Material nonimplication or abjunction (Latin ab = "away", junctio= "to join") is the negation of material implication. That is to say that for any two propositions ${\displaystyle P}$ and ${\displaystyle Q}$, the material nonimplication from ${\displaystyle P}$ to ${\displaystyle Q}$ is true if and only if the negation of the material implication from ${\displaystyle P}$ to ${\displaystyle Q}$ is true. This is more naturally stated as that the material nonimplication from ${\displaystyle P}$ to ${\displaystyle Q}$ is true only if ${\displaystyle P}$ is true and ${\displaystyle Q}$ is false.

It may be written using logical notation as ${\displaystyle P\nrightarrow Q}$, ${\displaystyle P\not \supset Q}$, or "Lpq" (in Bocheński notation), and is logically equivalent to ${\displaystyle \neg (P\rightarrow Q)}$, and ${\displaystyle P\land \neg Q}$.

falsehood-preserving: The interpretation under which all variables are assigned a truth value of "false" produces a truth value of "false" as a result of material nonimplication.

The symbol for material nonimplication is simply a crossed-out material implication symbol. Its Unicode symbol is 219B₁₆ (8603 decimal): ↛.

Bitwise operation: A&(~B)

Logical operation: A&&(!B)

• Implication
• Boolean algebra
• Set difference