Thinking, Guessing, and Believing

1. Introduction

There are many questions about belief on which there is little consensus. Is belief closed under entailment? What is the relationship between belief and credence? How about belief and knowledge? Can one believe a proposition without being disposed to assert it or use it as a premise in deliberation? Is belief sensitive to pragmatic or moral factors?

But if there’s one thing that has been a matter of consensus—or near enough anyway¹—it’s that it is rational to believe a proposition p only if one’s evidence for p is a good amount better than one’s evidence for ¬p. The question of how much better is debated. But the details aside, the general thought is widely regarded as a simple platitude.

That is until recently. Hawthorne et al. (2016) have argued compellingly that the evidential requirements on belief are much weaker than epistemological orthodoxy has taken them to be.² Not only can one believe that p while knowing that one isn’t in a position to know that p, one can believe that p even while knowing that one’s evidence makes it more likely than not that p is false.

We will see the details of some of their arguments shortly. But the train of thought underlying them is roughly this: (i) belief is what is denoted by the ordinary expression ‘believes’; (ii) ‘thinks’ and ‘believes’ are synonymous; (iii) the attitude denoted by ‘thinks’ has very weak evidential requirements; so (iv) belief has very weak evidential requirements.

This paper takes no stance on the soundness of this argument. In particular, it takes no stance on the truth of either (i) or (ii). This is because the aim of the paper is not to defend a theory of belief per se, but to defend a theory of thinking—i.e., the attitude denoted by ‘thinks’ in sentences of the form ‘S thinks that p’. Of course, if thinking just is believing—if one thinks that p iff one believes that p—then this paper is also defending a theory of belief. But it will be better to develop a theory of thinking from a position of neutrality on these matters. This is not so much because the arguments for (i) or (ii) are particularly complex or controversial. Whether (i) is true or false seems to be mostly just a matter of stipulation. And though (ii)’s truth or falsity clearly isn’t a matter of stipulation, the case for it is very strong.

Here is the short version.³ If thinking weren’t believing, then you’d expect to be able to imagine circumstances in which, for some agent S and proposition p, it would be natural to think or say that S thinks that p without believing it (or vice versa). But evidently we cannot imagine such circumstances:

• (1) ✗ I think it’s raining, but I wouldn’t say I believe it is.
• ✗ I’m not sure whether Jane thinks Federer will win Wimbledon, but I know she doesn’t believe he will.
• ✗ My friends think I’m a good person, but my mom believes I am.

It is difficult (if not impossible) to recover a coherent interpretation of these sentences. But replace ‘believe’ with (e.g.) ‘is sure’ or ‘is certain’ and they sound perfectly fine. If thinking is believing then it is entirely unsurprising why this is so. But if thinking isn’t believing—if one can think that p without believing that p (or vice versa)—then the existence of these judgments is a mystery. There is thus a strong case to be made that a theory of thinking just is a theory of the attitude that is the denotation of ‘believe’. So supposing that the attitude that is the denotation of ‘believe’ just is belief, it follows that there is a strong case to be made that a theory of thinking just is a theory of belief.

That said, some epistemologists are happy to help themselves to a notion of belief other than the one expressed by ‘believe’ in ordinary language.⁴ And although the arguments for the view that thinking is believing seem to me quite strong, it’s not as if they are unchallengeable.⁵ And so I think it would be useful if we approached our central topic—What does it take to think that p?—without prejudging whether in doing so we are also investigating the nature of belief.

Until the paper’s concluding section, then, we will be concerned exclusively with thinking, with the aim of answering the following two questions:

The descriptive question: What is it to think that p?

The normative question: Under what conditions is it rationally permissible to think that p?

For conspicuously absent from both Hawthorne et al. (2016) and Rothschild’s (2020)’s discussions is anything resembling a full account of thinking’s metaphysics and norms. Dorst (2019), to his credit, develops a substantive view, but we will soon see reasons to think his cannot be correct. So regardless of one’s stance on the connections between thinking and believing, or between the attitude that is the denotation of ‘believe’ and the object of conventional epistemological study, there is a lacuna that deserves to be filled.

And on that matter, this paper defends the view that to think that p is to guess that p is the answer to the question at hand, and that to think that p rationally is for one’s guess to be in a certain sense non-arbitrary. Some theses that will be argued for along the way include: that thinking is question-sensitive and, correspondingly, that ‘thinks’ is context-sensitive; that it can be rational to think that p while having arbitrarily low credence that p; that, nonetheless, rational thinking is closed under entailment; that thinking does not supervene on credence; and that, in many cases, what one thinks on certain matters is in a very literal sense a choice. Finally, if, as it seems, thinking just is believing, then all this goes for belief as well.

2. A point about felicity judgments

The main aim of this paper is to give a theory of the attitude that is the denotation of ‘thinks’. I will not be making any prior assumptions about this attitude’s theoretical roles, and so the primary source of evidence for the theory will be judgments about natural language sentences involving ‘thinks’—judgments that for the most part will simply be taken at face value.

One might wonder how intuitions about natural language sentences containing ‘thinks’ could reveal facts about the rational requirements on thinking. But the idea is entirely familiar, even if not under this guise. There is something obviously strange about an utterance of a sentence like:

• (2) ✗ I think that it’s raining and that it isn’t.

This is what the ‘✗’ is meant to indicate. We could try to explain the strangeness by giving a semantics for ‘thinks’ (i.e., a theory of thinking simpliciter) that makes ‘S thinks that p’ entail ‘It is not the case that S thinks that ¬p’. But that is not what the evidence warrants. If prior to asserting (2) I had declared that I tend to have irrational patterns of beliefs about the weather, there would be no impression that (2) expresses something that couldn’t be true. Indeed, it would be reasonable to cite the truth of (2) as evidence that I was irrational. Our judgments reveal something about the norms on thinking, rather than its metaphysics.

