Editing Laws of Form (section)

====Two-element Boolean algebra 2====
The ''primary algebra'' is an elegant minimalist notation for the [[two-element Boolean algebra]] '''2'''. Let:
* One of Boolean [[join (mathematics)|join]] (+) or [[meet (mathematics)|meet]] (×) interpret [[concatenation]];
* The [[Complement (order theory)|complement]] of ''A'' interpret [[Image:Laws of Form - not a.gif]]
* 0 (1) interpret the empty Mark if join (meet) interprets [[concatenation]] (because a binary operation applied to zero operands may be regarded as being equal to the [[identity element]] of that operation; or to put it in another way, an operand that is missing could be regarded as acting by default like the identity element). 
If join (meet) interprets ''AC'', then meet (join) interprets <math>\overline{\overline{A |} \ \ \overline{C |}  \Big|}</math>. Hence the ''primary algebra'' and '''2''' are isomorphic but for one detail: ''primary algebra'' complementation can be nullary, in which case it denotes a primitive value. Modulo this detail, '''2''' is a [[model theory|model]] of the primary algebra. The primary arithmetic suggests the following arithmetic axiomatization of '''2''': 1+1=1+0=0+1=1=~0, and 0+0=0=~1.

The [[Set (mathematics)|set]] <math>\ B=\{</math>[[Image:Laws of Form - cross.gif]] <math>,</math> [[Image:Laws of Form - double cross.gif]]<math>\ \}</math> is the [[Boolean domain]] or ''carrier''. In the language of [[universal algebra]], the ''primary algebra'' is the [[algebraic structure]] <math>\lang B,-\ -,\overline{- \ |},\overline{\ \ |} \rang</math> of type <math>\lang 2,1,0 \rang</math>. The [[functional completeness|expressive adequacy]] of the [[Sheffer stroke]] points to the ''primary algebra'' also being a <math>\lang B,\overline{-\ - \ |},\overline{\ \ |}\rang</math> algebra of type <math>\lang 2,0 \rang</math>. In both cases, the identities are J1a, J0, C2, and ''ACD=CDA''. Since the ''primary algebra'' and '''2''' are [[isomorphic]], '''2''' can be seen as a <math>\lang B,+,\lnot,1 \rang</math> algebra of type <math>\lang 2,1,0 \rang</math>. This description of '''2''' is simpler than the conventional one, namely an <math>\lang B,+,\times,\lnot,1,0 \rang</math> algebra of type <math>\lang 2,2,1,0,0 \rang</math>.

The two possible interpretations are dual to each other in the Boolean sense. (In Boolean algebra, exchanging AND ↔ OR and 1 ↔ 0 throughout an equation yields an equally valid equation.) The identities remain invariant regardless of which interpretation is chosen, so the transformations or modes of calculation remain the same; only the interpretation of each form would be different. Example: J1a is [[Image:Laws of Form - (A)A=().png|80px]]. Interpreting juxtaposition as OR and [[Image:Laws of Form - cross.gif|30px]] as 1, this translates to <math>\neg A \lor A = 1</math> which is true. Interpreting juxtaposition as AND and [[Image:Laws of Form - cross.gif|30px]] as 0, this translates to <math>\neg A \land A = 0</math> which is true as well (and the dual of <math>\neg A \lor A = 1</math>).

===== operator-operand duality =====
The marked state, [[Image:Laws of Form - cross.gif]], is both an operator (e.g., the complement) and operand (e.g., the value 1). This can be summarized neatly by defining two functions <math>m(x)</math> and <math>u(x)</math> for the marked and unmarked state, respectively: let <math>m(x) = 1-\max(\{0\}\cup x)</math> and <math>u(x) = \max(\{0\} \cup x)</math>, where <math>x</math> is a (possibly empty) set of boolean values.

This reveals that <math>u</math> is either the value 0 or the OR operator, while <math>m</math> is either the value 1 or the NOR operator, depending on whether <math>x</math> is the empty set or not. As noted above, there is a dual form of these functions exchanging AND ↔ OR and 1 ↔ 0.