Pragmatic conceptual analysis

Why think there is one definition?

One major thing philosophers try to do is to analyze concepts. Epistemologists try to say what it means to “know” something or for a belief to be “justified”; ethicists try to say what it is for someone to be “morally responsible” for an act; metaphysicians try to say what it is to “exist”. One method of conceptual analysis often used in the “analytic” tradition in philosophy is to propose some potential necessary and/or sufficient conditions for a concept to apply, and then search for counterexamples - either cases where the concept applies but a putative necessary condition doesn’t, or cases where the concept doesn’t apply but a putative sufficient condition does.

For instance, Edmund Gettier argued that having a belief that is justified, and also true, isn’t sufficient to have knowledge, by thinking of cases of justified, true belief, where we intuitively say the person doesn’t have knowledge - maybe someone has a justified belief that the Democrat will win the election, because they see polls showing the Democrat up in Georgia, North Carolina, and Arizona, but the polls turn out to be misleading, and the Democrat wins by carrying Wisconsin, Michigan, and Pennsylvania instead. Harry Frankfurt argued that the possibility of doing otherwise isn’t necessary for a person to be morally responsible for an act, by thinking of cases where we intuitively say the person is morally responsible even though they couldn’t have done otherwise - maybe there’s an evil wizard who would cast a spell to make them kick the puppy if they look like they aren’t going to, but they kick the puppy without the wizard needing to act.

This method is fun, but it tends to end up with lots and lots of counterexamples, and no successful analyses. Non-philosophers can get a sense of both the fun and the futility by trying to come up with an analysis of what it is to be a “sandwich”, and seeing how bogged down you get trying to find a definition that properly includes or excludes examples like hamburgers, hot dogs, tacos, open-faced sandwiches, bruschetta, pizza, etc.

John Rawls proposed a modification of known as the method of “reflective equilibrium”, where you can revise judgments about particular cases rather than just revising the analysis. Other philosophers have instead sometimes taken their relevant concept to be an unanalyzable primitive that can still be used in other philosophical theories. But most of these still work from the presumption that there is one concept under study (or a few specific ones) that has many different applications. There is one concept of “knowledge” that is appropriate for governing assertion, that we aim for in our belief, and that is most relevant for explaining behavior. There is one concept of “free will” that plays a role in moral responsibility and decision theory, and distinguishes humans from automata, and so on.

One pragmatist idea I have come to embrace is that we would do better to think of a different concept for each application. In some cases there isn’t even a clear range of definitions - just a paradigm case, and many different ways that other cases can be more or less like that one.

Mathematical concepts

I get much of my thinking about this from mathematics. There are some cases in which many different attempted mathematical definitions all end up being the same (for instance, the various definitions of “computable” that mathematicians gave in the early 20th century), and other cases in which many related mathematical definitions are not identical, but all fall under a single unifying concept (for instance, various attempts to define “prime” in number theory, algebra, and other areas).

But sometimes there are just importantly different versions of a concept for different purposes. For instance, the concept of “dimension” is defined in related, but distinct, ways for vector spaces, for manifolds, and for algebraic rings. These often coincide when there’s a simple relationship between the objects, but each extends to different cases that the others don’t apply to. Hausdorff’s definition of “dimension” turned out to be useful for understanding fractals as mathematical objects whose dimension is not an integer.

In other cases, mathematicians don’t even attempt a definition. We often start our mathematical education thinking that we know what a number is, but after moving from the natural numbers we use for counting to the positive and negative integers, and then the fractions, and then the real numbers we use for measuring, we start getting weirder generalizations. Thinking about algebraic properties of the reals and rationals, we move to the algebraic number fields, and then go further to complex numbers, and quaternions, and octonions. But Cantor showed how to generalize the counting uses of the natural numbers to infinite sets in two different ways, giving the ordinals and cardinals. And in trying to think about the infinitely small, Abraham Robinson came up with the hyperreals, and John Conway generalized this to the surreals. Mathematicians don’t worry about whether we’ve got the “right” concept of “number” - it’s just a term that is sometimes useful for pointing to certain objects that can be used for counting or measuring or something else like that. As long as you know what you are trying to do with numbers, you should be able to figure out which set is right for your purpose.

