According to Tversky and Kahneman (1983), ‘A conjunction can be more representative than one of its constituents, and instances of a specific category can be easier to imagine or to retrieve than instances of a more inclusive category’. This has been explained by Tversky and Kahneman with the most basic qualitative law of probability known as the conjunction rule. The conjunction rule has been explained as follows: ‘The probability of a conjunction, P(A&B), cannot exceed the probabilities of its constituents, P(A) and .P(B), because the extension (or the possibility set) of the conjunction is included in the extension of its constituents’. This means that individual components already include components of the conjunction as properties of constituents would also be included in properties of the conjunction.
This thesis goes further into the Jonnson & Hampton journal article – The Inverse Conjunction Fallacy (2006) and considers the article as a starting point for further research. The discussion uses empirical and research evidence to examine the various dimensions of the inverse conjunction fallacy.
The article by Jonnson and Hampton begins with the following point – ‘If people believe that some property is true of all members of a class such as sofas, then they should also believe that the same property is true of all members of a conjunctively defined subset of that class such as uncomfortable handmade sofas’.
Yet according to the authors, a series of experiments could actually demonstrate that such assumptions and logical constraints may not be necessarily true and this condition is termed as the inverse conjunction fallacy. Generalisations may be true and we tend to believe in certain general assumptions, but specific subsets of such beliefs may not be immediately assumed. Thus, even if we have a general belief that all charities are beneficial, it may not be true that a specific charity located in the US is necessarily equally good or beneficial. As far as people’s acceptance of beliefs are concerned, people in general tend to accept the validity of the more general belief than the specific ones and when they accepted both the general and specific beliefs, they naturally gave more importance to the general rather than the specific one. This could be a demonstration in fallacious or faulty reasoning and alternative accounts and conditions could be considered in terms of intentional reasoning or where the reasoning is guided by one’s intentions.
The concepts of intentional and fallacious reasoning could then be compared with extensional and intuitive reasoning and the truths or validity of all these types of reasoning could be asserted or examined. At the moment, for the purposes of this research, however, the inverse conjunction fallacy has been considered in terms of all types of reasoning and the cases have been taken from the article by Johnson and Hampton and, following the direction of the article, the experiments and especially experiment 4 has been taken further to understand the significance, if any, of the inverse conjunction fallacy.
The main research question is whether the inverse conjunction fallacy would translate into negated or particular forms, or whether it is specific to the universal affirmative. We then explore this question in accordance with the findings of the study and the different explanations of the result.
This research project consisted of 74 participants, from 18 to 48 years, with a mean of 23 years and all of them were undergraduate students at City University, London. Fifty six participants were female, ten were male and eight participants did not state their gender. Each participant was given one booklet consisting of a cover sheet of instructions and three pages containing 64 statements. The 64 sentences in each of the eight booklets consisted of 40 target & 24 filler sentences. The numbers 1 through 10 appeared to the right of each statement. Participants were instructed to circle a number indicating how likely to be true they thought each statement was (1 = very unlikely; 10 = very likely). Fifteen minutes was allotted for the task, but most participants finished in 10 minutes. Modifiers were chosen so that they did not affect the predicate, and the statement stayed conceivably true. Within each booklet, half the target statements were modified and half unmodified.
There are various explanations of the conjunction fallacy than has opened up debates on the rationality of human reasoning and the limitations of reasoning. A satisfying account or explanation of the conjunction effects or conjunction fallacy is yet not attained and probability judgments are typically guided by confirmation relations of Bayesian theory. Crupi et al (2007) have suggested that the conjunction effect is not a fallacy and provided references to confirmation theoretic accounts.
Some researchers have however distinguished between the conjunction effect and the fallacy bringing out psychological dimensions of human reasoning and the associated flaws. On the other hand, alternative arguments have been provided by researchers as well and Sides et al (2001) have suggested that ‘attributing higher “probability” to a sentence of form p-and-q compared to p is a reasoning fallacy only if (a) the word “probability” carries its modern, technical meaning, and (b) the sentence p is interpreted as a conjunct of the conjunction p-and-q’ (Sides et al 2001).
