Naturalness and induction

David Lewis’s notion of the naturalness of predicates may seem at first sight like just the thing to solve Goodman’s new puzzle of induction: unlike green, grue is too unnatural for induction with respect to grue to be secure.

But this fails.

Roughly speaking, an object is green provided its emissivity or reflectivity as restricted to the visible range has a sufficiently pronounced peak around 540 nm. But in reality, it’s more complicated than that. An object’s emissivity and reflectivity might well have significantly different spectral profiles (think of a red LED that is reflectively white, as can be seen by turning it off), and one needs to define some sort of “normal conditions” combination of the two features. Describing these normal conditions will be quite complex, thereby making the concept of green be quite far from natural.

Now, it is much easier to define the concepts of emissively black (eblack) and emissively white (ewhite) than of green (or black or white, for that matter) in terms of the fundamental concepts of physics. And emeralds, we think, are eblack (since they don’t emit visible light). Then, just as Goodman defined grue as being observed before a certain date and being green and or being observed after that date and being blue, we can define eblite as existing wholly before 2100 and being eblack or existing wholly after 2100 and being ewhite. And here is the crucial thing: the concept of eblite is actually way more natural, in the Lewis sense of “natural”, than the concept of green. For the definition of eblite does not require the complexities of the normal conditions combination of emissivity and reflectivity.

Thus, if what makes induction with green work better than induction with grue is that greenness is more natural than grueness, then induction with eblite (over short-lived entities like snowflakes, say) should work even better than induction with green, since ebliteness is much more natural than grueness. But we know that we shouldn’t do induction with eblite: even though all the snowflakes we have observed are eblite, we shouldn’t assume that in the next century the snowflaskes will still be eblite (i.e., that they will start to have a white glow). Or, contrapositively, if eblite is insufficiently natural for induction, green is much too unnatural for induction.

Moreover, this points to a better story. Lewisian unnaturalness measures the complexity of a property relative to the properties that are in themselves perfectly natural. But this is unsatisfactory for epistemological purposes, since the perfectly natural properties are ones that we are far from having discovered as yet. Rather, for epistemological purposes, what we want to do is measure the complexity of a property relative to the properties that are for us perfectly natural. (This, of course, is meant to recall Aristotle’s distinction between what is more understandable in itself and what is more understandable for us.) The properties that are for us perfectly natural are the directly observable ones. And now the in itself messy property of greenness beats not only grue and eblite, but even the much more in itself natural property of eblack.

This can’t be the whole story. In more scientifically developed cases, we will have an interplay of induction with respect to for us natural properties (including ones involved in reading data off lab equipment) and in themselves natural properties.

And there is the deep puzzle of why we should trust induction with respect to what is merely for us natural. My short answer is it that it is our nature to do so, and our nature sets our epistemic norms.

Disjunctive predicates

I have found myself thinking these two thoughts, on different occasions, without ever noticing that they appear contradictory:

1. Other things being equal, a disjunctive predicate is less natural than a conjunctive one.

2. A predicate is natural to the extent that its expression in terms of perfectly natural predicates is shorter. (David Lewis)

For by (2), the predicates “has spin or mass” and “has spin and mass” are equally natural, but by (1) the disjunctive one is less natural.

There is a way out of this. In (2), we can specify that the expression is supposed to be done in terms of perfectly natural predicates and perfectly natural logical symbols. And then we can hypothesize that disjunction is defined in terms of conjunction (p ∨ q iff ∼(∼p ∧ ∼q)). Then “has spin or mass” will have the naturalness of “doesn’t have both non-spin and non-mass”, which will indeed be less natural than “has spin and mass” by (2) with the suggested modification.

Interestingly, this doesn’t quite solve the problem. For any two predicates whose expression in terms of perfectly natural predicates and perfectly natural logical symbols is countably infinite will be equally natural by the modified version of (2). And thus a countably infinite disjunction of perfectly natural predicates will be equally natural as a countably infinite conjunction of perfectly natural predicates, thereby contradicting (1) (the De Morgan expansion of the disjunctions will not change the kind of infinity we have).

Perhaps, though, we shouldn’t worry about infinite predicates too much. Maybe the real problem with the above is the question of how we are to figure out which logical symbols are perfectly natural. In truth-functional logic, is it conjunction and negation, is it negation and material conditional, is it nand, is it nor, or is it some weird 7-ary connective? My intuition goes with conjunction and negation, but I think my grounds for that are weak.

