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Boolean algebras aren't algebras (to the best of my understanding).

So why are they called algebras?

Wouldn't it make more sense to call them a "Boolean system" or a "Boology" or something else like that?

Chill2Macht
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2 Answers2

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Because Boole himself introduced the word "algebra" into the subject.

The term "algebra of logic" appears in Boole's 1854 book on Laws of Thought:

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. The laws, the axioms, and the processes, of such an Algebra will be identical in their whole extent with the laws, the axioms, and the processes of an Algebra of Logic. Difference of interpretation will alone divide them. Upon this principle the method of the following work is established.

Boole strongly emphasized the relation between logic and algebra. References to algebra and its correspondence with logic permeate the book.

Other writers continued to use "algebra of logic" for Boole's system and its later simplification to what is now called Boolean algebra. For example, MacFarlane Principles of the Algebra of Logic (1874), C.S. Pierce "On the Algebra of Logic" (1880), and E. Schroeder Algebra der Logik (1890).

In addition to the analogy that Boole had observed with ordinary algebra, there is an equivalence of Boolean algebras with rings satisfying $x^2=x$ for all $x$, which are equivalent to some algebras (in the modern sense) over the 2-element field.

zyx
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  • When you say "ordinary algebra" and "the relation between logic and algebra", you mean the middle school stuff, right? So Boole was drawing an analogy with symbolic manipulation? Also do you have a reference for the last statement (equivalence with idempotent rings)? I probably wouldn't understand it, but it sounds interesting to at least look at. -- Actually that does make some sense intuitively, since Boolean algebra is binary logic, so that it would be in some equivalent to the 2-element field kind of makes sense. Thanks so much for your detailed answer! – Chill2Macht May 16 '16 at 04:54
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    Yes, yes and yes ( http://en.wikipedia.org/wiki/Boolean_ring#Relation_to_Boolean_algebras ) . – zyx May 16 '16 at 04:55
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    Though Boole was making an analogy more precise than both systems involving symbolic expressions. – zyx May 16 '16 at 05:15
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    The first of Boole's pubblications was The Mathematical Analysis of Logic: Being an Essay Towards a Calculus of Deductive Reasoning (1847); see Introduction: "THEY who are acquainted with the present state of the theory of Symbolical Algebra,...". – Mauro ALLEGRANZA May 16 '16 at 11:42
  • Boole (1847) calls it a "Calculus of Logic" and writes that there can be different calculi with different rules and interpretations of the symbols. The terminology changed to "algebra of logic" in his 1854 book. @MauroALLEGRANZA – zyx May 16 '16 at 14:14
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They are algebras, in the sense of universal algebra (where "algebra" is basically synonymous with "first-order structure", except that it requires the language to have no relation symbols).

In fact, I think this notion of algebra = algebraic structure long preceded the definition of algebra as (roughly) a module with multiplication. And it was in this context that Boolean algebras were so named - that is, just because of their "algebraic" nature.

Noah Schweber
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  • wait so how is it a first-order structure/algebraic structure? sorry for being dumb – Chill2Macht May 16 '16 at 03:58
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    @William Well, it's a set equipped with a few operations - that's a first-order structure! The signature varies, but is usually taken to consist of two binary function symbols (for join $\vee$ and meet $\wedge$) and one unary function symbol (for complementation). – Noah Schweber May 16 '16 at 04:15
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    I think Boology has a future – Will Jagy May 16 '16 at 04:19
  • Then a probability space would be a triple consisting of an event space, a probability measure, and a $\sigma-$Boology! – Chill2Macht May 16 '16 at 04:20
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    They can also be seen as special commutative rings with unity, the Boolean rings, that satisfy $x + x = 0$ for all $x$. Then we can define $x \land y = xy$, $x \lor y = x + y - xy$, $\lnot x = 1 + x$ and all axioms for a Boolean algebra check out. – Henno Brandsma May 16 '16 at 04:34
  • "Universal algebra" appeared decades later than Boole's "algebra of logic". – zyx May 16 '16 at 04:51
  • @NoahSchweber Wouldn't the constants $0$ and $1$ also be taken to be built in? – bof May 16 '16 at 04:54
  • @HennoBrandsma I feel like $\vee$ is the wrong symbol to use for exclusive OR, which you've described. – Joey Eremondi May 16 '16 at 06:18
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    @jmite $+$ is "exclusive or" in a Boolean ring (hence the $x+x = 0$) $x \lor y $ is also equal to $x + y + xy$ (as $-z = z$) and so corresponds to the "union" if $+$ is exclusive or. – Henno Brandsma May 16 '16 at 07:02
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    I don't think "algebra = algebraic structure" existed in Boole's time either. What he was getting at was simply "algebra = the idea of manipulating symbolic expressions by rules" rather than by ad-reasoning about their meaning. – hmakholm left over Monica May 16 '16 at 09:39