Tversky and Kahneman dealt exclusively with the concepts of possibilities and uncertainty and suggested that uncertainty is an unavoidable condition as we tend to base our choices on our beliefs, and uncertain events could be closely related to intuitive inferences. The paper also considered probability theory in the following manner – ‘if A is more probable than B then the complement of A must be less probable than the complement of B’. The laws of probability formally derive from extensional considerations as a probability measure is defined on the basis of a family of events. In a probability scenario, each event is construed as a set of possibilities or having some unique components and these components add up to form a probability set, such as the three ways of getting the value of 10 on a throw of a pair of dice. As Sides have suggested the word probability means approvable opinion and this is done with appeal to authority or evidential support.
The probability of an event would thus equal the sum of the probabilities of the outcomes or which can be defined as its components. Although probability theory was essentially developed to explain chance processes, the theory has been successfully applied to explain unique events and favourable outcomes. Probability theory would suggest that the set of possibilities associated with a conjunction A&B is included in the set of possibilities associated only with A or only with B, the same principle can also be expressed by the conjunction rule /•(A&B) < P(B): A conjunction cannot be more probable than one of its constituents. As the authors have suggested, people do not normally tend to think of daily events as separate or distinctive lists of possibilities but rather as unique entire events and also do not evaluate compound probabilities by aggregating or adding up elementary ones. Thus small incidents do not necessarily mean that larger events would necessarily occur or that larger events are necessarily made up of smaller ones. We have schemata and prototypes according to which we store and process information and events have representative mental models and this may be an essential aspect of understanding probability. If a sample of individuals has the same attributes, they are seen as similar. Representativeness would also relate to frequency so something that is found in abundance would be more representative. If in America the maximum number of men is 5 ft 10 in tall, then it would be assumed that such men are representative of the male population in America. However, this concept could be controversial. To explain the concept of conjunction, Tversky and Kahneman give the following example of two personality sketches – Bill and Linda – to differentiate between constituents and conjunctions: As given in their 1983 paper:
“Bill is 34 years old. He is intelligent, but unimaginative, compulsive, and generally lifeless. In school, he was strong in mathematics but weak in social studies and humanities.
Bill is a physician who plays poker for a hobby.
Bill is an architect.
Bill is an accountant. (A)
Bill plays jazz for a hobby. (J)
Bill surfs for a hobby.
Bill is a reporter.
Bill is an accountant who plays jazz for a hobby. (A&J)
Bill climbs mountains for a hobby.
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.
Linda is a teacher in elementary school.
Linda works in a bookstore and takes Yoga classes.
Linda is active in the feminist movement. (F)
Linda is a psychiatric social worker.
Linda is a member of the League of Women Voters.
Linda is a bank teller. (T)
Linda is an insurance salesperson.
Linda is a bank teller and is active in the feminist movement.
The description of Bill was constructed to be representative of an accountant (A) and unrepresentative of a person who plays jazz for a hobby (J). The description of Linda was constructed to be representative of an active feminist (F) and unrepresentative of a bank teller (T)”.
Hertwig and Chase (1998) highlighted forty years of experimental inferences, the problem of categorizing and class and probabilistic explanations of human reasoning and the conjunction fallacy is simply seen as an extension of this rather than as a new and unique problem. Hertwig and Chase discussed the Bill and Linda problem and suggested that class inclusion or categorizing according to certain similar or common features and attributes is especially characteristic of the personality sketches by Tversky and Kahneman. The probability estimates and ranks could explain differences in the response mode although Hertwig and Chase try to explain when and why people adhere to class inclusion and how participant influences matter.
These examples show how components and conjunctions could be related and how predictive facts or preconceptions determine expectations.
This phenomenon associated with the summation of probability of components is demonstrated in a variety of contexts including estimation of word frequency, personality judgment and possible behaviour, medical prognosis and chances of illness, decisions making processed and that under risk, suspicion of criminal acts and detection of crime, and political forecasting and election predictions. Systematic violations of the conjunction rule are observed in the judgments of both laymen and of experts (Tversky and Kahneman1983).
Experimental tests that would help decipher the conjunction rule could be: indirect tests, direct-subtle tests and direct-transparent tests. In the indirect tests, one group of subjects evaluates the probability of the conjunction, and a second group of individuals evaluates the comparative probability of the constituents that make up the conjunction. With an adequate idea of conjunction and probability of events and its constituents, it is useful to move to the next stage of inverse conjunction fallacy as described by Jonsson and Hampton.
Sloman (1998) suggested that rational inference as happens in probabilistic conditions could be associated with category inclusion. However, people may not apply this principle when evaluating categorical arguments and Sloman discussed non explainable predicates – all electronic equipment has parts made of germanium, therefore all stereos have parts made of germanium.
In most cases the category inclusion may not be applied despite the fact that stereos are electronic equipment and judgments are in proportion of similarity between premise and end categories.
Mosconi and Macchi (2001) began by discussion the validity of the conjunction fallacy of Tversky and Kahneman and suggested the importance of conversational rules or what is being said or spoken or how an idea is communicated. If conversational rules are not followed, even logical answers don’t seem acceptable. Uninformative questions could involve a comparison between inclusive and included classes although such questions are useful only under certain specific conditions. If the context is marked or specified in some manner, the conjunction fallacy does not then remain valid. Mosconi and Macchi also described other critical approaches considering a classical pragmatic view and a frequentist view.
In a relevant paper on conjunction errors and causes for these errors, Crandall et al (1986) provided a different explanation. They assessed conjunction errors in 166 undergraduates and tried to understand the reasons for these errors and whether they were statistical or linguistic confusions. Subjects were trained either in linguistic or statistical rules or even given placebo and the results indicated that statistical training reduced errors although linguistic training had no effect. Although linguistic training could improve recognition accuracy with conjunctions, (but not performance on conjunction problems), so linguistic training could still be considered as effective. Crandall et al have suggested that conjunction errors are largely dependent on statistical errors rather than linguistic functions (Crandall et al 1986).
Fisk (1996) argued that several researchers have suggested that the conjunction fallacy may not be a real fallacy although it may be a case of judgmental problem of reverse probabilities P(A&B|X) or P(B|X). This is a violation of conjunction rule but in accordance with the law of probability and the results showed that there was a drop in conjunction fallacy scenarios and participants could somehow distinguish between conjunction fallacies and probabilistic law. Despite this, however, the fallacy was still noted among participants and many participants’ judgment of P(X|A&B) > P(X|B) probabilistic instances may not be consistent with other probabilistic theorems such as that of Bayes.
According to Osherson et al (1990), Tversky and Kahneman’s pioneering work could lead to a defined prediction: ‘The judged probability that an instance belongs to a category is an increasing function of the typicality of the instance in the category’. This means the closer is an instance or a component to a category and fits into the other features of the category, the higher is the likelihood of it being judged as belonging to the category. In an experiment, subjects rated how typical was a person of a certain category and some other subjects rated the probability of a person belonging to a category and in certain cases, 1) subjects rated a particular person as more typical of a conjunctive category (a conjunction effect); (2) in some cases subjects rated that it is more probable that the person would belong to the conjunctive category (a conjunction fallacy); and (3) the magnitudes of the conjunction effect and fallacy were found to be highly correlated so when there’s the conjunction effect, high conjunctive fallacy was expected. This is where Osherson et al also discussed the significance of inclusion fallacy or including falsely to people in certain categories on the basis of some preconceived notions.
