Laws of Form - The Primary Arithmetic (Chapter 4)

The Primary Arithmetic (Chapter 4)

The syntax of the primary arithmetic (PA) goes as follows. There are just two atomic expressions:

• The empty Cross ;
• All or part of the blank page (the "void").

There are two inductive rules:

• A Cross may be written over any expression;
• Any two expressions may be concatenated.

The 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 of the unmarked or marked states to the other. We can now state the "arithmetical" axioms 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:

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:

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 of logical equivalence partitions all primary arithmetic expressions into two equivalence classes: 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 of order 1 with a null alphabet;
• The simplest context-free language in the Chomsky hierarchy;
• A rewrite system that is strongly normalizing and confluent.

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