Combining Probabilities with Full Beliefs

Abstract

One of the major problems in formal epistemology is the difficulty of combining probabilistic and dichotomous (all-or-nothing) beliefs in one and the same formal framework. Based on the properties of actual human belief systems, a set of ten desiderata is proposed for belief system models that contain both probabilistic and full beliefs. Previously proposed combined belief systems are shown to fall short of these desiderata. A proof of concept is provided to show that they can be satisfied in a formal model. In the proposed model, the amnesia problem of probabilistic updating is solved by introducing infinitesimal probabilities as carriers of memories of beliefs that have been given up. In the resulting system of probability revision, contingent propositions can be transferred in both directions between full beliefs and beliefs held to a lower degree, in a way that mirrors real-life acquisitions and losses of full beliefs. The subsystem consisting of full beliefs has a pattern of change that constitutes a credible system of dichotomous belief change.

As human reasoners, we operate with two major types of factual beliefs. We have full beliefs, which we provisionally treat as certain, and currently do not doubt. We also have uncertain beliefs, often called credences, which we hold to various degrees, short of certainty. Shifts in both directions between these two types of beliefs are common, and we often need to make such shifts in order to respond rationally to new evidence or new insights.

In formal epistemology, these two types of beliefs are usually represented in separate models. Uncertain beliefs are commonly represented by probabilities, and changes in uncertain beliefs by probabilistic updates. This use of probabilities is of course an idealization, since only few of our actual uncertain beliefs are held to degrees that can be expressed in exact numerical terms (Halpern, 2003, p. 24; Titelbaum, 2013, p. 27). A person’s full beliefs are usually represented by a set of propositions that is closed under deduction. Changes in full beliefs are represented by belief change operations, such as revision operators that assimilate a new belief and when necessary remove a selection of previous beliefs in order to avoid loss of consistency. These models are also idealized, not least since they are based on deductive inference, whereas real-life reasoning makes abundant use of non-deductive inference.

Neither of these two types of models can easily be extended to cover the class of beliefs that the other represents. Models of full belief, such as belief change models, lack resources for expressing beliefs coming in lower degrees. In probabilistic models, full beliefs can be represented as propositions with probability 1, but this construction has a serious problem: When a proposition is assigned probability 1, it passes a point of no return. No subsequent probabilistic update or series of such updates can ever bring it back to a lower probability. Conversely, once a proposition is assigned probability zero, it is lost in a “black hole” and can never be regained.

We will assume that formal models in epistemology stand in the same relation to their objects as formal models in other areas of science: A formal model is a simplified representation in formal language of its object. It mirrors some features of the object that have been selected for their importance or prominence, but there are usually other features that it does not reflect adequately (Hansson, 2000, 2018e; Yap, 2014).¹ The coexistence of credences and full beliefs is an important and prominent feature of actual belief systems, and so is the possibility of transfers in both directions between these two types of belief. Therefore, we need reasonably simple models that reflect these features of actual beliefs systems. One could expect this to be achievable in a combined model that contains both a probability function representing uncertain beliefs and a deductively closed set of propositions, representing full beliefs. However, the construction of such a combined model has turned out to be a difficult task, and previously proposed such constructions have been shown to have implausible properties.

In this contribution, we will propose a set of major desirable properties for models aimed to adequately represent human belief systems. We will show that previous proposals fall short of these requirements, and introduce the outlines of a model that satisfies them. Section 1 introduces some key features of actual human belief systems that should be matched by corresponding properties in formal models of such belief systems. Section 2 discusses what logical properties the set of full beliefs in a formal model should be required to have. In Sect. 3, the recommended features of formal models of human belief systems are summarized as ten desiderata. In Sect. 4, previously proposed models are analyzed. They are all shown to violate some of the desiderata. In Sect. 5, a model that satisfies the desiderata is presented. Section 6 concludes.

1 Human Belief Systems

The construction of formal models in epistemology should be based on careful considerations of the structure of the human beliefs and belief systems that these models are intended to represent.

The term “belief” has many meanings, but here it will refer to empirical beliefs, beliefs about what the world we live in is like. Such beliefs are propositional attitudes, mental states referring to propositions. We also have other types of propositional attitudes. For instance, you can believe that there is a flock of wolves in the forest where you plan to spend your holidays, but you can also wish, fear, like or dislike that there are wolves in that forest. Belief differs from these other propositional attitudes in being purely factual (Engel, 2013, 2018). From the mere fact that you believe there are wolves in the forest, it cannot be deduced whether you like or dislike that this is so. It is a fundamental feature of our mental constitution that we hold and express purely factual, per se value-neutral, attitudes to the world. In other words, the fact-value distinction is an essential and universal pattern of human thought. At an early stage of their cognitive development, children know the difference between the factual statement that there is ice-cream on the table and various other statements about ice-cream on the table, such as statements expressing wishes or demands (Kuhn et al., 2000; Wainryb et al., 2004).

There is an obvious explanation why the purely factual mode of thinking about states of affairs is so fundamental and universal in our species: It comes with considerable evolutionary advantages. In order to survive we need to adjust our actions to the real world, not confusing it with our surmises, hopes or fears. As a social species, we also need to communicate factual information in ways that clearly distinguish it from our expressions of non-factual propositional attitudes (Arbib, 2005; Wilkins & Griffiths, 2012; Hansson, 2018a).

The value-neutrality of beliefs, once formed, does not exclude a role for values in the process leading to their formation. For instance, the level of evidence that we require to adopt a full belief that an elevator cable carries its maximum load without breaking may be higher for a passenger lift than for a lift that will only carry goods. However, the influence of values on belief acquisitions has to be kept within rather strict limits in order not to jeopardize the representational accuracy that the belief system must have to provide reliable action guidance and interpersonal communication (Hansson, 2014, 2017).

Two main features of our factual beliefs and the ways in which we express them are particularly important for their formal representation. First, factual beliefs come with different degrees of confidence or certainty.² You can believe that perhaps there is ripe fruit in the trees on the other side of the river, or you can be sure that this is so. Such differences are often important for how we choose to act, and we need to communicate them to others in order to make up joint, well-informed, plans. In other words, in our thoughts and communications we need to distinguish between different strengths of beliefs.

The other important feature follows from our cognitive limitations. We cannot justifiedly entertain the same type of strong certainty about any empirical statement as about simple mathematical or analytical truths such as $1+1=2$. It could therefore seem appropriate never to treat any empirical claim as certain. Even if you just came back from the fruit trees, something might have happened to the trees or the fruits in the short time since you saw them. A huge flock of hungry birds can have emptied the trees in a jiff, or a flash of lightning can have put the trees to fire. However, if you have no reason to believe that something like that has happened, you will not say to your friends that it is “highly likely” or “almost certain” that there is ripe fruit in these trees. You just say that there is ripe fruit in the trees, and you consider yourself to know this to be the case.

This example highlights something we do all the time, both in our thoughts and in our communications with others: When something is sufficiently close to certain, we treat it as certain (Lance, 1995; Weisberg, 2020, pp. 12–17). This is a highly simplifying practice. The alternative would be to keep everything open, and spend thoughts and preparations on all kinds of unlikely possibilities.³ To avoid this cognitive burden, we often refrain from doubt even when we do not have full certainty. That which we do not doubt can be called our “full beliefs” (or “outright beliefs”). Without full beliefs, our everyday thoughts and communications would have been much more complicated.

