Oaksford and Chater challenge the logicist conception of human rationality. In its place, they advocate Bayesian rationality, which offers a framework for reasoning in the face of uncertainty. Bayesian rationality involves probabilistic reasoning. Here, probability describes an agent’s degrees of belief and is thus considered qualitative/subjective, not numerical.

 

1. Logic and the Western conception of mind

Oaksford and Chater describe an early approach to rationality, known as the logicist conception of the mind, according to which inferential relations maintain absolute certainty. Oaksford and Chater proceed to use syllogisms in demonstrating that logical arguments are truth preserving: To believe the premises of a logical argument is to believe its conclusion. So, denial of that conclusion is incoherent. O&C note that logical certainty prevents the addition of contingent facts (70). They then introduce two contemporary theoretical accounts of human reasoning—the mental logic and mental models theories. The mental logic view assumes reasoning involves logical calculations over symbolic representations; the mental models view takes reasoning to involve concrete representation of situations (71). In concluding this section, O&C introduce Bayesian rationality as the approach that best deals with the discovery of theory-refuting data.
2. Rationality and rational analysis

This section seeks to demonstrate why a Bayesian perspective is better than a logical one. O&C begin by outlining the six steps of rational analysis (71–72), which includes normative theory in its larger account of empirical data concerning thought and behavior. Rational analysis aims to understand the structure of the problems facing the cognitive system and takes into account the relevant environmental and processing constraints. O&C address two caveats of Bayesian rationality—that rational analysis isn’t a theory of psychological processes and that it doesn’t measure performance on probabilistic or logical tasks (72). The authors assure us, though, that these caveats are not to be of any inconvenience.

 

3. Reasoning in the real world: How much deduction is there?

This section challenges the use of the logic calculus in everyday reasoning. In reasoning about the everyday world, we usually have only bits of knowledge, some of which we believe only partially and/or temporarily. Moreover, the non-monotonicity of commonsense reasoning means that we can overturn virtually any conclusion upon learning additional information (72). Non-monotonic inferences cannot be accounted for by conventional logic. Importantly, classical logic fails to deal with the notorious “frame problem,” which refers to the difficulty of representing the effects of an action without having to enumerate obvious “non-effects” (73). BR, by creating non-monotonic logics, thus addresses this “mismatch” between logic-based and commonsense reasoning.

 

4. The probabilistic turn

O&C offer probabilism as an approach that best deals with the uncertainty of everyday reasoning. The authors provide the example of court cases decided by jury. In these situations, new pieces of evidence can modify one’s degree of belief regarding the guilt of the defendant. Here, probability is determined subjectively (74). In the remaining paragraphs, O&C present a modified version of the familiar conditional If A then B. This version, embraced within the cognitive sciences, takes B to be probable, if A is true (74). The authors conclude this section by noting the shift towards probability theory across a number of domains, including philosophy of science, AI, and cognitive psychology.

 

5. Does the exception prove the prove the rule? How people reason with conditionals

This section deals with the first of three core areas of human reasoning—conditional inference. The authors identify four conditional inference patterns (Fig. 1): (1) Modus Ponens, (2) Modus Tollens, (3) Denying the Antecedent, and (4) Affirming the Consequent. Of the four, two are logical fallacies—Denying the Antecedent and Affirming the Consequent.

Figure 2, Panel A presents data from experiments that asked people if they endorse each of the four inference patterns. The observed results diverge from the predictions of the standard logical model. Logicists attempt to account for the divergence by allowing people the pragmatic biconditional interpretation, though it is logically invalid (75).

The Bayesian approach, however, appeals only to probability theory. The probabilistic account of conditional inference entails four key ideas (75):

  1. P(if p then q) = P(q|p), aka “The Equation;”
  2. Probabilities are interpreted as degrees of belief, and this allows for belief updating;
  3. The Ramsey Test determines conditional probabilities; and
  4. By conditionalization (i.e., when the categorical premise is certain, not supposed), our new degree of belief in q is equal to our prior degree of belief in pq. Quantitatively: If P0(q|p) = 0.9, and P1(p) = 1, then P1(q) = 0.9. The takeaway here is that, from a probabilistic account, we are able to update our degrees of belief in q upon learning p is true without making too strong of a claim.

The remainder of this section deals with biases observed in conditional inference. The first of these are the inferential asymmetries (that MP is more widely endorsed than MT and that AC is more widely endorsed than DA). Though the probabilistic account can explain inferential asymmetries without invoking pragmatic inferences or cognitive limitations, it distorts the magnitudes of the MP–MT and DA–AC asymmetries (Panel C). Learning that the categorical premise is true can alter one’s degree of belief in the conditional, and this constitutes a violation of the rigidity condition. This violation lowers our degree of belief in P0(q|p), and this lower estimate, when included in calculations of probabilities for DA, AC, and MT, in turn explains the resulting magnitudes of the asymmetries (76).

The second of these biases is the negative conclusion bias—that people endorse DA, AC, and MT more often when the conclusion contains a negation. Since the probability of an object being red, for instance, is lower than the probability of it not being red, P0(p) and P0(q) take on higher values when p or q is negated. So, a seemingly irrational negative conclusion is simply attributed to a “high probability conclusion effect” (77).

The authors conclude the section with an overview of a small-scale implementation of the Ramsey test and question whether future implementations can explain the full range of empirical observations in conditional inference.

 

6. Being economical with the evidence: Collecting data and testing hypotheses

This section deals with the second of three core areas of human reasoning—data selection. Recall the Wason selection task. In testing the hypothesis if there is an A on one side of a card, there there is a 2 on the other, one should seek out falsifying examples (i.e., p, not q cases). Accordingly, one should select the A and 7 cards. This does not seem to be the case, however: Participants more often select cases that confirm the conditional (confirmation bias) (77).

Bayesian hypothesis testing is comparative—not falsifying. The optimal data selection (ODS) model assumes that people compare the dependence hypothesis (HD)—that P(q|p) is higher than the base rate of q—with an independence hypothesis (HI), according to which P(q|p) is the same as the base rate of q (78). Initially, people are thought to be equally uncertain about which hypothesis is true. The goal of the selection task, then, is to reduce this uncertainty. Using Bayes’ theorem (see Note 2), one can calculate her new degree of uncertainty about HD upon discovering p→q.

The ODS model is based on information gain, and participants in Wason’s task base their decisions on expected information gain. The ODS model also assumes the rarity of the properties that belong to the antecedent and consequent. So, the expected informativeness of the q card is greater than that of the not q card since almost certainly we would learn nothing about our hypothesis if we investigated not q cases. This approach is at odds with the falsification perspective but agrees with the empirical data (78). The ODS model thus suggests that performance on Wason’s task is in fact consistent with rational hypothesis testing behavior.

The authors also address the apparently non-rational matching bias, as a result of which participants match values named in the conditional. Given the rule if A then not 2, people tend to make the falsifying response: They select the A and 2 cards (rather than the 2 and 7 cards). Because of the rarity assumption, however, not q is a high probability category, and a high probability consequent thus warrants the falsifying response (79).

In the remainder of the section, the authors discuss deontic selection tasks (which involve conditionals that express rules of conduct, not facts about the world). In such instances, people do select the “logical” cards (the p and not q cards). Here, it is not a hypothesis that is tested but a regulation; it is useless to confirm or disconfirm how people should act. Rather, participants seek out violators of the rule (79). Moreover, in such selection tasks, people select cards to maximize expected utility, and because only the p and not q cards have positive utilities, these are the cards chosen (80). This model has also been extended to rules with emotional content.

