Affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true

conditional Conditional (if then) may refer to: * Causal conditional, if X then Y, where X is a cause of Y * Conditional probability, the probability of an event A given that another event B has occurred *Conditional proof, in logic: a proof that asserts a ...

statement (e.g., "If the lamp were broken, then the room would be dark"), and invalidly inferring its converse ("The room is dark, so the lamp is broken"), even though that statement may not be true. This arises when a consequent ("the room would be dark") has other possible antecedents (for example, "the lamp is in working order, but is switched off" or "there is no lamp in the room"). Converse errors are common in everyday thinking and communication and can result from, among other causes, communication issues, misconceptions about logic, and failure to consider other causes. The opposite statement, denying the consequent, ''is'' a valid form of argument ( modus tollens).

Formal description

Affirming the consequent is the action of taking a true statement

P \to Q

and invalidly concluding its converse

Q \to P

. The name ''affirming the

consequent A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if ''P'' implies ''Q'', then ''P'' is called the antecedent and ''Q'' is called t ...

'' derives from using the consequent, ''Q'', of

P \to Q

, to conclude the antecedent ''P''. This fallacy can be summarized formally as

(P \to Q, Q)\to P

or, alternatively,

\frac

. The root cause of such a logical error is sometimes failure to realize that just because ''P'' is a ''possible'' condition for ''Q'', ''P'' may not be the ''only'' condition for ''Q'', i.e. ''Q'' may follow from another condition as well. Affirming the consequent can also result from overgeneralizing the experience of many statements ''having'' true converses. If ''P'' and ''Q'' are "equivalent" statements, i.e.

P \leftrightarrow Q

, it ''is'' possible to infer ''P'' under the condition ''Q''. For example, the statements "It is August 13, so it is my birthday"

P \to Q

and "It is my birthday, so it is August 13"

Q \to P

are equivalent and both true consequences of the statement "August 13 is my birthday" (an abbreviated form of

P \leftrightarrow Q

). Of the possible forms of "mixed hypothetical syllogisms," two are valid and two are invalid. Affirming the antecedent ( modus ponens) and denying the consequent ( modus tollens) are valid. Affirming the consequent and