Much of traditional natural theology offers causal explanations- e.g. for the universe's existence and ability to host our sort of life. But a less-remarked strand offers ontological explanations, claiming that theories involving God are the best answers to ontological questions. Leibniz, for instance, wrote in the Monadology that

Though Leibniz barely gestures at the argument, what he wants to claim is that the ontology of modality requires God, and this is reason to believe that God exists. Nor does he treat God as a cause of possibility; he thinks instead that God's nature makes possibility claims true, offering an ontological explanation (an account of what sort of entity makes a sort of claim trueFootnote ²) rather than a causal one.Footnote ³ Again, Alan Rhoda and Dean Zimmerman offer God as a response to a truthmaker problem arising on presentist views of time.Footnote ⁴ Augustine made a similar move on the problem of universals. In an early work, he puts forward the premise that if there is anything superior to human reason, there is a God.Footnote ⁵ He then argues that ‘immutable’ (i.e. necessary) mathematical truth is superior to human reason.Footnote ⁶ It emerges that such truth has its foundation in a realm of Platonic mathematical objects, which turn out to be somehow identical with the Platonic Forms.Footnote ⁷ So at one level, Augustine's premises are that

1. 1. if something is superior to human reason, God exists, and

2. 2. something is superior to human reason.

But given that his case for (2) is really a case for a Platonist account of attributes, at another level, we can see Augustine's premises as

1. 1*. if a Platonist account of attributes is true, God exists, and

2. 2*. a Platonist account of attributes is true.

I hold no brief for Platonism, but Augustine's idea of arguing God's existence via the claim that the true theory of attributes involves God seems to me worth pursuing. The theist theory of attributes I defend could be called nominalist or Platonist: nominalist in that it invokes only concrete particulars, Platonist in that it invokes something (God) outside spacetime and is structurally similar to Plato's theory. My full case for this theory has three broad parts. One shows that it can in fact do the full range of work one wants from an attribute-theory.Footnote ⁸ One argues that non-theist forms of nominalism either do not do some of this work or do some of it at greater ontological cost. The last rejects non-theist forms of realism because they generate unacceptable metaphysics of concrete substances. If I'm right, the theist view gives one overall the best account of substance and attribute and ought therefore to be believed. My full case, then, is matter for a book, not a paper. Here I can only offer one piece of it, which is why all I do here is take one step toward God. I begin by explaining some problems to which Platonism was an answer, what Plato's answer was, and how Augustine's answer improved on it. This shows inter alia that inclusion of a theist element can in fact improve a theory of attributes. I then state a theist account of attributes which modifies Augustine's, begin my argument for it by addressing one non-theist form of nominalism, then answer three objections to the theist account.

Attributes, predication and Platonism

Predication raises some of the oldest ontological questions. If I say that an apple is red, I attribute being red to the apple. Being red, then, is something I attribute, an attribute. Questions that arise here include just what if anything an attribute is and what it is to have an attribute. Since a predication is true just if its subject(s) has (have) the attribute we predicate, these amount to the question of what makes predications true.

Plato gave the first theory of attributes and predication: that corresponding to most predicates is an abstract entity, a Form, and if there is a Form for the predicate ‘F’, things to which ‘F’ applies are Fs by standing in a relation, participation, to the Form of F.Footnote ⁹ Plato did not make clear what ‘participation’ is.Footnote ¹⁰ But it is at least likeness grounded on some sort of real dependence: anything's being F consists in its depending on the paradigm F, The F, for a likeness to The F.

There are three sorts of reason we are not all Platonists. Plato's theory has internal difficulties: the Third Man argument, the difficulty of understanding participation, the strangeness of the Forms. Plato's theory is uneconomical: it adds hosts of abstract entities to our ontology. Forms are also suspect on broadly epistemological grounds. As Platonism developed, middle- and Neo-Platonists used God or gods to replace the Forms. Plutarch replaced them simply with God.Footnote ¹¹ Alcinous replaced some Forms with ideas in the mind of God.Footnote ¹² So did Plotinus.Footnote ¹³ Augustine combined the moves, replacing with God the Forms for attributes God can share with creatures and with God's concepts the Forms for attributes creatures alone can have.Footnote ¹⁴