By the same token, the fact that a sentence of the form ‘S thinks that p’ is felicitous in certain circumstances will often be pretty good evidence that S can rationally think that p in those circumstances. For example, if I know that Jones has an extremely low chance of winning the upcoming lottery and you ask me how I think he’ll fare, it would be perfectly appropriate for me to respond:

• (3) ✓ I think he’ll lose.

There is no impression that I represent myself as irrational in using (3) to answer your question about what I think. And we know that we often do detect when a person making a ‘thinks’-report would have to be irrational for the report to be true. So absent strong reasons to think our judgments about sentences like (3) are generally confused, we should think that a rational agent can speak from knowledge in asserting it.⁶

With that, we can now turn to answering the two central questions: What does it take to think that p? And what does it take to think that p rationally? As it turns out, getting clear on the first question is much easier once we’ve gotten clear on the second. So for the next few sections our concern will be with the theory of rational thinking, rather than thinking simpliciter.

3. Thinking is weak

We’ll start with a thesis defended in detail by Hawthorne et al. (2016). It is that thinking is weak. Somewhat more precisely:

Weakness

The evidential requirements on thinking that p are weaker than the evidential requirements on asserting that p, using p as a premise in deliberation, or being sure that p.

Whether the evidential requirements on assertion, deliberation, and surety are equivalent is not a question we need to settle here. All that the proponent of weakness is committed to is that the strength of evidence needed to rationally think a proposition is less than the strength of evidence needed to do any of these other things—i.e., that there are circumstances in which it is appropriate for an agent to think that p, despite the fact that it would be inappropriate for her to be sure of it, assert it, or use it as a premise in deliberation.

Since the arguments for weakness have already been given a fair bit of attention in the recent literature, our treatment of them here will be brief.

One style of argument focuses on simple judgments about ‘thinks’-reports. For starters, it is perfectly felicitous to assert that one thinks p but is unsure whether p:

• (4) ✓ I think it will rain, but I’m not sure it will.

It is also perfectly felicitous to assert that one thinks but does not know whether p:

• (5) ✓ I think it will rain, but I don’t know that it will.

And it is also possible to felicitously assert that one thinks that p but knows that there is a substantial chance that ¬p:

• (6) ✓ I think it will rain, but I know there’s a substantial chance it won’t.

Since there are circumstances in which conjunctions like (4)–(6) can be uttered felicitously, it must be possible for a rational agent to speak truly in asserting them. Since it is possible for a rational agent to speak truly in asserting them, it must be possible for a rational agent to think it will rain while knowing there is a substantial chance that it won’t. But of course it is not possible for a rational agent in these circumstances to be sure that it will rain, assert it will rain, or use the proposition that it will rain as a premise in deliberation. Hence weakness.

We can make a similar argument for weakness by thinking about cases rather than sentences. Suppose A has purchased one of the 100 tickets to an upcoming lottery. She has no special information about what the outcome will be. Does A have enough information to be sure that she’ll lose the lottery? Of course not. Can she assert that she’ll lose, or take for granted that she’ll lose in (e.g.) deciding whether to sell the lottery ticket for a penny? No—this would be a failure on A’s part to give proper weight to her evidence. But does A have enough information to think that she’ll lose? Certainly. There is nothing irrational about thinking you’re not going to win the lottery.

Indeed, examples that illustrate the weakness of thinking are legion. We think things about the weather, upcoming elections, unsolved murders, and mathematical conjectures even when we know full well that our evidence on these matters is far from decisive. But we do not act as if we are sure of these things. We do not assert them outright, and we do not treat them as the kind of propositions whose possible falsity can be ignored when engaging in practical deliberation. The evidential requirements on thinking are weak.

4. Thinking is extremely weak

Alright—but how weak? The answer this section defends is: extremely weak. One can rationally think that p despite the fact that one’s evidence makes the probability that p arbitrarily close to zero.⁷

It will be helpful going forward to make some brief quick technical stipulations. I will model an agent S’s evidence (hereafter ‘E_S’) as a proposition: in particular, the proposition that is the conjunction of all the propositions S rationally permitted to be sure of. I will associate with each such body of evidence a rational credence function (hereafter ‘C_S(·)’), which takes propositions to real numbers on the unit interval. I will assume (i) that rational credence functions obey the axioms of the probability calculus, and (ii) that the rational credence function associated with a certain agent’s evidence takes p to 1 iff that agent’s evidence entails p. In saying ‘S has rational credence x that p’ (i.e., C_S(p) = x), I will mean that the rational credence function associated with S’s evidence takes p to x, rather than that S in fact has credence x that p and is rational for doing so. As a reasonable gloss: S’s rational credences are the credences S would have were S rational.⁸

With this terminology in place, here is a precise characterization of the claim that thinking is extremely weak:

Extreme Weakness

There is no positive number x such that: necessarily, for any agent S and proposition p, if C_S(p) ≤ x, then S is not rationally permitted to think that p.

The argument for extreme weakness is fairly simple. But to warm up to it, I’ll first argue for the following more moderate principle:

Substantial Weakness

Possibly: for some agent S and proposition p such that C_S(p) < .5: S is rationally permitted to think that p.

In other words: it is possible to rationally think p while having rational credence less than .5 that p.

Suppose I tell you that an upcoming lottery has 100 tickets, that A has purchased 48 of them, and that the remaining 52 have been distributed evenly among 52 other people (B, C, D… etc.). So A has a 48% chance to win, everyone else 1%. Question: Who do you think will win?

