Conjunction elimination

From HandWiki
Conjunction elimination
TypeRule of inference
FieldPropositional calculus
StatementIf the conjunction [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] is true, then [math]\displaystyle{ A }[/math] is true, and [math]\displaystyle{ B }[/math] is true.

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:

[math]\displaystyle{ \frac{P \land Q}{\therefore P} }[/math]


[math]\displaystyle{ \frac{P \land Q}{\therefore Q} }[/math]

The two sub-rules together mean that, whenever an instance of "[math]\displaystyle{ P \land Q }[/math]" appears on a line of a proof, either "[math]\displaystyle{ P }[/math]" or "[math]\displaystyle{ Q }[/math]" 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:

[math]\displaystyle{ (P \land Q) \vdash P }[/math]


[math]\displaystyle{ (P \land Q) \vdash Q }[/math]

where [math]\displaystyle{ \vdash }[/math] is a metalogical symbol meaning that [math]\displaystyle{ P }[/math] is a syntactic consequence of [math]\displaystyle{ P \land Q }[/math] and [math]\displaystyle{ Q }[/math] is also a syntactic consequence of [math]\displaystyle{ P \land Q }[/math] in logical system;

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

[math]\displaystyle{ (P \land Q) \to P }[/math]


[math]\displaystyle{ (P \land Q) \to Q }[/math]

where [math]\displaystyle{ P }[/math] and [math]\displaystyle{ Q }[/math] are propositions expressed in some formal system.


  1. David A. Duffy (1991). Principles of Automated Theorem Proving. New York: Wiley.  Sect., p.46
  2. Copi and Cohen[citation needed]
  3. Moore and Parker[citation needed]
  4. Hurley[citation needed]

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