Fundamentality and Structure

The last article argued that essence has a direction which necessity flattens. It is necessary that Socrates belongs to his singleton, if he exists at all, and yet belonging to a set is no part of what Socrates is, whereas having Socrates as its sole member is exactly what the singleton is. Real definition, like grounding before it, is a notion of priority. Grounding says what holds in virtue of what; real definition says what a thing is in terms of what.

Priority raises two further questions, and this article takes them in turn. The first is where it stops. If facts about tables hold in virtue of facts about molecules, and those in virtue of facts about particles, perhaps there is a last stage, a body of facts that hold in virtue of nothing further. The second question sounds quite different. Physics describes the world in terms of mass, charge and spin. Indefinitely many other vocabularies could be used to write down truths about the same world, most of them useless. Is the physicist's choice merely convenient, or does it get something right about the world that the others miss? The answers David Lewis and Theodore Sider gave to the second question turn out to bear on the first.

What it is to be fundamental

Grounding orders facts. The fact that the postbox is scarlet grounds the fact that it is red. The fact that it is raining in Leeds grounds the fact that it is raining in Leeds or seven is even. On the standard view, set out in the grounding article, the relation is irreflexive and transitive, so it has a direction, from the more basic to the less. The notion of the fundamental is what you get by following that direction to its end.

Two features of the grounding version matter here. First, it is purely negative: it says that a fundamental fact lacks grounds, and nothing about what such facts are like, including whether they concern small things. Whether the basic facts concern particles, fields or the universe as a whole is left open.

Second, fundamentality is not a modal notion, and the last two articles' lesson applies to it directly. That a particular electron is negatively charged is contingent, since the electron need not have existed, and it is a good candidate for being fundamental. That Socrates is either wise or not wise is necessary, and it is not fundamental: a disjunction is grounded in whichever of its disjuncts is true. A necessary fact can be derivative and a contingent fact can be basic, so no account of fundamentality in terms of necessity will be correct.

The definition makes no promise that anything is fundamental. The familiar picture combines two claims that should be kept apart. One is that there are fundamental facts. The other is that the fundamental facts are complete: everything that is not fundamental is grounded, perhaps through many stages, in facts that are. Each claim can fail without the other. A world might contain some ungrounded facts and also some facts whose chains of grounds never reach them, and a world might contain no ungrounded facts at all.

Is there a bottom level?

The familiar image is of levels: societies above organisms, organisms above cells, then molecules, atoms and the particles of physics. Each level is composed of the one below and holds in virtue of it. The image adds that the series stops at a lowest level of things without parts, whose properties ground everything above.

Jonathan Schaffer asked, in "Is There a Fundamental Level?" (2003), what reason we have to believe that the series stops. The history of physics is not encouraging. Atoms were named for being indivisible and turned out to have nuclei and electrons; nuclei turned out to have protons and neutrons; these turned out to have quarks. Electrons and quarks are currently treated as having no parts, but the track record gives little confidence that the current list is final, and nothing in physics rules out a world in which every level has a level beneath it. Schaffer argued that the evidence for a fundamental level is inconclusive and that infinite descent, levels all the way down, is a live possibility. His concern was with metaphysics more than with physics. A great many theses are stated in terms of the bottom level, such as the thesis that everything depends on the fundamental physical facts, and those theses inherit all the uncertainty about whether there is one.

Against infinite descent there is an old argument.

The third premise does the work, and it rests on an image of borrowing. A chain of loans in which every lender has borrowed what they lend, with no one at the start who simply has the money, is a chain in which nobody has anything.

Even if the regress argument succeeds, it does not show that there is a bottom level. It shows that chains of grounds must end somewhere, and they need not end at the smallest things. Schaffer's later work took this route. In "On What Grounds What" (2009) he held that every chain of grounds terminates, and he went on to argue that since endless descent is possible, the terminus cannot be guaranteed to lie in the parts, and so lies in the whole, with the cosmos as the one fundamental thing. That view has its own article later in this tier. For now the point is that fundamentality and smallness are different ideas, and the possibility of infinite descent pulls them apart.

Lewis's naturalness

The second question was about vocabulary, and it began as a question about properties. The beginner tier noted the dispute between sparse and abundant conceptions: whether every meaningful predicate answers to a property, or only those that carve nature at its joints. David Lewis took properties to be as abundant as possible. On his view a property is simply a class of possible things, so any class whatever is a property. There is the property of being green, and equally the property of being grue, which Nelson Goodman defined as being green and examined before a fixed future date, or not so examined and blue. There is a property had by exactly this postbox, the Eiffel Tower and every left shoe.

Abundance has a consequence that Lewis stated plainly in "New Work for a Theory of Universals" (1983). Abundant properties discriminate nothing. Any two things share infinitely many of them and fail to share infinitely many others, so the bare fact that two things share a property tells you nothing about whether they are alike. Two electrons resemble each other in a way that an electron and a left shoe do not. Lewis concluded that an adequate theory must recognise an objective distinction within the abundant properties: an elite minority whose sharing makes for genuine resemblance.

The intermediate article on universals met one way of drawing that line. Armstrong's universals are sparse, discovered by physics and not read off from predicates, and having a universal in common is what makes things alike. Lewis's point was that the line is needed whether or not you believe in universals. He left open whether to draw it with Armstrong's universals, with a primitive distinction among classes, or with primitive resemblance among particulars, and argued only that one of these must be done. The last article met the idea briefly, in the sparse repair to the modal account of essence.

The perfectly natural properties are few, but Lewis held that they are enough: two possible worlds alike in how the perfectly natural properties and relations are distributed across them are alike in every qualitative respect. The facts about that distribution are therefore Lewis's candidate for what everything else rests on. He stated the connection in terms of supervenience, and the grounding article showed why supervenience records covariation without recording which side is prior. A grounding theorist would put the same thought as a claim of priority: the distribution of the perfectly natural properties is the fundamental fact, and everything else holds in virtue of it.

