Editing Laws of Form (section)

==The primary algebra (Chapter 6)==

===Syntax===
Given any valid primary arithmetic expression, insert into one or more locations any number of Latin letters bearing optional numerical subscripts; the result is a primary algebra [[formula]]. Letters so employed in [[mathematics]] and [[logic]] are called [[Variable (mathematics)|variables]]. A primary algebra variable indicates a location where one can write the primitive value [[Image:Laws of Form - cross.gif]] or its complement [[Image:Laws of Form - double cross.gif]]. Multiple instances of the same variable denote multiple locations of the same primitive value.

===Rules governing logical equivalence===
The sign '=' may link two logically equivalent expressions; the result is an [[equation]]. By "logically equivalent" is meant that the two expressions have the same simplification. [[Logical equivalence]] is an [[equivalence relation]] over the set of primary algebra formulas, governed by the rules R1 and R2. Let "C" and "D" be formulae each containing at least one instance of the subformula ''A'':
*'''R1''', ''Substitution of equals''. Replace ''one or more'' instances of ''A'' in ''C'' by ''B'', resulting in ''E''. If ''A''=''B'', then ''C''=''E''.
*'''R2''', ''Uniform replacement''. Replace ''all'' instances of ''A'' in ''C'' and ''D'' with ''B''. ''C'' becomes ''E'' and ''D'' becomes ''F''. If ''C''=''D'', then ''E''=''F''. Note that ''A''=''B'' is not required.
'''R2''' is employed very frequently in ''primary algebra'' demonstrations (see below), almost always silently. These rules are routinely invoked in [[logic]] and most of mathematics, nearly always unconsciously.

The ''primary algebra'' consists of [[equations]], i.e., pairs of formulae linked by an [[infix operator]] '='. '''R1''' and '''R2''' enable transforming one equation into another. Hence the ''primary algebra'' is an ''equational''  formal system, like the many [[algebraic structures]], including [[Boolean algebra (structure)|Boolean algebra]], that are [[variety (universal algebra)|varieties]]. Equational logic was common before ''Principia Mathematica'' (e.g. {{harvp|Johnson|1892}}), and has present-day advocates ({{harvp|Gries|Schneider|1993}}).

Conventional [[mathematical logic]] consists of [[Tautology (logic)|tautological]] formulae, signalled by a prefixed [[Turnstile (symbol)|turnstile]]. To denote that the ''primary algebra'' formula ''A'' is a [[Tautology (logic)|tautology]], simply write "''A'' =[[Image:Laws of Form - cross.gif]] ". If one replaces '=' in '''R1''' and '''R2''' with the [[biconditional]], the resulting rules hold in conventional logic. However, conventional logic relies mainly on the rule [[modus ponens]]; thus conventional logic is ''ponential''. The equational-ponential dichotomy distills much of what distinguishes mathematical logic from the rest of mathematics.

===Initials===
An ''initial'' is a ''primary algebra'' equation verifiable by a [[decision procedure]] and as such is ''not'' an [[axiom]]. ''LoF'' lays down the initials:

{|
|-
| 
*J1:
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
|
{|
|-
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| A
|}
| A
|}
|}
| = .
|}
The absence of anything to the right of the "=" above, is deliberate.

{|
|-
| 
*J2:
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
|
{|
|-
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| A
|}
|
{|style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| B
|}
|}
|}
| C
| =
|
{|style="border-top: 2px solid black; border-right: 2px solid black;"
|-
|
{|
|-
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| A C
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| B C
|}
|}
|}
|.
|}

'''J2''' is the familiar [[distributive law]] of [[sentential logic]] and [[Boolean algebra (structure)|Boolean algebra]].

Another set of initials, friendlier to calculations, is:

{|
|-
| 
*J0:
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
|
{|
|-
|
|}
|}
|}
| A
| =
| A.
|}

{|
|-
|
*J1a:
|
{|
|-
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| A
|}
| A
|}
| =
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
|
{|
|-
|
|}
|}
|.
|}

{|
|-
| 
*C2:
| A
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| A B
|}
| = 
| A
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|-
| B
|}
|.
|}

It is thanks to '''C2''' that the ''primary algebra'' is a [[lattice (order)|lattice]]. By virtue of '''J1a''', it is a [[complemented lattice]] whose upper bound is [[Image:Laws_of_Form_-_cross.gif]]. By '''J0''', [[Image:Laws_of_Form_-_double_cross.gif]] is the corresponding lower bound and [[identity element]]. '''J0''' is also an algebraic version of '''A2''' and makes clear the sense in which [[Image:Laws_of_Form_-_double_cross.gif]] aliases with the blank page.

T13 in ''LoF'' generalizes '''C2''' as follows. Any ''primary algebra'' (or sentential logic) formula ''B'' can be viewed as an [[ordered tree]] with ''branches''. Then:

'''T13''': A [[formula|subformula]] ''A'' can be copied at will into any depth of ''B'' greater than that of ''A'', as long as ''A'' and its copy are in the same branch of ''B''. Also, given multiple instances of ''A'' in the same branch of ''B'', all instances but the shallowest are redundant.

While a proof of T13 would require [[mathematical induction|induction]], the intuition underlying it should be clear.

'''C2''' or its equivalent is named:
*"Generation" in ''LoF'';
*"Exclusion" in Johnson (1892);
*"Pervasion" in the work of William Bricken.
Perhaps the first instance of an axiom or rule with the power of '''C2''' was the "Rule of (De)Iteration", combining T13 and ''AA=A'', of [[Charles Sanders Peirce|C. S. Peirce]]'s [[existential graph]]s.

''LoF'' asserts that concatenation can be read as [[commutativity|commuting]] and [[associativity|associating]] by default and hence need not be explicitly assumed or demonstrated. (Peirce made a similar assertion about his [[existential graph]]s.) Let a period be a temporary notation to establish grouping. That concatenation commutes and associates may then be demonstrated from the:
* Initial ''AC.D''=''CD.A'' and the consequence ''AA''=''A''.{{sfnp|Byrne|1946}} This result holds for all [[lattice (order)|lattices]], because ''AA''=''A'' is an easy consequence of the [[absorption law]], which holds for all lattices;
* Initials ''AC.D''=''AD.C'' and '''J0'''. Since '''J0''' holds only for lattices with a lower bound, this method holds only for [[bounded lattice]]s (which include the ''primary algebra'' and '''2'''). Commutativity is trivial; just set ''A''=[[Image:Laws_of_Form_-_double_cross.gif]]. Associativity: ''AC.D'' = ''CA.D'' = ''CD.A'' = ''A.CD''.
Having demonstrated associativity, the period can be discarded.

The initials in {{harvp|Meguire|2011}} are ''AC.D''=''CD.A'', called '''B1'''; '''B2''', J0 above; '''B3''', J1a above; and '''B4''', C2. By design, these initials are very similar to the axioms for an [[abelian group]], '''G1-G3''' below.

===Proof theory===
The ''primary algebra'' contains three kinds of proved assertions:
* ''Consequence'' is a ''primary algebra'' equation verified by a ''demonstration''. A demonstration consists of a sequence of ''steps'', each step justified by an initial or a previously demonstrated consequence.
* ''[[Theorem]]'' is a statement in the [[metalanguage]] verified by a ''[[Mathematical proof|proof]]'', i.e., an argument, formulated in the metalanguage, that is accepted by trained mathematicians and logicians.
* ''Initial'', defined above. Demonstrations and proofs invoke an initial as if it were an axiom.

The distinction between consequence and [[theorem]] holds for all formal systems, including mathematics and logic, but is usually not made explicit. A demonstration or [[decision procedure]] can be carried out and verified by computer. The [[Mathematical proof|proof]] of a [[theorem]] cannot be.

