This is the first derived statement in the Principia Mathematica, which proves it by taking the disjunction of the modus ponens and modus tollens to yield "p implies q equals not p or q," then considering what happens if q is allowed to be p.
Using this rule allows one to add additional terms to the left-hand sides of hypotheses in the deduction theorem; in that case, the form of this principle used is known as Pierce's law. The so-called consequentia mirabilis relies only on this result. This is the first derived statement in the Principia Mathematica, which proves it by taking the disjunction of the modus ponens and modus tollens to yield "p implies q equals not p or q," then considering what happens if q is allowed to be p. Polyvalent logics are ones lacking this rule, and N-valued logics have versions of this rule applying to N minus one disjunctions of mutual exclusions. Sometimes confused with the principle of bivalence, this law was first stated in Aristotle's On Interpretation as the principle of non-contradiction. For 10 points, name this law of classical logic according to which "p or not p" is true for any p, since no other result is possible.
Using this rule allows one to add additional terms to the left-hand sides of hypotheses in the deduction theorem; in that case, the form of this principle used is known as Pierce's law.
law of the excluded middle [or law of the excluded third; accept Pierce's law before read; accept principle of non-contradiction before read]