 

7. An uncertain quantity: How people reason with syllogisms

This section deals with the last of three core areas of human reasoning—syllogistic inference, which relates two quantified premises, of which there are four types: all, some, somenot, and none. Of the 64 possible syllogisms, 22 are logically valid (Table 1).

The Probabilistic Heuristics Model (PHM) employs the probabilistic approach to syllogisms. PHM’s most important feature is that it also applies to generalized quantifiers, like most and few (82). PHM assigns probabilistic meanings to terms of quantified statements (81). For instance, the meaning of the universally quantified statement All P are Q can be given as P(Q|P) = 1. Similarly, the generally quantified statement Most P are Q can be understood as 0.8 < P(Q|P) < 1, for instance.

These interpretations are then used to build simple dependency models of quantified premises, and these models can be parameterized to determine which inferences are probabilistically valid (81).

The PHM model also assumes that, in general, because the probabilistic problems encountered by the cognitive system are very complex, people employ simple and effective heuristics to reach “good enough” probabilistic solutions (83).

There are two background ideas to keep in mind regarding heuristics (81): (1) the informativeness of a quantified claim and (2) the probabilistic entailment between quantified statements. A claim is informative in proportion to how surprising (unlikely) it is. No P are Q, which is very likely to be true, is thus an uninformative statement; All P are Q is the most informative. Regarding the second idea, the quantifier All probabilistically entails (p-entails) Some; Some and Somenot are mutually p-entailing.

There are two types of heuristics for syllogistic reasoning (82)—the generate heuristics (produce candidate conclusions) and the test heuristics (evaluate the plausibility of candidate conclusions).

There are three generate heuristics: (G1) the min-heuristic, (G2) p-entailments, and (G3) the attachment-heuristic.

The two test heuristics are (T1) the max-heuristic and (T2) the some_not-heuristic. In general, where there is a probabilistically valid conclusion, these heuristics identify it successfully. The authors offer experimental data in support of this claim.

 

8. Conclusion

Taken together, the empirical data provided support O&C’s probabilistic approach to human rationality. In sum, the cognitive system is best understood as building qualitative probabilistic models of the uncertain world.

 

Questions:

  1. Does Bayesian rationality actually account for the uncertainty of everyday situations that logical methods ignore?
  2. Can logical rationality and Bayesian rationality exist in unison?
  3. If classical logic is indeed inadequate in its explanation of human rationality, what becomes of the normative/descriptive gap?
  4. How might a program like ECHO accommodate the Bayesian perspective, if at all?

11 thoughts on “Oaksford & Chater (2009), “Précis of Bayesian Rationality: The Probabilistic Approach to Human Reasoning” — Steven Medina & Deniz Bingul

  1. Oaksford and Chater (2009) state, “The core objective of rational analysis, then, is to understand the structure of the problem from the point of view of the cognitive system, that is to understand what problem the brain is attempting to solve” (72).

    Does this presuppose a transformative approach to epistemology?

    1. Actually, I’m most curios regarding the concept of “Monotonicity” – In other words, what distinguishes the “non-monotonicity” of everyday thinking from inductive arguments?

  2. I appreciate Oaksford & Chater’s discussion of human logic as non-monotonic, as it probes at a certain disquieting intuition that has frequently come up in class. Classical logic seems defensible: There are certain arguments that seem unquestionably valid, with conclusions that are impossible to deny if the premises are correct. However, we keep running into this issue of framing, of possibly infinite implied caveats. (Are we in barn county? Barn facade nation? Will the professor be inexplicably struck dead on the way to buy donuts? Do these unlikely counterexamples affect the validity of the argument itself?) Technically, I might argue that most of these are implied constraints on the premises, not the argument itself. Yet effectively it does make basically all classical syllogisms contingent at best. Even seemingly ironclad conclusions are ultimately open to revision after new information is gleaned. I appreciate that Bayesian rationality attempts to capture this element of reasoning.

    I believe that classical logic may be containable in a Bayesian system of rationality. To me, it seems that classical logic is best described as one of those useful heuristics that the authors admit people employ in their probability, finding a more easily calculable approximate solution for the sake of efficiency. In any syllogism, the premises or relation between them may have a probability very slightly less than one (even, for example, that all dogs are mortals and all mortals eventually die, so all dogs must eventually die; perhaps we will eventually find a zombie or vampire dog that would cause us to revise our certainty on all dogs dying). However, for our evolutionary purposes, it is most prudent to consider these things as certain. The chances of there being some caveat we have not considered is extremely small, and the mental calculations of inference are much simpler if we’re assuming a probability of 1 instead of 0.9999999…. So it seems to me that classical inference rules may simply be heuristics of probability calculus that treat things as certain once they are sufficiently close, simply for ease of computing.

    Finally, it does seem that Bayesian rationality should fit very well in some kind of computer model. It wouldn’t look exactly like ECHO, and I’m unsure what the programming details would be, but the basic machinery would seem to have strong similarities to ECHO. After all, ECHO was designed to calculate relations between propositions and allow the strength of those relations to change one another, ultimately settling into a state which best accommodates all the various activation levels. This is exactly what Bayesian rationality does with probabilities in revising beliefs.

  3. Oaksford & Chater’s proposed model of reasoning appeals to me, as well. With the inclusion of Bayesian rationality and probability, it seems that this model can account for the uncertainty of everyday life, if used properly and as proposed amongst the human population. I think that I would also agree that faults in human reasoning appear more in qualitative reasoning and that “it is perhaps not surprising that people are not good at explicit reasoning with probabilities – indeed, they fall into probabilistic fallacies just as readily as they fall into logical contradictions” (84). I’m not sure if it is humanly possible to take into account all possibilities and probabilities, or if maybe this is a subconscious process. If it is not a probability analysis that we are aware of, are we being rational according to Bayesian rationality? If at a higher level, is the mind capable of such complex rationality as O & C proposed, or do we see it on a spectrum of complexity?

    I was also concerned with the tests of rationality Oaksford and Chater used. The Ramsey test and Wason test, especially, seem to me to have certain faulst in testing human reasoning. Perhaps this is a lack of understanding on my part, but the Wason test and Ramsey test, especially, seem to draw out the expected answer from subjects. I like that the Ramsey test is more applicable to real life scenarios and is easier to understand, but it seems to me that it doesn’t leave room for alternative paths of reasoning, and is therefore an invalid test for Oaksford or Chater to use. We discussed the Wason test more thoroughly in class, and its applications seem more viable. Again, this could be my misunderstanding, but does anyone else have an issue with either of these tests?

  4. I think that Oaksford and Chater’s (2009) probabilistic model is an interesting approach to the problem of certainty in logic. I agree with Carly that certainty seems almost (if not entirely) impossible in a world about which we have such relatively limited knowledge; there is almost no situation in which the addition of more information could not change the outcome. As O&C say, “almost any conclusion can be overturned, if additional information is acquired” (72). So how do we deal with this? The probabilistic model seems to be a good approach, although I do have some misgivings. For example, wouldn’t it take a huge amount of time and computational power to calculate the probability of each outcome? By assigning numeric values to different p and qs, we complicate the process of how we reason considerably. Do the benefits of the improved model outweigh the additional cost?
    In response to question 2, I think that they can exist in unison. In fact, is that not exactly what O&C do? They take constructions of logic, such as if p then q, and assign different weights to different statements based on various factors (such as whether the less likely statement is negated, or how reliable the source of information is, etc). In this way, they seem to be combining probability with classical logic. Is this more beneficial than just a logical approach? I would argue that it is, because a purely logical approach leaves so much unanswered, and suggests that humans are largely irrational much of the time. But perhaps we are simply acting based on the limited information that we have in a rather practical way, based on the relative probabilities of various factors and outcomes?