The magical

Magical things happen. The glorious glow of the sunset, the elegant glide of the turkey vulture or the delight of conversation with friends. Such experiences of the magical are what gives life its zest.

To experience these things as magical is to experience the events as going over and beyond the merely natural. Thus, if naturalism is true, such experiences are all deceptive. And that makes naturalism a very dour doctrine indeed.

Yet even if naturalism is false, how can these experiences be of events going beyond the merely natural? A sunset is, after all, just light refracted in the atmosphere as the part of the earth on which one stands turns away from the sun. So there is something more going on than the physics describes. This something more could be intrinsic or relational or both. Perhaps the sunset reflects something much deeper beyond it. Or perhaps there is more in the very sunset than the physics describes.

Vague mental states

I've been thinking through my intuitions about vagueness and mental states, especially conscious ones. It certainly seems natural to say that it can be vague whether you are in pain or itching, or that it can be vague whether you are sure of something or merely believe it strongly. But I find very plausible the following mental non-vagueness principle:

• (MNV) Let M be a maximally determinate mental state. Then it cannot be vague whether I am in M.

MNV is compatible with the above judgments. For if I am in a borderline case between pain and itch, it is not vague that I have the maximally determinate unpleasant conscious state U that I have. Rather, what is vague is whether U is a pain or an itch. Intuitively, this is not a case of ontological vagueness, but simply of how to classify U. Similarly, if I am borderline between sureness and strong belief, there is a maximally determinate doxastic state D that I have, and I have it definitely. But it's vague whether this state is classified as sureness or strong belief.

Interestingly, though, MNV is strong enough to rule out a number of popular theories.

The first family of theories ruled out by MNV is just about any theory of diachronic personal identity that allows personal identity to be vague. Psychological continuity theories, for instance, are going to have to make personal identity be vague (on pain of having a very implausible cut-off). More generally, I suspect any theory of personal identity compatible with reductive materialism will make personal identity be vague. But suppose it's vague whether I am identical with person B who exists at a later time t. Then likely B has, and surely could have, a maximally determinate mental state M at t that definitely nobody else has at that time. Then if it's vague at t whether I am B, it's vague at t whether I have M, contrary to MNV.

I suppose one could weaken MNV to say that it's not vague whether something is in M. I would resist this weakening, but even the weakened MNV will be sufficiently strong to rule out typical (i.e., non-Aristotelian) functionalist theories of mind. For suppose that my present maximally determinate mental state M is constituted by computational state C. But now imagine a sequence of possible worlds, starting with the actual, and moving to worlds where my brain is more and more gerrymandered. Just replace bits of my brain by less and less natural prosthetics, in such ways that it becomes more and more difficult to interpret my brain as computing C. (For instance, at some point whether something counts as a computational state may depend on whether it's raining on a far away planet.) Suppose also nothing else computing C is introduced. Then there will be a continuum of worlds, at one end of which there is computation of C and at the other of which there isn't. But it would be arbitrary to have a cut-off as to where M is exemplified. So it's vague whether M is exemplified in some of these worlds, contrary to MNV.

Moral reasons

What are moral reasons?

I don't want to say "moral reasons trump". That's misleading. Deontic constraints do trump, by definitions, but not all moral reasons are deontic constraints. Imperfect duty reasons are "moral" but don't trump.

One can roughly delineate a family of reasons that roughly corresponds with the ordinary notion of "moral reasons". Some of the reasons in the family always trump (namely, the deontic constraints). Others don't (e.g., imperfect duties; promises on some views). When I think about the variety of reasons to put in this family (e.g., reasons arising from promise-like speech acts, needs to prevent the pain of others, deontic constraints, etc.), I really doubt that there is a natural kind that covers these reasons.

There are some natural distinctions among reasons:

• Those that always trump and those that don't.
• Those that concern the flourishing of other persons and those that don't.
• Those that concern the flourishing of persons and those that don't.
• Those that concern the flourishing of conscious things other than self and those that don't.
• Those that concern the flourishing of conscious things and those that don't.

But while these distinctions cut at the joints, none of them has the property that all and only the members of the messy family lie on one side.

Here's another argument. A reason is something that connects an action with a good, as furthering the good, respecting the good, detracting from the good, etc. One would expect natural kinds of reasons to be delineated in one of two ways (or a combination): by kind of good and by the nature of the relation to the good. But no natural delineation of either variety sorts the reasons into the moral and non-moral, as per the ordinary notion. And I bet no combination does either.

