Conjunction Elimination

In propositional logic, conjunction elimination (also called ''and'' elimination, ∧ elimination, or simplification)Hurley is a valid immediate inference, argument form and rule of inference which makes the inference that, if the conjunction ''A and B'' is true, then ''A'' is true, and ''B'' is true. The rule makes it possible to shorten longer proofs by deriving one of the conjuncts of a conjunction on a line by itself.

An example in English:
:It's raining and it's pouring.
:Therefore it's raining.

The rule consists of two separate sub-rules, which can be expressed in formal language as:

:$\frac$

and

:$\frac$

The two sub-rules together mean that, whenever an instance of "$P \land Q$" appears on a line of a proof, either "$P$" or "$Q$" can be placed on a subsequent line by itself. The above example in English is an application of the first sub-rule.

Formal notation

The ''conjunction elimination'' sub-rules may be written in sequent notation:

: $(P \land Q) \vdash P$
and
: $(P \land Q) \vdash Q$

where $\vdash$ is a metalogical symbol meaning that $P$ is a syntactic consequence of $P \land Q$ and $Q$ is also a syntactic consequence of $P \land Q$ in logical system;

and expressed as truth-functional tautologies or theorems of propositional logic:

:$(P \land Q) \to P$
and
:$(P \land Q) \to Q$

where $P$ and $Q$ are propositions expressed in some formal system.

References

{{reflist

Rules of inference
Theorems in propositional logic

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