Trends in Cognitive Sciences
Opinion‘If’ and the problems of conditional reasoning
Introduction
When John F. Kennedy addressed the Irish parliament, he said:
If this nation had achieved its present political and economic stature a century or so ago, my great-grandfather might never have left New Ross ..
Like Kennedy, you can think hypothetically. You understand ‘if’ and draw inferences from the conditional assertions that it makes, and neither task seems difficult. Yet for millennia, no agreement has existed about the meaning of conditionals or about their logical properties. Now, following intensive experimental and theoretical efforts, cognitive scientists seem closer to discovering the truth about ‘if’. Here, we outline seven problems that any theory of conditionals must solve, and we outline the solutions offered by each of three main sorts of theory. They are theories based on formal logic, on mental models of the world and on the idea that ‘if’ invites you to make a supposition. We argue that the mental model theory is most likely to solve the puzzle of conditionals: it explains the factors that affect reasoning from conditionals, what they mean, their truth or falsity, their counterfactual use, their denials and their probabilities.
Section snippets
Problem 1: why are some conditional inferences difficult?
Psychologists study four standard inferences exemplified in Table 1. Consider this one:
If he spoke then she laughed.
He spoke.
What follows?
The conclusion is obvious: she laughed. This affirmative inference is ‘valid’, that is, its conclusion holds in every possibility in which its premises hold [1], and so if its premises are true then its conclusion is too.
By contrast, this negative inference is harder:
If he spoke then she laughed.
She didn’t laugh.
What follows?
A typical error is to respond,
Problem 2: what do conditionals mean?
Conditionals mean many things [3]. Suppose someone tells you: if the shelf collapsed then someone put a heavy object on it. You infer that the event in the then-clause occurred ‘before’ the event in the if-clause. Other conditionals convey the opposite temporal order. Formal rule theories (Box 1) take the meaning of conditionals to be implicit in the inferences that follow from them. However, their rules yield too few sorts of inference (e.g. they do not account for the inference above). One
Problem 3: in what cases are conditionals true?
In what cases is this conditional true: if there is a circle then there is a triangle? The obvious answer is when there is both a circle and a triangle. Likewise, it is false when there is a circle but no triangle. But, is it true or false when there is not a circle? Many people say: neither, the case is irrelevant 3, 6. This pattern of evaluations is known as the ‘defective truth table’, and it does not square with the cases that individuals spontaneously list as possible (see the previous
Problem 4: why is reasoning with if and only if easier?
The phrase, ‘if and only if’, seems like something that only a lawyer or logician would say, but consider this inference [3]:
If and only if he spoke then she laughed.
She didn’t laugh.
What follows?
You should find it easier to infer that he did not speak than the negative inference that you made in the first section. Similarly, consider this inference:
He yawned only if she did.
She didn’t yawn.
What follows?
Again, you should find it easy to infer that he did not yawn. What explains these two robust
Problem 5: what makes counterfactuals special?
Kennedy's conditional at the start of this article refers to a possibility that did not actually occur – Ireland was still struggling for independence a century before he spoke. Such ‘counterfactual’ conditionals are common in everyday discourse 15, 16, 17, 18, but they differ in meaning and inferential consequences from the conditionals that we have discussed so far. Counterfactuals, as the model theory postulates, refer to two sorts of possibility 3, 18. One possibility is factual: Ireland
Problem 6: what denies a conditional?
Individuals have no clear idea about how to ‘negate’ an assertion, but they do know how to deny one. Some individuals deny a conditional such as, ‘if he spoke then she laughed’, by asserting: if he spoke then she didn’t laugh; and others deny it categorically: he spoke and she didn’t laugh[27]. But, which denial is correct? The answer is clear for a general claim, such as: if any man speaks then a woman laughs. Its denial is at least one man speaks and a woman does not laugh (i.e. one
Problem 7: what is the probability of a conditional?
The principle to which we have just alluded also explains the answer that individuals give to the following sort of question:
They interpret the question to mean:What is the probability that if the nickel is heads then the dime is heads?
If the nickel is heads then what's the probability that the dime is heads?
They estimate, not the probability of the conditional, but the conditional probability of its consequent given its antecedent [9]. Formal rule theories do not seem to have addressed this
Conclusions
We have described seven problems about conditionals that concern their meanings, their truth or falsity, their counterfactual use, their denials, their probabilities and the factors that affect reasoning from them. They are not the only problems, of course (Box 4), but any adequate theory of conditionals has to solve them. Table 2 summarizes the solutions of the three main theories.
Formal rule theories focus on the proofs of given conclusions. Their biggest problem is the lack of any
Acknowledgement
This research was supported in part by National Science Foundation Grant SES 0844851 to the second author to study deductive and probabilistic reasoning.
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