Changes

Part the task of the remaining discussion is gradually to formalize the promissory notes that are represented by these terms and stipulations and to see whether their casual comprehension can be converted into an explicit subject matter, one that depends on grasping the corresponding collection of almost wholly, if still partially formalized conceptions.  To this we now turn.

Part the task of the remaining discussion is gradually to formalize the promissory notes that are represented by these terms and stipulations and to see whether their casual comprehension can be converted into an explicit subject matter, one that depends on grasping the corresponding collection of almost wholly, if still partially formalized conceptions.  To this we now turn.

The ''binary domain'' is the set <math>\mathbb{B} = \{ 0, 1 \}</math> of two algebraic values, whose arithmetic follows the rules of <math>\operatorname{GF}(2).</math>

The ''binary domain'' is the set <math>\mathbb{B} = \{ 0, 1 \}</math> of two algebraic values, whose arithmetic operations obey the rules of <math>\operatorname{GF}(2),</math> 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 <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> of two logical values, whose elements can be read as "false" and "true", or as "falsity" and "truth", respectively.

The ''boolean domain'' is the set <math>\underline\mathbb{B} = \{ \underline{0}, \underline{1} \}</math> 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 roughly or abstractly 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 <math>\underline\mathbb{B}</math> 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 <math>\mathbb{B},</math> and operations that are isomorphic to the rest of the boolean operations in <math>\underline\mathbb{B}</math> can always be built on the binary basis of <math>\mathbb{B}.</math>

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 <math>\underline\mathbb{B}</math> 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 <math>\mathbb{B},</math> and operations that are isomorphic to the rest of the boolean operations in <math>\underline\mathbb{B}</math> can always be built on the binary basis of <math>\mathbb{B}.</math>

Of course, as sets of the same cardinality, the domains <math>\mathbb{B}</math> and <math>\underline\mathbb{B}</math> 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 present context appears to be a matter more of grammar than an issue of logical or mathematical substance, namely, so that the signs <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \underline{1} ^{\prime\prime}</math> can appear with some 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 <math>^{\backprime\backprime} 0 ^{\prime\prime}</math> and <math>^{\backprime\backprime} 1 ^{\prime\prime},</math> 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 that remains to be determined.

Of course, as sets of the same cardinality, the domains <math>\mathbb{B}</math> and <math>\underline\mathbb{B}</math> 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 present context appears to be a matter more of grammar than an issue of logical and mathematical substance, namely, so that the signs <math>^{\backprime\backprime} \underline{0} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \underline{1} ^{\prime\prime}</math> can appear with some 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 <math>^{\backprime\backprime} 0 ^{\prime\prime}</math> and <math>^{\backprime\backprime} 1 ^{\prime\prime},</math> 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 that remains to be determined.

The ''negation'' of a value <math>x\!</math> in <math>\underline\mathbb{B},</math> written <math>^{\backprime\backprime} \underline{(} x \underline{)} ^{\prime\prime}</math> or <math>^{\backprime\backprime} \lnot x ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} \operatorname{not}\ x ^{\prime\prime},</math> is the boolean value <math>\underline{(} x \underline{)} \in \underline\mathbb{B}</math> that is <math>\underline{1}</math> when <math>x\!</math> is <math>\underline{0}</math> and <math>\underline{0}</math> when <math>x\!</math> is <math>\underline{1}.</math>  Negation is a monadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \to \underline\mathbb{B},</math> as shown in Table&nbsp;8.

The ''negation'' of a value <math>x\!</math> in <math>\underline\mathbb{B},</math> written <math>^{\backprime\backprime} \underline{(} x \underline{)} ^{\prime\prime}</math> or <math>^{\backprime\backprime} \lnot x ^{\prime\prime}</math> and read as <math>^{\backprime\backprime} \operatorname{not}\ x ^{\prime\prime},</math> is the boolean value <math>\underline{(} x \underline{)} \in \underline\mathbb{B}</math> that is <math>\underline{1}</math> when <math>x\!</math> is <math>\underline{0}</math> and <math>\underline{0}</math> when <math>x\!</math> is <math>\underline{1}.</math>  Negation is a monadic operation on boolean values, that is, a function of the form <math>f : \underline\mathbb{B} \to \underline\mathbb{B},</math> as shown in Table&nbsp;8.