Saturday, December 04, 2004

By Request: Reasoning

In my most recent plea for requests, Richard asked:
How does reasoning work?
What a question! You could pretty much reword it as, "How does cognition work?" Clearly, I'm not prepared or qualified, much less capable of answering that question sufficiently. Fortunately for me, Richard got more specific, writing:
For example, from beliefs that P, and that P->Q, what causes us to then believe Q?
Other stuff about the gap between normative logic and how people *actually* reason would be interesting too - ya know, all those fallacies and such that we fall for.
He also referenced this good little post from Jonathan Ichikawa, which raises a whole other set of issues. So, my task isn't an easy one, but I'm going to try to post on reasoning as best I can. The different topics that these short questions envoke are numerous, and I don't think I can write about them all, at least not in the near future. Obviously, there's the difference between deductive and inductive reasoning, then there are issues and subissues like domain-general vs. domain-specific processes, mental models (which Jonathan's post alludes to, consciously or not), causal and counterfactual reasoning, heuristics and biases, analogical reasoning, categorical inference, probabilistic reasoning, conditional reasoning, etc. It's hard to know exactly how to touch on all of these, or even most of them, in a blog post (or series of posts), and I've spent some time trying to decide how I should do it. The solution I've come up with is to basically wing it. However, I have decided on a definite starting point. The starting point is Richard's last request, the one about errors in reasoning. I think that if we start here, we'll start to see many of the other issues emerging, and the ones that don't emerge we can get to eventually. So, let's get started.

Wason Selection Task

An interesting and informative place to start, when looking at errors in human reasoning, is the Wason selection task1. Imagine you are given the following two-sided cards:

Posted by Hello

You are then given the this rule: If there is a vowel on one side of a card, then there is an odd number on the other side. If you are asked to select the minimum number of cards, which cards would you need to check to determine if this rule is true? Almost everyone picks the "E" card, when given this task. However, what other card(s) should you check? Ninety percent of participants select either only "E," or select "E" and "7." Almost no one selects the R. Those of you with a background in logic will immediately notice the problem with both of those elections. Yes, you should check "E," because if there is an even number on the other side of "E," then the rule is false. This is a simple instantiation of modus ponens. Also, participants are correct in not selecting the "R." The rule says nothing about what should be on the other side of a consonant. However, "7" is irrelevant. If there is a vowel on the other side, then the rule can still be true, but if there is a consonant on the other side, then we haven't learned anything. Participants who select the "7" are committing the common fallacy of affirming the consequent. Instead, people should select the "2," because if there is a vowel on the other side of the "2" card, then the rule is false. This is the rule modus tollens.

Why are people so bad at this task? A common answer, when this task was first used, involved modus ponens being an easier rule to learn than modus tollens, a fact to which anyone who has ever taught a logic course can attest. Some even hypothesized that modus ponens is an innate rule, while modus tollens is not. However, the picture gets muddier. Imagine the problem had been given with the same structure, but different content. For instance, imagine you had been given the following cards instead:

Posted by Hello

Then you receive the following rule: If a person is under 21, she cannot drink beer. Given that you have to select the fewest possible cards to make sure that the rule is being followed, which cards would you select? When given this task, almost all participants select "18" and "Beer," and very few select "21" or "Coke." Why is it that, when the content is changed to a familiar rule, participants are able to select the correct cards, utilizing both modus pollens and modus tollens? This certainly calls the learnability and innateness hypotheses into question.

Some have argued that the differences in performance on the two versions of the Wason selection task indicate a need for a focus on domain-specific theories of reasoning, rather than the traditional domain-general theories that are embodied in the abstract rules of deductive reasoning. For instance, Cosmides and her colleagues2 have theorized that in the sort of thematic version of the Wason task in the second example above, a cheater detection module, developed through evolution, is activated. Under their view, people are innately programmed to detect deception in situations where cheating is likely, and when this ability is activated in the Wason task, people are easily able to reason correctly. However, this view has been heavily criticized from several directions. Some3 have argued that Cosmides and her colleagues misunderstand the Wason task, and the second version (ages and beer) is actually a different type of task altogether. Furthermore, research has shown that even when cheater detection scenarios are not involved, people perform much better at the Wason task when the content is thematic, or familiar, than when it is abstract or unfamiliar4. This suggests that instead of activating a domain-specific module (e.g., a cheater detection model), the thematic Wason tasks activate background knowledge, and this makes the task much easier. Still, the exact reason for the differences in performances is still being hotly debated, and touching on the debate foreshadows a lot of what I will talk about in future posts.