Full beliefs reduce the cognitive workload in at least two ways: Since the negations of the adopted full beliefs are excluded, the number of possibilities that have to be considered is reduced (Staffel, 2019, pp. 937 and 940). In addition, full beliefs allow “reasoning directly with propositions, without having to constantly consult the underlying probabilities” (Lin & Kelly, 2012a, p. 961; cf. Harman 1986, p. 47). In such reasoning, the conclusions that we draw by propositional inference from our full beliefs will in their turn function as full beliefs in the sense that they can be used as premisses in further inferential reasoning. The usual way to represent this feature in a formal model is to have the set of full beliefs closed under propositional inference. This is an immensely simplifying but also strikingly unrealistic idealization, about which more will be said in Sect. 2.

Furthermore, for the set of full beliefs to be meaningfully action-guiding, it should not contain, or inferentially support, conflicting beliefs in any matter that the agent may consider. The usual means to represent this property in a formal model is to assume that the set of propositions representing full beliefs is (deductively) consistent. This is a simple condition on formal systems, which is standardly applied in both probability theory and belief change theory.⁴

In summary, our belief systems allow us to think and communicate both in terms of uncertain beliefs, i.e., beliefs coming in different degrees of strength, short of certainty, and in terms of full beliefs, which are provisionally treated as certain, but can be given up in the light of new information or new deliberations. The reason why we cannot do with only uncertain beliefs is that it would be too cognitively demanding to hold everything open.

In order to make full use of the advantages that full beliefs provide us with, the two categories have to be mutually exclusive.⁵ The cognitive burden would not be reduced by having full belief in a claim if you at the same time treat it as having some less-than-certain degree of belief that you need to keep track of. This can explain why we would find it strange if someone signalled full belief in a proposition and at the same time assigned a non-unit probability to it:

“I am sure there will be no serious accident in this nuclear reactor during its lifetime. The probability that there will be no such accident is 99.97%.”

The mutual exclusivity of credences and full beliefs is supported by empirical evidence, summarized by Jonathan Weisberg, which indicates that “no proposition is treated as both a given and a mere probability at the same time. At any given time, [a proposition] is either taken as an assumption or else it is weighted according to its subjective probability, not both” (Weisberg, 2020, p. 10).

The simplification and cognitive relief obtained with full beliefs come at a price. Now and then, a full belief turns out to be wrong, and we have to give it up (Levi, 2008). For example, if you go to work by bus every workday, then you certainly “know” where the nearest bus stop is. This is a most practical propositional attitude. If you considered the location of the bus stop to be an open issue, then you would have to spend time and effort every morning making a fresh decision where to go. However, it may well happen that one day, the bus stop has been moved overnight. It would not be a good idea to defiantly wait for the bus at the place of the old bus stop. You will have to give up your full belief about the location of the bus stop, and form a new one. As this example shows, the revisability of full beliefs is necessary for a system of full beliefs to fill its purpose. On the one hand, we do not doubt our empirical full beliefs. They are undoubted, since we currently see no reason to doubt them. On the other hand, they are doubtable, since reasons to doubt and perhaps revise them may arrive at some later point in time.

Actual human agents add, retract, and modify full beliefs many times a day. It is not uncommon for a proposition to be subject to a series of several such changes. This requires that after each change in beliefs, the mechanisms of change are not only intact, but also adjusted to the change that has just taken place. In the terminology of belief change theory, an account of iterated belief change is needed.

In order for full beliefs to fulfil their simplifying and alleviating function in the belief system, they have to be all-purpose beliefs, rather than contextual. The formation of a full belief takes some cognitive effort. If that effort had to be reiterated for each new use of the belief, then the reduction of cognitive load would be small or nil.⁶ For instance, if you have convinced yourself that a jug does not leak when filled with water, then you will assume that it does not either leak when filled with milk or juice. But obviously, the decontextualization that is usually involved in the formation of a full belief can sometimes lead us wrong. If experience has convinced you that a plank lying across a brook does not break when you use it as a footbridge, then that conclusion cannot safely be generalized to several persons walking or standing on the plank at the same time.

We have two major methods to reduce the risk of including fallacious claims among our full beliefs. The first of these is high entrance requirements for the inclusion of a proposition in the set of full beliefs. For instance, you would presumably not adopt a full belief that the plank will not break if it sagged ominously when you walked on it. We only adopt a proposition as a full belief if we have strong reasons to do so. Thus, if you consider it to be only slightly more likely that a proposition is true than that it is false (preponderance of evidence, representable as a probability just above 0.5), then you do not adopt it as a full belief. To do that, you would typically require evidence convincing you that the proposition is almost certain to be true (representable as a probability fairly close to 1). High demands on the plausibility of propositions to be adopted as full beliefs will create a margin of safety that reduces the risk that applications in new contexts will lead to false beliefs.

The second method to avoid false full beliefs is expeditious exits. This means that we retract or modify a full belief when we become aware of reasons to doubt it. Such reasons can emerge either in its original areas of application or when it is used in a new way.⁷ For instance, reflection on the previously unconsidered possibility of several persons passing over the plank at the same time can make us replace the previous belief “the plank is safe to walk on” by “the plank is safe to walk on, one person at a time”. Notably, the new full belief is an all-purpose belief, just like the previous one. In other cases, a new application or a reconsideration of a full belief may lead us to replace it with an uncertain belief (credence). I was previously convinced that one of my colleagues is a leftist, but after hearing him complain about high income taxes I am now uncertain about his political leanings.

The formation of a full belief in a proposition is a mechanism for cognitive simplification. Therefore, it is reasonable to let the expected practical usefulness of such a simplification have an influence on whether or not we adopt a particular proposition as a full belief. The same applies to the contravening usefulness of taking the uncertainty concerning that proposition into account. For instance, it would presumably take stronger evidence to convince us that an underwater vehicle resists the pressure at a depth of 100 m if it will carry school children learning about the ocean than if it will only be used for unmanned research explorations. The probability that the vehicle resists the pressure will have to be higher to support the formation of a full belief in the former case than in the latter. This means that whether we replace uncertainty concerning a proposition (representable by a non-unit probability) by a full belief in that proposition will depend not only on the prior degree of uncertainty (level of probability) but also on the values at stake.⁸ Consequently, it is more accurate to model the formation of a full belief as a separate operation than as a mere consequence of the assignment of certain probability values.⁹ It also follows that although full beliefs are non-contextual in the sense of being intended for application in all contexts, the formation and abandonment of full beliefs is contextual in the sense of depending on the situations in which it can be foreseen that they will be applied.

