Outright Belief

Ralph Wedgwood

1. Outright belief and levels of confidence

Sometimes, we think of belief as a phenomenon that comes in degrees – that is, as consisting in the many different levels of confidence that a thinker might have in various propositions. Sometimes, we think of belief as a simple two-place relation that holds between a thinker and a proposition – that is, as what I shall here call outright belief.

Both ways of thinking of belief seem to be in evidence in everyday folk-psychological discourse. It is perfectly natural to say, for example, you are considerably more confident that Dublin is the capital of Ireland than that Dushanbe is the capital of Tajikistan. But it is also natural to say that you simply believe that Dublin is the capital of Ireland, without giving any further qualifications about how confidently or strongly you believe it. In general, we move back and forth easily between talking about what one simply believes, and talking about how confident one is of various propositions.

This raises the question: How are outright beliefs and levels of confidence related to each other? Philosophers have defended many different answers to this question. A few philosophers – such as Richard Jeffrey (1970) and Patrick Maher (1993, 152–5) – have taken an eliminativist stance towards outright beliefs, claiming that strictly speaking, outright beliefs do not really exist. A larger number of philosophers have attempted some kind of reductive program. Thus, many philosophers – including Scott Sturgeon (2008) and Richard Foley (2009) among others – maintain that outright beliefs are reducible to levels of confidence, while other philosophers – such as Mark Kaplan (1999) and Brian Weatherson (2005) – claim that outright beliefs are reducible to facts about the relevant agent’s preferences or utilities together with facts about the agent’s levels of confidence. Others – most notably, Gilbert Harman (1986) – maintain that levels of confidence are reducible to facts about outright beliefs. Finally, a few recent philosophers – especially, Jacob Ross and Mark Schroeder (unpublished) and Jonathan Weisberg (unpublished) – maintain that neither of these two kinds of belief is reducible to the other.

In this essay, I shall propose yet another answer to this question. This answer has two main components. The first component is the thesis that there are in fact two different kinds of levels of confidence, which I shall call “practical credences” and “theoretical credences” respectively. The second component is the thesis that outright beliefs are reducible to facts about our practical credences – but not to facts about our theoretical credences.

2.     Outright belief and levels of confidence: Some distinguishing marks

To begin our investigation of outright beliefs and levels of confidence, I shall enumerate some of the distinguishing marks of these two types of belief. I shall start by listing some of the distinguishing marks of outright belief.

First, it seems that when we say simply that a thinker “believes” a proposition p, without any explicit qualifications, we are normally saying that the thinker has an outright belief in p. For example, if I say simply that Peter Strawson “believed” that David Hume was born in 1711, I would normally be saying that Strawson had an outright belief in the proposition that Hume was born in 1711.

Secondly, we typically assume that a sincere assertion of a proposition p expresses an outright belief in p. If you meet someone who asserts that the capital of Uzbekistan is Samarkand, you will typically assume that if the speaker is sincere, then the speaker has an outright belief that the capital of Uzbekistan is Samarkand.^[1]

Finally, there seems to be a fundamental connection between outright belief and knowledge. Whenever a thinker knows a proposition p, the thinker must have an outright belief in p. If you know that the capital of Uzbekistan is Tashkent and not Samarkand, then you must have an outright belief that the capital of Uzbekistan is Tashkent and not Samarkand.^[2]

I shall now list some of the distinguishing marks of levels of confidence. Your levels of confidence create an (at least partial) ranking of the propositions that you have beliefs towards. For example, you might have more confidence that 1 + 1 = 2 than that Dushanbe is the capital of Tajikistan; so your levels of confidence rank the proposition that 1 + 1 = 2 above the proposition that Dushanbe is the capital of Tajikistan.

We can illustrate this point by imagining an (at least partially) ordered collection of boxes: your levels of confidence in effect sort these propositions into these boxes. For example, the proposition that 1 + 1 = 2 will presumably be sorted into the box at the very top of this ordered collection; the proposition that Dushanbe is the capital of Tajikistan is sorted into a box that comes somewhat lower in this ordering; the proposition that Dushanbe is the capital of France is sorted into a box that comes much lower down in this ordering; and the proposition that 1 + 1 = 5 comes in the box at the very bottom of the whole ordered collection, along with other obviously false and absurd propositions.^[3]

In fact, it seems that our levels of confidence may do more than simply rank propositions into an ordering of this sort. We can also make sense, not just of the thought that you are more confident of p than of q, but also of the thought that you are much more confident of p₁ than of q₁, or that you are only slightly more confident of p₂ than of q₂. In effect, we can at least sometimes compare differences in levels of confidence. For example, we can ask: Is the degree to which you are more confident of p₁ than of q₁ greater or smaller than the degree to which you are more confident of p₂ than of q₂?

This makes it plausible that it may be possible to measure a thinker’s levels of confidence, by means of some kind of credence function, which assigns real numbers to the various propositions that the thinker has beliefs towards.^[4] However, even if the thinker’s levels of confidence can be represented by means of such a credence function, we should not assume that this credence function will be a probability function. There are at least two ways in which the thinker’s credence function may fall short of being a full-blown probability function.

