Elsevier

Cognitive Science

Volume 26, Issue 4, July–August 2002, Pages 425-468
Cognitive Science

Strategies in sentential reasoning

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Abstract

Four experiments examined the strategies that individuals develop in sentential reasoning. They led to the discovery of five different strategies. According to the theory proposed in the paper, each of the strategies depends on component tactics, which all normal adults possess, and which are based on mental models. Reasoners vary their use of tactics in ways that have no deterministic account. This variation leads different individuals to assemble different strategies, which include the construction of incremental diagrams corresponding to mental models, and the pursuit of the consequences of a single model step by step. Moreover, the difficulty of a problem (i.e., the number of mental models required by the premises) predisposes reasoners towards certain strategies. Likewise, the sentential connectives in the premises also bias reasoners towards certain strategies, e.g., conditional premises tend to elicit reasoning step by step whereas disjunctive premises tend to elicit incremental diagrams.

Introduction

Logical reasoning is central both to the development of science and mathematics and to the solution of problems in daily life. Naı̈ve individuals can grasp that a set of propositions logically implies a conclusion. The term naı̈ve here refers to individuals who have no explicit mastery of formal logic or any other cognate discipline. It does not impugn their intelligence. What underlies their logical ability, however, is controversial. Theorists have proposed that it depends on a memory for previous inferences (e.g., Kolodner, 1993), on conditional rules that capture general knowledge (e.g., Newell, 1990), on “neural nets” representing concepts (e.g., Shastri & Ajjanagadde, 1993), or on specialized innate modules for matters that were important to our hunter–gatherer ancestors (Cosmides, 1989). But, none of these accounts readily explains the ability to reason about matters for which you have no general knowledge. Suppose, for instance, that you know nothing about computers but you are given the following premises:

If the software is right and the cable is correct then the printer works.

The software is right, but the printer does not work.

You are able to infer the conclusion:

The cable is not correct.

This inference is valid, that is, its conclusion must be true granted that its premises are true. It is an example of a major class of logical deductions, sentential inferences, which hinge on negation and sentential connectives, such as “if,” “or,” and “and,” and which are captured in an idealized way in the branch of logic known as the “sentential” or “propositional” calculus.

Theorists have two alternative views about how naı̈ve individuals make sentential inferences (see Baron, 1994, for a review). Originally, they thought that naı̈ve individuals rely on formal rules of inference akin to those of logic (e.g., Braine & O’Brien, 1998; Inhelder & Piaget, 1958; Rips, 1994). The discovery that content influences reasoning (see Wason & Johnson-Laird, 1972), coupled with the need to account for the mental representation of discourse, led to a different view of reasoning. Individuals grasp the meaning of premises, and they use this meaning to construct mental models of the possibilities that the premises describe. They evaluate an inference as valid if its conclusion holds in all their mental models of the premises (Johnson-Laird & Byrne, 1991; Polk & Newell, 1995).

The controversy between the two competing views has been fruitful. It has led to the development of explicit computer models of reasoning, and to more stringent experiments. But, it has focused on simple inferences. Most studies of sentential reasoning have examined inferences based on no more than two premises (for reviews, see Evans, Newstead, & Byrne, 1993; Garnham & Oakhill, 1994). They have aimed to reveal the hidden nature of inferential mechanisms: do they rely on formal inference rules or on mental models? In this paper, however, we want to go beyond the usual investigation of the basic inferential mechanisms and to examine an aspect of reasoning that has often been neglected—the strategies that individuals develop to make complex inferences (see also Schaeken, De Vooght, Vandierendonck, & d’Ydewalle, 2000).

We propose the following working definition:

A strategy in reasoning is a systematic sequence of elementary mental steps that an individual follows in making an inference.

We refer to each of these mental steps as a tactic, and so a strategy is a sequence of tactics that an individual uses to make an inference. We illustrate this terminology with the following problem, which you are invited to solve:

There is a white pill in the box if and only if there is a green pill.

Either there is a green pill in the box or else there is a red pill, but not both.

There is a red pill in the box if and only if there is a blue pill.

Does it follow that:

If there isn’t a white pill in the box then there is a blue pill in the box?

Like most people, you probably responded correctly to this problem:

Yes, if there isn’t a white pill in the box, then there is a blue pill in the box.

But, how did you solve the problem? What kind of tactical steps did you carry out and how were they organized? These are the questions that we want to address in this paper.

One possible strategy is to use a supposition, i.e., an assumption for the sake of argument. Thus, you might have said to yourself:

Suppose that there isn’t a white pill in the box. It follows from the first premise that the box does not contain a green pill, either. It then follows from the second premise that there is a red pill in the box. And it follows from the third premise that there is a blue pill too. So, if there isn’t a white pill in the box, then it follows that there is a blue pill in the box. The conclusion in the question is correct.

Hence, in this strategy, you made a supposition corresponding to a single possibility and followed up its consequences step by step. One tactic in the strategy is to make a supposition, and another is to draw a conclusion from the supposition and the first premise. Strategies are therefore the molar units of analysis, tactics are the molecular units, and the inferential mechanisms underlying tactics are the atomic units. Whereas the nature of inferential mechanisms is the topic of several hundred papers in the literature, strategies and tactics have not yet been investigated for sentential reasoning.

We use the term “strategy” in much the same sense as Bruner, Goodnow, and Austin (1956), who described strategies in concept attainment. Like them, we do not imply that a strategy is necessarily conscious. It may become conscious as reasoners try to develop a way to cope with a problem. But, as we will see, individuals describe each tactical step they take in an inference rather than their high-level strategy. We can infer their strategies from these descriptions (cf. Bruner et al., 1956, p. 55). Miller, Galanter, and Pribram (1960, p. 16) defined a “plan” as “any hierarchical process in the organism that can control the order in which a sequence of operations is to be performed.” We could have used the term “plan” instead of “strategy,” but it has misleading connotations, particularly in artificial intelligence. It suggests that people first plan how they will make an inference and then carry out the plan. Reasoners, however, do not seem to proceed in this way. Instead, they start reasoning at once, and one tactical step leads to another, and so on, as their strategy unfolds spontaneously. Tactics are also akin to the component processes that Sternberg postulated in his analysis of analogies, numerical series problems, and syllogisms (see e.g., Sternberg, 1977, Sternberg, 1983, Sternberg, 1984). We likewise follow in the tradition of analyzing cognitive tasks into their component processes, or mechanisms, within the information-processing methodology (Hunt, 1999).

As Fig. 1 shows, our account distinguishes four levels in a hierarchy of thinking and reasoning. Each level in the hierarchy is a level of organization, but it depends on the level below for its implementation, much as a programming language is a level of organization that depends for its implementation on the lower level of machine language. In other words, the levels are not independent of one another, and the organization at one level has to be underpinned by what happens at a lower level. At the highest level, there is metacognitive thinking, i.e., thinking about thinking, usually in order to develop a solution to a problem. Thinking at this level often occurs in solving complex problems such as the tower of Hanoi, and in “game-theoretic” situations such as the prisoner’s dilemma (Berardi-Coletta, Buyer, Dominowski, & Rellinger, 1995). When it is not obvious how to solve a problem, individuals may think about thinking in this self-conscious way. But, as we will see, they do not seem to develop reasoning strategies at the metacognitive level. At the second level are the strategies in thinking. They unfold in a series of actions without the individual necessarily having a conscious awareness of an overall strategy. At the third level are each of the tactics in a strategy, such as making a supposition, or combining it with a premise to make an inference. At the lowest level are the cognitive mechanisms that underlie the tactical steps, e.g., the construction of a mental model if the model theory is correct, or the application of a rule of inference if formal rule theories are correct. Our aim in the present paper is to delineate the nature of the strategies and tactics underlying sentential reasoning.

The levels of thought are not independent of one another: strategies depend on tactics, and tactics depend on inferential mechanisms. Hence, we need an account of inferential mechanisms in order to explain tactics. We propose that inferential mechanisms are based on mental models (Johnson-Laird, 2001), and we will try to show how such mechanisms at the lowest level can compose the tactics one level up, which in turn make up strategies at the second level (see Section 5). It is conceivable that inferential mechanisms are based on formal rules of inference instead of mental models, and we return to this possibility in Section 7.

The phenomena occurring at the strategic and tactical levels are comparable to the control procedures and the logical components in the implementation of a theorem prover in artificial intelligence. Logic itself is not enough (Stenning & Oaksford, 1993). It cannot specify the procedure for proving theorems. Hence, nonlogical decisions are necessary to obtain a practical implementation of a theorem prover. Typically, these procedures are designed to avoid a combinatorial explosion in demands on resources. For instance, PROLOG, a programming language based on the analogy between programs and proofs, uses heuristic tools such as “backward-chaining” to implement an effective theorem prover. Similarly, inferential mechanisms at the lowest level of thought do not determine how human reasoning proceeds. They are the basic tools. The ways they are put to use is a matter of strategies and tactics in reasoning.