Rather than taking the messy family to be the "moral reasons", it strikes me as a better way to talk to say that all reasons are moral reasons. Each reason has the property that one fails in the love of God when one knowingly fails to follow the reason in the absence of sufficient countervailing reason. To fail act in accordance with a reason, absent sufficient countervailing reason, is to be bad qua person. If I make something ugly, when at insignificantly higher cost I could have made it prettier, I thereby failed to glorify God in creation as I should have. And so I failed as a person.

Reference magnetism and anti-reductionism

According to reference magnetism, the meanings of our terms are constituted by requiring the optimization of desiderata that include the naturalness of referents (or, more generally, by making the joints in language correspond to joints in the world, as much as possible) and something like charity (making as many real-world uses as possible be correct).

Suppose we measure naturalness by the complexity of expression in fundamental terms—terms that correspond to perfectly natural things. (In particular, we can't talk of what cannot be expressed in fundamental terms, since reference magnetism would presumably not permit reference to what is infinitely unnatural.) Consider the reductionist thesis that the vocabulary of microphysics is the only fundamental vocabulary about the natural world. If this thesis is true, then our ordinary terms like "conscious" or "intention" or "wrong" are going to be cashed out in terms of extremely complex sentences, often of a functional sort. But I suspect that once these expressions are sufficiently complex, then there will be many non-equivalent variants of them that will fit our actual uses about as well and are about as complex. Consequently, we should expect that the meaning of terms terms like "conscious", "intention" and "wrong" to be highly underdetermined.

If we have reason to resist this underdetermination, we need to embrace an anti-reductionism on which the terms of microphysics are not the only fundamental ones, or else have another measure of naturalness.

Induction, naturalness and physicalism

Something is grue provided that it is now before the year 3000 and it is green or it's the year 3000 or later and it's blue. From:

1. All observed emeralds were grue

we should not infer that all emeralds will be grue. But from

2. All observed emeralds were green

we should infer that all emeralds will be green. A standard thought (e.g., Sider in his Book book) is that the relevant difference between (1) and (2) is that "green" carves reality more at the joints, is more natural, than "grue".

Suppose that we understand naturalness in a Lewisian way: a concept is more unnatural the longer its expression in a language whose bits refer to perfectly natural stuff. And suppose we think that among the sciences only the terms of fundamental physics refer to perfectly natural stuff. Now consider:

3. All observed electrons were nesitively charged

where an object is nesitively charged provided it's negatively charged and it's before the year 3000 or it's positively charged and it's 3000 or later. We had better not infer that all electrons will be nesitively charged. But "nesitively charged" is an order of magnitude more natural than "green". Consider this beginning of an account of "green":

4. in electromagnetic radiation of the 484-789 THz range, reflecting or transmitting primarily that in the 526-606 THz range.

And this account is not finished. To make this be in terms of the perfectly natural stuff, we'd need to specify the units (terahertz) in microphysical terms, presumably in terms of Planck times or something like that, and we'll get quite messy numbers. Moreover, we need an account of reflection and transmission. I suspect that we can more easily give an account of nesitive charge: "positive" and "negative charge" seem to already be perfectly natural or close to it; the year 3000 is a bit tricky, but we can count it (or maybe just some other "neater" date) in Planck times from the Big Bang.

If naturalness then correlates with brevity of microphysical expression, "green" is not more natural, and probably is less natural, than "nesitive charge". And so we had better not base induction on naturalness.

I think the lesson of this is that we either shouldn't think of degrees of unnaturalness as distance from the perfectly natural, or we shouldn't limit the perfectly natural (even in the concrete realm) to the microphysical. The latter gives us reason to accept some kind of antireductionism about the special sciences and ordinary language.

The neural prosthetic argument against naturalism

While it is unclear whether my mental functioning could survive my getting getting a prosthetic brain, surely it could survive my getting a prosthetic brain part:

1. For any 0.5 centimeter cube in my brain and any machine that functions in exactly the same way with respect to inputs and outputs on the cube boundaries as the neural matter did, it is possible that replacing the cube with the machine would not change my mental functioning.

Claim (1) strengthened by removing "it is possible that" is in fact a key argument for functionalism: roughly, one repeats application of the strengthened claim until the whole brain has been replaced by a functional isomorph. So claim (1) certainly doesn't beg the question against functionalism. And it's pretty plausible.

