Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 40

ferent logical functions that are possible with N propositions (arguments).
Since we know that N two-valued arguments produce n in different states,
each of which again has two values, true or false, the total number of logical
functions is, with eq. (17):

2 2 N .

n

(18)

n

I ==

2

LF

For two arguments ( N = 2) we have precisely 16 logical functions.
Another symbolism in use is that proposed by Russell and Whitehead
(1925), and Carnap (1925) who employ the signs “•”, “v”, “Æ”, “-” for the
logical “and”, “or”, “implies”, “non” respectively. It can be shown that all
other logical functions can be represented by a combination of these
functions.
Finally, we wish to mention still another form for representing logical
functions, with the aid of a formalized Venn diagram. Venn, in 1881,
proposed to show the relation of classes by overlapping areas whose
various sections indicate joint or disjoint properties of these classes (fig. 11).
McCulloch and Pitts (1943) dropped the outer contours of these areas, using
only the center cross as lines of separation. Jots in the four spaces can rep-
resent all 16 logical functions. Some examples for single jots are given in
fig. 11. Expressions with two or more jots have to be interpreted as the
expressions with single jots connected by “or”. Hence,

·
·

= (

) ( )

“ neither nor or and AB AB ”,

which, of course, represents the proposition “ A is equivalent to B ”. The sim-
ilarity of this symbol with the greek letter chi suggested the name “chias-
tan” symbol. The advantage of this notation is that it can be extended to
accommodate logical functions of more than two arguments (Blum, 1962).

FIGURE 11. Development of the
Chiastan symbol for logical functions
from Venn's diagrams.