// laws and rules · Fallacy
Fallacy of affirming the consequent_
Affirming the consequent is the most common formal fallacy because it looks so much like modus ponens. The difference is crucial: here the conclusion of the implication is affirmed in order to deduce its condition, and that is invalid. The truth table proves it: the formula is a contingency, not a tautology.
Example
((p ⇒ q) ∧ q) ⇒ p
What the variables mean
- ▸ p: “It is raining”
- ▸ q: “The street is wet”
In plain words
“If it rains, the street gets wet. The street is wet. Therefore it rained.” False: the street-cleaning truck may have passed.
Truth table
| p | q | p ⇒ q | (p ⇒ q) ∧ q | ((p ⇒ q) ∧ q) ⇒ p★ |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | F | T |
| F | T | T | T | F |
| F | F | T | F | T |
Classification: Contingency · 4 rows
Statement of the fallacy
Premise 1: p ⇒ q. Premise 2: q. Conclusion (invalid): p. As a single formula the argument reads ((p ⇒ q) ∧ q) ⇒ p.
An argument is valid only if its formula is a tautology. On evaluation the calculator classifies this one as a contingency, which is enough to declare the reasoning invalid.
The row that refutes it
The counterexample sits in the row p = F, q = T. There p ⇒ q is T (a false antecedent makes the implication true) and q is T, so the conjunction of premises is T. Yet the conclusion p is F, and the final implication comes out F.
That single row invalidates the argument: there is a situation where both premises are true and the conclusion false. In the other three rows the formula gives T, which is why the overall result is a contingency rather than a contradiction.
Why it is confused with modus ponens
Modus ponens affirms the antecedent: from p ⇒ q and p it concludes q, and it is valid. The fallacy affirms the consequent: from p ⇒ q and q it tries to conclude p. The visual structure is almost identical, which is what makes it so persuasive.
The root of the error is treating the conditional as if it were a biconditional. Had the premise been p ⇔ q, concluding p from q would indeed be legitimate: in that case the argument becomes valid.
Real-world examples
Medicine: “If you have the flu, you will have a fever. You have a fever. So you have the flu.” Fever has many possible causes; the reasoning mistakes a symptom for a diagnosis.
Software debugging: “If there is a memory leak, the process grows. The process is growing. So there is a memory leak.” It could be a legitimate cache. This fallacy is a frequent source of wrong diagnoses.
A related legitimate use: in science, confirming a prediction (q) does not prove the hypothesis (p), it only supports it. That is why we speak of corroboration rather than proof.
How to avoid it
Always ask whether some other cause could produce q. If one exists, p cannot be deduced from q.
To reason backwards validly use modus tollens: from p ⇒ q and ¬q conclude ¬p. Denying the consequent is valid; affirming it is not.
Try it yourself
Edit the expression in the calculator and watch how every step of the table changes.
Open in the calculator →Related operators
Frequently asked questions
Why does the calculator say contingency rather than contradiction? ▼
Because the formula is false in only one of the four rows (p = F, q = T) and true in the other three. A contradiction would be false in all of them.
When can I legitimately conclude p from q? ▼
When the premise is a biconditional p ⇔ q rather than a conditional. A biconditional does license reasoning in both directions.
How does it differ from modus ponens? ▼
Modus ponens affirms the antecedent (from p ⇒ q and p conclude q) and is valid. This fallacy affirms the consequent and improperly reverses the implication.