Logical 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, a binary truth functions 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 F, 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.
| p | ~p | |||
|---|---|---|---|---|
| T | T | F | T | F |
| F | T | T | F | F |
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 possible n-ary truth functions. The sixteen possible binary truth functions are listed in the table below.
| p | q | T | NAND | → | ~p | ← | ~q | ↔ | NOR | ∨ | XOR | q | N← | p | N→ | & | F |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| T | T | T | F | T | F | T | F | T | F | T | F | T | F | T | F | T | F |
| T | F | T | T | F | F | T | T | F | F | T | T | F | F | T | T | F | F |
| F | T | T | T | T | T | F | F | F | F | T | T | T | T | F | F | F | F |
| F | F | T | T | T | T | T | T | T | T | F | F | F | F | F | F | F | F |
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.
| Conventional Symbol | Matrix | Logic Alphabet Shape |
|---|---|---|
| T |  |
|
| NAND |  |
|
| → |  |
|
| ~p |  |
|
| ← |  |
|
| ~q |  |
|
| ↔ |  |
|
| NOR |  |
|
| ∨ |  |
|
| XOR |  |
|
| q |  |
|
| N← |  |
|
| p |  |
|
| N→ |  |
|
| & |  |
|
| F | |
|