# What is the origin of the negation ( ¬ ) operator from logic?

I'm curious as to what the rationale was, and who the idea occurred to, for the ¬ symbol. I'll grant that more common mathematical symbols like +, −, × and ÷ are also likely unknown, but they seem to vaguely relate to their meaning in an onomatopoetic sense, or are at least clearly derived from the next-simplest related operation (like × to +). Even in set theory, ∪ denotes union—fair enough, probably from a language that began the word with "u"—and ∩ denotes intersection, its logical opposite hence upside-down. Even conjunction and disjunction (∧, ∨) seem to derive from this standard.

After all of that, I have to admit, for "¬" to imply predicate-negation makes no intrinsic sense to me. When I first saw the symbol in college, even as a philomath, I had absolutely no idea what it was trying to say to me.

Does anyone know where it originated and why?

• Incidentally, I believe ∨ has the strongest etymological claim, probably deriving from Latin “vel”. The others may follow from there. Commented Aug 4, 2022 at 12:39
• ÷ makes perfect sense if you think about what a fraction looks like, a horizontal line with one thing above it and one below. Commented Aug 4, 2022 at 15:36
• @DarrelHoffman Or perhaps even without thinking about fractions... Two items that have been separated by the line. Commented Aug 4, 2022 at 23:13
• @CarstenS A fair point, I only encountered it in set theory. I changed the tag. Commented Aug 6, 2022 at 0:37

If vague resemblance is enough, then "$$¬$$" resembles "$$-$$", which denotes negation in arithmetic. Lambert in Sechs Versuche einer Zeichenkunst in der Vernunftlehre (1782) and Boole in Laws of Thought (1854) just used "$$-$$" for negation, and "$$+,×$$" for "$$\lor,\land$$". The latter explicitly cited the arithmetical analogy:

"Signs of operation, as $$+,-,×$$ standing for those operations of the mind by which the conceptions of things are combined or resolved so as to form new conceptions involving the same elements... And these symbols of Logic are in their use subject to definite laws, partly agreeing with and partly differing from the laws of the corresponding symbols in the science of Algebra."

Peano modified "$$-$$" to "$$\sim$$" in Studii di logica matematica (1897), possibly because, unlike Lambert and Boole, he was using logical symbols alongside arithmetic ones. Russell and Whitehead followed suit in Principia (1910-13). Heyting introduced "$$¬$$" in Die formalen Regeln der intuitionistischen Logik (1930).

For more information and references to the original works on this and other symbols, see Tou, Math Origins: The Logical Symbols and Earliest Uses of Symbols of Set Theory and Logic.

• And ~ lives on as the bitwise NOT operator in many programming languages. Commented Aug 4, 2022 at 14:49
• I have to admit that I always thought that it was derived from Frege's symbol for negation en.wikipedia.org/wiki/Begriffsschrift. en.wikipedia.org/wiki/Begriffsschrift#/media/… Commented Aug 4, 2022 at 15:23
• @Barmar of course ~ has the advantage over ¬ for programming languages, since ~ is already on the keyboard. Commented Aug 4, 2022 at 16:03
• @GeraldEdgar: Depends on your country. UK has ¬ on the keyboard. Commented Aug 4, 2022 at 16:18
• @RBarryYoung Frege's work was largely unknown until Russell popularized it, but he did not adopt Frege's notation. Not even Frege himself ever used it after Begriffsschrift. Most of Principia's notation comes from Peirce with additions and modifications by Schröder and Peano, see Anellis, How Peircean was the "'Fregean' Revolution" in Logic? Commented Aug 4, 2022 at 16:30