Rule of Implication/Proof Rule

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.

As a proof rule it is expressed in the form:

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.

It can be written:

$\ds {\begin {array} {|c|} \hline \phi \\ \vdots \\ \psi \\ \hline \end {array} \over \phi \implies \psi} \to_i$

Let $\phi$ and $\psi$ be two well-formed formulas in a tableau proof.

The is invoked for $\phi$ and $\psi$ in the following manner:

			Pool:					The pooled assumptions of $\psi$
			Formula:					$\phi \implies \psi$
			Description:
			Depends on:					The series of lines from where the assumption $\phi$ was made to where $\psi$ was deduced
			Discharged Assumptions:					The assumption $\phi$ is discharged
			Abbreviation:					$\text{CP}$ or $\implies \II$

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 $\phi \implies \psi$.

The Rule of Implication is sometimes known as:

This is a rule of inference of the following proof systems:
- Definition:Natural Deduction

When invoking the in a tableau proof, use the {{Implication}} template:

{{Implication|line|pool|statement|start|end}}

where:

line is the number of the line on the tableau proof where the is to be invoked

pool is the pool of assumptions (comma-separated list)

statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters

start is the line of the tableau proof where the antecedent can be found

end is the line of the tableau proof where the consequent can be found

1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 3$
1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $2$ Conditionals and Negation
1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 7$: Conditional proof
2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction