for the relativity of space and time (Leibniz, 1714(Leibniz, / 1989) and Spinoza's denial of free will (Spinoza, 1632(Spinoza, /1985. Further, versions of the PSR arguably appear in western philosophy since the presocratic era. For example, Leucippus seems to have appealed to PSR in his argument for determinism when he asserts that "Nothing happens at random but everything for a reason and by necessity" (Diels & Freeman, 1983). And, although the PSR lost its philosophical luster during the 20th century (e.g., Bennett, 1984;Van Inwagen, 1983) it has seen renewed interest in recent years (e.g., Amijee, 2021Amijee, , 2022Dasgupta, 2016;Della Rocca, 2010).
Thus, the PSR plays an important role in abstract metaphysical arguments. Yet, it is also often taken to be a deep and commonsense conviction about the nature of the world. Leibniz coined the term "Principle of Sufficient Reason", but didn't think that he had invented the idea. He maintained that the PSR has guided philosophy for centuries and it is widely accepted in everyday reasoning. In a letter to Clarke, he writes, "Has not everybody made use of this principle upon a thousand occasions? " (1989). This is an empirical claim that, if true, would yield important insights about our explanatory cognition and may even inform our philosophical reasoning. For example, William Rowe (2007, p. 32) notes that if it were true that we all presuppose the PSR, then "to be consistent we should accept the Cosmological Argument." However, Rowe also notes that "no one has succeeded in showing that PSR is an assumption that most or all of us share" (Rowe, 2007, p. 32).
Indeed, it is our goal in the present research to examine this issue empirically: is the PSR just an analytic tool for metaphysicians, or might a PSR-like presumption also be present in ordinary thought?

The PSR in psychology
In order to lay out the scope of the hypothesized PSR presumption, and compare it to other components of explanatory cognition, it is helpful to introduce a more precise formulation of the PSR. In abstract form, the PSR can be stated as follows: PSR: for every x, if x is a fact, then there is a y such that y explains x.
Or, even more schematically: ∀x(if x is a fact, ∃y(xRy)).
One can ask many questions of this principle. One may ask about the explanans y: what counts as a proper explanation? (cf. Joo, Yousif, & Keil, 2021;Lewry & Lombrozo, 2022). Similarly, one can ask about the explanatory relation R: what is it to explain something? (Keil, 2019;Lombrozo, 2012). In the present studies, we limit our scope to test people's judgments about x, the explanandum (the "fact-to-beexplained"). Specifically, we aim to explore the extent to which people judge that a series of putative facts across scientific, ordinary, and supernatural domains must have an explanation.
There are important precedents in cognitive and developmental psychology that bear on the examination of the PSR in ordinary cognition. A consistent theme in this work is that people in general-and children in particular-have an abiding drive for explanation (cf. Gopnik, 1998), at least among Western populations examined to date (cf. Henrich, Heine, & Norenzayan, 2010). For example, children are prone to generate explanations for unexplained facts and reject answers that leave salient facts unexplained (Woolley & Cornelius, 2017;Woolley & Dunham, 2017). Likewise, recent studies show that adults evaluate candidate explananda on a wide variety of dimensions, including: whether the fact "demands" explanation (Liquin, Metz, & Lombrozo, 2020), whether science can possibly explain the fact (Gottlieb & Lombrozo, 2018), and whether explaining the fact would achieve desirable moral or social ends (Davoodi & Lombrozo, 2022).
These studies show that people can appraise candidate explananda in sophisticated ways, but they do not provide strong evidence that a PSRlike principle guides our explanatory judgment. One critical limitation of the extant work is that it merely demonstrates that adults and children expect or want explanations for various facts. But the PSR is committed not just to the expectation of an explanation-it says that every fact must have an explanation. This is a claim about the necessity of an explanation, not merely the expectation or value of an explanation. For example, one may think it is not possible to know why the universe exists, but still think there has to be an explanation. Or, one may not want to know why Steve had a tuna sandwich for lunch last Tuesday, but still think there has to be an explanation. Hence, judgments in accordance with a metaphysical, PSR-like principle with strong modal force should predictably diverge from related epistemic and value judgments. Further, note the PSR's distinctive scope. According to the PSR, every fact must have an explanation. Judgments in accordance with a PSR-like principle should apply to facts in general-not just the facts we want to explain or expect that we will explain.
In philosophy, the PSR is said to yield proof of the existence of God, the truth of causal determinism, and more because every fact must be explained and not simply because we expect to find explanations, or we value having explanations. This stronger sense of explanation is what we call here a metaphysical sense of explanation. In our studies, we aim to measure people's metaphysical judgments about whether there has to be an explanation for every fact.
However, the scope and modal force of the PSR raises a methodological challenge. The PSR is supposed to apply to every fact, but to discern whether ordinary people have a PSR-like presumption we will need items that elicit both positive and negative ratings on our measures. Otherwise, our measures run the risk of confounding evidence of a PSR-like presumption with evidence of a simple positive response bias. To address this issue, we opted to measure participants' judgments about facts and also mere coincidences, which are typically not considered to be apt for explanation (cf. Lando, 2017;Sober, 2012;Strevens, 2008, p. 433;Woodward, 2000, p. 197;Bhogal, 2020). Hence, we expect that facts will elicit high ratings, and coincidences will elicit low ratings on our measures. Since this fact/coincidence distinction plays a central role in our studies, next we discuss it in detail.

