An Introduction to Symbolic Logic

Logic is a branch of science that studies correct forms of reasoning. It plays a fundamental role in such disciplines as philosophy, mathematics, and computer science. Like philosophy and mathematics, logic has ancient roots. The earliest treatises on the nature of correct reasoning were written over 2000 years ago. Some of the most prominent philosophers of ancient Greece wrote of the nature of deduction more than 2300 years ago, and thinkers in ancient China wrote of logical paradoxes around the same time. However, though its roots may be in the distant past, logic continues to be a vibrant field of study to this day.

Modern logic originated in the work of the great Greek philosopher Aristotle (384–322 BCE), the most famous student of Plato (c.427–c.347 BCE) and one of the most influential thinkers of all time. Further advances were made by the Greek Stoic philosopher Chrysippus of Soli (c.278–c.206 BCE), who developed the basics of what we now call propositional logic.

Logician Alfred North Whitehead, co-author of Principia Mathematica (Source: Convergence Portrait Gallery)

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For many centuries the study of logic was mostly concentrated on different interpretations of the works of Aristotle, and to a much lesser degree of those of Chrysippus, whose work was largely forgotten. However, all the argument forms were written in words, and lacked formal machinery that would create a logical calculus of deduction with which it would be easy to work.

The great German philosopher and mathematician Gottfried Leibniz (1646–1716) was among the first to realize the need to formalize logical argument forms. It was Leibniz’s dream to create a universal formal language of science that would reduce all philosophical disputes to a matter of mere calculation by recasting the reasoning in such disputes in this language.

The first real steps in this direction were taken in the middle of the nineteenth century by the English mathematician George Boole (1815–1864). In 1854 Boole published An Investigation of the Laws of Thought, in which he developed an algebraic system for discussing logic. Boole’s work ushered in a revolution in logic, which was advanced further by Augustus De Morgan (1806–1871), Charles Sanders Peirce (1839–1914), Ernst Schröder (1841–1902), and Giuseppe Peano (1858–1932).

Logician Bertrand Russell, co-author of Principia Mathematica (Source: Wikimedia Commons)

The next key step in this revolution in logic was made by the great German mathematician and philosopher Gottlob Frege (1848–1925). Frege created a powerful and profoundly original symbolic system of logic, as well as suggested that the whole of mathematics could be developed on the basis of formal logic, which resulted in the well-known school of logicism.

By the early twentieth century, the stage was set for Bertrand Russell (1872–1970) and Alfred North Whitehead (1861–1947) to give a modern account of logic and the foundations of mathematics in their influential treatise Principia Mathematica. Published in three volumes between 1910 and 1913, Principia was a culmination of work that had been done on logic and the foundations of mathematics in the preceding century, and had a tremendous influence on further development of the subject in the twentieth century.

Our project, An Introduction to Symbolic Logic, uses the primary source Principia Mathematica to provide students with basics of propositional and predicate logic. The Latex source is also available for instructors who may wish to modify the project for students. The comprehensive “Notes to the Instructor” presented next are also appended to the project itself. Our primary source project module for students is part of a larger collection published in Convergence, and an entire introductory discrete mathematics course can be taught from various combinations selected from these projects. For additional projects, see Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science.

Notes to the Instructor

This project is a self-contained treatment of the topics from propositional and predicate logic typically covered in a first course in discrete mathematics. It may be used as a text for the logic unit of a standard one-semester course at the freshman or sophomore level, and should require approximately three/four weeks of class time to complete. Very few prerequisites are assumed. Students with a year course in calculus are more than prepared for the material contained here, and for the majority of exercises no more background than college algebra is required.

The exercises form a particularly important part of the project, and were designed to simultaneously provide students with practice applying the ideas discussed and to extend the discussion to new material. Several exercises introduce concepts from logic, number theory, the theory of relations, and other topics that are not normally covered until later in a discrete mathematics course. All such exercises are elementary, and may be taken as a stand-alone opportunity to study the primary material, or as an invitation to explore more advanced concepts. Because it is customary to cover logic at the beginning of a discrete mathematics course, the instructor may wish to begin with the material here, and use these exercises as a way of connecting logic to the material covered later in the course.

Several exercises (e.g., Exercise 2.8(c), the last question of Exercise 3.8, and Exercise 3.15(e)) are slightly open-ended. In our opinion, this stimulates independent thinking, as well as provides an opportunity for further in-class discussion. In our experience, such discussions enhance students’ understanding of the material.

Developing the logical skills necessary to read and write mathematical proofs is emphasized throughout. The instructor may wish to discuss the material covered here and proceed to introduce basic proofs using number theory or naive set theory. The instructor may even use exercises that discuss concepts from number theory or the theory of relations as a jumping off point to assign outside exercises on reading or writing proofs before the material here is completed. This provides an extremely quick way of exposing students to proofs.

Download the project An Introduction to Symbolic Logic as a pdf file ready for classroom use.

Download the modifiable Latex source file for this project.

For more projects, see Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science.

Acknowledgment



The development of curricular materials for discrete mathematics has been partially supported by the National Science Foundation's Course, Curriculum and Laboratory Improvement Program under grants DUE-0717752 and DUE-0715392 for which the authors are most appreciative. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.