What is logic? What is an “arbitrary variable”? What is a consequence relation? What is deduction? Why should the excluded middle be rejected? Are proofs mental constructions or are they Platonic entities out there somewhere? These are the sorts of questions I associate with the philosophy of logic. Other questions, related to the properties of logics (compositionaliy, soundness, etc), are just mathematics, often still pursued in philosophy departments, moving increasingly to computer science and pure mathematics.
I think sociophilosophy is a way to get back to the philosophy of logic. How does the process of constructing new logics actually work? From where do the ideas come? What impact does this have on the status of concepts like proof?
Think how the debates work, in research groups, between student and supervisor, at workshops. (Here follows a rapid sketch of Lakatos.) Conjectures suggested, refuted, adapted, proofs sketched, fixed, attacked, repaired, interactively… until a final proof is produced. The final proofs must have non-formal bits — otherwise there’d be infinite regress — so how do groups of mathematicians decide when something has really been proved? Why does proof work so much better than empirical methods even though there’s a social component? (Though sometimes things fall apart, e.g., the serious problem with Frege’s logic, discovered by Russell, or more recently the problem with Martin-Löf’s type theory logic, discovered by Girard.)
I think it’s revealing when you present a class with a bunch of sentences which just have in common that they are if-thens. Not everyone sees the same structure. And, has history as shown, different logicians and linguists also see different things in the very same natural language descriptions of situations.
Though inventing logics is a human activity, there’s something (social and physical?) Out There constraining what pops out. How does what’s out there do the constraining?
Now over to psychology.
Increasingly it’s seen as necessary (and obvious) that psychologists of reasoning take ideas from logic (many still resist). Though not just off the shelf. Logicians don’t care about psychology. One of the earliest discoveries, the figural effect, is bizarre when approached from logic. Look at how automated theorem provers work, for instance: miles away from how people do things. Obviously the maths logicians produce will not immediately model cognition. But there are important ideas in there. And, trivially, given a fixed interpretation of a task, logics give the right answers to the questions — but that’s the least interesting part, I think.
The relationship between psychologists and logicians is very similar to the relationship between biologists who care about how birds fly and aeronautical engineers who just want to build things that fly — by whatever means necessary. The well-rehearsed story goes somewhat like this. Biologists can learn about flight from the physicists and engineers, e.g., lift, drag, air pressure, and so on. They have no need to learn the details of the most recent jet engines. The engineers learned a bit by looking at the phenomena of flight. Wings still feature a lot, and nature got there first with that idea. However flapping the wings in a bird-like fashion was given up very early… And so on.
I think that when people wonder, why the hell should logic care about psychology, they’re in the same mode of thinking. And I agree. Engineers building reasoning systems have no reason to care about psychology, but they might get ideas — very important though primitive ideas which are very quickly generalised, polished, and forgotten.
But issues related to the philosophy of logic can be informed by psychology. For instance, how do people see and represent the world? How do they analyse what they see? How do constraints related to cognitive control and memory representations influence logical analyses. How do processes of abstraction work? How do groups of logicians get together to overcome their individual cognitive failings. What is the relationship between amateur (often implicit) and professional logic, i.e., the role of specialisation and expertise.