The fact/coincidence distinction
What is a "mere coincidence"? And in what ways do mere coincidences differ from facts? To provide our discussion with concrete details, we will introduce an example of a mere coincidence (cf. Sober, 2012): Lotto1&Lotto2: Bob won a fair lottery with ticket #437, and, two years later, Bob won the same lottery with ticket #6810. 1 The details of the arguments vary among authors and historical contexts.
The role of PSR in rationalist philosophy was anticipated by the importance of cosmological arguments in medieval philosophy (see, Aquinas, 1265Aquinas, /1975Hammond et al., 2013;Scotus, 1987). Importantly, medieval cosmological arguments were developed with direct influence and discussion of works in Islamic philosophy that also relied on something like the PSR (Avicenna, 1027(Avicenna, / 2005Craig, 1980;Fakhry, 1957).
2 Of course, this opens the questions about whether God explains His own existence or is somehow exempt from the PSR. What matters for our purposes, though, is that the PSR gets the cosmological argument off the ground. One may, for instance, follow Leibniz in thinking that, unlike contingent truths, the explanation of necessary truths is explained by the fact that their falsehood entails contradiction.
Lotto1&Lotto2 describes a pair of coinciding observations: the same person won the same lottery twice! But not all coinciding observations are mere coincidences. Some coinciding observations owe to a causal connection. For example, yearly fluctuations of temperature in Boston are correlated with yearly ice cream sales in New York City, because the Earth's position relative to the Sun is a common cause of both. 3 By contrast, we know that the coinciding observations in Lotto1&Lotto2 do not have a causal connection, since we know that the lottery was fair. That is, the process that selected ticket #437 in the first lottery was independent from the process that selected ticket #6810 in the second lottery, despite the fact that Bob happened to be the person who purchased both tickets. Since Lotto1 and Lotto2 do not have a causal connection, by putting them together we have conjoined counterfactually independent facts. If ticket #437 hadn't won the first lottery, ticket #6810 would have still won the second lottery (and vice-versa).
What we call "mere coincidences" are such arbitrary conjunctions of facts, in the sense that each conjunct is counterfactually independent of the others. Here are some examples that we use in our studies: Darwin&Lincoln: Charles Darwin and Abraham Lincoln were born on the same day, February 12, 1809. Coolidge&Macarena: Calvin Coolidge became president of the United States (1923) and "The Macarena" became the #1 song in the US pop charts (1996) on the same day of the year (August 3). soccer&surroundings: the words 'soccer' and 'surroundings' begin with the same letter (s).
To be sure, each conjunct of these coincidences can be explained separately as unique events. Take Lotto1&Lotto2. No doubt something happened inside the machine that selected ticket #437 to win, and, two years later, something else happened to select ticket #6810. But the selection of ticket #437 and the selection of ticket #6810 has no explanation beyond the explanation of each conjunct (i.e., the something and the something else that happened two years later). Likewise, Darwin's birth and Lincoln's birth have separate explanations as unique events, but the conjunction of the two events does not have a further explanation. If all relevant explanations to Darwin&Lincoln are either explanations of Darwin or explanations of Lincoln, then, ipso facto, there are no explanations of Darwin&Lincoln. Hence, a key a difference between mere coincidences and their constituent facts is that mere coincidences do not have explanations in themselves.
An important role of our commonsense and scientific theories about the world amounts to teasing apart causally unified facts from spurious coincidences that do not call for an explanation. In part this is because we are interested in generating explanations that are modally robust and explain facts across a wide range of similar circumstances (Strevens, 2008;Woodward, 2000). However, since coincidences are themselves modally fragile (they could have easily not happened), they do not have robust explanations. The events that led to Bob winning in Lotto1 and the events that led to Bob winning in Lotto2 do not explain why people in general or even Bob in particular would win lotteries across a range of similar circumstances. Relatedly, we want explanations that provide good predictions about the world. But since coincidences are modally fragile, they cannot provide the basis for good predictions. Conjoining an explanation of Lotto1 and an explanation of Lotto2 will not be of any help in predicting who will win the lottery next.
To recap: what we call "coincidences" are arbitrary conjunctions of counterfactually independent facts. Crucially for the purpose of measurement validation, mere coincidences are not themselves apt for explanation, whereas the constituent facts are apt for explanation. Hence, we expect participants to give metaphysical judgments that affirm facts must have explanations and deny coincidences must have explanations.
Does this mean that coincidences are counterexamples to PSR? Not necessarily. If facts are the kinds of counterfactually robust occurrences that are to be explained by our commonsense and scientific theories, then coincidences are not facts. Hence, coincidences would fall outside of the principle's scope-for it only refers to facts-and do not count as the kind of thing that would falsify it. Views of this sort have precedence in philosophy (see Mulligan & Correia, 2021, sect. 2.2.). For example, Bertrand Russell (1918) argued that it is unnecessary to suppose that conjunctive propositions like Lotto1&Lotto2 correspond to facts, since the truth value of any conjunctive proposition could be entirely derivative on the truth-value of its constituents (see Klement, 2020 for discussion). Say, however, that one wants to insist that coincidences are facts. After all, coincidences are proper objects of phrases like "it is a fact that [x]". If so, to make sense of why they do not call for explanation in themselves, we may appeal to the following recursive construal of the PSR: PSR*: for every x, if x is a fact, either there is a y such that y explains x, or x is composed of facts, each of which has an explanation.
In this rendering, coincidences fall under the second disjunct of the principle. Although they are facts, they do not call for explanations in themselves. Instead, what calls for explanation are the individual facts that ultimately compose coincidences. Crucially, PSR* is still committed to the strong conclusion that there are no brute facts, i.e., there are no unexplainable facts that bottom out the chain of explanations.
Under any of these renderings of the PSR, we would expect that a measure that genuinely tracks a PSR-like presumption would systematically yield lower ratings for coincidences than for any other putative fact that is not an arbitrary conjunction of modally independent occurrences. Hence, for our purposes, coincidences are ideal stimuli to test the validity of our measures and ensure they are tracking a PSR-like presumption and not just reflecting a positive response bias.

The present research
To show evidence of a PSR-like principle in ordinary thought, we should establish that participants' judgments meet the following requirements: Convergence: judgments in accordance with a PSR-like presumption should show agreement on two measures that both purport to measure a PSR-like presumption. Divergence: judgments in accordance with a metaphysical, PSR-like presumption should predictably diverge from related epistemic and value judgments. Generality: judgments in accordance with a PSR-like presumption should apply to facts in general.
The present research 4 investigates whether people's PSR-conforming judgments indeed meet these requirements. In Study 1, we test for Convergence by examining whether participants' judgments about facts across a wide range of domains show agreement on two separate measures. In Study 2 and 3, we test for Divergence by examining participants' judgments about a curated set of explananda that we predict are likely to elicit differences in metaphysical, epistemic, and evaluative judgments. In Study 4 and 5, we test for Generality by examining participants' judgments about a large, comprehensive set of explananda that were sampled from randomly selected Wikipedia entries.

Study 1: Evidence for convergence
In Study 1 we collect participants' ratings on two measures that we expect will show (i) agreement and (ii) evidence of the hypothesized PSR-like presumption. In doing so, we measure participants' judgments about candidate explananda that span a wide range of domains (e.g., scientific, mathematic, supernatural, etc.) and also coincidences. We expect participants to judge that facts must have explanations and deny that coincidences must have explanations.
Since measures of the metaphysical judgment relevant to PSR are not available in the extant psychological literature, we devised two novel measures for Study 1. The words "explanation" and "reason" both translate the technical term "explanatory relation" (or, "sufficient reason") into ordinary English reasonably well, though their application tends to be context sensitive. 5 To capture this general notion of explanation, we designed a simple measure that uses strong modal language and a disjunctive phrasing (scale: 1-Strongly disagree, 7-Strongly agree): Simple: There must be an explanation or reason why [balloons lose helium].
Simple is ecumenical with respect to "explanation" and "reason." But it does not explicitly differentiate an epistemic interpretation (i.e., there must be a knowable explanation) from a metaphysical interpretation (i. e., there must be an explanation, independently of whether anyone can know it). Given this possibility, we designed another measure that explicitly contrasts these readings: In so far as Simple and Explicit produce similar ratings, then we have shown that our key measures track the metaphysical features relevant to the hypothesized PSR-presumption. In particular, we expect participants will give high ratings for facts from a wide range of domains, and low ratings for coincidences.

Participants
N = 390 participants were recruited from Prolific to complete a survey for modest compensation. Participation was restricted to adults living in the United States who had completed at least 50 prior tasks with a minimum approval rating of 95%. According to pre-registered exclusion criteria, 21 participants were excluded for failing to complete the survey and 50 participants were excluded for failing a basic attention check. Thus, n = 319 participants (m age = 37.8 years, sd = 14.8 years; 57% female) were included in the final analyses.