  5. In their discussion of the earliest western conceptions of logic, our ability to determine which inferences are absolutely certain and which are uncertain, Oaksford and Chater discuss Aristotle’s proposal that the logical certainty of an argument depends purely on its structure, rather than conviction or confidence in the validity of facts upon which the argument is based (p. 70). They provide the example that we may be confident in the validity of the statement “all woman are mortal” because it logically follows from our belief that all living things are mortal, and women are living things. They then argue, however, that these considerations do not provide logical certainty because they are dependent on “contingent facts” which “are not themselves certain”. For me, this brings several questions to the table:

    If the logical soundness of an argument depends purely on its structure, does the validity of facts play no role? For example, if I argue that: 1. All humans have green skin. 2. My dog is a human. Conclusion: My dog has green skin. Structurally, this argument is sound, but would Aristotle or Oaksford and Chater take any issue with its obvious factual incorrectness?

    Secondly, Oaksford and Chater argue that even a scientifically indisputable claim such as “all women are mortal” is not logically certain without a logical system of syllogisms, as Aristotle proposed. This, they argue, is because facts by themselves are not invariably certain. Is there a standard for determining which factual “considerations” are reliable/certain and which are not? Does it matter which are certain and which are not, as long as the structure of the argument is sound?

  6. I would like to discuss a possible shortcoming of Oaksford and Chater’s (2009) probabilistic model. Consider how Oaksford and Chater discuss the nonmonotonicity of inferences: “if it is raining and I go outside, then I will get wet” is an uncertain inference because the rain might stop or you might take an umbrella (73). According to this logic, you could argue that an indefinite amount of conditions could refute any inference, including conditions of which we are unaware. Do you this probabilistic model could be used to contradict itself? Do you think the mental logic model is a stronger model because it assumes monotonicity?

    On a separate note, I really appreciate that Oaksford and Chater (2009) provide a realistic model of human reasoning that accounts for the inherent uncertainty in our everyday lives. I am left wondering: is this probabilistic model rooted in psychology or philosophy? If the answer is psychology, then does the mental logic model still outline how humans should aspire to reason? (this is similar to Steven and Deniz’s third question)

  7. It seems that what O&C are saying is that having absolute certainty is irrational in everyday reasoning because of the limited amount of knowledge available, thus “everyday reasoning seems to be more a matter of tentative conjecture, rather than water-tight argument” (72). However, this motivation for Bayesian rationality only seems to work if humans are only exposed to limited knowledge, but in a technology-driven world in which we are constantly bombarded with information, isn’t everyday reasoning evolving as a result of modernization to become a matter of water-tight arguments?

    Is certainty achievable? If certainty can be achievable as the level of knowledge acquired increases, how does Bayesian rationality change over the course of an individual’s development? I ask this because I’m assuming that a child’s reasoning in an uncertain world would be different from an adult’s reasoning, but it seems that O&C might argue that regardless of age and experience, humans are always doomed to uncertainty.

  8. I found O&C’s account of rationality very appealing, because it aims to take into account the lack of certainty that we face when making decisions in our everyday lives. As they presented their initial description of Bayesian Rationality, however, I had similar misgivings as Timmy. O&C state that “in the context of cognitive science, probability […] describes a reasoner’s degrees of belief” (69). Does BR allow, then, for any kind of normative model of rationality, if this subjective analysis of probability is so integral to so-called rational decision making? In a similar vein, can a person with false beliefs or a skewed perception of different probabilities still be rational? I believe that according to O&C they would be, which I find to be problematic.

    I think that question 2 that Steven and Deniz bring up at the end of their summary is an interesting one as well, concerning the “coexistence” of logical and bayesian rationality in human cognition. As I understand it, Aristotle’s logical syllogisms necessarily include certainty. For example, the classic “All men are people, all people are mortal -> all men are mortal” syllogism does not involve probabilistic statements, and comes to an irrefutable conclusion. I guess my responding question to Steven and Deniz would be: Do humans even have the capacity for logical rationality? If we can be absolutely of very few things (including our existence as entities undisturbed by evil demons), my first inclination is to say no.

