Destructive dilemmaMoore and Parker is the name of a valid

rule of inference Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the Logical form, logical structure of Validity (logic), valid arguments. If an argument with true premises follows a ...

propositional logic The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...

. It is the

inference Inferences are steps in logical reasoning, moving from premises to logical consequences; etymologically, the word '' infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinct ...

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 Conditional (if then) may refer to: *Causal conditional, if X then Y, where X is a cause of Y *Conditional probability, the probability of an event A given that another event B *Conditional proof, in logic: a proof that asserts a conditional, a ...

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 In propositional logic, ''modus tollens'' () (MT), also known as ''modus tollendo tollens'' (Latin for "mode that by denying denies") and denying the consequent, is a deductive argument form and a rule of inference. ''Modus tollens'' is a m ...

''. The disjunctive version of ''

modus ponens In propositional logic, (; MP), also known as (), implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "''P'' implies ''Q.'' ''P'' is true. Therefore, ''Q'' must ...

'' is the constructive dilemma. The destructive dilemma rule can be stated: :

\frac

where the rule is that wherever instances of "

P \to Q

", "

R \to S

", and "

\neg Q \lor \neg S

" appear on lines of a proof, "

\neg P \lor \neg R

" can be placed on a subsequent line.

Formal notation

The ''destructive dilemma'' rule may be written in

sequent In mathematical logic, a sequent is a very general kind of conditional assertion. : A_1,\,\dots,A_m \,\vdash\, B_1,\,\dots,B_n. A sequent may have any number ''m'' of condition formulas ''Ai'' (called " antecedents") and any number ''n'' of ass ...

notation: :

(P \to Q), (R \to S), (\neg Q \lor \neg S) \vdash (\neg P \lor \neg R)

where

\vdash

is a

metalogic Metalogic is the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a lo ...

al symbol meaning that

\neg P \lor  \neg R

is a syntactic consequence of

P \to Q

R \to S

, and

\neg Q \lor \neg S

in some

logical system A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in math ...

; and expressed as a truth-functional tautology or

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

of propositional logic: :

(((P \to Q) \land (R \to S)) \land (\neg Q \lor \neg S)) \to (\neg P \lor \neg R)

where

P

Q

R

and

S

are propositions expressed in some

formal system A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in ma ...

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

Example proof

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

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. Overview The assumed antecedent of a conditional proof is called the condi ...

(CP) and

reductio ad absurdum In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical argument'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absur ...

(RAA) in the following way:

References

Bibliography

* Howard-Snyder, Frances; Howard-Snyder, Daniel; Wasserman, Ryan. The Power of Logic (4th ed.). McGraw-Hill, 2009, , p. 414.

External links

*http://mathworld.wolfram.com/DestructiveDilemma.html {{DEFAULTSORT:Destructive Dilemma Rules of inference Dilemmas Theorems in propositional logic