Here is a perfectly reasonable answer: A. After all, you know she is 48 times more likely to win than anyone else. But you also know her chances of winning are a mere .48. So rational thinking does not require rational credence greater than .48. Hence substantial weakness.

Similar judgments can be elicited in other domains. Consider questions like:

• Which horse do you think will finish in first?

• Who do you think will get the Democratic nomination?

• How many people do you think were responsible for the “Jack the Ripper” murders?

You might only have credence .35 that horse A will win, or that Harris will get the nomination, or that the Jack the Ripper murders were performed by no more than one person. But this is no principled barrier to offering these up as your answers. The question is about what you think, not what you know. Thinking isn’t just weak, it’s substantially weak.

Of course, an argument for substantial weakness is not yet an argument for extreme weakness. But since the argument for extreme weakness is really just a generalization of the argument for substantial weakness, it is worth pausing to consider how this intermediate conclusion might be rejected. I’ll consider two quick objections.

First, one might (correctly) point out that questions of the form ‘Wh-F do you think Gs?’ are standardly taken to presuppose that there is some F that you think Gs. So, for example, in asking you ‘Who do you think will win the lottery?’, I presuppose—possibly incorrectly—that there is some person you think will win. So perhaps in the circumstances of the 48 ticket case, your answering ‘A’ is merely your best attempt to accommodate my question’s presupposition, rather than a genuine attempt to report what you actually think.

I think this response has some serious problems. For one, the fact that it is so easy to accommodate the presuppositions of the question ‘Who do you think will win the lottery?’ is itself good evidence that there is no barrier to thinking propositions in which one’s rational credence is a mere .48. Why? Because supposing that being in a position to rationally think that p is compatible with having rational credence .48 that p, it is no surprise that the question’s presuppositions are easy to accommodate. But supposing these things are not compatible, it is much less obvious why we would go about accommodating the question’s presuppositions. Notice that the appropriate answer to questions like ‘Who is it that you are sure will win the lottery?’ or ‘Who do you have greater than .5 confidence will win?’ is ‘No one’, not ‘A’. This is good evidence that, contrary to the imagined response, we do not blithely represent ourselves as irrational in order to try to accommodate the presuppositions of the questions we have been asked.

Another issue with this objection is that one doesn’t even have to be asked a question to felicitously report oneself as thinking that A will win the lottery. In the circumstances of the 48 ticket lottery scenario, it is perfectly acceptable to assert outright: ‘I think A will win’, or ‘I think the winner will be A’, etc. Placing stress on ‘A’ makes the true readings of these sentences crystal clear. Likewise, if you were to overhear someone else say any of these things (knowing they have the same evidence as you), the natural conclusion to draw would not be that that person is irrational; rather, the natural conclusion to draw would be that that person is doing the perfectly normal thing of expressing the thought that the overwhelming favorite to win will, well, win. The fact that such reports are so readily elicited by questions is beside the point.

A different objection to the argument for substantial weakness focuses on the apparent optionality in how one may choose to respond to questions about what one thinks. More concretely, though it can be appropriate to answer the lottery question with ‘I think A will win’, it seems it can also be appropriate to answer agnostically: say with an ‘I don’t know’, ‘I’m not sure’, ‘There isn’t anyone I think will win’, or what have you.

Why would the availability of this alternative response cast doubt on the probative force of the data? I myself find it less than perfectly clear. But presumably it would have to involve the view that, for any given rational credence function and proposition p, rationality permits exactly one of the following attitudes: thinking that p, thinking that ¬p, or agnosticism toward p (i.e., neither thinking that p nor thinking that ¬p). It would then follow that at least one of the two seemingly appropriate kinds of answers (‘I think A will win’ vs. ‘I don’t have a view’) fails to track the underlying facts about rational thinking. And since the details of the case are such that the opinionated ‘I think A will win’ is more surprising than the agnostic ‘There isn’t anyone I think will win’—at least from the perspective of conventional epistemology—one might take this all to be reason to regard it with suspicion.

I will return to the issue of optionality at some length in §10; for now my treatment of it will be brief. The view that the laws of rationality associate any given rational credence function with no more than one coarse-grained doxastic attitude is controversial and contestable. Though its theoretical appeal may count as some evidence that the natural language judgments ought to be regarded with suspicion, it would be a dubious methodological practice to dismiss them outright on such grounds.

It is worth noting, for example, that the exact same phenomenon seems to arise in more mundane lottery cases. Suppose we modify the details of the lottery so that A has 99 of the 100 tickets, rather than 48. Now consider again the question ‘Who do you think will win?’. Though I expect many will be inclined to answer ‘A’, it is not at all clear that it would be irrational to answer agnostically instead. Responses like ‘I don’t know’ or ‘I have no particular view on who will win’ seem just fine. So there is optionality here too. But in this version of the case it seems especially implausible that the presence of optionality implies that we don’t report what we think when we say that we think the person who has a 99% chance of winning will win.

Consequently, I believe we ought to take our intuitions about the answers to questions like ‘Wh- F do you think is G?’ seriously. And since there is nothing intuitively problematic about reporting that one thinks that the person with 48 of the 100 tickets to a lottery will win that lottery, we have good reason to believe that substantial weakness is true.

From there the argument for extreme weakness follows quickly.⁹ For any real number 0 < x ≤ 1, we just need to come up with a lottery in which: (i) some person A has the best chance of winning and (ii) that chance is x. For then we will have found a scenario in which it can be rational to think that A will win the lottery despite the fact that one’s rational credence in that proposition is no higher than x.

For x = .01, just imagine a 1,000 ticket lottery in which A has 10 tickets and the remaining 990 are distributed evenly among 990 other entrants. If asked ‘Who do you think will win the lottery?’ the answer ‘A’ remains appropriate. Again, this is not to say that it is mandatory. An ‘I don’t know’ or ‘There isn’t anyone I think will win’ is fine too. But one who answers ‘I think A will’ needn’t be irrational.

The same goes regardless of whether A’s ticket count is 2 rather than 10 or if the total count is 10,000 rather than 1,000. Granted, if you shrink the gap between A’s chances of winning and the next highest person’s, or if you lower A’s absolute chances of winning—or both—then the agnostic response to the question ‘Who do you think will win?’ may start to seem more compelling. But again: making the agnostic response more compelling is not the same thing as making the opinionated response unacceptable. And I contend that no matter how bad or close the odds happen to be, it will remain permissible to answer the question ‘Who do you think will win the lottery?’ with ‘The person who is most likely to’. So it is possible to rationally think that p even when one’s rational credence that p is arbitrarily close to zero. Hence extreme weakness.

5. Thinking is non-monotonic

The kinds of cases in which it seems appropriate for S to think that p despite having very low credence that p seem to be those in which S’s evidence makes p more likely to be true than any other proposition in a class of relevant alternatives. This insight will eventually form the basis of this paper’s theory of rational thinking. But for now I want to put it to a different purpose, which is to argue that the relationship between rational thinking and rational credence is non-monotonic:

Non-Monotonicity

There can be two agents (or the same agent at different times) S₁ and S₂ such that: , yet S₁ is rationally permitted to think that p while S₂ is not.

In other words: increasing the strength of the evidence will sometimes decrease the strength of the conviction. One can start off rationally thinking that p, acquire evidence that increases the likelihood that p, but—because the evidence increases the likelihood of a relevant alternative q even more—one can thereby become rationally prohibited from thinking that p.

The case for non-monotonicity looks to be at least as strong as the case for extreme weakness. Consider a horse race with three entrants: A, B, and C. Suppose S₁ has been told by an expert horse bettor that the horses’ chances of winning are 40%, 35%, and 25%, respectively, while S₂ has been told by a different expert horse bettor that the horses’ chances of winning are 45%, 50%, and 5%. What is the rational thing for each of S₁ and S₂ to think about the outcome of the race?

Intuitively, S₁ should think A will win (or perhaps have no opinion) and S₂ should think B will win (or perhaps have no opinion). If you were to ask S₁ who she thinks will win, she would do just fine answering ‘A’, whereas she would seem to represent herself as irrational in answering either ‘B’ or ‘C’. Likewise, if you were to ask S₂ who he thinks will win, he would do just fine answering ‘B’, whereas he would seem to represent himself as irrational in answering either ‘A’ or ‘C’. And all of this is despite the fact S₂ has strictly greater rational credence that A will win than S₁ has. That is to say: S₂ would be less surprised than S₁ if A were to win, would be willing to take worse bets on A’s winning, would have reason to think S₁ is under-confident about A’s prospects, and so on. So we have a case in which there are two agents such that the first is rationally permitted to think that p while the second is not, and yet the second’s rational credence that p is higher. And of course the same point can be made with one agent instead of two: just modify the case so that the two sets of rational credences are both S₁’s, only relativized to different times. Either way we have non-monotonicty.

6. Thinking and closure

This section will provide yet further evidence that rational thinking is about according what one thinks to what’s most likely to be true given one’s evidence. It will do so by considering the vexed question of whether rational thinking is closed under entailment—i.e., whether the following principle is valid:¹⁰

Closure

If a set of propositions Γ is such that S is rationally permitted to think every member of it, and Γ entails p, then S is rationally permitted to think that p.¹¹

There is a strong case to be made for closure. For one thing, it is highly plausible that rational agents can come to think that a proposition is true by deducing it from other propositions they already think are true. This seems to be exactly what we do when we try to reason through things. For another, we seem to ordinarily think and talk in a way that takes closure for granted. Notice how puzzling it is to speak as if the principle had false instances:

• (7) ✗ I think B will be at the party. I also think C will be at the party. But I wouldn’t say I think both B and C will be at the party.
• ✗ I think B will be at the party. And I think that if B will be at the party, C will be there too. But it’s not fair to say that I think C will be at the party.

As is well known, however, closure also happens to face a number of putative counterexamples. Take Kyburg’s (1961) famous lottery puzzle, for instance. You know that each of the 100 entrants to an upcoming lottery has a 1% chance of winning. Consequently, for each entrant it seems rationally permissible to think that that entrant won’t win. But of course you know that someone has to win, and are thus not rationally permitted to think that none of the entrants will. And therein lies the problem: if for each entrant you’re rationally permitted to think that that entrant won’t win, yet you are not rationally permitted to think that none of the entrants will win, then rational thinking cannot be closed under conjunction.

The standard explanation for why closure allegedly fails in these sorts of cases is in terms of epistemic risk aggregation. Taken individually, each of the lottery propositions about specific entrants has a very high chance of being true, and is thus rationally thinkable. But each also has some chance of being false. As one conjoins these propositions the chances aggregate, eventually resulting in a proposition that is guaranteed to be false. And this is allegedly why, contrary to closure, one can rationally think each member of a set of propositions without being entitled to rationally think their conjunction.

One point I think is worth emphasizing is that even if we do take cases like Kyburg’s to undermine closure, we should reject these sorts of risk-theoretic diagnoses of its invalidity. After all, we know from §4’s discussion of extreme weakness that there is no principled barrier to rationally thinking a proposition in which one’s rational credence is arbitrarily close to zero. So it’s not obvious why increasing the number of conjuncts in the relevant lottery proposition should automatically make it less fit as an object of rational thought. More to the point, though: if we’re willing to take the “rational thinking is about thinking most likely” slogan seriously, then we should expect to find even more striking putative counterexamples to closure—ones that have nothing in particular to do with the aggregation of risk.

