# Formal Methods in Philosophy: Initial Thoughts and an Interactive Event (guest post by Liam Kofi Bright)

*Plausible answers as to the nature of our mission as philosophy educators gives us no unique reason to focus on logic as the mathematical tool of interest to philosophers.*

The following is a guest post* by Liam Kofi Bright (London School of Economics) about the justifications philosophers offer for requiring their students to have instruction in logic, over other formal methods, and about his role facilitating worldwide participation in an upcoming event on this topic.

### Formal Methods in Philosophy

by Liam Kofi Bright

Every year a great many philosophy departments force both graduate and undergraduate students to learn at least some mathematical logic. For these departments, some basic ability to deploy mathematical reasoning is part of the normatively expected skill set of the philosopher. What is more, we do not tend to insist on knowledge of other formal theories in the same way—logic is picked out as an especially relevant branch of mathematics. Why is that? There are two things I want to suggest about this. First, the justifications I have heard of for this would mandate making instruction in other formal tools or theories besides just logic obligatory. Second, the available justifications for this reflect deep and abiding disagreements concerning what constitutes good philosophy.

The first and most frequent justification one hears for our logic teaching is that we are bound to carry on the philosophical tradition wherein logic has played a big role. What’s more, given the realities of bureaucratic institutional inertia, probably this tradition does indeed have an outsized causal role in ensuring logic still gets taught. It is certainly true that logic has a long and storied history in philosophy, with logical traditions found not just among the Aristotelians and the Stoics but also sophisticated theories of argumentation developed in classical Indian and ancient Mohist philosophy.

However, the philosophical tradition also contains much reasoning in and about geometry (at least since the Meno!) and—since developments in early modern European philosophy—has contained plenty of probabilistic theorizing, too. If we are willing to bring the boundaries of the tradition closer to the present, then at least in the analytic tradition one may say that decision theory can fairly be traced back to pioneers of that tradition. Even setting that aside, if we assume that Plato and Pascal are firmly established in the philosophical tradition, it is not obvious that logic should get pride of place above these other mathematical fields. At the least, our focus on teaching logic as the one expected formal field of training in philosophy represents an elevation of some aspects of our tradition over others, and we might reasonably wonder why this is.

Another argument one often hears is that logic is an unusually useful field. Sometimes the claim is that logic is marketable in some broad sense: it looks good to employers, or draws in students from other departments, or trains people to recognize fallacies and so makes them better citizens. But sometimes the claim is that it is useful to philosophy more particularly: in philosophy we must offer arguments, and logic trains the mind to do this well.

With all such purported pragmatic benefits of logic, I see them more often asserted than proven; but even setting that aside, it is not clear why an emphasis on usefulness picks out logic in particular, for either philosophical specifically or broader marketability. Actual arguments one encounters in both philosophy and broader life, and the sort of mathematical skills which it is useful to have both as citizen and employee, are frequently probabilistic or inductive rather than strictly deductive. It is very rare thing indeed that philosophers establish arguments which are sufficiently complex that formal logical training might be useful for understanding them, yet which at the same time even purport to be deductively valid arguments. Much more often we are in the business of offering suggestive considerations, or amalgamating assorted facts which together render plausible one’s conclusion. These are more naturally read as probabilistic or inductive arguments than deductive ones, and where they suggest some practical course of action they are perhaps most naturally analyzed with the tools of decision theory. In life beyond academic philosophy, it will frequently be claims couched in statistical garb that one will encounter in the newspapers and the workplace. If we want to help students understand and construct the sort of arguments we actually make and encounter, they need to know more than just deductive logic.

Finally, people sometimes pick out logic as being especially generative of deep philosophical issues. It connects up naturally to questions about the nature of truth and rationality, for instance, as well as coming with a host of paradoxes to sink one’s teeth into. While there might inevitably be a subjective element to such judgments of interestingness, it is once again not clear that logic is alone in this. Probability theory may claim a similar intellectual worthiness, given the connection to deep philosophical questions mentioned above, and has its own paradoxes (e.g.) to play around with.