First, especially if the agent is less than perfectly rational, the thinker’s credence function may not be probabilistically coherent. For example, there may be cases in which the thinker assigns different levels of confidence to two propositions p and q, even though in fact p and q are logically equivalent. Indeed, later in this essay, I shall assume for the sake of argument that a perfectly rational thinker’s levels of confidence would be probabilistically coherent in this way; but I shall not be able to defend that assumption here. At all events, it is clear that most actual thinkers are not perfectly rational; and so we should not assume that the credence functions of actual thinkers must be probabilistically coherent.

Secondly, a probability function is defined over an infinite propositional algebra. (For example, for every pair of propositions in this algebra, the algebra must also contain their conjunction and their disjunction, and so on.) We should also not assume that any real believer has a level of confidence in every proposition in a complete propositional algebra of this sort. There may be many “gaps” – many propositions towards which the believer is totally “attitudeless”, and so has no level of confidence at all.

Many philosophers hold that human believers like you and me will inevitably have a set of levels of confidence that is “gappy” in this way. Indeed, given that the core realization of our minds is found in the physical state of our brains, and given the limited information density of matter, it may be that it would actually be impossible for us to have a belief of some kind in every proposition in an infinite propositional algebra. I shall not attempt to address this issue here. However, even if it is possible for some human believers to have attitudes towards infinitely many propositions in this way, this essay will focus exclusively on finite believers – that is, on thinkers who only have attitudes towards finitely many propositions.

3.     The functional role of these kinds of belief

To give an adequate account of the relationship between outright beliefs and levels of confidence, we need to get beyond these distinguishing marks, and to investigate the essential features of these two kinds of belief. I shall suppose here that one essential feature of each type of mental state is its functional role – the role that the mental state is disposed to play in our thinking and reasoning.^[5]

There is fairly widespread agreement among philosophers about the essential functional role of outright belief. If you have an outright belief in p, you will simply take p for granted, treating p as a starting point for further reasoning (including both practical and theoretical reasoning). You have a picture of the world that you rely on in all such further reasoning, and this proposition p is part of that picture of the world.

So far, this account of the functional role of outright belief is largely metaphorical. A complete account might take a long time to state in less metaphorical terms. But it seems plausible that this functional role involves at least the following two elements – one of them concerning the role of outright belief in one’s theoretical reasoning (that is, in the process of forming and revising one’s beliefs and the like) and the other concerning the role of outright beliefs in one’s practical reasoning (that is, in the process of forming and revising one’s intentions about how to act).

On the theoretical side, the essential functional role of outright beliefs involves a disposition towards deductive cogency. Deductive cogency itself has two components. First, it involves a kind of multi-premise closure: if you deduce a conclusion from a certain set of premises, and (even after carrying out the deduction) you have an outright belief in each of those premises, then to be deductively cogent, you must have an outright belief in the conclusion as well. (The distinctive feature of this kind of multi-premise closure is that it allows for two different ways of responding to one’s having carried out such a deduction – either by having an outright belief in the conclusion or by not having an outright belief in each of the premises. It is only if one has an outright belief in each of the premises even after carrying out the deduction that this kind of closure requires an outright belief in the conclusion.)

Secondly, deductive cogency involves a kind of overall consistency. Specifically, for you to be deductively cogent, the propositions in which you have outright beliefs must form a logically consistent set. It is part of the functional role of outright beliefs that they are disposed to conform to the requirements of this sort of deductive cogency.^[6]

On the practical side, having an outright belief in p involves what John Hawthorne (2003, 29) might call “using p as a premise in one’s practical reasoning”. As Robert Stalnaker (1984, 15) puts it, “to believe [p] is to be disposed to acts in ways that would tend to satisfy one’s desires, whatever they are, in a world in which p (together with one’s other beliefs) were true.” I shall assume that this means something like the following: if the set of propositions in which you have outright beliefs is P, and together with your values, P entails that the optimal options for you to choose constitute a certain subset {A_k, … _m} of the available options {A₁, … A_n}, then you must be disposed to choose an option that belongs to that subset {A_k, … A_m}. So, in particular, if you have an outright belief that a particular option A is the best available, then you must be disposed to choose A.

What is the functional role of levels of confidence? This is a more controversial question. For the purposes of this essay, a particular answer to this question will simply be assumed, at least for the sake of argument. Broadly speaking, we shall assume a version of the orthodox Bayesian view of the essential functional role of levels of confidence – with the crucial amendment that it is not assumed here that the believer must have a level of confidence in every proposition in a whole propositional algebra.

As I shall interpret it, this Bayesian view is based on an account of what it is for a believer’s levels of confidence to be rational. The central idea is that the essential functional role of levels of confidence is that they are disposed to approximate to conforming to these requirements of Bayesian rationality. This Bayesian account of rationality has two components: one component is an account of theoretical rationality; and the other component is an account of practical rationality. (It will turn out to be important for our purposes to keep these two components clearly separated from each other.)