Procedure
In a fully within-subjects design, participants made judgments about 30 candidate explananda. Twenty-six of these explananda (7 scientific, 7 health-related, 3 mathematical, 3 psychological, 3 supernatural, and 3 religious) have been in used in previous research on explanatory judgment 6 . Participants also made judgments about 4 coincidences that we devised for this study in particular. Table 1 gives characteristic examples, and the full set of explananda can be viewed at OSF.
Our reasons for using explananda from Liquin et al., 2020 are twofold. First, the candidate explananda span a fairly wide range of domains, from scientific (e.g., "the earth's plates move") to supernatural (e.g., "demons are powerful") facts. Since the hypothesized PSR-like presumption is meant to apply universally, it is important that we examine participants judgments across a wide range of domains. Using the candidate explananda from Liquin et al., 2020 thus provides a clean "first pass" way of addressing this desideratum (which we address more directly with "stimulus sampling" methods in Studies 4-5). Second, Liquin et al. (2020, p. 6) found that participants judged that supernatural and religious facts did not "demand" explanation, and participants were more comfortable accepting supernatural and religious facts as mysteries. The hypothesized PSR-like presumption requires that even mysterious facts and facts that do not "demand" explanation (in the epistemic sense) must have an explanation (in the metaphysical sense). Given this, using the explananda from Liquin et al., 2020 is also beneficial for a "first pass" examination of whether participants' PSRconforming judgments diverge from the pattern of judgments about explanatory demand seen in  The procedure had three main parts. In Part 1, participants first judged whether each candidate explanandum was true on a 7-point scale (e.g., Please rate your agreement with the following: It is true that [balloons lose helium]. 1 -Strongly disagree, 7 -Strongly agree). If participants gave a truth rating >4 (the scale midpoint), they then gave a rating on the Simple measure.
Participants' judgments were 'truth-piped' in this manner since, for our purposes, PSR-relevant judgments apply only to facts that people judge to be true. After going through this piping procedure for all 30 phenomena (order randomized), participants proceeded to Part 2. In Table 1 Summary statistics for highest-and lowest-rated explananda. Part 2, participants rated all 30 explananda on the Explicit measure, with the order randomized and different names assigned to the disagreeing parties in each case. In Part 3, participants made judgments about general formulations of the PSR. We included these measures to gauge whether people endorse the PSR as a general principle, and, if so, how this tendency is related to their judgments about specific events. We developed four measures (presented in random order within 'happens' and 'exists' blocks, all scales: 1 -Strongly disagree, 7 -Strongly agree):

Reason -Happens: "
To what extent do you think there has to be a reason for anything that happens?" Explanation -Happens: "To what extent do you think there has to be an explanation for anything that happens?" Reason -Exists: "To what extent do you think that for anything that exists there has to be a reason for why it exists?" Explanation -Exists: "To what extent do you think that for anything that exists there has to be an explanation for why it exists?" Finally, participants completed a religiosity inventory (Pennycook, Cheyne, Seli, Koehler, & Fugelsang, 2012) collected for exploratory purposes, and answered standard demographic questions.

Measurement reliability for metaphysical judgments
To assess the reliability of the Simple and Explicit measures, we fit our proposed measurement model via confirmatory factor analysis using the R package lavaan (Rosseel, 2012). The model posits three latent constructs to explain the variance in participants' judgments across the PSR-relevant measures: a tendency to presuppose the PSR in judgments about specific explananda (Specific), a tendency to endorse the general formulations of the PSR as it pertains to reasons (Reason), and a tendency to endorse the general formulations of the PSR as it pertains to explanations (Explanation). This 3-factor model displayed excellent fit across all absolute fit indices (RMSEA = 0.01, SRMR = 0.005, CFI > 0.99 TLI > 0.99), and outperformed a 2-factor model that collapsed Reason and Explanation (3-factor: AIC = 130,657, BIC = 130,772; 2-factor: AIC = 132,448, BIC = 132,549).
From the fitted measurement model, we can compute the composite reliability between Simple and Explicit to assess whether these measures are internally consistent. Composite reliability is a metric of the shared variance, relative to the total scale variance, among the observed variables that indicate a latent construct (Bacon, Sauer, & Young, 1995;Raykov, 1997). The composite reliability for Simple and Explicit was CR = 0.892, which corresponds to Simple and Explicit sharing approximately 79% of the total scale variance. Thus, insofar as the Explicit measure reliably tracks people's metaphysical judgments, the Simple measure also reliably tracks people's metaphysical judgments.
In addition, ratings for Scientific items were higher than ratings for Religious (t(769.8) = 15.99, p < .001) and Supernatural items (t(439.9) = 17.53, p < .001). This result is consistent with previous findings by , who found that the Scientific items received higher ratings than the Religious and Supernatural items when participants were asked whether the item "demands" explanation. A notable difference, however, is that Liquin et al. (2020, p. 6) found that the Religious and Supernatural items received ratings below the scale mid-point for their "demand" DV, whereas we found that these items received ratings above the scale mid-point with our measures that elicit distinctively metaphysical judgments.
In addition to analyzing the average ratings for explananda in each domain, we can also analyze what rating an explananda from a given domain was most likely to receive. To examine this, we fitted an ordinal regression model with rating as the dependent variable and domain as the independent variable. Fig. 2 shows the fitted model's estimates for the probability of each scale rating, given the explananda domain. A scale rating of '1' was most probable for coincidences, and a scale rating of '6' or '7' was most probable for facts across all domains.
So far, we have found large differences in how participants rated facts and coincidences. However, showing a difference in the aggregate is not the same as showing that everyone makes this distinction. To assess individual differences, we fit a mixed-effects linear regression model with rating as the dependent variable and domain (fact or coincidence) as a fixed effect independent variable, with random slopes by participant for domain and a random intercept term for participant. From the fitted model, we extracted the coefficients for the domain term for each participant. As shown in Fig. 3, 99% (316/319) of participants have coefficients above zero, meaning the model predicts the vast majority of participants will draw the fact/coincidence distinction (with 94% (299/ 319) of participants having 95% CIs that do not contain zero). At the same time, there is notable variation among participants (mean b = 3.79, sd = 1.43). Hence, although the model predicts that participants will draw the fact/coincidence distinction, the model also predicts substantial variation among participants in the degree to which this distinction manifests in judgment.
In our analysis of the data collected for exploratory purposes, we found that average ratings on each of the measures that elicited judgments about general formulations of PSR were above the scale mid-point .001). Participants' ratings on both of the "Reason" measures exhibited positive partial correlations with participants' average ratings on the Religiosity inventory (Reason -Happens: r = 0.17, p = .003; Reason -Exists: r = 0.18, p < .001). Participants ratings on the "Explanation" measures did not display significant partial correlations with Religiosity scores (Explanation -Happens: r = − 0.11, p = .06; Explanation -Exists: r = − 0.07, p = .24). Overall, these patterns of results remain consistent in an aggregated analysis of participants' responses from across all five studies (see Supplementary materials). Since we collected ratings on these measures for exploratory purposes, however, we do not discuss them further in subsequent studies.

