The Formal and Informal Nature of Logic by Rogelio G. Dela Cruz, Ph. D (January, 2008)

Logic is the study of the principles and criteria of valid inference and demonstration. Logic arose from a concern with correctness of argumentation. The conception of logic as the study of argument is historically fundamental, and was how the founders of distinct traditions of logic, namely Plato and Aristotle, conceived of logic. As a formal science, logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and through the study of arguments in natural language. The field of logic ranges from core topics such as the study of fallacies and paradoxes, to specialized analysis of reasoning using probability and to arguments involving causality. Logic is also commonly used today in argumentation theory.

Traditionally, logic is studied as a branch of philosophy, one part of the classical trivium, which consisted of grammar, logic, and rhetoric. Since the mid-nineteenth century formal logic has been studied in the context of foundations of mathematics, where it was often called symbolic logic.

In 1903 Alfred North Whitehead and Bertrand Russell attempted to establish logic formally as the cornerstone of mathematics with the publication of Principia Mathematica. However, the system of Principia is no longer much used, having been largely supplanted by set theory. As the study of formal logic expanded, research no longer focused solely on foundational issues and the study of several resulting areas of mathematics came to be called mathematical logic. The development of formal logic and its implementation in computing machinery is the foundation of computer science.

Form is central to logic. It complicates exposition that form in formal logic is commonly used in an ambiguous manner. Symbolic language is just one kind of formal logic, and is distinguished from another kind of formal logic, traditional Aristotelian syllogistic logic, which deals solely with categorical propositions. On the other hand, informal logic is the study of natural language arguments. The study of fallacies is an especially important branch of informal logic wherein the dialogues of Plato are a major example.

Furthermore, formal logic is the study of inference with purely formal content, where that content is made explicit. Formal logic is often used as a synonym for symbolic logic, where informal logic is then understood to mean any logical investigation that does not involve symbolic abstraction; it is this sense of 'formal' that is parallel to the received usage coming from formal languages or formal theory. In the broader sense, however, formal logic is old, dating back more than two millennia, while symbolic logic is comparatively new, only about a century old.