Before we move on, there's one more interesting fact about performance on the Wason selection task that is worth mentioning. Imagine the first version of the selection task, with the letters and numbers, but with a "reduced array." In this case, the two choices that participants have no problem with, namely the "E," which they almost always select, and the "R," which they almost never select, are removed. In this case, participants do very well, almost always selecting the "2." One explanation for this result is that participants aren't viewing the task as an abstract, deductive reasoning task at all. Instead, they are viewing it as a categorization task. Here is a more thorough description of this explanation, from Margolis:
This odd, even bizarre, improvement can be explained if subjects are seeing the cards not as particular cards but as indicating categories of cards. If explicitly asked, subjects understand the intended meaning of the question. But their responses make logical sense only with respect to a drastic misreading of the question. The question is misread as being about which categories of cards should be examined; for example, any cards with a [vowel] on either side; rather than about the particular card shown with a [vowel] on its upside.
This misperception, Margolis argues, is a product of the "pragmatics of language," writing:
In everyday conversation, even logicians rarely use phrasing like "if and only if" (iff) to distinguish this "if/then" relation from "if but not only if" (if). Distinguishing between "if" and "iff" is almost always left to context. But the basic Wason problem provides so little context that if/then here could be interpreted
either way.
Another explanation comes from decision theory. For instance, some have argued that using a Bayesian decision making process, participants' selections make sense in terms of information gained5. Here's a description of this view:
Care selected on the basis of their informativeness. The informativeness of each card is a function of the probability of the items mentioned in the rule. Under an assumption of rarity, i.e. that P(p) and P(q) are low, the A and 3 cards are most informative, the 7 card is less informative and the D card provides no information whatsoever. The approach captures the intuitive judgement that it is more sensible to look for ravens or black things when testing the rule "All ravens are black" in the classic Ravens Paradox.
The message to take away from both these views is that contrary to previous explanations, people aren't actually behaving irrationally when they select the "incorrect" answers in the original Wason task. These views are still not widely heald among cognitive scientists, but they are certainly worth noting.

Heuristics and Biases

Throughout history, humans have been seen as rational beings. This ability to use reason, it has long been believed, is one of the major distinctions between humans and the other animals. Most economic theory has traditionally been based on this assumption. However, in the 1940s and 50s, researchers began to notice that people didn't always behave rationally (and in fact, when rationality is equated with optimality, as it usually is in economics, they rarely behaved rationally)6. Several different theories were derived to attempt to account for human irrationality, without leaving behind the assumption that humans really are rational (e.g., Bayesian decision theories and the still popular bounded rationality perspective). Then came a couple guys by the name of Amos Tversky and Daniel Kahneman. They argued that the old view of human rationality could not be salvaged, given all the examples of suboptimality. Instead, it was time to take an entirely different approach to bounded rationality. As Gilovich and Griffin put it:
Although acknowledging the role of task complexity and limited processing capacity in erroneous judgment, Kahneman and Tversky were convinced that the processes of intuitive judgment were not merely simpler than rational models demanded, but were categorically different in kind. Kahneman and Tversky described three general-purpose heuristics – availability, representativeness, and anchoring and adjustment – that underlie many intuitive judgments under uncertainty. These heuristics, it was suggested, were simple and efficient because they piggybacked on basic computations that the mind had evolved to make. Thus, when asked to evaluate the relative frequency of cocaine use in Hollywood actors, one may assess how easy it is to retrieve examples of celebrity drug-users – the availability heuristic piggybacks on highly efficient memory retrieval processes. When evaluating the likelihood that a given comic actor is a cocaine user, one may assess the similarity between that actor and the prototypical cocaine user (the representativeness heuristic piggybacks on automatic pattern-matching processes). Either question may also be answered by starting with a salient initial value (say, 50%) and adjusting downward to reach a final answer.

In the early experiments that defined this work, each heuristic was associated with a set of biases: departures from the normative rational theory that served as markers or signatures of the underlying heuristics. Use of the availability heuristic, for example, leads to error whenever memory retrieval is a biased cue to actual frequency because of an individual’s tendency to seek out and remember dramatic cases or because of the broader world’s tendency to call attention to examples of a particular (restricted) type. Some of these biases were defined as deviations from some “true” or objective value, but most by violations of basic laws of probability. (p. 3)
Thus, instead of reasoning with abstract logical rules, people are using heuristics and their associated biases, which can, under some circumstances, lead to suboptimality. Since the heuristics and biases program was first formulated, research on the different heuristics and biases has formed an extremely large literature, spanning several disciplines. It would be silly to try to describe all of this research, so, given the original request (all those fallacies and such), I'm going to focus on a particular example, the conjuction fallacy. This fallacy occurs when people assign a higher probability to the conjunction of two events than than they assign to the events individually. The often-cited version of this fallacy from the literature goes something like this7:
Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.
After reading this, participans were presented with the following possibilities, and asked which is more probably:
(1) Linda is a bank teller.
(2) Linda is a bank teller and is active in the feminist movement.
Given the above description of Linda, participants overwhelmingly selected (2), despite the fact that this is a conjunction of (1) and another property. Thus, they rated the conjunction as being more probably than one of the instances alone, thereby committing the conjunction fallacy. Tversky and Kahneman explain this by reference to one of their heuristics, the representativeness heuristic. They argue that participants see the properties attributed to Linda as being representative of feminists, and therefore are more likely to believe that Linda is a feminist. The use of this heuristic causes them to commit what is a logical error, given the formal nature of probabilities, but which is likely to be a practical belief nonetheless.

