The correct answer to the card problem above is the two cards A and 7. Many people answer A and 4. The B card is clearly not useful, because it cannot prove or disprove the rule regardless of what is on the other side. Surprisingly, however, the same is true for the 4 card: even if it has an A on the other side, it supports but neither proves nor disproves the rule that any card with an A on one side has a 4 on the other side. In contrast, flipping the 7 card does test the rule, because the rule would be disproved if the other side is an A.
Many philosophers of science interpret the A & 4 answer as evidence of a confirmation bias: the chooser of the 4 card is seeking a result that confirms the hypothesis, rather than choosing the 7 card and potentially disproving the hypothesis. Scientists, in contrast, may justify choice of the 4 card as a search for patterns where they are most likely to be found. Not choosing the 7 card, however, is a failure to consider deductively the importance of potential results.
Two problems can involve identical deductive logic yet differ in difficulty. How a deductive problem is posed can affect the likelihood of correct results. Concrete examples are easier to solve than are the same problems expressed in symbols. For example, the success rate on the problem above was increased from 10% to 80% [Kuhn et al., 1988] when the problem was recast: given an envelope that may or may not be sealed and may or may not have a stamp on it, test the hypothesis, ‘if an envelope is sealed, then it has a 5-pence stamp on it’.
Our greater facility with the concrete rather than with abstract deductions challenges the very basis of this decision-making. Possibly we do not even make decisions based on learned rules of formal logic [Cheng and Holyoak, 1985], but instead we recognize conceptual links to everyday experience [Kuhn et al., 1988]. The problem must seem real and plausible if there is to be a good chance of a successful solution; thus the postage problem is easier than the 4-card problem. In deductive logic, a similar strategy is often useful: recast the problem so that the logical structure is unchanged but the terms are transformed into more familiar ones. This technique, known as substitution, is one that we shall employ later in this chapter.
The four-card problem illustrates several points:
- prior thought can prevent needless experiments;
- sketches can be valuable in avoiding error;
- the same problem is more likely to be solved correctly if in familiar terms than if in abstract terms;
- confirmation bias is present in science, but to some extent it is a normal consequence of our pervasive search for patterns; and
- many people’s ‘deductive thinking’ may actually be inductive pattern recognition of a familiar deductive form.
Logic
Logic means different things to different people. To Aristotle (384-322 B.C.), the ‘Father of Logic’, it was a suite of rules for deductive evaluation of syllogisms. To Peter Abelard (1079-1142) and William of Occam (1285-1349), Aristotelian logic was a useful launching point for development of a more comprehensive logic. G. W. Leibniz (1646-1716) sought to subsume all types of arguments within a system of symbolic logic. During the last century, symbolic logic has been the focus of so much study that it almost appeared to be the only type of logic. A notable exception was John Stuart Mill’s Canons of inductive logic (Chapter 3).
