A logical-mathematical comparison… legally relevant.

The processing of “law” by the computer requires a two-way mechanism. It needs a tool capable of simultaneously translating calculation into logic and vice versa. Particularly relevant are the studies and thoughts of brilliant minds that, here, with apologies, can only be mentioned: Gottfried Wilhelm von Leibniz and George Boole, Konrad Zuse, Claude Elwood Shannon, Herbert Alexander Simon, John von Neumann, Alan Mathison Turing and many others.

Leibniz’s contribution, although of milestone importance for the subject, does not reach the level of a true algebraic representation of ‘reasoning’.

The concrete feasibility of transforming logical thought into calculus belongs to the English mathematician George Boole. Boole succeeded in transferring the scientific rigour of the methodologies underlying algebraic research to the study of logic. Boole created a new and revolutionary language through which to create algorithms that could be applied to an infinite number of hypotheses. The fortunes of Boolean algebra are inextricably linked to the thinking of the British mathematician Alan Mathison Turing. Turing imagined a ‘machine’ capable of executing any algorithm. Leibniz’s dream (a symbolic calculus with which all kinds of problems could be solved automatically) “…materialized in calculators no longer made of flesh and blood but of silicon copper”[1].

1] At the basis of Boole’s logical-mathematical system is the adoption of the binary system as a means of representing arithmetical operations and the typical processes of human reasoning. Boole’s task was to disguise logic with a mathematical dress[2] and to keep mental and algebraic operations separate[3].

The historical and cultural steps necessary to place Boole’s algebra at the basis of computer science and computers can be symbolically summarised as follows:

(a) the adoption of the binary system as the basic scheme of computers[4];

b) the intuition of the use of the computer to process not only numerical but also logical operations[5];

c) the advent of a universal computer[6].

Boole, in fact, “…devised a system of formal hypothetical logic which, thanks to the adoption of BITs (of Arabic origin), served equally well to perform arithmetical operations with lightning speed and, at the same time…to reproduce the mechanism of reasoning.”[7].

[1] M. Davis, Il calcolatore universale. Da Leibniz a Turing, Adelphi, 2012 (Traduttori G. Rigamonti e A. La Rana).

[2] G. Boole, L’analisi matematica della logicaBollati Boringhieri, 1993. La versione in inglese è disponibile nella edizione del 1847 su Google Libri. A.Albertelli, Il pensiero logico di George Boole, in Le Scienze (Scientific American), n.146, ottobre 1980.

[3] In materia di rapporti tra matematica, logica e diritto si rinvia a quanto scritto da G. Danzi, Sull’evoluzione del diritto: un lungo cammino tra logica e matematica fino alla “giustizia predittiva” del Prof. Luigi Viola, in L.Viola (a cura di), Giustizia Predittiva e Interpretazione della Legge con Modelli Matematiciop. cit., 68 e ss.

[4] In 1939, Konrad Zuse built the Z1, based on the studies of his predecessors. This was the first in an innovative series of binary-based, programmable electromechanical computers, which operated first with mechanical memories and then with relays (Z2, Z3). Then, in 1939, John Vincent Atanasoff and Clifford E. Berry of Iowa State University designed and built the ‘Atanasoff Berry Computer’ (called the ABC). It was the first fully electronic digital computer. The Atanasoff-Berry Computer was a novelty in the field of computers, as it used binary numbers. However, the first binary-based, fully programmable computer was Konrad Zuse’s Z3, which he built in 1941.

[5] Già nel 1938 Claude Elwood Shannon riuscì a dimostrare con la tesi “A symbolic analysis of relay and switching circuits” (MIT Libraries https://dspace.mit.edu/handle/1721.1/11173 ” Analogue with the Calculus of Propositions. We are now in a position to demonstrate the equivalence of this calculus with certain elementary parts of the calculus of propositions. The algebra of logic originated by George Boole, is a symbolic
method of investigating logical relationships. The symbols of Boolean algebra admit of two logical interpretations. If interpreted in terms of classes, the variables are not limited to the two possible values
o and 1. This interpretation is known as the algebra of classes..”) che nello scorrere di un segnale elettrico attraverso una rete di interruttori (le cui uniche variabili possibili possono essere On/Off – Acceso/Spento) si possono riprodurre le regole dell’algebra di Boole anch’essa basata sui valori dicotomici (Vero e Falso) propri della logica simbolica. Alle conclusioni di Shannon si aggiunse, qualche decennio dopo, l’intuizione di Herbert Alexander Simon, economista e informatico statunitense, Premio Nobel per l’economia “per le sue pioneristiche ricerche sul processo decisionale nelle organizzazioni economiche” (1978). Simon, riallacciandosi ad un’idea e al pensiero di studiosi del passato, riscoprì portandola a nuova luce, «…quanto Boole e Babbage avevano già intravisto quasi un secolo prima e che, cioè, il computer è in grado di elaborare non solo numeri, ma anche simboli (ancorché espressi in BIT) e, quindi, di svolgere operazioni non solo aritmetiche, ma anche logiche» (tratto da R. Borruso, S. Russo, C. Tiberi, L’informatica per il giurista, op.cit., 84).

[6] Neumann and Turing were responsible for milestone contributions in fields such as set theory, functional analysis, cryptanalysis, topology, quantum physics, economics, computer science, game theory, fluid dynamics and many other areas of mathematics. The computer was supposed to ‘learn’ from software, as Alan Turing had brilliantly foreseen with the design of the so-called ‘universal machine’: A. M. Turing, Computing machinery and intelligence, Volume LIX, Issue 236, October 1950, Pages 433–460.

[7] R. Borruso, S. Russo, C. Tiberi, L’informatica per il giurista, Dal Bit ad Internet, III ed., Milano, 2009,84.