Let ''A'' and ''B'' be ''primary algebra'' [[formula]]s. A demonstration of ''A''=''B'' may proceed in either of two ways:
* Modify ''A'' in steps until ''B'' is obtained, or vice versa;
* Simplify both [[Image:Laws of Form - (A)B.png|50px]] and [[Image:Laws of Form - (B)A.png|50px]] to [[Image:Laws of Form - cross.gif]]. This is known as a "calculation".
Once ''A''=''B'' has been demonstrated, ''A''=''B'' can be invoked to justify steps in subsequent demonstrations. ''primary algebra'' demonstrations and calculations often require no more than '''J1a''', '''J2''', '''C2''', and the consequences [[Image:Laws of Form - ()A=().png|80px]] ('''C3''' in ''LoF''), [[Image:Laws of Form - ((A))=A.png|80px]] ('''C1'''), and ''AA''=''A'' ('''C5''').

The consequence [[Image:Laws of Form - (((A)B)C)=(AC)((B)C).png|170px]], '''C7''' in ''LoF'', enables an [[algorithm]], sketched in ''LoF'''s proof of T14, that transforms an arbitrary ''primary algebra'' formula to an equivalent formula whose depth does not exceed two. The result is a ''normal form'', the ''primary algebra'' analog of the [[conjunctive normal form]]. ''LoF'' (T14–15) proves the ''primary algebra'' analog of the well-known [[Boolean algebra (logic)|Boolean algebra]] theorem that every formula has a normal form.

Let ''A'' be a [[formula|subformula]] of some [[formula]] ''B''. When paired with '''C3''', '''J1a''' can be viewed as the closure condition for calculations: ''B'' is a [[Tautology (logic)|tautology]] [[if and only if]] ''A'' and (''A'') both appear in depth 0 of ''B''. A related condition appears in some versions of [[natural deduction]]. A demonstration by calculation is often little more than:
* Invoking T13 repeatedly to eliminate redundant subformulae;
* Erasing any subformulae having the form [[Image:Laws of Form - ((A)A).png|50px]].
The last step of a calculation always invokes '''J1a'''.

''LoF'' includes elegant new proofs of the following standard [[metatheory]]:
* ''[[Completeness (logic)|Completeness]]'': all ''primary algebra'' consequences are demonstrable from the initials (T17).
* ''[[axiom|Independence]]'': '''J1''' cannot be demonstrated from '''J2''' and vice versa (T18).
That [[sentential logic]] is complete is taught in every first university course in [[mathematical logic]]. But university courses in Boolean algebra seldom mention the completeness of '''2'''.

===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;
|
{| 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.

====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}}.

===An example of calculation===
The following calculation of [[Gottfried Wilhelm Leibniz|Leibniz]]'s nontrivial ''Praeclarum Theorema'' exemplifies the demonstrative power of the ''primary algebra''. Let C1 be <math>\overline{\overline{A |} \Big|}</math> =''A'', C2 be <math>A \ \overline{A \ B |} = A \ \overline{B |}</math>, C3 be <math>\overline{\ \ |} \ A = \overline{\ \ |}</math>, J1a be <math>\overline{A |} \ A = \overline{\ \ |}</math>, and let OI mean that variables and subformulae have been reordered in a way that commutativity and associativity permit.

{|
| [(''P''→''R'')∧(''Q''→''S'')]→[(''P''∧''Q'')→(''R''∧''S'')].
| ''Praeclarum Theorema''.
|-
|
{|
|
{| 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;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
| S
|}
|}
|}
|
{| 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;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|}
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|}
| .
|}
| ''primary algebra'' translation
|-
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
| S
|}
|
{| 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;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|}
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|}
| .
|}
| C1.
|-
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
| S
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|}
| .
|}
| C1.
|-
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
| S
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|}
| .
|}
| OI.
|-
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
| S
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|}
| .
|}
| C2.
|-
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
| S
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|}
| .
|}
| OI.
|-
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|}
| .
|}
| C2.
|-
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|}
| .
|}
| OI.
|-
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|}
| .
|}
| C2.
|-
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
| S
| .
|}
| C1.
|-
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| S
|}
| S
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
| .
|}
| OI.
|-
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <span style="color:white;">B</span>
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
| .
|}
| J1a.
|-
|
{|
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <span style="color:white;">B</span>
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| P
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| Q
|}
|
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| R
|}
|.
|}
| OI.
|-
| 
{| style="border-top: 2px solid black; border-right: 2px solid black;"
| <span style="color:white;">B</span>
|}
| C3. <math>\square</math>
|}