According to the Bayesian account of theoretical rationality, rational levels of confidence can be represented by a unique real-valued credence function, and this function must meet the following two requirements: first, must be probabilistically coherent; and secondly, it must evolve in response to experience by means of some kind of conditionalization.^[7] (According to this account of theoretical rationality, this credence function may be partial or “gappy” – that is, it may not assign a credence to every proposition in the whole propositional algebra. However, it is assumed that a perfectly rational thinker’s levels of confidence can be represented by a unique credence function: that is, for every proposition p that the thinker has any attitude towards, all credence functions that can represent the thinker’s levels of confidence must assign the very same credence to p.)^[8]

According to the Bayesian account of practical rationality, rationality requires every agent’s levels of confidence to constrain her choices and intentions in a certain distinctive way. Let us say that whenever one makes a certain choice, or has a certain intention, the object of one’s choice or intention is an “option”. Then according to this conception of practical rationality, there is some sort of “value” that options might exemplify such that the rational agent’s choices and intentions will always be for options that maximize the expectation of this sort of value – where the relevant “expectation” is defined in terms of these levels of confidence. There are several tricky technical questions about how best to understand this sort of “expectation”, but the key idea is this: there is an appropriate “partition” of “states”, such that we can make sense of the precise degree to which each of the relevant options exemplifies the relevant value in each of these states; then, if the agent’s levels of confidence can be interpreted as assigning a probability to each of these states, an option’s “expected value” can be identified with the probability-weighted sum of the option’s value in each of these states, weighting the value that the option has in each of these states by the relevant probability of the state.^[9]

This account of the functional roles of these types of belief seems to create insuperable difficulties for the view that levels of confidence are reducible to outright beliefs. How could we reconcile the view that levels of confidence are reducible to outright beliefs with the idea that it is part of the essential functional role of levels of confidence to approximate to these broadly Bayesian requirements of rationality? It seems that the only way to achieve this reconciliation would be by identifying the state of having level of confidence n in a proposition p with an outright belief in the proposition that p has a probability of n.

However, this identification seems to imply that few if any human thinkers have any levels of confidence at all. The very idea of precisely measuring the probability of propositions did not emerge until the 17^th century.^[10] Brilliant thinkers such as Plato and Aristotle and the ancient Greek mathematicians never had any beliefs whatsoever in any proposition of the form ‘Proposition p has a probability of n’. So, on this view, these thinkers never had any levels of confidence at all. Since this seems to be an incredible result, we should conclude that if this account of the functional role of these types of belief is correct, then levels of confidence simply cannot be reduced to outright beliefs. Levels of confidence are either more fundamental than outright beliefs, or the two kinds of belief are equally fundamental and irreducible. At all events, from now on I shall simply set aside all attempts to reduce levels of confidence to outright beliefs; I shall turn to evaluating other theories of these types of belief instead.

4.     A dilemma: Trivialization or incoherence

There is in fact a glaring problem with the account of the functional role of outright belief that I gave in the previous section. According to this account, the functional role of outright belief in p seems to require having the maximum possible level of confidence in p. If your level of confidence in p were even slightly less than the maximum possible, the kind of belief that you have in p would not be generally disposed to conform to the requirements of deductive cogency, and you would also not be generally disposed to use p as a premise in your practical reasoning.

According to my account, deductive cogency implies believing the conjunction of every pair of propositions that one believes – at least whenever one deduces the conjunction from the propositions in question, and one still believes those propositions even after performing the deduction. If the functional role of outright belief in p does not require having the maximum level of confidence in p, there will presumably be perfectly normal cases in which one has outright beliefs in a very large set of propositions p₁, … p_n such that one’s level of confidence in each proposition p_i in this set is less than the maximum possible. But there are surely also normal cases in which the conjunction of these propositions ‘p₁ &…&  p_n’ is a proposition in which one definitely does not have an outright belief (even if one actually deduces this conjunction from the premises p₁, … p_n, and one’s attitude towards those premises does not change as a result of one’s performing this deduction). So it seems that if one’s level of confidence in these propositions p₁, … p_n is even slightly less than the maximum possible, the kind of belief that you have in these propositions will not generally be disposed to conform to the requirements of deductive cogency.

Similarly, according to my account, the practical side of the functional role of outright beliefs involves basing one’s choices simply on one’s outright beliefs, and not on any levels of confidence that one has in any propositions that are incompatible with what one believes. Yet if one’s level of confidence in p is ever so slightly less than the maximum possible, there will be perfectly normal situations in which one would base one’s choices on one’s level of confidence in propositions that are incompatible with p – such as a situation in which the available courses of action promise virtually no advantages or disadvantages if p is true, but differ immensely in their costs or benefits if p is false. So again, if one’s level of confidence in p is even slightly less than the maximum possible, the kind of belief that one has in p will not be disposed to play the functional role that my account assigns to outright belief.

This is not because there was anything idiosyncratic about my account. On the contrary, the phrases which philosophers typically use to describe the state of having an outright belief in a proposition p – phrases such ‘taking p for granted’ or ‘treating p as though it were true’ – all seem to imply that having an outright belief in p involves treating p as though it were (for all intents and purposes) certain – that is, treating p as though one had credence 1 in it.^[11]

Yet we all seem to have outright beliefs in propositions of which we are not maximally confident. For example, you can surely have an outright belief both in ‘Dushanbe is the capital of Tajikistan’ and in ‘1 + 1 = 2’, even if you have less confidence in the former than in the latter. But if you have less confidence in ‘Dushanbe is the capital of Tajikistan’ than in ‘1 + 1 = 2’, you clearly cannot have maximum confidence in the former.

