Modus ponendo tollens

Modus ponendo tollens (MPT; Latin: "mode that denies by affirming") is a valid rule of inference for propositional logic. It is closely related to modus ponens and modus tollendo ponens.

Overview
MPT is usually described as having the form: For example:
 * 1) Not both A and B
 * A
 * 1) Therefore, not B
 * 1) Ann and Bill cannot both win the race.
 * 2) Ann won the race.
 * 3) Therefore, Bill cannot have won the race.

As E. J. Lemmon describes it: "Modus ponendo tollens is the principle that, if the negation of a conjunction holds and also one of its conjuncts, then the negation of its other conjunct holds."

In logic notation this can be represented as:
 * 1) $$ \neg (A \land B)$$
 * 2) $$ A$$
 * 3) $$ \therefore \neg B$$

Based on the Sheffer Stroke (alternative denial), "|", the inference can also be formalized in this way:
 * 1) $$ A\,|\,B$$
 * 2) $$ A$$
 * 3) $$ \therefore \neg B$$

Strong form
Modus ponendo tollens can be made stronger by using exclusive disjunction instead of non-conjunction as a premise:
 * 1) $$ A \underline\lor B$$
 * 2) $$ A$$
 * 3) $$ \therefore \neg B$$