k 2.7.20 Assuming the soundness and completeness of H, prove the soundness and com-pleteness of each of ST, ND, and RES, by using Proposition 76.. 2.7.21 Assuming the soundness and comp
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2.7.20 Assuming the soundness and completeness of H, prove the soundness and
com-pleteness 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
com-pleteness 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 phi-losophy at the University of Lwów where he achieved a PhD in
1902 under the supervision of Kazimierz Twardowski for a
disser-tation 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, Stanis-law 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 dur-ing 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 Founda-tions 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