In the decades around the turn of the 20th century, the science of logic underwent a remarkable transformation. The mathematical revolution overthrew traditional Aristotelian logic and established the modern tradition of mathematical logic. Driven by the desire to ground arithmetic in pure logic, Frege and Russell reshaped logic by applying mathematical concepts and methods. To serve as the basis of arithmetic, logic needed to be just as precise and rigorous. Natural languages were regarded as misleading for logical purposes, so Frege and Russell constructed artificial formal languages designed to make the logical forms of sentences explicit. This new approach transformed the field and led to the development of powerful formal systems. The key theoretical claims of mathematical logic became implicit assumptions, often taken for granted even in the philosophy of logic.
However, the success of the technical machinery of mathematical logic left many philosophical issues unresolved. According to natural logic, the main task of logic is to give an account of reasoning in natural languages. Since the 1950s, proponents of this approach have argued that the standard formal languages of mathematical logic fail to accurately capture natural language reasoning. Furthermore, they maintain that valuable features of traditional logic were lost during the mathematical revolution and ought to be revived. Key contributors to natural logic include Peter Strawson, Fred Sommers, Johan van Benthem, Lawrence Moss and Hanoch Ben-Yami. This loose tradition involves a synthesis of elements of traditional Aristotelian logic with modern linguistic evidence. Despite a shared goal and many areas of convergence, it so far lacks a unifying philosophical framework.
Project overview
The arguments for natural logic provoke critical reflection on the assumptions underlying modern mathematical logic. If we find the arguments to be sound, we have two options: modify the existing framework or develop a new one. But when we examine the philosophical foundations of mathematical logic, we find gaps and deep disagreements. Key concepts, such as logical form and logical consequence, remain obscure. We therefore have reason to return to first principles to ensure that the science of logic is properly grounded.
This project has two main aims. The first is to develop a philosophical framework that places natural logic on secure foundations. The second is to construct a theory of natural logic on those foundations. This theory incorporates insights from other systems of natural logic, including van Benthem’s monotonicity calculus and Ben-Yami’s quantified argument calculus (QUARC). These insights are embedded in a philosophical account of the subject matter and epistemology of logic.
The general approach is to revive the traditional idea that logic is concerned with concepts in the minds of individual thinkers. The resulting system is called cladian logic – “cladian” refers to the branching structures between concepts (from the Greek kládos, branch). Schopenhauer’s account of logic serves as a starting point for exploring what the traditional syllogistic theory got right. In his view, logic is concerned with the relations between abstract concepts in the faculty of reason. As he put it, “concepts are always the primary material of thought, and the forms of logic are always related to these.” In developing this idea to address the limitations of traditional logic, cladian logic incorporates a theory of concepts grounded in empirical evidence. This is a form of psychologism, which is generally thought to have been refuted by Frege and Husserl. The project engages with their anti-psychologistic arguments and reassesses logic’s relationship to both psychology and mathematics.
About
I’m Campbell Edgar, an independent researcher based in Melbourne, Australia. I studied philosophy at the University of Melbourne and completed an Honours thesis under the supervision of Dr Laura Schroeter. The thesis, Identity and Distinctness in Singular Thought, defends Strawson’s account of informative identity statements as a solution to Frege’s puzzle. My current research interests include logical consequence, concepts, consciousness, the foundations of logic, and the common sense philosophy of Thomas Reid.
