Semantic coherence and fallacies in estimating joint probabilities

In three experiments on joint probability estimation, gist representations were manipulated with analogies, and the suboptimal strategy of ignoring relevant denominators was counteracted with training in using 2 × 2 tables to clarify joint probability estimates. The estimated probabilities of two events, as well as their conjunctive and disjunctive probabilities, were assessed against two benchmarks, logical fallacies and semantic coherence—a constellation of estimates consistent with the relationship among sets. Fuzzy‐trace theory (FTT) predicts that analogies will increase semantic coherence, and a table intervention affecting denominator neglect will both increase semantic coherence and reduce fallacies. In all three experiments, analogies increased semantic coherence. In both experiments training participants to use 2 × 2 tables, such tables reduced fallacies and increased semantic coherence. As the relations among sets in the problem materials progressed in cognitive complexity from identical sets, mutually exclusive sets, and subsets to overlapping sets, fallacies generally increased, and semantic coherence generally decreased. These findings indicate that denominator neglect is pervasive, but that it can be remedied with a straightforward intervention that clarifies relations among sets. Further, intuitive gist‐based probability estimation can be improved through the use of simple analogies. Copyright © 2009 John Wiley & Sons, Ltd.     

Assessing semantic coherence and logical fallacies in joint probability estimates

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.     

Are People Naïve Probability Theorists? A Further Examination of the Probability Theory + Variation Model

Two experiments tested predictions derived from the Probability Theory + Variation (PTV) model. PTV model assumes that judgments follow probability theory, but systematic errors arise from noise in the judgments. Experiment 1 compared the PTV model to a configural weighted averaging model in joint probability judgment and found more support for the PTV model in diagnostic cases. Specifically, noise was negatively correlated with semantic coherence and conjunction and disjunction fallacies increased when order effects produced more noise in conjunctions and disjunctions. Consistent with both models, judgments adhered stochastically to the addition law. Contrary to the integration rules of the PTV model, we failed to find increased noise in disjunctions compared to conjunctions. Experiment 2 tested predictions of the PTV model for conditional probability judgment. Consistent with the PTV model, noise was negatively correlated with semantic coherence in conditional probabilities and judgments adhered stochastically to Bayes' theorem. Conversion errors were generally more prevalent than conditional reversals, a finding that is not fully consistent with the PTV model. In general, the quantitative fit of the PTV model was relatively better for overlapping and subset problems compared to identical, independent and mutually exclusive problems, especially for semantic coherence. Copyright © 2014 John Wiley & Sons, Ltd.     

Explaining High Conjunction Fallacy Rates: The Probability Theory Plus Noise Account

The conjunction fallacy occurs when people judge the conjunctive probability P(A ∧ B) to be greater than a constituent probability P(A), contrary to the norms of probability theory. This fallacy is a reliable, consistent and systematic part of people's probability judgements, attested in many studies over at least 40 years. For some events, these fallacies occur very frequently in people's judgements (at rates of 80% or more), while for other events, the fallacies are very rare (occurring at rates of 10% or less). This wide range of fallacy rates presents a challenge for current theories of the conjunction fallacy. We show how this wide range of observed fallacy rates can be explained by a simple model where people reason according to probability theory but are subject to random noise in the reasoning process. Copyright © 2016 John Wiley & Sons, Ltd.     

四卡问题解决中的视角效应

视角效应是指在四卡问题解决中,当被试从不同的角度来对条件规则进行检验时,其选择偏向会在卡片P,-Q和-P,Q之间发生稳定变化的现象。到目前为止,义务和非义务推理中都发现了该效应的存在,不同的推理理论都试图对此作出解释,却一直存在分歧,其焦点在于视角效应的形成机制是领域特殊性的还是领域一般性的。该文评述了不同的演绎推理理论对视角效应发生机制作出的解释,并且最后总结了该领域目前研究存在的主要争论。     

Disjunction and conjunction fallacies in episodic memory

It has recently been found that episodic memory displays analogues of the well-known disjunction and conjunction fallacies of probability judgement. The aim of the present research was, for the first time, to study these memory fallacies together under the same conditions, and test theoretical predictions about the reasons for each. The focus was on predictions about the influence of semantic gist, target versus context recollection, and proactive versus retroactive interference. Disjunction and conjunction fallacies increased in conditions in which subjects were able to form semantic connections among list words. In addition, disjunction fallacies were increased by manipulations that minimised proactive interference, whereas conjunction fallacies were increased by manipulations that minimised retroactive interference. That pattern suggests that disjunction fallacies are more dependent on target recollection, whereas conjunction fallacies are more dependent on context recollection.     

The ‘conjunction fallacy’ revisited: how intelligent inferences look like reasoning errors

Findings in recent research on the ‘conjunction fallacy’ have been taken as evidence that our minds are not designed to work by the rules of probability. This conclusion springs from the idea that norms should be content‐blind—in the present case, the assumption that sound reasoning requires following the conjunction rule of probability theory. But content‐blind norms overlook some of the intelligent ways in which humans deal with uncertainty, for instance, when drawing semantic and pragmatic inferences. In a series of studies, we first show that people infer nonmathematical meanings of the polysemous term ‘probability’ in the classic Linda conjunction problem. We then demonstrate that one can design contexts in which people infer mathematical meanings of the term and are therefore more likely to conform to the conjunction rule. Finally, we report evidence that the term ‘frequency’ narrows the spectrum of possible interpretations of ‘probability’ down to its mathematical meanings, and that this fact—rather than the presence or absence of ‘extensional cues’—accounts for the low proportion of violations of the conjunction rule when people are asked for frequency judgments. We conclude that a failure to recognize the human capacity for semantic and pragmatic inference can lead rational responses to be misclassified as fallacies. Copyright © 1999 John Wiley & Sons, Ltd.     

Fallacies in probability judgments for conjunctions and disjunctions of everyday events

The conjunction fallacy occurs when people judge a conjunction B‐and‐A as more probable than a constituent B, contrary to probability theory's ‘conjunction rule’ that a conjunction cannot be more probable than either constituent. Many studies have demonstrated this fallacy in people's reasoning about various experimental materials. Gigerenzer objects that from a ‘frequentist’ standpoint probability theory is not valid for these materials, and so failure to follow the conjunction rule is not a fallacy. This paper describes three experiments showing that the conjunction fallacy occurs as consistently for conjunctions where frequentist probability theory is valid (conjunctions of everyday weather events) as for other conjunctions. These experiments also demonstrate a reliable correlation between the occurrence of the conjunction fallacy and the disjunction fallacy (which arises when a disjunction B‐or‐A is judged less probable than a constituent B). This supports a probability theory + random variation account of probabilistic reasoning. Copyright © 2008 John Wiley & Sons, Ltd.     

Inconsistent probability estimates of a hypothesis: the role of contrasting support

This paper studies consistency in the judged probability of a target hypothesis in lists of mutually exclusive nonexhaustive hypotheses. Specifically, it controls the role played by the support of displayed competing hypotheses and the relatedness between the target hypothesis and its alternatives. Three experiments are reported. In all experiments, groups of people were presented with a list of mutually exclusive nonexhaustive causes of a person's death. In the first two experiments, they were asked to judge the probability of each cause as that of the person's decease. In the third experiment, people were asked for a frequency estimation task. Target causes were presented in all lists. Several other alternative causes to the target ones differed across the lists. Findings show that the judged probability/frequency of a target cause changes as a function of the support of the displayed competing causes. Specifically, it is higher when its competing displayed causes have low rather than high support. Findings are consistent with the contrastive support hypothesis within the support theory.     