Till now, we have focused on individual belief systems. There are also collective belief systems, reflecting what groups of humans, or humans in general, agree on. The most prominent collective system of empirical beliefs is that of science. In most studies of the scientific belief system, the focus has been on the set of jointly held full beliefs, which is usually called the scientific corpus. It consists of the statements that there is, according to the consensus among scientists, currently no reason to doubt. It should be noted, though, that agreement on credences can also have an important role in science. Agreement that an hypothesis is reasonably plausible can lead to joint efforts to test that hypothesis. Scientific agreement that there is significant although insufficient evidence that a chemical exposure is harmful can, if communicated to decision-makers, result in measures that reduce or eliminate that exposure (Hansson, 2018d). But generally speaking, scientific discussions usually have a strong focus on what can currently be considered to be known, that is, what is contained in the scientific corpus. Therefore, the set of full beliefs has a dominant role in the scientific belief system, possibly more dominant than in individual belief systems.

Both the full beliefs and the credences that have been agreed upon in science are all-purpose beliefs. Context-dependence would be even more problematic in science than in individual belief systems. We expect scientific statements to have general validity. For instance, if we discovered that analytical and synthetic chemists ascribe different structures to one and the same substance, then we would not accept this as an adequate expression of contextuality. Instead, we would conclude that something has gone wrong and that a correction is needed.

The social role of science is to provide, at each point in time, the most reliable information about the subject-matter that falls within its domain (Bunge, 2011; Hansson, 2018b, pp. 404–405). To achieve this, science employs both of the above-mentioned methods to avoid erroneous full beliefs in the corpus. In the application of the first of these methods, high entry requirements, it is essential that science is a collective process. We humans are much better at criticizing ideas put forward by others than our own ideas. Therefore, the introduction of a new statement into the scientific corpus tends to require more convincing evidence than what individuals usually require to adopt a full belief. For the second method, expeditious exit of claims that have become subject to reasonable doubt, it is essential that inclusion and retainment in the corpus require (near-)consensus among the scientists with the relevant expertise. At least when science functions according to its established core values (Merton, 1942, p. 126) doubt expressed by a minority of the experts can remove an assertion from the corpus and make it an open issue in need in further investigation.

Thus, the collective belief system of science has the same basic structure as that of individual belief systems. In particular, it contains both uncertain and full beliefs, and the latter have the same function, namely to reduce the cognitive workload. Probably, the most important differences concern the processes leading to changes in the corpus (set of full beliefs). As already indicated, these processes will be treated here as external to the respective belief systems. In what follows, the focus will be on individual belief systems, but much of the analysis will also apply to collective belief systems such as that of science.

2 The Logical Properties of the Set of Full Beliefs

Most of the desirable properties of a model of full beliefs that were mentioned in Sect. 1 can fairly straight-forwardly be reformulated as desiderata that are precise enough to be used in the evaluation of such models. This reformulation will be performed in Sect. 3. However, one of the issues brought up in Sect. 1, namely the logical properties of the set of full beliefs, requires a somewhat more thorough preparative discussion.

As mentioned in Sect. 1, logical closure is the dominant requirement on the set of full beliefs, both in belief change theory and in other branches of epistemic logic (Alchourrón et al., 1985; Fermé & Hansson, 2018).¹⁰ It is also assumed in standard probability theory, where the set of propositions that are assigned probability 1 by a probability function is logically closed. However, logical closure is also connected with some undesired properties of formal models.

It follows from the logical closure of the set of full beliefs that the epistemic agent is assumed to believe in all logically true propositions that are expressible in the language. For simple logical truths, this does not lead to any counterintuitive consequences. For instance, suppose that you have no idea whether or not Mozart was in Vienna on his 10th birthday. Then neither the statement $v$ (“Mozart was in Vienna on his 10th birthday”) nor its negation $\lnot v$ is one of your full beliefs. Nevertheless, in a formal model you are assumed to fully believe in the statement $v\vee \lnot v$ (“either Mozart was in Vienna on his 10th birthday, or he was not”). This is not much of a problem; presumably you would consent to that tautology. However, things grow worse if we turn to more complex logico-mathematical truths. We will then find ourselves considering an agent who knows all mathematical truths, including truths that would require more resources than can ever be available in the physical universe to prove or even just to express. This is known as the problem of logical omniscience (Égré, 2020). It has usually been discussed in relation to epistemic logic and dichotomous models of belief change, but it applies to probabilistic models as well. The true logical and mathematical statements that are expressible in the language of a probabilistic model will necessarily have to be assigned probability 1. This is problematic if the probability function represents an individual’s degrees of belief.¹¹

The problem of logical omniscience is unavoidable in models that have a logically closed set of full beliefs. But on the other hand, models in which logical omniscience does not hold tend to have the opposite problem of “logical ignorance”, i.e., the logical inferences that the agent is able to make are not easily represented (Duc, 1997). Since our focus here is on empirical beliefs and their combinations, logical ignorance seems to be the most important problem. With the exception of some scientific inferences, most conclusions that can be drawn from empirical beliefs appear to be within ordinary human power. We can therefore require of a formal model that elementary inferences from empirical full beliefs are included in its set of full beliefs. A simple way to achieve this would be to require that set to be logically closed, but in order to avoid requiring logical omniscience, it is better to look for a weaker and less controversial condition. In that search, we can have use for the well-known fact that if the logic is compact, then a non-empty set of full beliefs is logically closed if and only if both of the following two conditions hold:

$$ \begin{aligned}{} & {} \text {If }a_1\text { and }a_2\text{ both represent a full belief, then so does }a_1 \& a_2\text { (conjunctive closure)}. \end{aligned}$$

(1)

$$\begin{aligned}{} & {} \text {If }a\text { represents a full belief and }a\text { logically implies }d, \text { then }d\text { represents a full belief}\nonumber \\{} & {} \text {(single-premiss consequence).} \end{aligned}$$

(2)

Conjunctive closure refers to an elementary type of inference, which we can expect human agents to be able to perform. In contrast, since any premiss logically implies all tautologies, with single-premiss consequence all the mathematics that is expressible in the language will be included in the set of full beliefs. Therefore, in our search for an inference-allowing condition that models of human belief systems should undeniably satisfy, conjunctive closure may seem to be a plausible candidate.

However, conjunctive closure can be applied repeatedly any number of times. We can form the conjunction of any finite number of propositions. When we do this, the uncertainty increases with the number of empirical conjuncts. I am able to list from memory twenty food items that I am certain that I have in my kitchen, but the probability that some item on the list is missing will be much higher than if I only mention one of the items. This is the multi-premiss problem, which is also illustrated in two classical probability paradoxes, the lottery paradox (Kyburg 1961, p. 197) and the preface paradox (Makinson 1965, p. 205). To make sure that our condition can reasonably be expected of a model of human belief systems, we should therefore limit the number of conjunctive steps. The most drastic way to do that is the following:

$$ \begin{aligned}&\text {If the language in which full beliefs are expressed consists of two atomic sentences and their}\nonumber \\&\text {truth-functional combinations, then it holds that if each of the two (not necessarily atomic)}\nonumber \\&\text {sentences } e_1 \text { and }e_2\text { represents a full belief, then so does } e_1 \& e_2 \text { (diatomic conjunctive closure) }. \end{aligned}$$

(3)

Diatomic conjunctive closure is a remarkably weak condition, but also a highly plausible one. If a model of human belief systems violates this condition, then it surely has a problem with logical ignorance. We will therefore use this property as one of our desiderata for formal models of human belief systems.

