University of Alberta Dictionary of Cognitive Science: Abecedarium

Jevons (1870) revised Boole’s(1854) algebra to create 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 in which 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 8 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 each defines an intersection of three different classes.

Jevons (1870) demonstrated how the abecedarium could be used as an inference engine. First, he used his set notation to define concepts of interest – in one 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 rewritten as “A is B”, which in Boole’s algebra becomes AB; “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” (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 19^th century invention (Buck & Hunka, 1999).

References:

Boole, G. (1854/2003). The Laws of Thought. Amherst, N.Y.: Prometheus Books. (Originally published in 1854).