Constructive dilemma is a name of a valid rule of inference of propositional logic. It is the inference that, if P implies Q and R implies S and either P or R is true, then Q or S has to be true.

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

${\displaystyle P\rightarrow Q}$

${\displaystyle R\rightarrow S}$

${\displaystyle P\lor R}$

${\displaystyle \therefore Q\lor S}$.

In logical operator notation with two premises^[1]

${\displaystyle (P\rightarrow Q)\land (R\rightarrow S)}$

${\displaystyle P\lor R}$

${\displaystyle \therefore Q\lor S}$.

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.

1.	${\displaystyle (P\rightarrow Q)\land (R\rightarrow S)\land (P\vee R))}$
2.	${\displaystyle \Leftrightarrow (\neg P\vee Q)\land (\neg R\vee S)\land (P\vee R)}$
3.	${\displaystyle \Rightarrow (\neg P\vee (Q\vee S))\land (\neg R\vee (Q\vee S))\land (P\vee R)}$	(addition)
4.	${\displaystyle \Rightarrow (\neg P\vee (Q\vee S))\land (\neg R\vee (Q\vee S))}$	(simplification)
5.	${\displaystyle \Leftrightarrow (Q\vee S)\vee (\neg P\land \neg R)}$	(distribution)
6.	${\displaystyle \Leftrightarrow (Q\vee S)\vee \neg (P\vee R)}$	(DeMorgan's Law)
7.	${\displaystyle \Leftrightarrow (Q\vee S)\vee False}$	(from assumption)
7.	${\displaystyle \Leftrightarrow Q\vee S}$