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FIGURE 10. Threshold defining the function com-
puted by a McCulloch formal neuron.
Returning to our example, we now tabulate, for a particular threshold
value q, the output F
q
(
A
,
B
) which has been computed by this element for
all input states, i.e., all combinations of
A
,
B
belong ON or OFF. For zero
threshold (q=0), this element will always be in its ON state, hence, in the
column q=0 we insert “one” only. We raise the threshold one unit (q=1)
and observe that our element will fire only if either
A
or
B
, or both
A
and
B
are ON. We proceed in raising the threshold to higher values until a
further increase in threshold will produce no changes in the output
functions, all being always OFF.
In order to see that for each threshold value this element has indeed
calculated a logical function on the propositions
A
and
B
, one has only to
interpret the “zeros” and “ones” as “false” and “true” respectively, and the
truth values in each q-column, in conjunction with the double column rep-
resenting the input states, become a table called—after Wittgenstein
(1956)—the “truth table” for the particular logical function. In the example
of fig. 10, the column q=0 represents “Tautology” because F
o
(
A
,
B
) is
always true (1), independent of whether or not
A
or
B
are true: “
A
or not
-
A
, and
B
or not -
B
”. For q=1 the logical function “
A
or
B
” is computed;
it is false (0) only if both
A
and
B
are false. q=2 gives “
A
and
B
” which,
of course, is only true if both
A
and
B
are true, etc.
Today there are numerous notations in use, all denoting these various
logical functions, but based on different reasons for generating the appro-
priate representations, which all have their advantages or disadvantages.
The representation we have just employed is that of Wittgenstein's truth
table. This representation permits us to compute at once the number of dif-
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