Counterexamples and the Suppression of Inferences

Journal of Memory and Language 40, 347–373 (1999) Article ID jmla.1998.2622, available online at http://www.idealibrary.com on

Counterexamples and the Suppression of Inferences Ruth M. J. Byrne Trinity College, University of Dublin, Dublin, Ireland

and Orlando Espino and Carlos Santamaria University of La Laguna, Tenerife, Spain Reasoners resist even the valid inferences, such as If Lisa met her friend then she went to a play, Lisa met her friend, therefore she went to a play, when they are given an additional condition, e.g., If Lisa had enough money then she went to a play. Four experiments test alternative accounts of this suppression effect. The first experiment shows that inferences are suppressed even when reasoners produce their own conclusions, based on the first or second conditional. The second experiment shows that suppression occurs even when the premises are expanded to rule out the converse. The third experiment shows that inferences are suppressed more when the background conditions are expressed in a biconditional. The fourth experiment shows that both valid and fallacious inferences can be suppressed when the premises refer to alternatives as well as to additional conditions. We suggest that suppression rests on the availability of counterexamples. © 1999 Academic Press

Imagine you are working in a nuclear power plant during an emergency and your job is to work out what is going wrong. Suppose you know the general rule, “if the test is to continue, the turbine must be rotating fast enough to generate emergency power,” and suppose the computer monitoring system informs you, “the turbine is not rotating fast enough.” What should you infer? The valid modus tollens inference is to infer the conclusion, “the test cannot continue.” But people have difficulty making some inferences. The power plant workers in Chernobyl did not infer that they should discontinue We are deeply indebted as always to Phil Johnson-Laird for helpful discussions on the model theory. We are grateful to Denise Cummins, Mike Oaksford, David O’Brien, David Over, and Kristien Dieussaert for helpful comments on an earlier draft. Thanks also to Mark Keane, Simon Handley, and Walter Schaeken for discussions of the suppression effect and to David Chan for providing us with his data. Some of the research was supported by a grant from the Consejeria de Educacion del Gobierno de Canarias. Some of the experimental results were presented at the International Workshop on Reasoning Strategies, Belgium, 1998. Address correspondence to Ruth Byrne, Psychology Department, University of Dublin, Trinity College, Dublin 2, Ireland. Fax: 353-1-6712006. E-mail: [email protected]; http://www.tcd.ie/Psychology/People/Ruth_Byrne/. 347

the test in time, and as a result partly of this inferential difficulty, the Chernobyl nuclear disaster occurred, with worldwide implications (Johnson-Laird, 1994a; Medvedev, 1991). If we are to avoid repeating the mistakes of the past, we need to examine those mistakes carefully and establish how they came about. The competent development of inferential aids, such as artificial intelligence reasoning tools to overcome detected human inferential frailties, depends on an adequate explanation for why people have difficulty making some inferences. The modus tollens inference may be difficult for many reasons, including limitations of working memory and vagaries in the interpretation of rules (see e.g., Evans, Newstead, & Byrne, 1993, for a review). It may even be suppressed because reasoners bring to bear their own unstated assumptions: The workers at Chernobyl may not have made the inference because they considered additional assumptions, such as whether the safety procedures had to be followed to the letter in such a case (Byrne, 1997a). In this paper we will examine the suppression of valid inferences that conflict with back0749-596X/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.

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BYRNE, ESPINO, AND SANTAMARIA TABLE 1 Four Inferences from a Conditional Modus ponens If Lisa met her friend, she went to a play. Lisa met her friend. Therefore, she went to a play. Affirmation of the consequent If Lisa met her friend, she went to a play. Lisa went to a play. Therefore, she met her friend.

ground assumptions (Byrne, 1989). Our suggestion is that the suppression of valid inferences occurs because background knowledge results in the explicit mental representation of a counterexample to a putative conclusion. Human rationality may depend on the semantic principle that a conclusion is valid if there are no counterexamples to it, that is, there are no states of affairs in which the premises are true but the conclusion is false (Johnson-Laird, 1983; Johnson-Laird & Byrne, 1991). A considerable body of evidence addresses the idea that people attempt to keep in mind alternative states of the world, or mental models, when they reason deductively (e.g., Johnson-Laird & Byrne, 1991). But, it has proved more difficult to examine experimentally the search for counterexamples, and it remains the case that little is known about how counterexample search is carried out (Johnson-Laird & Byrne, 1991). We will suggest that the suppression of inferences provides one way to begin to examine the use of counterexamples, especially when they are made explicitly available. THE SUPPRESSION OF INFERENCES Four main sorts of inferences can be constructed for a conditional, such as “If Lisa met her friend then she went to a play,” by affirming or negating the first component, the antecedent, and the second component, the consequent, as Table 1 shows. Two of the inferences are valid, the modus ponens and tollens inferences in Table 1, and two are fallacies, the affirmation of the consequent and denial of the antecedent inferences, according to the “material implication” interpretation of the conditional (e.g., Jef-

Modus tollens If Lisa met her friend, she went to a play. Lisa did not go to a play. Therefore, she did not meet her friend. Denial of the antecedent If Lisa met her friend, she went to a play. Lisa did not meet her friend. Therefore, she did not go to a play.

frey, 1981). On this interpretation, the antecedent “Lisa met her friend” is sufficient but not necessary for the consequent “She went to a play.” The conditional is true when Lisa met her friend and went to a play, false when Lisa met her friend and did not go to a play, true when Lisa did not meet her friend and went to the play, and true when she did not meet her friend and did not go to a play, as Table 2 shows. According to the “material equivalence” interpretation of “if” as a biconditional “if and only if,” all four inferences are valid (e.g., Jeffrey, 1981). On this interpretation, the antecedent “Lisa met her friend” is both sufficient and necessary for the consequent “She went to a play,” as Table 2 shows. The conditional and biconditional interpretations have received considerable attention in research on conditional inference, but there are also two further interpretations, less well researched, and they are relevant to the empirical phenomena that we will examine in this paper. One of these interpretations is the “reversed conditional” interpretation, in which the antecedent, “Lisa met her friend,” is not sufficient but is necessary for the consequent “She went to a play” (see Staudenmayer, 1975). As Table 2 shows, on this interpretation the affirmation of the consequent and denial of the antecedent inferences are supported, and the modus ponens and tollens inferences are not. The final interpretation is a “nonconditional” one, where the antecedent is neither sufficient nor necessary for the consequent (see Staudenmayer, 1975). As Table 2 shows, on this interpretation none of the inferences are supported. In experiments, almost all participants tend to

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COUNTEREXAMPLES AND INFERENCES TABLE 2

Different Interpretations of “If” Illustrated for the Assertion “If Lisa Met Her Friend Then She Went to a Play”

Interpretation

Models

Inferences supported

Antecedent sufficiency/necessity

Conditional

friend not-friend not-friend

play not-play play

MP,MT (not DA, AC)

Sufficient, not necessary

Biconditional

friend not-friend

play not-play

MP, MT, DA, AC

Sufficient, necessary

Reversed Conditional

friend not-friend friend

play not-play not-play

DA, AC (not MP, MT)

Not sufficient, necessary

Nonconditional

friend not-friend not-friend friend

play not-play play not-play

not MP,MT nor DA, AC

Not sufficient, not necessary

make the modus ponens inference, but fewer people make the modus tollens, denial of the antecedent, and affirmation of the consequent inferences (see Evans et al., 1993). However, the frequency with which they make these inferences can be influenced by manipulating the interpretation they reach of the conditional, and one dramatic demonstration of this phenomenon is the suppression of inferences. The fallacies of the denial of the antecedent and affirmation of the consequent can be suppressed by providing alternative conditions that could bring about the consequent (Rumain, Connell, & Braine, 1983). Many people make the denial of the antecedent fallacy (see Evans, et al., 1993) when they are asked to select the valid conclusion from among a set of conclusions for premises of the sort “If Lisa met her friend then she went to a play, Lisa did not meet her friend.” The correct conclusion is (c) from among the set (a) she went to a play, (b) she did not go to a play, (c) she may or may not have gone to a play, but people frequently choose (b). The inference can be suppressed by providing information about alternatives, such as “If Lisa met her friend then she went to a play, if Lisa met her brother then she went to a play, Lisa did not meet her friend.” For such premises, people choose (c) from the set of conclusions (e.g. Rumain, Connell, & Braine, 1983). In this case,

“Alternative antecedents may or may not be mutually exclusive, but each is sufficient for the consequent” (Byrne, 1989, p. 66). There is a similar suppression of the related fallacy of the affirmation of the consequent inference. The fallacies can even be suppressed implicitly, by requiring participants to generate alternatives themselves (Markovits, 1984, see also Staudenmayer, 1975). People endorse the two valid inferences but refrain from the fallacies for conditionals that contain alternatives, and this pattern is consistent with the material implication interpretation of the conditional. The valid inferences can be suppressed in a similar manner to the suppression of the fallacies (Byrne, 1989). Most participants when asked to select the correct conclusion for the premises, “If Lisa met her friend then she went to a play, Lisa met her friend”, make the valid modus ponens inference and choose (a) from the set (a) she went to a play, (b) she did not go to a play, (c) she may or may not have gone to a play. The inference can be suppressed by providing information about additional background conditions, such as “If Lisa met her friend then she went to a play, if Lisa had enough money then she went to a play, Lisa met her friend.” For such premises, people choose (c) from the set of conclusions (Byrne, 1989). In this case, “Additional conditions are jointly necessary for

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the outcome” (Byrne & Johnson-Laird, 1992, p. 93). There is a similar suppression of the related valid inference of modus tollens. The valid inferences can also be suppressed implicitly, by requiring participants to generate additional background conditions themselves (Cummins, Lubart, Alksnis, & Rist, 1991), and the more additional conditions participants can think of, the greater the suppression of the inferences (Elio, 1997; see also Thompson, 1994, 1995). People endorse the two fallacies but refrain from the valid inferences for conditionals that contain additional conditions, and this pattern is consistent with the “reversed conditional” interpretation. We suggest that the inferences people make are influenced by the explicit availability of counterexamples in their mental representations of the information in the premises.

