Rule of Material Implication

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

$\vdash \paren {p \implies q} \iff \paren {\neg p \lor q}$

That is:

statement $p$ implies statement $q$

is logically equivalent to:

statement $p$ is false or statement $q$ is true

As a definition

$p \implies q := \neg p \lor q$

Also known as

The Rule of Material Implication 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:

• Modus Ponendo Tollens
• Modus Tollendo Tollens

• Definition:Formal Implication

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): $2$ Arguments Containing Compound Statements: $2.2$: Conditional Statements
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): material implication
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): material implication