Purely psychological principles like limitations of cognitive capacity guide psychological theories. However, a priori rationality norms provide powerful roadmaps for the development of psychological theories. They play essential roles in the context of discovery and in the context of justification. In the context of discovery, they guide the research questions, the tasks, and the evaluation of the results. Wason's selection task, for example, was developed within the normative framework of classical logic. From a probability logical point of view, this task does not distinguish between the conditional event and the material conditional interpretation, as both provide the same psychological predictions. In the context of justification, rationality norms are used to rationally reconstruct reasoning processes and the empirical data. The probability propagation rules of the modus tollens, for example, are formally much more complex than those of the modus ponens, which explains experimental data (Pfeifer & Kleiter Reference Pfeifer and Kleiter2009). Moreover, a formal theory provides a language that makes psychological theorizing precise. Therefore, I do not believe that “theories of higher mental processing would be better off freed from normative considerations” (target article, Abstract).
Elqayam & Evans (E&E) argue that conditional elimination inferences are single-norm paradigms. This is true in the framework of classical logic. However, this does not hold in the framework of probability logic. There are at least two norm conflicts in the context of conditional elimination inferences (modus ponens, modus tollens, affirmation of the consequent, and denial of the antecedent). The probability propagation rules depend on the interpretation of the conditional. As an example, consider the probabilistic modus ponens. If the conditional is interpreted as a conditional event, then the probabilistic modus consists of the following inference:
If the conditional is interpreted as a material conditional, then the probability propagation rule is a different one:
There are similar norm conflicts for the other three conditional syllogisms (Pfeifer & Kleiter Reference Pfeifer and Kleiter2005). Therefore, these conditional elimination inferences are not examples of single-norm conflicts.
The authors argue, citing Evans (Reference Evans1982), that “we should desist from the practice of reporting logical accuracy in reasoning tasks, and instead report what people actually did” (sect. 6, para. 7). I agree that empirical studies should report what people actually did. However, I argue that reporting (dis)agreement with rationality norms is important and informative. The basic question is the choice of appropriate rationality norms. After about a decade of reasoning research within the normative framework of classical logic, we can safely state that it is high time to consider extensions or alternative normative frameworks.
Coherence-based probability logic is one example (see, e.g., Pfeifer & Kleiter Reference Pfeifer, Kleiter, Leitgeb and Schurz2002; Reference Pfeifer and Kleiter2009; Reference Pfeifer, Kleiter, Oaksford and Chater2010). In this framework, the focus is not on whether or how people draw logically correct inferences about the logical validity of certain argument forms. Rather, the tasks instruct the participants to transmit the uncertainty of the premises to the conclusion. Thus, the focus is on the participant's degree of belief in the conclusion. Probability logic allows for making psychological predictions precise and offers new psychological explanations of the inferences that people draw. The conditional introduction inference from “B” to “If A, then B”, for example, is not probabilistically informative under the conditional event interpretation of indicative conditionals (P(B|A)) and this is the reason why most people claim that one cannot infer a conditional from its consequent (Pfeifer & Kleiter Reference Pfeifer, Kleiter, Manktelow, Over and Elqayam2011). Contrary to standard approaches to probability, this even holds in the special case where the premise is given for sure (P(B)=1). Probability logic tells us which argument forms are probabilistically informative and which ones are not (Pfeifer & Kleiter Reference Pfeifer and Kleiter2006; Reference Pfeifer and Kleiter2009). “Probabilistic informativeness” is not an empirical term, it is a criterion derived within the normative framework. I argue that patterns of inferences beyond the conditional syllogisms should be investigated. Psychological plausible principles, like the ability of withdrawing conclusions in the light of new evidence and the defeasibility of everyday inferences, should be investigated. I agree that reasoning experiments should not focus on logical validity. Rather, the degrees of belief in the conclusions should be investigated.
E&E suggest there are asymmetric relations between formal theories and psychological theories and data (see their Figure 2). Indeed, formal theories inspire psychological theories. However, psychological data can inspire formal theories as well. Ford (Reference Ford2005), for example, investigates how experimental data informs artificial intelligence systems and has developed a formal system of nonmonotonic reasoning, which is inspired by psychological data (Ford Reference Ford2004). Moreover, not only psychological data arbitrate between formal theories: the converse holds as well. In the field of nonmonotonic reasoning, for example, there are many competing systems. However, System P (Kraus et al. Reference Kraus, Lehmann and Magidor1990) is an example of a common denominator of rationality principles any system of nonmonotonic reasoning should satisfy. It makes sense psychologically to use such a system to arbitrate between formal theories in this field.
Another way of arbitrating between formal theories is to require psychologically plausible but minimal principles for rationality, such as coherence (see, e.g., Coletti & Scozzafava Reference Coletti and Scozzafava2002). Coherence requires only avoiding bets that lead to sure loss. This criterion is much weaker than, for example, requiring maximizing expected utility. In my opinion, the relation between normative and descriptive components in a psychological theory of reasoning is a genuinely interactive one.
I agree with the authors that learning should be included in reasoning research. Moreover, there seems to be a consensus among reasoning researchers that the interpretation of the task material remains the same within participants. However, this is not the case in general. If participants solve several items of the probabilistic truth table task, the number of conditional event responses tend to increase from about 40% at the beginning of the experiment to about 80% at the end. Thus, the participant's responses converge on the competence answers (Fugard et al. Reference Fugard, Pfeifer, Mayerhofer and Kleiter2011b). Assuming appropriate bridge laws that connect “is” and “ought” inferences, I avoid committing the is–ought fallacy, if I claim that rational reasoners should converge to the conditional event response.