Discussion
Study 1 made two steps toward validating our measures of a PSR-like principle in ordinary judgment. First, we established that a simple measure of metaphysical judgment displays excellent internal consistency with a measure that explicitly rules out non-metaphysical interpretations. Second, we showed that participants' judgments on the proposed measures are broadly sensitive to the relevant theoretical predictions. In general, participants gave higher ratings for facts and lower ratings for coincidences. Notably, we observed PSR-conforming judgments even about religious and supernatural facts that participants in a previous study were more likely to judge did not "demand" explanation and were more comfortable accepting as mysteries .
In analysis of individual differences, we also found that the vast majority of participants draw the fact/coincidence distinction that is also observed in the aggregate. In addition, however, we found that there is substantial variation in the degree to which the fact/coincidence distinction is reflected in participants' judgments. An intriguing possibility is that this variation systematically owes to an unexamined factor. For example, perhaps education or advanced training in science fosters a stronger degree of conviction in PSR. In follow-up work, we have started to investigate whether individuals with little or no formal education also exhibit a PSR-like principle in judgment. With the present research, however, we have opted to focus primarily on the findings that relate to the effects that are observed in aggregate.
Next, we build from these findings by further demonstrating how participants' metaphysical judgments predictably diverge from epistemic judgments (Study 2) and value judgments (Study 3).

Study 2: Metaphysical-epistemic divergence
In Study 2, we assess whether participants' metaphysical judgments about explanation predictably diverges from their epistemic judgments  Error bars correspond to standard error. As shown, a '6' or '7' was the most probable rating for all facts, and a '1' was the most probable rating for coincidences. about explanation. Epistemic judgments have to do with the reach of our knowledge. For instance, people might accept that it will forever remain a mystery why Stonehenge was built, or more dramatically, why the universe exists. Whether decisive evidence evades the fossil record or the form of a candidate explanation is difficult to grasp, the key point here is that knowledge of certain explanations is relatively inaccessible to human beings. Nevertheless, people may still think these facts must have explanations. People also routinely find that many explanations are straightforwardly accessible in an epistemic sense-see, e.g., any wellestablished explanation in science. If people presuppose PSR, they should also think these explanations are necessary in a metaphysical sense. Hence, we can make use of the distinction between accessible and inaccessible explanations to tease apart PSR-conforming judgments from epistemic judgments. Our prediction is that participants' epistemic judgments will substantially vary across explanada with accessible (e.g., 'balloons lose helium') and inaccessible (e.g., 'The universe exists') explanations, whereas participants' metaphysical judgments will vary to a lesser degree, as PSR states that both accessible and inaccessible explanations must exist.

Participants
N = 127 participants were recruited from Prolific to complete a survey for modest compensation. Participation was restricted to adults living in the United States who had completed at least 50 prior tasks with a minimum approval rating of 95%. According to pre-registered exclusion criteria, 6 participants were excluded for failing to complete the survey, 13 participants were excluded for failing a basic attention check, and 4 additional participants were excluded for completing the survey in under 5 min. Thus, n = 104 participants (m age = 31.3 years, sd = 10.5 years; 61% female) were included in the final analyses.

Procedure
In a fully within-subjects design, participants made judgments about 32 candidate explananda. We pre-registered the six science-and six health-related explananda from Study 1 as the 'epistemically accessible' subset (Accessible), the three religious and three supernatural explananda from Study 1 with eight new explananda as the 'epistemically inaccessible' subset (Inaccessible), and the four coincidences from Study 1 as 'coincidences' (Coincidence).
As in Study 1, participants were presented with the explananda in a randomized order, and, for each explanandum, participants were first asked to judge whether they thought the explanandum was true. If the participant gave a truth rating above 4 (the scale midpoint), they next provided a metaphysical and epistemic judgment about the explanandum (order randomized between-subjects, both scales: 1 -Strongly disagree, 7 -Strongly agree): Metaphysical: There must be an explanation or reason why [ancient people built the monuments at Stonehenge]. Epistemic: It is possible for us to know why [ancient people built the monuments at Stonehenge].
After going through this piping procedure for all 32 explananda, participants provided responses to the same general measures and religiosity inventory from Study 1 (both collected for exploratory purposes).

Results
To assess whether participants' ratings of the Accessible and Inaccessible explananda differed across measures in the predicted manner, we used a mixed-effects linear model with scale ratings as the dependent variable. We included explananda type (Inaccessible, Accessible) and measure type (Epistemic, Metaphysical) as fixed-effect independent variables. We also included random slopes for explananda type and measure type, and we included a random intercept term for participant. The model was specified as follows: rating ~ measure type * explananda type + (1 + measure type * explananda type | participant) As predicted, the model results indicate a significant measure type x explananda type interaction (F(1,130) = 76.93, p < .001). Post-hoc tests confirmed the predicted pattern of judgments. Compared to the difference in Metaphysical ratings, participants gave lower Epistemic ratings to Inaccessible explananda than Accessible explananda (b measure x type = − 0.90, se = 0.10, p < .001, d = − 1.54; summaries, Inaccessible: m Epistemic = 4.64, sd Epistemic = 1.77, m Metaphysical = 5.52, sd Metaphsical = 1.46; Accessible: m Epistemic = 6.39, sd Epistemic = 0.74, m Metaphysical = 6.39, sd Metaphysical = 0.80) (see Fig. 4a). Table 2 reports summary statistics for select items of interest.
As with Study 1, we obtained ordinal regression estimates to see what scale ratings are most likely for each type of explananda (see Fig. 4b). For Accessible explananda, the fitted model predicts that ratings of '7' are most likely for both the Epistemic and Metaphysical measure. For Inaccessible explananda, a scale rating of '5' or '6' was most likely for the Epistemic measure, and a scale rating of '6' or '7' was most likely for the Metaphysical measure.
Next, we assessed whether participants' metaphysical judgments of Accessible and Inaccessible explananda differed from their ratings of Coincidence explananda. Here, we used a mixed-effects linear model with Metaphysical ratings at the dependent variable. We included explananda type (Coincidence, Inaccessible, Accessible) as a fixed-effect independent variable with a random slope, and we included a random intercept term for participant. As predicted, we replicated a key result from Study 1 whereby Coincidence explananda received lower scale ratings than both Accessible (b = 2.75, se = 0.08, p < .001) and Inaccessible (b = 1.90, se = 0.09, p < .001) explananda.

Discussion
Study 2 demonstrates that participants' metaphysical judgments diverge predictably from participants' epistemic judgments. Explanations that were judged to be accessible and inaccessible in the epistemic sense were all judged to be necessary in the metaphysical sense, as required by PSR. However, it is also worth noting the two measures are also related in a statistical sense (r = 0.64, p < .001): items that received higher Metaphysical ratings also tended to receive higher Epistemic ratings. Together, these results raise some intriguing questions about the relation between metaphysical judgments and epistemic judgments in ordinary cognition-we return to this point in the General Discussion. For now, it is worth noting that the Epistemic measure we deployedwhich asks about the possibility of knowing the explanationis among the strongest tests for metaphysical-epistemic divergence. There are many other cognitive attitudes which are properly considered epistemic judgments, for example: "we are confident we know why x," "we have good evidence for knowing why x," and so on. Since these sorts of epistemic judgments that do not explicitly prompt any modal thought, it is likely that they would display even stronger divergence with the metaphysical judgment measured here.

