## Inductive Reasoning Dr Steve Abel

Reasoning, as one of the oldest research areas in psychology, has concerned itself with a key question about human nature: "Are human beings rational?". Philosophers have tended to answer this question with a resounding "yes", with arguments that the laws of logic are the laws of thought (Boole, 1854; Mill, 1843). This basic idea has been used, albeit in more sophisticated forms, in the psychology of reasoning. The psychology of reasoning covers both deductive and inductive reasoning. When people carry out deductive reasoning they usually determine what conclusion, if any, necessarily follows when certain statements or premises are assumed to be true. In inductive reasoning, people make a generalised conclusion from premises that describe particular instances (see e.g. Chapters 10 and 15).

Johnson-Laird and Byrne (1991, p. 3) point out that deductive reasoning is a central intellectual ability, which is necessary:

in order to formulate plans; to evaluate alternative actions; to determine the consequences of assumptions and hypotheses; to interpret and formulate instructions, rules and general principles; to pursue arguments and negotiations; to weigh evidence and to assess data; to decide between competing theories; and to solve problems. A world without deduction would be a world without science, technology, laws, social conventions and culture.

Deductive reasoning research makes central use of logical systems—especially the propositional calculus— to characterise the abstract structure of reasoning problems. So, we will review the logic of deduction in some detail before considering the psychological research. If you are not familiar with logic then you may find that a little extra effort with this section will improve your understanding of later sections.

In this chapter, we focus on deductive reasoning with conditionals or, more simply, reasoning with "if". The propositional calculus and reasoning based on it involve the use of a number of logical operators: or, and, if... then, if and only if (see Evans, Newstead, & Byrne, 1993b, for a full account of propositional reasoning). In research on conditional reasoning the question "Are people rational?" is re-cast as "Are they logical?". In other words, do people conform to the logical interpretation of if...then, make valid inferences and reject the invalid inferences dictated by the propositional calculus (see section on logic). As we shall see later on, the simple answer is "No".

### The use of logic in reasoning research

Newell and Simon's problem-space theory uses the notion of an idealised problem space to characterise the abstract structure of a problem, quite independently of any psychological proposals (see Chapters 14 and 15). In reasoning research, some logics—usually, the propositional calculus—have been used in a similar manner. These logics are used to characterise the abstract structure of reasoning problems and to determine categories of responses (i.e., incorrect and correct answers). So, a clear understanding of this sub-section is essential to make sense of large portions of reasoning research.

In mathematical systems we use symbols to stand for things (e.g., let hi be the height of the Empire State building and h2 be the height of the Eiffel Tower) and then apply mathematical operators to these symbols to manipulate them in various ways to produce a new statement (the combined height of both buildings should be hi plus h2, where plus is the operator used). In an analogous fashion, logical systems use symbols to stand for sentences and apply logical operators to them to reach conclusions. So, in the propositional calculus, we might use P to stand for the verbally expressible proposition "It is raining" and Q to stand for "Alicia gets wet", and then use the logical operator if... then to relate these two propositions: if P then Q. It is very important to remember that even though logical operators use common words (such as or, and, if. then) in logic these terms have very different meanings. The logical meaning of the conditional (i.e., if. then) is well specified and differs markedly from everyday conceptions of the words "If. then". In the next subsection, we attempt to explain how logicians specify the meaning or semantics of these operators.

The propositional calculus has a small number of logical operators: not, and, or, if... then, if and only if. In this logical system, propositions can only have one of two truth-values, they are either true or false. For instance, if P stands for "it is raining" then P is either true (in which case it is raining) or P is false (it is not raining). The calculus does not admit any uncertainty about the truth of P (where it is not really raining but is so misty you could almost call it raining), although there are other multi-valued logics that do.

Logicians use a system of truth tables to lay out the possibilities for a proposition (i.e., whether it is true or false) and to explain how a logical operator acts on that proposition. For example, a single proposition P can be either true or false. In truth tables, this is notated by putting P as a heading and showing the two values of it, as follows:

Truth tables and the "meaning" of logical operators

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