3 Desiderata for a Combined Model

Based on the above exposition of human belief systems, we can now summarize some properties of a formal model that will make it conform to important properties of real-life belief systems. This will be done in the form of ten desiderata. The first two of these are concerned with the relationship between the two types of beliefs. This is followed by four desiderata on the static properties of full beliefs and finally four that concern the patterns of change. It should be emphasized that the ten desiderata are based on the account of human belief systems that was presented in Sect. 1, with its emphasis on how these systems are adapted to make efficient use of our limited cognitive capacities. Other pictures of human belief systems, such as Jeffrey’s (1970), which abstracts from our cognitive limitations, would give rise to different desiderata.

We assume that beliefs refer to propositions, that the set of available such propositions is closed under classical truth-functional consequence, and that there is a function from the set of available propositions to numbers in the interval [0, 1], such that uncertain beliefs (credences) have values in the subinterval (0, 1). That function is usually a probability function.

1.:

Both belief types. The model contains representations of both (i) full empirical beliefs and (ii) uncertain empirical beliefs coming in different degrees.

2.:

Mutual exclusivity. The two types of belief are mutually exclusive in the sense that the agent does not at the same time have full belief in a proposition and treat it as uncertain (i.e., assign a non-unit probability to it).

3.:

Contingent full beliefs. The belief system allows for full beliefs in the types of contingent empirical propositions that humans usually have full belief in.

4.:

Non-contextuality. Although the formation of full beliefs may be influenced by contextual factors, once formed the full beliefs are intended for application in all contexts in which the proposition they refer to is relevant.

5.:

Diatomic conjunctive closure. If the language in which beliefs are expressed consists of two atomic sentences and their truth-functional combinations, then: If each of $e_1$ and $e_2$ represents a full belief, then so does $e_1 \& e_2$. (See Sect. 2.)

6.:

Consistency. No logical contradiction can be derived from the set of full beliefs or from the belief system as a whole.

7.:

Iterability. Operators that change the belief system can be performed an unlimited number of times after each other.

8.:

Bidirectional transfer. There are operations for transferring any contingent proposition in both directions between uncertain belief and full belief.

9.:

Non-functionality. The formation of full beliefs is a separate operation, whose outcome is not completely determined by the (prior) assignment of probabilities.

10.:

Convincing evidence. The formation of full belief in a proposition requires convincing evidence that the proposition is true.

4 Previously Proposed Models

We are now going to evaluate major previously proposed models of belief systems against the ten desiderata introduced in Sect. 3.

4.1 Standard Probabilities

In elementary textbooks, empirical propositions (often referring to the outcomes of a random process) are routinely accepted through Bayesian updates, and consequently assigned the probability 1. We can call this the standard probability model.¹² If the propositions with probability 1 are identified as the set of full beliefs, then this is actually a model containing representations of both probabilistic and full beliefs, although it is usually not presented as such.

Such a model satisfies both belief types and mutual exclusivity, since full beliefs have probability 1 and uncertain beliefs have probabilities in the range (0, 1). It also satisfies contingent full beliefs, non-contextuality, diatomic conjunctive closure (and also logical closure), consistency, and iterability. However, it does not satisfy bidirectional transfer. This is because standard (Bayesian) probability revision is irreversible. If a sentence has been assigned probability 1, then there is no way to reduce that probability to a lower value. To see this, let $\mathfrak {p} (a)=1$, and suppose that there is some sentence b such that after revision by b, we obtain a new probability function $\mathfrak {p}'$ with $\mathfrak {p}'(a)\ne 1$. According to the Bayesian revision formula, this is equivalent to $\mathfrak {p} (a \& b)/ \mathfrak {p} (b)\ne 1$, which is impossible since it follows from $\mathfrak {p} (a)=1$ that $\mathfrak {p} (a \& b)= \mathfrak {p} (b)$. Thus, any revision or series of revisions of $\mathfrak {p}$, whatever the inputs are, will result in a probability function that also assigns probability 1 to a. Consequently, the overall pattern of standard (Bayesian) probability revision is monotonically cumulative. Every change induced by an update creates an additional full belief, but no full belief can ever be given up. It does not seem possible to avoid this monotonic cumulativeness by replacing the Bayesian revision formula by some other revision formula that assigns probability 1 to the input sentence. The reason for this is that standard probability theory has an amnesia problem: If $\mathfrak {p} (a)=1$, then $\mathfrak {p}$ does not carry any information about what probabilities other propositions would have if the probability of a were not 1. Most notably, for all sentences x, we have $\mathfrak {p} (x \& \lnot a)=0$, so we are completely uninformed about probabilities given $\lnot a$.

In this model, full belief in a proposition results from an operation of change if and only if that operation assigns probability 1 to the proposition. Therefore, non-functionality is not satisfied. Since the criteria for the formation and selection of inputs is external to the model, it does not contain any guarantee that convincing evidence is satisfied. However, under the intended interpretation, probability 1 will only be assigned to propositions for which there is strong evidence, which means that this desideratum will in practice be satisfied.

4.2 Jeffrey’s Model

Richard Jeffrey (1956) maintained that full beliefs have no place in science, and later he expressed skepticism about the usefulness of full beliefs more generally (Jeffrey, 1970). Similar views have been expressed by Churchland (1981), Christensen (2004, esp. pp. 96–105) and others.¹³ We will use the term “Jeffrey’s model” for a probability model in which contingent (empirical) propositions are never assigned probability 1. Since the only operation of change in conventional probability theory, namely Bayesian updating, always assigns probability 1 to some sentence, this approach requires the use of some other operation of change. We will assume the use of Jeffrey conditionalization (Jeffrey 1983, pp. 171–172), which is the standard method to change the probability of an input proposition from a value in the interval (0, 1) to some other value in that same interval.

This model is based on intuitions and ideals that are very different from those presented in Sect. 1. In particular, it allows only logically true sentences to be full beliefs. It satisfies both belief types and mutual exclusivity, with the restriction that the full beliefs coincide with the logical truths. It does not satisfy contingent full beliefs, but it satisfies non-contextuality, diatomic conjunctive closure (and logical closure), consistency, and iterability. It violates bidirectional transfer, since Jeffrey conditionalization cannot assign a probability other than 1 to a logical truth, i.e., to one of the full beliefs in this model. It also violates non-functionality, since the full beliefs are exactly the sentences that are assigned probability 1. Since the evidence for logical truths is convincing, it satisfies convincing evidence.

4.3 Dichotomous Belief Change

In dichotomous models of belief change, the object of change is the set of full beliefs, which is called the belief set (Alchourrón et al., 1985; Fermé & Hansson, 2018). Uncertain beliefs are represented by propositions a, such that neither a nor $\lnot a$ is an element of the belief set. There is no grading of uncertain beliefs in terms of their uncertainty or plausibility.¹⁴ This means that dichotomous models have insufficient resources for providing a full representation of a human belief system, as outlined in Sects. 1–3. However, since these models possess some of the features missing in probabilistic models, it is of interest to evaluate dichotomous belief change, and in particular the dominant AGM model (Alchourrón et al., 1985) in terms of the desiderata presented in Sect. 3.

