Editing Laws of Form (section)

===Interpretations===
If the Marked and Unmarked states are read as the [[two-element Boolean algebra|Boolean]] values 1 and 0 (or '''True''' and '''False'''), the ''primary algebra'' [[interpretation (logic)|interprets]] '''[[two-element Boolean algebra|2]]''' (or [[sentential logic]]). ''LoF'' shows how the ''primary algebra'' can interpret the [[syllogism]]. Each of these [[interpretation (logic)|interpretations]] is discussed in a subsection below. Extending the ''primary algebra'' so that it could [[interpretation (logic)|interpret]] standard [[first-order logic]] has yet to be done, but [[Charles Sanders Peirce|Peirce]]'s ''beta'' [[existential graph]]s suggest that this extension is feasible.

====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.

====Sentential logic====
Let the blank page denote '''False''', and let a Cross be read as '''Not'''. Then the primary arithmetic has the following sentential reading:

:::&nbsp;= &nbsp; '''False'''

::[[Image:Laws of Form - cross.gif]] &nbsp;= &nbsp;'''True''' &nbsp;= &nbsp;'''not False'''

::[[Image:Laws of Form - double cross.gif]] &nbsp;= &nbsp;'''Not True''' &nbsp;= &nbsp;'''False'''

The ''primary algebra'' interprets sentential logic as follows. A letter represents any given sentential expression. Thus:

::[[Image:Laws of Form - not a.gif]] interprets '''Not A'''

::[[Image:Laws of Form - a or b.gif]] interprets '''A Or B'''

::[[Image:Laws of Form - if a then b.gif]] interprets '''Not A Or B'''  or   '''If A Then B'''.

::[[Image:Laws of Form - a and b.gif]] interprets '''Not (Not A Or Not B)'''
:::::or '''Not (If A Then Not B)'''
:::::or '''A And B'''.

{|
| &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
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|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <big><big><big>a</big></big></big>
|}
| <big><big><big>b</big></big></big>
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <big><big><big>a</big></big></big>
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <big><big><big>b</big></big></big>
|}
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|,
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <big><big><big>a</big></big></big>
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <big><big><big>b</big></big></big>
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|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <big><big><big>a&nbsp;b</big></big></big>
|}
|}
| both interpret '''A [[if and only if]] B''' or '''A is [[logical equivalence|equivalent]] to B'''.
|}

Thus any expression in [[sentential logic]] has a ''primary algebra'' translation. Equivalently, the ''primary algebra'' [[interpretation (logic)|interprets]] sentential logic. Given an assignment of every variable to the Marked or Unmarked states, this ''primary algebra'' translation reduces to a primary arithmetic expression, which can be simplified. Repeating this exercise for all possible assignments of the two primitive values to each variable, reveals whether the original expression is [[Tautology (logic)|tautological]] or [[Satisfiability|satisfiable]]. This is an example of a [[decision procedure]], one more or less in the spirit of conventional truth tables. Given some ''primary algebra'' formula containing ''N'' variables, this decision procedure requires simplifying 2<sup>''N''</sup> primary arithmetic formulae. For a less tedious decision procedure more in the spirit of [[Willard Van Orman Quine|Quine]]'s "truth value analysis", see {{harvp|Meguire|2003}}.

{{harvp|Schwartz|1981}} proved that the ''primary algebra'' is equivalent — [[syntax|syntactically]], [[Semantics of logic|semantically]], and [[proof theory|proof theoretically]] — with the [[Propositional calculus|classical propositional calculus]]. Likewise, it can be shown that the ''primary algebra'' is syntactically equivalent with expressions built up in the usual way from the classical [[truth value]]s '''true''' and '''false''', the [[logical connective]]s NOT, OR, and AND, and parentheses.

Interpreting the Unmarked State as '''False''' is wholly arbitrary; that state can equally well be read as '''True'''. All that is required is that the interpretation of [[concatenation]] change from OR to AND. IF A THEN B now translates as [[Image:Laws of Form - (A(B)).png|50px]] instead of [[Image:Laws of Form - (A)B.png|50px]]. More generally, the ''primary algebra'' is "self-[[Duality (mathematics)|dual]]", meaning that any ''primary algebra'' formula has two [[sentential logic|sentential]] or [[two-element Boolean algebra|Boolean]] readings, each the [[Duality (mathematics)|dual]] of the other. Another consequence of self-duality is the irrelevance of [[De Morgan's laws]]; those laws are built into the syntax of the ''primary algebra'' from the outset.

The true nature of the distinction between the ''primary algebra'' on the one hand, and '''2''' and sentential logic on the other, now emerges. In the latter formalisms, [[Logical complement|complementation]]/[[negation]] operating on "nothing" is not well-formed. But an empty Cross is a well-formed ''primary algebra'' expression, denoting the Marked state, a primitive value. Hence a nonempty Cross is an [[Operator (mathematics)|operator]], while an empty Cross is an [[operand]] because it denotes a primitive value. Thus the ''primary algebra'' reveals that the heretofore distinct mathematical concepts of operator and operand are in fact merely different facets of a single fundamental action, the making of a distinction.

====Syllogisms====
Appendix 2 of ''LoF'' shows how to translate traditional [[syllogism]]s and [[polysyllogism|sorites]] into the ''primary algebra''. A valid syllogism is simply one whose ''primary algebra'' translation simplifies to an empty Cross. Let ''A''* denote a ''literal'', i.e., either ''A'' or <math>\overline{A |}</math>, indifferently. Then every syllogism that does not require that one or more terms be assumed nonempty is one of 24 possible permutations of a generalization of [[syllogism|Barbara]] whose ''primary algebra'' equivalent is <math>\overline{A^* \ B |} \ \ \overline{\overline{B |} \ C^* \Big|} \ A^* \ C^* </math>. These 24 possible permutations include the 19 syllogistic forms deemed valid in [[Aristotelian logic|Aristotelian]] and [[medieval logic]]. This ''primary algebra'' translation of syllogistic logic also suggests that the ''primary algebra'' can [[interpretation (logic)|interpret]] [[monadic logic|monadic]] and [[term logic]], and that the ''primary algebra'' has affinities to the [[Boolean term schema]]ta of {{harvp|Quine|1982|loc=Part II}}.