A Theory of Friction (For Bayesian Epistemology)

I. Here I expound and defend probabilism (a.k.a., Bayesianism) in its capacity as an epistemological theory. First, I outline what is meant by probabilism and in what sense it is an epistemological theory. Following this, I outline what I take to be a substantial obstacle the theory faces to success: various problems of 'subjectivity.' Next, I reply to these associated objections and outline why I think that not only is this concern not a worry for Bayesians, but that probabilist epistemology gives us reason to be optimistic about the state of knowledge in the world. Finally, I argue that that the internalism about rationality this account implies is not a bad thing. In an important way, I think this can be read as a partner piece to my previous post "The Tendency to Know", where my 'theory of friction' in this essay is basically my theory of the state of knowledge in a system applied to what is considered a serious problem for a Bayesian epistemology.

II. This section outlines probabilism as a theory of probability and the formal aspects of a probabilist epistemology.

Probabilism is a subjective theory of probability. On this interpretation, probabilities are not a measure of something ‘in the world,’ but rather a measure of the confidence, degree of belief, or uncertainty – however you want to slice it – of the agent at a given time. This confidence is measured on a scale from 0 to 1, from impossibility to certainty. For any given event, the agent assigns it a probability between these numbers based on how likely they think it will occur. This includes paradigmatic cases of situations in which we think we understand the objective ‘in-the-world’ probability of something. For example, if I flip a coin, it does not have a 0.5 probability of landing on heads (like we would typically think); it is just that my confidence that it will land on heads is 0.5. This causes some problems for the account that I will go over later, but for now, it is just important to note how radically internal this account is to the agent. The benefit of the account is that it avoids all the pitfalls of having to formulate an objective theory of probability, which is highly metaphysically elusive. 

How likely we think something will occur, traditional probabilists often suppose, can be measured roughly by our willingness to act on some belief, given a (probabilistically consistent) appraisal of the expected utility of the situation.[1] For example, I would happily bet $1 on a coin to land on heads if it were paying out $10 because I know that there is a good chance it would benefit me much more than I would lose by betting on it.[2] Conversely, someone who erroneously thought it impossible that coins could land on heads would assign the probability to 0 and would thus act on their belief by never taking such a bet. The strength of some belief is measured by the strength of one’s disposition to act in any given situation. I have no room to defend this claim here, but it is a relatively intuitive notion, especially in the case of expected utility (broadly construed, not just in terms of bets, because money’s value scales non-linearly for most people).

The probabilistic theory can be extended to be a complete epistemological theory. A theory of probabilistic epistemology says basically three things:

(1) All beliefs are partial or uncertain beliefs, where the extent to which we think each belief is true is measured by the relative confidence we assign to it.

(2) All beliefs ought to be consistent with the laws of probability as a coherence constraint on belief formation.

(3) All beliefs, when faced with new evidence, ought to be updated according to conditionalisation, a rule of probabilistic inference.

Per (1), probabilists disagree whether all beliefs should be subjected to the probabilistic treatment of being treated merely as probably true rather than absolutely true. Some probabilists think we can rationally accept certain statements as absolutely true, according to special principles (such as high confidence, a priori, and analytic statements). In contrast, others think the notion of belief must be given up entirely in favour of degrees of belief. However, nothing in the present essay will hinge on this disagreement for now. It should be clear that (1) is merely an extension of the logic of interpreting probability subjectively, to beliefs in general. Thus, I need to explain (2) and (3).

That all beliefs ought to be consistent with the laws of probability is the second aspect of a probabilistic epistemology. It is a view about how we should arrange our current set of partial beliefs. Since we are now taking our beliefs to be probabilistic, the argument goes, we ought to make them consistent in the same way we demand probabilities be consistent. Thus, beliefs should be consistent with the laws of probability. The most important probabilistic law is, firstly, that all probabilities are between zero and one (which corresponds to our confidence). Secondly, that logical truths have a probability of 1. Thirdly, that the probability of any two mutually exclusive event types, P and Q, is the sum of their individual probabilities. Finally, from this, that the probability of a sentence and its negation sum to 1 (they are mutually exhaustive).

That all beliefs ought to be ‘updated’ according to conditionalisation is the final aspect of a probabilistic epistemology. This is a rule of probabilistic inference that a probabilist thinks we ought to accept. It is a view about how to rationally change one’s beliefs in light of new evidence. Thus one is rational according to the probabilist only insofar as their beliefs are consistent (per (1)) and they conditionalise them properly. Once again, since we are taking our degrees of belief to be probabilistic, the argument goes, we ought to treat belief revision as we treat conditional probabilities. Conditionalization is the process of taking one’s current beliefs and ‘updating’ their assigned probabilities to values that are ‘conditional’ on (that is, take into account) the new evidence. Thus, if some belief Q is assigned a probability of P(Q), then the conditional probability of Q given new evidence E will be P(Q/E) where P(Q/E) = P(Q&E)/P(E). A new value will be given by ‘conditionalising’ belief Q, which, according to a probabilist epistemology, ought to be the new value assigned to Q. I Follow with an example of this.

Suppose you believe Q: “in flipping this coin, it will land on heads” with a confidence of 0.5. However, you find out that the coin you are using is somehow weighted to always land on heads, and you are certain of this new evidence E. You would update your initial credence for your belief by conditionalising on this new information to reach a new probability assignment of Q. It would go as follows: P(Q/E) = P(Q&E)/P(E) = 0.5/0.5 = 1. Thus, we have confirmed Q, and rationally, we ought to be certain of it.

Of course, this is a highly artificial example, and it is not at all clear that we would ever be in a position of certainty regarding new evidence – that is, to assign it a probability of 1. It is more likely that we would be at least partially sceptical of such evidence. For example, news outlets are often wrong or biased, and we ascribe a certain attenuated level of confidence to its proclamations. Alternatively, in scientific discovery, theories thought to be certain are often overturned. Thus, this has led some to factor this uncertainty about the evidence into the conditionalisation process by assigning confidence values to the evidence received and calculating the conditional probability of any statement that includes the possibility of the relevant evidence turning out to be false. Taking this route means that nothing can be logically certain and thus mirrors the debate between those that want to keep fully-fledged beliefs versus those that want to dispense with them. These differences do not matter for the content of this essay, especially here in the expository part. It only needs to be noted that all probabilists think that conditionalisation in most contexts is the process that ought to govern the way we revise our beliefs if we are to be rational. The specific constraints one puts on this process will be important in determining which problems each probabilist theory will face, but only contingently to the problem I will be concerned with.

A probabilist epistemology says three things. Firstly, at least some (or all) of our beliefs are partial beliefs and are thus probabilistic. Secondly, those partial beliefs ought to be formed according to the laws of probability. Finally, rational agents ought to update their beliefs by conditionalising their current beliefs in light of new ones.

III. There are many potential problems with a probabilistic epistemology. For example, postulate (3) entails that a completely rational agent would have to conditionalise every single belief in light of any arbitrary new evidence, which could be as banal as seeing something. Since probabilists also think that if we are to be rational, we must follow (3), it seems to follow that we ought to conditionalise every single belief in light of every bit of evidence. We do not need to test anything to know that this is not something any human could possibly undertake, and if ought implies can, then (3) must be false. Prima facie, this is a pretty strong argument with plausible premises. I think this is the biggest problem with the theory (and not one I can dispute here), or more generally that it seems to entail in inhuman task to actually follow these rules. Perhaps one could plausibly weaken the norm for rational agents only to include the need for revising those beliefs possible or relevant to one’s situation, the stronger norm remaining an ideal.

Another problem is in justifying the norms (2) and (3). This is often done by appealing to ‘Dutch Book Arguments’, which show that we have good prudential or pragmatic justification for conforming our beliefs to the probability calculus (because otherwise we could always be ripped off). They purport to show that if we fail to do this, we will make decisions that seem obviously irrational and lead us to ruin. As far as norms go, these seem as rationally acceptable as any standard norms or rules of inference we find in classical logic prescribing consistency and good inference. Thus, I am happy to accept them prima facie. One might worry that the prudential justification is not enough, but I do not, at least as far as developing a deductive logic for partial beliefs go.

