Conjunction elimination

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In propositional logic, conjunction elimination (also called and elimination, ∧ elimination,[1] or simplification)[2][3][4] 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{P \land Q}{\therefore P}


\frac{P \land Q}{\therefore Q}

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


(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


(P \land Q) \to Q

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


  1. David A. Duffy (1991). Principles of Automated Theorem Proving. New York: Wiley.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> Sect., p.46
  2. Copi and Cohen[citation needed]
  3. Moore and Parker[citation needed]
  4. Hurley[citation needed]

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