Editing Laws of Form (section)

==Equations of the second degree (Chapter 11)==
Chapter 11 of ''LoF'' introduces ''equations of the second degree'', composed of [[recursion|recursive]] formulae that can be seen as having "infinite" depth. Some recursive formulae simplify to the marked or unmarked state. Others "oscillate" indefinitely between the two states depending on whether a given depth is even or odd. Specifically, certain recursive formulae can be interpreted as oscillating between '''true''' and '''false''' over successive intervals of time, in which case a formula is deemed to have an "imaginary" truth value. Thus the flow of time may be introduced into the ''primary algebra''.

{{harvp|Turney|1986}} shows how these recursive formulae can be interpreted via [[Alonzo Church]]'s Restricted Recursive Arithmetic (RRA). Church introduced RRA in 1955 as an axiomatic formalization of [[finite automata]]. Turney presents a general method for translating equations of the second degree into Church's RRA, illustrating his method using the formulae '''E1''', '''E2''', and '''E4''' in chapter 11 of ''LoF''. This translation into RRA sheds light on the names Spencer-Brown gave to '''E1''' and '''E4''', namely "memory" and "counter". RRA thus formalizes and clarifies ''LoF''{{'}}s notion of an imaginary truth value.