Religious motives may have played a part in this. But I think the Platonists got all godly about it for largely philosophical reasons: using God and His concepts to take the Forms' place makes progress on Platonism's philosophical problems. We don't have a good grip on participation. But if God replaces some Forms, participation in these cases becomes depending really on God for a likeness to God. If God can act in the world, it is clear at one level what this comes to: it is being helped by God to be in some way or degree Godlike. At a deeper level, how clear this is depends, of course, on how well we understand what it is for God to act in the world. But it is at least clear that this is efficient causation, if of an unusual sort. We hadn't even this much grip on what Plato meant by participation. So the God-move gains us something philosophically. If God's concepts replace other Forms, participation in these cases becomes depending really on God for being such as to satisfy His concepts. If God can act in the world, it is also clear at one level what this comes to, and so again, the God-move replaces Plato's cloudy picture with one a bit clearer. Again, while Forms are sui generic, gods are persons of a sort, albeit an unfamiliar sort. We're already in the business of offering a non-naturalist theory, if we're Platonists; the concept of God is not less intelligible than that of a Form; and the allied mysteries lessen a bit if God takes the Forms' place.

Again, while Forms are abstract; God is concrete, being a cause. One might reply that God's concepts are abstract, and so no gain is made in moving Forms into His mind. Plotinus, oddly, would've disagreed: as he saw it, each divine idea was a minor god.Footnote ¹⁵We can instead pare divine concepts away, in favor of divine mental events. Again, Forms are myriad; Augustine pared deities down to one. He had of course religious reasons to do so, but the move gets strong support from Ockham's Razor. One God can do all the philosophical work other versions of Platonism did with many gods, so there is just no philosophical basis to multiply them. Again, Forms don't seem to permit a good epistemology. A God able to cause events in the world can cause us to have certain sorts of knowledge. Finally, consider the Third Man.Footnote ¹⁶ Participation involves likeness: anyone's being a man involves likeness to The Man. Now likeness is symmetrical. So if men are like The Man by being men, The Man is like men in respect of being a man – The Man is a man. So one must ask what it is for The Man to be a man. Plato reasoned that if The Man is a man, it has something in common with other men. So there is a Form, Man, in which they all participate. But Plato also held that no Form can participate in itself. If so, there is another Man, common to the initial Man and all other men – and so ad infinitum. For theistic Platonism, though, there is in place of The Man a divine concept (in my variant, a divine mental event). A concept (or event) is not a man. So the Third Man dissolves, at least for Forms replaced by divine concepts. I suggest a line on Forms replaced by God below.

There thus was good philosophical reason for classical Platonists to take a theist – and even monotheist – turn.Footnote ¹⁷ So we might see behind Augustine's (1*) the premises that

1. 3. if Platonism is true, its best version is true, and

2. 4. The best version of Platonism includes theism.

(3) is plausible. If Platonism is true, we must decide between its versions to see what the whole truth is. We do so by seeing which version has the greatest theoretical virtues – scope, simplicity, parsimony, etc. This (we hope) selects out a theory it is most rational to believe: Platonism's best version. And though the move is hardly beyond controversy, we take it that what is most rational to believe is most likely true. I have suggested why a classical Platonist might incline to (4).

I now begin my case for a theist solution to the problem of attributes, and hence for God's existence. I first outline my theist story about predication. I then begin my case against other theories of attributes by comparing this view with one version of nominalism. I do not argue first that theism is possibly true. One way to show that Ks are possible is to display an actual K; an argument that God actually exists is ipso facto one that He possibly exists.