In his book, Proofs and Refutations, Imre Lakatos gives an explanation for how this multitude of definitions usefully arises in math. He considers the mathematical concept of a “polyhedron”, whose paradigmatic examples are the platonic solids:

Looking at these five, we can let V be the number of vertices (the “points”), F be the number of faces (the flat surfaces on its border), and E be the number of edges (the “lines” where faces meet), and see that for all of them, V-E+F=2. Near the beginning of the book is a proof that this formula always holds (roughly: poke a hole in some face and “pull it out until it’s flat”, then remove edges one by one, each time also either pulling two faces into one, or removing a vertex, so V-E+F stays the same during the process, and at the end you have V=1, E=0, F=1, so the total must have been 2 throughout the entire process). But then he quickly comes up with counterexamples:

For this one, there is a solid cube, with a cubic hole in the middle, so all the numbers are doubled compared to a normal cube, so V-E+F=4.

Lakatos calls this one a “picture frame” - its entire surface is connected, but we can count that V=16, E=32, F=16, so V-E+F=0.

Obviously, neither of these was “what we meant” when we thought of a polyhedron. Some might advocate just tacking on some property into the definition of “polyhedron” that rules them out. But Lakatos notes that it’s more productive to just see which step in the proof fails, and add that step as part of the definition. In both of these cases, the proof fails at the first step, when you punch a hole and try to fold it out flat.

He eventually considers more complicated counterexamples, like the one on the cover of the book:

(This one is not a counterexample if we think of the visible triangles as the faces - but he notes that each of these triangles sits in the same plane as four others, and we can think of the “faces” as being “pentagons” like this:

with only five vertices and five edges - though the edges cross through each other at points that look like vertices, and the faces pass through each other as well. This gives a different count on which the number doesn’t add up.)

Each of these examples ends up leading to a different way of generalizing the concept of “polyhedron”, that allows a proof of a different version of the theorem, useful for different purposes. Importantly, he argues that we get better concepts if we define each concept to fit the theorem we want to use it for, rather than if we just arbitrarily add conditions to the definition to rule out the particular counterexamples we have seen.

Ordinary concepts

We can see this same sort of thing go on with concepts in non-mathematical domains as well.

Consider what goes on when you try to measure two people’s height to see who is taller. If one person has really high heels, or a thick hairdo, we sometimes feel like it’s “wrong” to count those things when seeing who is taller. But I think we should consider why we care about who is taller.

If we’re in a crowded space, and want to know who can venture out into the crowd and still be visible from far away, then their shoes and hair are a perfectly good part of what we want to measure! If we want to know who will still be able to see us after they venture out into the crowd, then we should count their shoes but not their hair (in fact, we should just measure up to their eyes, not even the top of their head). If we’re wondering whether they can fit into a particular spacesuit, then we shouldn’t count their shoes, but probably should count their hair (at least, compressed - unless they plan to shave). There’s no reason that one single definition of “height” should be what matters.

This sort of distinction gets formalized in lists of the world’s tallest buildings. At the moment, the Burj Khalifa in Dubai is the tallest under any definition - but before it was completed, there were several candidates, depending on whether one counted the highest point of any part of the building, the highest architectural element (not counting removable spires), the roof of the highest occupied floor, or the highest occupied floor. There were also several candidates depending on whether we only count buildings that are usable most of the way up, or also count structures like the Köln Cathedral and the CN Tower that are unoccupied at most elevations, and also whether the building had to hold itself up or if we include radio towers supported by wires and deep-sea oil platforms supported by the water. In this case, it seems to me that there are fewer reasons to pick one definition rather than another - if you’re an aviator trying to fly over the building, or a tourist trying to find the highest observation deck, you have reasons to prefer one definition over another, but many of these others just seem like fruitless precisifications that are only relevant for bragging rights (and none of which is clearly better than another for these purposes).

Philosophical concepts

Returning to philosophical concepts, my thought is that all the proposed analyses of concepts like “justification” or “meaning” or “intention” get at something meaningful. But argument about which is the correct analysis is missing the point. There is no fact about which concept we were talking about before we started doing philosophy - likely on different occasions, we were emphasizing different aspects of what is relevant for the concept, and used the same term because it was close enough. There might be some cases in which there is a single analysis that is robustly more natural than all the others in the vicinity, like with “prime number”. But there will also be some where there are a small number of distinct nearby natural concepts (like ≤ vs <, or natural numbers including or excluding zero). And there will be some with a broad mass of concepts in the vicinity, with varying degrees of naturalness and relevance, like with “height” or “dimension”.