    Finally, I want to address the applicability of BR to ECHO. In Thagard (1989), ECHO is presented as being able to take the weights of different pieces of information into account when making decisions (440). Thagard describes this weighting process as a numerical account of reliability, which seems comparable to our own heuristic assessments of probability when making decisions. For us, however, these assessments are not programmed into us by an external force (as is the case with ECHO), but are informed by our own beliefs. BR, it would seem, is only as reliable as our previously held beliefs. Can true rationality really exist is this context?

  9. Throughout this paper, I was really drawn to the idea that it is hard, if not impossible, to have absolute certainty with anything in this world. The quote, “one can never be certain that a scientific hypothesis is true in the light of observed evidence, as the very next piece of evidence one discovers could be a counterexample”, emobodied the “jist” that I got from this entire paper (77). I wonder, is it ever possible to have absolute certainty about something? Or will you always be waiting for that next piece of evidence to prove everything wrong? The importance of non-monotonicity, “almost any conclusion can be overturned if additional information is acquired”, perpetuated this conclusion and made me realize that there are major issues with logic based reasoning. My skepticism lies within the fact that monotonicity is inherent with logic, and I have a hard time accepting this. Besides mathematics, it seems as if it is always possible to aquire new information and change your viewpoints or conclusions. Although pessimistic, it made me appreciate the theory of Bayesian reasoning and the importance of being able to update your degree of belief (75). It seems reasonable to have differing degrees of uncertainties.

    O&C discuss expected information gain when people perform Wason’s task. I found it very interesting that while the “expected informativeness of the q card is higher than that of the ~q card,” people still oftentimes chose the ~q card (78). Instead of seeking falsifying information (by selecting the ~q or ~p cards), people chose cards that would confirm the conditional (by selecting the q and p cards). Obviously, humans are not perfect replicates of the optimal data selection (ODS) model. This makes me wonder why our instincts surrender to confirmation bias, when in fact this is not an optimal reaction? Furthermore, if ODS does not model actual human response, is it indeed the optimal model?

  10. Oaksford and Chater (2009) start by discussing the definition of probability and its correlation with degrees of belief. They state, “… probability refers not to objective facts… but rather, it describes a reasoner’s degrees of beliefs… The perspective is the subjective, or Bayesian view of probability” (69). If probability is subjective and human rationality is defined by probability, how can we physically define human rationality as human thoughts are always changing?

    In this article, irrationality was seen in different perspectives. When discussing irrationality, the authors state, “all illogical performance resulted from misunderstandings and from the faulty way in which the mind might sometimes apply logical rules” (71). Henle also believes that errors cannot be associated with faulty reasoning (71). Do you think that this is actually true? Are errors really from our interpretation? Can errors actually help develop new ideas?

    Oaksford and Chater (2009) state, “The core objective of rational analysis, then, is to understand the structure of the problem from the point of view of the cognitive system, that is to understand what problem the brain is attempting to solve” (72). Does this imply more top-down or bottom-up processing? The first part seems like it pertain more to bottom-up while the second part factors more into top-down.

    Based on nonmonotonicity, how is it possible to know any facts are true or learn new facts? This pertains to the example about football and philosophical facts and knowledge.

    Do you think that the different types of heuristics are more beneficial in probability? Lastly, Oaksford and Chater (2009) state, “Instead, we suggest, the mind is a qualitative probabilistic reasoner… As we have stressed, this does not imply that the mind is a probabilistic calculating machine (although it may be); still less does it imply that the mind can process probabilistic problems posed in a verbal or mathematical format” (84). Do you think that the mind could be a probabilistic calculating machine? Is there a formula for everything or is qualitative data and cooperation essential?

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