Experimental Insights into the Influence of Logic and Pragmatics on Conditional Argument Evaluation

A research link between conditional reasoning and mathematics has been reported only for late adolescents and adults, despite claims about the pivotal importance of conditional reasoning, i.e., reasoning with if–then statements, in mathematics. Secondary students’ problems with deductive reasoning in mathematics have been documented for a long time. However, evidence from developmental psychology shows that even elementary students possess some early conditional reasoning skills in familiar contexts. It is still an open question to what extent conditional reasoning with mathematical concepts differs from conditional reasoning in familiar everyday contexts. Based on Mental Model Theory (MMT) of conditional reasoning, we assume that (mathematical) content knowledge will influence the generation of models, when conditionals concern mathematical concepts. In a cross-sectional study, 102 students in Cyprus from grades 2, 4, and 6 solved four conditional reasoning tasks on each type of content (everyday and mathematical). All four logical forms, modus ponens (MP), modus tollens (MT), denial of the antecedent (DA), and affirmation of the consequent (AC), were included in each task. Consistent with previous findings, even second graders were able to make correct inferences on some logical forms. Controlling for Working Memory (WM), there were significant effects of grade and logical form, with stronger growth on MP and AC than on MT and DA. The main effect of context was not significant, but context interacted significantly with logical form and grade level. The pattern of results was not consistent with the predictions of MMT. Based on analyses of students’ chosen responses, we propose an alternative mechanism explaining the specific pattern of results. The study indicates that deductive reasoning skills arise from a combination of knowledge of domain-general principles and domain-specific knowledge. It extends results concerning the gradual development of primary students’ conditional reasoning with everyday concepts to reasoning with mathematical concepts adding to our understanding of the link between mathematics and conditional reasoning in primary school. The results inspire the development of educational interventions, while further implications and limitations of the study are discussed.

In deductive reasoning, people are asked\nto infer the truth of an argument’s conclusion given a set of premises. Research\ninto the processes underlying deduction\nhas focused on examining how well people discriminate between logically valid\nand invalid arguments, and how irrelevant factors such as one’s prior beliefs\ninterfere with the ability to reason logically (Evans et al., 1983). This normative approach to validity has traditionally\ninformed both practice and theory in the\nliterature. However, its critics argue that\n“normativism” often leads investigators to\nbiased or misleading interpretations of\nphenomena (Elqayam and Evans, 2011).

The effect of premise order in conditional reasoning: a test of the mental model theory

Conditional reasoning by mental models: chronometric and developmental evidence

Five experiments investigated two theories of conditional reasoning. The pragmatic schema theory posits that conditional reasoning is mediated by context-sensitive inference rules. According to the contextual cuing theory, inferences are based on a mental model that represents necessity and sufficiency relations. Both schematic relations and necessity relations predicted responses on Wason's four-card selection task. In contrast, after the effects of perceived necessity had been partialled out, schematic relations did not predict responses to either a conditional arguments task, or a task in which subjects judged the similarity of and only statements. These findings question the assumption that reasoning is mediated by schematic rules, which presumably apply regardless of task. However, there was evidence to suggest that both schematic variables and the availability of counter-examples may be important in evaluating necessity relations, suggesting an alternative role for schematic-based interpretations in reasoning.Conditional reasoning involves drawing inferences about situations in which the occurrence of one event is conditional upon the occurrence of another event (e.g., if the car runs out of gas, then it stalls). Despite the apparent simplicity of this task, two decades of research have revealed that people reason inconsistently: They sometimes draw invalid inferences; at other times, they fail to draw valid inferences. The purpose of this research was to evaluate two current explanations for this variability in conditional reasoning performance. Cheng and Holyoak's (1985) pragmatic schema theory proposes that conditional reasoning performance is mediated by context-specific inference rules, and predicts that inference patterns will vary as a function of the reasoning schema that is evoked. According to the contextual cuing theory, the types of inferences made will vary as a function of the necessity and sufficiency of the conditional relation.These approaches are similar in that both propose a means of integrating the interpretation of conditional relations with the underlying inferential processes. However, they emphasize different interpretational variables (pragmatic context vs necessity/sufficiency), and propose different mechanisms for the generation of deductive inferences (abstract rules vs counter-examples). The purpose of this paper is to test predictions that contrast the two views, andto develop a theoretical position that integrates these two perspectives into a single framework.Conditional Reasoning PatternsConditional reasoning involves drawing inferences from conditional relations that are usually phrased p, then q, and are typically assessed using one of two tasks. The Wason selection task requires subjects to find the information necessary to determine whether a conditional relation is true or false (Wason, 1968). For example, if the rule was a card has a vowel on one then it has an even number on the other side, then finding an odd number opposite a vowel violates the rule. The conditional arguments task, on the other hand, requires subjects to evaluate the validity of inferences derived from a conditional rule.For a conditional arguments task, the subject is presented with a conditional relation, and is then asked to evaluate the validity of four logical questions or arguments, termed Modus Ponens (MP), Modus Tollens (MT), Denying the Antecedent (DA), and Affirming the Consequent (AC). Each argument involves an affirmation or negation of the antecedent (p) or consequent (q), as is illustrated below:1) If this tree is a spruce, then it has needles.MP: This tree is a spruce. Does it have needles? (YES)DA: This tree is not a spruce. Does it have needles? (MAYBE)AC: This tree has needles. Is it a spruce? (MAYBE)MT: This tree does not have needles. Is it a spruce? (NO)According to standard, propositional logic, the correct responses to the MP, DA, AC, and MT questions are yes, maybe, maybe, and no respectively. …

The Role of Logical Structure and Premise Believability in Belief Revision Dustin P. Calvillo (calvillo@psych.ucsb.edu) Department of Psychology, University of California Santa Barbara, CA 93106-9660 USA Russell Revlin (revlin@psych.ucsb.edu) Department of Psychology, University of California Santa Barbara, CA 93106-9660 USA Belief Revision Belief revision occurs when one moves from one belief state to another after encountering some data that are inconsistent with oneis initial belief set. Experiments in belief revision have demonstrated that the initial logical structure of an argument affects how reasoners revise their beliefs. When arguments for changing beliefs are made in a logical form, the typical finding is that the major premise is revised more frequently than the minor premise. This is evident when the modus ponens (MP) inference is contradicted (if p then q; p; therefore, q), while there is no clear preference when the modus tollens (MT) inference is contradicted (if p then q; not q; therefore, not p) (Dieussaert, Schaeken, De Neys, & diYdewalle, 2000; Elio & Pelletier, 1997; Politzer & Carles, 2001). Others have reported a different finding: reasoners revise belief in the major and minor premises equally often in MP problems, but prefer to disbelieve the minor premise in MT problems (Revlin & Calvillo, 2002; Revlin, Cate, & Rouss, 2001). In three experiments, we explore possible explanations for these two different patterns of results. Three possible explanations for the inconsistent results are the types of major premises, the revision alternatives presented to participants, and the prior believability of the major premises. The major premises used by Elio and Pelletier (1997), Dieussaert et al. (2000), and Politzer and Carles (2001) were conditional (if p then q) and somewhat neutral in believability. Participants in these experiments were allowed to express uncertainty toward premises. The major premises used by Revlin et al. (2001) and Revlin and Calvillo (2002) were universal quantifiers (all p are q) and considerably more believable. Participants in these experiments were forced to decide, with certainty, to disbelieve the major or minor premise. In Experiment 1, we assigned 80 introductory psychology students from the University of California, Santa Barbara into four groups. Logical structure (MP or MT) and type of major premise (conditional or quantifier) were between- participants variables. The major premise revision rates are presented in Table 1. The rates for both quantifiers and conditionals were similar to those found by Revlin et al. (2001). Logical structure had a significant effect, type of major premise did not, and the two variables did not interact. This ruled out the use of different major premise types as an explanation for the different previous findings. In Experiment 2, we assigned 50 participants to two groups and presented them with MP and MT problems like in Experiment 1, but gave them the revision alternatives used by Politzer and Carles (2001). As seen in Table 1, logical structure had a reliable effect on revision rates and the major premise revision rates were similar to those of Revlin et al. (2001), ruling out the use of different revision alternatives as an explanation for the inconsistent results. Table 1: Major premise revision rates by logical structure. Experiment 1: Conditional Experiment 1: Quantifier Experiment 2 Experiment 3: Low believability MP MT In Experiments 1 and 2, the major premises used were highly believable. In Experiment 3, we gave 47 participants either MP or MT problems with major premises of low- believability. The results, as seen in Table 1, were similar to those of Elio and Pelletier (1997). There was a preference to revise belief in the major premise in both MP and MT problems and there was no effect of logical structure. Experiments 1 and 2 ruled out the use of different types of major premises and revision alternatives explanations for the varying results in the literature. Experiment 3 showed that believability of the major premise is a likely source of the different patterns of results, demonstrating the need for models of belief revision to include initial premise believability to account for how reasoners revise beliefs. References Dieussaert, K., Schaeken, W., De Neys, W., & diYdewalle, G. (2000). Initial belief state as a predictor of belief revision. Cahiers de Psychologie Cognitive/Current Psychology of Cognition, 19, 277-288. Elio, R., & Pelletier, F.J. (1997). Belief change as prepositional update. Cognitive Science, 21, 419-460. Politzer, G., & Carles, L. (2001). Belief revision and uncertain reasoning. Thinking and Reasoning, 7, 217-234. Revlin, R., & Calvillo, D.P. (2002). Stages in counterfactual reasoning. Unpublished Manuscript. Revlin, R., Cate, C.L., & Rouss, T.S. (2001). Reasoning counterfactually: Combining and rending. Memory & Cognition, 29, 1196-1208.

