##  [Conjunction Elimination](/conjunction-elimination-0) 

 Definition

An inference rule allowing one to infer either conjunct from a conjunction A ∧ B; formally from A ∧ B derive A (and symmetrically derive B).

 

 

 

 

 

 





## Principle

Principle

A conjunction asserts both of its conjuncts; therefore each conjunct is individually entailed by the conjunction as a premise in a proof.

 

 

 

 

 





## Demonstration

Demonstration

From a derivation of A ∧ B, apply conjunction elimination to derive A. Example: from 'It is raining ∧ the ground is wet' derive 'It is raining.'

 

 

 

 

## Misapplication

Misapplication

Applying conjunction elimination to a formula that is not a genuine conjunction (e.g., to a material implication or to a disjunction), or assuming elimination can produce conjuncts that were not present (e.g., inferring B from A alone).

 

 

 

 

 





## Consequence

Consequence

Permits extraction and local use of specific facts contained inside a compound assertion, thus enabling modular reasoning and focused subproofs.

 

 

 

 

## Reversal

Reversal

The inverse operation is conjunction introduction (composition): whereas elimination breaks a compound into parts, introduction composes parts into a compound; confusing them leads to unsupported inferences.

 

 

 

 

 





## Boundary

Boundary

Holds in classical, intuitionistic and most standard deductive systems; may be restricted in logics where conjunction has a different meaning (e.g., linear logic with resource tracking) or when the conjunction is defined extensionally rather than proof-theoretically.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension arises with distribution or with composite connectives whose surface syntax resembles conjunction but carry additional structure; deciding when elimination is valid requires attention to the connective’s semantics.

 

 

 

 

 





## Synthesis

Synthesis

Conjunction elimination is the rule that lets proofs access the individual components guaranteed by a conjunction, turning a joint assertion into usable individual premises for subsequent reasoning.