# Material nonimplication Venn diagram of $P\nrightarrow Q$ Material nonimplication or abjunction (Latin ab = "from", junctio =–"joining") is the negation of material implication. That is to say that for any two propositions $P$ and $Q$ , the material nonimplication from $P$ to $Q$ is true if and only if the negation of the material implication from $P$ to $Q$ is true. This is more naturally stated as that the material nonimplication from $P$ to $Q$ is true only if $P$ is true and $Q$ is false.

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

## Definition

### Truth table

 $P$ $Q$ $P\nrightarrow Q$ 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.

 $P\nrightarrow Q$ $\Leftrightarrow$ $\neg (P\rightarrow Q)$  $\Leftrightarrow$ $\neg$  In classical logic, it is also equivalent to the negation of the disjunction of $\neg P$  and $Q$ , and also the conjunction of $P$  and $\neg Q$

 $P\nrightarrow Q$ $\Leftrightarrow$ $\neg ($ $\neg P$ $\lor$ $Q)$ $\Leftrightarrow$ $P$ $\land$ $\neg Q$  $\Leftrightarrow$ $\neg ($  $\lor$  $)$ $\Leftrightarrow$  $\land$  ## 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)