‘If’ and the problems of conditional reasoning

doi:10.1016/j.tics.2009.04.003

Trends in Cognitive Sciences

Volume 13, Issue 7, July 2009, Pages 282-287

https://doi.org/10.1016/j.tics.2009.04.003 Get rights and content

‘If’ is a puzzle. No consensus has existed about its meaning for over two thousand years. Here, we show how the main psychological theories deal with the seven crucial problems that it raises. These competing explanations treat ‘if’ as though it was a term in a formal logic, or as eliciting the construction of a mental model of the world, or as an instruction to suppose that a proposition holds. The solution to ‘if’ would be a major step towards understanding how people reason, and towards implementing a computer program that can reason in a human way. We argue that the mental model theory is closer to resolving the puzzle of ‘if’ than its competitors.

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:

What is the probability that if the nickel is heads then the dime is heads?

They interpret the question to mean:

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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Trends in Cognitive Sciences

Opinion‘If’ and the problems of conditional reasoning

Introduction

Section snippets

Problem 1: why are some conditional inferences difficult?

Problem 2: what do conditionals mean?

Problem 3: in what cases are conditionals true?

Problem 4: why is reasoning with if and only if easier?

Problem 5: what makes counterfactuals special?

Problem 6: what denies a conditional?

Problem 7: what is the probability of a conditional?

Conclusions

Acknowledgement

Cognit.

Cognit. Psychol.

Trends Cogn. Sci.

J. Mem. Lang.

Cognit. Psychol.

Cogn. Sci.

Trends Cogn. Sci.

Formal Logic: Its Scope and Limits

How We Reason

Conditionals: a theory of meaning, inference, and pragmatics

Psychol. Rev.

Human Reasoning and Cognitive Science

If

Hypothetical Thinking

General propositions and causality

The mental model theory of conditionals: A reply to Guy Politzer

Topoi

Conditionals and directionality: on the meaning of if versus only if

Q. J. Exp. Psychol.

On the nature of mental models of conditional: The case of If

If then and Only if. Think. Reason.

A critique of Oaksford, Chater and Larkin's (2000) conditional probability model of conditional reasoning

J. Exp. Psychol. Learn. Mem. Cogn.

The Handbook of Imagination and Mental Simulation

The Rational Imagination: How People Create Alternatives to Reality

A Philosophical Guide to Conditionals

Philosophical knowledge and knowledge of counterfactuals

Opinion
‘If’ and the problems of conditional reasoning