Logic alphabet

The logic alphabet was developed by Shea Zellweger in the 1950s while working as a switchboard operator. It constitutes a novel set of symbols systematically representing the sixteen possible binary truth functions.

Truth functions
Truth functions are functions from sequences of truth values to truth values. A unary truth function, for example, takes a single truth value and maps it onto another truth value. Similarly, a binary truth function maps ordered pairs of truth values onto truth values, while a tenary truth function maps ordered triples of truth values onto truth values, and so on.

In the unary case, there are two possible inputs, viz. T and F, and thus four possible unary truth functions: one mapping T to T and F to F, one mapping T to F and F to F, one mapping T to T and F to T, and finally one mapping T to F and F to T, this last one corresponding to the familiar operation of logical negation. In the form of a table, the four unary truth functions may be represented as follows.

In the binary case, there are four possible inputs, viz. (T,T), (T,F), (F,T), and (F,F), thus yielding sixteen possible binary truth functions. Quite generally, for any number n, there are $$2^{2^n}$$ possible n-ary truth functions. The sixteen possible binary truth functions are listed in the table below.

The logic alphabet
Shea Zellweger's logic alphabet offers a systematic way of representing each of these sixteen binary truth functions. The idea behind the logic alphabet is to first represent the sixteen binary truth functions in the form of a square matrix rather than the more familiar tabular format seen in the table above, and then to assign a letter shape to each of these matrices on the basis of the distribution of 'T's in the matrix. The square matrix corresponding to each binary truth function, as well as its corresponding letter shape, are displayed in the table below.

Significance
The interest of the logic alphabet lies in its aesthetic qualities. For example, by reflecting the symbol for NAND (viz. 'h') across the vertical axis we produce the symbol for ←, whereas by reflecting it across the horizontal axis we produce the symbol for →, and by reflecting it across both the horizontal and vertical axes we produce the symbol for ∨. Similar geometrical transformation can be obtained by operating upon the other symbols. Indeed, Zellweger has constructed intriguing structures involving the symbols of the logic alphabet on the basis of these symmetries. The considerable aesthetic appeal of the logic alphabet has led to exhibitions of Zellweger's work at the Museum of Jurassic Technology in Los Angeles, among other places.

The value of the logic alphabet for the field of logic itself is questionable. Systems of natural deduction, for example, generally require introduction and elimination rules for each connective, meaning that the use of all sixteen binary connectives would result in a highly complex proof system. It should also be noted that various subsets of the sixteen binary connectives (e.g. {∨,&,→,~}, {∨,~}, {&, ~}, {→,~}) are themselves functionally complete in that they suffice to define the remaining connectives. In fact, both NAND and NOR are sole sufficient operators, meaning that the remaining connectives can all be defined solely in terms of either of them.