Why are there two different notations for negation in boolean logic?

Question

For the boolean variable $x$, there are two notations for its negation: $\neg x$ and $\bar x$. So why are there two different notations?

Answer 1

The "reason" is purely historial: the use of symbols changes during time.

The "overline" symbol for negation it seems is due to C.S.Peirce.

See e.g.:

• Charles Sanders Peirce, On the Algebra of Logic (1880), page 18:

This we shall write $P_1 \overline < C_1$, a dash over any symbol signifying in our notationt the negative of that symbol.

and:

• Charles Sanders Peirce (editor), Studies in Logic (1883): ON THE ALGEBRA OF LOGIC, by Christine Ladd, page 18:

The negative of a term or a proposition or a symbol is indicated by a line drawn over it. $\overline a =$ what is not $a$.

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The sign was used in mathematical logic also by:

• David Hilbert & Wilhelm Ackermann, Grundzüge der theoretischen Logik (1928); see Engl.transl.(1950) of the 2nd German ed.,1937, page 3:

1. $\overline X$ (read "not $X$") stands for the opposite or contradictory of $X$, that is, for that sentence which is true if $X$ is false and which is false if $X$ is true.

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The symbol $\lnot$ for negation is quite recent; according to the site Earliest Uses of Symbols of Set Theory and Logic [but see also the references to Peano and Russell for the use of $\sim$]:

it was first used in 1930 by Arend Heyting in “Die formalen Regeln der intuitionistischen Logik,” Sitzungsberichte der preußischen Akademie der Wissenschaften, phys.-math. Klasse, 1930, p. 42-65.

Compare with:

• Arend Heyting, Intuitionism: An Introduction (1956, 3rd ed., 1971), page 102:

The negation $\lnot$ is the strong mathematical negation which we have already discussed. [...] Then $\lnot p$ can be asserted if and only if we possess a construction which from the supposition that a construction $p$ were carried out, leads to a contradiction.