Information seeking on Bayesian conditional probability problems: A fuzzy-trace theory account

Recently, the ‘heuristics and biases’ approach to the study of decision making has been criticized, with a call for better integrated theory. Three experiments stemming from fuzzy-trace theory addressed information seeking on probability problems, and the cognitive representation of hit-rates, base-rates, and the contrapositive. As predicted by the fuzzy-trace principle of ‘denominator neglect’, many subjects exhibited ‘conversion errors’, confusing the hit-rate, P(A|B), with the answer, P(B|A). These subjects sought base-rates less often than other subjects. On causal problems, more subjects correctly represented base-rates, sought base-rates more often, and produced more accurate estimates than on non-causal problems. Subjects tutored on the meaning of the hit-rate sought the base-rate more often, and were more accurate than control subjects. Results are explained by fuzzy-trace theory principles of gist extraction, fuzzy processing preference, denominator neglect, and output interference.     

The probability of causal conditionals

Conditionals in natural language are central to reasoning and decision making. A theoretical proposal called the Ramsey test implies the conditional probability hypothesis: that the subjective probability of a natural language conditional, P(if p then q), is the conditional subjective probability, P(q/p). We report three experiments on causal indicative conditionals and related counterfactuals that support this hypothesis. We measured the probabilities people assigned to truth table cases, P(pq), P(p notq), P( notpq) and P( notp notq). From these ratings, we computed three independent predictors, P(p), P(q/p) and P(q/ notp), that we then entered into a regression equation with judged P(if p then q) as the dependent variable. In line with the conditional probability hypothesis, P(q/p) was by far the strongest predictor in our experiments. This result is inconsistent with the claim that causal conditionals are the material conditionals of elementary logic. Instead, it supports the Ramsey test hypothesis, implying that common processes underlie the use of conditionals in reasoning and judgments of conditional probability in decision making.     

Perspective effects in nondeontic versions of the Wason selection task

Perspective effects in the Wason four-card selection task occur when people choose mutually exclusive sets of cards depending on the perspective they adopt when making their choice. Previous demonstrations of perspective effects have been limited to deontic contexts--that is, problem contexts that involve social duty, like permissions and obligations. In three experiments, we demonstrate perspective effects in nondeontic contexts, including a context much like the original one employed by Wason (1966, 1968). We suggest that perspective effects arise whenever the task uses a rule that can be interpreted biconditionally and different perspectives elicit different counterexamples that match the predicted choice sets. This view is consistent with domain-general theories but not with domain-specific theories of deontic reasoning--for example, pragmatic reasoning schemas and social contract theory--that cannot explain perspective effects in nondeontic contexts.     

The consistency of disjunctive assertions

In two experiments, we established a new phenomenon in reasoning from disjunctions of the grammatical form either A or else B, where A and B are clauses. When individuals have to assess whether pairs of assertions can be true at the same time, they tend to focus on the truth of each clause of an exclusive disjunction (and ignore the concurrent falsity of the other clause). Hence, they succumb to illusions of consistency and of inconsistency with pairs consisting of a disjunction and a conjunction (Experiment 1), and with simpler problems consisting of pairs of disjunctions, such as eIther there is a pie or else there is a cake and Either there isn't a pie or else there is a cake (Experiment 2), that appear to be consistent with one another, but in fact are not. These results corroborate the theory that reasoning depends on envisaging models of possibilities.     

Exploring the conjunction fallacy within a category learning framework

The literature presents two major theories on the cause of the conjunction fallacy. The first attributes the conjunction fallacy to the representativeness heuristic. The second suggests that the conjunction fallacy is caused by people combining p(A) and p(B) into p(A&B) in an inappropriate manner. These two theories were contrasted in two category‐learning experiments. As predicted by the latter theory, data showed that participants that could assess p(A&B) directly made fewer conjunction fallacies than participants who had to compute p(A) and p(B) separately and then combine them into p(A&B). Least conjunction fallacies were observed in the cases where the representativeness heuristic was applicable. Overall, data showed that an inability to appropriately combine probabilities is one of the key cognitive mechanisms behind the conjunction fallacy. Copyright © 2008 John Wiley & Sons, Ltd.     

