Why are the symbols $\cdot$ and $+$ used for the operations of Boolean algebra?

Question

The symbols $\cdot$ and $+$ are often used to denote Boolean product and sum, but they make some of the system's properties, like distributivity over $\cdot$, counter-intuitive: $$a+(b\cdot{c})=(a+b)\cdot(a+c)$$

Why are these particular symbols (and names) used?

Answer 1

I came upon Florian Cajori's A History of Mathematical Notations (1928-1929), which gives a lot of historical details on this.

First is George Boole (Mathematical Analysis of Logic, 1847; Laws of Thought, 1854), who "uses the symbols of ordinary algebra, but gives them different significations." Boole:

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.

... 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.

So Boole is aware that his use of the symbols does not necessarily align with how they're used in "ordinary" algebra, but opts for familiarity. They do align with certain properties like identity, if you treat the variables as either 0 or 1:

Let us conceive, then, of an Algebra in which the symbols x, y, z, etc., admit indifferently of the values 0 and 1, and of these values alone.

Ernst Schroder expands on Boole's work (Der Operaiionskreis des Logikkalkuls, 1877; Vorlesungen uber die Algebra der Logik, 1890). According to Cajori, "it was Schroder who gave to the algebra of logic 'its present form,' in so far as it may be said to have any universally fixed form":

Schroder represents the identical product and sum (logical aggregate) of a and b in his logic by ab (or a.b) and a+b. If a distinction is at any time desirable between the logical and arithmetical symbols, it can be effected by the use of parentheses as in (+), (⋅) or by the use of a cross of the form ✣, or of smaller, black, or cursive symbols.

Platon Poretsky develops a similar notation independently of Schroder (Sept Lois fondamentales de la théorie des égalités logiques, 1899), but it's not well-known.

Giuseppe Peano goes on a very different route from Schroder (Calcolo geometrico 1888; Formulaire de mathematiques, 1895):

...he states that, in order to avoid confusion between logical signs and mathematical signs, he substitutes in place of the signs ×, +, A₁, 0, 1, used by Schroder in 1877, the signs ◠, ◡, −A, ○, ●.

Peano is very popular at the time, but Alfred North Whitehead adopts a notation more similar to Schroder's (Treatise on Universal Algebra, 1898), which he continues to develop with Bertrand Russel (Principia mathematica, 1910).

Then comes Ludwig Wittgenstein, who takes cues from both Peano, Whitehead and Russell (Tractatus Logico-Philosophicus, 1920).

Cajori's history ends here, having been published just eight years after the Tractatus. He ends the chapter with the following:

No topic which we have discussed approaches closer to the problem of a uniform and universal language in mathematics than does the topic of symbolic logic... No group of workers has been more active in the endeavor to find a solution of that problem than those who have busied themselves with symbolic logic... their mode of procedure has been in the main individualistic. Each proposed a list of symbols, with the hope, no doubt, that mathematicians in general would adopt them. That expectation has not been realized. What other mode of procedure is open for the attainment of the end which all desire?

So the answer is that, like in other realms of mathematical convention, what we use today is the just the last note in a long correspondence between many mathematicians, each with his own intuitions and perceptions of familiarity and ease. It's not necessarily the easiest notation to adjust to, but it is in a sense a "happy medium" which emphasizes usability and internal correctness.

Answer 2

This is the algebraic notation used for the Boolean semiring ${\Bbb B} = \{0, 1\}$, which is the simplest example of a semiring that is not a ring. It is an idempotent semiring, that is, it satisfies $x + x = x$ for all $x$. Mathematically speaking, it is a very convenient notation, which allows for natural extensions such that Boolean matrices, polynomials and formal power series over $\Bbb B$, etc.

Answer 3

It appears that George Boole himself started using these symbols in his book An Investigation of the Laws of Thought

He doesn't use the dot notation though. He writes just, for example, $xy$.