Affirming the consequent

Formal logical fallacy, in which it is (falsely) assumed that a logical consequence can be the premise of a converse proposition.

Example:

If A lives in London, then A lives in the United Kingdom.
A lives in the United Kingdom.
Therefore, A lives in London.

Even if the major premise is true (and we ignore other places called “London” in other countries), this does not mean that its converse (“if A lives in the United Kingdom, then A lives in London”) is also true. In fact, there are many other places where A could live in the UK.

Name

Other names

• Commutation of conditionals
• Converse error
• Fallacy of the consequent
• [Illicit] inductive conversion

Origin of the name

In a subjunction, i.e. a logical statement of the form “if A, then B” (`A → B`), we refer to A as antecedent (or condition) and B as consequent (or consequence).

Different to the (valid) form modus ponens, which uses an affirmative antecedent as the second premise, this fallacy is based on an affirmative consequence in this position, which leads to an invalid conclusion.

Explanation

This fallacy arises from incorrect application of the modus ponens, possibly in conjunction with incorrect mixing with the modus tollens.

For comparison, the following table contrasts the two valid forms of inference with the fallacy:

When are such conclusions valid?

There are certain circumstances in which a valid conclusion can be derived by affirming the consequence:

1. Tautological statements

If antecedent and consequent are identical (e.g. “If it rains, it rains”), obviously it is always possible to invert the statement. This is equivalent to a modus ponens.

This also applies to statements in which the tautology exists only indirectly, for example through the definitions of the terms used. Good examples for this situation can be found in mathematics - e.g. “if a number is divisible by 2, it is even” and “if a number is even, it is divisible by 2” are both true, since “even” and “divisible by 2” can be used synonymously by definition.

Such tautologies can also hide in relatively complex definition structures, as the following example shows:

If today is Monday, then the day after tomorrow is Wednesday.
The day after tomorrow is Wednesday,
Therefore: Today is Monday.

The term “Wednesday” can be defined as “the day after Tuesday”, and Tuesday in turn as “the day after Monday”. Thus, by definition, Wednesday is “two days after Monday”. Therefore, the statement in the first line is tautological and what appears like an affirmation of the consequence is actually a valid conclusion.

2. Empty complementary set

Even outside of true tautologies, A and B can describe (extend to) identical groups. In set theory, this can be described as follows: `𝔸 ∖ 𝔹 = ∅` (the difference set of A with B is empty).

Since in this case A and B denote identical sets, one can also conclude from “if A then B” that “if B then A”.

Examples

The examples normally used to explain this fallacy are either very abstract or seem far-fetched. This gives the impression that this fallacy is purely an academic matter that has nothing to do with real life.

On the contrary, though, affirmation of the consequence is one of the fallacies that is actually very commonly observed in daily life, often taking the shape of one of the more general errors of reasoning described in the other sections of this site.

Price comparison

The following false reasoning was probably committed by everyone (unconsciously) at least sometimes when shopping:

To get a cheaper price [per unit] for a product, I have to buy a larger quantity of the product.
There is a larger pack of a product that I want to buy.
Therefore, the bigger pack is cheaper per unit than the smaller one.

In fact, retailers routinely use such misconceptions to entice customers to buy larger quantities of a product, without actually giving reductions

For example (the author has actually seen this in a supermarket!):

A regular pack of cat food with 12 portions of 100g each for € 10.19.
A large pack of cat food with 24 portions of 100g for € 26.09.