Logic is one of the oldest intellectual disciplines in human history. It dates back to Aristotle. It has been studied through the centuries by people like Leibniz, Boole, Russell, Turing, and many others. And it is still a subject of active investigation today.
We use Logic in just about everything we do. We use the language of Logic to define concepts, to encode constraints, to express partial information. We use logical reasoning to derive conclusions from these bits of information. We use logical proofs to convince others of our conclusions.
And we are not alone! Logic is increasingly being used by computers – to prove mathematical theorems, to validate engineering designs, to diagnose failures, to encode and analyze laws and regulations and business rules.
Logic is also becoming more common at the interface between man and machine, in “logic-enabled” computer systems, where users can view and edit logical sentences. Think, for example, about email readers that allow users to write rules to manage incoming mail messages – deleting some, moving others to various mailboxes, and so forth based on properties of those messages. In the business world, eCommerce systems allow companies to encode price rules based on the product, the customer, the date, and so forth.
Moreover, Logic is sometimes used not just by users in communicating with computer systems but by software engineers in building those systems (using a programming methodology known as logic programming).
This chapter is an overview of Logic as presented in this book. We start with a discussion of possible worlds and illustrate the notion in an application area known as Sorority World. We then give an informal introduction to the key elements of Logic – logical sentences, logical entailment, and logical proofs. We then talk about the value of using a formal language for expressing logical information instead of natural language. Finally, we discuss the automation of logical reasoning and some of the computer applications that this makes possible.
Logic is the systematic study of reasoning and inference. It is a fundamental branch of philosophy and mathematics that aims to understand and analyze the principles and patterns of valid reasoning. Logic provides a framework for evaluating arguments, distinguishing between sound and unsound reasoning, and establishing the truth or falsity of statements.
The primary goal of logic is to develop formal methods for determining the validity of arguments. An argument consists of a set of statements called premises, which are intended to support or justify another statement known as the conclusion. By applying logical principles and rules, we can determine whether the conclusion logically follows from the premises or not.
In logic, we use symbols and formal languages to represent statements and logical relationships. These symbols include logical operators such as conjunction (AND), disjunction (OR), negation (NOT), implication (IF…THEN), and equivalence (IF AND ONLY IF). With these symbols, we can construct logical expressions and analyze their logical properties.
Two major branches of logic are propositional logic and predicate logic. Propositional logic deals with the study of simple statements and their logical relationships, while predicate logic extends this to include more complex statements involving variables, quantifiers, and predicates.
Propositional logic focuses on the analysis of truth values (true or false) and the interplay between logical operators. It allows us to represent arguments and reason about their validity using truth tables, logical equivalences, and rules of inference.
Predicate logic, on the other hand, introduces variables to represent objects and quantifiers to specify the range of those objects. It enables us to reason about properties, relationships, and generalizations. Predicate logic allows for the formalization of mathematical reasoning and the study of more complex arguments.
In addition to propositional and predicate logic, there are other specialized branches such as modal logic, which deals with necessity and possibility, and fuzzy logic, which handles degrees of truth rather than strict binary values.
The study of logic has practical applications in various fields, including mathematics, computer science, philosophy, linguistics, and artificial intelligence. It provides a foundation for critical thinking, problem-solving, and rigorous analysis of arguments and reasoning processes.
Overall, logic plays a vital role in our quest for understanding and knowledge by providing us with a systematic and formal framework for reasoning and inference. It helps us identify fallacies, construct valid arguments, and make informed decisions based on sound reasoning.
