94 Logic as a Tool 2.7.20 Assuming the soundness and completeness of H, prove the soundness and completeness of each of ST, ND, and RES, by using Proposition 76 2.7.21 Assuming the soundness and completeness of ST, prove the soundness and completeness of each of H, ND, and RES by using Proposition 76 2.7.22 Assuming the soundness and completeness of ND, prove the soundness and completeness of each of ST, H, and RES by using Proposition 76 2.7.23 Assuming the soundness and completeness of RES, prove the soundness and completeness of each of H, ST, and ND by using Proposition 76 Jan Leopold Łukasiewicz (21.12.1878–13.02.1956) was a Polish logician and philosopher who introduced mathematical logic in Poland and made notable contributions to analytical philosophy, mathematical logic, and history of logic Łukasiewicz studied first law and then mathematics and philosophy at the University of Lwów where he achieved a PhD in 1902 under the supervision of Kazimierz Twardowski for a dissertation On induction as the inverse of deduction He taught at the University of Lwów before WW I then joined the University of Warsaw in 1915, where he held the position of rector in 1922–23 and 1931–32 He also served as a minister of education in 1919 Together with another prominent logician, Stanislaw Lesniewski, Łukasiewicz founded the world-famous Warsaw School of Logic Alfred Tarski, a student of Lesniewski but also strongly influenced by Łukasiewicz, also contributed to the reputation of the school Łukasiewicz fled from Poland during WW II In 1946 he was appointed Professor of Mathematical Logic at the Royal Irish Academy in Dublin, where he worked until his retirement in 1953 Łukasiewicz did important work in modernizing formal logic He developed propositional logic and its implicational and equivalential fragments, for all of which he obtained some elegant short axiomatizations Notably, he introduced many-valued logics (partly as an alternative to the Aristotelian 2-valued logic) in 1917 Łukasiewicz also introduced the Polish notation which allowed logical formulae to be written unambiguously without the use of brackets For instance, the formula (p → (¬p → q )) is written in Polish notation as CpCN pq He also developed a theory of axiomatic rejection, wrote a book on Logical Foundations of Probability Theory, and conducted very important work in the history of logic by studying and popularizing both Aristotle’s syllogistic and Stoic’s propositional logic