The work naturalness does

Lewis's case for naturalness was not a direct argument. It was a list of jobs, each of which the abundant properties cannot do and one distinction can.

Similarity is the first. Two things are duplicates, Lewis proposed, when they have exactly the same perfectly natural properties and their parts stand in the same perfectly natural relations. An intrinsic property is then one that never differs between duplicates. This defines the distinction the beginner tier tried to capture with the lonely test, and it does so without appeal to what a thing would be like on its own.

Laws are the second. Lewis's own account of laws, which a later article treats properly, takes them to be the general statements in the best systematisation of everything that happens, the one that best combines simplicity with strength. The trouble is that simplicity depends on vocabulary. Introduce a predicate that applies to a thing just in case the world it inhabits is exactly like ours, and the one-line axiom "everything has it" entails every truth about the world. Every regularity would then come out as a law. Lewis's remedy was to require that the systematisation be stated in perfectly natural terms, which rules the gerrymandered predicate out.

Meaning is the third. Saul Kripke's reading of Wittgenstein, in Wittgenstein on Rules and Private Language (1982), presents a sceptic who asks what makes it the case that by "plus" you have always meant addition. Suppose you have never added numbers as large as 57. Then every sum you have worked out fits quaddition, a function that agrees with addition when both arguments are below 57 and gives 5 otherwise. The sceptic argues that nothing in your past behaviour, your dispositions or your instructions to yourself settles which function you meant. Lewis's response was that the facts about use do not have to settle the matter alone. Addition is a far more natural function than quaddition, and meaning goes to the most natural candidate that fits well enough with use. The idea, often called reference magnetism, makes naturalness part of what fixes content.

Induction is the fourth. Every emerald examined so far has been green, and so, while the date in Goodman's definition is still in the future, every emerald examined so far has been grue. The evidence fits "all emeralds are green" and "all emeralds are grue" equally well, and the two predict different colours for emeralds first examined after the date. We project green and not grue, and the obvious explanation is that green marks a genuine respect of similarity and grue does not. W. V. O. Quine had already tied projectibility to a sense of similarity in "Natural Kinds" (1969). A friend of naturalness gives that sense an objective basis.

Set out, the case has the shape of an inference to the best explanation.

The argument is strongest taken whole. Each job on its own might be done some other way: by facts about human psychology for induction, by a different account of laws, by a theory of meaning that tolerates more indeterminacy. The case is that one posit does them all.

From naturalness to structure

Sider's Writing the Book of the World (2011) takes Lewis's distinction and generalises it. Lewis's naturalness is a feature of properties, and so of predicates, and Sider asks why joint-carving should be confined to predicates. Quantifiers, logical connectives and other operators also figure in our best descriptions of the world, and it can be asked of them too whether they reflect the world's divisions or only our conventions.

The motivation runs through ideological commitment, which needs a brief recap. Quine distinguished a theory's ontology, the things its variables range over, from its ideology, the primitive notions it uses without defining. Just as a theory is ontologically committed to whatever it must say exists, it is ideologically committed to the primitive notions it cannot do without, and those commitments can be counted and weighed as entities are. Sider adds a further step: accepting a theory gives you reason to think that the notions it takes as primitive carve at the joints. Our best physical theories take charge and mass as primitive, and their success is reason to think those notions carve. They also take the quantifiers and connectives of logic as primitive, and the same reasoning applies to those.

The image in the book's title is of a description of the complete truth about the world written using only notions that carve at the joints. Such a book would contain no talk of tables, money or grue, though everything true of them would hold in virtue of what it says. The thesis that reality has a privileged description is the thesis that some such book is uniquely correct: not merely true, since many books are true, but written in the world's own terms. Sider adds a principle he calls purity: the fundamental truths involve only fundamental notions. A truth stated partly in terms of tables is not fundamental, however basic it seems.

Purity has a consequence for grounding. A grounding fact, such as the fact that facts about particles ground the fact that there is a table, mentions tables, so by purity it cannot be fundamental. Sider is wary in any case of treating grounding as basic, and in the book he builds his account of the fundamental from structure instead. This marks a real fork in the subject. On one approach, fundamentality belongs to facts and consists in their having no grounds. On the other, it belongs to notions and consists in their carving at the joints, and which facts are basic is settled by which notions are. A world of endless descent is a hard case for the first approach and less obviously for the second, since a vocabulary might carve at the joints at every level of a world that has no lowest one.

Structure in logic matters most for what it does to ontology. Eli Hirsch has argued that the quantifier "there is" could be given several meanings, all equally good, on some of which "there is a table" comes out true and on others of which only particles exist. If so, the dispute over whether tables exist is verbal, since each side speaks truly in its own sense. Sider's reply is that if the quantifier carves at the joints, one candidate meaning is privileged, and the question asked with that meaning has an answer that does not depend on a choice of words. Whether ontological disputes are substantive then turns on whether there is quantificational structure. A later article on deflationism takes this exchange further.

Doubts about structure

Sider's view is clear and ambitious, and the objections to it are mostly about the price of the ambition.

That question leads to the sharpest recent objection.

Next

This article treated fundamentality as a feature of facts and of notions: a fact is basic when nothing grounds it, a notion basic when it carves at the joints. Many questions of priority are posed about things instead. A set seems to depend on its members, a smile on a face, a hole on the thing it is in. The next article sorts the ways one thing can depend on another, existentially, essentially, rigidly and generically, and shows that modal definitions of dependence misfire for the reason the article on essence would predict.