===Relation to magmas===
The ''primary algebra'' embodies a point noted by [[Edward Vermilye Huntington|Huntington]] in 1933: [[Boolean algebra (logic)|Boolean algebra]] requires, in addition to one [[unary operation]], one, and not two, [[binary operation]]s. Hence the seldom-noted fact that Boolean algebras are [[magma (algebra)|magmas]]. (Magmas were called [[groupoid]]s until the latter term was appropriated by [[category theory]].) To see this, note that the ''primary algebra'' is a [[commutative]]:
*[[Semigroup]] because ''primary algebra'' juxtaposition [[Commutative property|commute]]s and [[associative property|associates]];
*[[Monoid]] with [[identity element]] [[Image:Laws of Form - double cross.gif]], by virtue of '''J0'''.

[[group (mathematics)|Groups]] also require a [[unary operation]], called [[Group (mathematics)#Definition|inverse]], the group counterpart of [[Boolean algebra (logic)|Boolean complementation]]. Let [[Image:Laws of Form - (a).png|20px]] denote the inverse of ''a''. Let [[Image:Laws of Form - cross.gif]] denote the group [[identity element]]. Then groups and the ''primary algebra'' have the same [[signature (logic)|signatures]], namely they are both <math>\lang - \ -, \overline{- \ |}, \overline{\ \ |} \rang</math> algebras of type 〈2,1,0〉. Hence the ''primary algebra'' is a [[list of algebraic structures|boundary algebra]]. The axioms for an [[abelian group]], in boundary notation, are:
* '''G1'''. ''abc'' = ''acb'' (assuming association from the left);
* '''G2'''. [[Image:Laws of Form - ()a=a.png|80px]]
* '''G3'''. [[Image:Laws of Form - (a)a=().png|80px]].
From '''G1''' and '''G2''', the commutativity and associativity of concatenation may be derived, as above. Note that '''G3''' and '''J1a''' are identical. '''G2''' and '''J0''' would be identical if &nbsp;&nbsp;[[Image:Laws of Form - double cross.gif|25px]]&nbsp;=&nbsp;[[Image:Laws of Form - cross.gif|20px]]&nbsp;&nbsp; replaced '''A2'''. This is the defining arithmetical identity of group theory, in boundary notation.

The ''primary algebra'' differs from an [[abelian group]] in two ways:
*From '''A2''', it follows that [[Image:Laws of Form - double cross.gif]] ≠ [[Image:Laws of Form - cross.gif]]. If the ''primary algebra'' were a [[group (mathematics)|group]], [[Image:Laws of Form - double cross.gif]] = [[Image:Laws of Form - cross.gif]] would hold, and one of &nbsp;&nbsp;[[Image:Laws of Form - (a).png|20px]]&nbsp;''a''&nbsp;=&nbsp;[[Image:Laws of Form - double cross.gif|30px]]&nbsp;&nbsp; or &nbsp;&nbsp;''a''&nbsp;[[Image:Laws of Form - cross.gif|30px]]&nbsp;=&nbsp;''a''&nbsp;&nbsp; would have to be a ''primary algebra'' consequence. Note that [[Image:Laws of Form - cross.gif|20px]] and [[Image:Laws of Form - double cross.gif|25px]] are mutual ''primary algebra'' complements, as group theory requires, so that <math>\overline{\ \overline{\ \overline{\ \ |} \ \Big|} \ \Bigg|} = \overline{\ \ |}</math> is true of both group theory and the ''primary algebra'';
*'''C2''' most clearly demarcates the ''primary algebra'' from other magmas, because '''C2''' enables demonstrating the [[absorption law]] that defines [[lattice (order)|lattices]], and the [[distributive law]] central to [[Boolean algebra (structure)|Boolean algebra]].
Both '''A2''' and '''C2''' follow from ''B''{{'}}s being an [[ordered set]].