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The "binary domain" is the set !B! = {!0!, !1!} of two algebraic values,
−whose arithmetic operations obey the rules of GF(2), the "galois field"
−of exactly two elements, whose addition and multiplication tables are
−tantamount to addition and multiplication of integers "modulo 2".
−The "boolean domain" is the set %B% = {%0%, %1%} of two logical values,
−whose elements are read as "false" and "true", or as "falsity" and "truth",
−respectively.
−At this point, I cannot tell whether the distinction between these two
−domains is slight or significant, and so this question must evolve its
−own answer, while I pursue a larger inquiry by means of its hypothesis.
−The weight of the matter appears to increase as the investigation moves
−from abstract, algebraic, and formal settings to contexts where logical
−semantics, natural language syntax, and concrete categories of grammar
−are compelling considerations. Speaking abstractly and roughly enough,
−it is often acceptable to identify these two domains, and up until this
−point there has rarely appeared to be a sufficient reason to keep their
−concepts separately in mind. The boolean domain %B% comes with at least
−two operations, though often under different names and always included
−in a number of others, that are analogous to the field operations of the
−binary domain !B!, and operations that are isomorphic to the rest of the
−boolean operations in %B% can always be built on the binary basis of !B!.
−Of course, as sets of the same cardinality, the domains !B! and %B%
−and all of the structures that can be built on them become isomorphic
−at a high enough level of abstraction. Consequently, the main reason
−for making this distinction in the immediate context appears to be more
−a matter of grammar than an issue of logical and mathematical substance,
−namely, so that the signs "%0%" and "%1%" can appear with a semblance of
−syntactic legitimacy in linguistic contexts that call for a grammatical
−sentence or a sentence surrogate to represent the classes of sentences
−that are "always false" and "always true", respectively. The signs
−"0" and "1", customarily read as nouns but not as sentences, fail
−to be suitable for this purpose. Whether these scruples, that are
−needed to conform to a particular choice of natural language context,
−are ultimately important, is another thing I do not know at this point.
−
<pre>
<pre>
−The "negation" of x, for x in %B%, written as "(x)"
The "negation" of x, for x in %B%, written as "(x)"
and read as "not x", is the boolean value (x) in %B%
and read as "not x", is the boolean value (x) in %B%
