A Completeness Theorem in Modal Logic (paper) by Kripke Saul

By Kripke Saul

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Of course, various ancient peoples studied argumentation—debate, reasoning, controversy, disputation, refutation, and deliberation—but these studies always mixed logical force with rhetorical force. Only Aristotle made logical force a subject of study in isolation from an argument’s other features. Other thinkers also studied the difference between genuine knowledge and mere belief, but this, too, is different from studying validity. (An argument can be valid even if none of its premises count as genuine knowledge, and an invalid argument can have genuinely known elements.

Again, why did the most important work on induction and scientific method only appear during a much later period—after the wars of religion in Europe in the sixteenth and seventeenth centuries and then, in a further burst of effort, in the years surrounding the First and Second World Wars? This time the answer lies in political turmoil provoked by the growth of trade. The rising commercial classes of early modern Europe instigated devastating fanatical violence in the sixteenth and seventeenth centuries, so much so that intellectuals like René Descartes undertook a new search for the rational foundations of belief.

The geography of trade is, in fact, the fundamental determinant in all periods of logic’s development; trade shapes the types of audiences that the logicians of any age will find waiting for them. And this was no less true for early modern Europe—in contrast to China—than for classical Greece. Europe was like China in trying to develop trade across the seas, but for China the consequences were different. During the fifteenth century, Chinese navigators sought to expand their commerce across the entire Indian Ocean, but in the end they found the effort unprofitable.

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