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

ewritten as “A is B,” which in Boole’s algebra becomes AB, and “metal is element”
becomes BC. Third, given a set of premises, Jevons removed the terms that were
inconsistent with the premises from the abecedarium: the only terms consistent
with the premises AB and BC are ABC, aBC, abC, and abc. Fourth, Jevons inspected
and interpreted the remaining abecedarium terms to perform valid logical inferences.
For instance, from the four remaining terms in Jevons’ example, we can conclude
that “all iron is element,” because A is only paired with C in the terms that
remain, and “there are some elements that are neither metal nor iron,” or abC. Of
course, the complete set of entities that is elected by the premises is the logical sum
of the terms that were not eliminated.
Jevons (1870) created a mechanical device to automate the procedure described
above. The machine, known as the “logical piano” because of its appearance, displayed
the 16 different combinations of the abecedarium for working with four different
classes. Premises were entered by pressing keys; the depression of a pattern
of keys removed inconsistent abecedarium terms from view. After all premises had
been entered in sequence, the terms that remained on display were interpreted.
A simpler variation of Jevons’ device, originally developed for four-class problems
but more easily extendable to larger situations, was invented by Allan Marquand
(Marquand, 1885). Marquand later produced plans for an electric version of his
device that used electromagnets to control the display (Mays, 1953). Had this device
been constructed, and had Marquand’s work come to the attention of a wider
audience, the digital computer might have been a nineteenth-century invention
(Buck & Hunka, 1999).
With respect to our interest in the transition from Boole’s work to our modern
interpretation of it, note that the logical systems developed by Jevons, Marquand,
and others were binary in two different senses. First, a set and its complement (e.g.,
A and a) never co-occurred in the same abecedarium term. Second, when premises
were applied, an abecedarium term was either eliminated or not. These binary characteristics
of such systems permitted them to be simple enough to be mechanized.
The next step towards modern binary logic was to adopt the practice of assuming
that propositions could either be true or false, and to algebraically indicate these
states with the values 1 and 0. We have seen that Boole started this approach, but
that he did so by applying awkward temporal set-theoretic interpretations to these
two symbols.
The modern use of 1 and 0 to represent true and false arises later in the nineteenth
century. British logician Hugh McColl’s (1880) symbolic logic used this
notation, which he borrowed from the mathematics of probability. American logician
Charles Sanders Peirce (1885) also explicitly used a binary notation for truth
in his famous paper “On the algebra of logic: A contribution to the philosophy of
notation.” This paper is often cited as the one that introduced the modern usage
26 Chapter 2