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## On Definitions in Philosophy

When trying to define a term, we think generally of providing a set of necessary and sufficient conditions: a recipe for including or excluding a thing in a particular category of existence. For instance, an even number (definitions tend to work best in the mathematical arena, since definitions there can be as precise as possible) is definable as an integer that when divided by 2 does not leave a remainder. It is easy, given this definition, to ascertain whether or not a given number is even. Divide it by two and see if it leaves a remainder. If it does, then it’s not even; if it doesn’t, then it is. We have here a clear test for inclusion or exclusion in the set of even numbers.

Outside of mathematics, things get trickier. (Inside mathematics, things can be tricky as well. Imre Lakatos‘ excellent book Proofs and Refutations details some of the problems here. If you are mathematically and philosophically inclined, this is a must-read book.)

In Ludwig Wittgenstein‘s Philosophical Investigations, he famously talks about the travails of defining the term “game”. Is there a set of necessary and sufficient criteria that will let us neatly split the world into games and non-games? For instance, do all games have pieces? (No, only board games have these.) Winners and losers? (There are no winners in a game of catch.) Strategy? (Ring-around-the-rosie has no strategy.) Players? (Well, since games are a particularly human endeavor, it would be an odd game that had no human participants. But, of course, some games have only one player.) There seems to be no single set of characteristics that spans across everything we’d like to call a game. Wittgenstein’s solution was to say that games share a “family resemblance” — “a complicated network of similarities overlapping and criss-crossing”. A great many games have winners and losers, and so share this family trait; and then there are games that have pieces, and this is another trait that can be shared. Many (but not all) of the games with pieces also have winners and losers, and so there is significant overlap here. Games with strategy span another vast swath of the game landscape, and many of these games have winners and loses, many of which also have pieces. But not all. And so a networks of resemblances between games is found — not a single boundary that separates games from non-games, but a set of sets that is overlapping and more or less tightly connected.

This is a brilliant idea, but one that often leaves analytical philosophers with a bad taste in their mouths. If you try to formalize family resemblances (and analytical philosophers love to formalize things), you run up against the same problems as you had with more straightforward definitions. Where exactly do you draw the line in including or excluding a resemblance? Games are often amusing, for instance. But so are jokes. So jokes share one resemblance with games. But jokes are often mean-spirited. And so are many dictators. And dictators are often ruthless. As are assassins. So now we have a group of overlapping resemblances that bridges games to assassins. And if you want to detail the conditions under which this bridge should not take us from one group of things (games) to the other (assassins), you are back to specifying necessary and sufficient conditions.

Wittgenstein, I imagine, would have laughed at this “problem”, telling us that we just have to live with the vague boundaries of things. Which is all well and good, but is easier said than done.

## Knowledge

The defining of knowledge gives us a great example of definitions at work and their problems. For those of you who haven’t been indoctrinated in the workings of epistemology, it turns out that a good working definition for knowledge is that it is justified true belief.

Is Knowledge Justified True Belief
I take it as axiomatic as can be that something has to be believed to be known. If you have a red car but you don’t believe that it’s red, you don’t have knowledge of that fact. But, clearly, belief isn’t sufficient to define something as knowledge. For instance, if I believe that my red car is actually blue, I still don’t have any knowledge of its actual color. So we have to bring truth into the picture. If I believe that my car is red, and it is actually red, I’m certainly closer to having a bit of knowledge. But, again, this isn’t sufficient. What if my wife has bought me a red car that I haven’t seen yet. I believe it’s red because I had a dream about a red car last night. Do I have knowledge of my car’s color? I’d say not. We need a third component: Justification. If I believe that my new red car is indeed red because I’ve seen it with my own eyes (or analyzed it with a spectrometer, if the worry of optical illusions bugs you), then we should be able to say I do indeed have a bit of knowledge here.

In 1963, Edmund Gettier came up with a clever problem for this definition — one that presents a belief that is justified and true, but turns out to not be knowledge. Here is the scenario:

• Smith and Jones work together at a large corporation and are both up for a big promotion.
• Smith believes that Jones will get the promotion.
• Smith has been told by the president of the corporation that Jones will get the promotion.
• Smith has counted the number of coins in Jones’ pocket, and there are 10.

The following statement is justified:

(A) Jones will get the promotion and Jones has 10 coins in his pocket.

Then this statement follows logically (and is therefore also justified):

(B) The person who will get the promotion has 10 coins in his pocket.

But it turns out that the president is overruled by the board, and Smith, unbeknownst to himself, is actually the one will be promoted. It also turns out that Smith, coincidentally, has 10 coins in his pocket. Thus, (B) is still true, it’s justified, and it is believed by Smith. However, Smith doesn’t have knowledge that he himself is going to get promoted, so clearly something has gone wrong. Justification, truth, and belief, as criteria of knowledge, let an example of non-knowledge slip into the definitional circle, masquerading as knowledge.

## More Games

Let’s get back to the problem of defining games, and say that, contrary to Wittgenstein, you’re sure you can come up with a good set of necessary and sufficient conditions. You notice from our previous list of possible necessary traits that games certainly have to have players. Let’s call them participants, since “player” is something of a loaded word here (a player presupposes a game, in a way). And now you also take a stand that all games have pieces. Board games have obvious pieces, but so, you say, do other games. Even a game of tag has objects that you utilize in order to move the game along. (In this case, you’re thinking of the players’ actual hands.) So let’s add that to the list, but let’s call it what it is: not pieces so much as tools or implements. And perhaps you are also convinced that all games, even games of catch, have rules. Some are just more implicit and less well-defined than others. So let’s stop here, and see where we are. We have participants, implements, and rules.

And now we begin to see the problem. If we leave it at that, our definition is so loose as to allow under the game umbrella many things that aren’t actually games. A group of lab technicians analyzing DNA could fall under the conditions of having participants, implements, and rules. But if we tighten up the definition, we run the risk of excluding real cases from being called games. For instance, if we tighten the definition to exclude our lab workers from the fun by saying that games also have to have winners and losers we immediately rule out as games activities like catch and ring-around-the-rosie.

Lakatos coined two brilliant phrases for these definitional tightenings and loosenings: “monster-barring” and “concept-stretching”. Monster-barring is an applicable strategy when your definition allows something repugnant into the category in question. You have two options as a monster-barrer: do your utmost to show how the monster doesn’t really satisfy your necessary and sufficient conditions, or tweak your definition to keep the monster out.

Concept-stretching allows one to take a definition and run wild with it, applying it to all sorts of odd cases one might not have previously thought to. For instance, perhaps we should expand entry into the realm of games to include our intrepid DNA lab workers. What would that mean for our ontologies? And what would it mean for people who analyze games? And for lab technicians?

Philosophers love to define terms; they also love to find examples that render definitions problematic. It’s a trick of the trade and a hazard of the business.