# Material nonimplication

Venn diagram of ${\displaystyle P\nrightarrow Q}$

Material nonimplication or abjunction (Latin ab = "from", junctio =–"joining") 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 ${\displaystyle P\land \neg Q}$.

## Definition

### Truth table

 ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle P\nrightarrow Q}$ T T F T F T F T F F F F

### Logical Equivalences

Material nonimplication may be defined as the negation of material implication.

 ${\displaystyle P\nrightarrow Q}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \neg (P\rightarrow Q)}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \neg }$

In classical logic, it is also equivalent to the negation of the disjunction of ${\displaystyle \neg P}$  and ${\displaystyle Q}$ , and also the conjunction of ${\displaystyle P}$  and ${\displaystyle \neg Q}$

 ${\displaystyle P\nrightarrow Q}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \neg (}$ ${\displaystyle \neg P}$ ${\displaystyle \lor }$ ${\displaystyle Q)}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle P}$ ${\displaystyle \land }$ ${\displaystyle \neg Q}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \neg (}$ ${\displaystyle \lor }$ ${\displaystyle )}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle \land }$

## Properties

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.

## Symbol

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

"p but not q."

## Computer science

Bitwise operation: A&(~B)

Logical operation: A&&(!B)