• Source: Conjunction elimination

In propositional logic, conjunction elimination (also called and elimination, ∧ elimination, or simplification) 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:







P
∧
Q


∴
P





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


and







P
∧
Q


∴
Q





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


The two sub-rules together mean that, whenever an instance of "



P
∧
Q


{\displaystyle P\land Q}

" appears on a line of a proof, either "



P


{\displaystyle P}

" or "



Q


{\displaystyle 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
∧
Q
)
⊢
P


{\displaystyle (P\land Q)\vdash P}


and




(
P
∧
Q
)
⊢
Q


{\displaystyle (P\land Q)\vdash Q}


where



⊢


{\displaystyle \vdash }

is a metalogical symbol meaning that



P


{\displaystyle P}

is a syntactic consequence of



P
∧
Q


{\displaystyle P\land Q}

and



Q


{\displaystyle Q}

is also a syntactic consequence of



P
∧
Q


{\displaystyle P\land Q}

in logical system;
and expressed as truth-functional tautologies or theorems of propositional logic:




(
P
∧
Q
)
→
P


{\displaystyle (P\land Q)\to P}


and




(
P
∧
Q
)
→
Q


{\displaystyle (P\land Q)\to Q}


where



P


{\displaystyle P}

and



Q


{\displaystyle Q}

are propositions expressed in some formal system.




References

Kata Kunci Pencarian:

• Holokaus
• Conjunction elimination
• Logical conjunction
• Conjunction (grammar)
• List of rules of inference
• Dead-code elimination
• Natural deduction
• Fitch's paradox of knowability
• Paraconsistent logic
• Principle of explosion
• Quantifier elimination