What is more, on reflection each of these answers hooks up rather naturally to a theory of what it is we are trying to do as philosophy educators. Maybe you think we are custodians of a tradition seeking to pass on knowledge of a canon; in that case, logic’s place in the tradition will seem salient to you. Or if what you instead think we do is provide people with intellectual tools, the utility of logic will seem the more natural thing to emphasize. Or finally maybe you think we are just folk with some interesting or deep issues that we can encourage students to think through and maybe even point towards good answers thereto. Then logic being a source of such questions will be what one wishes to highlight. Of course there is no suggestion that these are exhaustive options; I only wish to point out that these are not merely ad hoc claims about what we might do with logic. They tie in to deep theories about what it is we are doing in philosophy. Hence it seems that some plausible answers as to the nature of our mission as philosophy educators gives us no unique reason to focus on logic as the mathematical tool of interest to philosophers.

These are only hastily sketched answers to some difficult questions about the place and role of formal methods in philosophy. What raised them to salience for me is an upcoming event. Samuel C. Fletcher (University of Minnesota) has organized a two-day workshop that will “bring together diverse philosophers into conversation about the present and future status of formal and mathematical methods in philosophy, their institutionalization in graduate (and undergraduate) pedagogy, and how these changes now reflect and will engender evolving relationships between philosophy and other disciplines”. You can check out the details **here**.

At this workshop, which begins tomorrow, September 20^{th}, participants will be going into far greater depth about what it is we hope to achieve with formal methods and what their role is or should be in philosophy. My part in all this will be to play the hype man, with live tweets on my feed and retrospective blog posts to continue the discussion. In fact, my hope is that if anybody finds the topics interesting as I live tweet them, they can suggest questions to me on Twitter that I will ask the speaker and on which they can give real-time feedback! So if you would like to know just why it is we teach logic, and whether we ought perhaps to update or augment this focus with other formal tools, then I hope you will join in the conversation over the next few days, and check out the forthcoming blog posts.

Related: “Formal Methods Training for Philosophy Graduate Students“, “Stats Courses For Philosophers“, “What Should Philosophers Teach in Quantitative Reasoning Courses?“, “New Open Access Text On Probability & Decision”

I suspect that one motivation for the emphasis of logic and the de-emphasis of, say, probability theory, is that many philosophy professors/instructors don’t feel competent to teach the latter.

Ah that could well be, but I hope that if this were all there were too it that might be agreed to be an argument in favour of our slowly retraining ourselves to do better by our students.

Yes – Travis’s answer – which I suspect is a practical reason in many cases, even if not well articulated – seems like a variant of the inertia response. And, given the realities of how much time people have to learn things that may or may not help their careers, it may be rational for philosophers who can teach logic, but not probability theory, to stick with logic. But I wouldn’t see this as a reason for someone who can, or wants to, teach probability theory to teach logic instead, or for a department to insist on students studying logic instead of probability theory, or something else.

I mostly agree. The basics of probability theory (including Bayes’ theorem) aren’t that difficult, and can be taught as a section of a standard critical thinking course. Those basics can also enhance discussion of the lottery and related paradoxes.

Once you get past the basics, though, the material gets mathematically sophisticated quickly. Go calculate the values of a z- or t-table!

I thought one reason for philosophers to offer courses in logic–as opposed to geometry, probability theory, or statistics–is that university students can typically find excellent courses on the technical aspects of the latter subjects in other departments. But I imagine it’s pretty rare for other departments to teach an intro to logic or philosophical logic course.

This consideration is a far cry from establishing that logic should be a requirement for philosophy students. But it does seem to suggest that, given the current structure of the university, there may be some justification in philosophy departments offering technical courses in logic rather than the other subjects listed.

But there are two distinct questions here.