And indeed such examples are not hard to come by. Suppose we’re trying to track down James Bond. Our sources tell us there is a 40% chance he is hiding in London and a 20% chance he is hiding in each of Berlin, Frankfurt, and Munich. In light of this information, are we in a position to think that Bond is hiding in London? Well, suppose we are asked ‘Which city do you think Bond is in?’. By my lights, ‘We’re not sure, but we think he’s in London’ is a perfectly appropriate response. So we must be rationally permitted to think that Bond is in London. But are we rationally permitted to think that Bond is in the United Kingdom? It’s not so clear. If asked ‘Which country do you think Bond is in?’, the most natural response would be ‘We’re not sure, but we think he’s in Germany’, or perhaps just the agnostic ‘We’re not sure’. If instead we were to answer ‘We’re not sure, but we think he’s in the United Kingdom’, it would be reasonable to object that we are significantly more confident that he is in Germany. So despite the fact that we appear to be rationally permitted to think that Bond is in London, we don’t appear to be rationally permitted to think that Bond is in the United Kingdom. And this is despite the fact that we are certain that London is in the United Kingdom, and thus have no less rational credence that he’s in the United Kingdom than we do that he is in London.

What to make of closure then? At this point it is hard to say. But this much is clear: if the putative counterexamples to the principle are genuine, we are owed a debunking explanation of the theoretical arguments in favor of the principle as well as the intuitions to the effect that speeches like (7) are defective. And if the putative counterexamples are merely putative, then we are owed a debunking explanation of at least one of the intuitive judgments driving them.

I will ultimately leverage the “rational thinking is about thinking most likely” slogan to argue in favor of the second approach. In particular, I will take closure to be valid, and I will try to explain away the recalcitrant data in terms of illicit shifts in the underlying class of alternatives against which the relevant ‘thinks’-reports are assessed. But first it will be good to take stock.

7. Against existing views

So far I’ve argued in favor of weakness, extreme weakness, and non-monotonicity, and have made an equivocal case for closure. Since it will be useful in what’s to come, I will now make explicit the existence of a consistency requirement on rational thinking, which is the natural principle underlying the badness of speeches like (2, ‘I think that it’s raining and that it isn’t’):¹²

Consistency

S is not rationally permitted to be such that: S thinks that p and S thinks that ¬p.

The aim now is to find a theory of rational thinking that can make sense of all of these principles. I’ll start by ruling out the two main existing theories of rational thinking: Cartesianism and Lockeanism.

8. Rational thinking as thinking most likely

The goal is to give a theory of rational thinking that does justice to the intuitive thought that rational thinking is about thinking true the proposition most supported by one’s evidence. Since whether a proposition is most supported by one’s evidence depends on the alternatives to which it is compared, alternative-sensitivity is going to have to be built into the theory of rational thinking. I’ll thus start by briefly complicating the picture of thinking simpliciter.

9. Situating the view

That said, I do not think best guess is the true answer to the normative question. This is for reasons that will be explored in the next section, where the paper’s final theory of rational thinking will be presented. But before getting into that, I think it will be helpful to explain the relationship between best guess and more familiar “contrastivist” theories of knowledge and belief. I will also discuss a popular objection stemming from the view’s use of question-based context-sensitivity.

10. The optionality in what one thinks

With these points aside, I now want to turn to what I take to be a more serious issue for best guess, which is its inability to fully account for the phenomenon of optionality raised to salience in §4’s discussion of extreme weakness. Reflection on these points will lead to the final theory of rational thinking, while also placing some striking constraints on the space of possible theories of thinking simpliciter—the topic of the next section.

To help reintroduce the optionality phenomenon, I’ll focus on a horse race with four entrants—A, B, C, and D—where S’s rational credences that each will win are [a: .35, b: .30, c: .20, d: .15]. Suppose S is asked ‘Who do you think will win the race?’. We know she does just fine in answering in terms of her best guess:

• (11)   ✓ A.

We also know (back from the discussion in §4) that she also does just fine in answering agnostically, as in:

• (12) ✓ I don’t know.
• ✓ There isn’t any horse in particular I think will win.

It thus appears that the norms of rationality leave S with options. If S goes the opinionated way, she can defend herself by pointing out that A has the best chances of winning. And if instead she goes the agnostic way, she can defend herself by pointing out that none of the horses is sure to win. In neither case would S seem to be making a mistake.

So far none of this is a problem for best guess, for best guess tells us that rational agents are permitted to think_Q whatever is entailed by their best guess to Q^?. It doesn’t say that they must think it. The problems for best guess emerge when we consider some of the more subtle patterns in the possible answers to questions about what we think.

We can imagine the possible answers to the question ‘Who do you think will win the race?’ as falling on a spectrum. On one end of the spectrum we have the four maximally opinionated answers: ‘A’, ‘B’, ‘C’, and ‘D’. On the other we have the maximally agnostic answers: ‘I don’t know’, ‘One of A, B, C, or D’, etc. Between these two extremes we have “mixed” answers that are opinionated in some respects, agnostic in others: ‘I’m not sure about A, but I do think it’ll be A or B’, ‘B or D’, etc. Given that S’s rational credences are [a: .35, b: .30, c: .20, d: .15], it is felicitous for her to give voice to some but not all of the possible mixed answers. These, for instance, are felicitous:

• (13) ✓ There is no horse in particular I think will win, but I do think it will be either A or B.
• (14) ✓ All I can say is that I think it won’t be D.

These, for instance, are not:

• (15) ✗ I think B or C will win.
• (16) ✗ There is no horse in particular I think will win, but I do think it will be either A or C.

Intuitively both (15) and (16) have the feel of wishful thinking on S’s part. (15) flies in the face of the fact that S knows that A has the best chance of winning, while (16) flies in the face of the fact that if S is only willing to narrow down her view on who will win to two horses, she should be thinking that the winner will be A or B, not that it will be A or C.

Given best guess, it’s no surprise why (13) and (14) are fine: S is rationally permitted to think_{a,b,c,d} whatever is entailed by her best guess to the question ‘Who will win the race?’. And both of these answers are indeed entailed by her best guess. It is also no surprise why (15) is infelicitous: the proposition that B or C will win the race is not entailed by her best guess to the question of who will win the race, and so S is not rationally permitted to think_{a,b,c,d} it.

The kind of judgment that poses a problem for best guess is the one raised to salience by (16). S’s best guess to the question of who will win the race is that A will. This proposition entails that A or C will. So best guess says S is rationally permitted to think_{a,b,c,d} that A or C will win. But again, best guess only tells us when an agent is rationally permitted to think that a proposition is true, not when they must think it. So for all best guess is concerned, S is perfectly rational in thinking_{a,b,c,d} that the winner will be one of A or C while not having a view_{a,b,c,d} on whether A in particular will win. So best guess wrongly predicts that there should be nothing wrong with (16).

As I see it, the lesson to take from the infelicity of (16) is that what one is rationally permitted to think relative to a question Q^? depends not only on the distribution of one’s rational credences in its answers, but also on what one in fact thinks the answer to that question is. If S happens to think_{a,b,c,d} that A will win (in accordance with her best guess), then she is indeed rationally permitted to think_{a,b,c,d} that A or C will win. But if S doesn’t think_{a,b,c,d} A will win—which is what she says in uttering (16)—then S isn’t rationally permitted to think_{a,b,c,d} that A or C will win. At most she can rationally think_{a,b,c,d} that which is entailed by the proposition that A or B will win.

Now to turn this idea into a theory. I’ll start by defining S’s guess to Q^? as the strongest answer to Q^? such that S thinks_Q that answer is true.²⁷ By assumption, then, ‘guess’ denotes a three-place relation between agents, questions, and unions of complete answers to those questions. If the question is {a, b, c, d}, S’s guess might be maximally strong, as when it’s a, b, c, or d; or it could be maximally weak, as when it’s a ∨ b ∨ c ∨ d; or it could be of middling strength, as when it’s something like a ∨ b.

With this notion of guessing in place, here is a schematic theory of rational thinking:

Rational Thinking Schema

S is rationally permitted to think_Q that p just in case: (i) the conjunction of E_S and S’s guess to Q^? entails p, and (ii) S’s guess to Q^? is rational.

The rest is just a matter of spelling out the details of (ii). And here the idea will be that S’s guess to Q^? is rational if and only if it is cogent, where a proposition p is cogent for an agent S relative to a question Q^? if and only if:

• (1)   p is a union of complete answers to Q^?
• (2)   If there is a complete answer to Q^?, r, such that r doesn’t entail p, then there is no other complete answer to Q^?, r*, such that: r* entails p, but C_S(r) ≥ C_S(r*).

Although the statement of cogency is a bit of a mouthful, an example should make the idea intuitive. The question {a, b, c, d} has four complete answers: a, b, c, and d. A union of its complete answers will either be a single complete answer (say a) or some disjunction of complete answers (say a ∨ b). So for S’s guess to {a, b, c, d} to satisfy condition (1), it better be either a single complete answer or some disjunction of complete answers. To satisfy condition (2), S’s guess needs to have the following property: for each of the complete answers included in S’s guess (i.e., for each of the guess’s disjuncts), S must have higher rational credence in that complete answer than in any of the complete answers excluded from the guess. So, for example: if S’s guess to the question {a, b, c, d} is a ∨ b, then S’s guess satisfies (2) iff S’s rational credences are such that: (i) C_S(a) > C_S(c) and C_S(a) > C_S(d), and (ii) C_S(b) > C_S(c) and C_S(b) > C_S(d). Abstracting from the example, the idea is that a guess is cogent with respect to a question just in case in the process of building the guess, one doesn’t “skip” any answers that are equal or better than the ones that have been included.

With the notion of a cogent guess in hand, here is the paper’s official theory of rational thinking:

Cogent Guess

S is rationally permitted to think_Q that p just in case: (i) the conjunction of E_S and S’s guess to Q^? entails p, and (ii) S’s guess to Q^? is cogent.

That is to say: S is rationally permitted to think that p relative to Q^? just in case p is entailed by the strongest thing S thinks relative to Q^?, and the strongest thing S thinks relative to Q^? is cogent.

cogent guess correctly predicts that ‘thinks’-reports like (16) should be infelicitous. Given that S’s rational credences in the answers to the question ‘Who will win the race?’ are [a: .35, b: .30, c: .20, d: .15], only the following guesses are cogent for S: a, a ∨ b, a ∨ b ∨ c, and a ∨ b ∨ c ∨ d. This means that relative to the question ‘Who will win the race?’, S is rationally permitted to be in all and only the following states: thinking that A will win, thinking merely that A or B will win, thinking merely that A, B, or C will win, or thinking merely that some horse or other will win. Thus, the only way for S to rationally think_{a,b,c,d} that A or C will win is if she rationally thinks_{a,b,c,d} that A will. So there is no way S can speak truly in uttering (16) while being rational.

