Traditionally, logic was regarded as a theory of correct reasoning. In this sense, logicians did not attempt to describe reasoning as it occurs in everyday life, but rather to establish the laws that any reasoning must follow in order to be valid. In other words, logic was expected to determine the principles that enable us to deduce truths from true premises.
However, in the second half of the nineteenth century, logic underwent a revolution that granted it a foundational role in mathematics, particularly in arithmetic. While the systematisation of correct reasoning was not entirely neglected, logic became a tool through which the fundamental principles of arithmetic could either be regimented or deduced.
Frege, Schröder, Peano, Russell, and Dedekind are key figures in this revolution; each conceived their own approach to understanding the relationship between logic and mathematics. Some systematised the basic principles of logic, viewed as an abstract theory, while others used these principles to define fundamental mathematical notions and derive core mathematical laws.
Despite the evolution of logic in the twentieth century and the first quarter of the twenty-first century, contemporary discussions regarding the nature of the basic components of logic and its role in modern mathematics continue to reference pioneers of modern logic. Historical work on the contributions of mathematicians such as Frege and Peano can be fruitfully applied to contemporary philosophical debates. A notable example is logicism, which has recently been reinvigorated after the seminal works of Frege and Russell, as a project to uncover the logical foundations of arithmetic. The history of late nineteenth-century logic, particularly the rational reconstruction of the works of its most representative figures, seeks to understand the nature of their logical contributions by considering their motivations, the philosophical and mathematical contexts they addressed, and the impact of their technical achievements. The development of formal languages — including a systematic way to express quantification, the separation of the content expressed by mathematical theories from their truth (which highlights the notion of a model), and the subsequent development of model theory are just a few notable examples of the contributions of nineteenth-century logic without which contemporary logic would not be the mature discipline it is today.
Philosopher
Currently researcher at the University of Lisbon