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