My normal tendency is to write long posts that I never finish. I’ll start off this series with small posts to see if I can break the habit.

The idea of Horn clauses emerged from model theory, so I will begin there. Model theory considers ideas that can expressed in very limited languages. You begin with a small vocabulary of constants, function symbols and relation symbols, known as the signature. For example, you can express the theory of groups in terms of two function symbols and one constant: the group product, the inverse function, and a constant that represents the unit. The theory of directed graphs can be described by a single relation symbol R(x,y) that expresses whether a directed edge begins at x and ends at y.

When coupled with first-order logic, even a very limited language can be very expressive. Set theory itself, for example, can be expressed using a single relation symbol (set membership). Horn clauses are special first-order logical statements that are not nearly as expressive, but still cover very many cases, as we shall see.

Previous post here. Next post here.

I was musing about the foundations of mathematics the other day, when it occurred to me that you could make a pretty good case that the key foundational idea of mathematics is that of Horn clauses (also known as universal Horn sentences). Horn clauses, despite begin obscure outside certain areas, are ubiquitous. Many (perhaps most?) basic mathematical objects can be described by Horn clauses. Fundamental category theoretic notions have Horn clause interpretations. Even first-order logic, which contains Horn clauses as a special case, can be viewed as having inference rules in the form of Horn clauses applied at the level of proofs.

I thought I’d spend a couple of posts describing Horn clauses, and laying out the case for their ubiquity.

Next post here.