The AGM model fails to satisfy both belief types, since that desideratum requires the representation of different degrees of uncertain belief. However, it satisfies mutual exclusivity, since no proposition a can at the same time be fully believed ($a\in K$, where K is the belief set) and uncertain ($a\notin K$ and $\lnot a\notin K$). It satisfies contingent full beliefs, non-contextuality, and diatomic conjunctive closure (and logical closure). It does not satisfy consistency, since AGM revision by an inconsistent proposition results in an inconsistent belief set. However, this is a feature of the original framework that has been criticized, and models of dichotomous belief change that satisfy consistency have been proposed (Hansson et al., 2001; Fermé & Hansson, 2018, pp. 65–68; Hansson, 2023a). This requires only small changes in the original AGM framework.

The original AGM framework did not satisfy iterability, since the selection function that is the key element of the mechanism for change was only defined for the original belief set, not for the new belief sets that are outcomes of performing an operation of change. However, it was later found that iterability can be obtained by a simple extension of the domain of the selection function, without imposing any new properties on the operations of change (Hansson, 2012, p. 152). Several other constructions that make AGM operations iterable have been proposed (Fermé & Hansson, 2018, pp. 59–64; Rott, 2009).

AGM revision satisfies bidirectional transfer, since it has mechanisms both for acquisitions and removals of full beliefs. It satisfies non-functionality in a vacuous way, since it does not have a probability function. In the original AGM model and most of its successors, all input sentences are accepted. Thus convincing evidence is not satisfied by mechanisms internal to the model. However, several models of non-prioritized belief change have been proposed. In these models, the input does not necessarily have priority over the information already present in the belief state (Fermé & Hansson, 2018, pp. 65–68). This leaves room for satisfying the desideratum of convincing evidence.

4.4 The Lockean Thesis

Under the assumption that high probability is required for the formation of a full belief, one obvious option would be to identify the full beliefs with the propositions that have a probability above a certain threshold $t\in [0,1)$, which should be close to but not equal to 1:

$$\begin{aligned} a \text { is a full belief if and only if }\mathfrak {p}(a)>t \end{aligned}$$

(4)

This approach was defended by Henry Kyburg (1961, pp. 196–198). Following Foley (1993, pp. 140–141), it is now commonly called the Lockean thesis.¹⁵

A model based on this thesis satisfies both belief types, but it violates mutual exclusivity, since a proposition with a probability in the interval (t, 1) retains its non-unit probability while at the same time being fully believed. Contingent full beliefs and non-contextuality are satisfied. However, diatomic conjunctive closure is not satisfied (Leitgeb 2017, pp. 44–45). To see this, let the language consist of the atoms $a_1$ and $a_2$ and their truth-functional combinations. Let $\mathfrak {p}(a_1 \& a_2)=0.94$ and $\mathfrak {p}(a_1 \& \lnot a_2)=\mathfrak {p}(\lnot a_1 \& a_2)=0.02$, and let $t=0.95$. Then both $a_1$ and $a_2$ are full beliefs, but $a_1 \& a_2$ is not.

It has long been recognized that the use of a fixed non-unit probability limit for full belief will lead to violation of logical closure. Kyburg explicitly rejected logical closure and proposed its replacement by closure under single-premiss consequence. It would of course be possible to replace Eq. (4) by a definition that implies logical closure, such as the following:

$$\begin{aligned}&a \text { is a full belief if and only if it follows logically from a set }\{b_1,\dots b_n\}\text { of propositions such that}\nonumber \\&\mathfrak {p}(b_k)>t\text { for all }b_k\in \{b_1,\dots b_n\}. \end{aligned}$$

(5)

This, however, would create another problem: Consistency is satisfied with Eq. (4), but not with Eq. (5).¹⁶

In this model iterability is satisfied, but bidirectional transfer is not. Transfers of propositions between the probability ranges (0, t] and (t, 1) are possible, but if only Bayesian update is used, then each update will turn its input proposition into a permanent full belief, unchangeable in future updates. This can be remedied by introducing another update operation, such as Jeffrey conditionalization.

Non-functionality does not hold, since the value of $\mathfrak {p}(a)$ determines whether a is a full belief or not. Provided that t is close to 1, as intended, convincing evidence is satisfied.

4.5 Lockean Thesis with Exception Clauses

In order to solve the lottery paradox, Pollock (1990, p. 81) proposed that the Lockean thesis should be provided with an exception clause. For a proposition a to be accepted, he required both that $\mathfrak {p}(a)>t$ and that there is no set S of propositions such that $a\in S$, all elements of S have probabilities above t, and S is inconsistent. Douven (2002) modified this proposal by replacing Pollock’s criterion that S is inconsistent by the criterion that for all $s\in S$, there is some $S'\subseteq S$ such that $\mathfrak {p}(s\mid \textit{ \& }S')\le t$.¹⁷ Ryan (1996, p. 130) proposed an exception clause in somewhat more informal terms, which can be formalized as there being no set S of “competing statements” with $a\in S$, all elements of S having probabilities above t, and $\mathfrak {p}(\textit{ \& }S)\le t$.

All these three proposals avoid the lottery paradox, but none of them avoids the preface paradox. This is for three different reasons. For Pollock’s proposal, it is that in the preface paradox, the claim that all statements in the book are true is not inconsistent, although it has too low probability to be believed. Douven’s proposal fails to solve the preface paradox in the plausible variant that refers to a set S of statements in the book, all of which are probabilistically independent of each other. In that case, it holds for all non-empty $S'\subset S$ and all $s\in S{\setminus } S'$ that $\mathfrak {p}(s\mid \textit{ \& }S')=\mathfrak {p}(s)> t$. Ryan’s proposal fails to solve the preface paradox since the truth-claims referring to individual statements in the book need not be “competing statements” in any reasonable interpretation of that phrase.

These attempted solutions all have an ad hoc character. To make the exception clause approach at all viable, it would be necessary to exclude the general pattern exhibited in the lottery and preface paradoxes, namely the multi-premiss problem referred to in Sect. 2, not only specific instances of it. In formal terms, the multi-premiss problem consists in the existence of a set $A=\{a_1,\dots , a_n\}$ of propositions, such that $\mathfrak {p}(a_k)>t$ for all $a_k\in A$ and $\mathfrak {p}(a_1 \& \dots \& a_n)\le t$. Such a set A can be constructed in different ways. It can be based on negations of mutually exclusive propositions, as in the lottery paradox. It can also consist of logically independent propositions, as in the above-mentioned version of the preface paradox or in an alternative lottery paradox, in which there is a separate draw for each ticket, and each ticket has a 1 in 10,000 chance of winning, independently of the outcomes for the other tickets (Levi 1967, p. 40). Furthermore, A can be a set of high-probability propositions with mixed degrees of dependencies, but enough independence to make it highly improbable that they are all true. This is exemplified by the usual variant of the preface paradox, but perhaps more importantly, such a set A can be formed from everyday beliefs. Suppose that your kitchen table and coffee room conversations in the last few months have all been surreptitiously recorded, and a list has been compiled of all statements that you made with phrases such as “I am sure that”, “there can be no doubt that”, or other markers of full belief. It would be no surprise if a paradox with the same structure as the preface paradox can be based on that list.¹⁸