“not” is a propositional-like tag to indicate negation, and separate models are represented on separate lines (Johnson-Laird, Byrne, & Schaeken, 1992). The three models correspond to a “material implication” interpretation of the premise, but people who interpret “if” as “if and only if” will consider the premise consistent with the first two models only, which correspond to a biconditional or “material equivalence” interpretation. Because of the constraints of working memory, the initial representation of the premise may contain explicit information corresponding only to the elements mentioned, perhaps because they are considered most relevant (e.g., Sperber, Cara, & Girotto, 1992) and the rest of the information may be represented implicitly: friend

play

COUNTEREXAMPLES IN MODELS

...

One existing theory of reasoning proposes that counterexample availability is central to the deductive process, and we will outline this theory before we report the results of our experiments on the availability of counterexamples and the suppression of inferences. According to the model theory of deduction, people make inferences according to the semantic principle that a conclusion is valid if there are no counterexamples to it, that is, if there are no situations in which the premises are true but the conclusion is false (Johnson-Laird & Byrne, 1991). The model theory proposes that the underlying mechanism of inference relies on rules that guide the construction of mental models (e.g., Johnson-Laird, 1983; Johnson-Laird & Byrne, 1991). Mental models correspond to the way the world would be if the premises were true. A conditional, such as “If Lisa met her friend then she went to a play,” is consistent with three alternative situations in the world. People represent what is true in their models, as indicated in the following diagram:

The three dots indicate a wholly implicit model, that is, reasoners keep in mind that there may be alternatives to the first model but they do not represent these alternatives explicitly. The initial set of models can be fleshed out to be more explicit and models may contain “mental footnotes” to indicate how they are to be fleshed out, e.g.,

friend not-friend not-friend

play not-play play

where “friend” indicates that Lisa met her friend, “play” indicates that she went to a play,

[friend]

play ...

where the square brackets indicate that the idea of Lisa meeting her friend has been exhaustively represented with respect to her going to a play, that is, meeting her friend will not occur in any other model without going to a play (JohnsonLaird & Byrne, 1991). A more recent computer simulation of the theory implements the idea of exhaustivity not by square brackets, but by the equivalent technical device of a mental footnote of the negation of the antecedent in the implicit model (see Johnson-Laird & Savary, 1995): friend

play . . . not-friend,

The mental footnote indicates that when the implicit model is fleshed out to be explicit it must contain “not friend” rather than “friend,” but footnotes may be rapidly forgotten (John-

351

COUNTEREXAMPLES AND INFERENCES

son-Laird & Savary, 1995). The different levels of representation in the diagrams above—the fully explicit set of models, the initial set of models with implicit models but no mental footnotes, and the initial set of models with implicit models and mental footnotes (whether represented by the exhaustivity symbol or the negation of the antecedent)—may even correspond to different levels of sophistication in reasoning (Johnson-Laird, Byrne, & Schaeken, 1994; O’Brien, Braine, & Yang, 1994). The suppression of inferences may depend on the availability of counterexamples (Byrne, 1991, p. 77): . . .one of the main points of the suppression experiments, a point well appreciated by philosophers and artificial intelligence workers, but often ignored by psychologists who support the formal rule view. . .is this: In daily mental life, there are always background conditions necessary for an outcome that can be called into question; and there are always alternative conditions sufficient for an outcome that can be introduced.

For example, additional requirements, such as “If Lisa met her friend then she went to a play, if she had enough money then she went to a play,” are interpreted with background knowledge to mean that the antecedents refer to additional conditions for the same outcome, and so the antecedents are represented in a single model: friend

money

and the resulting set of models for the pair of conditionals: friend

money ...

play

makes available a counterexample to the valid inferences. Table 3 presents the output of a recent computer program developed by Phil JohnsonLaird to represent assertions of this sort according to various compositional rules for constructing mental models for compound assertions (see Johnson-Laird & Savary, 1995). As it shows, the initial set of models with implicit models but no footnotes distinguishes between “if she met her friend and she had enough money then she went to the play” and a purely conjunctive assertion such

as “she met her friend and she had enough money and she went to the play” because the presence of the implicit model indicates that there may be alternatives to the first model. The initial set of models with no footnotes does not distinguish between “if she met her friend and she had enough money then she went to the play” and the related assertion, “if she met her friend then she had enough money and she went to the play,” as Table 3 shows. However, the next level of representation, with implicit models and footnotes, distinguishes these two assertions, and the footnotes indicate that the implicit models for “if she met her friend and she had enough money then she went to the play” can be fleshed out to include the negated antecedent only: friend

money ...

play not-friend friend not-friend

not-money not-money money

The fully explicit set of models for the pair of conditionals includes seven models which is likely to far exceed working memory constraints. But people might keep in mind the situation corresponding to the explicit initial model, in which Lisa meets her friend, has enough money, and goes to a play, and they might also keep in mind that there are alternatives to this situation, and in one of those alternative situations, Lisa meets her friend and does not have enough money. Hence they do not endorse the conclusion that Lisa went to a play, because of the availability of a counterexample, that is, a situation where Lisa met her friend but she did not go to the play (because she did not have enough money). Likewise, alternative antecedents, e.g., “If Lisa met her friend then she went to a play, if she met her brother then she went to a play,” are interpreted with background knowledge to mean that the two antecedents refer to alternative conditions which are represented in separate models, e.g., friend brother

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BYRNE, ESPINO, AND SANTAMARIA TABLE 3

The Output of a Computer Program That Implements the Mental Model Theory for Various Compound Assertions Level 1 (implicit models no footnotes)

Level 2 (implicit models plus footnotes)

Level 3 (explicit models)

P QR . . . -P Q P -Q -P -Q

PQR -P-Q-R -P Q-R P-Q-R -P-Q R P-Q R -P Q R

If p and q then r PQR ...

p and q and r PQR

P QR

PQR

If p then q and r PQR ...

P QR . . . -P

PQR -P-Q-R -P Q-R -P-Q R -P Q R

If p or q (or both) then r PR QR ...

If p or q (but not both) then r PR QR ...

P QR P-QR -P QR . . . -P -Q

PQR P-Q R -P Q R -P-Q-R -P-Q R

P-QR -P QR . . . -P -Q P Q

P-Q R -P Q R -P-Q-R P Q-R PQR -P-Q R

Comments

Model (1) for TATC Models (3) for FAFC

Models (3) for FATC

Model (1), true conjuncts Model (1) for TATC Models (3) for FAFC

Model (1) for FATC Models (3) for TATC

Model (1) for FAFC Model (1) for FATC Models (2) for TATC Models (2) for FAFC Models (2) for FATC

Note. TA, true antecedent; TC, true consequent; FA, false antecedent; FC, false consequent; (-) not.

and the resulting set of models for the pair of conditionals: friend brother ...

play play

enables the discovery of a counterexample to the fallacies (Byrne, 1991), as Table 3 shows. For example, people do not endorse the conclusion that Lisa did not go to a play, because of the availability of a counterexample, that is, a situation where Lisa did not meet her friend but she did go to the play (because she met her brother). Background knowledge can affect the construction, fleshing out, and elimination of

models (Johnson-Laird & Byrne, 1991), and the interpretative component is crucial to a theory of reasoning if it is to explain the suppression effect (Fillenbaum, 1993). Our key suggestion is that the provision of an alternative or additional antecedent enables people to represent counterexamples to conclusions explicitly. We will test several alternative explanations of the suppression effect in a series of four experiments. EXPERIMENT 1: IS SUPPRESSION PREMISE DOUBT? The counterexample availability explanation of the suppression effect suggests that people do

COUNTEREXAMPLES AND INFERENCES

not consider a simple categorical conclusion, such as “Lisa goes to a play,” to be a valid conclusion for pairs of conditionals that contain additional antecedents, such as “If Lisa meets her friend then she goes to a play, if she has enough money then she goes to a play,” when only one of the antecedents is affirmed in the categorical premise “Lisa meets her friend,” because they can readily think of a counterexample to it. An alternative possibility is that the suppression of the valid inferences arises because the additional condition undermines the first premise and leads people to no longer believe its truth (George, 1995; O’Brien, 1993; Politzer & Braine, 1991). Politzer & Braine (1991 p. 105) suggest that the suppression effect “is caused by the subjects being brought to doubt the truth of the first conditional premiss.” Their emphasis is on the lack of truth of the conditional, as O’Brien (1993, p. 124) puts it, “The expanded premise set does not serve as a set of assumptions, as the first conditional is open to doubt.” When the antecedent of this doubted first premise is affirmed people do not make the modus ponens inference because they no longer have confidence in the truth of the conditional, given its inconsistency with the second conditional. Stevenson and Over (1995, p. 626) similarly emphasize doubt in the premises, suggesting that “The use of the second conditional premise in the additional condition can make subjects doubt the first premise. . .” and they propose that “The impact of the additional premise in Byrne’s study was to induce uncertainty in the first premise” (Stevenson & Over, 1995, p. 616). In a conclusion selection task, the conclusion option that incorporates this uncertainty is not the definite conclusion, e.g., “she went to a play” but the conclusion with modal operators, e.g., “she may or may not have gone to a play,” and so it is this latter option that participants choose. What does it mean to doubt the truth of a premise? In the two-valued logic of a formal inference rule system, a premise can be either true or false. There is no mechanism to calculate degrees of certainty or uncertainty in existing rule theories (e.g., Braine & O’Brien, 1991;

353

Rips, 1994) and as Stevenson and Over (1995, p. 638) point out: [Mental logic] cannot tell us on its own what qualified conclusions to infer from beliefs in which we do not have total confidence. One can supplement a mental logic with the probability calculus for this purpose, but presumably no one would suggest that our natural mental system also contains an axiomatization of the probability calculus. Even relatively simple derivations in that system are surely too technical for ordinary people.

