Is Classical Logic a Subset of Symbolic and Formal Logic?

Classical logic, often considered as the backbone of modern formal logic systems, holds a unique position within the broader categories of formal and symbolic logic. Understanding the relationship between these logical systems is crucial for anyone interested in the foundations of logic and its applications. In this article, we will explore the definitions of formal logic, symbolic logic, and classical logic, and how classical logic fits into these systems as a subset.

Formal Logic

Formal logic is a vast and inclusive category that encompasses various systems of reasoning that use formal languages and symbols to express logical forms and relationships. This category includes systems such as propositional logic, predicate logic, modal logic, and others. The term 'formal' in this context refers to the fact that these logics adhere to rigorous rules and structures, facilitating the investigation of logical principles in a systematic manner.

Symbolic Logic

Symbolic logic is often used interchangeably with formal logic. However, it typically emphasizes the use of symbols and formulas to represent logical expressions. This focus on notation and representation allows for the development of precise and unambiguous expressions of logical principles, making it easier to analyze and manipulate logical arguments. In the realm of classical logic, its principles are frequently expressed through symbols and formulas, thus falling under the category of symbolic logic.

Classical Logic

Classical logic refers specifically to traditional systems of logic, such as propositional and predicate logic, which adhere to principles like the law of excluded middle and non-contradiction. These principles are central to the structure of classical logic and distinguish it from other logical systems. Classical logic, therefore, is a foundational type of logic that can be expressed using symbolic notation, making it a subset of both formal and symbolic logic. As such, it is often referred to as Aristotelian logic, named after the Greek philosopher Aristotle, who initiated this study of logic.

Mathematical Logic and Misconceptions

There is a common misconception among some scholars and practitioners that the term "classical logic" refers to a part of mathematical logic as it is standardly used by mathematicians. This usage is profoundly misleading, as it directly contradicts the Aristotelian origins of classical logic. According to this perspective, classical logic would be in conflict with Aristotle's syllogistic and would not fall under the broader category of mathematical logic.

However, there are notable exceptions. For instance, Irving M. Copi, a prominent logician and professor, clearly delineates between classical logic and modern logic in his work. In Introduction to Logic, he defines classical logic, also known as Aristotelian logic, and modern logic, also called modern symbolic logic, as two distinct but related bodies of theory. His distinction is unequivocal:

The first is called classical logic or Aristotelian logic, after the Greek philosopher who initiated this study. The second is called modern logic or modern symbolic logic, developed mainly during the nineteenth and twentieth centuries.

This perspective is further supported by Copi's extensive work in the field of logic, which clearly demonstrates that classical logic is a semi-formal logic derived from Aristotle's original syllogistic. Despite this, it could theoretically be made entirely formal and symbolic, although this has never been fully realized in practice.

Conclusion

In summary, classical logic is indeed a subset of both symbolic and formal logic, adhering to the strictures and symbols of formal logic systems while maintaining the traditional principles that Aristotle introduced. The clear distinction made by logicians like Irving M. Copi emphasizes the importance of correctly defining and applying these logical systems. Understanding the nuances between these terms is crucial for any student of logic and serves as a foundation for more advanced study in the field.