With respect to the earlier principles of interest, cogent guess makes the same predictions as best guess. One’s best guess to Q^? will be cogent no matter one’s absolute rational credence in it, so cogent guess predicts extreme weakness. Likewise, the relationship between the cogency of one’s possible guesses to Q^? and one’s rational credences is non-monotonic, so the view predicts non-monotonicity. Since cogent guess is a Hintikkan analysis of ‘is rationally permitted to think’, it gets closure for free. And since it evokes the same kind of question-sensitivity as best guess, it gets the same debunking explanation of the putative counterexamples to closure. Finally, since cogent guesses never entail contradictions, cogent guess is guaranteed to preserve consistency.

We thus have our answer to:

The normative question: Under what conditions is it rationally permissible to think that p?

And since for many agents S and questions Q^? there are a range of possible cogent guesses to Q^?, it is predicted that the norms of rationality are permissive with respect to thinking:²⁸

Permissivism

It is not the case that: for any body of evidence E and proposition p, there is a unique doxastic attitude toward p that is consistent with being perfectly epistemically rational and having E as one’s evidence.

Given the optionality in how one chooses to answer questions about what one thinks, this is exactly what we should expect to be true. Rational thinking is about thinking cogently. Thinking whatever is most likely to be true is one way to think cogently, but it is not the only way.

11. Thinking simpliciter

At last we turn to:

The descriptive question: What is it to think that p?

And once again I’ll start with a schematic answer:

Thinking schema

S thinks_Q that p just in case: the conjunction of all the propositions S is sure of and S’s guess to Q^? entails p.

Having earlier defined S’s guess to Q^? as the strongest proposition S thinks relative to Q^?, thinking schema is, of course, a circular analysis of thinking. This is not to say it is uninformative. It entails that thinking is question-sensitive. It also entails that holding context fixed, the inference from ‘S is sure that p’ to ‘S thinks that p’ is valid. And it also entails that thinking has some nice closure properties.²⁹ But ideally we would have a sense for what makes it the case that S’s guess is one thing rather than another. This is what I take to be the heart of the descriptive question.

I’ll start by saying something negative: whatever it is that determines whether one’s guess to Q^? is p rather than p*, it isn’t one’s credences. To see why, suppose that S₁’s actual and rational credences in the answer to the question ‘Who will win the race?’ are [a: .35, b: .30, c: .20, d: .15]. Suppose also that S₂ is a credal duplicate of S₁ and that all this is common knowledge between them. We know from the previous section that a person whose rational and actual credences are [a: .35, b: .30, c: .20, d: .15] does just fine in answering ‘Who do you think will win the race?’ in any of the following ways:

• (17) ✓ A.
• ✓ One of A or B.
• ✓ One of A, B, or C.
• ✓ I have no idea.

What I want to suggest is that because someone with these credences could answer in any of these ways, it seems as if nothing guarantees S₁ and S₂ will answer in the same way. S₁ might answer with (17a), S₂ with (17d). And since S₁ and S₂ can (and in fact might) answer in different ways despite being credal duplicates, the natural hypothesis is that they can think different things despite being credal duplicates.

This point can be sharpened by imagining things from S₁’s perspective. You know what your credence function says about the various possible outcomes of the horse race, and you know that S₂ has the same credence function as you. You also know how you will answer the question ‘Who do you think will win the race?’. But do you know how S₂ will answer it? I don’t see how you could. For all you know she could answer with any of (17a)–(17d).

More generally, knowing a person’s credences just doesn’t seem to suffice for knowing how they’ll answer questions like ‘Who do you think will get the Democratic nomination?’ or ‘Where do you think Bond is hiding?’. They might answer in accordance with their best guess. But they might also answer agnostically. And until they answer, you simply won’t know. So knowing a person’s credences does not suffice for knowing what they think.

But if not our credences, then what does determine the facts about what we think? Is it our behavioral dispositions? That seems unlikely. Regardless of the fact that S₁ and S₂ might think different things about who will win the race, neither would use the proposition that A will win the race as a premise in theoretical or practical reasoning. Nor would either be willing to assert that A will win the race. Nor would either take a bet on the outcome of the race that the other wouldn’t. Indeed, it seems that the only relevant behavioral difference that could arise between S₁ and S₂ concerns the first-personal ‘thinks’-reports they’d be willing to make about themselves. But clearly that can’t be what grounds the differences in what they think. They make the ‘thinks’-reports they do because of what they think, not the other way around.

I thus want to suggest a rather different account of the metaphysics of thinking. In particular, I want to suggest that the facts about what we think are determined by our choices—that is, by certain kinds of pure acts of the will. I think it is because S₁ chose as her answer A will win and S₂ chose as her answer One of the entrants will win that S₁ thinks_{a,b,c,d} that A will win while S₂ merely thinks_{a,b,c,d} that some horse or other will.

I doubt an analysis can be given of the notion of ‘choice’ involved here, but the basic idea can be grasped through some paradigm cases. The kind of choice involved in deciding what to think about a question seems to me akin to the kind of choice involved in picking whether to go left or right at a fork, or in figuring out how to compose the next sentence of an email, or—perhaps most relevantly to the present discussion—in choosing between heads or tails when asked to guess how a coin will land.³⁰ Your credences don’t determine whether you guess heads or tails. Nor do your broader behavioral dispositions. What determines your guess is how you make up your mind. I claim something similar goes for thinking.

Although I won’t say more to substantively characterize the kind of choices involved in thinking, I will assume they abide by two structural constraints. First, they are question-directed: you are choosing an answer to a question. And second, when presented with a question Q^?, what one chooses is the union of a subset of Q^?’s answers: you are choosing between collections of the question’s maximally specific answers.

The hypothesis, then, is that for p to be one’s guess to Q^?—for p to be the strongest thing one thinks relative to Q^?—is for p to be one’s choice of answer to Q^?. This gives the answer to the descriptive question:

Doxastic Choice

S thinks_Q that p just in case: the conjunction of all the propositions S is sure of and S’s choice of answer to Q^? entails p.

And combining doxastic choice with cogent guess, the following tidy picture of thinking and its norms emerges: to think that p is for p to be one’s choice of answer to the question at hand; to rationally think that p is for one’s choice to be cogent.

Two clarificatory points before wrapping up. First, just because thinking involves choosing does not mean that one always consciously comes to think what one does. Choosing an answer to a question can be as automatic and subconscious as choosing which parts of one’s environment to attend to, or how to execute the various steps in a complex physical or cognitive task—say, hitting a serve, playing an instrument, choosing one’s gestures or words in speech, and so on. A choice that is subconscious can be a choice all the same.

Second, one might wonder whether a theory like doxastic choice implies that, at the level of metaphysics, what one thinks about a question is a purely voluntary matter. That is: for any question Q^?, is it in my powers to think_Q any of Q^?’s possible answers? Not necessarily. Or at least doxastic choice does not by itself imply this. For one, doxastic choice says that we can’t help but think any proposition we’re sure of. In normal cases that will cover much of what we think on the basis of perception, memory, and testimony. Further, it is an open empirical question the extent to which people are capable of choosing to think non-cogently.³¹ Speaking for myself, I know that as much as I might want it to be true that I think that a particular divine being exists (say for Pascalian reasons), I can’t bring myself to do so (at least not on the basis of a wager). But that said, the possibility of wishful thinking suggests that some agents are capable of making these kinds of doxastic choices. And so it seems an open question whether non-cogent thinking is metaphysically impossible, psychologically impossible, or merely just psychologically very difficult. I see it as a virtue of a theory like doxastic choice that it does not prejudge these issues.

12. Thinking and believing

Let’s take stock. Thinking is a question-directed attitude. There are two ways one can come to think a proposition p relative to a question Q^?. One can either be sure that p, or one can choose as one’s answer to Q^? a proposition that, together with the rest of what one is sure of, entails p.

Since thinking is a choice, it follows that a strong form of doxastic voluntarism is true.³² It also follows that two agents with identical credence functions can think different things relative to the same question. This means that (rational) thinking does not supervene on (rational) credence. It also explains why two rational agents with the same evidence can come to different conclusions about whether p without either of them having made a mistake.

Of course, if thinking is believing—if ‘S thinks that p’ is true in context iff ‘S believes that p’ is—then everything we’ve said about thinking is true of belief as well. And this would mean that belief cannot play many of the theoretical roles with which philosophers have associated it. If I know that A has only a 10% chance of winning the upcoming race, then it is not rational for me to take for granted that A will win the race or to assert that A will win the race; nor is it rational for me to take an even-money bet that A will win the race (let alone a 3:1 or 4:1 bet). But it can be rational for me to believe that A will win the race. So belief is not the attitude we hold toward the propositions we rely on in theoretical or practical reasoning; nor is it the attitude we hold toward the propositions we are willing to assert; nor is it even the attitude we hold toward propositions that we find highly likely to be true.

The lesson to take from this is that if thinking is believing, then just about every existing theory of belief is false. Or, more conservatively, if existing theories of belief are theories of the attitude that is the denotation of the ordinary expression ‘believe’, then just about every existing theory of belief is false.

Many theorists of “belief” might happily accept this conditional while denying its antecedent. Perhaps all along they’ve taken themselves to be giving theories of the distinct attitude of acceptance, where, as a matter of stipulation, to accept a proposition p is to do some number of the following: treat p as a premise in deliberation; be willing to assert p; be the internal duplicate of one who knows p; be disposed to feel surprise upon learning that ¬p; and so on. It is worth noting that many of the questions this paper has been focused on retain much of their theoretical interest when restated in terms of this technical notion. So it’s not clear how much is lost in conceding that one’s theory of acceptance is a theory of something other than belief.³³ And for those who think that having a home in natural language is a precondition on an attitude’s being a worthwhile object of epistemological study, note that all the claims just made about acceptance seem to lose neither plausibility nor theoretical interest when stated in terms of being sure. Perhaps that’s the attitude epistemologists have been theorizing about all along.³⁴

Defenders of traditional conceptions of belief who are uncomfortable with the idea of restating their theories in terms of technical terminology or the folk notion of being sure are in a more difficult situation. If thinking is a kind of choosing but believing isn’t, then thinking can’t be believing. But the evidence that thinking is believing is very strong. Again, if there are coherent interpretations of sentences like

• (1a) ✗ I think it’s raining, but I wouldn’t say I believe it is.
• (1b) ✗ I’m not sure whether Jane thinks Federer will win Wimbledon, but I know she doesn’t believe he will.
• (1c) ✗ My friends think I’m a good person, but my mom believes I am.

they appear to be elusive. And readers can see for themselves whether their judgments about the paper’s example sentences are affected in any substantial way when ‘believes’ is substituted for ‘thinks’. I myself detect no meaningful difference.

This raises the question: If thinking isn’t believing—indeed, if thinking really isn’t anything like believing—then why does natural language seem to treat the two as if they were the same?

Whether those who defend traditional conceptions of belief can give a satisfying answer to this question is not something I will try to resolve here. But the conditional point remains: if thinking is believing, then believing isn’t about being sure, or even about being sufficiently sure. It’s about choosing. And your evidence only settles what you ought to choose when it is decisive on the matter. Otherwise the choice is yours. So long as you choose cogently, you can take comfort knowing that your belief will be rational.