Indeed, virtually all philosophers who have explored the relationship between levels of confidence and outright belief assume that it must be possible to have outright beliefs in propositions of which one is not maximally confident. They have good reason for assuming this. In everyday speech, ‘I believe that p’ seems to be a distinctly weaker claim that ‘I am certain that p’. Moreover, it seems that unless it is possible to have an outright belief in propositions of which one is not maximally confident, the notion of an outright belief will effectively be trivialized: it would simply be the notion of the highest possible level of confidence, not a notion that plays any distinctive and interesting role of its own in our thinking about agents.

However, this creates a dilemma for all existing theories of outright belief. On the one hand, theorists who give reductive accounts of outright belief are forced to deny that the functional role of outright belief really is what it seems to be. On the other hand, those who recognize both outright belief and levels of confidence as equally fundamental and irreducible types of belief seem committed to the view that non-trivial outright beliefs must involve a form of incoherence – in effect, the incoherence of simultaneously treating the relevant propositions as though they both were and were not certain.

The best-known attempts to reduce outright beliefs to levels of confidence all seem to be impaled on the first horn of the dilemma. This is particularly clear with versions of the threshold view of philosophers such as David Christensen (2004) and Scott Sturgeon (2008) – that is, the view that to have an outright belief in a proposition p is to have a level of confidence in p that exceeds a certain threshold r. These views more or less explicitly imply that the functional role of outright beliefs is not, strictly speaking, the functional role of credence 1.

Brian Weatherson (2005) takes a slightly different approach, attempting to reduce outright beliefs to facts about the agent’s preferences as well as her levels of confidence. Weatherson’s account comes closer to implying that outright belief has the functional role that it seems to have: his account implies that an agent who has an outright belief in p has the same preferences over options as she would if she had credence 1 in p; and at least if certain (in my view, rather questionable) assumptions about the agent’s preferences are true, the agent’s outright beliefs will also conform to a certain closure principle.

It does not follow, however, that Weatherson’s account vindicates the thesis that outright belief really has this functional role. The functional role of a type of belief is a matter of the dispositions that go along with this type of belief, and the notion of a disposition is a causal or explanatory notion. According to Weatherson’s account, the explanation of why an agent has the preferences that she has does not appeal to the agent’s disposition to use the propositions that she has outright beliefs in as premises in her practical reasoning. Instead, this explanation appeals to the functional role of preferences and levels of confidence, and to the fact that the agent has levels of confidence both in the propositions that she has outright beliefs in and in other incompatible propositions as well. Similarly, the explanation of why the agent’s outright beliefs conform to the closure principle does not appeal simply to the disposition to conform to the requirements of deductive cogency, but rather to certain assumptions about the agent’s preferences, as well as to the functional roles of preferences and levels of confidence.^[12] So Weatherson’s approach also fails to vindicate the thesis that the functional role of outright belief is what it seems to be.

On the other hand, the theorists who accept that outright belief has the functional role of the maximal level of confidence are impaled on the second horn of this dilemma – they seem to be imputing a kind of incoherence to all thinkers who have outright beliefs in any propositions of which they are not maximally confident. Indeed, the threat of incoherence is obvious: by having an outright belief in a proposition p of which one is not maximally confident, one is in a way simultaneously treating p as though it were certain and also treating p as though it were not certain.

It is this apparent incoherence that lies behind the notorious lottery paradox of Henry Kyburg (1961). Suppose that one carries out a multi-premise deduction. Then treating the premises as though they were certain would commit one to treating the conclusion as though it were also certain – while treating the premises as though they were not certain would in many cases require having less confidence in the conclusion than in any of the premises. It might be claimed, however, that a rational agent can always avoid this incoherence – if the agent responds to having carried out this deduction by ceasing to have an outright belief in at least one of the premises, instead of by having an outright belief in the conclusion.

However, the incoherence that concerns us will arise even in cases of single-premise deduction. According to my account of the functional role of outright belief, if you deduce q₁ from p, and even after performing this deduction, you still have an outright belief in p, you will be disposed to have an outright belief in q₁ as well. But “deducing” q₁ from p is effectively inferring q₁ from p in a way that is sensitive to the fact that the conditional probability of q₁ given p is 1. We should also consider how you will respond to inferring a propositionq₂ from p if you have an outright belief in p, while the conditional probability of q₂ given p is slightly less than 1.

