# 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

 $\displaystyle{ P }$ $\displaystyle{ Q }$ $\displaystyle{ P \nrightarrow Q }$ True True False True False True False True False False False False

### 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 219B16 (8603 decimal): ↛.

## Natural language

"p minus q."

"p without q."

### Rhetorical

"p but not q." "q is false, in spite of p."

## Computer science

Bitwise operation: A&(~B)

Logical operation: A&&(!B)