2 Deductive Reasoning in Propositional Logic This chapter presents the deductive side of propositional logic: deductive systems and formal derivations of logical consequences in them First, I explain the concept and purpose of a deductive system and provide some historical background I then introduce, discuss, and illustrate with examples the most popular types of deductive systems for classical logic: Axiomatic Systems, Semantic Tableaux, Natural Deduction, and Resolution In a supplementary section at the end of the chapter I sketch generic proofs of soundness and completeness of these deductive systems In another supplementary section I discuss briefly the computational complexity of the Boolean satisfiability problem and the concept of NP-completeness The deductive systems introduced in this chapter are specifically designed for classical propositional logic, but the concept is universal and applies to almost all logical systems that have been introduced and studied In Chapter I extend each of these deductive systems with additional axioms and rules for the quantifiers, so that they also work for first-order logic 2.1 2.1.1 Deductive systems: an overview The concept and purpose of deductive systems The fundamental concept in logic is that of logical consequence It extends the concept of logical validity and is the basis of logically correct reasoning Verifying logical consequence in propositional logic is conceptually simple and technically easy (although possibly computationally expensive), but this is no longer the case for full first-order logic, which I introduce in Chapter 3, or for the variety of non-classical logics which I not discuss here In fact, verifying logical consequence (in particular, logical validity) in first-order logic is usually an infinite task, as it generally requires checking infinitely many possible models rather than a finite number of simple truth assignments A different approach for proving logical validity and consequence, which is not based on the semantic definition, is therefore necessary Such a different approach is provided by the notion of a deductive system, a formal, mechanical – or, at least mechanizable – procedure Logic as a Tool: A Guide to Formal Logical Reasoning, First Edition Valentin Goranko © 2016 John Wiley & Sons, Ltd Published 2016 by John Wiley & Sons, Ltd