Yesterday I argued that if functionalism is true, basic mental states are perfectly natural. In comments, Brian Cutter offered some excellent criticisms (though I responded back), but even if Cutter's criticisms are right, we still have:

2. If functionalism is true, the realizers of basic mental states have to be at least fairly natural.

But if we replace a cube of neural matter whose state is a part of the functional realizer of a basic mental state M by a sufficiently complex prosthetic while keeping fixed edge interaction, we can make the corresponding realizer as messy as we like. By (1), mental functioning could be unchanged by this, while (2) tells us that if functionalism is true, we'd have to lose mental state M. So we've argued that

3. If (1) and (2) are true, functionalism is false.

Now, it is actually pretty plausible that:

4. If naturalism is true, functionalism is true.

The naturalistic alternatives to functionalism just don't seem great. So, we have an argument against naturalism based on the possibility of neural prostheses.

Anyway, probably any naturalistic alternative to functionalism will be heavily biological in nature. It will tie mental functioning to organic rather than functional features of our brains. And in so doing, it is apt to violate (1) as well. Or at least it will violate a strengthened version of (1) which says that (1) necessarily holds for any mental being whose cognitive organs have the same kind of functional density that our brains have. For the replacement of a cube by a prosthetic need not change functional density, and then one could do a second replacement, and continue. Finally, by S4 one would conclude that it is possible that mental functioning could continue after total prosthetization of the brain, which would violate the organicity of our naturalistic alternative to functionalism.

So, surprisingly, gradual replacement considerations may favor dualism, not functionalism.

Functionalism, biological antireductionism and dualism

According to functionalism, a mental state such as a pain is characterized by its causal roles. But if one physical state plays the causal role of pain, so do many others and so the characterization fails. For instance, if neural state N plays the causal role of pain in me, so does the conjunction of N with my having blue eyes. One could require minimality of the state, but that won't help. First, plausibly, there is no minimal state that plays the role: if a state plays it, so does that state minus a particle. Second, even if there is one, it is very unlikely to be unique. There is likely to be redundancy, and there will be many ways of getting rid of redundancy.

The solution to this problem in the spirit of Lewisian functionalism is to restrict one's quantifiers to natural states. There are two ways of doing this. First, we could restrict the quantifiers to states which are sufficiently natural, whose degree of unnaturalness is below some threshold. (An obvious way to measure unnaturalness is to measure the length of the shortest linguistic expression taht expresses the state in terms that are perfectly natural.) But this is unlikely to work. If mental states have degreed unnaturalness, presumably there will be a lot of variation in the degree of unnaturalness. Some mental states will, for instance fall far below the threshold. Those states could then be made slightly more complicated while still staying below the threshold, so once again we would have a problem.

So we better restrict quantifiers to perfectly natural states, at least in the case of the basic mental states (or maybe protomental states—I won't distinguish these) out of which more complex ones are built. Thus we have our first conclusion:

1. If functionalism is true, basic mental states are perfectly natural.

This has an interesting corollary. Presumably no macroscopic state of a purely physical computer is perfectly natural. Thus:

2. If functionalism is true, a purely physical computer has no basic mental states, and hence no mental states.

Thus, the only way a computer could have mental states is if it wasn't purely physical (Richard Swinburne once suggested to me that if a computer had the right functional complexity, God could create a soul for it.)

What about organisms? Well, if organisms are purely physical, then their mental states will be biological states (subject to evolution and the like). So:

2. If functionalism is true, then some of the biological states of a minded purely physical organism are perfectly natural.

This is an antireductionist conclusion. Thus,

3. Functionalism implies that all minded organisms have non-physical states (dualism) or some minded organisms have perfectly natural biological states (antireductionism) (or both).

Moreover, our best account of naturalness is that it is fundamentality. If that is the right account, then our antireductionism is pretty strong: it says that some biological states are fundamental.

Moreover, functionalism is the only tenable version of physicalism (I say). Thus:

4. Physicalism implies biological antireductionism.

Deflation of predicates

Some deflationary theories take some predicate, such as "is necessary" or "is true", and claim that there is really nothing in the predicate for philosophical investigation—the predicate is not in any way natural, but just attributes some messy, perhaps even infinite, combination of more natural properties.

But I know only three candidates for a way that we could come to grasp a meaningful predicate. One way is by ostension to a natural property. Here's a rough idea. The predicate "is circular" might be introduced as follows. We are shown a bunch of objects, A₁,...,A_n, and told that each "is circular", and a bunch of other objects, B₁,...,B_m, and told that each "is not circular." The predicate "is circular" is then grasped to indicate some property that all or almost all of the As have and all or almost all of the Bs lack. But there may be many (abundant) properties like that (for instance, being one of A₁,...,A_n). Which one do we mean by "is circular"? Answer: The most natural of the bunch.

The second way depends on a non-natural view of mind. It could be that our minds, unlike language, can directly be in contact with some properties. And it may be that a predicate tends to be used in circumstances in which both speaker and listener are directly contemplating a particular property, and that makes the predicate mean that property.

The third way is by stipulation. I just say: "Say that x is frozzly if and only if x is frozen and green."

The predicates, like "is true" and "is necessary", that are the subjects of these deflationary theories are not introduced in the first way if the theories are correct to hold that the predicates do not correspond to a natural property. Are they introduced in the third way? That is very unlikely. I doubt there was a first user of "is necessary" who stood up and said: "I say that p is necessary if and only if...." That leaves the second way, the non-naturalistic way. Therefore:

1. If these deflationary theories are correct, naturalism is false.

Which is interesting since the motivation for the theories is sometimes naturalistic (e.g., Hartry Field in the case of "is true"[note 1]).

But in any case, the following is very plausible. Any properties we are in direct non-natural cognitive contact with are either innately known or natural. So, the deflated predicates must refer to innately known predicates. I doubt, however, that we innately know any entirely non-natural predicates. And that leaves little room for these theories.

More generally, the above considerations make it difficult to see how we could have any genuine non-natural, non-stipulative predicates. Thus, if we have good reason to think that P does not indicate a natural property, and is not stipulative, we have good reason to have an error theory about P.

Concepts of artifacts appear to be a counterexample. "Is a chair" is neither natural nor stipulated. My inclination is to say that it is not really a predicate ("Bob is chair" expresses some sentence about Bobbish reality being chairwise arranged, or something like that), which makes for a kind of error theory.

Natural mathematical structures

This post is inspired by Heath White's comment here.

There are lots and lots of different kinds of mathematical structures. Here's an operation on the real numbers: a#b = a^3b+7. You can study this operation heavily, but chances are that you won't get anything very interesting (but maybe you will!).

But on the other hand, take something like addition or multiplication (or both). These have many beautiful properties, and lend themselves to many kinds of abstraction: groups, fields, rings, monoids, etc. When this happens, it is evidence that the structure one was studying is somehow natural. While in some way any coherent set of coherent axioms might be fruitfully studied, there both seem to be particularly natural axioms for a structure--like, commutativity--and particularly natural clusters of axioms--like those defining a group or a ring--that seem worthy of study. Anyway, around a particularly natural structure there springs up a wealth of mathematics.

Some of the most creative mathematics seems to be the identification and introduction of natural structures. For instance, one of the things I learned in my recent work in formal epistemology is that classical probability is a very natural structure. On the other hand, hyperreal-valued probabilities of the sort that some philosophers like seem to be quite an unnatural structure--one doesn't get the same wealth of neat results. The more one plays with hyperreal probabilities, the more they look like Frankenstein's monster. (On the other hand, the R(I) monoid I discuss in a recent post is rather more natural, though it may not seem that way initially.)

What is this naturalness of structure? David Lewis took natural properties to be more basic, and unnatural ones to be constructions from the more basic ones. That is not the case for natural mathematical properties. If we consider mathematics set-theoretically, all the properties--both the natural and the unnatural ones--we are studying are constructions out of set-theoretic properties. A natural cluster of axioms might be no simpler than an unnatural cluster of axioms. Moreover, the naturalness seems independent of the foundational grounding. Suppose one day we have a better foundation for mathematics than set theory. (Not unlikely!) Group theory and probability theory will still be studying something natural.

What, then, makes a mathematical structure natural? Is it purely extrinsic, with the natural properties and clusters of axioms being those that are mathematically fruitful? Or maybe there is no distinction: Maybe if the amount of effort that has gone into analyzing addition were put into analyzing the # operation I gave at the beginning of this post, we would find just as beautiful mathematics? Maybe such deflationary stories are the whole story about mathematical naturalness. But maybe there something deeper about the natural properties and clusters of axioms. Aquinas thinks all creation in some way reflects God. Perhaps the more natural properties--whether empirical or mathematical--are those that somehow more deeply reflect God's mind?

