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:
- give conditions that are necessary and sufficient but not explanatory
- give conditions that are necessary and sufficient in some limited setting
- give necessary but not sufficient conditions, or vice versa.