Destructive dilemma
Transformation rules 

Propositional calculus 
Rules of inference 
Rules of replacement 

Predicate logic 
Destructive dilemma^{}^{} is the 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 Q is false or S is false, then either P or R must be false. In sum, if two conditionals are true, but one of their consequents is false, then one of their antecedents has to be false. Destructive dilemma is the disjunctive version of modus tollens. The disjunctive version of modus ponens is the constructive dilemma. The rule can be stated:
where the rule is that wherever instances of "", "", and "" appear on lines of a proof, "" can be placed on a subsequent line.
Formal notation
The destructive dilemma rule may be written in sequent notation:
where is a metalogical symbol meaning that is a syntactic consequence of , , and in some logical system;
and expressed as a truthfunctional tautology or theorem of propositional logic:
where , , and are propositions expressed in some formal system.
Natural language example
 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.
Proof
Proposition  Derivation 

Given  
Given  
Material implication  
Transposition  
Hypothetical syllogism  
Simplification  
Hypothetical syllogism  
Material implication 
Example proof
The validity of this argument structure can be shown by using both conditional proof (CP) and reductio ad absurdum (RAA) in the following way:
1.  (CP assumption)  
2.  (1: Simplification)  
3.  (2: simplification)  
4.  (2: simplification)  
5.  (1: simplification)  
6.  (RAA assumption)  
7.  (6: DeMorgan's Law)  
8.  (7: simplification)  
9.  (7: simplification)  
10.  (8: double negation)  
11.  (9: double negation)  
12.  (3,10: modus ponens)  
13.  (4,11: modus ponens)  
14.  (12: double negation)  
15.  (5, 14: disjunctive syllogism)  
16.  (13,15: conjunction)  
17.  (616: RAA)  
18.  (117: CP) 
References
Bibliography
 HowardSnyder, Frances; HowardSnyder, Daniel; Wasserman, Ryan. The Power of Logic (4th ed.). McGrawHill, 2009, ISBN 9780073407371, p. 414.
External links
 http://mathworld.wolfram.com/DestructiveDilemma.html