A constellation of joint probability estimates is semantically coherent when the quantitative relationship among estimates of P(A), P(B), P(A and B), and P(A or B) is consistent with the relationship among the sets described in the problem statement. The possible probability estimates can form an extremely large number of permutations. However, this entire problem space can be reduced to six theoretically meaningful patterns: logically fallacious (conjunction or disjunction fallacies), identical sets (e.g., water and H2O), mutually exclusive sets (e.g., horses and zebras), subsets (e.g., robins and birds), overlapping sets (e.g., accountants and musicians), and inconsistent overlapping sets. Determining which of these patterns describes any set of probability estimates has been automated using Excel spreadsheet formulae. Researchers may use the semantic coherence technique to examine the consequences of differently worded problems, individual differences, or experimental manipulations. The spreadsheet described above can be downloaded as a supplement from http://brm.psychonomic-journals.org/content/supplemental.
How do we know whether a probability estimate is any good? Researchers have developed a number of metrics to address this question. Hammond (2000) identified two general strategies for assessing the quality of judgments such as probability estimates: correspondence and coherence (Adam & Reyna, 2005). Correspondence is satisfied when a person's judgments achieve empirical accuracy, and coherence is satisfied when a person's judgments achieve logical, mathematical, or statistical consistency (Hammond, 2000, p. 53).
Benchmarks for assessing probability judgments by employing a correspondence standard include empirical accuracy (Reyna & Adam, 2003; Yates, Lee, Shinotsuka, Patalano, & Sieck, 1998) and calibration (Keren, 1991). Correspondence benchmarks have a number of attractive features, including their potential as measures of expertise (Wright, Bolger, & Rowe, 2002) and their implications for successful behavior in contemporary and evolutionary contexts. Unfortunately, in many settings, it is difficult or impossible to independently assess the empirical accuracy of probability estimates. Moreover, in some cases, correspondence is of relatively little utility for understanding psychological processes, as when external factors such as luck play a large role in determining accuracy.
Coherence benchmarks typically include measures of logical fallacies-patterns of responses that are internally inconsistent with respect to the axioms of probability theory and logic (Tversky & Kahneman, 1983). Logical fallacies have been found consistently in the study of probability judgments (e.g., Gilovich, Griffin, & Kahneman, 2002). A conjunction fallacy is committed when a conjunction of events (A and B) is judged to be more probable than one of its constituents (e.g., A). A disjunction fallacy occurs when a disjunction of events (A or B) is judged to be less probable than any of its constituents (Bar-Hillel & Neter, 1993; Young, Nussbaum, & Monin, 2007).
Recently, Wolfe and Reyna (2010) developed a new metric for joint probabilities, semantic coherence, which is a more stringent coherence standard than whether a logical fallacy has been committed. This metric describes whether a constellation of nonfallacious probability judgments is consistent with the relationship among the sets in the problem statement. For any two sets or classes of objects where the relationship between the two is stipulated, there are four possible relationships: identical sets, where all A are B and all B are A; mutually exclusive sets, where sets A and B have no members in common; subsets, where all members of set A belong to B but B also has members that do not belong to A; and overlapping sets, where some A are B and some B are A, but also some A are not B and some B are not A. A constellation of joint probability estimates can be said to be semantically coherent when the quantitative relationship among estimates of P(A), P(B), P(A and B), and P(A or B) is consistent with the stipulated relationship among sets (e.g., one set is a subset of the other set).
This approach can be used to distinguish among the consequences of different theoretically motivated interventions designed to improve performance. For example, Wolfe and Reyna (2010) examined the effectiveness of pedagogic analogies and teaching the logic of the 2 � 2 table with respect to logical fallacies and semantic coherence. They found that both interventions improved semantic coherence but, generally, only the table intervention reduced fallacies, confirming predictions arising from fuzzy-trace theory (FTT). Other researchers may use the semantic coherence technique described here to examine the properties of differently worded problems, study individual differences in working memory, or test manipulations derived from other theories. For example, this approach could be extended to the study of causal reasoning (Lien & Cheng, 2000) by manipulating the wording of problems to represent genuine causes or noncausal covariation and examining semantic coherence under varying experimental conditions. Assessing semantic coherence is not dependent on any particular theoretical framework.
