Rule of Implication

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.

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.

The can be symbolised by the sequent:

$\ds \paren {p \vdash q}$

$\ds $

$\ds \vdash \ \ $

$\ds p \implies q$

$\ds $

The 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 $\phi \implies \psi$.

The is sometimes known as:

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