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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?

MJD
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Richard Ambler
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    See this post: $A \not\rightarrow B$ – Mauro ALLEGRANZA Sep 17 '21 at 09:49
  • Thanks @MauroALLEGRANZA for the comments. Just to clarify, I'm aware that we commonly just cross out a symbol to signify its opposite meaning, and that crossing out a symbol often leads to a unique symbol. For example, $\downarrow$ and $\uparrow$ appear to be stylized versions of a crossed out $\lor$ and $\land$ respectively (and they feel much nicer than the $\not\lor$ and $\not\land$ they appear to be inspired by). I thus thought there would similarly be a unique symbol for $\lnot(p\rightarrow q)$, possibly inspired by $\not\rightarrow$. (If it were up to me, I'd go with $\nearrow$.) – Richard Ambler Sep 17 '21 at 13:34

3 Answers3

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$\require{cancel}$

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.

poetasis
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  • Not the answer I was hoping for, but most likely the correct one. :) – Richard Ambler Sep 18 '21 at 05:11
  • @Richard Ambler Please check out my update. It may be what you are looking for. – poetasis Sep 18 '21 at 07:42
  • I've upvoted your answer for its helpfulness but @Idéophage has provided a symbol with the meaning described that has been used in at least some mathematics literature. Thanks for your input! – Richard Ambler Sep 18 '21 at 10:06
  • @Richard Ambler I am glad you got a good answer. A lot has changed since I learned boolean algebra $50$ years ago. :) – poetasis Sep 18 '21 at 10:16
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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 $⊕$.

Dabouliplop
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  • I was aware of the $\setminus$ operator, but thought it only acted on sets. In your link, I see the authors have it defined for use on propositions too, but they seem to be explicitly defining it in order to create a bi-Heyting algebra from a Boolean algebra. – Richard Ambler Sep 18 '21 at 09:57
  • @RichardAmbler The symbol of negation is also used: see here and here. – Dabouliplop Sep 18 '21 at 10:40
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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 $\mathbb{F}$.

M. Logic
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