It must be the case that to doubt the truth of a premise, e.g., “If Lisa met her friend then she went to a play,” in a two-valued (true or false) formal inference rule system, is to believe the premise to be false, “It is false that if Lisa met her friend then she went to a play.” Participants may be unwilling to make any inferences whatsoever from premises that they consider untrue (Braine & O’Brien, 1991). A more plausible suggestion may be that to doubt the truth of a premise is to consider it uncertain. It may be the case that people judge the conditional probability of the consequent given the antecedent, and their judgment is altered by the provision of additional requirements (Stevenson & Over, 1995). Participants may be willing to make inferences only to conclusions that indicate a degree of uncertainty, e.g., by using modal operators such as “may.” One prediction from this premise doubt account is that, when participants draw their own conclusions from premises in response to an open-ended “what, if anything, follows?” question, they should tend to qualify their conclusions by including modal auxilliaries such as “may,” “might,” “maybe,” and so on. Our counterexample availability explanation of the suppression effect leads to another prediction. It suggests that people disbelieve, not the truth of the premises, but the validity of the conclusion, because they can readily think of a counterexample to it. On our account, people do not doubt the truth of either of the conditional premises: A conditional that has an antecedent that is insufficient but necessary for the consequent is no more uncertain or doubted than a conditional which has an antecedent that is sufficient but not necessary. The truth conditions of

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a “reversed conditional” may be different from the truth conditions of a material implication conditional, but they can be calculated with certainty nonetheless, as Table 2 shows. The process of inference is likely to require procedures that not only construct and combine mental representations, but also procedures that describe the resultant mental representations. According to the model theory, the procedures that describe models refer to the elements in the remaining model or models by using whichever logical connective is best suited to the structure of the model and, for parsimony, by not repeating elements asserted in the categorical premises (Johnson-Laird & Byrne, 1991, Chap. 9). One prediction from this theory is that, when participants draw their own conclusions, they should tend to produce conclusions that refer to the additional condition that has not been affirmed or denied in the argument. We tested these alternative predictions by requiring participants to produce their own conclusions, asking them, “What, if anything, follows?” This conclusion production task allows us to examine the nature of the conclusions that people spontaneously produce (e.g., Johnson-Laird & Byrne, 1991). A second aim of the experiment was to examine the frequency of inferences and their suppression when the minor premise referred to the antecedent in the first conditional or the second conditional. The arguments about premise doubt have focused solely on the first premise (Politzer & Braine, 1991; Stevenson & Over, 1995), although it is plausible that the arguments can apply to either premise. According to our counterexample availability explanation of the suppression effect, modus ponens suppression should occur equally for both sorts of minor premises. For a pair of conditionals, such as “If Laura cooks a meal then she has a pleasant dinner, if there are good ingredients in the cupboard then she has a pleasant dinner,” a counterexample is available whether the modus ponens inference is based on affirming that she cooks a meal or on affirming that she has good ingredients. In the experiment we test this suggestion by examining the inferences participants make from pairs of conditionals for which

the minor premise is based on the first conditional’s antecedent (MP-1): If Laura cooks a meal then she has a pleasant dinner. If there are good ingredients in the cupboard then she has a pleasant dinner. Laura cooks a meal.

and we compare them to the inferences they make from pairs of conditionals for which the minor premise is based on the second conditional’s antecedent (MP-2): If Laura cooks a meal then she has a pleasant dinner. If there are good ingredients in the cupboard then she has a pleasant dinner. There are good ingredients in the cupboard.

Method Design and materials. We constructed two sorts of arguments, one set consisting of two conditionals that contained additional requirements and a second set consisting of a single conditional. We gave each set of arguments to a different group of participants in a betweenparticipant design. We constructed three sorts of inferences for the additionals arguments based on a modus ponens (MP) inference that affirmed the antecedent of the first conditional (MP-1), a modus ponens inference that affirmed the antecedent of the second conditional (MP-2), and a modus tollens (MT) inference that denied the consequent shared by both conditionals. Each participant carried out nine inferences: three MP-1, three MP-2, and three MT. The matched inferences for the single conditionals were six MP and three MT inferences. We assigned three different contents at random to each of the three inference forms to construct a different set of nine problems for each participant (see the Appendix). The problems were presented to the participants in their native language of Spanish. Their task was to construct a conclusion in response to the question “What, if anything, follows?” We gave each participant the nine arguments in a different random order. Procedure. We tested the participants in groups. We gave each participant a booklet and the first page contained instructions which explained the task with reference to a single-conditional modus ponens argument as an example.

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COUNTEREXAMPLES AND INFERENCES TABLE 4 The Percentages of Inferences Endorsed in the First Experiment MP-1

MP-2

MT

Conclusions:

If p then q (If r then q) p q

(If p then q) If r then q r q

If p then q (If r then q) not-q not-p/not-r

Additional conditionals Single conditionals Total

51 98 75

57 95 76

52 84 68

Premises given:

Total

53 92 73

Note. Premises in parantheses were presented to the additionals argument group only. MP, modus ponens; MT, modus tollens.

In line with the instructions used in previous experiments on suppression (e.g., Byrne, 1989; Stevenson & Over, 1995), participants were asked to assume that the premises were true. Their task was to draw the conclusion that they thought followed from the premises. They were asked to read each problem carefully and to work from beginning to end at their own pace, without changing any responses or skipping any items. Participants. The 54 participants who took part in the experiment were undergraduate students at the University of La Laguna, Tenerife, Spain. They were randomly assigned to one of two groups (n 5 27 in each). None of the participants in this, or any of the subsequent, experiments reported having received formal training in logic. Results and Discussion The results show that inferences are suppressed when participants generate their own conclusions. We carried out a two by three analysis of variance on the frequency of correct categorical responses with the between-participant factor of argument type (additional-antecedent arguments, single conditional arguments) and the within-participant factor of inference type (MP-1, MP-2, MT). It showed a main effect of argument type, F(1,52) 5 66.7, MSE 5 0.09, p , .0001, indicating a difference in the frequency of inferences made from single conditionals (92%) and from additional-ante-

cedent arguments (53%), as Table 4 shows. There was no main effect of the type of inference, F(2,104) 5 1.12, MSE 5 0.09, p 5 .33, indicating that there were no reliable differences between the overall frequency of MP-1 (75%), MP-2 (76%), or MT (68%). The results also show that modus ponens inferences are suppressed whether the inference is based on the first antecedent of an additionalantecedent argument or the second antecedent. We carried out planned comparisons on the nonsignificant interaction, F(2,104) 5 0.88, MSE 5 0.09, p 5 .41, between the two factors (see Winer, 1971, for the legitimacy of such comparisons). Participants made fewer modus ponens inferences from additional-antecedent arguments based on the affirmation of the first antecedent (MP-1: 51%) than from the matched inference from single conditional arguments, 98%, F(1,52) 5 36.03, MSE 5 0.08, p , .0001. Likewise, they made fewer modus ponens inferences from additional-antecedent arguments based on the affirmation of the second antecedent (MP-2: 57%) than from the matched inference from single conditional arguments, 95%, F(1,52) 5 34.22, MSE 5 0.06, p , .0001. They made fewer modus tollens inferences from additional-antecedent arguments (52%) than from single conditional arguments, 84%, F(1,52) 5 11.33, MSE 5 0.12, p , .005. Given the negation of the consequent in a modus tollens inference, there were as many conclusions based on the negation of the antecedent of the first con-

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BYRNE, ESPINO, AND SANTAMARIA TABLE 5 Percentages of Different Sorts of Responses to Experiment 1

Premises given: Conclusions: Additional arguments Categorical endorsements q r p not-p not-r not-q Modal endorsements can-q/might-q Answers referring to the other/both antecedents if p then q if r then q if not-r then q not-p and not-r not-p or not-q not-p because not-r Answers to the effect: not enough information Single conditional q not-p p if r then q Not enough information Adding further information

MP-1

MP-2

MT

If p then q (If r then q) p q

If p then q (If r then q) r q

If p then q (If r then q) not-q not-p/not-r

51 1 — — — 1

57 — 7 — — —

— — — 20 32 —

4

—

—

1 17 1 — — — 23

11 1 — — — — 23

— 1 — 6 9 1 31

98 — — 1 — 1

95 — — — 5 —

1 84 1 — 12 1

Note. MP, modus ponens; MT, modus tollens. The endorsements of modus ponens and modus tollens are in bold.

ditional (20%) as the second conditional (32%), as Table 5 shows. Participants produced a rich variety of conclusions to the additional-antecedent arguments, as Table 5 shows. The classification of their conclusions was carried out according to four simple principles: (a) Conclusions were categorized according to which component was mentioned. Given an argument of the form, if p then q, if r then q, and the minor premise p, r, or not-q, conclusions were categorised according to whether they mentioned either p, q, r, not-p, not-q, or not-r. (b) Conclusions were categorized according to whether the component was asserted with certainty or not. Some conclusions contained modal auxillaries such as

“may,” “might,” and “maybe” which qualified the conclusion. Hence, conclusions such as “q happened,” “q must have happened” were classified as endorsements, whereas conclusions such as “q might have happened,” “q could have happened” were classified as endorsements containing modals. (c) Conclusions were categorized according to whether they mentioned a single component or more than one component and in the latter case, according to which connective was used, e.g., if, because, and, or. (d) Conclusions which were nonpropositional in nature, that is, they did not refer to any of the components, and which indicated that no valid conclusion could be deduced were classified as “not enough information to know.” Answers in

