Philosophical Logic and Mathematical Logic in the PGR

Last week, Brian Leiter posted about possibly re-drawing the dividing lines between the specialty areas ranked in the Philosophical Gourmet Report

In a comment, Cian asks:

Philosophical Logic and Mathematical Logic. While there is a fair amount of divergence between the two rankings, I also see, for example, that NYU gets ranked at 4.5 in mathematical logic even though as far as I can see, almost everything the relevant people have written about logic is more naturally classified as philosophical logic than as mathematical logic. Is that a sign that the borderline between these two areas is too wide and fuzzy for the distinction to be worth making by the PGR?

If one thinks of mathematical logic as “formal logic motivated by mathematical concerns”—roughly, this is the conception according to which mathematical logic consists of model theory, set theory, recursion theory, and proof theory–then it is indeed puzzling why NYU gets ranked at the top of group 2 in the mathematical logic ranking, in addition to the top in the philosophical logic ranking. But one might also think of mathematical logic the way it’s defined in the AMS Mathematics Subject Classification (03). Now how do we take “philosophical logic”? But if we take the now-standard (at least in North America) definition of “philosophical logic”, then it’s that part of formal logic that paradigmatically includes: “various versions of modal, temporal, epistemic, and deontic logic; constructive logics; relevance and other sub-classical logics; many-valued logics; logics of conditionals; quantum logic; decision theory, inductive logic, logics of belief change, and formal epistemology; defeasible and nonmonotonic logics; formal philosophy of language; vagueness; and theories of truth and validity” (from editorial description of the Journal of Philosophical Logic). Almost all of that is included in MSC 03Bxx! Clearly, for the purpose of the PGR at least, it would be better to define “mathematical logic” as “formal logic, but not philosophical logic”.

I don’t know if the evaluators for the PGR “mathematical logic” and “philosophical logic” categories get instructions on the intended scope of the category. It probably also makes a big difference–bigger than in other specialty areas–if faculty with appointments in the mathematics (or computer science) departments get included in the PGR faculty lists. As the note at the bottom of the ranking says, “much work in mathematical logic goes on in Mathematics and Computer Science departments.”

Then there’s also the confusion between the definition of “philosophical logic” as “formal logic motivated by philosophy” and the older (British) use of “philosophical logic” to mean “philosophy motivated by logic” (and including philosophical study of notions such as reference, necessity, truth, analyticity, etc.) Maybe a better term for that is “philosophy of logic”. Leiter’s proposed restructuring would have a category “philosophy of language & logic” (but no philosophical logic category).

I don’t think that “philosophical logic” and “mathematical logic” should be combined in the PGR ranking. But as it currently stands–with the scope of these categories so unclear–the rankings aren’t particularly informative. I’m not sure what would be more informative, but getting clarity on the definitions would be one step. Maybe it wouldn’t be such a bad idea to include “philosophy of logic” in the “philosophy of language” category, and reserve “philosophical logic” for the formal work you find in the JPL or the Review of Symbolic Logic. Maybe it wouldn’t even be a bad idea to merge mathematical logic into the philosophy of mathematics category. What do others think?

6 thoughts on “Philosophical Logic and Mathematical Logic in the PGR

  1. I am pleased to see discussion on this topic. I briefly comment on three of the raised issues. A. I think that “philosophical logic” should not be subsumed under “phil lang and logic” any more than “phil lang” should be subsumed under “metaphysics”. The points of similarity/overlap are not enough to justify blurring over the points of difference. If current categories must be combined (I’m not convinced that they should be), then “phil of logic” could be combined with “philosophical logic”, even though there are philosophers working in each area who rarely (if ever) cross streams. B. I’m not sure what to do with “mathcal logic”. Again, there are definitely points of overlap but there are serious points of difference, particularly wrt motivation, typical research areas, etc. Again, if we /must/ remove some current categories (and I don’t see why we must), I suppose that “mathcal logic” could be dropped **with a clear note for students to check with the departments about availability of mathcal logic**. (E.g., we have a nice group here where an increasing number of students in philcal logic take classes/seminars in mathcal logic. This sort of stuff can be noted somewhere on a dept page. (We are just now getting around to actually putting the ‘official’ site up: – though this is still bare.) C. I think that Cian’s question highlights the real issue (one on which Richard and I briefly corresponded a year or so ago): what do evaluators /mean/ by the respective categories?! I am convinced that evaluators are using different enough meanings that it skews the rankings (i.e., that it over-inflates and under-inflates). Fortunately, I think that there’s a fairly simple solution: the specialty evaluators should simply agree on a definition. In the case of “philcal logic”, what Richard indicates is right, viz. “formal logic motivated by philosophical concerns (e.g., modal, dynamic, relevant, whatnot)” (and similarly for “mathcal logic”). This isn’t perfect, but it’ll do. NB: it’s probably also important to note that “logic”, at least in “philcal logic”, generally involves an account of what follows from what in the given language(s). So, e.g., in truth theories, one gives an account of what follows from what in a language(s) with “true” etc. Things are not quite like this in “mathcal logic”. (Pursuing degrees of unsolvability is perfectly normal activity in mathcal logic, but it isn’t exactly giving an account of consequence over a language, etc.) Moreover, “phil of logic” need not be either formal or concerned with consequence. Still, it is clearly different from much of mainstream phil language, at least in essential points of focus.