One is what philosophy departments should offer. And it’s a very good point that this question should be affected by thinking of our place in a larger system.

The other is what philosophy departments should require for progress through the degree, either at grad or undergrad level. A model where philosophy departments only offered logic courses, but the ‘logic’ requirement could be satisfied by taking a statistical methods course would probably satisfy a lot of the folks who want change around here.

Good point Brian, and also good point about the system thinking, Joe. Right now I don’t have any clear answers to these things, but I hope that by the end of this conference my ignorance and indecision will at least be a little bit better informed.

At least at some universities it could be institutionally problematic to allow classes other than logic, as taught be philosophers, to stand in for required classes, if this lead to decreased enrollment in philosophy classes. (At the university where I was an undergrad the math department offered a very good class that covered lots of formal logic and set theory, but it wouldn’t typically count towards a philosophy major or degree requirements.) Instrumental reasons like this are, perhaps, not very philosophically pleasing, but given the realities of most universities these days, I’d think that keeping student enrollment up would be something to consider. (Of course, if these “alternative to logic” classes can plausibly be taught by philosophers, that may be different, but there might again be questions about enrollment numbers and worries about stepping on the turf of other departments.)

This is very related to the broader question of what sort of formal methods requirements universities should have. In the United States, it has been traditional to require all undergraduates in all majors to complete a calculus class. It’s quite clear why this might be relevant for certain types of engineers. But it’s not clear why this should be the general requirement.

I believe many universities in the United States have broadened their requirements so that a number of different math classes might count. When talking to mathematician colleagues, they have often been easy to convince that a basic class on simple logic and statistical reasoning would serve much more of the general liberal arts role that mathematical reasoning is meant to play in producing some sort of educated citizenry.

For philosophy in particular, we presumably want something more than just the basic amount of literacy that an educated citizen is supposed to have. But the two questions seem quite parallel (though we have more control over the institutionally validated answer to one question than the other).

” In the United States, it has been traditional to require all undergraduates in all majors to complete a calculus class.”I think you might be surprised to learn that in quite a lot of universities this is not a requirement – and probably a good thing, too, as it would lead to much lower graduation rates. (I would be very surprised, in fact, if this was a strict requirement for even half of the university students in the US.)

Also, thanks for posting that amazing link to the gifs of Bertrand’s Paradox!

Haha I know right, glad you liked!

Like many other philosophy PhDs, I was required to take a logic course that culminated in Gödel’s incompleteness proofs.

Looking back now as a full and relatively accomplished member of the profession, that course turned out to be the least useful one I was required to take in grad school.

Worse yet, I think that course contributed to the imposter syndrome I was experiencing at the time more than any other.

Sorry to hear that Ed 🙁 I will try and ensure the overall question “But actually why have any formal requirements at all?” remains on the table at the conference – in fact I am chairing a roundtable discussion, so I can ensure it gets discussed by fiat!

Sounds good.

I think there’s a place for formal requirements. My hang up concerns formals requirements as they’re typically found in phil phd programs. What’s to be gained by studying the metalogic required to work through incompleteness for the typical philosopher? Vey little, in my experience.

The modal logic course I had to take probably had a similar effect on me: my lack of competence and preparation was far more obvious here than in other grad school classes, and required a lot more effort to become passable at it. For me at least, however, I think that’s all to the good. It gave me a certain respect for a branch of philosophy whose difficulty I hadn’t appreciated; it forced me to confront weaknesses of my own (both philosophical and personal); it reminded me that being good at writing papers or talking in seminars isn’t all there is to the skillset.

Bottom line: I doubt modal logic did much to train me for my eventual research, but it was more humbling than anything else on offer at my department.