Study 3: Metaphysical-value divergence
Study 2 demonstrated that participants' metaphysical judgments diverge predictably from their epistemic judgments. In Study 3, we examine whether the same holds true for participants' value judgments. Value judgments have to do with goodness or badness. For example, you may think that it is really good to know why insulin injections help patients who have diabetes. In contrast, you may think that is not at all worth the effort to find out why your neighbor Steve ate a tuna sandwich for lunch last Tuesday. Whether or not actually having an explanation is good or bad, the PSR states the explanation must exist. Hence, PSRconforming judgments should also diverge from value judgments.
Even when value judgments systematically differ, PSR-conforming judgments should remain stable.
To show this, we make use of a common distinction between token explanation and type explanations (cf., Wetzel, 2018). A type event picks out a class of particular token instances: "Steve enjoys holding this chihuahua" describes a token instance of the type "people enjoy holding dogs." You might value knowing why people enjoy holding dogs in general, but, unless you're friends with Steve, you might not value knowing why Steve enjoys holding a particular chihuahua. Nevertheless, a disinterested observer may still believe there has to be an explanation for why this person enjoys holding this dog. Thus, our prediction is that participants' value judgments will vary according to whether the candidate explanandum requires a token explanation or type explanation, whereas participants' metaphysical judgments should remain similar across both cases. If so, this finding would also help guard against worries about a "value" confound in the earlier studies. It is possible that people judge a fact must have an explanation only because they have a local interest in knowing what that explanation is (contra PSR, which is supposed to apply globally).

Participants
N = 315 participants were recruited from Prolific to complete a survey for modest compensation. Participation was restricted to adults living in the United States who had completed at least 50 prior tasks with a minimum approval rating of 95%. According to pre-registered exclusion criteria, 15 participants were excluded for failing to complete the survey, 36 participants were excluded for failing a basic attention check, and 10 additional participants were excluded for completing the survey in under 5 min. Thus, n = 254 participants (m age = 32.03 years, sd = 11.53 years; 50% female) were included in the final analyses.

Procedure
Participants were randomly assigned to either the Type condition or the Token condition. In the Type condition, participants made judgments about explananda regarding types (e.g., 'people enjoy holding dogs'). In the Token condition, participants made judgments about explananda regarding token instances matched to the types in the Type condition (e.g., 'this woman enjoys holding this dog'). Within each condition, participants made a series of judgments about 20 explananda  (order randomized within-subjects). Each explananda was presented with a corresponding image sourced from free stock photo libraries on the internet. 7 To construct this set, first we randomly sampled 20 nouns from a list of over 6700 English nouns. 8 We used each noun as a search string at the stock photo library and selected an image from the first page of search results. Lastly, we annotated each image to pick out a token and type event depicted in the image (see OSF for full image set Participants also provided ratings for the Metaphysical and Epistemic measures from Study 2 (judgment order randomized withinsubjects). After going through this procedure for all 20 explananda, participants provided responses to the same general measures and religiosity inventory from the earlier studies (both collected for exploratory purposes).

Results
To assess whether participants' metaphysical judgments diverged from their value judgments, we used a mixed-effects linear regression model with scale rating as the dependent variable. We included condition (Token, Type) and measure type (Metaphysical, Normative, Value, Motivational) as independent variables. 9 We also included a random slope for measure type, since measure type is within-subjects (cf. Barr, 2013), and a random intercept term for participant. The model was specified as follows: rating ~ condition * measure type + (1 + measure type | participant) The results showed a significant measure x condition interaction (F(3, 371.78) = 2.74, p = .04). Post-hoc follow up tests confirmed the predicted pattern of judgments (see Fig. 5a). Metaphysical judgments and value judgments showed a greater difference in the Token condition than in the Type condition (b measure=Normative x condition=Type = 0.29, se = 0.13, p = .02, d = 0.30; b measure=Value x condition=Type = 0.34, se = 0.13, p = .005, d = 0.35; b measure=Motivational x condition=Type = 0.29, se = 0.13, p = .02, d = 0.29).
In addition, scale ratings on all three value measures showed significant, positive partial correlations with each other (Normative-Value: r = 0.40, p < .001; Normative-Motivational: r = 0.52, p < .001; Motivational-Value: r = 0.41, p < .001), and near-zero partial correlations with the Epistemic and Metaphysical ratings (see Fig. 9). All else equal, if a person gave a high scale rating on Normative (we should try to answer why p), they were also more likely to give a high scale rating on Value (it would be good for us to know why p) and Motivational (it would be worth the effort to find out why p). But giving a high scale rating on Normative (or Value, or Motivational) had nearly zero unique association with a person's rating on Metaphysical (there must be a reason or explanation for why p).
As throughout Studies 1 and 2, participants gave high absolute scale ratings for the Metaphysical measure (Token: m = 5.58, sd = 1.31; Type: m = 5.96, sd = 1.17), with ratings in both conditions significantly above the sale midpoint (Token: t(128) = 22.15, p < .001; Type: t(124) = 30.98, p < .001). Once again, ordinal regression estimates show that '6' or '7' was the predicted rating for Metaphysical (see Fig. 5b). Thus, these findings also provide evidence that a PSR-like presupposition applies more generally beyond the stimuli used in previous experimental research. Importantly, the high responses on token facts provides evidence that people's judgments conform to PSR even for token facts.

Discussion
Study 3 demonstrates that participants' value judgments predictably diverge from their metaphysical judgments. In contrast with the evidence of epistemic divergence-which showed epistemic and metaphysical judgments are separable, but statistically related-the evidence of value divergence is rather stark. Not only are metaphysical judgments and value judgments separable, but individual value judgments showed nearly zero unique statistical association with metaphysical judgments. This result shows that participants' metaphysical judgments are tracking a value-independent feature of candidate explananda-that there must be an explanation, whether or not we value knowing it.

