Charles S. Peirce is generally regarded today as one of the most out-standing philosophers in American history and more particularly as the inventor of pragmatism. He also discovered several of the most important concepts of twentieth century mathematical logic, including the quantifiers, the interpretation of first-order logic by means of relations, and the concept of logical consequence. There is very little general knowledge of Peirce's influence on the development of mathematical logic and almost total ignorance of the details of that influence. The main technical results of this book establish that Peirce laid down the main elements of a framework for the model-theoretic line of development in mathematical logic.