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

meaning “no time” and “the whole of time” respectively. While this usage differs
substantially from our modern approach, it has been viewed as the inspiration for
modern binary logic (Post, 1921).
Boole’s work inspired subsequent work on logic in two different ways. First,
Boole demonstrated that an algebra of symbols was possible, productive, and
worthy of exploration: “Boole showed incontestably that it was possible, by the aid
of a system of mathematical signs, to deduce the conclusions of all these ancient
modes of reasoning, and an indefinite number of other conclusions” (Jevons, 1870,
p. 499). Second, logicians noted certain idiosyncrasies of and deficiencies with
Boole’s calculus, and worked on dealing with these problems. Jevons also wrote
that Boole’s examples “can be followed only by highly accomplished mathematical
minds; and even a mathematician would fail to find any demonstrative force
in a calculus which fearlessly employs unmeaning and incomprehensible symbols”
(p. 499). Attempts to simplify and correct Boole produced new logical systems that
serve as the bridge between Boole’s nineteenth-century logic and the binary logic
that arose in the twentieth century.
Boole’s logic is problematic because certain mathematical operations do not
make sense within it (Jevons, 1870). For instance, because addition defined the
“exclusive or” of two sets, the expression x + x had no interpretation in Boole’s
system. Jevons believed that Boole’s interpretation of addition was deeply mistaken
and corrected this by defining addition as the “inclusive or” of two sets. This produced
an interpretable additive law, x + x = x, that paralleled Boole’s multiplicative
fundamental law of thought.
Jevons’ (1870) revision of Boole’s algebra led to a system that was simple
enough to permit logical inference to be mechanized. Jevons illustrated this with
a three-class system, in which upper-case letters (e.g., A) picked out those entities
that belonged to a set and lower-case letters (e.g., a) picked out those entities
that did not belong. He then produced what he called the logical abecedarium,
which was the set of possible combinations of the three classes. In his three-class
example, the abecedarium consisted of eight combinations: ABC, ABc, AbC, Abc,
aBC, aBc, abC, and abc. Note that each of these combinations is a multiplication
of three terms in Boole’s sense, and thus elects an intersection of three different
classes. As well, with the improved definition of logical addition, different terms of
the abecedarium could be added together to define some set of interest. For example
Jevons (but not Boole!) could elect the class B with the following expression:
B = ABC + ABc + aBC + aBc.
Jevons (1870) demonstrated how the abecedarium could be used as an inference
engine. First, he used his set notation to define concepts of interest, such as
in the example A = iron, B = metal, and C = element. Second, he translated propositions
into intersections of sets. For instance, the premise “Iron is metal” can be
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