Editing Laws of Form (section)

==The primary arithmetic (Chapter 4)==
The [[syntax]] of the primary arithmetic goes as follows. There are just two [[atomic formula|atomic expressions]]:
* The empty Cross [[Image:Laws of Form - cross.gif]] ;
* All or part of the blank page (the "void").
There are two inductive rules:
* A Cross [[Image:Laws of Form - cross.gif]] may be written over any expression;
* Any two expressions may be [[concatenation|concatenated]].
The [[Semantics of logic|semantics]] of the primary arithmetic are perhaps nothing more than the sole explicit [[definition]] in ''LoF'': "Distinction is perfect continence".

Let the "unmarked state" be a synonym for the void. Let an empty Cross denote the "marked state". To cross is to move from one value, the unmarked or marked state, to the other. We can now state the "arithmetical" [[axiom]]s A1 and A2, which ground the primary arithmetic (and hence all of the Laws of Form):

"A1. The law of Calling". Calling twice from a state is indistinguishable from calling once. To make a distinction twice has the same effect as making it once. For example, saying "Let there be light" and then saying "Let there be light" again, is the same as saying it once.  Formally:

::[[Image:Laws of Form - cross.gif]] [[Image:Laws of Form - cross.gif]] <math>\ =</math>[[Image:Laws of Form - cross.gif]]

"A2. The law of Crossing". After crossing from the unmarked to the marked state, crossing again ("recrossing") starting from the marked state returns one to the unmarked state. Hence recrossing annuls crossing. Formally:

::[[Image:Laws of Form - double cross.gif]] <math>\ =</math>

In both A1 and A2, the expression to the right of '=' has fewer symbols than the expression to the left of '='. This suggests that every primary arithmetic expression can, by repeated application of A1 and A2, be ''simplified'' to one of two states: the marked or the unmarked state. This is indeed the case, and the result is the expression's "simplification". The two fundamental metatheorems of the primary arithmetic state that:
* Every finite expression has a unique simplification. (T3 in ''LoF'');
* Starting from an initial marked or unmarked state, "complicating" an expression by a finite number of repeated application of A1 and A2 cannot yield an expression whose simplification differs from the initial state. (T4 in ''LoF'').
Thus the [[relation (mathematics)|relation]] of [[logical equivalence]] [[partition of a set|partitions]] all primary arithmetic expressions into two [[equivalence class]]es: those that simplify to the Cross, and those that simplify to the void.

A1 and A2 have loose analogs in the properties of series and parallel electrical circuits, and in other ways of diagramming processes, including flowcharting. A1 corresponds to a parallel connection and A2 to a series connection, with the understanding that making a distinction corresponds to changing how two points in a circuit are connected, and not simply to adding wiring.

The primary arithmetic is analogous to the following formal languages from [[mathematics]] and [[computer science]]:
* A [[Dyck language]] with a null alphabet;
* The simplest [[context-free language]] in the [[Chomsky hierarchy]];
* A [[rewrite system]] that is [[strongly normalizing]] and [[confluence (abstract rewriting)|confluent]].

The phrase "calculus of indications" in ''LoF'' is a synonym for "primary arithmetic".

===The notion of canon===
While ''LoF'' does not formally define canon, the following two excerpts from the Notes to chpt. 2 are apt:

<blockquote>The more important structures of command are sometimes called ''canons''. They are the ways in which the guiding injunctions appear to group themselves in constellations, and are thus by no means independent of each other. A canon bears the distinction of being outside (i.e., describing) the system under construction, but a command to construct (e.g., 'draw a distinction'), even though it may be of central importance, is not a canon. A canon is an order, or set of orders, to permit or allow, but not to construct or create.</blockquote>

<blockquote>...the primary form of mathematical communication is not description but injunction... Music is a similar art form, the composer does not even attempt to describe the set of sounds he has in mind, much less the set of feelings occasioned through them, but writes down a set of commands which, if they are obeyed by the performer, can result in a reproduction, to the listener, of the composer's original experience.</blockquote>

These excerpts relate to the distinction in [[metalogic]] between the object language, the formal language of the logical system under discussion, and the [[metalanguage]], a language (often a natural language) distinct from the object language, employed to exposit and discuss the object language. The first quote seems to assert that the ''canons'' are part of the metalanguage. The second quote seems to assert that statements in the object language are essentially commands addressed to the reader by the author. Neither assertion holds in standard metalogic.