This project has been based on Jonnson and Hampton’s paper on the inverse conjunction fallacy. Another paper on modifier effects by Jonsson and Hampton shows that the modifier effect results in the reduction in judged likelihood of a generic statement about the property of a concept modified. The authors argue that the mutability of the property of an unmodified concept tends to affect its truth as a modified concept. The modifier effect was also found to be associated with pragmatic knowledge-based reasoning
In certain cases subjective probability judgments could be judged such that conjunction of two events is judged to be more likely than probability that either of the events will be occurring separately. According to Besch and Fiedler (1999) conjunction effects have usually been explained on the basis of representativeness or psychological relations between constituents although the contemporary approach stresses on inferred information. The explicit information as it relates to any stimulus is changed to implicit mental models and as such a mental models approach rather than component events approach has been found to be responsible for probability judgments. Mental models could be explained or fit around conjunctions and a change in conjunction effects judgments could be noted.
To understand reasoning and probabilistic thinking, the experiment was done in the following way – each of the 74 participants was given one booklet consisting of a cover sheet of instructions and three pages containing 64 statements. The 64 sentences in each of the eight booklets consisted of 40 target & 24 filler sentences. The numbers 1 through 10 appeared to the right of each statement. Participants were instructed to circle a number indicating how likely to be true they thought each statement was (1 = very unlikely; 10 = very likely).
- A) Modifier effect generated an inverse conjunction fallacy with all sentences, as before.
All N are P 7.3
All MN are P 6.3
This result suggests that when one constituent or component is considered as true or believed to have certain attributes, it is naturally assumed that its conjunctive effect with another constituent will also continue to have the same attribute or remain true or valid.
When we then looked at the converse of these sentences, which should be false as the others are true: i.e.
Some N are not P 5.1
Some MN are not P 5.2
The modifier effect disappears. The responses are closer to the middle of the scale, so maybe confidence is having less effect. The converse of the sentences that if a constituent is not valid, its conjunctive effect is also not valid gets more approval from the participants than the positive statement that when a constituent is valid, the conjunctive will also have to be valid which in some cases have met with scepticism.
- B) we then looked at false predicates, generated from true ones by finding an opposite (e.g. jeans are delicate). So the “No N are Q” sentences should be true, and “Some N are Q” would be false
No N are Q 5.8
No MN are Q 5.5
The modifier has a reduced effect here, but still significant.
Modifiers like these would indicate false predicates and true events could be found by examining the opposite so if no constituents are valid then the conjunction of constituents would also not be valid. It also goes that if no constituents are valid, then the proposition that some constituents are true would also not be valid. If no dogs have five legs then a statement that some dogs have five legs would be false.
The converse of these were
Some N are Q 4.5
Some MN are Q 4.1
Modifiers reduced truth again here – even though the sentences were more false than true. So modifiers made partly false sentences more clearly false. This clearly shows how the validity of components is determined as when none of the components are true, some of them would also not be true.
The result (A) suggests that the likelihood of a counterexample (some ravens are not black) is the SAME for unmodified and modified sentences. So the difference for ALL is not owing to it being more likely that some MN are not P than that some N are not P.
The results in (B) are harder to understand – there is a reduced modifier effect (.3 rather than 1.0) that is the same in both versions.
The “fallacy” here is in supposing that No N are Q is more likely than No MN are Q. That comes out as significant, so the fallacy is also found with false statements and negation. The Some N are Q sentences are by contrast logically consistent – if Some MN are Q, it follows that Some N are Q, but not vice versa, so the unmodified should be more likely.
The differences in modifier and quantifier effects in case of No components are true and some components are true are shown here.
|UNMOD||7.3 (1.4)||5.1 (1.7)||6.2|
|MOD||6.3 (1.0)||5.2 (1.0)||5.8|
This table shows that usually for a universal affirmative that if A is C then AB is C is usually true in higher cases than the negative propositions that if A is not C then AB will also be not C. Overall scores of modifier effects are lower and universal affirmative and positive statements are higher suggesting that people are more drawn towards the possibilities of occurrences of positive and affirmative sentences and instances rather than the negative instances and events. This is especially true for positive sentences and propositions.
|UNMOD||5.8 (1.5)||4.5 (1.5)||5.15|
|MOD||5.5 (1.4)||4.1 (1.2)||4.8|
For negative propositions, the universal negative statements are more in sync with expectations and thus are considered more likely than the affirmative statements so the scores reported here are higher for negative possibilities rather than affirmative possibilities. The overall scores are lower with modifier effects.
