The destructive dilemma rule may be written in sequent notation:

${\displaystyle (P\to Q),(R\to S),(\neg Q\lor \neg S)\vdash (\neg P\lor \neg R)}$ 

where ${\displaystyle \vdash }$  is a metalogical symbol meaning that ${\displaystyle \neg P\lor \neg R}$  is a syntactic consequence of ${\displaystyle P\to Q}$ , ${\displaystyle R\to S}$ , and ${\displaystyle \neg Q\lor \neg S}$  in some logical system;

and expressed as a truth-functional tautology or theorem of propositional logic:

${\displaystyle (((P\to Q)\land (R\to S))\land (\neg Q\lor \neg S))\to (\neg P\lor \neg R)}$ 

where ${\displaystyle P}$ , ${\displaystyle Q}$ , ${\displaystyle R}$  and ${\displaystyle S}$  are propositions expressed in some formal system.

If it rains, we will stay inside.

If it is sunny, we will go for a walk.

Either we will not stay inside, or we will not go for a walk, or both.

Therefore, either it will not rain, or it will not be sunny, or both.

The validity of this argument structure can be shown by using both conditional proof (CP) and reductio ad absurdum (RAA) in the following way:

• Howard-Snyder, Frances; Howard-Snyder, Daniel; Wasserman, Ryan. The Power of Logic (4th ed.). McGraw-Hill, 2009, ISBN 978-0-07-340737-1, p. 414.

• http://mathworld.wolfram.com/DestructiveDilemma.html