material nonimplication

#LaTeX

Material nonimplication is the negation of material implication. Its logical connective is represented $⇏$ , written \nRightarrow.

Boolean material nonimplication is an operation which takes as input two logical truth values; it returns a value of true if and only if both its antecedent is true and its consequent is false, and false otherwise.

Visually,

\documentclass{standalone}
\usepackage{tikz}
\usepackage{amssymb}
%
\definecolor{fillcolor}{HTML}{8A5CF5}
%
\tikzset{truthy/.style n args={1}{%
    fill=#1,draw,%
    inner sep=0,minimum size=1cm}}
%
\begin{document}
\begin{tikzpicture}
	\path (-2,-2) rectangle (2,2);
	\node at (0,0.5) {Material Nonimplication};
	\node at (0,-0.5) {$P\nRightarrow Q$};
\end{tikzpicture}
\begin{tikzpicture}
	\node at (-1.5,0.5) {$P$};
	\node at (0.5,2.5) {$Q$};
	\node at (-1,0) {\small$1$};
	\node at (-1,1) {\small$0$};
	\node at (0,2) {\small$0$};
	\node at (1,2) {\small$1$};
    \node[truthy={fillcolor}] at (0,0) {$1$}; % bottom left
    \node[truthy={none}] at (1,0) {$0$}; % bottom right
    \node[truthy={none}] at (0,1) {$0$}; % top left
    \node[truthy={none}] at (1,1) {$0$}; % top right
\end{tikzpicture}
\hspace{1cm}
\begin{tikzpicture}
	\path (-3,-2) rectangle (3,2);
	% P and not Q
	\begin{scope}[even odd rule]
		\clip (-1,0) circle (1.5);
		\fill[fillcolor] (-1,0) circle (1.5) (1,0) circle (1.5);
	\end{scope}
	%
    \draw (-1,0) circle (1.5);
    \draw (1,0) circle (1.5);
    \node at (-1.25,0) {$P$};
    \node at (1.25,0) {$Q$};
\end{tikzpicture}
\end{document}

Notation

The material nonimplication of propositions from $P$ to $Q$ is generally written $P ⇏ Q$ .