Deductive Reasoning in Propositional Logic 49 development of elementary geometry based on several simple assumptions about points and lines (such as “every two different points determine exactly one line”, “for every line there is a point not belonging to that line”, etc.) Using these postulates and some informal logical reasoning, other geometric facts are derived and thus the entire body of Euclidean geometry is eventually built The logical concept of the deductive system gradually emerged much later, notably through the ideas of Gottfried Leibniz (1646–1716), one of the greatest philosophers and mathematicians of all times, of a characteristica universalis (universal language) and a calculus ratiocinator (calculus of reasoning) The first formal deductive system for modern logic, however, was only constructed in the late 19th century by the mathematician and philosopher Gottlob Frege (1848–1925) in his seminal book Begriffsschrift (1879), where he developed the prototype of classical first-order logic Later, David Hilbert (1862–1943), one of the leading mathematicians of the late 19th and early 20th century, reworked Euclid’s system of geometry developed in the Elements into a rigorous and mathematically precise treatment that resulted in his book Foundations of Geometry (1899) The mathematician Giuseppe Peano (1858–1932) developed a formal system of arithmetic, still known as Peano’s axiomatic system, in his most important work Formulario Mathematico The concept of formal deductive system was further developed by the philosophers Bertrand Russell (1872–1970) and Alfred North Whitehead (1861–1947) in their three-volume book Principia Mathematica (1910–1913), and by David Hilbert and Wilhelm Ackermann (1896–1962) in their book Principles of Mathematical Logic (1928) These books were most influential for the development of logic and foundations of mathematics in the first half of the 20th century In particular, Hilbert strongly promoted the idea of building the whole body of mathematics as a formal axiomatic system, a deductive system mainly based on axioms and on very few and simple rules of inference Hilbert was the leading proponent of the development of the axiomatic approach in mathematics, meant to replace the semi-formal notion of mathematical proof by the completely formalized notion of derivation in an axiomatic system A major purpose of the formal axiomatic approach was to avoid the occurrence of any paradoxes and contradictions in mathematics by performing all mathematical proofs within such a formal system of deduction which has been proved to be consistent, that is, free from contradictions Such a system, based on first-order logic, was proposed by Hilbert and Ackermann in Principles of Mathematical Logic, and Hilbert’s ultimate goal was to prove that that system was both consistent and complete, that is, capable of deriving every mathematical truth, but not deriving any contradictions Hilbert’s idea of a deductive system was further developed by several logicians, including Jacques Herbrand, Emil Post, Alfred Tarski, and others In the late 1920s–early 1930s the concept of a deductive system was extended in the works of Gerhard Gentzen and Stanisław Ja´skowski by adding the possibility of introducing and withdrawing assumptions in derivations, that is, formulae that are not axioms and have not been derived Thus, the more intuitive and practically more efficient rule-based type of deductive system, called Natural Deduction and the closely related Sequent Calculus, were developed, essentially founding the major field of logic known as proof theory In the 1950s–1960s, the the refutation-based system of Semantic Tableaux emerged, implementing the idea of a systematic and exhaustive search for a falsifying model of the assumptions plus the negation of the desired conclusion, where a derivation consists of an established failure to construct such a falsifying model The idea of refutation-based deductive systems led to the development in the 1960s–1970s of the method of resolution, which turned out to be very suitable for computer-aided automated deduction