Theism: the basic story

To give my theist story, I return to Plato. Plato need not have gotten in Third Man trouble. A Platonist can say that The Man and other men are men in related ways. For them all, being men includes being like The Man (contentfully, rather than in virtue of sharing such transcendental properties as self-identity). Men other than The Man depend really on The Man for this likeness. The Man is contentfully like The Man. So it is a man. But it does not have that likeness by depending really on itself. It has that likeness by being itself. It is like itself because it is identical with itself: what makes the Man a man is that it is the Man. Thus there is no Form in which The Man and other men all participate. This is a ‘no ontology’ view of The Man's being a man. We get this for free given that it is The Man. Thus the Third Man evaporates. This strategy applies too to Forms God eventually replaced. On this approach, at an abstract level, the Platonist story about predication involving Forms becomes that

That is, the paradigm F is F per se, just by being itself. For other Fs, being F is bearing a relation to the paradigm F.Footnote ¹⁸ What makes F-predications true is always of one or the other sort. This account is not outré. We don't jib at

Augustine adopts (P), adding that

1. 5. if it can be the case that God is F and some concrete non-divine being is F, God is the standard/paradigm FFootnote ²⁰, and

2. 6. if God can be F and no concrete non-divine being can be F, then if God is F, God is F per se Footnote ²¹, and

3. 7. if some concrete non-divine being can be F and God cannot be, God's F-concept provides the standard.

(7) raises the specter that God's concept of evil will provide the standard evil. Augustine's doctrine that evils are lacks of goods provides an answer to this; Aquinas used this to say in effect that God has no concept of evil as such, merely concepts of goods whose absence He notes – and since evils are just absences, that is all He need do.Footnote ²²

I alter Augustine: I parse divine concepts away. I say that talk of them is fiction. If I can make this claim good, I can provide the explanatory power of Augustine's theist Platonism without positing abstract entities of any sort. What makes something a cat on my account is its dependence not on a divine concept but on certain bits of God's mental life. These are just primitively events with a particular causal role in God's mental economy, such that when appropriately involved in the execution of a divine intention, cats result. This is no worse than a realist's saying that a universal or Form, cathood, is just primitively cathood – that it does not have to instance another property, being cathood, to have this character. Explanation runs out somewhere; these seem reasonable stopping-points.

I also make it part of the theory that

1. 8. every substance is either God or a concrete non-divine particular, and for all x and t, if x is a concrete non-divine particular, if x exists at t, God conserves x in existence at t.

(8) is standard theism: for all x and t, if x is concrete and not God and exists at t, God wills that x exist at t, and this contributes causally to x' existence at t. If x has existed for some period open toward t, this is contributing to causing x to continue to exist. This does not imply that only God causes concrete creatures to continue; it could be that God provides a necessary but insufficient condition for continuation. If x has not previously existed, God's willing x to exist at t constitutes creating x at t.Footnote ²³ This might be a simple case of creation ex nihilo. But God can also cause x to begin to exist via creaturely intermediaries; God can conserve or create by willing there to be created causal chains that begin or sustain creatures' existence. So my preferred theist theory consists of (P), (5), (6), (8) and

1. 9. if some concrete non-divine being can be F and God cannot be, being F consists in being conserved in dependence on those divine mental events which lie behind fictional talk of God's having an F-concept.Footnote ²⁴

God conserves Fs as Fs just if those events figure in the right way in the intention which directs His conserving power. This employment gives rise to the right sort of causal dependence on them. This kind of constant conservation threatens occasionalism, the doctrine that only God causes any event: if first God conserves me as a boy and then He conserves me as a man, it might seem to follow that by switching how He conserves me, God causes me to become a man. But the most that would follow here is overdetermination or part-causation- perhaps both God and my biology account for this. I do not think it is overdetermination, either, because (I submit) it can be the case that causation among creatures accounts for what it is that God conserves. It might e.g. be that causal processes in some water account for the fact that by conserving the water-molecules individually, first God conserves some liquid and then He conserves some ice. For reasons of space, though, I cannot pursue this further.

Theism vs. class nominalism

One recommends a theory of attributes by rebutting arguments against it, showing its rivals' problems and showing that it can do the work one wants of a theory of attributes. Here I can offer only one comparison, with one current view, which holds that attributes are sets. I raise six problems for the set theory, and show that the theist view does not face them. There are also questions one can raise for the theist view which do not arise for the set theory. I answer some below, and answer more elsewhere.Footnote ²⁵ If some such answers work and no worse problems arise for my view, then if my work does at least as much metaphysical work as class nominalism, class nominalism's problems suggest that my view has an edge on it. And that it does at least the work is clear, since theists can easily eliminate sets in favor of certain divine mental activities.Footnote ²⁶ I argue below that my view in fact does more of the ontological work one wants out of a theory of attributes than the set view. If I'm right about this, my view faces fewer problems than the set view and does more ontological work. I also suggest that it does this view at less ontological cost. If correct, all this constitutes a fairly strong case that my view is preferable to the set view.

