Conjunction elimination

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.

Conjunction elimination
TypeRule of inference
FieldPropositional calculus
StatementIf the conjunction and is true, then is true, and is true.
Symbolic statement

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:

and

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

Formal notation

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

 

and

 

where   is a metalogical symbol meaning that   is a syntactic consequence of   and   is also a syntactic consequence of   in logical system;

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

 

and

 

where   and   are propositions expressed in some formal system.

References

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  1. ^ David A. Duffy (1991). Principles of Automated Theorem Proving. New York: Wiley. Sect.3.1.2.1, p.46
  2. ^ Copi and Cohen[citation needed]
  3. ^ Moore and Parker[citation needed]
  4. ^ Hurley[citation needed]