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