Finally, a third problem could be that the acceptance of a probabilistic epistemology and the consequent idea that it is possible that nothing being certain jeopardises the status of the full belief of anything other than the norms supposed to govern probabilism. This would mean giving up beliefs entirely in favour of partial beliefs, which is quite the price to pay. This is also a bullet I can bite, if I were forced to take this position, and no hybrid formulations worked.

What I am more concerned about here is, if you accept probabilism, by its own lights, can it do the work we need a theory of knowledge to do when it comes to knowing things about the world. This problem will take up the rest of the essay.

IV. There is a slew of problems with the probabilist theory of knowledge that I am worried about, which I would call problems of subjectivity. They include the problem of priors and the problem of inductive content. This section goes over these in turn, followed by a brief discussion of convergence.

The former refers to the problem that probabilism offers no rational constraint on adopting one’s initial beliefs (before conditionalisation) beyond them coherently adhering to the probability calculus. The choice of priors is subjective (in that one can choose whatever they want). This means that any old arbitrary beliefs can be taken up and taken up rationally. For example, someone could (seemingly) erroneously believe that a coin has a 0.8 chance of landing on heads. This seems false, but probabilism, as described so far, says nothing about this as long as they also think the chance of it landing on tails is 0.2 (as long the beliefs are probabilistically coherent). This is a problem for the epistemological theory because an epistemological theory ought not only to tell us how to be rational but how this process can let us know things about the world. Thus, if rational adherence to the tenets of probabilism can produce beliefs that are systematically false and decohered across multiple (if not all) agents, this is a problem for the theory.

The latter problem of inductive content follows from the previous problem. Probabilistic epistemology purports itself to be a formalisation of inductive logic. Thus it purports to show how to confirm theories and how to make true (or at least accurate) predictions about the world. However, as we have just noted, the values assigned to any given belief cannot just be plugged into the agent directly by the world. The agent has to come up with them – and they could be arbitrary. We have a plausible deductive formalism for organising and updating our partial beliefs in conditionalisation, but that does not logically guarantee the inductive efficacy of the content arrived at by following such formalism. We could be perfectly rational according to probabilism, and yet our belief scheme would be, in McDowell’s memorable phrase, “a frictionless spinning in the void.”[4] Thus the theory, in order to be successful, needs to secure some friction on the world.

Both of these concerns are also concerns that rational Bayesian agents would not converge both with other people and on truths about the world. This is because initial probabilities would differ in a way that no possible conditionalisation is ever likely to yield overlapping posterior beliefs, to levels rationality would seem to require. We seem to need such convergence to communicate, co-operate, and ultimately justify our beliefs as convergence, at least in some cases and over the long term, is one of our best indications of truth.

There are broadly two approaches to these problems. The first can be called ‘subjective probabilism’ while the second can be called ‘objective probabilism.’ For my thesis, the specifics of the individual theories developed by philosophers along the continuum these poles constitute will not matter. I will just outline the limit cases of each to give one an idea of the conceptual space.

The maximal subjectivist would think there are no rational constraints on prior probabilities beyond the probability calculus itself, and prior probability assignments are entirely due to ‘irrational’ accidents such as culture, genes, or luck. Thus, the subjectivist bites the bullet on the possibility of no convergence between agents, the truth, and the world. The maximal objectivist, on the other hand, would think a uniquely rational and objective prior probability is determined in every case by some a priori principle. This solves the problem of convergence and the other problems of subjectivity by showing that insofar as the agent strays from the objectively correct probability assignments, they are irrational for doing so. Thus, on this picture, convergence is, by definition, rational. Rational divergence between agents is simply not possible – truth and reality are baked in. The former theory is easy to accept and specifies a minimal number of norms to follow and defend but sacrifices possible friction, while the other specifies an impossibly complex and unreachable number of norms to defend but has perfect friction.

Without getting into the weeds about what kind of rational constraints we can impose on adopting initial beliefs, my contribution is that I will argue that we have good reason to think that even a maximally subjective interpretation of a probabilistic epistemology would not be saddled with an objectionable level of subjectivity or arbitrary belief. I argue that the most pernicious possible problems of subjectivity simply cannot occur in normal humans and that most people’s beliefs will not only converge but also be true. And securing this only requires a few minimal assumptions. Thus, I attempt to vindicate the most vulgar form of subjective probabilism by explaining away the problems without simply biting the bullet (as our theoretical maximal subjectivist would).


V. In this section, I argue that even on a maximally subjective interpretation of probabilistic epistemology, mere coherence being satisfied is enough to guarantee both friction and convergence.[5] It is my theory of friction. I give three arguments for this claim. The first argument attempts to ameliorate the problem of priors and convergence, while the latter two attempt to ameliorate the problem of inductive content or friction.

The first is that just by virtue of being human and having to refer our beliefs through the instruments of our similarly constituted bodies (eyes to see with, hands to touch with), the range of possible initial beliefs of a typical[6] agent will be heavily constrained. The sceptic of subjective convergence rightly points out that there is an infinite number of possible coherent sets of beliefs that have no purchase on the world, but they are wrong to think that an agent could possibly hold one of them. What I am pointing out here is that the average constitution of an agent, whatever we are made up of, and barring extreme outliers, is such that the possible coherent sets of beliefs an agent could plausibly come to given their constitution is actually very low. That is, we have good reason to think that most people share most beliefs, most of the time.

This will sound incredulous if the beliefs you have in mind are political, aesthetic, or in some otherwise contentious domain, in which convergence does not seem ever to happen. But what I have in mind is the most basic of our beliefs – the beliefs we take for granted every day at almost every moment of our lives. These include, but are not limited to, motor-navigational beliefs and the shared conceptual structure of everyday experience. For example, when we are walking across the road, everyone (implicitly) assigns an extremely high probability that they will not fall through the ground or that we cannot walk through solid walls. Also, we all conceptualise the world as one carved up into objects to be interacted with, and we have a genuinely strong convergence about how we do this carving.

Even if other cultures have finer-grained distinctions for differentiating snow, we all know what is meant by snow in its generic sense, we all see it, and we all know what it would mean for “snow is white” to be true (given we understand the person is saying that snow is white). Indeed, returning to the erroneous coin, it is hard even to imagine what one would have to think about the world to rationally come to the belief that a coin has an 0.8 chance of landing on heads. They would have to have been through incredibly abnormal human circumstances for their priors to coherently believe such a proposition; it simply does not happen. Once you let yourself start thinking about beliefs of this kind, it is not hard to see that there is a deep well of firmly held and (almost) universal (barring some cases of extreme mental abnormality) human beliefs. In the most basic sense: we live in the same world, jointly constituted by these basic beliefs.

With regards to a probabilistic epistemology, this means that our human substrate guarantees convergence on a great many initial probability assignments. We cannot simply choose to believe that we can fly by flapping our arms or that setting ourselves on fire is a good long-term strategy for pleasure. Our presence in the world causes all of us to think otherwise. Humans simply cannot adopt priors that are both systematically false as well as communally divergent and at the same time rational. When new evidence is introduced, the possible direction of rational conditionalisation to posterior beliefs will always be heavily mediated by this shared ground of basic beliefs. Therefore, if someone does have their beliefs adhere to the probability calculus, as probabilism would have you do, given this basic constraint on possible belief sets a human could come to, coherence of beliefs alone would predict that convergence is reached among most people. Priors cannot be plucked out of thin air.

Thus, the worry about convergence is at least partially gone. This is because no matter what happens, the basic features of the world are always guaranteed. After all, a certain number of basic beliefs are impossible not to believe at the same time. Note that this argument relies at no point on postulating any rational or objective norms. It just makes a minimal descriptive claim about our shared psychological understanding of the world. That means it is compatible with the most hardcore subjectivist account.

One could reply to this that, even though we have convergence of belief in these basic ways, that still does not guarantee that the things converged on are truths about the world and not merely some consistent set of prudent outlooks that help humans navigate reality. In short: even if convergence is guaranteed by the consistent organisation of our beliefs according to the probability calculus, the truth of those beliefs is not. This community of beliefs could be systematically being deceived, even though they were arrived at rationally according to probabilism. This is just another statement of the problem of inductive content. I give two arguments that we have good reason to think that if our beliefs adhere to the probability calculus, they have some inductive content. 

Firstly, I think the sceptical rejection of our shared basic belief structure being true cannot be rationally accepted. I can obviously offer no decisive refutation of global scepticism here, nor can anyone else. I can only point out the irrational conclusions this objection entails. This response entails that all our basic beliefs are false and that at no stage during inquiry, even as a community, do we ever course-correct in the right direction. One must ultimately think not only that those individual beliefs are false, but that the most basic propositions of human knowledge that we all seem to share are also false.

All I can offer by way of argument is to appeal to one’s priors regarding scepticism. What seems more likely to be true: that all our beliefs are systematically and universally false, or that some basic beliefs regarding our own experience, existence, and presence in the world are true? If one honestly asks themselves, very few people truly have more confidence in the former than the latter. Thus, the sceptical premise should simply be abandoned in favour of a presumption of truth regarding our basic beliefs. If this answer is unsatisfactory because one’s condition for having any purchase on the world whatsoever is that the belief could not conceivably turn out to be false, it will be so for every other theory of knowledge, not just probabilism. Thus, qua epistemological theory, this is not a special concern for probabilism, lest we give up knowledge entirely. We have good reason to think, therefore, that our basic beliefs do have some inductive content.

Secondly, I think our distinctive form of collective sociality as humans indicates not only that our basic beliefs have some inductive content, but some of our best theories, rationally arrived at, do too. While our human constitution as individuals offers a restricted platter of initial probability assignments, our collective position as social creatures restricts the direction of conditionalisation by imposing communal standards on what is to be counted as worthy of belief.  As humans, we care about not only what is true individually, but what others think is true. As Peirce writes, in seeing that “men think differently from him…it will be apt to occur to him, in some saner moment, that their opinions are quite as good as his own, and this will shake his confidence in his belief.”[7]

We are constantly conditionalising our beliefs socially. Firstly on the fact that others believe different things to us and seem to do so rationally, and secondly on the alternative beliefs themselves seeming worthy of consideration. For example, take someone with a strongly held political belief but very little exposure to alternative points of view or to what is possible in public policy and in politics in general. They will start out relatively dogmatic on the issue. However, have one of their friends whom they trust (that is, assign a high probability to their evidence generally) earnestly share an alternative and discuss the workings of government with him, and he will be confronted with the arbitrariness of his own views. Consequently, he will conditionalise on his beliefs substantially. Even if this even-handed correction does not take place, one that cares enough might instead attempt to prove their own beliefs with more certainty in response to opposition and, in doing so, bring more true information and thus worldly friction to all belief sets. In fulfilling our impulse to know and to be correct about these things, we are cumulatively performing error correction on a given body of knowledge within a given social system. This arguably describes the process of an adequately systematised (though certainly idealised, but not alien) science. Ultimately, this process lifts our explanatory theories from the blind contingency of subjective choice and into the realm of socially tempered objective explanations of reality. It does this because, ultimately, we care, as humans, not merely about what seems true to us in the moment but what others think and especially what people ought to think, given the evidence (and we are happy to tell them so).

Part of this process involves enforcing the kind of consistency demanded by probabilism. We find it exceptionally psychologically dissatisfying to entertain contradictions within theories. But, more importantly, we find the same dissatisfaction with opposing candidate theories of particular phenomena. Thus, we try to stamp those contradictions out by seeking out confirmatory evidence and thus the required friction. In sum, the conditions of sociality make us care about consistency, completion, and the reality of our knowledge about the world and “unless we make ourselves hermits, we shall necessarily influence other’s opinions” in this way. My contention here is that the mere facts of sociality with basic beliefs as initial conditions, even absent some substantial objective norm of rationality, is enough to guarantee that rational agents following the formal tenets of the most austere subjective probabilism will run up against the world.

Thus, to sum, with the minimal assumptions that global scepticism is not true, that we share many basic beliefs, and that these beliefs have some purchase on the world, any rational conditionalisation of one’s beliefs will always be heavily constrained as to guarantee convergence. Also, these basic beliefs, which we cannot help but have about reality, are not merely subjective, but about the world, and largely true. Furthermore, standard conditions of social interaction give us reason to think that even our more tenuous and theoretical partial beliefs will have some predictive efficacy. Therefore, against the problems of subjectivity outlined above, we have good reason to think that any person with a (mostly) coherent set of beliefs (that is, adheres to the laws of probability) will have a set of beliefs that are also true to the facts (and not merely convergent with others), regardless of the subjective starting point. Probabilism, constrained by human nature, will always have inductive content.