How can we find out what reasoning strategies individuals develop? In our view, the first steps are to observe what they say as they think-aloud while they reason, and to give them paper and pencil and to see what they write down and what they draw. These data, however, are a controversial source of evidence. One problem is their validity. People can be unable to describe how they reached a certain decision or even be mistaken about why they acted as they did (see e.g., Greenwald & Banaji, 1995; Nisbett & Wilson, 1977). The need to think-aloud and to use paper and pencil may also change the nature of the thought process, and even impair it. Schooler, Ohlsson, and Brooks (1993) interrupted people who were trying to solve insight problems. After the interruption, those who had to make a retrospective report on their thinking solved fewer problems than those who carried out an unrelated task (a crossword puzzle). Other studies, however, report that thinking aloud enhanced performance (e.g., Berardi-Coletta et al., 1995). It can slow people down, but it often appears to have no other major effects (cf. Russo, Johnson, & Stephens, 1989). In general, it can be a reliable guide to the sequence of a person’s thoughts (Ericsson & Simon, 1980; Newell & Simon, 1972). Our view is that the use of “think-aloud” protocols and drawings and writings is indeed only a first step in the analysis of strategies. Bell (1999) has compared reasoning when individuals think-aloud and when individuals think to themselves in the usual way. The patterns of results were similar in the two conditions. The present paper makes no such comparisons, because its main goal is to delineate the variety of strategies that reasoners develop. Even if these strategies were unique to thinking aloud with pencil and paper, a major goal of psychology should be to give an account of them and of how individuals develop them.

The paper begins with an account of how psychologists have thought about reasoning strategies in the past (Section 2). It then presents a taxonomy of strategies in sentential reasoning based on experiments in which the participants thought aloud as they either evaluated given conclusions or drew their own conclusions from premises (Section 3). It outlines the core principles of the theory of mental models (Section 4), which it uses to formulate a theory of strategies in sentential reasoning (Section 5). It reports three experiments corroborating the theory’s account of how people develop strategies (Section 6). And it concludes with an appraisal of strategic and tactical thinking in reasoning (Section 7).

Section snippets

Previous studies of strategies in logical reasoning

The pioneering studies of reasoners’ strategies investigated relational “series” problems (Hunter, 1957, Huttenlocher, 1968, Piaget, 1921), such as:

  • John is taller than Pete.

  • Pete is smaller than Bob.

  • Who is the tallest?

When participants carry out problems based on five premises, they rapidly develop various “short cut” strategies (Wood, 1969; Wood, Shotter, & Godden, 1974). With premises that each contain the same relation, say, “taller than,” they look to see whether a term occurs only on the

A taxonomy of strategies in sentential reasoning

How can experimenters best observe the strategies that reasoners use in sentential reasoning? In our view, there are three desiderata. First, the inferential problems should be sufficiently time-consuming to force the participants to think, but not so difficult that they make many errors. Second, the experimental procedure needs to externalize strategies as much as possible. Third, the content of inferences should be neutral and unlikely to trigger general knowledge. Those materials that do

Reasoning with mental models

The taxonomy in Table 2 describes the strategies, but it does not explain them or their tactical steps. Our aim is to formulate a theory of strategies and tactics, and we proceed by first accounting for inferential tactics, and then for how they are integrated within strategies. Tactics include reading a single premise, writing it down, and drawing a diagram to represent it. Our concern, however, is with inferential tactics and with the mechanisms that underlie them. So we turn to the mental

The theory of reasoning strategies

In this part of the paper, we develop a theory of strategies and tactics. It derives from the theory of mental models, and from its application to earlier work on strategies in other sorts of reasoning (Bucciarelli & Johnson-Laird, 1999; Johnson-Laird & Byrne, 1990). We formulate the theory in terms of three main assumptions. Following Harman (1973), our first assumption is that reasoning is not a deterministic process that unwinds like clockwork:

  • 1.

    The principle of nondeterminism: thinking in

The development of strategies

How do reasoning strategies develop? The process might be idiosyncratic, but the evidence supports the occurrence of robust differences from one individual to another. People are likely to differ in their reasoning experiences, in the capacity of their working memories, and in their ability to employ complex inferential mechanisms. They are therefore likely to develop different strategies that reflect these differences. Yet, the principle of model-based tactics implies that everyone at the

General discussion

Current accounts of sentential reasoning have neglected strategies. Our aim has been to remedy the neglect and to advance a theory of strategies in sentential reasoning. The theory depends on three assumptions:

  • 1.

    Nondeterminism: thinking in general and sentential reasoning in particular are governed by constraints, but vary in ways that can be captured only in a nondeterministic account.

  • 2.

    Strategic assembly: naı̈ve reasoners assemble reasoning strategies bottom-up as they explore problems using

Supplementary data

. Appendix to Strategies in Sentential Reasoning.

Acknowledgements

This research was made possible in part by grants to the first and the third authors, respectively from the European Commission (Marie Curie Fellowship) and the National Science Foundation (Grant BCS-0076287) to study strategies in reasoning. We thank Fabien Savary for carrying out Experiment 1 and for transcribing its protocols. We are also grateful for the help of many colleagues, including Bruno Bara, Patricia Barres, Victoria Bell, Monica Bucciarelli, Ruth Byrne, Wim DeNeys, Kristien

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