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What precedes this point is intended as an informal introduction to the axioms of the primary arithmetic and primary algebra, and hopefully provides the reader with an intuitive sense of their motivation and rationale.

What precedes this point is intended as an informal introduction to the axioms of the primary arithmetic and primary algebra, and hopefully provides the reader with an intuitive sense of their motivation and rationale.

The next order of business is to give the exact forms of the axioms that are used in the following more formal development, devolving from Peirce's various systems of logical graphs via Spencer-Brown's ''Laws of Form'' (LOF).  In formal proofs, a variation of the annotation scheme from LOF will be used to mark each step of the proof according to which axiom, or 'initial', is being invoked to justify the corresponding step of syntactic transformation, whether it applies to graphs or to strings.

The next order of business is to give the exact forms of the axioms that are used in the following more formal development, devolving from Peirce's various systems of logical graphs via Spencer-Brown's ''Laws of Form'' (LOF).  In formal proofs, a variation of the annotation scheme from LOF will be used to mark each step of the proof according to which axiom, or ''initial'', is being invoked to justify the corresponding step of syntactic transformation, whether it applies to graphs or to strings.

===Axioms===

===Axioms===

The axioms are just four in number, divided into the ''arithmetic initials'' I₁ and I₂, and the ''algebraic initials'' J₁ and J₂.

The axioms are just four in number, divided into the ''arithmetic initials'', <math>I_1\!</math> and <math>I_2,\!</math> and the ''algebraic initials'', <math>J_1\!</math> and <math>J_2.\!</math>

{| align="center" border="0" cellpadding="10" cellspacing="0"

| [[Image:Logical_Graph_Figure_20.jpg|500px]] || (20)

|                                                           |

|-

| [[Image:Logical_Graph_Figure_21.jpg|500px]] || (21)

|-

|                  @        =         @                  |

| [[Image:Logical_Graph_Figure_22.jpg|500px]] || (22)

|                                                           |

|-

| [[Image:Logical_Graph_Figure_23.jpg|500px]] || (23)

|                                                           |

|}

|               ( ) ( )     =        ( )                  |

|                                                           |

o-----------------------------------------------------------o

| Axiom I_1.   Distract <--- | ---> Condense              |

 

|                                                           |

|                (( ))      =                            |

|                                                           |

o-----------------------------------------------------------o

| Axiom I_2.     Unfold <--- | ---> Refold                |

 

|                                                           |

|                 a o                  o                  |

|                a(a)        =        ( )                 |

|                                                           |

o-----------------------------------------------------------o

| Axiom J_1.     Insert <--- | ---> Delete                |

 

|                                                           |

|              ((ab)(ac))     =    a((b)(c))              |

| Axiom J_2.  Distribute <--- | ---> Collect                |

Here is one way of reading the axioms under the entitative interpretation:

Here is one way of reading the axioms under the entitative interpretation: