# Rule of Material Implication

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## Contents

## Theorem

The **rule of material implication** is a valid deduction sequent in propositional logic:

#### Formulation 1

- $p \implies q \dashv \vdash \neg p \lor q$

#### Formulation 2

- $\vdash \left({p \implies q}\right) \iff \left({\neg p \lor q}\right)$

That is:

is logically equivalent to:

## As a definition

- $p \implies q := \neg p \lor q$

## Also known as

This rule is sometimes seen referred to as the **definition of material implication**, as some sources use this rule as a definition of the conditional, so as to justify its semantics.

A **material implication** is sometimes expressed in the amplified form **implication in material meaning**.

## Also see

The following are related argument forms:

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S \text{II}.8$: Implication or Conditional Sentence - 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 2.3$: Basic Truth-Tables of the Propositional Calculus - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2.2$: Conditional Statements - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $3.2$: The Rule of Replacement - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic