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

(Ewald, 1996). Peirce extended Boole’s work on secondary propositions by stipulating
an additional algebraic law of propositions: for every element x, either x = 0 or x
= 1, producing a system known as “the two-valued algebra” (Lewis, 1918).
The two-valued algebra led to the invention of truth tables, which are established
in the literature in the early 1920s (Post, 1921; Wittgenstein, 1922), but were
likely in use much earlier. There is evidence that Bertrand Russell and his then student
Ludwig Wittgenstein were using truth tables as early as 1910 (Shosky, 1997). It
has also been argued that Charles Peirce and his students probably were using truth
tables as early as 1902 (Anellis, 2004).
Truth tables make explicit an approach in which primitive propositions (p,
q, r, etc.) that could only adopt values of 0 or 1 are used to produce more complex
expressions. These expressions are produced by using logical functions to
combine simpler terms. This approach is known as “using truth-value systems”
(Lewis & Langford, 1959). Truth-value systems essentially use truth tables to determine
the truth of functions of propositions (i.e., of logical combinations of propositions).
“It is a distinctive feature of this two-valued system that when the property, 0
or 1, of the elements p, q, etc., is given, any function of the elements which is in the
system is thereby determined to have the property 0 or the property 1” (p. 199).
Consider Table 2-1, which provides the values of three different functions (the
last three columns of the table) depending upon the truth value of two simple propositions
(the first two columns of the table):
p q pq p + q p(p + q)
1 1 1 1 1
1 0 0 1 1
0 1 0 1 0
0 0 0 0 0
Table 2-1. Examples of the truth value system for two elementary propositions
and some of their combinations. The possible values of p and q are given
in the first two columns. The resulting values of different functions of these
propositions are provided in the remaining columns.
Truth-value systems result in a surprising, simplified approach to defining basic or
primitive logical functions. When the propositions p and q are interpreted as being
only true or false, then there are only four possible combinations of these two propositions
that can exist, i.e., the first two columns of Table 2-1. A primitive function
can be defined as a function that is defined over p and q, and that takes on a truth
value for each combination of these variables.
Given that in a truth-value system a function can only take on the value of 0
or 1, then there are only 16 different primitive functions that can be defined for
Multiple Levels of Investigation 27