Changes

Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following:

Our observer might think to summarize the results of many such observations by introducing a label or variable to signify any shape of branch whatever, writing something like the following:

{| align="center" cellpadding="10"

| [[Image:Logical_Graph_Figure_17.jpg|500px]]

|}

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

o===========================================================o

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` ` ` ` ` |

Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra.

Observations like that, made about an arithmetic of any variety, germinated by their summarizations, are the root of all algebra.

Speaking of algebra, and having seen one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from Charles S. Peirce and G. Spencer Brown, respectively.

Speaking of algebra, and having seen one example of an algebraic law, we might as well introduce the axioms of the ''primary algebra'', once again deriving their substance and their name from Charles S. Peirce and G. Spencer Brown, respectively.

{| align="center" cellpadding="10"

| [[Image:Logical_Graph_Figure_18.jpg|500px]]

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

|-

| [[Image:Logical_Graph_Figure_19.jpg|500px]]

|}

| ` ` ` ` ` ` ` ` a @ ` ` ` ` = ` ` ` ` @ ` ` ` ` ` ` ` ` ` |

| ` ` ` ` ` ` ` ` a(a)` ` ` ` = ` ` ` `( )` ` ` ` ` ` ` ` ` |

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

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

| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |

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

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

The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems.  As it happens, the example that we noticed first, as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.

The choice of axioms for any formal system is to some degree a matter of aesthetics, as it is commonly the case that many different selections of formal rules will serve as axioms to derive all the rest as theorems.  As it happens, the example that we noticed first, as simple as it appears, proves to be provable as a theorem on the grounds of the foregoing axioms.