Purely psychological principles like limitations of cognitive capacity guide psychological theories. However, a priori rationality norms provide powerful roadmaps for the development of psychological theories. They play essential roles in the context of discovery and in the context of justification. In the context of discovery, they guide the research questions, the tasks, and the evaluation of the results. Wason's selection task, for example, was developed within the normative framework of classical logic. From a probability logical point of view, this task does not distinguish between the conditional event and the material conditional interpretation, as both provide the same psychological predictions. In the context of justification, rationality norms are used to rationally reconstruct reasoning processes and the empirical data. The probability propagation rules of the modus tollens, for example, are formally much more complex than those of the modus ponens, which explains experimental data (Pfeifer & Kleiter Reference Pfeifer and Kleiter2009). Moreover, a formal theory provides a language that makes psychological theorizing precise. Therefore, I do not believe that “theories of higher mental processing would be better off freed from normative considerations” (target article, Abstract).
Elqayam & Evans (E&E) argue that conditional elimination inferences are single-norm paradigms. This is true in the framework of classical logic. However, this does not hold in the framework of probability logic. There are at least two norm conflicts in the context of conditional elimination inferences (modus ponens, modus tollens, affirmation of the consequent, and denial of the antecedent). The probability propagation rules depend on the interpretation of the conditional. As an example, consider the probabilistic modus ponens. If the conditional is interpreted as a conditional event, then the probabilistic modus consists of the following inference:
If the conditional is interpreted as a material conditional, then the probability propagation rule is a different one:
There are similar norm conflicts for the other three conditional syllogisms (Pfeifer & Kleiter Reference Pfeifer and Kleiter2005). Therefore, these conditional elimination inferences are not examples of single-norm conflicts.
The authors argue, citing Evans (Reference Evans1982), that “we should desist from the practice of reporting logical accuracy in reasoning tasks, and instead report what people actually did” (sect. 6, para. 7). I agree that empirical studies should report what people actually did. However, I argue that reporting (dis)agreement with rationality norms is important and informative. The basic question is the choice of appropriate rationality norms. After about a decade of reasoning research within the normative framework of classical logic, we can safely state that it is high time to consider extensions or alternative normative frameworks.
Coherence-based probability logic is one example (see, e.g., Pfeifer & Kleiter Reference Pfeifer, Kleiter, Leitgeb and Schurz2002; Reference Pfeifer and Kleiter2009; Reference Pfeifer, Kleiter, Oaksford and Chater2010). In this framework, the focus is not on whether or how people draw logically correct inferences about the logical validity of certain argument forms. Rather, the tasks instruct the participants to transmit the uncertainty of the premises to the conclusion. Thus, the focus is on the participant's degree of belief in the conclusion. Probability logic allows for making psychological predictions precise and offers new psychological explanations of the inferences that people draw. The conditional introduction inference from “B” to “If A, then B”, for example, is not probabilistically informative under the conditional event interpretation of indicative conditionals (P(B|A)) and this is the reason why most people claim that one cannot infer a conditional from its consequent (Pfeifer & Kleiter Reference Pfeifer, Kleiter, Manktelow, Over and Elqayam2011). Contrary to standard approaches to probability, this even holds in the special case where the premise is given for sure (P(B)=1). Probability logic tells us which argument forms are probabilistically informative and which ones are not (Pfeifer & Kleiter Reference Pfeifer and Kleiter2006; Reference Pfeifer and Kleiter2009). “Probabilistic informativeness” is not an empirical term, it is a criterion derived within the normative framework. I argue that patterns of inferences beyond the conditional syllogisms should be investigated. Psychological plausible principles, like the ability of withdrawing conclusions in the light of new evidence and the defeasibility of everyday inferences, should be investigated. I agree that reasoning experiments should not focus on logical validity. Rather, the degrees of belief in the conclusions should be investigated.
E&E suggest there are asymmetric relations between formal theories and psychological theories and data (see their Figure 2). Indeed, formal theories inspire psychological theories. However, psychological data can inspire formal theories as well. Ford (Reference Ford2005), for example, investigates how experimental data informs artificial intelligence systems and has developed a formal system of nonmonotonic reasoning, which is inspired by psychological data (Ford Reference Ford2004). Moreover, not only psychological data arbitrate between formal theories: the converse holds as well. In the field of nonmonotonic reasoning, for example, there are many competing systems. However, System P (Kraus et al. Reference Kraus, Lehmann and Magidor1990) is an example of a common denominator of rationality principles any system of nonmonotonic reasoning should satisfy. It makes sense psychologically to use such a system to arbitrate between formal theories in this field.
Another way of arbitrating between formal theories is to require psychologically plausible but minimal principles for rationality, such as coherence (see, e.g., Coletti & Scozzafava Reference Coletti and Scozzafava2002). Coherence requires only avoiding bets that lead to sure loss. This criterion is much weaker than, for example, requiring maximizing expected utility. In my opinion, the relation between normative and descriptive components in a psychological theory of reasoning is a genuinely interactive one.
I agree with the authors that learning should be included in reasoning research. Moreover, there seems to be a consensus among reasoning researchers that the interpretation of the task material remains the same within participants. However, this is not the case in general. If participants solve several items of the probabilistic truth table task, the number of conditional event responses tend to increase from about 40% at the beginning of the experiment to about 80% at the end. Thus, the participant's responses converge on the competence answers (Fugard et al. Reference Fugard, Pfeifer, Mayerhofer and Kleiter2011b). Assuming appropriate bridge laws that connect “is” and “ought” inferences, I avoid committing the is–ought fallacy, if I claim that rational reasoners should converge to the conditional event response.