For example, suppose that you have an outright belief in p, and you infer q₂ from p, while being sensitive to the fact that the conditional probability of q₂ given p is 0.99. Moreover, suppose that even though you have an outright belief in p, you do not have a maximal level of confidence in p – instead, your level of confidence in p is just 0.9. Still, we are assuming that having an outright belief in p involves treating p as though it were (at least for all intents and purposes) certain. So, how can it be coherent for you to continue to have a confidence level of just 0.891 in q₂ when you are effectively treating p as though it were certain, and you are sensitive to the fact that the conditional probability of q₂ given p is 0.99? Surely you should treat q₂ as though your credence in q₂ were (for all intents and purposes) 0.99: after all, you treat q₁ as (for all intents and purposes) certain – just because the conditional probability of q₁ given p is 1. It is surely not coherent to treat q₁ and q₂ so differently in this way. So, on this view, having an outright belief in a proposition in which one has a non-maximal level of confidence seems to involve an unavoidable incoherence.

This clearly seems to be a serious problem with this view. Nonetheless, we have seen problems with all of the alternative approaches as well. Clearly, a new approach is called for. Somehow this new approach must allow that it to be perfectly coherent to have an outright belief in a proposition p, even if one is not maximally confident of p, and even if, in some crucial way, having an outright belief in p really does involve treating p as if it were (for all intents and purposes) certain.

5.      The solution to the dilemma

To solve this problem, I propose, we should recognize that the agent’s system of beliefs is best represented by two systems of credences – call them the agent’s theoretical credences and the agent’s practical credences. Both of these two kinds of credences come in degrees. The core distinction is this: theoretical credences represent the way in which the agent registers, or keeps track of, the strength of the justification that she has in favour of the relevant propositions, while practical credences are the credences on the basis of which the agent maintains and revises her intentions about how to act.

How exactly could it be that “theoretical credences represent the way in which the agent registers, or keeps track of, the strength of the justification that she has in favour of the relevant propositions”? Presumably, if you rationally have a greater theoretical credence in p than in q, that is the way in which you register the fact that you have more justification for p than for q – or as many philosophers would put it, the fact that your evidence gives more support to p than to q.

More precisely, I suggest, your theoretical credences are disposed to approximate to the requirements of theoretical rationality listed in Section 3 above. These credences are not only disposed to be probabilistically coherent; they are also disposed to change in response to experience; and the changes dictated by experience are propagated throughout the whole set of credences by means by some kind of conditionalization. These are the only kinds of changes in one’s theoretical credences that are involved in these theoretical credences’ essential functional role.

On the other hand, your practical credences represent the beliefs or doxastic states on the basis of which you maintain, form, and revise your intentions for action. These practical credences are disposed to approximate to the requirements of practical rationality listed in Section 3 above. That is, these practical credences are disposed to constrain the agent’s intentions, in such a way that those intentions tend to maximize the appropriate kind of expected value, when the relevant kind of “expectation” is defined in terms of these practical credences.

You can have one set of practical credences for the purposes of making one decision, and a different set of practical credences for the purposes of making another decision. For the purposes of making most decisions, I treat it as though it were quite certain that Dublin is the capital of Ireland. However, if I am offered a bet that pays $1 if Dublin is the capital of Ireland and results in my being tortured horribly to death if Dublin is not the capital of Ireland, I will probably base my decision on a different set of practical credences, in which it is not treated as certain that Dublin is the capital of Ireland. (It even seems possible for an agent who is engaged in “multi-tasking” to make two decisions at the same time, basing one decision on one set of practical credences, while basing the other decision on a second set of practical credences.)

When one has a practical credence of 1 in p, one is in effect treating p as though it were practically certain; when one has a theoretical credence of 1 in p, one is in effect treating p as though it were theoretically certain. On the face of it, there need be no incoherence in simultaneously treating a proposition p as though it were practically certain and also treating p as though it were not theoretically certain. The two kinds of doxastic or credal attitudes towards p do not conflict with each other, because they play fundamentally different roles: the state of being practically certain of p plays the role of guiding the way in which one maintains and revises one’s intentions about how to act, while the state of being theoretically uncertain of p plays the role in registering the fact that one does not have the highest possible degree of justification for p. This is the first step that we need to make to resolve the dilemma that confronted us in the previous section.

This distinction between theoretical credences and practical credences can now be used to develop an account of the nature of outright belief. An outright belief in p is the state of being stably disposed to have a practical credence of 1 in p, for at least all normal practical purposes.

It might be suggested that to believe a proposition p is simply to treat p as though it were practically certain. But that suggestion would overlook the fact that believing a proposition is an essentially dispositional state, while “treating” a proposition p in a certain way is an occurrent process of thinking or reasoning with p. This is why I propose instead that having an outright belief involves having a disposition to treat p as though it were practically certain.

Return to the example that we have just considered – the case in which you are offered a bet that will pay $1 if Dublin is the capital of Ireland, and will result in your being tortured to death if Dublin is not the capital of Ireland. In this case, it would be natural for you to say, “Look: I’m not going to take this bet. Even though I definitely believe that Dublin is the capital of Ireland, I’m not so fantastically confident of that to make it rational for me to take this insane bet!”^[13]

It seems to me that this is the natural response for you to make in this case because for all normal practical purposes, your practical credence in the proposition that Dublin is the capital of Ireland is 1; this is why you count as believing p. But for the highly abnormal purpose of deciding whether to take this bet, your practical credence is less than 1.