One advantage I think logic has over at least some other formal methods is that it’s more approachable for students. Even if the students come in with negligible background in formal methods, an advanced undergrad or beginning grad class in logic can get to proving some fairly hefty and important results, such as the first incompleteness theorem or the soundness and completeness theorems for first-order logic. The concepts involved don’t require substantial background to understand, and the proofs—while nontrivial (well, the soundness theorem is trivial :P)—are approachable without a lot of extra mathematical training. On the other hand, consider e.g. probability theory. Important results such as the central limit theorem require extra tools. If you want your students to understand a proof of the CLT, at minimum they will need a solid calculus background.

So with a logic course students can both learn a formal method and get a good amount of exposure to the deductive methods for proving things about formal systems. Whereas I think many other topics in formal methods aren’t quite so friendly to students.

I think this observation could be the basis for an argument in favor of the teaching of logic. Suppose we think that philosophy students should get some exposure to formal methods, even if we don’t care too much about which, because we think they should be exposed to this style of thinking, having some basic familiarity with reasoning about formal systems, etc. Then we have to pick which formal subject best meets these goals. And logic ends up being a good fit.

As a related point, I suspect that many of the difficulties students have with logic are actually difficulties with formal methods in general. (Which is completely reasonable and understandable—it’s hard to learn a new way of thinking.) To steal a comment from a colleague, a student who struggles with filling out truth tables will likely also struggle with calculating probabilities.

But how useful are those metalogical results? It’s (somewhat) important to know *that* propositional logic is sound and complete, but the actual details of the proof aren’t relevant to much outside formal logic itself. Similarly, students doing an elementary probability/statistics course need to know what the CLT says, but I don’t think they need to be able to prove it. (As a physics student I’m pretty sure I never saw a proof of the CLT, even though I used it a lot.)

The idea, as i touched on my second and third paragraphs, is that it’s good for students to have some exposure to how we reason about formal systems. It’s not the details of the proof of X that matter, but rather them being exposed to how results like X are justified. (Some students will go on to further study in these areas, and they should be prepared to be able to make their technical contributions. And for those students who won’t, it’s still good for them to be exposed to a diversity of methods.) From this perspective, it doesn’t matter too much what the specific results are; so to speak, it’s the process, not the destination. Logic enjoys having results that are important—so the students see real world examples, rather than artificial constructs of no real interest—while still being fairly approachable.

In mathematics education, people talk about the general familiarity and experience with formal methods as “mathematical maturity”, e.g. saying that students should have sufficient mathematical maturity before taking class Y. It’s not that Y is going to directly build on knowledge from class Z, but rather that students need to be sufficiently comfortable with the style of thinking that will happen in class Y. They could learn this from class Z, but also class W or …

It’s this sort of phenomenon I’m talking about. You may think that philosophy students should be exposed to formal methods so as to gain some experience and familiarity with that style of reasoning, but not think the specific details of the material is important. I think that if that’s where you’re coming from, there’s still good reason for logic to be the class to make students take. (And I think you could directly argue for the value of logic, but I’m setting that aside.)

Of course, you could instead disagree completely with the value of teaching formal methods. That’s a larger issue—one which it sounds like might be discussed at this meeting. My remark was directed toward the person who is already convinced that we should teach some formal methods, but now the question is which to teach.

I have to disagree that the proof itself is not relevant or at least not as important as the statement.

In fact, the proof itself is way more crucial to our understanding than the statement/problem itself.

Just like when you study a particular philosophical position in, let’s say, epistemology, you pay attention to specific arguments, instead of ignoring the “why” in favour of statements themselves.

Well, of course- if your goal is a deep understanding of the metalogic. But most of the comments here have proposed that the reason for studying logic is more as a tool for thinking, reasoning and doing philosophy, and for those purposes the details of the proof are low priority. (Just as someone doing a calculus course doesn’t need to prioritize learning how to prove the intermediate value theorem, and someone doing an applied probability course doesn’t need to prioritize proving the CLT.)

I would hazard a guess that if I dropped into a random room in the APA and asked who could, on the spot, prove the completeness of the propositional (let alone the predicate) calculus, a fairly small minority could do it, and mostly those would either be logic specialists or people who teach it a lot.

