Constructive dilemma

Rules of inference
Propositional calculus
Modus ponens (A→B, A ⊢ B) Modus tollens (A→B, ¬B ⊢ ¬A) Modus ponendo tollens (¬(A∧B), A ⊢ ¬B) Conjunction introduction (A, B ⊢ A∧B) Simplification (A∧B ⊢ A) Disjunction introduction (A ⊢ A∨B) Disjunction elimination (A∨B, A→C, B→C ⊢ C) Disjunctive syllogism (A∨B, ¬A ⊢ B) Hypothetical syllogism (A→B, B→C ⊢ A→C) Constructive dilemma (A→P, B→Q, A∨B ⊢ P∨Q) Destructive dilemma (A→P, B→Q, ¬P∨¬Q ⊢ ¬A∨¬B) Biconditional introduction (A→B, B→A ⊢ A↔B) Biconditional elimination (A↔B ⊢ A→B)
Predicate calculus
Universal generalization Universal instantiation Existential generalization Existential instantiation
v · d · e

In logic, a constructive dilemma is a formal logical argument that takes the form:

1a) P → Q.

b) R → S.

2) Either P or R is true.

Therefore, either Q or S is true.

In logical operator notation with three premises

$P \rightarrow Q$

$R \rightarrow S$

$P \lor R$

$\therefore Q \lor S$.

In logical operator notation with two premises^[1]

$( P \rightarrow Q ) \land ( R \rightarrow S )$

$P \lor R$

$\therefore Q \lor S$.

In sum, if two conditionals are true and at least one of their antecedents is, then at least one of their consequents must be too.

An example:

If I win a million dollars, I will donate it to an orphanage.

If my friend wins a million dollars, he will donate it to a wildlife fund.

Either I win a million dollars, or my friend wins a million dollars.

Therefore, either an orphanage will get a million dollars, or a wildlife fund will get a million dollars.

The dilemma derives its name because of the transfer of disjunctive operants.

Proof

1.	$(P \rightarrow Q) \and (R \rightarrow S) \and (P \vee R))$
2.	$\Leftrightarrow (\neg P \vee Q) \and (\neg R \vee S) \and (P \vee R)$
3.	$\Rightarrow (\neg P \vee (Q \vee S)) \and (\neg R \vee (Q \vee S)) \and (P \vee R)$	(addition)
4.	$\Rightarrow (\neg P \vee (Q \vee S)) \and (\neg R \vee (Q \vee S))$	(simplification)
5.	$\Leftrightarrow (Q \vee S) \vee (\neg P \and \neg R)$	(distribution)
6.	$\Leftrightarrow (Q \vee S) \vee \neg(P \vee R)$	(DeMorgan's Law)
7.	$\Leftrightarrow (Q \vee S) \vee False$	(from assumption)
7.	$\Leftrightarrow Q \vee S$

References

1. ^ Hurley, Patrick. A Concise Introduction to Logic With Ilrn Printed Access Card. Wadsworth Pub Co, 2008. Page 361

This logic-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Rules of inference Arguments Logic stubs Wikimedia Foundation. 2010. Игры ⚽ Поможем решить контрольную работу Anhalt-Dessau Cheongju Polytechnic College Look at other dictionaries: dilemma — The simplest form of a dilemma is an argument of the form: ‘If p then q, if not p then q, so in any event q .’ More complex forms were traditionally distinguished. A constructive dilemma is of the form: ‘If p then r, if q then r, but either p or… …   Philosophy dictionary Dilemma — For other uses, see Dilemma (disambiguation). Between a rock and a hard place redirects here. For other uses, see Between a Rock and a Hard Place (disambiguation). A dilemma (Greek: δί λημμα double proposition ) is a problem offering at least two …   Wikipedia Destructive dilemma — Rules of inference Propositional calculus Modus ponens (A→B, A ⊢ B) Modus tollens (A→B, ¬B ⊢ ¬A) …   Wikipedia Propositional calculus — In mathematical logic, a propositional calculus or logic (also called sentential calculus or sentential logic) is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules… …   Wikipedia Outline of logic — The following outline is provided as an overview of and topical guide to logic: Logic – formal science of using reason, considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and… …   Wikipedia List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia Modus ponens — Rules of inference Propositional calculus Modus ponens (A→B, A ⊢ B) Modus tollens (A→B, ¬B ⊢ ¬A) …   Wikipedia Conjunction introduction — Rules of inference Propositional calculus Modus ponens (A→B, A ⊢ B) Modus tollens (A→B, ¬B ⊢ ¬A) …   Wikipedia Disjunctive syllogism — Rules of inference Propositional calculus Modus ponens (A→B, A ⊢ B) Modus tollens (A→B, ¬B ⊢ ¬A) …   Wikipedia Disjunction introduction — Rules of inference Propositional calculus Modus ponens (A→B, A ⊢ B) Modus tollens (A→B, ¬B ⊢ ¬A) …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”