Studies 4-5: Evidence for generality
Study 1 showed evidence for Convergence: our key measures of the PSR-like presumption both showed (i) high ratings across a wide range of facts, and (ii) low ratings for coincidences. Studies 2 and 3 show evidence for Divergence: participants' PSR-conforming judgments predictably diverged from related epistemic and value judgments. The final requirement for showing evidence for a PSR-like presumption in ordinary thought is Generality: that people's metaphysical, PSR-conforming judgments apply in general to a widely sampled set of facts. In order to do so, we need to have greater confidence that our set of candidate explananda is representative of facts in general.
For this reason, we assembled a large set of facts that were selected from random Wikipedia entries. Using the same list of 6700 English nouns as we used in Study 3, we randomly sampled 100 words and used each as a search string on Wikipedia. On the resulting page, we selected up to three facts that met the following criteria. First, the fact had to be actual (i.e., it could not express a statement of possibility). Second, the fact had to be comprehensible (i.e., not excessively jargonistic or esoteric). Third, the fact had to be non-definitional. For example, if the search string was 'silver,' a selected fact could not be "Silver is the chemical element with the symbol Ag." Our rationale for these criteria was assembling a large set of facts that pertained to actual events or existents, thus keeping with the focus of the earlier studies. In total, we assembled a set of 230 facts from Wikipedia (for brevity, 'Wikipedia facts'), which was nearly ten times larger than the set of facts we had assembled from previous research (see OSF for the complete set of Wikipedia facts).
To have an appropriate contrast set, we also created a set of 150 coincidences that were either 'linguistic' (75 in total) or 'historical' (75 in total). The linguistic coincidences were constructed by randomly sampling a word from the list of 6700 English nouns and either matching it with (a) another word that began with the same letter (e.g., "the words 'sleet' and 'sunglasses' have the same first letter (s)") or (b) another word that had the same total number of letters (e.g., "the words 'slang' and 'roast' have the same number of letters (5)"). The historical coincidences were constructed by searching https://www.history.com/thi s-day-in-history for events that happened on the same date across different years (e.g., "Calvin Coolidge became president of the United 7 We used the following websites: https://www.pexels.com, https://pixabay. com, https://unsplash.com 8 Source: http://www.desiquintans.com/nounlist 9 In the main text we report results from a model with a maximal random effects structure. Initially, we pre-registered a model with only a random intercept term for participant that also included ratings on Epistemic. This preregistered model also showed a significant measure x condition interaction (F(4, 25,138) = 16.56, p < .001), and, as predicted, there were significant interactions between measure x condition for all value judgments individually (b measure=Normative x condition=Type = 2.95, se = 0.05, p < .001; b measure=Value x condition=Type = 3.12, se = 0.05, p < .001; b measure=Motivational x condition=Type = 3.40, se = 0.05, p < .001).
States (1923) and "The Macarena" became the #1 song in the US pop charts (1996) on the same day of the year (August 3)").

Study 4: Generality of the fact/coincidence distinction
With a larger, more representative set of facts in hand, in Study 4 we set out to generalize the key finding from Study 1 that established participants' metaphysical judgments are appropriately sensitive to the fact/coincidence distinction.

Participants
N = 375 participants were recruited from Prolific to complete a survey for modest compensation. Participation was restricted to adults living in the United States who had completed at least 50 prior tasks with a minimum approval rating of 95%. According to pre-registered exclusion criteria, 24 participants were excluded for failing to complete the survey, 27 participants were excluded for failing a basic attention check, and 1 additional participant was excluded for completing the survey in under 5 min. Thus, n = 323 participants (m age = 28.77 years, sd = 10.51 years; 80% female) were included in the final analyses.

Procedure
In a fully within-subjects design, 10 participants made judgments about 30 candidate explananda sampled randomly from the full set of 360 in total (230 Wikipedia facts and 150 coincidences). For each explananda, participants made a rating on the Truth and Explicit measures from Study 1.

Results
To assess whether participants' scale ratings depended on explananda type, we used a random-effects linear regression model with rating score as the dependent variable. We included explananda type (fact or coincidence) as fixed-effect independent variable, a random slope for explananda type, a random intercept for participant, and a random intercept for explananda (nested within type). The model was specified as follows: rating ~ explananda type + (1 + explananda type | participant) + (1 | explananda type: item) As predicted, model results indicate a significant main effect of explananda type (F(1) = 1359.8, p < .001). Post-hoc comparisons show the differences across domains follow the predicted pattern, with the Wikipedia facts receiving higher scores than the coincidences (b = 3.50, se = 0.09, p < .001, d = 3.50). Average ratings for coincidences were significantly below the scale midpoint (m = 2.31, sd = 1.85, t(3775) = − 55.87, p < .001), and the average ratings for the Wikipedia facts were significantly above the scale midpoint (m = 5.81, sd = 1.40, t(5913) = 99.09, p < .001) (See Fig. 6). Table 3 shows summary statistics for the three items with the highest and lowest ratings within each explananda type.
To analyze individual differences, next we extracted the coefficients for the domain term for each participant. The fitted model predicts that 98.4% (318/323) of participants will draw the fact/coincidence distinction (with 93% (302/323) of participants having 95% CIs that do not contain zero; see Fig. 7). Likewise with Study 1, there was considerable variation in participants' tendencies to draw the distinction (mean b = 3.50, sd = 1.39).
Next, we used Bayesian methods to further investigate the data. For our purposes, the main benefit of the Bayesian data analysis is multi-level posterior prediction (see McElreath, 2020, Chapter 13). This analysis technique provides a natural way to quantify and express uncertainty about the predicted scale response, given the explananda is a fact (or a 10 To control for the possibility of a response bias, we also ran a betweensubjects version of this study. In the between-subjects version of this study, N = 200 participants (pre-registered; US, Prolific) were assigned to rate a random sample of 10 coincidences or a random sample of 10 facts from Study 4. As predicted, we found a significant difference in ratings across conditions (b = 3.06, se = 0.10, p < .001) such that facts (m = 5.62, sd = 1.57) received higher ratings than coincidences (m = 2.58, sd = 2.11). Full details of the betweensubjects study are reported in the supplementary materials. All data and code for analysis is available at OSF. coincidence). 11 For this analysis, we used an ordered-logistic regression model, where scale rating was the dependent variable and explananda type (fact or coincidence), participant, and explananda item were included as independent variables. We included weakly-regularizing priors for the intercepts in the linear model and the logit-link to the intercepts: [probability of the rating] According to the fitted model (see Fig. 8), '7' is the most likely sale rating for facts (m = 0.44, 95% HDI: [0.21, 0.65]) and '1' is the most likely rating for coincidences (m = 0.54, 95% HDI: [0.36, 0.72]). Thus, the model predicts that people will strongly agree that facts must have an explanation and strongly disagree that coincidences must have an explanation.

Discussion
Study 4 demonstrates that people's judgments of metaphysical explanation are sensitive to the fact/coincidence distinction across a wide range of facts and coincidences. In addition, the Bayesian analysis shows that facts were most likely to receive a scale rating of 7 (i.e., participants strongly agreed the fact must have an explanation) and coincidences were most likely to receive a scale rating of 1 (i.e., participants strongly disagreed the coincidence must have an explanation). Thus, Study 4 provides compelling evidence that the earlier findings concerning the fact/coincidence distinction are indeed robust and generalize to a more comprehensive set of candidate explananda. Likewise, we also found a similar pattern of results in our analysis of individual differences: the vast majority of participants make the fact/ coincidence distinction, albeit there is substantial variation in the degree to which the distinction is reflected in participants' judgments.