In order to solve the multi-premiss problem with an exception clause, we would need to make sure that a proposition a is not believed in spite of having a probability higher than t if the following holds:

$$ \begin{aligned}&\text {There is some finite set }S\text { of contingent propositions, such that }\nonumber \\&a\in S\text {, } \mathfrak {p}(s)>t\text { for all }s\in S\text {, and }\mathfrak {p}(\textit{ \& }S)\le t\text {.} \end{aligned}$$

(6)

This, however, will exclude all contingent full beliefs. To see that, let a be any contingent proposition such that $t<\mathfrak {p}(a)<1$. Let $s_1,s_2,\dots$ be propositions that are logically independent of a and of each other, and let each of them have the same probability as a. They may for instance be outcomes of independent randomized processes. Then there is some lowest n such that $\mathfrak {p}(a)\times \mathfrak {p}(s_1)\times \dots \times \mathfrak {p}(s_n)\le t$. Let $S=\{a,s_1,\dots , s_n\}$. Then $\mathfrak {p}(s)>t$ for all $s\in S$ and $\mathfrak {p}(\textit{ \& }S)\le t$. Thus, a satisfies the exception clause and will not be believed. Since this construction is available for all empirical propositions a with above-threshold probabilities, the exception clause cannot solve the multi-premiss problem.

Let us nevertheless compare the exception clause approach to our ten desiderata. Since the version that solves the multi-premiss problem cannot sustain empirical full beliefs, it coincides with Jeffrey’s model, and we will therefore consider the variants that only solve the lottery paradox. In these variants, both belief types is satisfied, but mutual exclusivity fails for the same reason as for the (exceptionless) Lockean thesis. Contingent full beliefs holds (but it fails for the version that solves the multi-premiss problem). Non-contextuality is satisfied. Diatomic conjunctive closure fails for the same reason as for the (exceptionless) Lockean thesis.¹⁹Consistency holds for the same reason as for the (exceptionless) Lockean thesis, and so does iterability. Bidirectional transfer fails in the same way as for the (exceptionless) Lockean thesis, but here as well, the introduction of Jeffrey conditionalization can solve the problem. Non-functionality fails since probabilities determine which propositions are fully believed. Convincing evidence is satisfied, provided that t is sufficiently close to 1.

4.6 Lin and Kelley’s Model

Largely based on a proposal by Isaac Levi (1996, p. 286), Hanti Lin and Kevin Kelly (2012a; 2012b) have proposed a rule for deriving a set of full beliefs from a probability function $\mathfrak {p}$.²⁰ The domain of $\mathfrak {p}$ consists of a set $Q=\{h_1,\dots ,h_n\}$ of propositions and the disjunctions that can be formed from elements of that set. The elements of Q are mutually exclusive, and they form an exhaustive list of answers to some question. A ratio threshold r, with $0<r<1$ is assigned to Q. Then a proposition a is a full belief in the context of Q if and only if:

$$\begin{aligned} \bigvee \{h_i\in Q\mid \mathfrak {p}(h_i)\ge r \times \mathfrak {p}(h_{\small max}) \} \text { logically implies }a \end{aligned}$$

(7)

where $h_{\small max}$ is an element of Q that has the highest probability assigned by $\mathfrak {p}$ to any element of Q. In other words, the full beliefs are the propositions that follow logically from each of the answers that have sufficiently high probability in comparison to the most probable answer(s) to the question Q.

This model satisfies both belief types, since both uncertain and full beliefs are represented. It does not satisfy mutual exclusivity, since the contingent propositions adopted as full beliefs have non-unit probabilities. It satisfies contingent full beliefs, but it does not satisfy non-contextuality, since the full beliefs are only assumed to be held in the context of a particular question and a chosen set of answers to that question (Lin and Kelly 2012b, pp. 534 and 567–570). Diatomic conjunctive closure (and logical closure), consistency and iterability (through Bayesian updating) are satisfied. Bidirectional transfer fails since each Bayesian update makes its input a permanent full belief, but this can be remedied for instance by introducing Jeffrey conditionalization. Non-functionality is not satisfied since in each context (defined by Q and r), the probability function determines which propositions are full beliefs. Convincing evidence is not satisfied either. To see why, let $Q=\{h_1,h_2,h_3,h_4,h_5\}$, $\mathfrak {p}(h_1)=0.44$, and $\mathfrak {p}(h_2)=\mathfrak {p}(h_3)=\mathfrak {p}(h_4)=\mathfrak {p}(h_5)=0.14$. Furthermore, let $r=1/3$, which is a value used in an example presented by Lin and Kelly (2012a, pp. 969–970). Then equation (7) assigns full belief in $h_1$, although it is less probable than its negation.

4.7 Leitgeb’s Model

Hannes Leitgeb (2014; 2017) has proposed a model that is similar to Lin and Kelly’s in that the assignment of full beliefs is determined by numerical relations between the probabilities of maximally specified alternatives. Leitgeb assumes a set $W=\{w_1,\dots ,w_n\}$, where each $w_k$ is a maxiset, i.e., a maximal consistent subset of the language (“possible world”), and a probability function $\mathfrak {p}$ on W. A set K of propositions is an admissible set of full beliefs for W if and only if there is some $V\subseteq W$ such that:

$$\begin{aligned} K = \bigcap V \text { and }\mathfrak {p}(w)>\mathfrak {p}(W\setminus V)\text { for all }w\in V \end{aligned}$$

(8)

Thus, each maxiset w that is compatible with all the epistemic agent’s beliefs (i.e., $w\in V$) has to be more probable than the sum of the probabilities of all maxisets that are incompatible with the agent’s beliefs (i.e., more probable than the sum of the probabilities of all elements of $W\setminus V$). This proposal has an interesting connection with the Lockean thesis. If there is a (logically closed) set K of full beliefs such that equation (8) is satisfied, then K is also obtained with equation (4) for some threshold t. Conversely, if the set of full beliefs obtained from equation (4) is logically closed, then it also satisfies equation (8) (Leitgeb 2017, pp. 112–148; Hansson, 2018c, p. 277n). Depending on W and $\mathfrak {p}$, there may be one or several sets K that satisfy equation (8) (Leitgeb, 2017, pp. 116–117).

Leitgeb’s model satisfies both belief types, but not mutual exclusivity. It satisfies contingent full beliefs. However, it does not satisfy non-contextuality, since the partitioning of the alternatives represented by maxisets has a decisive impact on the epistemic agent’s full beliefs. For instance, let $W=\{w_1,w_2\}$, $\mathfrak {p}(w_1)=2/3$, $\mathfrak {p}(w_2)= 1/3$, $W'=\{w_1',w_1'',w_2\}$, $\mathfrak {p}'(w_1')=\mathfrak {p}'(w_1'')=\mathfrak {p}'(w_2)=1/3$, $a\in w_1\cap w_1'\cap w_1''$ and $a\notin w_2$. Then a is a full belief in the model based on W and $\mathfrak {p}$, but not in that based on $W'$ and $\mathfrak {p}'$. Leitgeb acknowledges the partition-sensitivity of his model and says that “[t]his doubly contextual approach, in view of the contextual determination both of [the probability limit] and of the partition, certainly needs proper defense” (Leitgeb 2013, p. 1387; cf. Leitgeb, 2017, pp. 7–8, 51, 100, and 130–148 and Schurz, 2019).