At all events, this seems to me the best account of the state of outright belief: it can explain why it seems that outright belief in p has the functional role of credence 1 – because outright belief really does have the functional role of (being disposed to) have (practical) credence 1 in . At the same time, it also makes it possible for an agent to have an outright belief in propositions of which she is not (theoretically) certain, without thereby incurring any of the troubling forms of incoherence that we explored in the previous section.

6.     Why finite agents need both practical and theoretical credences

Even if it is possible for some believers to have both practical and theoretical credences, of the sort that I described in the previous section, why should we think that any actual believers have two systems of credences in this way? In this section, I shall argue that finite agents – that is, agents whose belief systems are, as I put it above, “gappy” – can be rational only if they have both practical and theoretical credences. First, I shall argue that to be rational, a finite agent must have practical credence of 1 in some propositions for which she does not have the highest possible degree of justification – that is, propositions for which she definitely has less justification than for elementary logical truths and the like. Then I shall argue that to be rational, a finite agent must also have a theoretical credence of less than 1 in such propositions. If these arguments are correct, then rationality requires all finite agents to have both practical and theoretical credences.

My argument for the first step hinges on the definition of the decision-theoretic notion of a choice’s “expected value”. Within many forms of decision theory – including both causal decision theory and the “benchmarks theory” that I have defended elsewhere (Wedgwood 2011) – the notion of a choice’s “expected value” is defined in terms of a partition of states of nature.

A “state of nature” is a state of affairs that has the following two features: first, it is utterly beyond the agent’s control which state of nature actually obtains; and secondly, each state of nature is sufficiently detailed to determine exactly how good each of the available acts would be if it were performed in that state of nature. To fix ideas, I shall follow David Lewis (1981) in conceiving of these states of nature as conjunctions of non-backtracking counterfactuals – where each of these counterfactuals has the form ‘If I did act n, outcome O_m would result’. On this interpretation, states of nature are effectively propositions, and so we can talk of conjunctions, disjunctions, and negations of states of nature, and so on.

Admittedly, there is one well-known approach to decision theory – the evidential decision theory of Richard Jeffrey (1981) – that does not need to appeal to a partition of states of nature, and can define the notion of a choice’s “expected value” in terms of any partition whatsoever. I shall assume here that evidential decision theory is false, and that the correct decision theory (whatever exactly it is) will appeal to a partition of states of nature in this way.^[14]

The crucial point here is that to say that a set of states of nature forms a “partition” is to say that the relevant agent is certain that one and only one of these states of nature obtains. In other words, the relevant agent must be certain of the disjunction of these states of nature, and she must also be certain of the negation of the conjunction of any pair of these states of nature.

Intuitively, it seems clear, however, that a disjunction of such states of nature will never be the kind of proposition for which an agent whose system of beliefs is “gappy” has as high a degree of justification as for the most elementary logical truths.

In particular, the disjunction of these states of nature will not itself be a logical truth – not even an instance of the law of excluded middle ‘p ˅ ¬p’. The negation of a state of nature is not itself a state of nature: the negation of a conjunction of non-backtracking counterfactuals is not itself a conjunction of non-backtracking counterfactuals (it is effectively a disjunction of negations of non-backtracking counterfactuals, which is quite another thing). So, even though the agent has the highest possible degree of justification for the proposition that either the state of nature or its negation obtains, this is not enough for having such a high degree of justification for a disjunction of states of nature.

For example, consider a simple case, where I am deciding whether to travel from Oxford to London by bus or by train. I may just consider two states of nature: one state of nature S₁, in which the traffic on the roads is light, and so taking the bus would result in my arriving in London after a journey of about an hour and a half, and a second state of nature S₂, in which the traffic on the roads is heavy, and so taking the bus would take at least two hours. But I certainly do not have as much justification for the disjunction of S₁ and S₂ as I have for the simplest logical truths: after all, my evidence should make it quite clear that there is a non-zero chance that the bus will crash on the motorway, with the result that I never reach London at all. Nonetheless, for the purposes of deciding whether or not to catch the bus or the train, I do not consider the state of nature in which the bus crashes on the motorway. Indeed, since my belief system is “gappy” in the way that I have described, I may have no doxastic attitude whatsoever towards the proposition that the bus in question will crash: I may be totally “attitudeless” towards that proposition; and even if I do consider the possibility that the bus will crash, there will inevitably be other possibilities that I do not consider – such as that the bus will be kidnapped by terrorists or the like.

Thus, if the agent is ever to be capable of guiding her choices and intentions by this sort of expectation, the relevant partition will have to consist in a set of states of nature such that the agent is practically certain of the disjunction of these states of nature, even though she does not have the highest possible degree of justification for this disjunction.

Arguably, a similar issue arises at another point in the decision-making process. It is arguably necessary for agents to treat it as practically certain that each of the options that they are choosing between is practically available – roughly in the sense that if they choose the option, they will execute the choice and act accordingly. But the availability of an option is surely another proposition of which a finite agent will never have the maximum degree of justification. (There is always a remote possibility of suddenly dying, or being totally paralysed, in the time lag between making and executing the choice.) So here too it seems that rational agents will need to be practically certain of some proposition for which they have less than the highest possible degree of justification.

