Affirming the Consequent
Brett Gaul
If Sophia is in the Twin Cities, then she is in Minnesota. Sophia is in Minnesota. Therefore, she is in the Twin Cities.
John Doe
Affirming the consequent is a fallacious form of reasoning in formal logic that occurs when the minor premise of a propositional syllogism (an argument consisting of a general statement known as the major premise, a specific statement known as the minor premise, and a conclusion) affirms the consequent of a conditional statement.
A conditional statement is an “if-then” sentence that expresses a link between the antecedent (the part after the “if”) and the consequent (the part after the “then”). A conditional statement does not assert either the antecedent or the consequent. It simply claims that if the antecedent is true, then the consequent is also true. In the example, “Sophia is in the Twin Cities” is the antecedent and “she is in Minnesota” is the consequent. Affirming the consequent makes the mistake of assuming that the converse of an “if-then” statement is true. In other words, if “If p, then q” is true, then the converse, “If q, then p,” must also be true. However, the converse of an “if-then” statement isn’t necessarily true. Although affirming the consequent is an invalid argument form, it is similar to, and sometimes mistaken for, the valid argument form modus ponens(the mode of putting). While the valid argument form modus ponens asserts or affirms the antecedent of a conditional statement, the invalid argument form affirming the consequent asserts or affirms the consequent of a conditional statement.
| Modus ponens (valid) | Affirming the consequent (invalid) |
| If p, then q. | If p, then q. |
| p. | q. |
| Therefore, q. | Therefore, p. |
Modus ponens is a valid argument form because the truth of the premises guarantees the truth of the conclusion; however, affirming the consequent is an invalid argument form because the truth of the premises does not guarantee the truth of the conclusion.
Put another way, if an argument is in the form of modus ponens, the structure of the argument makes it impossible for the argument’s premises to be true and the conclusion to be false. Affirming the consequent is an invalid argument form, though, because the structure of that argument allows the premises to be true and the conclusion to be false.To see how modus ponens is valid, assume that p =“Sophia is in the Twin Cities” and that q = “she is in Minnesota.”
(1) If Sophia is in the Twin Cities, then she is in Minnesota. (If p, then q)
(2) Sophia is in the Twin Cities. (p)
(3) Therefore, Sophia is in Minnesota. (q)
Given the form of modus ponens, if Sophia really is in the Twin Cities, then it is impossible for her not to be in Minnesota, because if she is in the Twin Cities, then she is in Minnesota. In other words, being in the Twin Cities is a sufficient condition for being in Minnesota. If something is a sufficient condition, it guarantees something else. In this case, being in the Twin Cities is a sufficient condition for being in Minnesota, because being in the Twin Cities guarantees that one is in Minnesota. To deny that Sophia is in Minnesota if she is in the Twin Cities is to make a mistake in reasoning. The truth of the premises guarantees the truth of the conclusion. However, this is not the case in affirming the consequent.
(1) If Sophia is in the Twin Cities, then she is in Minnesota. (If p, then q)
(2) Sophia is in Minnesota. (q)
(3) Therefore, Sophia is in the Twin Cities. (p)
44 Brett Gaul
This argument form of affirming the consequent is invalid because if Sophia is in Minnesota, it is possible that she is somewhere other than the Twin Cities. Just being in Minnesota does not guarantee that one is in the Twin Cities. As Elliot D. Cohen puts it in Critical Thinking Unleashed (2009), the fallacy of affirming the consequent “confuses a necessary condition with a sufficient one” (40). The conditional statement in the example says that being in the Twin Cities is a sufficient condition for being in Minnesota and that being in Minnesota is a necessary condition, that is, one that is needed or required, for being in the Twin Cities.
However, it does not say that being in Minnesota guarantees that one is in the Twin Cities.The validity of modus ponens and the invalidity of affirming the consequent are confirmed by truth tables. In propositional logic, truth tables are used to represent the relation between any statement and its denial. For example, if p is true, then not p must be false. If p is false, not p must be true.
| p | Not p |
| True | False |
| False | True |
Truth tables are also used to represent the relationship between conjunctions (and statements), disjunctions (or statements), and conditionals (if- then statements). The truth table for conditionals looks like this:
| p | q | If p, then q |
| True | True | True |
| True | False | False |
| False | True | True |
| False | False | True |
The first line of the truth table indicates that if p is true and q is true, the conditional “If p, then q” is true. If p is true and q is false, then the conditional is false. If p is false and q is true, the conditional is true. Finally, if both p and q are false, the conditional is true.
For an argument to be valid, it must be impossible for the premises to be true and the conclusion to be false. Modus ponens is valid because there is no place on the truth table where all of the premises (If p, then q, and p) are true and the conclusion (q) is false. On the one line where both premises are true (the first line), the conclusion is also true.
Modus Ponens
| Premise | Premise | Conclusion |
| P | If p, then q | q |
| True | True | True |
| True | False | False |
| False | True | True |
| False | True | False |
Affirming the consequent is an invalid argument form, however, because the third row of the truth table below shows all true premises (If p, then q, and q) and a false conclusion (p).
The truth of the premises does not guarantee the truth of the conclusion.Affirming the Consequent
| Premise | Premise | Conclusion |
| q | If p, then q | p |
| True | True | True |
| False | False | True |
| True | True | False |
| False | True | False |
To return to the original example, given the conditional “If Sophia is in the Twin Cities, then she is in Minnesota,” if she is in Minnesota, it’s possible she is somewhere other than the Twin Cities. Simply being in Minnesota does not guarantee being in the Twin Cities. The fallacy of affirming the consequent is committed when one concludes that she is in the Twin Cities if she is in Minnesota. Avoid committing this fallacy by coming up with examples (p’s and q’s) to test the form of your argument. Is it true that the conclusion is true while the evidence is false? Is there another possible explanation? If Sophia is in Minnesota, could she be somewhere other than the Twin Cities? If you find that the argument does commit the fallacy, try to rework the structure of the argument to use modus ponens: If Sophia is in the Twin Cities, then she is in Minnesota.
Reference
Cohen, Elliot D. 2009. Critical Thinking Unleashed. Lantham, MD: Rowman & Littlefield.