主体关联性对贝叶斯推理概率估计的影响

用贝叶斯推理问题为实验材料,探讨了主体关联性对贝叶斯推理概率估计的影响。结果表明,当估计的事件与主体有关时,被试对消极事件概率估计较低,对积极事件概率估计值;当估计的事件与主体无关时,被试对消极事件和积极事件的概率估计无显著差异。反应时分析表明,被试对消极事件的概率估计比对积极事件的概率估计时间显著地长,当消极事件与主体有关时概率估计时间就更长;而对积极事件的概率估计,与主体有关和与主体无关时反应时差异不显著。这表明,被试对消极事件的概率估计(特别是消极事件与己有关时)更为慎重。     

The Generalized Quantum Episodic Memory Model

Recent evidence suggests that experienced events are often mapped to too many episodic states, including those that are logically or experimentally incompatible with one another. For example, episodic over‐distribution patterns show that the probability of accepting an item under different mutually exclusive conditions violates the disjunction rule. A related example, called subadditivity, occurs when the probability of accepting an item under mutually exclusive and exhaustive instruction conditions sums to a number >1. Both the over‐distribution effect and subadditivity have been widely observed in item and source‐memory paradigms. These phenomena are difficult to explain using standard memory frameworks, such as signal‐detection theory. A dual‐trace model called the over‐distribution (OD) model (Brainerd & Reyna, 2008) can explain the episodic over‐distribution effect, but not subadditivity. Our goal is to develop a model that can explain both effects. In this paper, we propose the Generalized Quantum Episodic Memory (GQEM) model, which extends the Quantum Episodic Memory (QEM) model developed by Brainerd, Wang, and Reyna (2013). We test GQEM by comparing it to the OD model using data from a novel item‐memory experiment and a previously published source‐memory experiment (Kellen, Singmann, & Klauer, 2014) examining the over‐distribution effect. Using the best‐fit parameters from the over‐distribution experiments, we conclude by showing that the GQEM model can also account for subadditivity. Overall these results add to a growing body of evidence suggesting that quantum probability theory is a valuable tool in modeling recognition memory.     

The Micro-Category account of analogy

Here, we investigate how activation of mental representations of categories during analogical reasoning influences subsequent cognitive processing. Specifically, we present and test the central predictions of the "Micro-Category" account of analogy. This account emphasizes the role of categories in aligning terms for analogical mapping. In a semantic priming paradigm, a four-word analogy task was compared to two other four-word tasks. Stimuli were identical in all tasks; only the instructions given to participants differed. Participants were instructed to identify analogy relations, category relations, or conventionalized semantic relations in the four-word sets. After each four-word set, a single target word appeared and participants named this word aloud. Target words that referred to category relations in the preceding four-word sets were primed as strongly when participants identified analogies as when participants identified categories, suggesting that activation of category concepts plays an important role in analogical thinking. In addition, priming of category-referent words in the analogy and category tasks was significantly greater than priming of these words when participants identified conventionalized semantic relations. Since identical stimuli were used in all conditions, this finding indicates that it is the activation of category relations, distinct from any effect of basic semantic association, that causes analogical reasoning to prime category-referent words. We delineate how the "Micro-Category" account of analogy predicts these phenomena and unifies findings from diverse areas of research concerning analogical reasoning.     

命题内容对青少年条件推理的影响

选取3个具体内容的条件命题作为实验材料,以小四、初一、高一、大三的学生为被试,探讨了命题内容对青少年条件推理的影响机制及其发展特点。结果表明:(1)对同一年级而言,不同内容的条件命题的相同推理(MP、MT、DA、AC)之间表现出显著的差异;对不同年级而言,相同内容的条件命题的四种推理之间也存在显著的差异。(2)青少年的条件推理过程似乎是一种基于对事件发生概率估计的直觉判断,这一判断过程主要取决于个体知识经验的增长和主体认知水平的提高,而用形式逻辑的标准来衡量个体条件推理能力的高低似乎并不妥当。