Study 5: Generality of metaphysical-value divergence
In Study 5, we set out to generalize a key finding from Study 3 Fig. 6. Mean ratings for coincidences (yellow) and facts (blue). Points correspond to ratings for individual explananda (in Study 1, 4 coincidences and 26 facts in total; in Study 4, 150 coincidences and 230 facts in total), and the shaded region corresponds to the density of ratings. Error bars correspond to 95% HDIs. Points are jittered horizontally for visual clarity. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 3 Facts and coincidences with highest and lowest mean ratings.  . 7. Extracted coefficients for the fact/coincidence distinction (x-axis) by participant (y-axis). Green dots correspond to participants with coefficients above 0. Error bars correspond to 95% CIs. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 11 Given our interest in the predicted scale response for out-of-sample items, the Bayesian approach is appropriate here since ratings in this study were generated from items that were sampled from a larger population of interest (i. e., facts on Wikipedia). In contrast, previous studies generated ratings from a "curated" set of items that themselves already constitute the population of interest. Hence why we apply Bayesian methods here but not in the earlier studies.
whereby participants' value judgments showed near-zero partial correlations with their metaphysical judgments. We chose to focus on the metaphysical-value divergence since this sort of divergence could plausibly apply to a wide range of explananda, whereas the metaphysical-epistemic divergence likely only applies to a narrower range of epistemically inaccessible explananda, which would have to be arbitrarily constructed.

Participants
N = 161 participants were recruited from Prolific to complete a survey for modest compensation. Participation was restricted to adults living in the United States who had completed at least 50 prior tasks with a minimum approval rating of 95%. According to pre-registered exclusion criteria, 12 participants were excluded for failing a basic attention check. Thus, n = 149 participants (m age = 39.49 years, sd = 14.05 years; 53% female) were included in the final analyses.

Procedure
Participants rated 20 candidate explananda sampled randomly from a set of 260 in total (230 Wikipedia facts and 30 coincidences). For each explananda, participants made a rating on the same measures as Study 3: Metaphysical, Epistemic, Normative, Value, and Motivational (order randomized).

Results
As in Study 3, scale ratings on the value measures showed significant, positive partial correlations with each other (Normative-Value: r = 0.43, p < .001; Normative-Motivational: r = 0.51, p < .001; Motivational-Value: r = 0.38, p < .001), and near-zero partial correlations with the epistemic and metaphysical ratings. Thus, a key result from Study 3 indeed replicates and generalizes when assessed using a more comprehensive set of stimuli. Fig. 9 shows the full partial correlation matrix across all measures, alongside the results from the "curated" set of items used to elicit metaphysical-value divergence in Study 3.
As throughout all studies conducted, the average rating for facts was high on the Metaphysical (m = 5.60, sd = 1.23; comparison to mid-point: t(2036) = 58.61, p < .001). Participants' ratings in this study provided an additional replication of the fact/coincidence distinction, with facts receiving higher ratings on the Metaphysical measure than coincidences (m = 3.26, sd = 1.98) (b = 2.36, se = 0.18, p < .001).

Discussion
Study 5 shows the metaphysical-value divergence extends to a wide range of explananda. Whether or not a person thinks we ought to seek out an explanation, values knowing an explanation, or believes it worth the effort to find the explanation has nearly zero association with whether that person thinks there must be explanation. At the same time, agreement that facts must have an explanation or reason was strong overall; even if a person does not value knowing an explanation, she is still likely to think there must be an explanation. Once again, this result indicates that people's PSR-conforming judgments track a valueindependent notion of explanation. Moreover, these results demonstrate that Metaphysical-Value divergence does not depend exclusively on the type/token distinction.

General discussion
People seek explanations. This is especially salient from children's incessant questions of "Why?" . Moreover, explanations provide us a primary means of understanding the world and predicting future events in both science and ordinary life. The present research indicates that there is a distinctively metaphysical aspect to our explanatory judgments that diverges from their epistemic and value dimensions. Across five studies, we found that participants consistently presupposed a PSR-like principle in their explanatory judgment. These judgments predictably tracked the metaphysical considerations relevant to the PSR (Study 1), predictably diverged from other epistemic judgments (Study 2) and value judgments (Study 3), and applied to a large set of facts selected from random Wikipedia entries (Studies 4-5). The consistency and range of metaphysical judgments about explanation suggests that participants presupposed a generalized PSR-like principle in their judgment: facts must have an explanation-even if we cannot know it or knowing it would not be valuable for us. Of course, the PSR is a universal principle, and we can hardly ask participants about every fact there is. Nonetheless, we have collected judgments across a wide range of facts, including supernatural and inaccessible items that would have seemed likely to yield judgments of inexplicability. And yet, from the fluid dynamics of party balloons to the existence of God and the universe, participants reliably judged that facts must have an explanation.

PSR and inquiry
Our experiments provide evidence that American adults have a PSRlike presumption. Given this fact, a further, thematically appropriate, question arises: Why? Why would people have a PSR-like presumption?
We propose that such a presumption plays a beneficial role in facilitating inquiry.
It is a familiar idea in cognitive science that explanations are valuable for creatures like us (Lombrozo, 2011). Perhaps the most direct benefit is that good explanations lead to future predictive success, which, in turn, is likely to confer a fitness advantage (Gopnik, 1998). Explanation also facilitates a host of other benefits for learning and inference (Lombrozo, 2016), including generalization (Williams & Lombrozo, 2010) and causal reasoning (Walker et al., 2017). Providing good explanations may also be prized in an individual's community (Davoodi & Lombrozo, 2022) and therefore help individuals to accrue social reward (cf. Williams, 2022). But this still doesn't tell us why having a PSR-like presumption would be advantageous.
To see why such a presumption would be advantageous, let's start with a simple case. Imagine Jane is considering the fact that pencils keep disappearing from her backpack. Suppose also that Jane dislikes losing all the pencils. If she presumes that there is an explanation for the disappearance, she is more likely to pursue an explanation than if she has no presumption at all about whether there is an explanation.
In the example above, Jane's interests made her value having an explanation for the disappearing pens. In general, whether an agent values having a particular explanation will depend on the agent's interests and circumstances. Given this, one might expect that the relationship between the PSR-like presumption and our explanationseeking practices could be explained by an interest-dependent account: Interest-dependent account: If having an explanation for x is valuable, we generate the presumption that there must be explanation for x.
On this picture, whenever we have an interest in having an explanation, we reliably generate a local expectation that there is an explanation about whatever it is that we value explaining-as opposed to a global PSR-like expectation that every fact must have an explanation. This amounts to a kind of motivated reasoning.
As we've seen, (Studies 3 and 5), participants' judgments about whether there must be an explanation often diverge from judgments about what explanations we are interested in. Hence, the interestdependent account does not accord with the data. Rather, we found that people's belief that there must be an explanation is interest-independent, as accords with the PSR. The interest-independence of the PSRpresumption plausibly has a benefit of efficiency over the interestdependent account. When faced with a fact for which an explanation is of interest, no extra step is required to generate the presumption that there is an explanation. The PSR-like presumption is part of the background that helps to facilitate the search for explanation even when one's interests change.
Of course, whether it is objectively beneficial to have a PSR-like presumption will depend on whether the candidate explanandum x actually has an explanation. In the extreme case, if a domain of inquiry is entirely comprised of brute facts, it is best to avoid conducting inquiry in the first place. In general, it is important that our explanatory presumptions are commensurate with how the world actually is. Thus, it is not necessarily beneficial for one's PSR-like presumption to take a completely unrestricted scope. Rather, having a PSR-like presumption is most beneficial when its scope encompasses domains comprised of mostly non-brute facts. For example, consider the following candidate domains of inquiry: CLASSICAL: x 1 : "objects fall to ground,"…, x n . CONSPIRACY: x 1 : "Apollo 11 was launched shortly after Stanley Kubrick filmed 2001: A Space Odyssey," …, x n . CLASSICAL is the domain of candidate explananda that concerns classical mechanics. Going by the explanatory success of classical mechanics, it is safe to assume that very few candidate explananda are brute in CLASSICAL. By contrast, CONSPIRACY is a domain of mere coincidences that relate to the moon landing. Again, if we go by track record, it would not be safe to assume that very few candidate explananda are brute in CONSPIRACY.
Suppose that we and others in philosophy of science are correct in thinking that mere coincidences do not have explanations. Given this, explanation-seeking, utility-maximizing agents should not have a PSRlike presumption that includes domains comprised entirely of mere coincidences. To a first approximation, our experimental results suggest that people's judgments conform to this distinction. Participants' judgments showed consistent agreement with the strong modal claim that "There must be an explanation or reason why x" for scientific facts but not coincidences.
The foregoing discussion sketches out the beginning of a functional analysis of the PSR-like presumption (cf. Marr, 1982): so long as one conducts inquiry in domains comprised of mostly non-brute facts, it's beneficial to have a PSR-like presumption. In a broad and uninteresting sense, this analysis boils down to saying that it is beneficial to have beliefs that correspond closely to how the world actually is-a conclusion that is hardly surprising. When applied to PSR, however, we think this analysis does point to something interesting about the nature and origins of our metaphysical convictions. One way to generate belief in an abstract, metaphysical principle is by assessing how well it fits within a system of other abstract principles. But an alternative route to metaphysical conviction may depend on how a belief in the principle helps us achieve our aims in the world. This latter possibility is largely neglected in philosophical discussion of PSR (see Amijee, 2022 for a notable exception), yet the foregoing analysis and experimental results together suggest that it is worth taking seriously.

