I've just come back from the Formal Epistemology Workshop! It was a lovely conference, and I highly recommend it to up and coming formal epistemology folk who want to get a sense of what's going on across the field. I was struck by the diversity of projects, and also by the interesting fact that multiple people said something like ``I feel like I am the least-formal formal-epistemologist here.'' So! I invented a taxonomy of projects in Formal Philosophy, which I'll present here with examples then comment on below.
About -- some formal system that touches upon matters of prior philosophical interest is either itself the object of study, or some feature of it or result therein is, or it is useful for stating/reformulating a prior philosophical problem. One does not reason within the system, but rather one either reflects on it, or some aspect of it, or draws out morals from it and reflects upon how they bear upon another problem.
Examples of work of this sort: Bertrand Russell's On Denoting, Plato's Meno, Audrey Yap's Idealization, Epistemic Logic, and Epistemology, Jason Stanley's Know How, David Lewis' On The Plurality of Worlds, Kenny Easwaran's Why Physics Uses Second Derivatives, Beall and Restall's Logical Pluralism, Danielle Wenner's The Social Value of Knowledge and the Responsiveness Requirement for Biomedical Research, Michael Weisberg's Who Is A Modeller?
Within -- the author(s) themselves use an established formal framework to prove results which are of philosophical interest. Perhaps they are taken to be interesting because of what they tell us about the formal system which is itself taken to be philosophically interesting, or how various such systems can be related, or perhaps because the result is itself intrinsically interesting or part of a family of results which collectively are taken to be interesting. The point is that Within projects gain whatever philosophical interest they have because of the relationship between a result the author has proven and something which is taken to be of philosophical interest.
Examples of work of this sort: Ruth C. Barcan's The Identity of Individuals in a Strict Functional Calculus of Second Order, Christian List and Philip Pettit's Aggregating Sets of Judgments: An Impossibility Result, Robert Stalnaker's On Logics of Knowledge and Belief, Catrin Campell-Moore's How To Express Self-Referential Probability, Cailin O'Connor's The Evolution of Guilt, Bertrand Russell's eponymous Paradox, the Marquis de Condorcet's Jury Theorem (and related results), Amartya Sen's The Impossibility of a Paretian Liberal, David Lewis' Probability of Conditionals and Conditional Probabilities, I. J. Good's On The Principle of Total Evidence, Harsanyi's Utilitarian Theorem (and related results).
Without -- the author invents or constructs a novel formal system that allows us to generate results or extract information about a new area of discourse not previously amenable to formal analysis, or which if there was a previous formal theory it took a markedly different form.
Examples of work of this sort: Aristotle's syllogistic, Russell and Whitehead's Principia Mathematica, Turing and Post on computability, Ruth Barcan's A Functional Calculus of First Order Based on Strict Implication, Kripke's A Completeness Theorem in Modal Logic, Carlos E. Alchourrón, Peter Gärdenfors and David Makinson's On The Logic of Theory Change, David Lewis' Convention, Frank P. Ramsey's Truth and Probability. Peter Spirtes, Clark Glymour, and Richard Scheines' Causation, Prediction, and Search.
Some comments on this, starting with remarks about the examples.
1 ) While I didn't put too much thought into constructing the example lists (which probably resulted in a demographic skew, alas, in what I highlighted -- on this see point (7) below) I did want to highlight a couple of points. Formal philosophy as a whole interacts with very diverse areas of philosophy and very diverse sets of formal tools. As has recently been discussed, logic gets the bulk of pedagogical attention in philosophy graduate programmes. But at a glance the above list contains work in ethics, social and political philosophy, metaphysics, philosophy of language, philosophy of physical, social, and biological sciences and mathematics, and epistemology. (I don't know of any formal aesthetics, but I would have liked to have been able to include that.) And the formal theories touched upon or deployed do indeed include logic, but also include probability theory, game and decision theory, statistical reasoning, calculus, social choice theory, and geometry.
2) I also wanted to use the examples to highlight that each of the sections contains work that would presumably be thought of as classical or canonical, as well as recent work by younger scholars.... This latter was a bit harder for the third section, since (for, I guess, reason discussed in (4) below) one hears about such work less. I decided that in the grand scheme of intellectual history though late 20th century is extremely recent philosophy, so it suffices to make my point. Which is that each of these modes of formal philosophy has shown itself both capable of making classic contributions and generating novel work. This is not a hierarchy of value, and none of these streams are yet dry. (When I reflected on my own work, I think I have some papers in the About category, and some papers in the Within section.) On the flip side, each of these is part of the grand tradition of formal philosophy, and there's no reason to think that some is more properly formal philosophy than the rest.
3) These are of course fuzzy categories. Gödel's incompleteness theorems are to some extent Within, but (if I understand the history correctly) the technique of Gödel-numbering was developed during these proofs and that was probably a significantly novel enough contribution to be its own Without work. For more recent work, I wanted to include more ethics, but couldn't decide whether this was About or Within. Is this Within or Without? Nothing, I think, really turns on subtleties here, I just wanted to acknowledge that there are plenty of edge cases. However, just to give people something to disagree with me about: I hereby claim that while this is fuzzy in the sense that some work can plausibly be in multiple categories, anything that could be called Formal Philosophy will recognisably fit into at least one of the categories.
4) While I don't think this is a hierarchy of value, my sense is that in terms of credit or repute the Without category is the high-risk high-reward category. It's the kind of work that is most likely to fail, but most likely to secure one's lasting glory if one can pull it off.
5) Work in the About category is probably the easiest sell to philosophers who don't work in formal philosophy. When formal philosophers are designing introductory lectures, outreach-y summer programmes, presentations for conferences in which there will be mixed company, or just in general interacting with a field that can be territorial and sceptical about things which fall outside the recognised boundaries, I think there is some reason to be cognisant of the distinction between About and Within work, and opt for About work. Nobody is going to argue that Plato's Meno isn't real philosophy.
6) I am less confident here, but I know there are metaphilosophical debates about what counts as experimental philosophy. I feel like a similar taxonomy would work there, with experimental philosophy being work that either is centrally based upon reflections on empirical work, is founded upon novel discoveries made by the authors, or comes up with a new way of testing things or generating results.
7) I didn't max out on demographic diversity in constructing the example lists, since it wasn't really to my point here. But I did find when making the lists that white blokes came to mind much more easily in all of the categories, and I guess especially dear to my heart given previous work -- I could scarcely think of any black folk! On reflection I can think think of more I didn't include -- for instance Kwasi Wiredu's Logic and Ontology (I cannae find a link!) could have gone in the About section, and I just met Lisa Cassell at the Formal Epistemology Workshop that sparked this very post. Still, even as I try hard not many brothers and sisters come to mind. This does not tell one much about the actual demographics of the field -- maybe I am just bad at remembering people, and I am myself very much trained in a certain tradition. But it is what it is.