Relational logic—an introduction

Moving beyond propositional logic to the logic of relations

In the preceding posts we have studied propositional logic, the logic of propositional assertions, which helped us to reason with these assertions and to express precisely the logical relations between them, while providing notions of truth and validity. It was interesting and fruitful, and it opened up fascinating further questions that we investigated.

Perhaps it is fair to say, however, that we weren't ever actually much confused about that kind of propositional reasoning, and we might frankly admit:

Propositional logic is primitive.

According to propositional logic, after all, the entire world and all that one can say about it amounts to a list of brute atomic facts—a row in a truth table—and every assertion is a finite Boolean combinations of these atomic propositional facts. This is too simple, and the limited expressive power of propositional logic consequently seems to miss the fuller richness we experience in other kinds of worlds. We can easily imagine worlds brimming with individuals exhibiting diverse properties and standing in various relations with one another; but the truths of such a world do not seem to reduce to trivial finite combinations of atomic propositional facts.

We seek instead a logic with a more robust capacity for reference to specific individuals and the ability to express that an individual has a certain property or does not, or that one individual stands in a certain relation to others. We should like to formalize predicative expressions such as “Every boy in the math class has a sister on the chess team” or “For any two distinct real numbers, there is a rational number between them.” We need a logic and language that can handle reference, predication, and quantification over the individuals in a domain of discourse.

This is precisely what predicate logic provides. So let us embark on a study of predicate logic, beginning in this chapter with the special case of relational logic, where we study the logic of relations, investigating the elementary logical properties that a given relation may or may not exhibit.