As I shall now argue, however, although a rational finite agent needs to be practically certain of some propositions of this sort, she also needs not to be theoretically certain of them. That is, she needs to register, or keep track of, the fact that she does not have the highest degree of justification for these propositions.

Suppose that you start by being practically certain of a proposition p, but then the practical stakes change – for example, because you are offered a bet that pays $1 if p is true, and results in your dying a horrible death if p is false. (Suppose that the proposition p is chosen at random, so that being offered this bet does not itself change the degree of justification that you have for p.) Then if you are rational, you will not accept the bet, because in making this decision, you will shift to a new set of practical credences, which do not involve being practically certain of p. But it seems that some kind of beliefs must guide you in shifting from one set of practical credences to another in this way – where these guiding beliefs must have been present before this shift in your practical credences, and must include some kind of belief in the very proposition p that is at issue. It seems that these guiding beliefs cannot be your pre-existing practical credences – since those credences involve treating p as if it were certain, and so cannot allow you to envisage the risks that would be involved in believing p if p is false. Instead, these guiding beliefs must include some non-maximal level of confidence in p itself.

Intuitively, for this sort of shift to be rational, these guiding beliefs must be a set of credences that keep track of the degree of justification that the agent has for each of the propositions involved. In effect, they must be what I have called theoretical credences. This seems to show that to be rational, finite agents need theoretical credences as well as practical credences: for finite agents like us, both practical and theoretical credences are indispensable.

7.     What practical credences will a rational agent have?

According to the argument that I have just given, the agent’s theoretical credences – presumably including credences about the practical stakes in the practical situation at hand – can somehow guide a rational agent to shift from one set of practical credences to another. To be confident that this argument is correct, we should give a more detailed account of how exactly a rational agent will shift from one set of practical credences to another in this way.

One suggestion that might be considered is that the rational agent will have a set of practical credences that maximizes expected value (when the relevant “expectation” is defined in terms of the agent’s theoretical credences). But this suggestion has a fatal problem. Practical credences that maximize expected value in this way might include a practical credence of 1 in a proposition p such that the agent is theoretically almost certain that p is false.

If you treat a proposition p as though it were certain, for the purposes of making a given decision, even though in fact you are theoretically almost certain that p is false, then your attitude towards p is surely not a state of belief at all. It seems rather to be the state that Michael Bratman (1992) has called “acceptance”. For example, for the purposes of planning your household budget, you might “accept” that the repairs that you have to carry out to the roof of your house will cost at least $25,000 – even though you are theoretically almost certain that the roof repairs will cost less than that. Or to take another example, which is due to Keith Frankish (2009, 86), a lawyer might “accept that their client is innocent for the purposes of defending them” – even if the lawyer is in fact theoretically almost certain that their client is in fact guilty.

In general, it seems that no rational believer could ever have a practical credence of 1 in a proposition p in which her theoretical credence is less than 0.5. This is a point that our account of which practical credences the rational agent will have must somehow explain.

In general, I propose, any kind of belief in a proposition p – including both theoretical and practical credences in p – is rationally required to be guided purely by certain distinctively epistemic values. Very roughly, being guided by these distinctively epistemic values involves pursuing the goals of believing p if p is true and not believing p if p is false. Practical considerations (such as the costs and benefits at stake in the agent’s situation) only make a difference to the practical credence that the agent has in p by making a difference to the way in which these two goals – believing p if p is true and not believing p if p is false – are balanced against each other.

For example, if the goal of believing p when p is true is weighted relatively heavily, while the goal of not believing p if p is false is not weighted particularly heavily, then this weighting seems to favour having practical credences that are more adventurous and extreme (i.e. closer to 0 and 1) than one’s theoretical credences. On the other hand, if the goal of not believing p is p is false is weighted very heavily, and the goal of believing p if p is true is not weighted especially heavily, then this alternative weighting seems likely to favour practical credences that are more cautious and non-committal (i.e. further from 0 and 1) than one’s theoretical credences.^[15]

We can articulate this idea more precisely by introducing the idea of the “epistemic disvalue” of a credence. A credence’s “epistemic disvalue” is a measure of the extent to which it falls short of achieving these two goals – by involving either (i) insufficient credence in a true proposition or (ii) excessive credence in a false proposition. In general, if p is true, then the higher one’s credence in p is, the lower the epistemic disvalue of one’s practical credence in p; while if p is false, the higher one’s credence in p, the greater the epistemic disvalue of one’s practical credence in p. More specifically, let us assume that the epistemic disvalue of one’s credence in p is an increasing function of the distance between that credence and p’s actual truth-value.^[16]

This notion of the “epistemic disvalue” of an agent’s credence in a particular proposition p can be generalized to obtain a more general notion of the epistemic disvalue of a whole set of credences. So long as we are only comparing different possible sets of credences in the same set of propositions, we may simply take the epistemic disvalue of each possible set of credences to be the sum of the epistemic disvalues of every particular credence in that set.^[17]

With this notion of the “epistemic disvalue” of credences in hand, we can now make sense of a notion of “practically modulated epistemic disvalue” (or “practical-epistemic disvalue” for short). Practical-epistemic disvalue always agrees with epistemic disvalue in its ranking of credences on the basis of their distance from the truth. So, for example, if p is true, it will always be better, in terms of practical-epistemic value, to have a credence of 0.9 in p than to have a mere credence of 0.5 in p. However, how much better (in terms of practical-epistemic value) it is, when p is true, to have a credence of 0.9 in p than to have a credence of 0.5 in p may depend on what is at stake in the relevant practical situation. In effect, what is at stake in the practical situation may determine which increasing function of the distance between the credence and the actual truth-value is the appropriate measure of practical-epistemic disvalue in that situation.

