| Summary: |
In this unit we present an introduction to propositional logic, a branch of science that is fundamental in the study of mathematics and computer science. The origin of logic dates back to the 3rd century BC and the Greek philosopher Aristotle who developed the earliest form of logical theory through rules for deductive reasoning. Modern mathematical logic is generally recognised as having started with the work of German mathematician Gottfried Leibniz in the 17th century. In the 19th century two English mathematicians, George Boole and Augustus De Morgan, are credited with extending the work of Leibniz and introducing symbolic logic. Other notable contributors to the development of propositional logic include the mathematicians, Gottlob Frege in Germany and Charles Pierce in the USA.
The unit begins with a brief overview of some of the terminology that features in propositional logic and the main logical operators (connectives) that are used in the construction of propositions are discussed in detail. Two special types of proposition known as tautologies and contradictions that are respectively always true or false are then described. The concept of logical equivalence is presented before we look at translating propositions from English to their corresponding symbolic form and vice-versa. The idea of a truth table, introduced earlier during the discussion on connectives, is then presented in further detail and we demonstrate how these tables can be used to prove properties such as logical equivalence. We then discuss how logical equivalence can be used to simplify propositions, identify tautologies and contradictions and prove identities. Next we look at how to determine whether a mathematical argument is valid or invalid based on how well the premises support the conclusion. To close the unit we briefly look at the role logic in computing, including simplifying expressions in computer programming and system specification. |
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