# Rule of Implication

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## Proof Rule

The **rule of implication** is a valid argument in types of logic dealing with conditionals $\implies$.

This includes propositional logic and predicate logic, and in particular natural deduction.

### Proof Rule

- If, by making an assumption $\phi$, we can conclude $\psi$ as a consequence, we may infer $\phi \implies \psi$.

- The conclusion $\phi \implies \psi$ does not depend on the assumption $\phi$, which is thus discharged.

### Sequent Form

The Rule of Implication can be symbolised by the sequent:

\(\ds \paren {p \vdash q}\) | \(\) | \(\ds \) | ||||||||||||

\(\ds \vdash \ \ \) | \(\ds p \implies q\) | \(\) | \(\ds \) |

## Explanation

The **Rule of Implication** can be expressed in natural language as:

- If by making an assumption $\phi$ we can deduce $\psi$, then we can encapsulate this deduction into the compound statement $\phi \implies \psi$.

## Also known as

The Rule of Implication is sometimes known as:

- The
**rule of implies-introduction** - The
**rule of conditional proof**(abbreviated $\text{CP}$).

## Also see

## Sources

- 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.4$: Statement Forms