Editing Laws of Form (section)

====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;
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| 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>
|}
| <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>
|}
|}
|}
|,
|
{|
|
{| 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>
|}
|}
|
{| 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.