In logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

, a constructive dilemma is a formal logical argument that takes the form:

1a) P → Q.

b) R → S.

2) Either P or R is true.

Therefore, either Q or S is true.

In logical operator notation with three premises.

In logical operator notation with two premises
.

In sum, if two conditionals are true and at least one of their antecedents is, then at least one of their consequents must be too.

An example:

If I win a million dollars, I will donate it to an orphanage.

If my friend wins a million dollars, he will donate it to a wildlife fund.

Either I win a million dollars, or my friend wins a million dollars.

Therefore, either an orphanage will get a million dollars, or a wildlife fund will get a million dollars.

The dilemma derives its name because of the transfer of disjunctive operants.

Proof

1.
2.
3.		(addition)
4.		(simplification)
5.		(distribution)
6.		(DeMorgan's Law)
7.		(from assumption)
7.