The Aristotelian logic, or its school, studies the forms of reasoning in everyday speech, and the terms of this which are needed to set out its laws. Mathematical logic, studies the forms of reasoning of the deductive sciences and especially mathematics, and from this copy the symbols which serve to enunciate its laws.
While the terms of common language often take on different meaning or values depending on the position they occupy in the context, because of the frequent exceptions to which the rules of grammar are subject, the symbols of mathematical logic retain the same meaning everywhere, the laws not being subject to exceptions to that which it was agreed they satisfy.
Moreover, while the terms of the common language give rise to their own combinations depending on the special structure of each language, the symbols of mathematical logic are a universal writing system independent of any language.
The characteristics now mentioned clearly separate the school of logic from mathematical logic, and give mathematical logic a degree of scientific rigour that is impossible to achieve with the former.
The merit of having given the first germs of mathematical logic, is indisputably in that vast mind of the mathematician and philosopher Gottfried Leibniz (Dissertatio de arte combinatoria, Leipzig, 1666). Leibniz was the first to enunciate the commutative and associative properties of the logical product and the laws of simplification. Leibniz was the first to conceive the grand plan to create a universal writing system, by which every idea could be expressed by means composed of simple ideas, each represented by a special sign. However, old age, as he himself says, prevented him from carrying out his plan, and translating into symbols all knowledge then known.
A solution of the problem of Leibniz has greatly contributed to the progress of mathematical logic. This took scientific form, for the first time, more especially in the work of the Englishman George Boole (An investigation into the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities, London, 1854) who has applied algebraic symbolism to logic. Ernst Schröder (Der Operationskreis der Logikkalkuls, Leipzig, 1872) has conveniently transformed the symbolism of Boole changing some definitions; he was responsible for, among other things, the law of duality.
Many writers have contributed to the progress of mathematical logic; refer, with its bibliography on this subject, to the massive and important work of Ernst Schröder, Algebra der Logik (1890-1910). We note, however, explicitly - since results make it clear that mathematical logic belongs to the field of the mathematician who is a philosopher - that these writers are all mathematicians and their work has been published in journals of mathematics.
In American Universities mathematical logic is part of the official teaching through the work of George Bruce Halsted, Charles Sanders Peirce, and others. Also Platon Sergeevich Poretsky a Russian lecturer from Kazan and Albino Nagy (1866-1900) a professor at the University of Rome.
Mathematical logic was introduced into Italy through the work of professor Giuseppe Peano (Calcolo Geometrico, secondo l'Ausdehnungslehre di H Grassmann, preceduto dalle operazioni della logica deduttiva, Turin, 1888) and Albino Nagy (Fondamenti del calcolo logico, Giornale di matematica 28 (1890), 1-35). Finally Peano, in his book "Arithmetices principia, nova methodo exposita, Rome, 1889", managed to completely expose symbolic axioms for the theory of the integers. This work was followed by many others, through the efforts of several people. Today the papers that are published in Peano's Rivista di matematica aim to treat completely the symbolic calculus in the various parts of mathematics.
This book contains the elements of Mathematical Logic, and is developed from a course of science lectures I've given in the current school year at the University of Turin. The works of which I made use of are those adopted for the course, just quoted, and numerous examples have been chosen in the field of elementary mathematics, to make possible the reading of this book, even for those who do not possess knowledge of higher mathematics.
Turin, March 1894.