A Boolean algebra is one in which there are only two variables, with two basic operations upon them. Due to the algebra's root in logic, these variables are conventionally denoted False and True, which are represented with the numerals 0 and 1, in the normal case.
Are either of these denotations arbitrary? They are not. In the former case, due to the connection to logic, in the latter, due to the connection to the natural numbers. Which of the 16 truth functions we consider basic is somewhat arbitrary, for historical and practical reasons we will choose AND and OR.
The use of FALSE and TRUE is easier to explain, and prior, logically and historically, to the use of O and 1. The relation between the two variables of a Boolean algebra does not require logic for either explanation nor instantiation of the properties of the algebra. For instance, we use an unpowered circuit to represent 0 and a powered circuit to represent 1. There is nothing false about an unpowered circuit, not inherently, and this decision is truly arbitrary, in that using a powered circuit to represent binary 0 would have no consequences for anyone but hardware engineers. Who would probably prefer we do something useful, like reverse the flow of current to match the movement of electrons; but I digress.
The reason that the two elements of the set are False and True is that Boolean algebra is the abstraction of Boolean logic, a powerful and enduring achievement. As Descartes created an algebra of geometry, so George Boole created an algebra of logic, showing that the classical logic, which was expressed formally but verbally, could be denoted using the tools of algebra and mapped precisely to numbers and their operations. As Boolean logic is the foundation of practical computation, all of mathematics has been subsumed into it in practice. The theory of mathematics is in flux with respect to the foundations thereof, but it seems safe to predict that logic will retain pride of place when the dust settles.
Logic is concerned with the consequences of truth and falsehood, not their correspondence to reality. Logicians, even classically, have no issue taking nonsense like "all cats are racist" and arbitrarily assigning it the value of being true. Logic is concerned with facts such as, if "all cats are racist" is true, and "all men are mortal" is also true, then the statement "all cats are racist and all men are mortal" is also true. While if "all cats are racist" is false, but "all men are mortal" is true, then "all cats are racist and all men are mortal" is necessarily a false statement, while "all cats are racist or all men are mortal" is a true one, as "all men are mortal" carries sufficient truth in itself.
Note that due to differences in meaning between grammatical particles of English and Greek, the OR we use in logic is inclusive, such that "all cats are racist or all men are mortal" remains true if both clauses are true. The English word 'or' tends in its meaning toward the exclusive or, what we call XOR in Boolean algebra, so this bears pointing out.
The key insight here is that, with AND as the operation, falsehood contaminates the truth, while with OR as the operation, the truth of the true proposition is unaffected. The falsehood may as well not exist; the evident fact that cats are not racist has no bearing whatsoever upon the frailty of our mortal coil.
What George Boole realized is that, if we choose to define FALSE and TRUE as 0 and 1, and further restrict ourselves to a group containing only those elements, that OR and AND correspond precisely to addition and multiplication, which we here denote in the usual programmer's style as + and *. He was also aware that, if we instead call 0 truth and 1 falsehood, then AND can be + and OR can be *. This formal correspondence remains true even when we expand our definition, saying (in the correct case) that 0 is falsehood and all the remaining natural numbers n represent truth. That is, we could reverse those definitions; but now, at last, we have a reason not to.
That reason is the very purpose of logic itself, which is to reason about the truth, the set of all facts which happen to be the case. It could be the case that all felines harbor racial animus, it might be that George Washington lead Britain throughout World War II, and it is possible that George Boole elucidated the eponymous system of logic; in reality, the question of feline political correctness makes no sense, Churchill was the Prime Minister during the war, and Boole did in fact discover Boolean logic.
Aristotle was well aware that logic could be misused, by assigning logical truth at variance with the facts of the matter. He was unimpressed by this, an opinion I share. For the consistency which logic brings to explanations can be maintained with imaginary facts, but the consistency breaks down if the logic is not assiduously followed, a result he derives from the evident fact that there are truths which derive that virtue from being also real.
One might write a story about brave General Washington, who won the election just after Kaiser Genghis Khan annexed Czechoslovakia, and who has a cat, Winston, who is as racist as any other cat. Prime Minister Washington might say "I beg your pardon Mr. Gandhi-ji, but Winston is sadly racist, as all of his species are. I assure you, we shall treat you with the same respect as any man holding your station, and we trust you will find that respect considerable. I shall have the housekeeper remove him at once". If one were later to introduce a cat, whose warm feelings of fellowship for all men are evident, we would have a quandary. As any American knows, George Washington cannot tell a lie; we must conclude that the author cares as little for logical consistency as he does for historical fidelity.
To Aristotle, the fact that we may detect a logical inconsistency even in a collection of truths we know to be false in reality, derives from the reality of truth and the further reality of the logical consistency of the set of facts which happen to be the case. Therefore the proper use of logic is in the pursuit of real truth, or at least in the pursuit of imaginary truth; in either case, falsehoods multiply without number and are to be discarded whenever they are found to be false.
