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*.

## 5 comments:

On (1) I think it's also the case that philosophers who do establish such limited equivalences have a strong incentive to give it an informal boost (i.e., not leave it at a limited equivalence, even if we can only give loose arguments). David Lyons's argument that rule utilitarianism reduces to act utilitarianism is a beautiful example of such a limited equivalence: it really does establish it, but only under a subsidiary condition (a particular principle governing how rule utilitarians use rules to comply with the principle of utility), and people who accepted as an unlimited argument just took it as plausible that that condition was universal among rule utilitarians. Part of this is, I think, exactly what you say; part of is that we often have targets in view and we want our arguments actually to hit the target, to show that it is false, rather than just rule out places where it might be, qualifying the ways in which it could be true. I think the method of counterexamples tends in philosophy to be a common area in which we tend to ignore subsidiary conditions, for instance, and I think this is the reason -- when we're giving counterexamples, we usually want counterexamples to establish that a conclusion is

falserather than just qualify the conditions under which it is true (assuming that the counterexample is not merely apparent, of course). When Aristotle (for instance) gives counterexamples, he's usually just trying to pin down what's required for a claim to be true, so it's not inherent to the method; it's just the way we tend to apply it.It occurs to me that there are two different ways of running the strategy in (1).

First, is to argue for equivalence GIVEN some unestablished assumption. E.g., proving that p1 and p2 are equivalent given some conjecture h.

Second, argue for the equivalence of two properties when they are conjoined with a third. I.e., instead of showing that P is equivalent to Q, show that P&R is equivalent to Q&R. The strategy of showing that some properties are equivalent in NORMAL cases is a version of this. E.g., in normal cases, knowledge and justified true belief are coextensive. I.e., necessarily, knowledge&normalcy is equivalent to justified-true-belief&normalcy.

Regarding (3), might not your points imply that perhaps some of our philosophical problems are just by-products of our language? For example, perhaps there exists a satisfactory analysis of the concept of knowledge, but we are unable to find it because our language lacks the expressive resources to state the concepts necessary for such an analysis.

You say in (3) that:

"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.)"Can we describe the real numbers in English? What about "the set of all cauchy sequences of rational numbers"? What if there were a countable "basis" of concepts that could be used to describe all concepts?

Mikhail:

That could be.

James:

One can describe the set of real numbers in English. E.g., like you did. But there are real numbers you can't describe in English, since any expression of English is finite in length, and there are only countably many expressions of finite length with a finite alphabet.

(Well, if gestures count, and one had perfectly well defined hands and a perfectly well defined unit of distance and perfect motor control, one could hold one's hands a certain distance apart and say "That exact distance apart" and then one would have described uncountably many. But even that's not just English.)

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