The model satisfies diatomic conjunctive closure (and logical closure), consistency and iterability. It does not satisfy bidirectional transfer, since Bayesian update makes the input sentence a permanent full belief, but this can be remedied if another mechanism for updates, such as Jeffrey conditionalization, is used. It satisfies non-functionality since, for a given probability function, there may be several sets of full beliefs that satisfy equation (8). Leitgeb describes the choice between these options as a matter of (context-dependent) epistemic cautiousness (Leitgeb 2017, pp. 94–100, 125, and 151). Finally, convincing evidence is not satisfied. This can be seen by letting $W=\{w_0, \dots ,w_{49}\}$, $\mathfrak {p}(w_0)=0.51$, $\mathfrak {p}(w_1)=\mathfrak {p}(w_2)=\dots =\mathfrak {p}(w_{49})=0.01$, $a\in w_0$ and $a\notin w_1\cup \dots \cup w_n$. Then equation (8) supports full belief in a, although its probability is only 0.51 (Hansson, 2018c, pp. 278–279).²¹

4.8 Primitive Dyadic Probability

As we saw in Sect. 4.1, standard probability theory has no means for changing the probability of a proposition that has been assigned probability 0 or 1. One of the best developed methods to avoid this problem is to let the belief state be represented by a primitive dyadic (“conditional”) probability function instead of a standard, monadic one. The dyadic probability function $\ddot{\mathfrak {p}}$ can be seen as a generalization of a monadic probability function $\mathfrak {p}$, such that $\mathfrak {p}(x)=\ddot{\mathfrak {p}}(x,{\scriptstyle \top })$ for all x. Revision of a dyadic probability function $\ddot{\mathfrak {p}}$ by a proposition a gives rise to a new monadic probability function $\mathfrak {p}*a$ such that $(\mathfrak {p}*a)(x) = \ddot{\mathfrak {p}}(x,a)$ for all x. The associated set of full beliefs is $\{x\mid \ddot{\mathfrak {p}}(x,a)=1\}$. Notably, this works even if $\mathfrak {p}(a)=\ddot{\mathfrak {p}}(a,{\scriptstyle \top })=0$. Such dyadic functions have often been called “Popper functions”.²² For an overview of the properties of primitive dyadic probability functions, see Makinson (2011).

With standard axioms for the dyadic function, this construction gives rise to a model that satisfies both belief types. It also satisfies mutual exclusivity, since empirical full beliefs have unit probability and uncertain beliefs have below-unit probabilities. Contingent full beliefs, non-contextuality, diatomic logical closure (and logical closure) and consistency are also satisfied. However, iterability is not satisfied in the available models using primitive dyadic probability functions. Although revision of $\ddot{\mathfrak {p}}$ by a gives rise to a new monadic probability function $\mathfrak {p}*a$ such that $(\mathfrak {p}*a)(x) = \ddot{\mathfrak {p}}(x,a)$ for all x, it does not give rise to a new dyadic probability function with which additional updates can be made (Arló-Costa, 2001, p. 585; cf. Lindström & Rabinowicz, 1989, p. 98).²³

Bidirectional transfer can be satisfied in these models, but is not necessarily so. For one direction, note that a proposition a with $0<\ddot{\mathfrak {p}}(a,{\scriptstyle \top })<1$ will receive unit probability after revision by a, since $\ddot{\mathfrak {p}}(a,a)=1$ (Makinson 2011, p. 126). For the other direction, let $\ddot{\mathfrak {p}}(a,{\scriptstyle \top })=1$. Then there may be some proposition c such that $\ddot{\mathfrak {p}}(c,{\scriptstyle \top })=0$ and $0<\ddot{\mathfrak {p}}(a,c)<1$.

Non-functionality is not satisfied since full beliefs are completely determined by the dyadic probabilities. Convincing evidence is not guaranteed by the formal structure, but will be satisfied under the intended interpretation, in which updates are only performed by propositions for which there is strong evidence.

5 A New Approach

The purpose of this section is to present a proof of concept, showing that the desiderata introduced in Sect. 3 can all be satisfied in one and the same formal model. Such a model will in these respects be more closely aligned with the structure of actual human belief systems as outlined in Sect. 1 than the models discussed in the previous section.

The new model is based on four rather simple components. The first of these is the use of infinitesimals to solve the amnesia problem of standard probability theory, i.e., to keep track of currently rejected but still reinsertable possibilities. Infinitesimals are additions to our standard number system (the real numbers). A positive infinitesimal is a number that is larger than zero but smaller than all positive real numbers. Infinitesimals are parts of the hyperreal number system, which also contains infinite numbers, but those are not used in this model. (For an introduction to infinitesimals and the hyperreal number system, see Keisler, 1986 or Keisler, 2022. For mathematical details concerning the hyperreal model discussed here, see Hansson, 2023a and Hansson, 2023b. On the model more generally, see also Hansson, 2020 and Hansson, 2022.)

Importantly, infinitesimals are used here for the sole reason that they have the structure needed to represent the adoption of beliefs previously treated as undoubtedly false, also in unlimitedly long sequences of updates. No metaphysical claims are attached to the infinitesimals. In particular, the use of infinitesimal probabilities in this model is not intended to imply that humans hold beliefs to infinitesimal degrees.²⁴ It is only implied that some aspects of our patterns of belief change have a structural similarity with a formal framework containing infinitesimals. The same effect could have been obtained by constructing a new formal framework with essentially the same structural properties, but the use of a well-known and thoroughly investigated mathematical framework has considerable advantages.

Standard probability theory can be transformed into hyperreal probability theory by extending its codomain, the closed real-valued interval [0, 1], to the closed hyperreal-valued interval [0, 1]. The latter consists of all numbers between 0 and 1 that are either real or the sum of a real number and a (positive or negative) infinitesimal. Previously, hyperreal probabilities have mainly been used to solve problems that arise when classical probability theory is applied to infinite domains (event spaces) (Benci et al., 2018; Wenmackers & Romeijn, 2016). Here, they will be used for track-keeping or memorizing purposes (Hansson, 2020, 2023a). The following example can be used to explain their use:

At a big party, you suddenly hear your old friend Sarah’s characteristic voice. There can be no doubt that it is her. You now fully believe that Sarah is at the party. But when you go and look for her, you cannot find her. It turns out that someone had played a recording of her voice, and in the noisy environment of the party it sounded just as if she were present.