This chapter focuses on the influence of pragmatic factors on reasoning — focusing on a prima facie puzzle for both logical and probabilistic accounts of reasoning: the asymmetry between modus ponens (MP) and modus tollens (MT) inferences in conditional reasoning. It discusses the account of the conditional developed by Adams. It shows that when applied to conditional inference, recent research in the normative literature (Sober, 2002; Sobel 2004; Wagner, 2004) is consistent with the account presented by Oaksford et al. (2000). These normative accounts introduce an important condition on probabilized MP and MT, called the rigidity condition which may explain the MP–MT asymmetry. It then argues that by exploiting this condition, far better fits to the data on conditional inference can be obtained.

Dual-process theories of conditional reasoning predict that relationships among four basic logical forms, and to intellectual ability and thinking predictions, are most evident when conflict arises between experiential and analytic processing (e.g., Stanovich & West, 2000). To test these predictions, 210 undergraduates were presented with conditionals for which the consequents were either weakly or strongly associated with alternative antecedents (i.e., WA and SA problems, respectively). Consistent with predictions, modus ponens inferences were not related to inferences on the uncertain forms (affirmation of the consequent, denial of the antecedent). On WA problems, modus tollens, affirmation of the consequent, and denial of the antecedent were related to each other and to verbal ability. Modus ponens was linked to verbal ability only when disabling conditions were activated. In accord with the predictions of Stanovich and West (2000), on most problems, thinking dispositions predicted variance in inferences independently from verbal ability. We argue that a largely automatic experiential processing system governs performance on modus ponens, unless disablers are activated. Consciously controlled analytic processing predominates on the uncertain forms and, under some conditions, on modus tollens.

An outstanding question for Hybrid dual process models of reasoning is whether both basic (e.g., modus ponens - MP) and more complex (e.g., modus tollens - MT) forms of conditional inference result from intuitive, type 1 processes. The present study considers whether a proclivity, ability, or capacity to engage in analytical (type 2) thinking might be more closely related to performance on MT than to performance on MP. Such a finding would suggest that the extent to which MT is intuitive for an individual is a function of analytical thinking level and that, in general, MT is not as intuitive an inference form as MP. The present study tested this prediction by way of a conditional reasoning task on which instructional set (belief or logic), congruency, and complexity of inference were manipulated. While results varied somewhat across experiments, it was generally the case that differences in performance between low and high levels of analytical thinking proclivity (AOT), ability (CRT), and capacity (Working Memory Span) were greater for MT problems than for MP problems suggesting that these inference forms may not be equally intuitive.

The Neural Basis of Conditional Reasoning with Arbitrary Content

The difference in spatiotemporal dynamics between modus ponens and modus tollens in the Wason selection task: An event-related potential study

Reasoning by a robot relates to commands it receives in the natural language in the form of incompletely stated arguments, i.e. enthymemes. An attempt is made by the robot to seek out missing premises or conclusions that will produce valid arguments on the basis of two inference rules, modus ponens and modus tollens. Component sentences of command arguments state a primary goal, an alternate goal, and a condition for achieving the primary goal. The number of valid command arguments that are valid by reason of the two inference rules is determined, and those that are plausible are established. These plausible command arguments are then grouped to convey plausible commands. Finally, the different plausible commands as enthymemes that can be supplied by the master are determined. Also determined are corresponding missing premises and conclusions that the robot seeks out in an attempt to achieve the primary goal. Theorems convey the results of the analysis. >

With p and q each standing for a familiar event, a disjunctive statement, "either p or q", seems quite different from its material conditional, "if not p then q". The notions of sufficiency and necessity seem specific to conditional statements. It is surprising, however, to find that perceived sufficiency and necessity affect disjunctive reasoning in the way they affect conditional reasoning. With B and C each standing for a category name, a universal statement, "all B are C", seems stronger than its logically equivalent conditional statement, "if B then C". However, the effects of perceived sufficiency or necessity were found to be as pronounced in conditional reasoning as in syllogistic reasoning. Furthermore, two experiments also showed that (a) MP (modus ponens)-comparable disjunctive reasoning was as difficult as MT (modus tollens)-comparable disjunctive reasoning, and that (b) MT-comparable syllogisms were easier to solve than MT problems in conditional reasoning.

Empirical evidence for the capacity to detect conflict between biased reasoning and normative principles has led to the proposal that reasoners have an intuitive grasp of some basic logical principles. In two studies, we investigate the boundary conditions of these logical intuitions by manipulating the logical complexity of problems where logical validity and conclusion believability conflict or not. Results pointed to evidence for successful conflict detection on the basic Modus Ponens (MP) inference, but also showed evidence for such a phenomenon on the more complex Modus Tollens (MT) inference. This suggests that both the MP and the MT inferences are simple enough for reasoners to have an intuitive grasp of their logical structure. The boundaries of logical intuition might thus reside in problems of greater complexity than these inferences. We also observed that on the invalid Affirmation of the Consequent (AC) and Denial of the Antecedent (DA) inferences, participants showed higher accuracy on the inference that was expected to be more complex (DA), and no evidence for successful conflict detection was found on these forms. Implications for the logical intuition framework are discussed.

The emergence of inferential rules: The use of pragmatic reasoning schemas by preschoolers