Deductive Reasoning in Propositional Logic 51 A deductive system D is adequate for a given semantics if it is both sound and complete for it, i.e.: A1 , , An |= C if and only if A1 , , An D C In particular: |= C if and only if D C The soundness of a deductive system can be guaranteed and proved easily in principle as long as the following two conditions hold: (i) All axioms (if any) must be true (ii) All rules of inference must be sound, that is, they must always produce true conclusions when applied to true assumptions The truth therefore propagates from the axioms to all theorems I present here some of the most popular types of deductive systems for classical logic, namely Axiomatic Systems, Semantic Tableaux, Natural Deduction, and Resolution.2 Each of these bears a different idea and has advantages and shortcomings compared to the others The use of each of these is illustrated with several examples and many more exercises, and in Chapter they will be put to real work for performing concrete mathematical reasoning References for further reading For further general discussion and historical notes on deductive systems, see the classical masterpiece Tarski (1965) and well as Nerode and Shore (1993), Jeffrey (1994), Fitting (1996), Barwise and Echemendy (1999), Hodges (2001), and Ben-Ari (2007) Aristotle (384–322 BC) was a great Ancient Greek philosopher, “the first genuine scientist in history” according to Encyclopaedia Britannica He was born in the Macedonian city of Stagira in the Chalkidiki peninsula, and joined Plato’s Academy in Athens at the age of 18 where he carried out scholarly work for around 20 years After the death of Plato he left Athens and, at the request of King Philip of Macedonia, became the private teacher of his son Alexander the Great during 356–323 BC It is believed that only about one-third of Aristotle’s original works have survived In these he made important contributions to just about every field of science and arts that existed in his time or was founded by him: philosophy, poetry, theatre, music, rhetoric, politics, government and ethics, physics, geology and biology Aristotle’s legacy, not only in philosophy, continues to be the object of active academic study today For instance, the idea of systematic classification of plants and animals, developed in the 18th century by Linnaeus, originate from him I not present Sequent Calculus in this book, but it is easily reducible to Natural Deduction See Ebbinghaus et al (1996) and Boolos et al (2007)