Investigating the effects of: a) sentence type (mod vs. unmod) b) kind of categorical proposition considering the contingency table, we can firstly conclude that unmodified statements were most likely to be judged truthful. It is seen from both the instances that unmodified statements are usually more likely to be considered as valid or true than modified statements and propositions. Thus categorical differentiation would exist in this case of modifiers as compared with unmodified propositions.
Means and standard deviations for the positive unmodified and modified sentences and overall can be seen in table 1.
There is a noticeable difference between the perceived truthfulness for the Modified and Unmodified statements, as well as between Universal Affirmative and Universal negative statements. The Standard Deviations of the scores are [low, indicating that variability among scores was relatively small]
Considering the contingency table, we can firstly conclude that unmodified statements were most likely to be judged truthful. Unmodified universal affirmative statements were judged as more truthful than unmodified particular negative statements (7.3 vs. 5.1) ->
Difference in the veracity with particular negative statements is – not significant, – negligible
Unmodified statements are/were judged more truthful than modified statements (6.2 vs. 5.8)
Universal affirmative statements are/were judged more truthful than particular negative statements (6.8 vs. 5.2). This indicates the power of influence of unmodified propositions when compared with modified ones and the positive sentences are generally more acceptable either as an affirmative or as a negative than the negative or converse propositions in general.
|UNMOD||7.3 (1.4)||5.1 (1.7)||6.2|
|MOD||6.3 (1.0)||5.2 (1.0)||5.8|
The means in Table 1 are illustrated for greater clarity in Figure 1.
Figure 1: Veracity of positive predicate statements as a function of quantifier type and sentence type
A two-way ANOVA was run with a quantifier type (universal affirmative vs. particular negative as between participants [and within items] and a type of the statement (modified vs. unmodified) within participants factors.
MAIN EFECT: Q -> F (1, 32) = 33.11, p < .001 KOLIKO DECIMALA?
M -> F (1, 32) = 11.4, p = .002
INTERACTION: M x Q -> F (1,32) = 4.91, p = .034 ?? KAKO NAPISATI? p < .05 ?
Q x M
The type of quantifier:
- a) affected the statements’ validity / veracity
- b) affected how the participants judge the statement’s validity
as in general universal affirmative statements were judged as more likely than particular negatives and the probabilistic laws as determining expectations in both these cases differed significantly.
Main effect of quantifier
The analysis confirmed a significant effect of a quantifier type (F(1, 32) = 33.11, p < .001), confirming the hypothesis that the reasoning in humans is significantly affected by affirmative and positive conditions as also by negative connotations and that quantifiers or affirmative versus negative quantifier types would determine how participants react to the proposition that certain events or statements are more likely to be true or valid.
The ANOVA output (appendix 1) shows that the factor of type of quantifier had a significant main effect (F(1,32) = 4.91, p = .034 ?? p < .05), indicating that the type of quantifier affected which type of statement will be validated/judged as more truthful.
The effect of quantifier on the veracity of the statement was dependent on whether the statements were modified or unmodified and the interaction of the quantifiers as seen from the graphs. In general, unmodified statements were found to be more acceptable than modified ones.
The perceived veracity of the statement is however also equally important and Unmodified universal affirmative statements were judged to be the most truthful (M = 7.3), whereas unmodified particular negative statements were pronounced the least truthful (M = 5.1)
The table shows that universal affirmative statements were validated/verified as more truthful than particular negative ones/statements (M = 5.2). Moreover, unmodified statements were judged to be more truthful (M = 6.2) than modified ones (M = 5.8)
Main effect of modifier
The main effect of modifier was also significant (F (1, 32) = 11.4, p = .002)
Interaction effect of type of modifier and type of statement on the perceived veracity of the statement or the proposition or the conjunction effect in general is seen here:
(F (1,32) = 4.91, p = .034)
|UNMOD||5.8 (1.5)||4.5 (1.5)||5.15|
|MOD||5.5 (1.4)||4.1 (1.2)||4.8|
The means in Table 2 are also shown for an ease of appreciation in graphical form in Figure 2.