More recently, another version of the conjunction fallacy has been demonstrated. This version is sometimes called the "disjunction fallacy," though it is logically identical to the conjunction fallacy. Here is a description of this fallacy, from the abstract of Bar-Hillel and Neter (1993)8:
This study demonstrates a violation of the rule in a context that justifies the label disjunction fallacy. Subjects received brief case descriptions, and ordered seven categories according to one of four criteria for including the case as a member of the category : 1. probability of membership ; 2. willingness to bet on membership ; 3. inclination to predict membership ; 4. suitability for membership. The category list included nested pairs of categories, such as Brazil and South American country, or Physics and A Natural Science. The more inclusive category was a union of basic level sets like the smaller category. From a normative standpoint, the first two criteria are equivalent, and either ranking a category as more probable than its superordinate, or betting on it rather than on its superordinate, is fallacious. On the other hand, inclination to predict may be guided by the desire to be maximize informativeness rather than merely likelihood of being correct, and suitability needs to conform to no formal rule. Hence, with respect to these two criteria, such a ranking pattern is not fallacious. In spite of this crucial difference, subjects in all four groups rendered highly similar judgments, and the ranking of categories higher than their superodinates was not lower when it amounted to a fallacy than when it did not.
The heuristics and biases program has, over the last decade or two, come under a great deal of criticism, but for the most part its hypotheses have stood up well under empirical pressure. One thing we can be sure of is that humans often reason not with logical rules, but with automatic and largely unconscious heuristics, and perhaps biases, that cause them to perform in ways that at least appear fallacious. Even if we can redescribe some of the problems the heuristic and biases program is meant to explain with, e.g., Bayesian probability, or some other version of bounded rationality, the fact remains that people do not always behave optimally, and this suboptimality can be at least partially explained by the failure to stick to the formal rules of logic or probability.

So, we've now got two different sets of reasons for human behavior failing to conform to normative reason. In the next few posts, we'll look at some of the issues human suboptimality and irrationality raise. In the next post, I'll talk about domain-general vs. domain-specific theories of reasoning in more detail. After that, we'll get into reasoning from background knowledge, which includes categorical and analogical reasoning. Finally, I'll try to summarize the mental models view of reasoning, which will touch on a wide range of types of reasoning (from deductive to causal). Stick around.

1 Wason, P.C. (1966) Reasoning. In B. M. Foss (Ed.) New Horizons in Psychology I. Penguin.

2 Cosmides, L. (1989). The logic of social exchange: Has natural selection shaped how humans reason? Studies with the Wason selection task. Cognition, 31, 187-276. See also Fiddick, L., Cosmides, L., & Tooby, J. (2000). No interpretation without representation: The role of domain-specific representations in the Wason selection task. Cognition, 77, 1-79.
3 E.g., Sperber, D. & Girotto, V. (2002). Use or misuse of the selection task? Rejoinder to Fiddick, Cosmides and Tooby. Cognition, 85(3), 277-290; Fodor, J. (2000). Why we are so good at catching cheaters. Cognition, 75(1), 29-32, and The Mind Doesn't Work That Way: The Scope and Limits of Computational Psychology by Jerry Fodor. For a rejoinder to Fodor, see Beaman, C. P. (2002). Why are we good at detecting cheaters? A reply to Fodor. Cognition, 83(2), 215-20.
4 E.g., Sperber & Girotto, 2002.
5 Oaksford, M & Chater, N. (1994). A rational analysis of the selection task as optimal data selection. Psychological Review, 101, 608-631, and Oaksford, M., Chater, N., Grainger, B. & Larkin, J. (1997). Optimal data selection in the reduced array selection task (RAST). Journal of Experimental Psychology: Learning, Memory & Cognition, 23, 441-458.
6 Notice that the belief that humans are rational (in the sense of optimality) is required for the Buridan's Ass problem discussed in previous posts to make sense. If humans generally behave suboptimality, the problem dissolves away.
7 Tversky, A. and Kahneman, D. (1983). Extension versus intuititve reasoning: The conjunction fallacy in probability judgment. Psychological Review, 90, 293-315.
8 Bar-Hillel, M. & Neter, E. (1993). How alike is it versus how likely is it: A disjunction fallacy in probability judgements. Journal of Personality and Social Psychology, 65, 1119–1131.


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