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.

A quick way to question conjunctive accounts

Suppose someone proposes a philosophical account of the form:

• x is F if and only if x is G₁ and x is G₂ and x is G₃.

There is a quick way to question this that I think works most of the time. Just query the proponent: "What if the three conditions on the right hand side are satisfied merely coincidentally?"

The proponent can only give one of two answers while maintaining the biconditional: "Yes, x is still F when the conditions are satisfied merely coincidentally" or "The conditions are of such a nature that they cannot be satisfied merely coincidentally."

But it is implausible that that a coincidental satisfaction of conditions should suffice for a natural concept. Thus, if coincidental satisfaction of the conditions is sufficient for x to be F, pretty likely Fness is a stipulative rather than natural concept.

On the other hand, if the proponent insists that the conditions were so crafted that they cannot be satisfied coincidentally, it is likely that one of two possibilities is the case. The first is that the proponent lacks philosophical imagination, and you just need to think a little bit about how to make the conditions be satisfied coincidentally, and then you'll have a counterexample on hand. Just reflect a bit on Gettier-type cases, and if you're clever you should be able to find something. The second possibility is that the conditions are weaselly by including something like "relevantly" or "non-aberrantly". Here is an example of weaselly conditions:

• x knows p if and only if p is true and x believes p and x is justified in believing p and the anti-Gettier condition is met for x with respect to p.

These conditions cannot be satisfied coincidentally because the anti-Gettier condition is telling us that the first three conditions are satisfied non-coincidentally. But of course this is weaselly, since we aren't told at all about the kind of non-coincidentally that's required. Every coincidence is a non-coincidence from some point of view. So, really, such weaselly conditions need to tell us not just that the conditions are satisfied non-coincidentally, but that they are satisfied relevantly non-coincidentally.

Moreover, in the above example there is a pretty good chance that the weaselly final condition entails the other three. For what it says is basically that the other three conditions are satisfied in an un-Gettiered way! I think this isn't uncommon with weaselly conditions.

Note: In the example, one could try to formulate the weaselly condition as the denial of Gettiering: "and x is not Gettiered with respect to p". Then the weaselly condition wouldn't entail the other three. But then the resulting conditions would be too strict. For suppose that x has two sources of data on p. One source gives knowledge. The other gives Gettiered knowledge. Then x knows p but x is Gettiered with respect to p.

Philosophical accounts whose right hand sides are of the form

• x is F if and only if ∃y(G₁(x,y) and G₂(x,y) and G₃(x,y))

can use the quantification to avoid coincidentality sometimes, but often are subject to a similar criticism.

Cheating at analytic philosophy

X knows p if and only if X and p stand in the most natural relation whose extension in the actual world closely overlaps with the extension of justified true belief.

A general form of philosophical argument

This is a bit cynical, but while reading Spinoza I was really struck by the prevalence of the following implicit line of philosophical argument, not just in Spinoza:

1. My theory cannot handle Xs.
2. So, there are no Xs.

It seemed obvious to me that the thing for Spinoza to do was not to conclude that there is no contingency, but to conclude that his theory was inadequate to handle contingency.

I use this form of argument myself.  Perhaps too much.  It takes wisdom to know when the thing to say is that the theory is inadequate to handle Xs and when to conclude that there are no Xs.