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

a band. However, its relationship to some other feature is constant—variations in
one feature are perfectly correlated with variations in another. Berkeley et al. (1995)
showed how definite features could be both objectively defined and easily discovered
using simple descriptive statistics (see also Dawson, 2005).
Definite features are always expressed in terms of the values of input unit activities.
As a result, they can be assigned meanings using knowledge of a network’s
input unit encoding scheme.
One example of using this approach was presented in Berkeley et al.’s (1995)
analysis of a network on the Bechtel and Abrahamsen (1991) logic task. This task
consists of a set of 576 logical syllogisms, each of which can be expressed as a pattern
of binary activities using 14 input units. Each problem is represented as a first
sentence that uses two variables, a connective or a second sentence that states a
variable, and a conclusion that states a variable. Four different problem types were
created in this format: modus ponens, modus tollens, disjunctive syllogism, and
alternative syllogism. Each problem type was created using one of three different
connectives and four different variables: the connectives were If…then, Or, or Not
Both… And; the variables were A, B, C, and D. An example of a valid modus ponens
argument in this format is “Sentence 1: ‘If A then B’; Sentence 2: ‘A’; Conclusion: ‘B’.”
For this problem, a network’s task is to classify an input problem into one of
the four types and to classify it as being either a valid or an invalid example of that
problem type. Berkeley et al. (1995) successfully trained a network of value units
that employed 10 hidden units. After training, each of these units were wiretapped
using the entire training set as stimulus patterns, and a jittered density plot was
produced for each hidden unit. All but one of these plots revealed distinct banding.
Berkeley et al. were able to provide a very detailed set of definite features for each
of the bands.
After assigning definite features, Berkeley et al. (1995) used them to explore
how the internal structure of the network was responsible for making the correct
logical judgments. They expressed input logic problems in terms of which band of
activity they belonged to for each jittered density plot. They then described each
pattern as the combination of definite features from each of these bands, and they
found that the internal structure of the network represented rules that were very
classical in nature.
For example, Berkeley et al. (1995) found that every valid modus ponens problem
was represented as the following features: having the connective If…then,
having the first variable in Sentence 1 identical to Sentence 2, and having the second
variable in Sentence 1 identical to the Conclusion. This is essentially the rule for
valid modus ponens that could be taught in an introductory logic class (Bergmann,
Moor, & Nelson, 1990). Berkeley et al. found several such rules; they also found
a number that were not so traditional, but which could still be expressed in a
176 Chapter 4