This post comes after spending over a week on some mathematical issues only to find today that I committed a subtle (perhaps only to me!) error at the beginning of the investigation, and almost all of the work has come to naught. This reminds me of the famous joke about dean talking to the physicist: "You always want money for more equipment! Why can't you be like the mathematicians? All they need are paper, pencils and garbage cans. Or better yet, why can't you be like the philosophers? They don't even need the garbage cans."

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.

Natural roles

I've been thinking about roles such as being married, being a parent, being a sibling, being a second cousin, being a firefighter, being a president or being a monarch. These roles come along with moral duties, permissions and rights. Each role has entry conditions, such that when one satisfies the entry condition, one falls under the role, and some of the roles have exit conditions. In some roles the entry conditions involve one's agreeing to enter the role—marriage, firefighting, presidency and monarchy are like that—but in some the entry conditions may have nothing to do with one. On the other hand, one need not do anything to become a sibling.

Here's what I am inclined to think about such roles. Our human nature specifies what one might call natural roles. Some of our natural roles directly give us roles. Thus, I think being married and being a parent are natural roles. I am not sure about being a sibling, and I quite doubt that being a second cousin, being a firefighter and being a president are natural roles.

Roles can have sub-roles which have more specific duties, permissions and rights. These sub-roles derive all their normative force from the full role and other conditions. For instance, the role of parent has sub-roles such as: parent of a small child and parent of an adult child. The moral duties, permissions and rights associated with the sub-role derive from the moral duties, permissions and rights associated with the full role together with the fact of the satisfaction of the condition.

My big conjecture is that all roles that are not natural are sub-roles of natural roles. Thus, there is no natural role of second cousin, but there is a natural role of distant relative, perhaps. The duties, permissions and rights of being a distant relative depend on multiple conditions, including the closeness of the relationship and the operative social conventions. So there may be such a thing as the role of second cousin in mid-twentieth century rural Pennsylvania. All the moral normative oomph in this sub-role comes from the moral normative oomph of the natural role of distant relative. It does this by means of the natural role having requirements that are conditional, in this case on closeness of relationship and the social conventions in place. For instance, the natural role of distant cousin may say: "Fulfill those non-immoral conventional duties associated with your degree d of relationship whose fulfillment does not exceed degree of onerousness c(d)", where c(d) is some function setting a cap for how onerous a conventional duty one is required to fulfill in the case of a degree d of relationship (presumably the closer d is, the higher the cap). In this way, conventional duties (which on my view are no more duties in themselves than rubber ducks are ducks) become moral duties when the natural role makes them so. Thus, a conventional duty to come to second cousins' weddings yields a moral rule that one should come to second cousins' weddings, when the onerousness does not exceed c(d), where d is the degree of relationship involved in being a second cousin.

We could have cases of non-natural roles whose moral normative oomph derives from two or more natural roles. In this case, we can say that the non-natural role is a sub-role of an aggregate of natural roles (we may or may not want to say that it is a sub-role of each natural role in the aggregate). Thus, while monarchy may simply be a sub-role of public authority, which is a natural role, being a fire-fighter and being a president may be sub-roles of public authority and being an employee.

There may also be merely conventional roles. Merely conventional roles are of no interest to me here. They do not role-ishly result in moral duties, permissions and rights, and non-moral duties, permissions and rights are of little more interest to me than rubber ducks are to ornithologists. :-)

A functional account of marriage vows

What does a couple have to validly promise each other, explicitly or implicitly, in order for those promises, when appropriately ratified by authority, to give rise to a marriage?

This is a hard question as to the specifics. But we can at least give a start of a functional characterization:

• Marriage vows are that complex of binding commitments that in fact makes it prima facie permissible for a couple to engage in intercourse.

One might want to add that the complex is natural, either in the David Lewis sense (in which case it is opposed to gerrymandered) or in the natural law sense (in which case it is that which is consonant with our nature).

Of course, this account can only have plausibility if uncommitted sex is wrong. I think that prior to the 20th century in the West, this functional characterization would have been seen as quite plausible, and I am still inclined to think it is correct.

A functional characterization is not, of course, a definition. Thus someone who disagrees with this characterization can still be talking about the very same thing when using the words "marriage vows" as someone who accepts this characterization.