This chapter introduces some core topics related to mathematical logic. We begin our discussion with propositional calculus. It involves statement formulas consisting of atomic propositions connected by logical operators such as “conjunction”, “disjunction”, or “negations”. The focus is on developing tools to study certain fundamental equivalences and implications. Next, we discuss first-order logic involving predicates and quantifiers. Our final aim is to discuss syllogisms that are used to arrive at a valid conclusion from a set of given premises by way of following a stepwise procedure, each known as a valid argument. In all that follows, syllogismsyllogism refers to an argument that is used as a rule of inference. Since ideas introduced in this chapter form the fundamentals of some important related topics such as natural language processing, it is strongly advised the reader pursuing computer science to grasp most of it. Also, it is equally important to attempt all the practice problems provided at the end of the chapter. The interested reader may like to read some advance concepts from excellent texts such as [1–4].