  2. A. Point taken, except that there’s no category “philosophy of logic” in the PGR, there’s only “philosophical logic”. So I’m guessing that in the minds of (some) evaluators and readers, “philosophical logic” and “philosophy of logic” already are combined in the PGR. But probably in the minds of some other evaluators and readers, “philosophical logic” in the PGR is just “philosophy of logic” in our definition, and “mathematical logic” is any kind of formal work, including what we call “philosophical logic” but also foundations of mathematics (proof theory, set theory). I also wonder where formal epistemology ends up in–a lot of it isn’t mathematical logic, but eg probablity theory. It would belong wih “philosophical logic” under our/the JPL’s definition, but not if you think of philosophical logic in the non-formal sense. There’s the “choice, decision, and game theory” category, but that might be too specific.

  3. After a bit of thought, I’m now fairly convinced that we should add “philosophy of logic” as a separate category, drop “mathcal logic” *if* a category need be dropped, and clarify “philcal logic” with a gesture towards the JPL/RSL or the like. (Of course, JSL/RSL seem also to be good places for philosophy of logic, but so too are Mind, Analysis, Australasian Journal of Philosophy, CJP, and other mainstream journals.) Departments that have active overlap with mathcal logic (e.g., active overlap with logicians in the Maths Dept or etc) can make this plain on their webpages. (E.g., Carnegie Mellon is obviously very good in mathcal logic, but they’re also good in philcal logic, and so they’d be on the specialty list and can advertise their mathcal strengths when students inquire.)D. Formal epistemology: I think that this is great example of a field that nicely intersects philcal logic and phil of logic. Good folks in the area often contribute to both philcal logic and phil of logic (witness Alan Hajek or Branden F. or the like). Still, I think that “formal epist” should also have its own category, and should, with “phil of logic”, be added to the list. (On the other hand, Branden seems to have an awfully broad account of “formal epist”, one such that “formal philosophy” might be better. There’s a worry about putting “formal phil” in the list of specialties, since this basically is a catchall for the more specific things we’re discussing — e.g., philcal logic, formal epist (on a narrower reading than I recall Branden giving at Banff). So, I advocate adding “phil of logic” and “formal epist”, taking away “mathcal logic” (if something needs to be taken away), and not adding “formal phil” since this is too broad. (Of course, I consider myself a “formal philosopher”, but the tag is way too broad to be of much use.)

  4. I am also quite pleased to see this topic being discussed, as the PGR categories could certainly use some fine tuning. As to the specifics of the issue, it seems to me that the AMS classification 03Bxx has a lot of merit, and that it rightly encompasses all that goes under the general heading of formal logic. The main distinction to be drawn concerns the methods, not the inspiration or the subject matter of the investigation. In this respect, “mathematical” logic and “philosophical” logic share the same method, their formal nature: definition are posited, consequences drawn in a purely formal fashion. The fact that the former is derived by predominantly mathematical inspiration whereas the latter by philosophical ones is less important in my view than the fact that they both proceed formally by establishing theorems. The more important distinction, then, from my point of view, is that between (philosophical) logic and philosophy of logic. The former is part of logic, the latter of philosophy, and they are distinguished by their methods.I would favor a separate PGR category for philosophy of logic (which I probably would not mind too much seeing grouped together with philosophy of language), as well as a category for “formal logic”, broadly construed, to include both mathematical and formal logic. Finally, I would like to call the readers’ attention to the memorable incipit of an old paper by Belnap and Grover:”Logic is many things: a science, an art, a toy, a joy. And sometimes a tool. […] Set theory has been developed for the mathematician, modal logic for the metaphysician, boolean logic for the computer scientist, syllogistic for the rhetorician; and the first order functional calculus for us all.”This seems to me to perfectly capture the correct notion of logic: it does not really matter what the application, the inspiration or the subject matter is, logic is characterized by its formal methods. This view enjoyed more currency at the beginning of the golden era of logic, when the ASL was founded in the 1930’s. Back then logic was just logic, as practiced by mathematicians and philosophers alike. That is why the association was called the Association for Symbolic Logic. Notice both the preposition (“for” instead of “of”) and the now somewhat quaint adjective “symbolic” (which we would probably now replace by “formal”). In my view, that is the spirit.

  5. This is just to say that, on the whole, I agree — I think — with Aldo’s position. The truth of the matter is a little difficult to figure out, but the practical line-drawing is probably most easily achieved per Aldo’s suggestion: open “formal logic” for both philcal logic and mathcal logic qua “formal logic”. In turn, open a separate category “philosophy of logic”. The one place where I (currently, tentatively) part ways with Aldo is whether “philosophy of logic” should go under/with “philosophy of language”. I think not. My reason: too much of current philosophy of language ignores issues in philosophy of logic, and some will probably (rightly) say the same of the other direction. I agree that some work in philosophy of language is hard to distinguish from philosophy of logic (and vice versa); however, there are enough points of difference to warrant — indeed, demand — different categories. One salient example: logical consequence. While philosophers of language should (and sometimes do) care about this topic, it is not seen as a central topic in the area. Wrt philosophy of logic, the topic of consequence (or perhaps proof, or etc) is of central importance. It seems to me a disservice to the fields to throw the two together. By way of synopsis, I repeat that I agree with Aldo on various issues, not least of which is the practical suggestion: let us add “formal logic” as a category to replace “philcal logic” and “mathcal logic”. In turn, let us add a category “philosophy of logic” to give the field its due, distinguishing it from phil lang and formal logic (i.e., philcal and mathcal logic).

  6. J.C., I am not at all opposed to keeping philosophy of logic and philosophy of language distinct, and concur that in spite of significant overlap they also have very distinct concerns.So looks like we are in agreement 🙂

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