I think there are very good reasons for requiring a logic course in philosophy. Most people are completely incapable of thinking things through patiently and sequentially, and most have very little or no experience being particularly attentive to anything. Many educated non-philosophers routinely get confused about the direction of if-then statements, and commit the fallacies of denying the antecedent or affirming the consequent on a regular basis. My own formal logic course, for one, works with students to help them avoid these errors and build up their strength at structured reasoning.

Are there other courses that might also have that effect? Sure, some mathematicians and even computer scientists teach things like that in their courses. For that matter, there are even some non-courses that teach that. Getting good at Sudoku puzzles or playing full-information games like chess, checkers or go can also help build up these skills and will even help people learn to avoid the formal logical fallacies. But is there a reason why philosophy departments should not teach that material, or should not require people to learn it? I don’t really see it.

A department may decide to just advise students to take similar courses in other departments, of course. I mean, such a move would generally cause a decline in departmental enrollments and might cost the department one or more faculty members. But if there’s a good reason for not requiring it — and I haven’t seen any such reason yet — then sure, follow your convictions and take the hit, I guess.

Second this. I’d just add that really good exercises in paraphrase give (in my experience) an additional benefit: they force students to think *very* carefully about what the sentences being paraphrased *say*. And getting good at that pays real dividends, when you’re doing philosophy! (Here’s a fun example, by the way, that shows that even sentential paraphrase can be usefully challenging: Paraphrase, in sentential form and using four sentence letters, “Billy will enroll in at least one of logic and Urdu, but only if he has the corresponding prerequisite.”)

Very good point, maybe logic comes with particular translational skills that are near unique to the field. I will try and find some opportunity to ask the participants in the conference on that, and see what they say!

I think part of the reason for the emphasis on formal logic is that philosophical questions were, for a long time, conceived of as conceptual questions. A conceptual relationship (say, between determinism and free will, or knowledge and true belief) will be a necessary truth or falsehood, not a statistical matter. So for arguing about classic philosophical (i.e. non-empirical) questions, you need classical logic and its offspring (counterfactuals, modal logic) not probability or statistics.

The conception of philosophical questions may be changing, though, and that might be a reason to revisit the curriculum. Certainly for practical everyday purposes, I think a decent statistical background is more useful than modal logic. (Related: knowledge of psychological biases is probably more useful than a list of classic informal fallacies.)

I think this is question of the relationship between the formal tools we use and substantial metaphilosophical beliefs about the nature of the field and the sort of claims we make is under-appreciated. I will try and find a way to ask about this too!

I think there may be other reasons for highlighting the formal aspects of philosophy, in particular logic and probability. As we know, the public’s general impression is that STEM fields are more relevant and financially lucrative than Humanities and the Liberal Arts. One of many ways that philosophers can dispel this notion is to be vocal about the ways in which our methods, not only overlap with those used in STEM fields, but may in fact provide students with a leg up over their peers who lack familiarity with logic. For example, learning translations involves many of the same skills needed for coding; and the relationship between decision theory, formal learning theory, and other aspects of probability theory are much needed at Tech companies like Amazon for calculating things such as the most profitable delivery routes. Finally, while many students are preparing to enter the job market with mostly technical skills, the critical and creative thinking skills involved in derivations and proving theorems provide students of philosophy with the much needed formal and informal knowledge that will help set them apart from the competition. Perhaps you have seen the many articles touting the hiring of philosophy majors at major Tech companies. Of course, this is also about the informal and inductive skills taught in our discipline, but we probably ought not undersell the importance of the formal aspects as well.

I think people are kinda unsure about how to respond to these sorta employability questions, but they’re definitely worth discussing! I will see if I can raise this over the course of the conference.

I agree completely with this post. Formal methods not only make students better thinkers, they make them more employable, and I think departments should expand there offerings of such courses (also, decision theory, game theory, etc. are super fun, and a lot of important work is being done as we speak.)

