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In
propositional logic Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...
, disjunction elimination (sometimes named proof by cases, case analysis, or or elimination), is the valid
argument form In logic, logical form of a Statement (logic), statement is a precisely-specified Semantics, semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly Syntactic ambiguity, ambiguous sta ...
and
rule of inference In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...
that allows one to eliminate a disjunctive statement from a logical proof. It is the
inference Inferences are steps in 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 distinction that in ...
that if a statement P implies a statement Q and a statement R also implies Q, then if either P or R is true, then Q has to be true. The reasoning is simple: since at least one of the statements P and R is true, and since either of them would be sufficient to entail Q, Q is certainly true. An example in
English English usually refers to: * English language * English people English may also refer to: Peoples, culture, and language * ''English'', an adjective for something of, from, or related to England ** English national ide ...
: :If I'm inside, I have my wallet on me. :If I'm outside, I have my wallet on me. :It is true that either I'm inside or I'm outside. :Therefore, I have my wallet on me. It is the rule can be stated as: :\frac where the rule is that whenever instances of "P \to Q", and "R \to Q" and "P \lor R" appear on lines of a proof, "Q" can be placed on a subsequent line.


Formal notation

The ''disjunction elimination'' rule may be written in sequent notation: : (P \to Q), (R \to Q), (P \lor R) \vdash Q where \vdash is a
metalogic Metalogic is the study of 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.Harry GenslerIntroduction to Logic Routledge, ...
al symbol meaning that Q is a syntactic consequence of P \to Q, and R \to Q and P \lor R in some logical system; and expressed as a truth-functional tautology or theorem of propositional logic: :(((P \to Q) \land (R \to Q)) \land (P \lor R)) \to Q where P, Q, and R are propositions expressed in some
formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A form ...
.


See also

*
Disjunction In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...
* Argument in the alternative * Disjunct normal form


References

{{DEFAULTSORT:Disjunction Elimination Rules of inference Theorems in propositional logic