COUNTEREXAMPLES AND INFERENCES

this category included conclusions such as “nothing follows,” “you can’t say what happened,” and so on. As Table 5 shows, 53% of conclusions were endorsements of the modus ponens and tollens inferences from additionals arguments. The remaining 47% of conclusions fell into two main categories: more than half of this 47% of conclusions were answers to the effect that there is not enough information to know (26%), and the frequency of this answer is greater to the additional-antecedent arguments than to the single conditionals (6%). This conclusion is consistent with either a counterexample availability account or with a premise doubt account. However, almost one-third of the 47% of conclusions referred to the antecedent of the unmentioned conditional for the modus ponens arguments or to both antecedents for the modus tollens arguments (16%), as expected by the counterexample availability account (see also Dieusseurt, Schaeken, Schroyens, & d’Ydewalle, 1998). Few of the answers contained modal operators (1%), such as “may” or “could,” contrary to the premise doubt account. The results show that when people are given the opportunity to generate their own conclusion, they frequently embrace the components of the unmentioned conditional. The results are also consistent with the recent finding that when people are given a wide selection of different possible answers, about half of their responses endorse a conclusion that refers to the consequent alone, and about one-third of their responses endorse a conclusion that refers to the consequent and the unmentioned antecedent (Dieussaert et al., 1998). Moreover, our results are also consistent with the recent finding that when participants are asked to indicate the revisions they would make to their initial belief set after a modus ponens inference has been contradicted, they very rarely judge the conditional itself to be invalid (Elio, 1998). Instead about half of all participants’ judgments demoted the conditional to a default rule, and another third of judgments indicated that the antecedent needed to be supplemented with a necessary condition or that the antecedent was present with a disabling condition (Elio, 1998).

357

The results suggest that suppression does not arise because people doubt the truth of one of the conditionals, but instead because they doubt the validity of a simple categorical conclusion to these compound arguments. EXPERIMENT 2: IS SUPPRESSION CONVERSION? The counterexample availability explanation proposes that the interpretation of a conditional, in the context of a second conditional that contains an additional antecedent, corresponds to a “reversed conditional” interpretation in which the antecedent is necessary but not sufficient. An alternative possibility is that people interpret pairs of premises that contain additional requirements to be inconsistent (Politzer & Braine, 1991). Politzer and Braine suggest that background knowledge ensures that people interpret the second premise in the pair of conditionals “If she has an essay to write then she will study late in the library, if the library stays open then she will study late in the library” to mean also “if she studies late in the library then necessarily the library stays open.” They suggest that knowledge ensures that the information in the new inferred conditional will be added to the premise set. (Note that the new conditional also is the converse of the second conditional in the pair.) The first conditional in the pair and the new conditional lead, they suggest, to the conclusion “If she has an essay to write then necessarily the library stays open.” They argue that “knowledge of libraries makes this conclusion false” (p. 105). In this way, people may perceive an inconsistency in the premises and decide they must reject one of the premises. This detection of inconsistency then leads to the suppression of inferences: people are naturally reluctant to base any inferences on premises that are inconsistent. According to the inconsistency account, the first crucial step in suppressing an inference is the step of adding extra information from background knowledge to the premise set. (Of course even from the obverse of the second conditional in the pair “If she does not study late in the library than the library does not stay open” it does not follow logically that “if she studies late in the library

358

BYRNE, ESPINO, AND SANTAMARIA

then necessarily the library stays open,” Oaksford & Chater, 1995. In fact, since rule theories do not include a rule for modus tollens, the inference from a conditional to its observe is in any case likely to be a difficult one). The extra new information added to the premise set from background knowledge in the example above corresponds to the converse of the second conditional (Politzer & Braine, 1991; O’Brien, 1993). The aim of the second experiment is to block the step of adding extra information from background knowledge to the premise set, that is, to block adding the converse of the second conditional to the premise set. The inconsistency account leads to the prediction that when this step is blocked the suppression should no longer occur: if people detect no inconsistency, they can make the inferences readily. In contrast, the counterexample availability account predicts that when the step of adding the converse is blocked, the inferences should still be suppressed, because there are still ready-made counterexamples to the inferences represented in the models. We pitted the two accounts against each other in an experiment in which we gave participants problems such as “If Alicia met her friend then she went to the cinema, if she had money for a cinema-ticket then she went to the cinema” and we “expanded” the premises explicitly to block the converse, that is, the premise asserted explicitly that its converse did not hold, “If she had money for a cinema-ticket then she went to the cinema, but if she went to the cinema she may or may not have had money for a cinematicket”. The technique of expanding premises explicitly to block the converse has been usefully employed in reasoning tasks by formal rule theorists (Rumain et al., 1983), and we relied on the same technique here. We constructed four versions of the problem to systematically expand either the first premise, “if Alicia met her friend then she went to the cinema, but if she went to the cinema than she may or may not have met her friend,” or the second premise, “if Alicia had money for a cinematicket then she went to the cinema, but if she went to the cinema then she may or may not

have had money for a cinema ticket,” or both premises. Method Design and materials. We constructed five sorts of arguments. Additional arguments with no expanded premises consisted of two additional conditionals of the form “if p then q, if r then q.” Additional arguments with the first premise expanded were of the form “if p then q but if q then p may or may not happen, if r then q.” Additional arguments with the second premise expanded were of the form “if p then q, if r then q but if q then r may or may not happen.” Additional arguments with both premises expanded were of the form “if p then q but if q then p may or may not happen, if r then q but if q then r may or may not happen.” The final set of arguments consisted of single conditionals. We gave each of the five sets of arguments to a different group of participants in a between-participants design. We constructed two sorts of inferences: the modus ponens inference was based on affirming the antecedent of the first conditional, and the modus tollens inference was based on denying the consequent shared by both conditionals. Each participant carried out six inferences: three modus ponens and three modus tollens inferences. We assigned three different contents at random to each of the two inference forms to construct a different set of six problems for each participant (see the Appendix). We also included four filler arguments based on two modus ponens and two modus tollens inferences from conditionals that contained alternative antecedents. Once again, the problems were presented to the participants in their native language of Spanish. Their task was to choose a conclusion from a set of three alternative conclusions. For the modus ponens inference the set of options were as follows: (a) q, (b) not-q, (c) q may or may not happen. For the modus tollens inference the set of options was as follows: (a) p, (b) not-p, (c) p may or may not happen. We gave each participant the 10 arguments in a different random order. Procedure. The procedure was similar to the first experiment. Once again, the participants

COUNTEREXAMPLES AND INFERENCES TABLE 6

Results and Discussion

The Percentages of Each Inference Endorsed in the Five Conditions of Experiment 2 MP

MT

Total

Minor premise: Conclusions:

p q

If p then q

90

80

85

If p then q If r then q

58

48

53

72

63

68

62

52

57

65

46

56

69

58

64

If p then q and if q then p may or may not If r then q If p then q If r then q and if q then r may or may not If p then q and if q then p may or may not If r then q and if q then r may or may not Total

359

not-q not-p

Note. MP, modus ponens; MT, modus tollens.

were tested in groups, we gave them the arguments in a single booklet with specific instructions printed on the front page, and the instructions explained the task with reference to a single conditional as an example. They were asked to assume that the premises were true and to “choose one of the conclusions, whichever you think follows from the sentences.” Participants were asked to read each item carefully and to work from beginning to end at their own pace, without changing any responses or skipping any items. Participants. The 100 participants who took part in the experiment were undergraduate students from the University of La Laguna, Tenerife, Spain. They were randomly assigned to one of five groups (n 5 20 in each). Nine participants were replaced because of their failure to complete more than three-quarters of the arguments, and these replacements were made during the course of the experiment, prior to any data analysis.

The results show that there is a similar level of suppression made in each of the four experimental conditions compared to the control condition, as Table 6 shows. We carried out a 5 by 2 analysis of variance on the frequency of correct conclusions, with the between-participant factor of argument type (no expanded premises, expanded first premise, expanded second premise, two expanded premises, and arguments based on a single conditional) and the within-participant factor of inference type (modus ponens and modus tollens). It showed a main effect of argument type, F(4,95) 5 6.55, MSE 5 0.10, p , .001, indicating differences in the frequency of inferences made from single conditionals (85%) and from additionals with no expanded premises (53%), an expanded first premise (68%), an expanded second premise (57%), or two expanded premises (56%). The main effect of inference type, F(1,95) 5 8.63, MSE 5 0.07, p , .005, indicated differences in the frequency of modus ponens (69%) and modus tollens (58%) inferences overall. We carried out planned comparisons on the nonsignificant interaction, F(4,95) 5 0.21, MSE 5 0.07, p 5 .93, between the two factors (see Winer, 1971). Participants made more modus ponens inferences from single conditionals (90%) than from additionals with no expanded premises, 58%, F(1,38) 5 12.27, MSE 5 0.08, p , .005; an expanded first premise, 72%, F(1,38) 5 6.86, MSE 5 0.05, p , .025; an expanded second premise, 62% F(1,38) 5 14.64, MSE 5 0.05, p , .0005; or two expanded premises, 65%, F(1,38) 5 9.23, MSE 5 0.07, p , .005. Similarly, they made more modus tollens inferences from single conditionals (80%) than from additionals with no expanded premises, 48%, F(1,38) 5 9.08, MSE 5 0.11, p , .005; an expanded first premise, 63%, F(1,38) 5 4.61, MSE 5 0.06, p , .05; an expanded second premise, 52%, F(1,38) 5 8.64, MSE 5 0.09, p , .005; or two expanded premises, 46%, F(1,38) 5 11.30, MSE 5 0.10, p , .005. There were no other reliable differences between the conditions for either inference.