Open questions for cognitive science
In addition to the possibility that the PSR-like presumption has a functional role in inquiry, another important question for cognitive science concerns the acquisition of the PSR-like presumption. One possibility is that there are innate biases that give rise to a PSR-like presumption. To evaluate this innateness hypothesis, it will be critical to explore the extent to which a PSR-like presumption is evident in early childhood. Similarly, it will also be important to examine whether a PSR-like presumption across domains is present across cultures.
Another possibility is that children learn to adopt the PSR. Interestingly, one of Leibniz's arguments for the PSR is inductive. The argument uses what he calls "the method of experimental philosophy, which proceeds a posteriori" (Leibniz, Clarke, & Ariew, 2000, p. 65). He argues as follows: I have often defied people to allege an instance against that great principle [of sufficient reason], to bring any one uncontested example wherein it fails. But they have never done it, nor ever will. 'tis certain, there is an infinite number of instances, wherein it suc-ceeds… From whence one may reasonably judge, that it will succeed also in unknown cases (Leibniz 2000, p. 65).
Leibniz's argument here provides a sketch for a naïve learning theory. If children entertain the possibility that every fact has to have an explanation, exposure to numerous cases of successful explanation might lead them to put credence in the PSR. Alternatively, explicit education (e.g., science education) might play a critical role in fostering credence in PSR. Again, to evaluate these possibilities, it will be important to examine whether and when children display a commitment to PSR. If one of these learning theories is correct, we might expect a developmental pattern in which older children will be more likely to give responses that conform to the PSR (cf. Woolley & Cornelius, 2017, p. 1594.
Further, if the PSR-like presumption is acquired or modulated by experience, this could help explain why we observe substantial variation in the degree to which participants draw the fact/coincidence distinction; presumably, not all participants have the same set of relevant experiences. In ongoing research, we are investigating whether individuals with little or no formal education exhibit the same PSR-like presumption as observed in the online convenience sample collected here. These studies and future research could speak to whether the adoption of the PSR presumption is connected to how people start to generate better explanatory theories about the world. For example, our explanatory theories may license better predictions once we learn to discriminate between facts to be explained and occurrences that should be dismissed as mere coincidences.
Another set of questions concerns the relationship between our epistemic, value, and metaphysical explanatory judgments. Throughout Studies 2-5, a consistent pattern of results was that participants' metaphysical and value judgments tended to be statistically independent, whereas participants' metaphysical and epistemic judgments tended to display a positive association. One possibility is that all of our various explanatory presumptions develop independently of each other. A more intriguing possibility is that their relationship is more complicated and intertwined. For instance, do we generate this metaphysical judgment because we have seen enough times that we can access explanations for certain facts? Or, alternatively, are our expectations of finding specific explanations couched or guided by our general sense that the world is explanatorily structured? A third possibility is that our metaphysical and epistemic judgments are mutually-reinforcing (cf. Dalege et al., 2016). Developmental and cross-cultural studies will be crucial for delineating between these possibilities.
The role of PSR judgments in religious thought is a further point of interest. There has been excellent psychological work bearing on the design argument for the existence of God (see, e.g., Evans, 2000, Kelemen, 2004). However, not much has been said about the cosmological argument (see de De Cruz & De Smedt, 2017 for a notable exception). 12 As we noted at the outset, modern rationalist philosophers argued the PSR serves as the crucial premise in the cosmological argument. If everything must have an explanation, then God must play a fundamental explanatory role in explaining the universe itself. Hence, research on the PSR-like presumption provides a direct entry point on the psychological bearings to the persuasive power of this argument.
Another intriguing question is whether a PSR-like presumption extends to the normative domain. So far, we have gathered participants' judgments about a wide range of descriptive facts about how the world is. Nonetheless, much of our psychology is also dedicated to normative judgments about how the world should be. It is an open question whether a PSR-like presumption would extend to these judgments. Consider prudential norms and moral norms. Does there have to be an explanation of why we should brush our teeth? Does there have to be explanation for why it is wrong to kill innocent people? Future research directed to the normative domain would illuminate the divergences or convergence of explanatory judgment with respect to normative claims. If people also hold PSR-conforming judgments with respect to ought statements, this would suggest that they interpret normative injunctions as normative facts about the world, governed by the same explanatory principles that govern descriptive statements.
In the meantime, our findings to date indicate that American adults presuppose facts must have an explanation, over and above whether it is possible to know, or would be good to know, the explanation. Given this, it seems that a metaphysical presumption that conforms with the 12 de Cruz & de Smedt note that developmental work reveals that even babies expect events to have causes, with a preference for agents as causes, and older children spontaneously seek causal explanations (2017,(63)(64)77). They suggest that this early emerging focus on cause and causal explanation contributes to the persuasive power of the cosmological argument.
PSR-an ancient and deep philosophical principle-may indeed hold a place in ordinary thought.

Credit statement
SN conceptualized project and obtained funding. Design of studies by SP, AV, and SN. Execution of research by SP. SP analyzed data. SP wrote the first draft of manuscript; all authors contributed to the writing of the final manuscript.

Declaration of Competing Interest
None.

Data availability
All data/code is available on the OSF page.