According to the traditional account of probability updates, when you assimilate the full belief s (“Sarah is at this party”), this is represented in the formal model by the assignment of probability 1 to s. As we saw in Sect. 4.1, this has the unwelcome consequence that the formal model cannot represent what happens afterwards, namely that you give up s and replace it by its negation $\lnot s$. According to the infinitesimal account, your acquisition of the belief that Sarah is at the party is instead represented by the assignment of a probability $1-\delta$ to s. Here, $\delta$ is either 0 or an infinitesimal number. It represents your willingness to give up s at some later stage. The smaller $\delta$ is, the less willing you are to give up s. Only if $\delta =0$ will it be impossible for you to do so,²⁵ The probability function assigns the probability $\delta$ to $\lnot s$, and it also assigns fractions of $\delta$ to various propositions that imply $\lnot s$. For instance, the probability of h (“Sarah is at home”) may be $0.5\times \delta$. This means not only that $\lnot s$ is reinsertable, but also that there are good indications of what probabilities other propositions should have after it has been reinserted. Given that $\mathfrak {p}(h)=\mathfrak {p}(h \& \lnot s)=0.5\times \delta$, it would seem reasonable, when reinserting $\lnot s$, to assign the probability 0.5 to the proposition that Sarah is at home.

The second component of this framework is a modified representation of the set of full beliefs. Instead of identifying the full beliefs with the propositions that have probability 1, the full beliefs are represented by the propositions whose probability is either 1 or infinitesimally smaller than 1. These propositions form a logically closed set (Hansson, 2020). In our example, this means that s was a full belief in the belief state you entered when you heard Sarah’s voice and concluded that she was present at the party, and that $\lnot s$ became a full belief when you found out that you had only heard a recording of her voice, and she was not there. Thus, with the help of infinitesimal probabilities, we have constructed a probabilistic model in which full beliefs can be removed, and do not accumulate.

The third component of the framework is a somewhat non-standard operation for revision by an input sentence. In the above example, revision by s was represented by the assignment of a probability $1-\delta$, rather than 1, to s. This means that standard Bayesian update cannot be used. It replaces the prior probability function $\mathfrak {p}$ by a new probability function $\mathfrak {p}'$ such that $\mathfrak {p}'(x)= \mathfrak {p} (x \& s)/ \mathfrak {p}(s)$ for all x, and consequently $\mathfrak {p}'(s)=1$, which is exactly what we have to avoid. Instead, we can use Jeffrey conditionalization (Jeffrey, 1983, pp. 171–172):

$$ \begin{aligned} \mathfrak {p}'(x)=(1-\delta )\times \dfrac{\mathfrak {p}(s \& x)}{\mathfrak {p}(s)} +\delta \times \dfrac{\mathfrak {p}(\lnot s \& x)}{\mathfrak {p}(\lnot s)} \end{aligned}$$

(9)

This means that revision by a sentence s can be performed in different ways, depending on the choice of the infinitesimal $\delta$. A smaller $\delta$ can make sentences implying $\lnot s$ less available in later revisions.

Finally, the fourth component is a restricted perspective that can be applied to a probability function, namely the perspective that only considers its set of full beliefs. We can think of each probability function $\mathfrak {p}$ as an iceberg, with the set of full beliefs as its tip. If we study only the changes on the tip of the iceberg, then what we see is a model of dichotomous belief change. It turns out, perhaps surprisingly, that what we will then see coincides (for logically consistent inputs) with what is usually regarded as the gold standard of dichotomous belief change, namely the AGM operation of belief revision (Alchourrón et al., 1985). In other words, an operation of dichotomous belief change on a set of full beliefs is an AGM operation if and only if there is a hyperreal probability function that has this operation as the “tip of the iceberg”. (For a proof, see Hansson, 2023a.)²⁶

For an illustration of how this model solves the amnesia problem, consider an agent whose belief state is first changed to accommodate the information that s holds, and then changed a second time to accommodate the information that $\lnot s$ holds.²⁷ In our representation of the first step, the original probability function $\mathfrak {p}$ is replaced by a new probability function $\mathfrak {p}'$ such that for all propositions x, for some positive infinitesimal $\delta '$:

$$ \begin{aligned} \mathfrak {p}'(x)=(1-\delta ')\times \dfrac{\mathfrak {p}(s \& x)}{\mathfrak {p}(s)} +\delta '\times \dfrac{\mathfrak {p}(\lnot s \& x)}{\mathfrak {p}(\lnot s)} \end{aligned}$$

(10)

Similarly, in the second step we obtain a new probability function $\mathfrak {p}''$ such that for all propositions x, for some positive infinitesimal $\delta ''$:

$$ \begin{aligned} \mathfrak {p}''(x)=(1-\delta '')\times \dfrac{\mathfrak {p}'(\lnot s \& x)}{\mathfrak {p}'(\lnot s)} +\delta ''\times \dfrac{\mathfrak {p}'( s \& x)}{\mathfrak {p}'( s)} \end{aligned}$$

(11)

It follows directly that $\mathfrak {p}''(\lnot s)=1-\delta ''$.

Finally, let us compare this model with the ten desiderata.²⁸Both belief types, mutual exclusivity, contingent full beliefs, non-contextuality, diatomic conjunctive closure (and logical closure), consistency, iterability and bidirectional transfer are all satisfied. The same applies to non-functionality, since propositions with high probability are not automatically included among the full beliefs. The formation of a full belief in a contingent proposition requires a separate operation, namely a Jeffrey revision that assigns to it a probability at most infinitesimally smaller than 1. Finally, convincing evidence requires that such an operation, making a proposition fully believed, is only performed if the previous probability function assigned a high probability to it. This is a requirement on what inputs are received by the model. Since the construction and selection of inputs is external to the model, this cannot be guaranteed by the model itself. However, this requirement will be satisfied if the model is used as intended.

6 Conclusions

Our conclusions are summarized in Table 1. All but one of the ten criteria outlined in Sect. 3 are violated in some of the studied models and satisfied in others. This confirms the descriptive relevance of these criteria. All the eight types of formal models discussed in Sect. 4 violate at least one of the criteria. In particular, all of them violate at least one of both belief types, mutual exclusivity, iterability and bidirectional transfer. Each of these is important for reasons explained in Sect. 1. Both belief types requires that both empirical full beliefs and graded uncertain empirical beliefs are represented in the model. Mutual exclusivity requires that we do not at the same time treat a proposition as a full belief and as having a non-unit probability. This is essential to ensure that full beliefs fill their function to reduce the cognitive load. Iterability requires that the model represents our ability to change beliefs more than once, and bidirectional transfer that changes can go in both directions between full and uncertain (probabilistic) beliefs.

Two of these models violate diatomic conjunctive closure. The others satisfy not only diatomic conjunctive closure, but also the much stronger and less plausible postulate of logical closure. This means that they have all the logical resources needed to represent inferences that humans can draw from empirical premisses. It also means that they can attribute superfluous logico-mathematical knowledge to epistemic agents, giving rise to the problem of logical omniscience discussed in Sect. 2.

We have also shown that it is possible for all ten desiderata to be satisfied in one and the same model, namely that outlined in Sect. 5.

The relevance and importance of these results depend on the realism and the normative adequacy of the informal account of human belief systems presented in Sect. 1. Some of the models discussed in Sect. 4 were constructed to represent a different picture of human belief systems, and they should of course also be assessed in relation to that picture. Hopefully, we can all agree that the development of formal models in epistemology requires close attention to the relationships between these models and actual human belief systems.

Table 1 The extent to which the investigated types of formal epistemic models satisfy the ten desiderata