Friday Fallacy: Affirming the Consequent

In my last Friday Fallacy post, I looked at the fallacy of  denying the antecedent. There I discussed conditional statements, statements of the form “if P then Q”. Examples would be statements such as “if it is raining then the grass will be wet” or “if the US had not shot Bin Laden then he would still be alive”. I noted that conditionals  have  an “antecedent” and a “consequent”. In the above examples the antecedent is the claim “it is raining” or “the US had not shot Bin Laden”. In a conditional statement one talks about what occurs if the antecedent is true. The consequent is what is said to be true if the antecedent is correct; in the examples above the consequent is the proposition “the grass will be wet” or “he would still be alive” respectively.

Conditionals are hypothetical statements; they affirm what would be the case,if something else were the case. They can be true even if the antecedent is false. Consider the claim that “if I had died yesterday, I would not be writing this blog post today.” This conditional is true, even though I did not die yesterday. So it is a mistake, albeit a common one, to reject a conditional because one thinks the antecedent is false. The issue is not whether it is false, it is what would be the case if it were true?

This week I want to look at another fallacy that occurs with conditionals. Whereas last time I looked at the fallacy of denying the antecedent, this week I want to look at the fallacy of affirming the consequent. To understand this we need to recap two valid argument forms I mentioned last time: modus ponensand modus tollens.

Modus Ponens

Modus Ponens has the following form:

1. If P then Q.

2. P;

Therefore:

3. Q.

To use our now belaboured example”

1. If it is raining then the grass will be wet.

2. It is raining;

Therefore:

3. The grass will be wet.

In a modus ponens one first asserts that a conditional statement is true and then affirms the antecedent of the conditional and concludes that the consequent true. As I noted, this is a valid argument. If the conditional “if P then Q” is true, and P is true, then Q must also be true.

Modus Tollens

Modus Tollens  has the form:

1. If P then Q.

2’ Not Q;

Therefore:

3′ Not P.

To take our example:

1. If it is raining then the grass will be wet.

2’ The grass is not wet;

Therefore:

3’ It is not raining.

In modus tollens one first asserts that a conditional statement is true and thendenies the consequent of the conditional and concludes that the antecedent is false. As I noted, this is also a valid argument. If the conditional “if P then Q” is true, and Q is false, then P must also be false.

Affirming the Consequent

This recap is necessary to focus on the fallacy I am going to discuss today. Asyou have probably guessed, the fallacy of affirming the consequent occurs when one affirms the consequent. More specifically it occurs when one first asserts that a conditional statement is true, affirms the consequent of the conditional and concludes  from this that the antecedent is true. It therefore has the form

1. If P then Q.

2’’ Q;

Therefore:

3’’ P.

Or using our example:

1. If it is raining then the grass will be wet.

2’’ The grass is wet;

Therefore:

3’’ It is raining.

On the face of it one  might think this argument is unobjectionable. However a little reflection shows that it is invalid. This is because it is possible for the premises to be true and the conclusion false. Consider the following example, it is a really hot summer day in Auckland, the children are sweltering so I turn on the sprinkler system and let them play outside on the grass.

In this situation, the conditional statement 1., is true. It is true that if it were raining, then the grass would be wet. Premise 2’’’ is also true; given I have the Sprinkler system on, the grass is wet. But 3’’’ is false; like I said, it is a hot summer day in Auckland and it is not raining.  Examples like this show that it is possible for the premises to be true and the conclusion false therefore this argument is invalid.

This example is also instructive as to what is wrong with this kind of reasoning. The conditional “if it is raining then the grass will be wet” tells us a relationship between two things ‘rain’ and ‘wet grass’. In this particular instance the relationship is a causal one, rain pouring from the sky causes grass to be wet. But note what the conditional does not say. It does not say that this is the only thing that causes wet grass. It is perfectly possible, and in fact actual, that several different things can cause grass to get wet: rain, sprinklers, hoses, buckets of water and so on. Therefore, the  existence of wet grass, by itself, does not provide grounds for thinking it has rained.

Arguments that do not Commit the Fallacy

Like with other fallacies, it is important to distinguish affirming the consequent with other arguments that appear similar to it but are not fallacious. I will look at two: bi-conditionals and certain types of probabilistic reasoning.

Bi-Conditionals
In the example I used above, rain causes grass to be wet but there are also other things such as sprinklers and hoses that can also cause the grass to be wet. The grass will be wet if it rains but it is not true that it will only be wet if it rains. This brings us to an example of an argument that looks like  the fallacy of affirming the consequent but which is not. These are  arguments that use bi-conditionals.

Consider the following argument:

1. X is a three sided figure if, and only if,  it is a triangle.

2. X is a triangle.

3. Therefore X is a three sided figure.

This argument is valid. The reason is that 1. is what is called a bi-conditional statement.

I noted above that conditional statements have the form:

“If P then Q”

Bi-conditional statements have the form:

“P, if, and only if, Q”

A bi-conditional statement differs from a normal conditional statement in that the “if” … “then” relationship goes both ways. It is true that if P is true then Q is, but it is also true that if Q is true then P is true. The language of if and only ifreflects this. The consequent will be true if the antecedent is and the consequent will only be true if the antecedent is. There is no other antecedent that is such that, if it were true, the same consequent would result.

When one is dealing with bi-conditional statements one can validly infer the truth of the consequent from the antecedent. This is because with bi-conditional statements the consequent can only be true if the antecedent is and so it is impossible for the consequent to be true and the antecedent false.

Probablistic Reasoning
Something like affirming the consequent also occurs in probabilistic reasoning.  In probabilistic reasoning people considering two hypotheses, H1 and H2, might compare them by considering the predictions they make and seeing if the predictions come true. If  the predictions of H1 come true and the predictions of H2 do not, this is taken to raise the probability of H1 relative to H2. An example might help illustrate this, suppose we are faced with two hypotheses, that it has just rained and also that a sand storm has just engulfed the back yard. We look outside and discover that the  green grass is soaking wet. We conclude, therefore, that it is more likely to have just rained than that a sand storm has just occurred.

In this kind of argument one argues in favour of the antecedent of a conditional by affirming the consequent of the conditional. We note that if it had rained then the grass would be wet and we take the wet grass as evidence in favour of it raining. But note, that in these instances the conclusion is that the antecedent is more probable given the truth of the consequent and it is more more probable than another antecedent which does not have the consequent in question. We think this wet grass makes the claim that it has just rained more probable than the claim that a sand storm has just occured. We do not reason that the antecedent is true solely because the consequent is.