Logic, mathematics, and general agency

TLDR: In this paper, the authors bring the two perspectives together in a light and preliminary manner, and make a distinction between mathematical logic and philosophical logic, arguing that mathematical logic should still be the hallmark of logic at a meta-level.

Abstract: If logic is the general study of a priori valid reasoning, then where is the paradigmatic area where we see this reasoning in its full glory? To some, this is clearly mathematics, where precision is relentless, and strings of inferences are taken to impressive lengths. But on another view, the highest form of reasoning is displayed in the ordinary world of common sense – say, when engaging in conversation about something that matters, where pure information is deeply intertwined with evaluation and goals, and where, crucially, we are surrounded by further agents like us that we must interact with. On the first view, to simplify things a bit, logic is about mathematical proof and related processes like computation, making mathematical logic and foundations of mathematics the heart of the field. Agency is not even needed, and no human aspects are modeled. On the second view (frankly speaking: my own), logic is about interactive agency and all that entails, making philosophical logic and much more equally central to the discipline. The purpose of this brief note is to bring the two perspectives together – though admittedly, only in a light and preliminary manner. But before I do, let me make sure that I am not setting up the wrong debate. First, from the viewpoint of agency, there is no competition. Mathematics is an important special form of human cognitive behaviour – and the fact that it has developed historically out of our daily social planning abilities does not detract from its power and importance. Any general logic of agency must come to terms with our mathematical activities. Moreover, one can even grant that agenda contraction and restriction to a subdomain can be a winning move in terms of scientific progress: the more specialized concerns of the foundations of mathematics have had immense benefits for logic in general. Also, a distinction needs to be kept in mind here. It might well be that mathematical logic should still be the hallmark of logic at a meta-level, in