### The *Logic Piano* of William Jevons

William Stanley Jevons (1835–1882) was an English economist and logician, a major figure, both in Britain and internationally, in the fields of political economy and social reform. Jevons is most often credited with being the first theorist to make economics a mathematical discipline, and he is regarded as one of the founders of the form of neo-classical economics, that dominates our current economic thinking and political discourse.

The most interesting for us aspect of his life is the designed in 1869 logical machine for doing logic inference, called *Logic Piano*, which was the best known logic machine of the nineteenth
century.

The work of Devons on *Logic Piano* was inspired by Stanhope's Demonstrator. The construction of the device was announced in his 1869 logic textbook, *Substitution of Similars*. It was the culmination of a long series of inventions and aids to the calculation of syllogisms: logical alphabet, logical slate, logical stamp, and logical abacus-all tools to write quickly the lines of a truth table in a logical argument.

The interest of Jevons to Logic began as yearly as in 1860, when he worked as an assayer at the Sydney Mint, Australia. Jevons wrote in his 1860 diary:

*As I awoke in the morning the sun was shining brightly into my room, there was a consciousness on my mind that I was the discoverer of the true logic of the future I felt a delight such as one can seldom hope to feel. I remembered only too soon though how unworthy and weak an instrument I was for accomplishing so great a work and how hardly I could expect to do it.*

His serious involvement and subsequent passion for Logic came about when, on his return to England, he met up with his former undergraduate teacher of mathematics, the famous logician Augustus de Morgan. It seems Jevons was one of the first in Britain to catch on to the importance of the newly developed formal logical systems of Boole and De Morgan.

The title page of Substitution of Similars of Jevons

Jevons read the *Mathematical Analysis of Logic* and *An Investigation of the Laws of Thought* of Boole and was fascinated. But he also saw problems with it, and by 1861 he was developing his own system of logic based on what he eventually called the *Substitution of Similars*, whereby *philosophy would be shown to consist solely in pointing out the likeness in things*. In 1863 he published his work first work on the subject, *Pure Logic*, which was hardly a success, with four copies sold in 6 months. But Jevons was a great one for persistence.

In his 1869 logic textbook, *The Substitution of Similars* he describes the *Logical Abacus*: a series of wooden boards with various combinations of true and false terms. It was intended that they be arranged on a rack and a ruler used to remove certain excluded combinations. This was the basic outline of the device that, with the addition of levers and pulleys, Jevons had a Salford clock maker construct for him in 1869. Fitted within a wooden case, and with a keyboard mounted on the front to operate the substitution mechanism, this was his *Logic Piano*.

The logic piano (see the nearby image) was a wooden box approximately 90 cm high. A faceplate above the keyboard displayed the entries of the truth table. Just like a piano, the keyboard had black-and-white keys, but here they were used for entering premises. As the keys were struck, rods would mechanically remove from the face of the piano the truth-table entries inconsistent with the premises entered on the keys.

The Logic Piano can deal with up to 4 terms. Jevons had in fact wanted to build a machine capable of dealing with up to 16 terms, but it would have been too large and taken up a whole wall in his office. The logic expressions are typed (or perhaps played) via the keys of the keyboard (see the nearby image), and hitting *full stop* removes all impossible combinations from the screen. The *copula* is the equals key, while the *finis* key resets the machine.

A truth-table for *n* proposition requires 2^{n} entries. The table for *n = 4* has 16 entries and is as follows, if we represent the truth of a proposition by an upper case letter, and its falsity by the same letter in lower case:

The truth-table for n=4 |
|||

PQRS |
PQRs |
PQrs |
PQrs |

PqRS |
PqRs |
Pqrs |
Pqrs |

pQRS |
pQRs |
pQrs |
pQrs |

pqRS |
pqRs |
pqrS |
pqrs |

The proposition *if P, then Q*, is true just in case P is false or Q is true. If this proposition were entered on the keyboard of the logic piano, the face would show:

The truth-table for proposition if P, then Q, is true just in case P is false or Q is true, when n=4 |
|||

PQRS |
PQRs |
PQrs |
PQrs |

(second line is removed) |
|||

pQRS |
pQRs |
pQrs |
pQrs |

pqRS |
pqRs |
pqrS |
pqrs |

As propositions were entered on the keyboard, representing additional premises that must be satisfied simultaneously, other inconsistent entries would disappear from the face.

The action of the *logic piano* actually did not result in a conclusion stated in the form of a proposition, but only in the truth table entries consistent with the conclusion. Jevons worked unsuccessfully to resolve this problem, which he termed the *inverse problem* and which he somewhat misleadingly associated with the process of mathematical induction.

As Jevons' adversary John Venn noted, the logic piano has no practical purpose, for there are no circumstances in which difficult syllogisms arise or in which syllogisms must be resolved repeatedly enough to justify mechanization of the process. Jevons countered that it was a convenience to his personal work and useful in his logic classes.