So, for example, there may be some situations in which, if p is true, the degree to which having a practical credence of 0.9 in p is better (in terms of practical-epistemic value) than merely having a practical credence of 0.5 in p is unusually great – whereas if p is false, the degree to which having a practical credence of 0.9 in p is worse than having a credence of 0.5 in p is unusually small. Intuitively, these are the situations in which the appropriate measure of practical-epistemic value favours having more adventurous or extreme practical credences.

More precisely, we may now propose the following account of which practical credences a rational agent will have. According to this account, a rational agent’s practical credences will always be a probabilistically coherent set of credences that minimizes expected practical-epistemic disvalue, according to the measure of practical-epistemic value that is appropriate for particular choice situation at hand – where the relevant “expectation” is defined in terms of the agent’s theoretical credences.

This approach can explain why the rational believer will never have a practical credence of 1 in a proposition in which her theoretical credence is 0.5 or less, if it is true that in all practical situations, the appropriate measure of practical-epistemic disvalue is a convex increasing function of the distance between one’s credence and the actual truth-value.^[18] Then the agent will never minimize expected practical-epistemic disvalue by having a practical credence of 1 in a proposition in which her theoretical credence is less than or equal to 0.5.

One famous measure of a credence’s disvalue is the Brier score – that is, the square of the distance between one’s credence in the proposition and the proposition’s actual truth-value. The Brier score has several well-known properties, which guarantee that in any situation in which the Brier score is appropriate measure of practical-epistemic disvalue, the practical credences that minimize expected disvalue will be those that are identical to the agent’s theoretical credences.^[19]

However, there may be other practical situations in which the appropriate measure of practical-epistemic disvalue differs from the Brier score. For example, in some practical situations, the appropriate measure might be the cube of the distance between one’s credence in the proposition and the proposition’s actual truth-value. With this “cubic” measure, the practical credences that minimize expected practical-epistemic disvalue will be more cautious and non-committal (that is, more clustered around the intermediate credences like 0.5) than one’s theoretical credences. So in any practical situation in which the appropriate measure of practical-epistemic disvalue is this “cubic” measure, a rational agent’s practical credences will be more cautious and non-committal in this way.

In some other practical situations, the appropriate measure of practical-epistemic disvalue might be the distance between the credence and the actual truth-value raised to the power of some real number n such that 1 < n < 2 – for example, the appropriate measure might be the distance between the credence and the actual truth-value raised to the power of 1.1. With this sort of measure, the practical credences that minimize expected practical-epistemic disvalue will be more extreme and adventurous (that is, closer to the extremes of 0 and 1) than one’s theoretical credences.

According to the arguments of Section 6, most practical situations will in fact be situations of this latter kind, in which the appropriate measure of practical-epistemic disvalue is one of these more “adventurous” measures. It is not clear that there are any situations in which the appropriate measure of practical-epistemic value is one of the “cautious” or “non-committal” measures that I have just described. Still, perhaps we should allow for the possibility-in-principle of situations of this sort. At all events, I propose the following two theses: first, there is an appropriate measure of practical-epistemic value for each practical situation that an agent is in; and secondly, a rational agent’s practical credences will always be a probabilistically coherent set of credences that minimizes expected practical-epistemic disvalue, according to the measure of practical-epistemic value that is appropriate to the particular practical situation at hand.

8.     Conclusion: Pragmatism vs. intellectualism

In conclusion, I shall note that this approach allows for a kind of reconciliation of the broadly pragmatist and intellectualist approaches in epistemology.^[20]

First, according to this approach, a resolute form of intellectualism is true of the theoretical credences. Which theoretical credences a rational believer will have is sensitive only to the degree of justification that she has in favour of the relevant propositions – that is, as most philosophers would put it, to the degree to which these propositions are supported by the believer’s evidence.

Secondly, a kind of pragmatism is true of the practical credences: which practical credences the rational believer will have is sensitive to practical considerations – such as the needs, costs and benefits that are at stake in the believer’s particular practical situation – since as I have explained, these practical considerations may make a difference to the measure of practical-epistemic value that is appropriate in that situation.

Finally, a qualified pragmatism is true of outright beliefs: which propositions the rational believer has an outright belief in is determined by the needs, costs, and benefits that are at stake in normal situations –though not necessarily to the needs, costs, and benefits that are at stake in the particular situation at hand.

This account of belief is undeniably complex and elaborate. However, I hope that the arguments given here have made it plausible that belief is itself a complex phenomenon, so that no less elaborate account could do justice to this phenomenon itself.^[21]

Notes

References

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