To illustrate the difference between logical fallacies and semantic coherence, consider the problem of a birdwatcher who is trying to determine whether a particular bird is a bluebird, an indigo bunting, or something else. Assume that P(bluebird) is the probability that the bird is a bluebird; similarly, P(indigo bunting) is the probability that the bird is an indigo bunting (bluebird and indigo bunting are mutually exclusive sets). If someone estimated P(bluebird) = .2, P(indigo bunting) = .2, and P(bluebird and indigo bunting) = .4, he or she would demonstrate a conjunction fallacy, because the probability of bluebird and indigo bunting cannot exceed the probability of indigo bunting alone. Consider the pattern of responding P(bluebird) = .2, P(indigo bunting) 5 .1, P(bluebird and indigo bunting) = 0, and P(bluebird or indigo bunting) = .3. This pattern is not fallacious and is consistent with the semantic description of mutually exclusive sets. Now consider the pattern of responding P(bluebird) = .6, P(indigo bunting) = .6, P(bluebird and indigo bunting) = .6, and P(bluebird or indigo bunting) = .6; then the birdwatcher has not committed a logical fallacy. However, this pattern is not semantically coherent with respect to the problem materials, because bluebirds and indigo buntings are mutually exclusive and, thus, P(bluebird and indigo bunting) should be 0. The issue is not the quality of any particular estimate (e.g., the probability of its being an indigo bunting is estimated to be .1); rather, it is the internal consistency of the whole set of responses.
Consider another example that involves subsets. For the problem of estimating the probability that the thing perched in a tree is a bird, a bluebird, or something else, the pattern of responding P(bluebird) = .2, P(bird) = .9, P(bluebird and bird) = .2, and P(bluebird or bird) = .9 is consistent with the semantic description of subsets, that bluebird is a subset of the set of birds. A semantically coherent constellation of responses, by definition, commits no logical fallacies of conjunction or disjunc tion. However, to be considered fully semantically coherent, a pattern of responses must match what is understood about the problem domain. Thus, the pattern of responding given above-P(A) = .2, P(B) = .1, P(A and B) = 0, P(A or B) = .3-is semantically coherent for problems in which the meaning of the problems describes mutually exclusive sets, but this same (equally nonfallacious) pattern of responses would not be semantically coherent with a problem describing subsets, because the conjunctive probability must equal the probability of the subset and the disjunctive probability must equal the probability of the larger, more inclusive set.
The task of asking people to estimate the probability of two events and their conjunctive and disjunctive probabilities allows researchers to consider the patterns of probability estimates that map onto two coherence benchmarks, logical fallacy and semantic coherence. Using integers between 0% and 100% to make probability estimates, there are over one hundred million possible permutations of two probability estimates and their conjunctive and disjunctive probabilities (although, of course, the number of permutations can be made larger or smaller, depending on the number of digits used in making each estimate-i.e., .1 or .001). This entire problem space can be reduced to six theoretically meaningful patterns: logical fallacy, identical sets, mutually exclusive sets, subsets, overlapping sets, or inconsistent overlapping sets. Thus, analyzing joint probability estimates in terms of logical fallacies and semantic coherence helps bring order to a wide range of behavior, revealing systematic patterns that would otherwise go unnoticed. The purpose of this article is to describe the logic behind these analyses and to share the specific Excel spreadsheet formula for making these calculations with the research community. The spreadsheet is available as a downloadable supplement for this article.
To illustrate, consider the following problem from Wolfe and Reyna (2010), where pilot testing and posttesting confirmed that the relationship between horses and zebras is generally perceived as one of mutually exclusive sets. Participants read a brief vignette about horses and zebras and then estimated P(horse), P(zebra), P(horse and zebra), and P(horse or zebra). Because making all estimates the same is a meaningful pattern, we asked the participants to make another, unlikely estimate to distinguish identical sets from a careless pattern of giving identical responses for each estimate. In this case, we asked the participants to estimate P(ostrich):
Valerie is a photographer for a famous magazine. She was on a photo shoot in an African wildlife park when she heard the sound of hoof beats in the distance. Not wanting to miss a shot, she grabbed for her camera and turned around.
Please rate the probability of each of these statements about the story above using a rating scale from 0% (impossible) to 100% (completely certain).