VI. To end, one might still be worried about those seemingly insoluble difficulties that arise from disagreements about politics, public policy, or morality. These things do not constitute part of our basic beliefs and also do not seem to be captured entirely by communal belief-forming practices. They are the primary source of frustration when it comes to appraising the rationality of others. When people fundamentally disagree with us politically (we are confident their beliefs are false), probabilism gives us no grounds for repudiating such beliefs (because they are still deemed rational). I do not think this is a problem, and argue that the kind of 'internalism' about rationality entailed by this account is good, actually.

Someone arriving at false beliefs through rational processes does not mean you cannot think they are wrong or that some kind of relativism is true. It just means that you disagree with their priors. Cases in which people are wrong yet rational are not only conceivable but ubiquitous. For example, someone stuck and tied up in a cave looking at shadows racing across the wall for all their living memory could only rationally conclude so much (maybe they figure out transcendental idealism). Similarly, someone in North Korea could only conclude so much about the wider world from their position of restricted information. Just because they are wrong that the cave or the party is all there is to reality does not mean we should not deem them rational. In cases such as these, the probabilist account of rationality not requiring truth and being saddled with the problems of subjectivity is a feature, not a bug of the theory. If this were not the case, we would have to think that every scientist and philosopher in history that we think was wrong (so most of them) were utterly irrational, and this is obviously false on any cursory reading of the history of ideas.

Thus, there is an acceptable level of divergence in individual beliefs that we should all accept. Probabilism is able to capture what that is perfectly: consistency. Rational processes can and should sometimes lead to beliefs that do not turn out to be true (indeed, this is what a prediction or betting often is). There should be a reasonable range of pluralism about what is rational to believe, not necessarily constrained by the truth because that excludes almost everyone. On top of this, it seems that there is some utility in having different intepretations of things, in that if certain points of view are falsified (or discredited, to be more precise), rational thought discovers and rules out more directions of empirical enquiry than a homogenous community of inquirers would. It covers more ground. Ultimately, people never ought to have perfectly convergent priors determinable by a priori laws, because there is no such thing. When different conclusions are reached about the interpretation of quantum mechanics, there is no one rational outcome: the correct one, where the others are irrational fools. There are individually rationally appropriate conclusions to draw based on the evidence presented, plus one’s subjective priors about how to weigh what is most important. This is simply postulate (2) and (3) of probabilism.

The fact that there is not perfect convergence of rationality to reality is not remotely a point against probabilism as a theory. As I have indicated, this does not result in a frictionless spinning in the void, or epistemological anarchy, because it is simply not possible that it could. Following the coherence constraints imposed on us by probabilism by itself gives us good reason to think, at least as a community in standard cases, that we are getting some true purchase on the world, that following its rules points us in the right direction, and finally that it leaves open a reasonable level of rational disagreement. Probabilism is not only strong as an epistemological theory; I think we have reasonable grounds to think that one of its biggest obstacles to success, the associated problems of subjectivity, are surmountable, even on the most austere reading of it’s norms.

This does not put aside the problem that probabilism must purport to govern and ultimately describe the actual reasoning processes rational agents must follow, when it is a highly idealised model that we could hardly even begin to follow formally. But it is a good start and gives a compelling (and I think convincing) model of how we should think.

Footnotes

[1] Jeffrey, Richard. Probability and the Art of Judgment. Cambridge: Cambridge University Press, 1992. 30-31.

[2] The expected utility of this bet would be as follows: (10 x 0.5) + (-1 x 0.5) = 4.5

[4] McDowell, John. Mind and World. London: Harvard University Press, 1996. 11.

[5] This strategy is somewhat similar to (and inspired by) Davidson’s in “A Coherence Theory of Truth and Knowledge” (2001), although he could be saying anything. It is closer in my mind to being inspired by Peirce’s “The Fixation of Belief” (1993).

[6] "Typical" can be read pretty widely and I still think this works.

[7] Peirce, Charles Sanders. “The Fixation of Belief.” In The Essential Peirce, Volume 1: Selected Philosophical Writings (1867-1893), 109-123. Bloomington: Indian University Press, 1993. 116.