Herbert Roseman

When two Kantian agents engage in inter-dependent transactions with one another, each agent has the responsibility to assure the morality of the other’s maxim of action and means to implement the maxim. This proposition is proven to be a logical consequence of Kant’s moral system and stated formally as the K-Completeness theorem. Transactions that satisfy the theorem are called “K-Complete.” The theorem is generalized to cover transactions with many agents. The paper then models multi-agent K-Complete transactions using mathematical graph theory and shows that such transactions are distinguished from Kant’s kingdom of ends. Finally, the paper shows that the extended morality provides Kant’s moral system with the resources to define an ethical community in such as the one in Religion within the Boundaries of Mere Reason without the introduction of God and a church.

Spinoza’s geometrical approach to his masterwork, the Ethics, can be represented by a digraph, a mathematical object whose properties have been extensively studied. The paper describes a project that developed a series of computer programs to analyze the Ethics as a digraph. The paper presents a statistical analysis of the distribution of the elements of the Ethics. It applies a network statistic, betweenness, to analyze the relative importance to Spinoza’s argument of the individual propositions. The paper finds that a small percentage of the propositions greatly exceed the majority in this importance. The paper then describes two logical structures that appear respectively in the Ethics and argues that they result in redundancy in the sense that about ten percent of the propositions could have been eliminated. The appendices list these structures and describes how the resources of the study can be made available to readers.