If the two elements of our Boolean logic were to be styled as Black and White, it is evident that either can be our falsehood, as long as we are consistent about the application of AND and OR and the rest of the construction. There is nothing about Black nor White which lets us choose between them; they have no structural distinction. If our two elements are 0 on the one hand, and the whole of the positive integers on the other, we have a distinction: 0 is singular, while the positive integers are infinite, yet discrete.
Let us be obstinate. Let us say, "Truth is singular! There is a single reality and that reality is all that is true. It is falsehood which has the property of infinite distinction". Can we make this work? Yes.
The natural numbers constitute a semiring, an algebraic structure which has two operations, each with an identity. The identity of addition is 0, that of multiplication is 1. 0 has a further property, that it annihilates a value under multiplication.
We can see that addition, in this contrary realm, has the AND property. Adding an identity to a value produces that value without modification, and if the value is itself the identity, the identity is produced. 0 + 0 = 0. 0 + n gives n, for any positive value of n, so the addition of any number of truths is true, but addition of any falsehood to any amount of truth results in that same falsehood.
We can see also that multiplication has the properties proper to OR, due to this very annihilation. We may multiply falsehood to our heart's content, pleasing Belial in this way, but a single multiplication of the product by 0 produces 0, as does the multiplication of 0 by itself.
But this is unsatisfying. For one thing, multiplication has an identity, but this identity plays no part in our logic. We use only 0 and n, addition and multiplication; why did we use an abstraction as powerful as the natural numbers, if we didn't need its properties? We would have been better off with Black and White. The isomorphism between + and AND on the one hand, and * and OR on the other, remains valid, but there is a piece missing nonetheless.
Adding 1 to 0 is just one of the ways to contaminate truth, while multiplying our falsehood by 0 is the only way to make truth from falsehood through an operation. It would be better if our identity elements had a necessary relation to our results.
Therefore let us follow the wisdom of our forebears, and of the inventor of this system, and assign 0 to falsehood and the positive integers each to truth of its own kind. We see, inexorably, that addition is now consonant with OR, and multiplication with AND. What also emerges is a role for 1: as our conventional representation of truth, we may AND it to itself, without changing its value, because it is the identity of the operation. Thus any distinct truth n may be multiplied indefinitely by truth itself, without changing in either verity nor value.
This is, happily, dual with 0 and falsehood. As 0 is the identity of addition, we may OR 0 against itself to our heart's content, accumulating a nullity of falsehood, until we encounter a truth; inexorably, the falsehoods vanish as though they never were, which indeed, they weren't. We are left with both the verity of the truth, and its value as well.
This has an additional property which I find pleasing, namely, if we represent each truth as some unique prime n, then the product of a chain of AND operations may be factored into each of the truths which constitute its several clauses. While OR does not share this property, the result of a series of OR operations on clauses which are all true, is equivalent in truth to those same clauses ANDed together. This is also true for a series of OR operations on clauses which are severally true and false, provided the false clauses are discarded. We also see that the sum of a series of true clauses is the same as the sum of a series containing those true clauses and any number of false ones, which may have some small utility.
Of course, our contrary schema allows us to enumerate and decompose falsehood in this way. But to what purpose? What sense would it make, to conjoin truths indefinitely, without distinguishing between them, and then contaminate them with a falsehood, to be left with... the value of the falsehood, and its falseness? We aim to discard falsehood and collect truth, or we should; we don't need the value of the falsehood, for it has none, beyond the fact that it is false.
So we say: all of truth is a single truth, and we denote that truth as 1. All of falsehood is also a single falsehood, which we call 0. All particular truths are distinct, and are to our best understanding quantized, discrete, at the fundamental level. These we may enumerate as our ingenuity allows.
Otherwise we shall speedily commit nonsense. "I have a fruit which does not exist", we say, "and it is called a Fnargle. There is another fruit, which also doesn't exist, and we call this the Argle-Bargle. The Fnargle is distinct in its qualities from the Argle-Bargle, as all men know, or at least suspect".
How could this claim be anything but false? Things which don't exist cannot be different in their qualities. On the other hand, things which do exist, must be different in their qualities, if they are different at all. An apple and an orange are different in all the familiar ways in which we may compare them, and even two apples from the same tree differ in the particulars of their shape, location, and the like.
We may give falsehood the shape of truth, and happily so, fiction is perhaps my greatest joy. But we have disposed of this condition once already. If I tell you that Argle-Bargles are never Fnargles, and later, that this Fnargle is also an Argle-Bargle, well. You might keep listening, I suppose. But I wouldn't blame you if you didn't.
This is because logic concerns itself with the truth, which derives from reality, which is the set of facts which happen to be the case. Therefore we properly represent False as 0, and Truth as 1, and all values greater. QED.
The alert reader may have noticed that my earlier claim about decomposition of the product of primes is perhaps optimistic. Given the prime clauses which make up the chain, we may easily verify a number to be the product of those primes. But given only the product, it can be quite challenging (absent a quantum computer capable of running Shor's Algorithm) to find the factors, if they are sufficiently large. This being the basis of public key cryptography, at least in its original form, we should be grateful it is so.