Figure 2: Veracity of anti-predicate statements as a function of quantifier type and sentence type
MAIN EFECT: Q -> F (1, 45) = 32.72, p < .001 KOLIKO DECIMALA?
M -> F (1, 45) = 6.29, p = .016
INTERACTION: M x Q -> F (1,45) = .03, p > .05
The main effect of quantifier
The main effect of modifier
The quantifier x modifier interaction was/is (F (1,45) = .03, p = .876) not significant.
The table shows that universal negative statements (M = 5.65) were validated/verified as more truthful than particular affirmative ones/statements (M = 4.3)
Moreover, unmodified statements were judged to be more truthful (M = 5.1) than modified ones (M = 4.8)
There was no significant interaction effect between the type of modifier and the type of quantifier (F (1,45) = .03, p > .05)
The investigation of means for Universal Negative and Particular affirmative statements for both types of modifier are shown in the table below (Table 2).
In this table, a score indicates that unmodified statements are/were judged more truthful than modified statements (5.1 vs. 4.8)
Unmodified universal negative statements are/were judged more truthful than unmodified particular affirmative statements (5.8 vs. 4.5)
This would require some explanation as to why unmodified negative statements were judged more truthful than unmodified affirmative statements. Any instance or proposition for example – that all birds do not fly seem to be considered more likely than most birds fly so negative statements are considered simply more plausible than affirmative ones.
Unmodified universal negative statements were judged to be the most truthful (M = 5.8), whereas modified particular affirmative statements were pronounced the least truthful (M = 4.1)
In fact the highest scores as being considered as most likely were the negative statements. For example if we suggest that the sun does not set in the east is considered more plausible than when we say that the sun sets in the west which is more of an affirmative statement. An analysis of this would mean that the human reasoning is more likely to accept negative statements than positive ones in case of universal truths as we may like to believe and assert that which cannot happen rather than the possibilities of something happening.
Jonsson and Hampton investigated the inverse conjunction fallacy using four experiments.
The first two experiments were designed to investigate whether people are actually prone to make inverse conjunction fallacies. The experiments were carried out with student participants and tested in two designs. In the between-subjects design (Experiment 1) the two versions of one statement were divided and presented to two groups of participants whereas in the within-subjects design (Experiment 2) both versions of a particular statement were presented for veracity to the same individuals. Results were analysed in terms of the Yes/No, Yes/Yes and No/No versions of the possible fallacies (Jonsson and Hampton 2006). In Experiment 3 the presumption that in many cases participants see the modified noun phrase as referring to a subset of the unmodified noun was tested and 66 sentences with yes/no propositions were given for consideration.
The fourth experiment studied by Jonsson and Hampton emphasised the significance of the word ‘All’. An alternative explanation for the fallacy is by interpreting the word ‘‘All’’ as mapping onto the universal quantifier in logic. All may signify some kind of universal quantifier, although Jonsson and Hampton have suggested that we may frequently use sentences starting with ‘‘All’’ which is habitually done and is a common way of communicating, yet this sort of word use may be strongly indicative of a generic rather than a strictly universal quantification.
In this project, Jonsson and Hampton’s fourth experiment has been taken further to understand not just the significance of all as a word used to specify universal quantifier conditions but rather the difference between universal and generic modes or propositions and the role of modifiers and quantifiers in suggesting likelihood of events and propositions. This study also goes a step further in trying to analyse the difference between unmodified and modified sentences and how these would affect participant decision on whether certain propositions are true, valid or more likely. The results of the study not only show the perception of generic quantifiers but also suggest the significance of unmodified sentences as having some influence in the determination of truth, validity or likelihood of events.
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