A quick heuristic for testing conjunctive accounts

Suppose someone proposes an account of some concept A in conjunctive form:

• x is a case of A if and only if x is a case of A₁ and of A₂ and ... of A_n.

It may seem initially plausible to you that anything that is a case of A is a case of A₁,...,A_n. There is a very quick and simple heuristic for whether you should be convinced. Ask yourself:

• Suppose we can come up with a case where it's merely a coincidence that x is a case of A₁,A₂,...,A_n. Am I confident that x is still a case of A then?

In most cases the answer will be negative, and this gives you good reason to doubt the initial account. And to produce a counterexample, likely all you need to do is to think up some case where it's merely a coincidence that A₁,A₂,...,A_n are satisfied. But even if you can't think of a counterexample, there is a good chance that you will no longer be convinced of the initial account as soon as you ask the coincidence question. In any case, if the answer to the coincidence question is negative, then the initial account is only good if there is no way for the conditions to hold coincidentally. And so now the proponent of the account owes us a reason to think that the conditions cannot hold coincidentally. The onus is on the proponent, because for any conditions the presumption is surely that they can hold coincidentally.

Consider for instance someone who offers a complicated account of knowledge:

• x knows p if and only if (i) x believes p; (ii) p is true; (iii) x is justified in believing p; (iv) some complicated further condition holds.

Without thinking through the details of the complicated further condition, ask the coincidence question. If there were a way for (i)-(iv) to hold merely coincidentally, would I have any confidence that this is a case of knowledge? I suspect that the answer is going to be negative, unless (iv) is something weaselly like "(i)-(iii) hold epistemically non-aberrantly". And once we have a negative answer to the coincidence question, then we conclude that the account of knowledge is only good if there is no way for the conditions to hold coincidentally. So now we can search for a counterexample by looking for cases of coincidental satisfaction, or we can turn the tables on the proponent of the account of knowledge by asking for a reason to think that (i)-(iv) cannot hold coincidentally.

Most proposed accounts crumble under this challenge. Just about the only account I know that doesn't is:

• x commits adultery with y if and only if (i) x or y is married; (ii) x is not married to y; (iii) x and y have sex.

Here I answer the coincidence question in the positive: even if (i)-(iii) are merely coincidentally true (e.g., x believes that he is married to y but due to mistaken identity is married to someone else), it's adultery.

Necessary and sufficient conditions

Both philosophers and mathematicians attempt to give nontrivial necessary and sufficient conditions for various properties. But philosophers almost always fail—the Gettier-inspired literature on knowledge is a paradigm case. On the other hand, mathematicians often succeed by the simple strategy of listing one or two necessary conditions and lucking out by finding the conditions are sufficient. And they do this, despite the fact that showing that the conditions are sufficient is often highly nontrivial.

Why do mathematicians luck out so often, while philosophers almost never do? Think how surprising it would be if you wrote down two obvious necessary conditions for an action to be morally wrong, and they turn out to be sufficient. And can philosophers learn from the mathematicians to do better?

1. Subsidiary conditions: Mathematicians sometimes "cheat" by only getting an equivalence given some additional assumption. A polygon has angles add up to 180 degrees if and only if it's a triangle, in a Euclidean setting. And such limited equivalences can still be interesting. While some philosophers accept such limited accounts, I know I often turn up my nose at them. I don't just want an account of knowledge or virtue that works for humans: I want one that works for all possible agents. Perhaps we philosophers should learn to humbly accept such incremental progress.

2. Different tasks: Philosophers often don't just ask for necessary and sufficient conditions. We want conditions that are prior, more fundamental, more explanatory. It may be true that a necessary and sufficient condition for an action to be wrong is that it is disapproved of by God, but that doesn't explain what makes the action wrong (assuming that the Divine Command theory is false). Moreover, sometimes we even want our necessary and sufficient conditions to work in impossible scenarios: we admit that God has to disapprove of cruelty, but we argue that if per impossibile he didn't disapprove of it, it would still be wrong (I criticize an argument like that here). This would be an absurd requirement in mathematics. "Granted, being a Euclidean polygon whose angles add up to 180 degrees is a necessary and sufficient for being a Euclidean triangle, but what if the Euclidean plane figure were a triangular circle?" The mathematician isn't looking to explain what a triangle is, but just to give necessary and sufficient conditions.

It is no surprise that if philosophers require more of their conditions, these conditions are harder to find. Again, I think we philosophers should be willing to accept as useful intellectual progress cases where we have necessary and sufficient conditions even when these do not satisfy the stronger conditions we may wish to impose on them, though I also think these stronger conditions are important.