Yay, cheers! 🙂 Couldn’t agree more!

There are some good reasons given here for requiring logic, but I’m less sure that they’re good reasons for logic *in the way we teach it*. If the idea of logic teaching is (e.g.) to avoid fallacies and build up strength at structured reasoning, that seems to be a reason to focus on relatively informal aspects and skip most of the metalogic. If I were designing a formal-methods course from scratch, I’d include bits of propositional and predicate logic, not too much about formal proofs, some propositional semantics, quite a lot of translation into and out of formal languages, and then some bits of probability and statistics.

(I do think one problem is that at least in research universities, elementary logic tends to be the sort of thing that gets done by graduate students or adjuncts, who don’t generally have the institutional standing to do serious from-scratch thinking about what ought to be covered.)

I think that propositional logic – the syntax and semantics of the language, the proof system – is a fine thing to teach to philosophers. Philosophers think by writing and I am convinced that the precision of my own writing (in natural language) has been increased by having been taught propositional logic. (Natural language semantics and translations between the language of quantifier logic and natural language has also been useful, I think.) These artificial languages are so transparent – and the sort of work one does when constructing derivations in them requires paying careful attention to syntax. When our work is accomplished by writing, and our topics are murky, it is good to pay this sort of attention to our own writing.

Caveats: This sort of care is unlikely to *require* a training in logic. Moreover, it is an empirical claim that philosophy students would typically benefit in this way – but it is plausible to me. Also: none of this is to suggest that philosophy students ought not learn the basics of probability theory, for example.

I’m skeptical on proof systems in particular. Most proof systems are selected with at least half an eye on metalogic and end up with some counterintuitively complicated proofs – think of the proof of Pv~P in ND, for instance.

Any rule system that is small enough to easily memorize is likely to make for some counter-intuitively complicated proofs. (Moreover, restrictions on the expressiveness of the language will tend to make representing certain things counter-intuitively complicated.) This counter-intuitiveness is a teaching opportunity – to help students understand the limits of the “universe” of objects and functions the system represents.

At any rate, I find that textbook proof systems for propositional logic strike a nice overall balance between the various proof-system-building considerations – including allowing for sufficiently many intuitive-enough proofs. Maybe you don’t. Is this a basically “aesthetic” matter?

As long as the textbook rule-set is intuitive enough, I expect it to serve the purpose I identified in my comment.

Sorry I don’t have much to add, but just wanted to say thanks for replying and interesting discussion!

One reason for our emphasis on formal logic that you don’t mention is that one needs to be somewhat fluent in logic to read a substantial portion of the work that is published in philosophy journals, across quite a few subfields. This claim does not apply equally to students at all levels or who are working in all philosophical traditions, of course. But at least for graduate students and undergraduates who plan to go to graduate school, and at least for those who want to be able to read analytic philosophy, knowing propositional, first-order, and modal logic is a basic literacy requirement.

This is not to say that knowing lots of other technical work isn’t also a good idea. I’m all for that. And for sure, there is some work that one can’t engage with without knowing probability theory or geometry. (Depending on what you’re specializing in, you might also need to know about dynamic semantics or quantum physics.) It just seems to me that logic literacy of the kind taught in most philosophy departments is table stakes for being able to understand and engage with broad areas of philosophy in a way that those other formal methods aren’t.

Good point! This is what Josh Knobe’s first talk was on! Check out the twitter thread here; https://twitter.com/lastpositivist/status/1175056066595446787?s=20

One direct way to connect to students about the importance of propositional logic is to demystify the 2-to-the-nth structure of the memory systems of their smartphones by teaching them the semantics of truth tables. The basic smartness of their phones is just computationally Basic 1/0 T/F stuff. Impress on them how much more complicated expressions in ordinary language can be built upon simpler expressions and how truth works throughout. Emphasize the centrality of if/thens in conveying all kinds of semantic dependencies and especially how false antecedents imply anything on the 1/0 T/F interpretation, and thus the limitation of the truth-functionality of that in getting those dependencies straight. But as others have said here–tie it into real language to see how propositional/predicate logic constitute the real bones holding up the flexible muscle of communication. The philosophical approach to teaching logic is indispensable–if it strives very hard to connect real life talk and writing to the Ts and Fs.