360

BYRNE, ESPINO, AND SANTAMARIA

The results show that even when the premises are explicitly expanded, the suppression of the valid inferences continues to be observed. This result casts doubt on the suggestion that background knowledge leads participants to add a further conditional from their background knowledge, which corresponds to the converse of the second premise, and on this basis to judge the premises to be inconsistent with one another. The expanded premise sets systematically expand the first or second or both premises, yet there are similar levels of suppression in each of these expanded premise sets as there are in the premise set with no expanded premises, compared to the single conditional arguments. Perhaps the expanded premises simply confused participants and made it difficult for them to understand the arguments? A glance at Table 6 shows that this possibility is unlikely: participants do not make fewer inferences from the expanded premise sets than from the additional conditionals premise set, as one would expect from confused participants, and their pattern of inferences is similar in each case to the additional conditionals premise set. The results remain consistent with our view of suppression as availability of counterexamples, since the expanded premise sets make no difference to the presence of a counterexample. As we predicted, the valid modus ponens and modus tollens inferences are suppressed even when the converse is blocked: the converse makes no difference to the presence of counterexamples to the valid inferences and so they are suppressed. EXPERIMENT 3: CAN REQUIREMENT STRENGTH DEPEND ON NUMBER OF MODELS? The counterexample availability explanation of suppression proposes that inferences are suppressed when people explicitly represent a counterexample to them. Are all counterexamples equally effective in suppressing inferences? The question arises because of the recent demonstration that people rate different additional requirements differently in terms of their perceived importance for the occurrence of the consequent (Chan & Chua, 1994). Given a conditional such as “If Steven is invited then he will

attend the dance party,” people rate some additional requirements, such as “Steven knows the host well” (R1), as less important than the antecedent of the conditional “Steven is invited,” and they rate other requirements, such as “Steven knows at least some people at the dance party” (R2) and “Steven completes the report tonight” (R3), as more important than the antecedent of the conditional. Moreover, there is only weak suppression of the valid inferences when the conditional is accompanied by a second one with a weaker antecedent, but there is much stronger suppression when it is accompanied by a second one with either of the stronger antecedents (Chan & Chua, 1994). 1 This differential suppression occurs for both “novices” (undergraduates) and “experts” (policeworkers) of a given domain (e.g., crime-oriented scenarios, Chan & Chua, 1994). 2 The suppression of the valid inferences also occurs for problems in reversed order, with the additional requirement conditional first, even with the weaker additional requirements (Chan & Chua, 1994). 3 Additional requirements may differ in the perceived strength of the connection between the antecedent and consequent (Chan & Chua, 1994, p. 221): For any given consequent “Q” two antecedents “R1” and “R2” may be given a different degree of importance because of the difference in their specific con1 Chan and Chua (1994) presented their data in terms of a log-linear modeling analysis, and they did not include in their paper the rates of endorsements of the inferences in their experiment. We are very grateful to them for providing us with their raw data, and we include them here for information: The percentages for 20 undergraduates to eight scenarios for the standard order of presentation for modus ponens was R1, 69%, R2, 50%, R3, 46%; and for modus tollens R1, 58%, R2, 34%, R3, 28%. 2 The percentages for 20 police to eight scenarios for the standard order of presentation for modus ponens was R1, 85%, R2, 42%, R3, 25%; and for modus tollens R1, 69%, R2, 18%, R3, 12% (Chan and Chua, 1994). 3 The percentages for 20 undergraduates to eight scenarios for the reversed order of presentation for modus ponens was R1, 53%, R2, 39%, R3, 36%; and for modus tollens R1, 39%, R2, 28%, R3, 26%. The percentages for 20 police to eight scenarios for the reversed order of presentation for modus ponens was R1, 64%, R2, 27%, R3, 20%; and for modus tollens R1, 51%, R2, 14%, R3, 12% (Chan and Chua, 1994).

361

COUNTEREXAMPLES AND INFERENCES tent despite the fact that both are construed as additional requirements.

It may even be the case that suppression depends on the saliency of additional requirements recruited by memory schemas that guide the reasoning process (Chan & Chua, 1994, p. 223): The saliency effect [can] be understood in terms of schematic representations. The basic idea is that people’s interpretations of a problem situation are guided by high-level mental representations known as schemas, each of which is a cognitive structure that encapsulates knowledge about things connected with a given type of situation.

Domain-specific accounts of reasoning attempt to incorporate knowledge directly into the reasoning process (e.g., Cheng & Holyoak, 1985; Cosmides, 1989). Chan and Chua suggest that suppression arises through the action of background knowledge stored in memory schemas which guides reasoning via a production system of condition-action rules which they list as follows (Chan & Chua, 1994, p. 234): Prod1: IF “R” OR “not-R” is unknown, THEN scale “R” according to assertion “P” (i.e., R/P) Prod 2: IF (R/P) , 1, THEN respond “Q,” OTHERWISE respond “don’t know.”

Can the suppression of fallacies and valid inferences be explained by recourse to the retrieval of memory schemas that contain specific knowledge about additional requirements? Chan & Chua (1994) showed that the perceived importance of the antecedent could be manipulated by altering the content of the antecedents, and their argument hinges on background knowledge of the relationship between the antecedent and consequent, which depends on content. Background knowledge may be represented in schemas or in some other sort of representation, and its retrieval from memory is undoubtedly important in the suppression of inferences, as Chan and Chua have shown. However, we do not think that memory schemas tell the whole story about suppression, and to illustrate our argument, our first initial aim in this experiment is to show that saliency or strength of antecedents can also be ma-

nipulated independently of content. One contentindependent way to manipulate the strength of an antecedent is through the phrasing of the premises as a conditional “if” or as a biconditional “if and only if.” Chan and Chua have demonstrated that suppression is affected when additional requirements have different strengths because of their content. In this experiment, we aim to demonstrate a similar effect but independently of content, that is, to show differential suppression when additional requirements have different strengths because of their phrasing, as “if” or “if and only if.” Our rationale is that a demonstration of strength of antecedent effects analogous to Chan and Chua’s, but independently of content, would cast some doubt on the suggestion that suppression arises solely because memory schemas guide production rules for reasoning. Our other, important, aim in this experiment is to test a unique prediction of the model theory about differential suppression and biconditionals. A biconditional is a semantically stronger assertion than a conditional—it rules out more states of affairs than a conditional (see, e.g., Johnson-Laird, Byrne, & Schaeken, 1992), as Table 2 shows. A biconditional is consistent with two alternative situations whereas a conditional is consistent with three alternative situations, and so the biconditional provides more semantic information than the conditional (e.g., Johnson-Laird & Byrne, 1991). Reasoners can readily understand biconditionals and their inferences from them show systematic similarities and differences to their inferences from conditionals (Johnson-Laird, Byrne, & Schaeken, 1992). Most importantly, given a conditional first premise and a biconditional second premise, “If Lisa meets her brother then she goes to a play, if and only if she has enough money then she goes to a play,” the biconditional emphasizes the importance of the enabling condition in the additional requirement conditional. It may even be represented explicitly by the models brother brother

money not-money ...

play not-play

which make explicit that the enabling condition is crucial for the consequent. A biconditional

362

BYRNE, ESPINO, AND SANTAMARIA

second premise should lead to more suppression than a conditional second premise. Method Design and materials. We constructed five sorts of arguments. One sort of argument consisted of two additional requirement conditionals of the form “if p then q, if r then q.” Arguments with a biconditional first premise were of the form “if and only if p then q, if r then q.” Arguments with a biconditional second premise were of the form “if p then q, if and only if r then q.” Arguments with two biconditional premises were of the form “if and only if p then q, if and only if r then q.” The final set of arguments consisted of single conditionals. We gave each of the five sets of arguments to a different group of participants in a betweenparticipant design. We constructed two sorts of inferences: modus ponens inferences were based on affirming the antecedent of the first premise, and modus tollens inferences were based on denying the consequent shared by both premises. Each participant carried out six inferences: three modus ponens and three modus tollens inferences. We assigned three different contents at random to each of the two inference forms to construct a different set of six problems for each participant (see the Appendix). We also included four filler arguments based on modus ponens and modus tollens inferences from conditionals that contained alternative antecedents. The problems were presented to the participants in their native language of Spanish. The participants task was to choose a conclusion from a set of three alternative conclusions similar to those in the second experiment. We gave each participant the 10 arguments in a different random order. Procedure and participants. The procedure and the instructions were the same as those in the second experiment. The 135 participants who took part in the experiment were undergraduate students from the University of La Laguna, Tenerife, Spain. They were randomly assigned to one of five groups (n 5 27 in each). Seven participants were replaced because of their failure to complete more than three-quar-

TABLE 7 The Percentages of Correct Deductions Made in the Five Conditions of Experiment 3 MP

MT

Total

Minor premise: Conclusions:

p q

not-q not-p

If p then q

91

82

87

If p then q If r then q

41

40

41

If and only if p then q If r then q

43

46

45

If p then q If and only if r then q

26

31

29

If and only if p then q If and only if r then q

27

32

30

Total

46

46

46

Note. MP, modus ponens; MT, modus tollens.

ters of the arguments, and these replacements were made during the course of the experiment, prior to any data analysis. Results and Discussion The results show that more suppression occurs when the second premise, which contains the additional requirement, is a biconditional than when the second premise is a conditional. As Table 7 shows there is suppression in each of the four experimental conditions compared to the control condition. We first carried out a 5 by 2 analysis of variance on the frequency of correct conclusions, with the between-participant factor of argument type (arguments with two conditionals, a biconditional first premise, a biconditional second premise, two biconditional premises, and single conditionals) and the within-participant factor of inference type (modus ponens and modus tollens). It showed a main effect of argument type, F(4,130) 5 18.26, MSE 5 0.17, p , .001, indicating differences in the frequency of inferences made from single conditionals (87%) and from arguments with two conditionals (41%), a biconditional first premise (45%), a biconditional second premise (29%), or two biconditional premises (30%).

363

COUNTEREXAMPLES AND INFERENCES TABLE 8 The Percentages of Correct Inferences Made in Experiment 3 as a Function of the Conditional or Biconditional Nature of the First and Second Premises Second premise If and only if r then q

If r then q

First premise If p then q If and only if p then q Totals Overall

Totals

MP

MT

MP

MT

MP

MT

Overall

41 43

40 46

26 27

31 32

34 35

36 39

35 37

42

43 43

27

32 30

35

38 73

Note. MP, modus ponens; MT, modus tollens.

There was no main effect of inference type, F(1, 30) 5 0.01, MSE 5 0.06, p 5 .93, as participants made the same frequency of modus ponens and modus tollens inferences (46% in each case), and the interaction was not reliable, F(4, 130) 5 0.87, MSE 5 0.06, p 5 .48. We carried out a second analysis of variance to examine the relative suppression in the twopremise arguments that contained a biconditional in the first premise or the second premise. The 2 3 2 3 2 analysis had one betweenparticipant factor of first premise (conditional, biconditional), another between-participant factor of second premise (conditional, biconditional), and a within-participant factor of inference type (modus ponens, modus tollens). As Table 8 shows, there was no main effect of first premise type, 35% versus 37%, F(1,104) 5 0.22, MSE 5 0.19, p 5 .64, but there was a main effect of second premise type, 43% versus 30%, F(1,104) 5 4.93, MSE 5 0.19, p , .05. There was no main effect of inference type, 35% versus 38%, F(1,104) 5 0.63, MSE 5 0.07, p 5 .42, there was no interaction of first premise with second premise, F(1,104) 5 0.07, MSE 5 0.19, p 5 .80, and neither first nor second premise type interacted with inference type, F(1,104) 5 0.07, MSE 5 0.07, p 5 .80, and F(1,104) 5 0.38, MSE 5 0.07, p 5 .53 respectively. The three-way interaction was not reliable, F(1,104) 5 0.07, MSE 5 0.07, p 5 .79.

The results show that more inferences are suppressed when the additional requirement is phrased as a biconditional rather than as a conditional. First, these results indicate that the strength of additional requirements does not depend solely on their content, and so it cannot depend solely on background knowledge in memory schemas. Background knowledge about the weakness of an additional requirement may lead to a weak requirement, such as “if Steven knows the host well then he attends the dance party” being represented in the following sorts of models: invited invited

host not-host ...

party party

where “invited” stands for “Steven is invited,” “host” stands for “Steven knows the host well,” and “party” stands for “Steven attends the dance party.” Moreover, the suppression of inferences varies when the content of the premises alters the perceived sufficiency and necessity of the antecedents (Neth, Beller, & Spada, 1998). Second, the content-independent demonstration of the effect of antecedent strength in our experiment can be accounted for in terms of the number of alternative models that are consistent with conditionals and biconditionals. Biconditionals are consistent with two explicit models that represent two alternative states of affairs.

364

BYRNE, ESPINO, AND SANTAMARIA

They provide more semantic information than conditionals which are consistent with three models. The explicit manipulation by using conditionals or biconditionals goes some way to support our suggestion that it is the number of models that affects the suppression and in particular whether there is a model that contains a counterexample to the putative conclusion. EXPERIMENT 4: SUPPRESSION DEPENDS ON COUNTEREXAMPLE AVAILABILITY Our suggestion is that inferences are suppressed when people explicitly represent counterexamples to them in their models. The valid inferences are suppressed by additional requirements because the interpretation of the first conditional within the context of the second conditional is as a “reversed conditional” rather than a conditional. The antecedent is considered necessary but not sufficient for the consequent, and as a result people make the denial of the antecedent and affirmation of the consequent inferences but they do not make the modus ponens and modus tollens inferences from the first conditional. The fallacies are suppressed by alternatives because the interpretation of the first conditional within the context of the second conditional in this case is as a conditional. The antecedent is considered sufficient but not necessary for the consequent and as a result people make the modus ponens and modus tollens inferences but they do not make the denial of the antecedent and affirmation of the consequent inferences from the first conditional. In everyday inference, “if” may be interpreted to express a conditional or a biconditional as previous research has emphasized, but it may sometimes be interpreted to express a reversed conditional or even nonconditional relation. It is to the nonconditional interpretation that we turn in our final experiment. If the suppression of inferences depends on the explicit availability of counterexamples, then it should be possible to provide counterexamples to all inferences (e.g., Cummins et al., 1991). It should be possible to provide both an additional and an alternative requirement and so to provide counterexamples to both the valid inferences and the fallacies as well. Consider

the following set of premises, “If Paul goes fishing then he has a fish supper, if Paul goes to the fishmarket then he has a fish supper, if Paul catches fish then he has a fish supper.” Given both an alternative and an additional requirement, the antecedent of the first conditional should be interpreted as neither necessary nor sufficient for the consequent. The first conditional is embedded in a set of conditionals that are coherent and consistent and yet, according to our account, it should not support any of the standard inferences, valid or fallacious. Our final experiment aims to examine whether it is possible to suppress all the standard inferences from a conditional by providing explicit counterexamples to them. Method Design and materials. We constructed four sorts of arguments. Additionals consisted of two conditionals with additional requirements, alternatives consisted of two conditionals with alternative antecedents, additional-and-alternatives consisted of three conditionals with additional and alternative antecedents, and the single conditionals consisted of one conditional. In the additional-and-alternatives set the order of the alternative and additional conditionals was counterbalanced. We gave each of the four sets of arguments to a different group of participants in a between-participant design. We constructed four sorts of inferences: modus ponens based on affirming the antecedent of the first conditional, modus tollens based on denying the consequent shared by the conditionals, denial of the antecedent based on denying the antecedent of the first conditional, and affirmation of the consequent based on affirming the consequent shared by the conditionals. Each participant carried out 12 inferences: three instances of each of the 4 sorts of inferences. We assigned three different contents at random to each of the four inference forms to construct a different set of 12 problems for each participant (see the Appendix). We also included eight filler arguments in each condition consisting of two instances of each of the four sorts of inference based on conditionals that contained the opposite sort of antecedent (e.g.,

365

COUNTEREXAMPLES AND INFERENCES TABLE 9 The Percentages of Inferences Made in the Four Conditions of Experiment 4 MP

MT

AC

DA

Total

Minor premise: Conclusions:

p q

not-q not-p

q p

not-p not-q

Single Alternatives Additionals Alternatives-and-additionals

99 94 47 58

91 67 32 46

92 26 58 20

91 18 44 21

93 51 45 36

Total

75

59

49

44

57

Note. MP, modus ponens; MT, modus tollens. AC, Affirmation of the consequent; DA, denial of the antecedent.

alternative antecedents for the additionals group). The problems were presented to the participants in their native language of Spanish. The participants task was to choose a conclusion from a set of three alternative conclusions similar to those in the second experiment. We gave each participant the 20 arguments in a different random order. Procedure and participants. The procedure and the instructions were the same as those in the second experiment. The 88 participants who took part in the experiment were undergraduate students from the University of La Laguna, Tenerife, Spain. They were randomly assigned to one of five groups (n 5 22 in each). Three participants were replaced because of their failure to complete more than three-quarters of the arguments, and these replacements were made during the course of the experiment, prior to any data analysis. Results and Discussion Additionals suppressed the valid inferences, alternatives suppressed the fallacies, and the combination of an additional and an alternative suppressed both the valid inferences and the fallacies, as Table 9 shows. We carried out a 4 by 4 analysis of variance on the correct conclusions with the between-participant factor of argument type (alternatives, additionals, alternatives-and-additionals, and single arguments) and the within-participant factor of inference type (modus ponens, modus tollens, denial of the antecedent, and affirmation of the conse-

quent). There was a main effect of argument type, indicating differences in the frequency of inferences made to the single conditionals (93%) and to the alternatives (51%), the additionals (45%), and the combination of the two (36%), F(3,84) 5 37.14, MSE 5 0.15, p , .001. There was a main effect of inference type indicating differences in the frequency of modus ponens (75%), modus tollens (59%), affirmation of the consequent (49%), and denial of the antecedent (44%) inferences, F(3,252) 5 22.34, MSE 5 0.07, p , .001. There was a reliable interaction between the two factors, F(9,252) 5 10.37, MSE 5 0.07, p , .001. Participants made fewer inferences of each of the four types from the additionals-andalternatives than from the single conditionals: they made fewer modus ponens inferences, 58% versus 99%, F(1,42) 5 17.69, MSE 5 0.10, p , .0001; fewer modus tollens inferences, 46% versus 91%, F(1,42) 5 24.23, MSE 5 0.09, p , .0001; fewer affirmation of the consequent inferences, 20% versus 92%, F(1,42) 5 87.33, MSE 5 0.07, p , .0001; and fewer denial of the antecedent inferences, 21% versus 91%, F(1, 42) 5 73.56, MSE 5 0.07, p , .0001. Participants made fewer of the valid inferences from the additionals than from the single conditionals: they made fewer modus ponens inferences, 47% versus 99%, F(1,42) 5 49.54, MSE 5 0.06, p , .0001, and fewer modus tollens inferences, 32% versus 91%, F(1,42) 5 57.34, MSE 5 0.07, p , .0001. However, they also made fewer affirmation of the consequent

