====== Destructive dilemma (logic) ======

A valid form of logical inference in propositional logic, which infers from two [[glossary:conditional|conditional]] and a //negative// [[glossary:disjunction|disjunct]] statement a new negative disjunct statement.

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Formally, the destructive dilemma has three premises, it looks as follows:

> Premise 1:   <span "if A, then B">''A ⟶ B''</span>   –   <q>if <var>A</var>, then <var>B</var></q>
> Premise 2:   <span "if C, then D">''C ⟶ D''</span>   –   <q>if <var>C</var>, then <var>D</var></q>
> Premise 3:   <span "not B or not D">''⌐B ∨ ⌐D''</span>   –   <q>not <var>B</var> //or// not <var>D</var> [//or neither//]</q>
> <span conclusio>Conclusion:   <span "not A or not C">''⌐A ∨ ⌐C''</span>   –   <q>not <var>A</var> //or// not <var>C</var> [//or neither//]</q></span>

A practical example could be the following:

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> //If// the sun shines tomorrow, [//then//] we will go to the beach.
> //If// it rains tomorrow, [//then//] we will go to the museum.
> Tomorrow we will //either// not got to the museum //or// not go to the beach [//or neither//].
> <span conclusio>Therefore it will //either// not rain //or// the sun will not shine [//or neither//].</span>
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===== Description =====

The //destructive dilemma// can be seen as a combination of two [[logic:inferences:modus_tollens|Modus Tollens]], which are connected by a //disjunct// statement. 

The term “dilemma” in this context should be understood as a “decision” between two conditionals.

The relationships between the various statements in a constructive dilemma can best be explained by showing them as a diagram:

<aside box><figure>{{:logic:inferences:destructive_dilemma.svg|Destructive dilemma}}<figcaption>
In this diagram, the two [[glossary:conditional|conditional statements]] are displayed as [[glossary:universal_quantification|universal quantifications]] to make it easier to understand how they are connected. This is possible because any (material) //conditional// can be transformed into a //universal quantification// (from “''when A, then B''” follows that “''all A are B''”).

It shows clearly that if the statement “''not-B //or// not-D''” holds //true//, then also “''not-A //or// not-C''” must be //true//.
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**Note:** All examples here (and indeed in most other places elsewhere) are explicitly based on [[glossary:adjunction|adjunctions]], i.e. they use an //inclusive// ‘or’. In fact, it is indeed also possible to construct a //destructive dilemma// based on a [[glossary:contravalence|contravalence]] instead. However, this can lead to certain complications, not least to the possibility of creating a “[[relevancy:false_dilemma|false dilemma]]”, which consist in postulating a //dichotomy// where none really exists.
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===== See also =====

  * [[logic:inferences:constructive_dilemma|Constructive Dilemma]]
  * [[logic:inferences:modus_tollens|Modus Tollens]]

===== More information =====

  * [[wp>Destructive dilemma]] on //Wikipedia//

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