Ha, I didn’t know this – cool!

Really interesting and important discussion here.

In asking which formal methods “we” should teach, there is an assumption that there should be some uniformity across departments. That seems justified, but we may have stronger reason to care about uniformity at an undergraduate level than at the graduate level.

At the undergraduate level, we, as representatives of an internationally recognized discipline, probably want to give students the impression that there are at least some unifying elements in the field. If you want to make the pitch that philosophy is worth studying, you need to present it as a coherent discipline, rather than an intellectual potpourri. From that perspective, Ned Hall’s point seems important. One thing that seems common to virtually all good philosophy is a heightened concern with getting the meanings of certain claims right. Learning a formal system for representing and paraphrasing natural language sentences imposes an acute awareness of truth conditions, and thereby deepens one’s appreciation for the subtleties involved in linguistic meaning. I take this line of thought to be one reason to maintain an undergraduate logic requirement. (Although, as a caveat, I’ll add that intro to logic classes are often badly designed, because they try to pursue three very different goals at once: (I) giving students and appreciation for the subtleties of NL semantics (ii) prep students for proof theory and meta-logic, and (ii) teach students how to argue better. In my view, these goals should be pursued independently, and the third is best accomplished by means of argument mapping rather than logic per se. )

At the graduate level, it seems more appropriate for departments to see themselves as representatives of a certain style of philosophy. Some styles are more formal; others less so. So, at the graduate level, we have less reason to impose standards across institutions.

The term “formal methods” hints at another assumption in this discussion that may be worth making explicit. It is that instruction in “mathy” fields should be undertaken in order to learn how to do some particular kind of philosophy. This seems like the right assumption to make about, for example, probability theory and formal epistemology. But one might want to learn some math, not because it contributes directly to paradigmatically philosophical issues, but because it helps you understand empirical work that might be relevant to your philosophical concerns. For example, a bioethics person might care about whether the evidential standards for bringing drugs to market are too high or too low. They should know some bio stats and experimental design. An epistemologist might be interested in climate change denial might need to better understand climate models. They should know something about differential equations. There are also mixed cases, where the formal apparatus is partially valuable because it addresses concerns traditionally considered within the purview of philosophy, and partially valuable because it improves one’s understanding of other fields. Here I think of decision theory, game theory, microeconomic modeling, etc.

In any case, if you see philosophy as at least partially a synthetic discipline, as I do, then it makes sense to encourage graduate students to get technical training outside of the philosophy department.

Thanks for the comment, many great points here! Just from my own experience the point about learning formal methods as a means of understanding stuff done elsewhere – my education was largely in philosophy of science, and I think that is basically the sub disciplinary norm nowadays. You just have to be at least comfortable reading and understanding the kind of arguments scientists (maybe: in the particular field you focus on) are making in their published work and talks and what not. The more integrated we are with other fields the more this sort of thing is going to become important throughout philosophy.

I don’t know what the major reqs are as I majored in computer science but 15 years ago at the University of Toronto probability and inductive logic was a standard 200 level course in philosophy along with symbolic logic and modes of reasoning (critical thinking with units on news, law, science, etc). While I didn’t take it because my major required a stats course (either the generic one or one geared towards computing) I imagine it’s conveying some of the same material, tuned to the problems of philosophy (maybe watered down compared to the stem supporting courses).

As a begining grad student, what top five books and/or articles wouold you reccomend getting (other than the usual logic textbooks) for training in formal methods in philosophy. Thanks much for your suggestions.

JJ