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inferences, 58% versus 92%, F(1,42) 5 15.66, MSE 5 0.09, p , .0005, and fewer denial of the antecedent inferences, 44% versus 91%, F(1,42) 5 33.80, MSE 5 0.07, p , .0001. The apparent suppression may be the result partly of the unusually high rate of denial of the antecedent and affirmation of the consequent inferences made to the single conditionals, which are almost at ceiling (92% and 91%, respectively). We will return to this result shortly. Participants made fewer of the fallacies from the alternatives than from the single conditionals: they made fewer affirmation of the consequent inferences, 26% versus 92%, F(1,42) 5 54.79, MSE 5 0.09, p , .0001, and fewer denial of the antecedent inferences, 18% versus 91%, F(1,42) 5 139.03, MSE 5 0.04, p , .0001. However, they also made fewer modus tollens inferences, 67% versus 91%, F(1,42) 5 9.27, MSE 5 0.07, p , .005, although not fewer modus ponens inferences, 94% versus 99%, F(1,42) 5 1.38, MSE 5 0.01, p 5 .25. Once again, the apparent suppression may be the result partly of the unusually high rate of modus tollens made to the single conditionals, which is almost at ceiling (99%). Given the high rate of all inferences to the single conditionals, a more stringent test of suppression in the alternatives-and-additionals group is to compare it directly to the additionals-alone group, and the alternatives-alone group, respectively. First, the comparison of the additional-andalternatives to the additionals-alone showed that, as expected, participants exhibited the same amount of suppression of the valid inferences from the additionals-and-alternatives as from the additionals, for both modus ponens, 58% versus 47%, F(1,42) 5 0.78, MSE 5 0.15, p 5 .38, and modus tollens, 46% versus 32%, F(1,42) 5 1.48, MSE 5 0.14, p 5 .23, Again, as expected, they made reliably fewer of the fallacies from the additionals-and-alternatives than from the additionals for both affirmation of the consequent, 20% versus 58%, F(1,42) 5 15.83, MSE 5 0.10, p , .0005, and denial of the antecedent inferences, 21% versus 44%, F(1,42) 5 4.68, MSE 5 0.12, p , .05. Second, the comparison of the additionaland-alternatives to the alternatives-alone showed that, as expected, participants exhibited the same amount of suppression of the fallacies from the

additionals-and-alternatives as from the alternatives, for both affirmation of the consequent, 20% versus 26%, F(1,42) 5 0.38, MSE 5 0.10, p 5 .54, and denial of the antecedent, 21% versus 18%, F(1,42) 5 0.11, MSE 5 0.09, p 5 .74. Again, as expected, they made fewer of the valid inferences from the additionals-and-alternatives than from the alternatives for modus ponens, 58% versus 94%, F(1,42) 5 12.60, MSE 5 0.12, p , .001, and marginally for modus tollens, 46% versus 67%, F(1,42) 5 3.52, MSE 5 0.14, p , .067. The results show that when an antecedent is interpreted in such a way that it is neither necessary nor sufficient for the consequent, both the valid and the invalid inferences are suppressed (e.g., Cummins et al., 1991). The suppression arises because the presentation of an alternative and additional requirement in the second and third conditional alters the interpretation of the first conditional. The suppression of valid inferences and fallacies indicates that people can interpret a conditional within a coherent set of conditionals as supporting none of the standard inferences whatsoever. A conditional can be interpreted as a “nonconditional,” that is, as containing an antecedent that is neither sufficient nor necessary for its consequent. Counterexamples to each of the four inferences are present in the models of the conditionals and so none of the categorical conclusions are supported. GENERAL DISCUSSION Suppression and Inference Rules The proper interpretation of the suppression of valid inferences has generated some debate. Some theorists have attempted to demonstrate that the suppression is not genuine (e.g., Braine & O’Brien, 1991; George, 1995; O’Brien, 1993; Politzer & Braine, 1991; Rips, 1990). They have argued that the suppression arises because an additional requirement leads people to disbelieve the truth of the first conditional and to judge that the premises are inconsistent. The first two experiments indicate that suppression is not caused by doubt in the premises or by premise inconsistency. The first experiment showed that when people spontaneously produce their own conclusions, their responses are consistent not with premise

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doubt but with doubt in the validity of the categorical conclusion. The second experiment shows that suppression occurs even when the conditional and the additional requirement are expanded to block any tendency to infer their converse. The results of these two experiments have implications for accounts of suppression based on inference rules. Although the premise doubt and premise inconsistency accounts of suppression attempt to show that the suppression of valid inferences is not genuine, the suppression of fallacies is maintained to be informative about the underlying mechanisms of human inference (Politzer & Braine, 1991). The suppression of the fallacies has been interpreted to mean that people misunderstand the conditional “If Lisa met her friend then she went to a play” to mean also its obverse, “If Lisa did not meet her friend then she did not go to a play” (Rumain et al., 1983). The fallacy of the denial of the antecedent inference is instead a valid modus ponens inference made from a misinterpretation of the conditional. An alternative, such as “She meets her brother,” blocks the misinterpretation, and so the fallacy is resisted. On this account, the mechanism of inference relies on a mental repertoire of formal inference rules which people access to construct a derivation or proof of a conclusion (e.g., Braine and O’Brien, 1991; Rips, 1994). The modus ponens inference is made by recovering its logical skeleton, “if p then q, p” and matching it to the rules in the mental repertoire, which include a rule for modus ponens: If p then q p therefore q

The inference is easy because the premises match a rule directly and the proof consists of a single step to the conclusion, q. The conclusion is then translated back to the content of the premises. The mental repertoire of rules contains elementary rules corresponding to valid inferences such as modus ponens, but it does not contain rules corresponding to the fallacies (e.g., Braine & O’Brien, 1991; Rips, 1994). The suppression of the fallacies has been viewed as support for the suggestion that there are no

formal rules corresponding to the fallacies in the mind (Rumain et al., 1983). Because the suppression of the fallacies was interpreted as evidence that there are no formal rules corresponding to the fallacies in the mind, by parity of argument, the suppression of the valid inferences could be interpreted as evidence that there are no formal rules corresponding to the valid inferences in the mind either (Byrne, 1989). The suppression of the valid inferences does not cast doubt on their validity but it may cast doubt on the idea that they are made with reference to formal rules of inference that apply regardless of content. Rule theorists have taken the possibility of the suppression of modus ponens as a serious challenge to their theories, asserting, e.g., (O’Brien, 1993, p. 124) “This claim [that modus ponens can be suppressed], if correct, would be a serious challenge to The [formal inference rule] Model which includes modus ponens as a core inference.” Politzer and Braine (1991) suggest that the suppression arises from inconsistent premises whose truth is thereby doubted. The second conditional in the set of premises i. If she has an essay to write then she will study late in the library ii. If the library stays open she will study late in the library. iii. She has an essay to write.

(if p then q) (if r then q) (p)

expresses a necessary condition which leads to the addition of the proposition iv. If she studies late then necessarily the library stays open.

(if q then r)

which happens to be, as noted earlier, the converse of the conditional in (ii). The first conditional in (i) and the new conditional in (iv) lead to the conclusion: v. If she has an essay to write then necessarily the library stays open.

(if p then r)

They argue that “knowledge of libraries makes this conclusion false” (p. 105) and that because (i) and (iv) lead to a wrong conclusion (v), one of them cannot be right (but see Oaksford &

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Chater, 1995). They suggest that (iv) cannot be questioned because it is a necessary truth, so “it follows that a rational subject has to question the literal truth of premiss [i]” (p. 105). The argument is reduced, they suggest, to: vi. That she has an essay to write is not sufficient for her to study late in the library. vii She has an essay to write.

and “a failure to conclude that she will study late in the library could hardly count as a suppression of modus ponens” (p. 106). The results of the first two experiments cast doubt on Politzer and Braine’s account, as we have seen. Reasoners do not indicate doubt in the premises when they spontaneously make inferences, and the suppression effect does not disappear when premises are expanded to block the converse of the conditionals. Moreover, it seems reasonable to infer that if a reasoner doubts the truth of a premise, then for the twovalued (true or false) logic of a formal rule system to capture this doubt, the premise must be represented as false. Perhaps we can infer that to say a reasoner disbelieves the truth of a premise within a two-valued rule system means that the reasoner adds to their premise set the negation of v: vi’. It is false that if she has an essay to write then necessarily the library stays open.

NOT (if p then r)

Because premise 1 and premise 2 lead to an inconsistency, Politzer and Braine say that participants disbelieve the truth of one of the premises, the first. Once again, perhaps we can infer that to capture the idea that reasoners disbelieve the truth of the first premise in a formal rule system, the premise set must include the negated premise vii’. It is false that if she has an essay to write then she will study late in the library

NOT (if p then q)

(Byrne, 1991). One possibility is that reasoners may make no further inferences at this point, on the assumption that they base inferences only on premises they consider true (Braine & O’Brien,

1991). But, there is clear evidence that reasoners are able to cope with making inferences about truth and falsity, and they are able to deduce what follows from false assertions (Byrne & Handley, 1997; Byrne, Handley, & Johnson-Laird, 1995). Formal rule theories need not be restricted to inferences from true assertions only (e.g., Rips, 1994). The conclusion in vii’ is equivalent to: viii’. She has an essay to write and the library does not stay open

(p and not q)

and given the minor premise in (iii) earlier, iii. she has an essay to write,

(p)

participants who reach (viii’) should conclude, by conjunction elimination from (viii’) and (iii): ix. she does not study late in the library,

(not q)

a conclusion that participants do not make. The step from (vii’) to (viii’) requires the negation of a compound expression which reasoners find difficult (e.g., Handley & Byrne, 1998), and it may require several inferential steps to derive in a formal rule system that does not contain direct rules for such negations (e.g., Braine & O’Brien, 1991). Overall, it remains unclear whether rule theories can accommodate the suppression of valid inferences and even more so whether they could explain the differential suppression by qualified affirmations, e.g., “Paul sometimes catches fish” (Stevenson & Over, 1995), or by weaker and stronger additional requirements (Chan & Chua, 1994). These gradations of suppression appear to lie beyond the grasp of such formal mechanisms (Stevenson & Over, 1995). Suppression and Models Theorists are agreed that suppression arises because background knowledge conflicts with the putative conclusion (Byrne, 1989; Byrne & Johnson-Laird, 1992; Chan & Chua, 1994; Fillenbaum, 1993; Politzer & Braine, 1991; Stevenson & Over, 1995), but they differ in their views about how background knowledge is called into play. The third experiment shows that the strength of antecedents can be manipulated independently of

COUNTEREXAMPLES AND INFERENCES

content through the phrasing of the premises as a conditional “if” or as a biconditional “if and only if.” There is even greater suppression of inferences when the additional requirement is stronger, that is, when it is expressed as a biconditional “if and only if,” than when it is expressed as a conditional. The content-independent manipulation of antecedent strength goes against the suggestion that suppression arises only because memory schemas guide production rules for reasoning (Chan & Chua, 1994). Background knowledge in memory is very important in reasoning and in the suppression of inferences. But an alternative view of the role of background knowledge in the suppression effect is that it provides ready-made counterexamples to putative conclusions. The fourth experiment shows that both the valid inferences and the fallacies can be suppressed for a single conditional when it is presented with both alternative and additional conditions. Counterexamples to each of the four inferences are available and so none of the categorical conclusions are supported. The results of the third and fourth experiments have implications for theories of the suppression effect based on models. There is a considerable body of evidence that addresses the suggestion that reasoners may make inferences by constructing mental models in the various domains of deduction, including reasoning about quantifiers (e.g., Johnson-Laird & Byrne, 1989; Johnson-Laird, Byrne, & Tabossi, 1989); relations (e.g., Byrne & JohnsonLaird, 1989; Schaeken, Johnson-Laird, & D’ydewalle, 1996); suppositions (e.g., Byrne & Handley, 1997; Byrne, Handley, & JohnsonLaird, 1995), and propositions (e.g., Girotto, Mazzocco, & Tasso, 1997; Johnson-Laird, Byrne, & Schaeken, 1992), and the model theory has been extended to related domains such as probabilistic thinking (Johnson-Laird, 1994b) and counterfactual thinking (Byrne, 1997b; Byrne & Tasso, in press). We suggest that the suppression of inferences may also depend on the construction of models, and it may result from counterexample availability. The model theory is also consistent with the other main phenomena of suppression. For example, people combine antecedents of the same form, “If p then q, If r then q,” in different ways

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when they are asked to paraphrase two conditionals using a single conditional. When the antecedents contain additional conditions they are combined conjunctively, “If she met her friend and she had enough money then she went to a play,” and when they contain alternatives they are combined disjunctively, “If she met her friend or she met her brother then she went to a play” (Byrne & Johnson-Laird, 1992). Adding an alternative or an additional antecedent changes the structure of the argument, and thereby the structure of the models people construct. People paraphrase two conditionals conjunctively when they contain additional requirements because they have represented both requirements in a single model; they paraphrase two conditionals disjunctively when they contain alternative requirements because they have represented the two requirements in two separate models (Byrne & Johnson-Laird, 1992). A second example is that given a pair of conditionals with additional requirements the modus ponens inference is suppressed from the minor premise “She met her friend,” but it is not suppressed from an expanded minor premise. For example, given “If Lisa met her friend then she went to a play, if Lisa had enough money then she went to a play, Lisa met her friend and she had enough money,” people choose (a) from the set (a) she went to a play, (b) she did not go to a play, (c) she may or may not have gone to a play (Byrne, 1989, Experiment 2). The suppression is released when both antecedents are jointly affirmed, because there are no longer any ready-made counterexamples to the conclusion in the set of models. Likewise, given a pair of conditionals with alternative antecedents the denial of the antecedent fallacy is suppressed from the minor premise “She did not meet her friend,” but it is not suppressed from the expanded minor premise in the argument “She did not met her friend or her brother” (Byrne, 1989). The result suggests that people do not gain a general insight into the idea that there may be alternatives or additional background conditions that are relevant to the inferences (e.g., Markovits, 1984; Politzer & Braine, 1991; Rumain et al., 1983). Third, when people are given premises of the

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sort “If Paul goes fishing then he will have a fish supper, if Paul catches a fish then he will have a fish supper, Paul is always lucky when he goes fishing, Paul goes fishing,” the inference is not suppressed and they infer that Paul will have a fish supper (Stevenson & Over, 1995). The premise “Paul is always lucky when he goes fishing” provides participants with the information that Paul always catches fish, and the premises’ meaning can be rendered as “If Paul goes fishing then he will have a fish supper, if Paul catches a fish then he will have a fish supper, Paul always catches fish, Paul goes fishing,” that is, the two antecedents are affirmed. An antecedent can be affirmed in a qualified manner, e.g., with “always” in “Paul is always lucky when he goes fishing,” or with “almost always,” “sometimes,” “rarely,” or “very rarely” (Stevenson & Over, 1995). The valid inferences are more suppressed the more the qualifying term introduces uncertainty about whether the antecedent (John catches fish) is affirmed (Stevenson & Over, 1995). Differential suppression occurs when one of the antecedents is partly affirmed, e.g., “Paul sometimes catches fish,” because the representation of the requirement captures as two separate alternatives that it may happen or it may not happen: catches not-catches

(see also Johnson-Laird, 1994b). Suppression and the Non-monotonicity or Defeasibility of Inferences The suppression of inferences may be a widespread phenomenon in everyday reasoning. An important characteristic of human inference is that people can withdraw a conclusion on foot of further information. Consider the premises “Molly has a bacterial infection. If a patient has a bacterial infection, then the preferred treatment for the patient is penicillin” (from Johnson-Laird & Byrne, 1991, p. 20). People can make the valid modus ponens inference “Therefore, the preferred treatment for Molly is penicillin.” But, if they discover that Molly is allergic to penicillin, then they can withdraw the

conclusion. This non-monotonicity or defeasibility of inference has been difficult to capture in logical and computational systems, but it is a common feature of everyday human reasoning (see Johnson-Laird & Byrne, 1991). Non-monotonic inferences may be captured within the model theory—the construction of mental models is constrained by available knowledge (Johnson-Laird & Byrne, 1993a), rather than by a logic (pace Chater, 1993). Oaksford & Chater (1995a) suggest that people may interpret natural language conditionals as “default” rules based on their prior world knowledge rather than as logical assertions. We suggest that prior world knowledge may guide what is represented explicitly in an initial mental model and the subsequent fleshing out of implicit models (Johnson-Laird & Byrne, 1991). Judgments about the relevance of information may be important to its explicit representation (e.g., Evans, 1993; Johnson-Laird, 1995; Keane, 1997; Oaksford & Chater 1995b). People may consider the most plausible model rather than all possible models (e.g., Garnham, 1993), and an account of how the most plausible model is chosen may require an account of information access in memory (Chater & Oaksford, 1993; Oaksford & Chater, 1991, 1993). When people engage in everyday reasoning, thinking of some alternatives to a scenario but overlooking many possibilities, they may even be using an intractable algorithm that is defeated by the magnitude of the task (see Johnson-Laird & Byrne, 1993; Oaksford, 1993). The model theory attempts to provide an account of one sort of non-monotonic reasoning, undoing default assumptions, but it remains an open question how people revise their beliefs when a conclusion contradicts them (see Elio & Pelletier, 1997; Johnson-Laird & Byrne, 1993). Conclusions In our experiments we explicitly gave people information that could be used to formulate counterexamples, and we showed that people use these counterexamples in predictable ways. In everyday reasoning people can not only use available counterexamples, but they can also search for and discover relevant

COUNTEREXAMPLES AND INFERENCES

counterexamples to conclusions they are considering. When background knowledge conflicts with current information, people may refrain from making an inference or withdraw a conclusion that would otherwise be endorsed. In the case of the Chernobyl nuclear disaster, if people place more importance on additional requirements from their background knowledge than on the rules provided, they may fail to draw otherwise valid inferences. Inferential aids to help avoid such mistakes may need to help people weigh accurately the relative strengths of their background knowledge and the rules they are required to follow so that they can assess whether an apparent counterexample to a conclusion is strong enough to suppress it val-

idly. The results of our experiments suggest that the suppression of valid inferences is a genuine effect that does not arise from premise doubt or premise inconsistency, but instead arises because of the availability of explicit counterexamples in the mental models of conditionals with additional requirements. Our emphasis has been on the influence of the availability of explicit counterexamples, and the results of our experiments may be a first step in one direction toward understanding how people use counterexamples. But the vital question remains of how people discover counterexamples or recover them from memory, and its answer requires further theoretical and empirical work.

APPENDIX Examples of the Materials Used in the Four Experiments Materials used Single Additional Single Additional Single Additional Single Additional Single Additional Single Additional Single Additional Single Additional Single Additional Materials used Single Alternative Additional Single Alternative Additional Single Alternative Additional a b

in If If If If If If If If If If If If If If If If If If in If If If If If If If If If

Experiment 1 Elisa has an essay to write then she studies in the library all evening. the library stays open then she studies in the library all evening. a,b Rosa meets her friend then she goes to the cinema. she has money for a cinema-ticket then she goes to the cinema. a,b it is raining then Alicia gets wet. Alicia forgets her umbrella then she gets wet. a Pablo takes the afternoon off work then he cuts the grass. Pablo gets the faulty lawnmower to work then he cuts the grass. b Juan knows how to gut fish then he cooks a fish dinner. he catches some fish then he cooks a fish dinner. Laura cooks a meal then she has a pleasant dinner. there are decent ingredients in the press then she has a pleasant dinner. there are boats available then Marcos goes rowing. the boats are in working condition then Marcos goes rowing. Jose turns the switch then the light comes on. the bulb is working then the light comes on. Maria studies hard then she gets a high grade in the examinations. she is intelligent then she gets a high grade in the examinations. Experiment 4 Susana met her friend then she went to the cinema. she met her family then she went to the cinema. she had enough money for a cinema-ticket then she went to the cinema. Rosa had an essay to write then she studied in the library all evening. she had some textbooks to read then she studied in the library all evening. the library stayed open then she studied in the library all evening. it rained then Alicia got wet. it snowed then she got wet. she went out for a walk then she got wet.

Materials also used in Experiment 2 (in the past tense). Materials also used in Experiment 3 (in the past tense).

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