Are there commonly used symbols for the negation of implications?

Question

Exhaustively, there are 16 possible truth tables for two propositions. If we interpret the values in the rows as binary digits, we can conveniently use the resulting hexadecimal digit as the respective name for each truth table, as follows:

		0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
$p$	$q$	$\mathbb F$	$\land$	?	$p$	?	$q$	$\veebar$	$\lor$	$\downarrow$	$\leftrightarrow$	$\lnot q$	$\leftarrow$	$\lnot p$	$\rightarrow$	$\uparrow$	$\mathbb T$
0	0	0	0	0	0	0	0	0	0	1	1	1	1	1	1	1	1
0	1	0	0	0	0	1	1	1	1	0	0	0	0	1	1	1	1
1	0	0	0	1	1	0	0	1	1	0	0	1	1	0	0	1	1
1	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1

Tables 0, 3, 5, A, C and F are not particularly interesting as the output is independent of at least one of the operands.

For the remaining tables, I'm aware of at least one commonly used symbol associated with the operation (examples shown above), but not for tables 2 and 4, which are the negation of sufficient, i.e. $\lnot(p\rightarrow q)$, and the negation of necessary, i.e. $\lnot(p\leftarrow q)$, respectively.

Are there symbols commonly used for these two operations?

Answer 1

There is the symbol $\setminus$ (\setminus in latex), used in set theory to denote set difference. The first question mark (2) could thus be $p\setminus q$. Note that it satisfies a property dual to the defining property of the implication $p ≤ q→r ⇔ p∧q≤r$, namely $p\setminus q ≤ r ⇔ p ≤ q∨r$. It could be called a "coimplication", cf. for instance here. See also here where it is called a subtraction (in the examples, they mention that $x\setminus y = x∧¬y$ in a Boolean algebra for instance).

I didn't know the symbol $\veebar$ for XOR, I only knew $⊕$.

Answer 2

$\require{cancel}$

The \not will put a diagonal slash through almost any single symbol as in

\not\gt or \not\in $\longrightarrow\space \not\gt \text{ or} \not\in.\quad$ The latter can also be coded as \notin.

Another loose interpretation of not is the diagonal $\space\text{\cancel{} or \bcancel{} }\space \longrightarrow \dfrac{x\cancel{(m-1)}}{y\cancel{(m-1)}} \quad\text{ or }\quad \dfrac{x\bcancel{(m-1)}}{y\bcancel{(m-1)}} $

For this to work, you must code at the beginning of the document

\$\require{cancel}\$ $\quad $in Mathjax

or

\usepackage{cancel} $\quad$ in LaTeX

I think there is similar coding for \strikethrough but I have never used it.

$\textbf{Update:}\quad$ I forgot about the NOT operator (!) used in Java etc. and in other boolean or logical environments.

Answer 3

OP, @poetasis 's answer is right, i.e., it's suitable to use $p\not\rightarrow q$ to express $\neg(p\to q)$ and $p\not\leftarrow q$ to express $\neg(p\leftarrow q)$, since in natural language we also do in the way equivalently.

By the way, we usually use $\dot{\vee}$ (the dot should be more lower) to denote incompatible disjunction instead of $\veebar$, use $\top$ to denote tautologies or validities instead of $\mathbb{T}$ and use $\bot$ to denote contradictions instead of $\mathbb{F}$.