The Increasing Heights of Philosophical Logic Requirements

Learn Logic, My Son

A while back, I read Andrew Bacon’s A Philosophical Introduction to Higher-Order Logics. The book is basically what it says: an introduction to higher order from a philosophical point of view.

A higher order logic contrasts with a first order logic. A first order logic is basically the logic of individuals and properties. You can represent statements like “Jack is a dog” as $D(j)$. You represent statements like “A dog exists” as $\mathrm{\exists }xD(x)$. But the big sticking point of first-order logic is that you cannot directly represent statements like “Jack has a property.” In higher order logic, this is simple. $\mathrm{\exists }P.P(j)$.

Bacon and a crew of other philosophers have made a lot of the significance of higher-order logic for metaphysics, lately. I can’t go into detail here, but one boilerplate view is that higher order logic allows us to engage in more rigorous philosophical argument about (for instance) the notion of a property.

One thing I’ve noticed, however, is that higher order logic requires more mathematical resources than traditional metaphysics has had. Higher order logic seems like a small change, but it’s actually a big change, from a mathematical point of view. The mathematics involved in really understanding a higher order system go beyond what most graduate students are likely to be taught.

While I found the book helpful, it does strike me that higher order logic really takes philosophy to a level of technicality that I think many philosophers are simply going to avoid. It’s interesting because math-avoidance isn’t something that people openly talk about in philosophy. In many cases, the barrier to reading or understanding a philosopher’s work is that it contains relatively difficult math, and no one wants to say that they weren’t smart enough to get the math. There’s a lot of math anxiety in the profession, and the application of math to philosophy can seem too much like science, and many philosophers don’t want to be scientists.

This is all to say that: despite having a book introducing philosophers to higher order logic, I think it will be a while before higher order logic is something that the average philosophy student can be quickly and easily trained up on.