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Conjunction elimination

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Title: Conjunction elimination  
Author: World Heritage Encyclopedia
Language: English
Subject: Logical conjunction, Simplification, Consensus theorem, Rule of inference, Biconditional introduction
Collection: Rules of Inference, Theorems in Propositional Logic
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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.

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}

and

\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

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

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