Symmetric relations and logic

Suppose Alice and Bob are friends, and that friendship is a fundamental relation. Consider the facts expressed by these two sentences:

1. Alice and Bob are friends.

2. Bob and Alice are friends.

It is implausible that these are different facts. For if they were different facts, they would both be fundamental facts (otherwise, which of them would be the fundamental one?), and we would be multiplying fundamental facts beyond necessity—only one of the two is needed in the totality of fundamental facts.

Furthermore, I think that the propositions expressed by (1) and (2) are the same. Here’s one reason to think this. Imagine a three-dimensional written language where plural symmetric predicates like “are friends” are written (say, laser-inscribed inside a piece of glass) with “and are friends” on one horizontal layer, with “Alice” on a layer below the “and” and “Bob” on a layer above it. If (1) and (2) express different propositions, we would have to ask which of them is a better translation of the three-dimensional language. But surely there is no fact about that.

If this is right, then First Order Logic (FOL) fails to accurately represent propositions about fundamental relations, by having two atomic sentences, F(a,b) and F(b,a), where there is only one fundamental fact. Moreover, FOL will end up having non-trivial proofs whose conclusion expresses the same proposition as the premise, since we will presumably have an axiom like ∀x∀y(F(x,y)→F(y,x)) that lets us prove F(b,a) from F(a,b). This is not the only example of this phenomenon. Take the proof that ∀xF(x) follows from ∀yF(y), even though surely the two express the same proposition, namely that everything is F.

In particular, the logic of sentences appears to differ from the logic of propositions, since the proposition that Bob and Alice are friends follows by reiteration from the proposition that Alice and Bob are friends if they are the same proposition, but sentence (2) does not follow from sentence (1) by reiteration (nor is this true for the FOL versions).

If we think there is a One True Logic, it presumably will be a logic of propositions rather than sentences, then. But what it will be like is a difficult question, to answer which we will have to a worked out theory of when we have the same proposition and when we have different ones.

An argument that there are no non-explanatory facts

A stronger naturalism says:

1. Every fact is natural.

A seemingly weaker naturalism only says:

2. Every explanatory fact is natural.

But now I will give an argument that (2) implies (1). I am suspicious of the argument but it is hard to put my finger on why it’s wrong. The argument proceeds by arguing for:

3. Every fact is explanatory.

(Here, I take “explanatory” to go with partial, not necessarily complete, explanation.)

Here is the argument. Suppose, for a reductio, that there is a non-explanatory fact F. Now, some philosophers believe that there are non-explanatory facts. Suppose Alice is such a philosopher. Since (by assumption) there are non-explanatory facts, Alice correctly believes there are non-explanatory facts. But the obtaining of F partially explains why Alice is correct in her belief that there are non-explanatory facts. Thus, F is explanatory, which is a contradiction. Hence, there are no non-explanatory facts.

Qualia and value-laden why-questions

[Note added later: A version of this argument was first discovered by Kahane.]

Consider the famous story of Mary, the neuroscientist raised in a monochrome environment who finally sees an instance of red. It has famously been argued that no matter how much science she knew before she saw red, she learned something new when she saw red, and hence there is something more to the mental life than what science says. I've always been rather sceptical of this line of argument: it just didn't seem to me that a fact was learned.

But I am now thinking--as a result of a social experience--that there is an interesting way to argue that at least in some cases like Mary's one is learning a fact when one experiences a new quale. To know the answer to a why-question is to know a fact. After all, the answer to a why-question encodes an explanation, and explanations are given by means of facts.

Now suppose that Mary instead of leading a monochrome life led a charmed life and never felt any pain. One day she stubs her toe. She learns something by stubbing her toe: what pain feels like. But again we ask: is there a fact that Mary has learned? Here then is an argument:

1. By learning what pain feels like, Mary learned why pain is bad.
2. One learns why something is the case only by learning a fact.
3. So learning what pain feels like is learning a fact.

I give the pain version of the argument not because I find it very plausible, but because I think some readers will find it plausible. I myself am not inclined to think that pain is intrinsically bad, and the reasons why pain is extrinsically bad were available to Mary prior to her stubbing the toe (she knew that pain distracts people from worthwhile pursuits, that it tends to go against people's desires, etc.) But even if I am not convinced by the pain case, I find it pretty plausible that there will be some value-based case where by learning what a quale is like one learns the answer to a why-question. I find particularly plausible aesthetic versions of this. Here's a case where I've had the relevant aesthetic experience: "Why is dark chocolate gustatorily valuable? Because it tastes like that!" Here's one where I haven't. Being largely insensitive to music (more a matter of the brain than the ears, I think), I don't experience music like other people do, and so I don't know why Beethoven is a great composer, though I know on the testimony of others that he is a great composer. But there are possible experiences--namely, those that normal people receive upon listening to Beethoven--such that the what-it-is-like of these experiences answers the question of why Beethoven is a great composer.

Of course, these examples won't help a value-nihilist. But why would anyone be a value-nihilist? (A question with a hook.)