Diagrams and the Concept of Logical System

In attempting to analyze the notion of a logical system, there are various approaches that could be taken. One would be to look at the things people have called logical systems and try to develop a natural framework which would encompass most or many of these, and then explore the consequences of the framework, seeing what else falls under the framework and what the consequences of the general notion happen to be. This was basically the approach taken, for example, in Barwise [1974], one of the early attempts to develop such a framework. This approach has much to recommend it, but it also has at least two serious drawbacks. It is too dependent on accidents of history, that is, on what particular systems of logic people happened to have developed. There is at least the theoretical possibility that biases of precedent and fashion have played a significant role in the way things have gone. If so, the abstraction away from practice has the danger of codifying these historicallycontingent biases, making them appear like necessary features of a logical system. The flip side of this problem is that there may well be some unnatural logical systems which contort the framework. But how else could one proceed in an attempt to get a principled notion of logical system? Another approach, the one we take here, is to look at the existing logical systems that people happened to have developed and to try to see what they were up to in more general terms. Our hope is to find some interesting natural phenomenon lurking behind these systems, a “natural kind,” if you will. If there is such a natural phenomenon, it could be used to guide the formulation of an abstract notion of logical system. If a characterization of logical systems could be found using this approach, it would have potentially two significant advantages over the more orthodox approach. First, it would provide a basis from which one could give a principled critique of existing systems claimed to be logical systems. Second, though, it would point out gaps, that is, logical systems which have yet to be developed.