3. Ordinary language is rich and poor: There are very few perfect synonyms within an ordinary language. There are subtle variations between the properties being picked out. Terms vary slightly in their meaning over time. But now necessary and sufficient conditions are very sensitive to this. Suppose that it were in fact true that x knows p if and only if x has a justified true belief that p. But now reflect on how many concepts there are in the vicinity of justification and in the vicinity of belief. Most of these concepts we have no vocabulary for. Some of these concepts were indicated by the words "justification" and "belief" in other centuries, or are indicated by near-synonyms in other other languages. If the English word "belief" were slightly shifted in meaning, we would most likely have no way of expressing the concept we now express with that word, and we would be unlikely to be able to give an account of knowledge. It can take great linguistic luck for us to have necessary and sufficient conditions statable in our natural language. Only a small minority of possible concepts can be described in English. (There are uncountably many possible concepts, but only countably many phrases in English.) What amazing luck if a concept can be described twice in different words!

I may be overstating the difficulty here. For sometimes the meanings of terms are correlated, in the way that vaguenesses can be correlated. Thus, "know" and "belief" may be vague, but the vaguenesses may neatly covary. And likewise, perhaps, "know" and "belief" can shift in meaning, but their shifts might be correlated.

Final remarks: The point here isn't that giving explanatory necessary and sufficient conditions won't happen, but just that it is not something we should expect to be able to do. And I should be more willing to accept as intellectual progress when we can do partial things:

1. give conditions that are necessary and sufficient but not explanatory
2. give conditions that are necessary and sufficient in some limited setting
3. give necessary but not sufficient conditions, or vice versa.

Conjunctive analyses

Sometimes we try to analyze a concept as a conjunction of two or more concepts. Thus, we might say that x knows p provided p is true and x justifiably believes p. Frequently, such proposed analyses founder on counterexamples—Gettier examples in this case.

I want to highlight one kind of failure. Sometimes analyzing x's being an F in terms of x's being a G and x's being an H, fails because to be an F, not only does x have to be a G and an H, but x's Gness and Hness have to be appropriately connected. While Gness and Hness are ingredients in Fness, their interconnection matters, just as one doesn't simply specify an organic compound by listing the number of atoms of each type in the compound, but one must also specify their interconnection.

I suspect this kind of connection-failure of conjunctive definitions is common. One way to see what is wrong with the justified true belief analysis of knowledge is to note that there has to be a connection between the justification and the truth and the belief. Specifying what the connection has to be like is hard (that is my understatement of the week).

Here's another case of the same sort. Suppose we say that an action is a murder provided it is a killing and morally wrong. Then we have a counterexample. Igor, who used to be a KGB assassin, has turned over a new leaf. As part of his turning over a new leaf, he has promised his wife that, no matter what, he will never kill again, no matter what. Maybe in ordinary cases that promise would be inappropriate. But given Igor's life history, it is quite appropriate. Now, Tatyana has just mugged Igor and is about to stab him to death so as not to leave any witnesses. Igor picks up a rock and kills her in self-defense. What he has done was a killing and it was morally wrong—it was the breaking of a promise. But it wasn't a murder because the connection between the fact that the action was a killing and the fact that the action was morally wrong wasn't of the right sort. (One might try to say that it was a killing and immoral, but wasn't immoral qua killing.)

When we hear a conjunctive analysis being given in philosophy, I think it's time to look for a connection-counterexample, a case where each conjunct is satisfied, but the satisfaction of the conjuncts lacks the right kind of interconnection. Sometimes, I think, one can intuitively tell that a proposed analysis is unsatisfactory for lack of such interconnection even without coming up with a counterexample. Here is a case in point. Consider the notion of "causal necessitation". A natural-sounding definition is this: an event E causally necessitates an event F provided that (i) it is nomically necessary that if E holds, then F holds; and (ii) E causes F. But even if it turns out that this is a correct characterization—that necessarily E causally necessitates F if and only if (i) and (ii) hold—I don't think it's a good definition. For it misses out the fact that one wants a connection between the necessitating and the causing—the co-presence of the two factors shouldn't be merely coincidental. But it's really hard to come up with an uncontroversial case where we have a difference between the two. (Interestingly, it may be possible to do so if Molinism is true.)

We are rightly suspicious of disjunctive analyses. I think we should have a similar, though weaker, suspicion of conjunctive ones.

There is a structural connection between the points in this post and Aristotle's Metaphysics H6. The point is also similar to Geach's discussion of the good. We cannot define a "good basketball player" as someone who is (i) good and (ii) a basketball player.

Analyses: a hypothesis

In philosophy journals, one occasionally sees things like this:

Necessarily, x is an F if and only if x satisfies each of the following n conditions:
(i) ...
(ii) ...
(iii) ...
(iv) ...
...

I hypothesize that every philosophical claim of this form that has ever been made in print by a Western philosopher with the number of conditions n greater than or equal to 4 is:

1. false, and/or
2. stipulative, and/or
3. circular, and/or
4. redundant.

By "circular" I mean that Fness is implicitly or explicitly found in the conditions. By "redundant" I mean that one of the conditions is entailed by the others.

My evidence for the hypothesis is inductive. I have never seen a correct, non-stipulative, non-circular and non-redundant set of necessary and sufficient conditions for anything philosophical where there are more than three conditions.

It could be that the hypothesis is false. Is there a counterexample?