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 destructive dilemma is any logical argument of the following form:

where represents the logical assertion

Logical assertion

A logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. It is equivalent to a sequent with an empty antecedent....

.

The argument can be read in this way:

1. If P, then Q
2. If R, then S
3. Not Q or not S
4. Therefore, not P or not R

And to once again restate the argument, one can turn this argument into a conditional, where if the first three premises, then not P or R:

The destructive dilemma is the disjunctive version of modus tollens

Modus tollens

In classical logic, modus tollens has the following argument form:- Formal notation :...

. The disjunctive version of modus ponens

Modus ponens

In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...

is the constructive dilemma

Constructive dilemma

In logic, a constructive dilemma is a formal logical argument that takes the form:Therefore, either Q or S is true.In logical operator notation with three premises P \rightarrow Q R \rightarrow S P \lor R \therefore Q \lor S ....

. Here is an example of the destructive dilemma in English:

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.

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

Example proof

The validity of this argument structure can be shown by using both conditional proof

Conditional proof

A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent....

 (CP) and reductio ad absurdum

Reductio ad absurdum

In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

 (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 Double negation In the theory of logic, double negation is expressed by saying that a proposition A is identical to not , or by the formula A = ~~A. Like the Law of Excluded Middle, this principle when extended to an infinite collection of individuals is disallowed by Intuitionistic logic... )
11.		(9: double negation)
12.		(3,10: modus ponens Modus ponens In classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments... )
13.		(4,11: modus ponens)
14.		(12: double negation)
15.		(5, 14: disjunctive syllogism Disjunctive syllogism A disjunctive syllogism, also known as disjunction-elimination and or-elimination , and historically known as modus tollendo ponens,, is a classically valid, simple argument form:where \vdash represents the logical assertion.... )
16.		(13,15: conjunction Conjunction Conjunction can refer to:* Conjunction , an astronomical phenomenon* Astrological aspect, an aspect in horoscopic astrology* Conjunction , a part of speech** Conjunctive mood , same as subjunctive mood... )
17.		(6-16: RAA)
18.		(1-17: CP)

External links

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