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Rule of inference of propositional logic
Destructive dilemmaType |
Rule of inference |
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Field |
Propositional calculus |
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Statement | If implies and implies and either is false or is false, then either or must be false. |
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Symbolic statement | |
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Destructive dilemma
[1]
[2] 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 destructive dilemma 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 truth-functional
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
Step
|
Proposition
|
Derivation
|
1 |
|
Given
|
2 |
|
Given
|
3 |
|
Given
|
4 |
|
Transposition (1)
|
5 |
|
Transposition (2)
|
6 |
|
Conjunction introduction (4,5)
|
7 |
|
Constructive dilemma (6,3)
|
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:
De Morgan'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. |
|
(6-16: RAA)
|
|
18. |
|
(1-17: CP)
|
References
-
^ Hurley, Patrick. A Concise Introduction to Logic With Ilrn Printed Access Card. Wadsworth Pub Co, 2008. Page 361
-
^ Moore and Parker
Bibliography
- 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.
External links