Mind, Body, World- Foundations of Cognitive Science, 2013a

combinations of the binary inputs p and q (Ladd, 1883). These primitive functions
are provided in Table 2-2; each row of the table shows the truth values of each function
for each combination of the inputs. An example logical notation for each function
is provided in the last column of the table. This notation was used by Warren
McCulloch (1988b), who attributed it to earlier work by Wittgenstein.
Not surprisingly, an historical trajectory can also be traced for the binary logic
defined in Table 2-2. Peirce’s student Christine Ladd actually produced the first five
columns of that table in her 1883 paper, including the conversion of the first four
numbers in a row from a binary to a base 10 number. However, Ladd did not interpret
each row as defining a logical function. Instead, she viewed the columns in
terms of set notation and each row as defining a different “universe.” The interpretation
of the first four columns as the truth values of various logical functions arose
later with the popularization of truth tables (Post, 1921; Wittgenstein, 1922).
p=0
p=1
p=0
p=1
Number
Notation
q=0 q=0 q=1 q=1
0 0 0 0 0 Contradiction
0 0 0 1 1 p·q
0 0 1 0 2 ~p·q
0 0 1 1 3 q
0 1 0 0 4 p·~q
0 1 0 1 5 p
0 1 1 0 6 p q
0 1 1 1 7 p q
1 0 0 0 8 ~p·~q
1 0 0 1 9 p q
1 0 1 0 10 ~p
1 0 1 1 11 p q
1 1 0 0 12 ~q
1 1 0 1 13 q p
1 1 1 0 14 p | q
1 1 1 1 15 Tautology
Table 2-2. Truth tables for all possible functions of pairs of propositions. Each
function has a truth value for each possible combination of the truth values
of p and q, given in the first four columns of the table. The Number column
converts the first four values in a row into a binary number (Ladd, 1883). The
logical notation for each function is taken Warren McCulloch (1988b).
28 Chapter 2