What is the probability that the hoof beats she heard were made by an ostrich?_____
What is the probability that the hoof beats she heard were made by a horse?_____
What is the probability that the hoof beats she heard were made by a zebra?_____
What is the probability that the hoof beats she heard were made by a zebra that is a horse?_____
What is the probability that the hoof beats she heard were made by either a zebra or a horse?_____
Mutually exclusive sets are sets with no overlapping members (e.g., apples and oranges). A constellation of four estimates is consistent with the semantic description mutually exclusive sets when (1) P(A) + P(B) < 1.0, (2) P(A) + P(B) = P(A or B), and (3) P(A or B) = 0. In cases in which P(A) = 0; P(B) > 0, P(A and B) = 0, and P(A or B) = P(B), we categorized the pattern as mutually exclusive sets, which was rare in the Wolfe and Reyna experiments. Other researchers may elect to handle these cases differently. Figure 1 shows a segment of the spreadsheet where Participant 1 has estimated P(ostrich) = 1%, P(horse) = 10%, P(zebra) = 80%, P(horse and zebra) = 0%, and P(horse or zebra) = 90%. The spreadsheet categorizes this pattern as mutually exclusive sets, because P(horse and zebra) = 0% and P(horse or zebra) = P(horse) + P(zebra).
A constellation of four estimates is consistent with the semantic description subset (e.g., roses and flowers) when (in the case in which A is a subset of B) (1) P(A) < P(B), (2) P(A) = P(A and B), and (3) P(B) = P(A or B). Figure 2 shows a segment of the spreadsheet where Par- ticipant 2 has estimated P(ostrich) = 5%, P(horse) = 90%, P(zebra) = 5%, P(horse and zebra) = 5%, and P(horse or zebra) = 90%. The spreadsheet categorizes this pattern as zebra is a subset of horse, because P(horse) > P(zebra), P(horse and zebra) = P(zebra), and P(horse or zebra) = P(horse). It is important to note that this is not a fallacious pattern of responses. However, it is not semantically coherent with respect to the problem materials, which call for mutually exclusive sets. This same pattern would be semantically coherent for another set of materials-for example, those describing flowers and roses.
When A and B are identical sets (e.g., water and H2O), P(A not B) = 0 and P(B not A) = 0. A constellation of four estimates P(A), P(B), P(A and B), and P(A or B) is consistent with the semantic description identical set when P(A) = P(B) = P(A and B) = P(A or B) ≠ 0. Figure 3 shows a segment of the spreadsheet where Participant 4 has estimated P(ostrich) = 0%, P(horse) = 10%, P(zebra) = 10%, P(horse and zebra) = 10%, and P(horse or zebra) = 10%. The spreadsheet categorizes this pattern as identical sets, because P(horse) = P(horse or zebra), P(horse) = P(zebra), and P(horse and zebra) = P(horse or zebra), indicating that all four estimates are the same. in this case, 10%. In our experiments (Wolfe & Reyna, 2010), an additional cell checks that the estimate for an unlikely event.in this case, P(ostrich).is not the same as the other four, thereby enabling us to distinguish between identical sets and a lazy pattern of responding by hitting the same key for each question.
There are three patterns of conjunction fallacies and three patterns of disjunction fallacies. Conjunction fallacies are (1) P(A and B) > P(A), (2) P(A and B) > P(B), and (3) P(A and B) < P(A) 1 P(B) 2 1.0. The most common, and most frequently studied, conjunction error is one in which the conjunction of two events P(A and B) is erroneously considered more probable than one of those events, P(A). This is depicted in Figure 4 as Participant = having estimated P(horse) = 2% and yet P(horse and zebra) = 90%. The following are disjunction errors: (1) P(A or B) < P(A), (2) P(A or B) < P(B), and (3) P(A or B) > P(A) + P(B). The most frequent and commonly studied disjunction error is when the disjunc- tion of two events, P(A or B), is erroneously considered less probable than one of those events. Figure = depicts this situation, where Participant 6 has estimated P(horse or zebra) = 75% and yet P(zebra) = 90%.
In some circumstances, it is fallacious to estimate P(A and B) too low or P(A or B) too high. For example, if one estimates that P(A) = .9, and P(B) = .9, it would be fallacious to estimate P(A and B) = .5, a conjunction fallacy. Figure 6 illustrates this kind of disjunction error, where P(A and B) is too low. Participant 8 has estimated P(horse) = 90%, P(zebra) = 90%, and yet P(horse and zebra) = 50%. The formula categorizes a constellation of responses to be a conjunction error if P(A and B) > P(A) + P(B) 2 1.0. In other words, the lower boundary of the conjunctive probability P(A and B) should be constrained when the sum of the constituents exceeds 1.0.
Turning to the case in which the disjunctive probability is too high, if one estimated P(A) = .1 and P(B) = .1, estimating the P(A or B) = .9 would be a disjunction fallacy. Figure 7 shows a situation in which Participant 7 has estimated P(horse) = 5%, P(zebra) = 60%, and yet P(horse or zebra) = 80%. The formula categorizes a constellation of responses as a disjunction fallacy if P(A or B) > P(A) + P(B). In other words, the probability of Event A or Event B should generally be equal to or greater than each of those constituents, but not greater than their sum. Thus, these Excel formulae can be used to study different kinds of conjunction and disjunction fallacies, depending on the goals and interests of the researcher.
The last two categories of responses are consistent and inconsistent overlapping sets. Our Excel spreadsheet handles these by first identifying total overlapping sets and then determining whether or not they are internally consistent. Total overlapping sets are simply those cases in which the spreadsheet has not shown a fallacy or mutually exclusive sets, subsets, or identical sets (i.e., the value of each of those cells is 0). To determine whether this overlapping set is consistent or inconsistent, it establishes whether the value of P(A or B) is identical to what would be predicted by the other three estimates P(A), P(B), and P(A and B). Figure 8 shows how the expected value of P(A or B) is determined as P(A and B) plus the unique contribution of A, P(A not B), plus the unique contribution of B, P(B not A). In the case depicted in Figure 8, Participant 7 has estimated P(horse) = 40%, P(zebra) = 60%, P(horse and zebra) = 15%, and P(horse or zebra) = 80%. Here, the expected value of P(horse or zebra) = (40% - 15%) + 15% 1 (60% - 15%) = 85%. As is illustrated in Figure 8, the deviation from internal consistency is the absolute value of the participant's estimated P(A or B) minus the expected value of P(A or B)-in this case, 5%. When participants' estimates are internally consistent, this value is 0. By definition, mutually exclusive sets, subsets, and identical sets are internally consistent, with no difference between the expected and observed P(A or B). Internally inconsistent overlapping sets are those cases in which total overlapping sets (1 or 0) multiplied by deviations from internal consistency is greater than 0. Thus, if total overlapping sets is 0 and/or deviation from internal consistency is 0, a constellation will not be categorized as inconsistent overlapping sets.
Consistent overlapping sets (e.g., feminists and bank tellers) are determined by the Excel spreadsheet to be all remaining cases in which the constellation of the four estimates P(A), P(B), P(A and B), and P(A or B) is shown not to be fallacious or to consist of subsets, identical sets, or mutually exclusive sets or cases in which the marginal totals of P(A) plus P(not A) fail to sum to one. Consistent overlapping sets are those remaining cases. The spreadsheet categorizes a constellation of responses as consistent overlapping sets if they are first labeled total overlapping sets but not inconsistent overlapping sets-specifically, if total overlapping sets (1 or 0) minus inconsistent overlapping sets equals 1.
Inconsistent overlapping sets occur when no first-order (or single) logical fallacy has been committed but when P(A or B) ≠ [P(A) - P(A and B)] + P(A and B) + [P(B) - P(A and B)]. For example, one of Wolfe and Reynafs (2010) participants estimated P(A) = .3, P(B) = .3, P(A and B) = .15, P(A or B) = .60. Either of the joint probability estimates is permissible by itself, but if P(A and B) is .15, P(A or B) should be .45, not .6. In these cases, no first-order fallacy has been committed, but the marginal totals do not sum to one.that is, P(A) 1 P(not A) ≠ 1.0. Some researchers may categorize these as fallacious patterns of responding, and others may prefer to categorize them separately, because no single estimate can be identified as fallacious in and of itself. When someone estimates P(A and B) > P(A), it is clear that the estimate of P(A and B) is fallacious. In the case of inconsistent overlapping sets, it is the pattern of all four estimates that is fallacious. An advantage of separating inconsistent overlapping sets from first-order fallacies is that it allows researchers to pursue different questions about semantic coherence and fallacies to examine these patterns individually or to combine them.
DISCUSSION
The techniques described above effectively reduce a very large problem space to a handful of meaningful patterns. The research reported in Wolfe and Reyna (2010) was theoretically motivated by FTT, a dual-process theory of memory and cognition with broad ramifications for the study of judgment and decision making (Reyna, 2004; Reyna & Brainerd, 1995; Wolfe, 1995). However, semantic coherence as a benchmark is not dependent on FTT or any particular theory of judgment and decision making, and researchers operating within other theoretical frameworks may fruitfully investigate conditions affecting semantic coherence using the techniques described here.
Although the Excel formulae described here have proved to be quite reliable and useful, they do suffer from a few shortcomings, and automating the process of categorizing patterns of responses does not substitute for inspecting one's data. There are a few patterns that cannot be uniquely interpreted. In particular, cases in which all responses are 0% or 100% can be conceived of in more than one way. For example, if someone estimated the probability that George Washington was an extraterrestrial visitor at 0 and that he was a sumo wrestler at 0, it would be problematic to attribute the constellation of estimates P(extraterrestrial) = 0, P(sumo wrestler) 5 0, P(extraterrestrial and sumo wrestler) = 0, and P(extraterrestrial or sumo wrestler) = 0 to mean that extraterrestrials and sumo wrestlers are identical sets.
It is also important to be cautious about misattributing semantic coherence in cases in which probabilities at different levels of a hierarchy are misaligned (Lagnado & Shanks, 2003). For example, in a case in which there are four job candidates, three men and one woman, it may be perfectly reasonable and semantically coherent to believe that Jane has a greater chance of being hired than Tom, Dick, or Harry and that it is more likely that a man will be hired than a woman. To illustrate, someone may attribute the probability of Jane's being hired at .4 and give each of the men a .2 chance, resulting in a .6 probability that the person hired will be a man.
The concept of semantic coherence can be expanded to other domains beyond conjunctive and disjunctive probabilities. One of our next steps is to apply these techniques to conditional probabilities-that is, P(A| B). The concepts of mutually exclusive sets, identical sets, and subsets can fruitfully be applied in these conditional probabilities (Reyna & Mills, 2007). For mutually exclusive sets, when people estimate P(A) and P(B), P(A| B) = 0 and P(B | A) = 0; the pattern is semantically coherent with respect to mutually exclusive sets. For subsets (A is a subset of B), when people estimate P(A) and P(B) > P(A), P(A| B) = P(A) /P(B) and P(B | A) = 1.0; the pattern is semantically coherent with respect to subsets. For identical sets, when people estimate P(A) and P(B) 5 P(A), P(A|B) = 1.0 and P(B|A) = 1.0; the pattern is semantically coherent with respect to identical sets. For overlapping sets, when people estimate P(A) and P(B), then estimate P(A| B) > 0 and < 1.0, and then estimate P(B | A) = [P(A| B) 3 P(B)] /P(A), the pattern is semantically coherent with respect to consistent overlapping sets. The concept of semantic coherence can also be applied to independent sets for which A and B are orthogonal and conditional probabilities provide no additional information. For example, the probability of drawing an ace from a deck of playing cards is independent of the probability of drawing a red card. Whether or not one has drawn a red card provides no additional information about whether or not an ace has been drawn. A pattern of responses is semantically coherent with respect to independent sets when P(A| B) = P(A) and P(B | A) = P(B).
Extending the notion of semantic coherence to conditional probabilities may have implications for the Bayesian or probabilistic approach to reasoning (Evans, 2007; Oaksford & Chater, 2007; Over, 2009). From this perspective, people judge the natural language proposition if A then B to be the conditional probability P(A| B) (Evans, 2007), and probability heuristics are applied to a range of reasoning phenomena. Employing semantic coherence within this new paradigm may help distinguish among models of reasoning (Oaksford & Chater, 2007), as well as models of probabilistic inference.
Semantic coherence provides another useful yardstick for assessing probability estimation. It is a higher standard than logical fallacies, yet, like fallacies, semantic coherence is a measure of internal consistency. We are only beginning to explore the conditions under which semantic coherence improves or is degraded. Although these conditions are not well understood, their investigation should shed new light on the processes of estimating probability.
[Sidebar]
SUPPLEMENTAL MATERIALS
An Excel spreadsheet for automating calculation of the probability space for any set of probability estimates may be downloaded from http://brm.psychonomic-journals.org/content/supplemental.
(Manuscript received November 11, 2009; revision accepted for publication January 25, 2010.)
[Reference]
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[Author Affiliation]
CHRISTOPHER R. WOLFE
Miami University, Oxford, Ohio
AND
VALERIE F. REYNA
Cornell University, Ithaca, New York
C. R. Wolfe, wolfecr@muohio.edu
[Author Affiliation]
AUTHOR NOTE
Correspondence concerning this article should be addressed to C. R. Wolfe, Department of Psychology, Miami University, Oxford, OH 45056 (e-mail: wolfecr@muohio.edu).