Flouted intuitions

Class nominalists like Quine, Quinton and David Lewis hold that attributes are sets of their instances.Footnote ²⁷ This creates metaphysical worries. We tend to think that attributes give their bearers their characters: what makes the apple red is that it has a color, redness. Class nominalism reverses this. A set is redness – the set of red things – because all and only the things in it are red.Footnote ²⁸ So the red things' being red helps make a particular set redness. (Had these items been in this set, but not been red, the set would not have been redness.) On class nominalism, then, bearers give attributes their characters, not vice-versa. But ceteris paribus it's best to preserve in our metaphysics as much as we can of our intuitive ideas about attributes. The theist view doesn't share this problem. The divine mental events of which we speak fictionally as concept-possession were what they were before God created anything. They do not derive from red things the character in virtue of which things appropriately related to them are red, and appropriate relation to them is a red-maker.

Again, on class nominalism, being redness turns out not to be an intrinsic property, for whether a set has it depends on whether anything other than its members is red. But this is odd: surely whether something is redness should depend on its own content alone. On the theist view, again, God's red-producing mental events had their full character when He alone existed. So having this character is intrinsic to them. This implies that their content is entirely narrow, but this is a claim without cost, for in the case of God's pre-creative mental events there is no pressure toward externalism.

A metaphysical commitment

Again, on class nominalism, if time really flows or passes,Footnote ²⁹ there are many rednesses over time, since many different sets are successively the red-set as new things come to be red and old ones cease to be red. We thus have to answer the question of what the red-sets all have in common that makes them all redness. This question looks hard. It will not do, for instance, to say that they are all members of a set, the set of things that are ever rednesses, because if time really passes, whenever there is a new red-set there is also a new set of rednesses. Even if time passes, this question does not arise on any other theories of attributes. Thus class-nominalism may be the sole view to require a theory on which time does not really pass. A large commitment about the nature of time is a cost: it is better to have a view neutral on this.

Being red vs. being in the red-set

Again, on class nominalism, ‘A is red’ predicates membership in the set of all and only red things. So what makes it true that A is red is that A belongs to the red-set. Which things are red determines which set this is. Red things' being red thus has to be something other than their belonging to the red-set, or it can't determine which set is the red-set. Putting this more carefully, on class nominalism,

But as we've seen, that A is red helps explain it that R is the red-set. So that A is red is logically or explanatorily prior to R's being the red-set. But what is prior to one conjunct's obtaining is prior to the conjunction's obtaining. So that A is red is logically or explanatorily prior to A ε R and R is the red-set. But what is prior to the conjunction's obtaining can't consist in its obtaining.

Approaching this another way, on class nominalism, what makes it true that A is red is that A belongs to the red-set. So for reasons seen, on class nominalism, A's being red is something other than that which on class nominalism makes it true that A is red. Yet it should be the case that A's being red makes it true that A is red. The theist view escapes this problem. Things' being red consists in their being appropriately related to certain divine mental events. This is also what makes it true that they are red. Now that last bit might strike some as odd: surely, they will say, what it consists in to be red is (say) being such as to reflect and absorb certain wavelengths of light. The theist needn't demur. There are two ways to take such ‘consists in’ statements. In one, they give the metaphysical constitution of having the property: this is the story involving God. All theories have such a story, involving universals, tropes, resemblances, classes, concepts, etc. The other sort of ‘consists in’ statement gives the content of the property. This is where light enters the picture.

Can we see that apples are red?

Again, what makes it true that the apple is red is that the apple is red. This is observable. So what makes it true that the apple is red is observable. But on class nominalism, what makes it true that the apple is red is that the apple belongs to the red-set. This is not observable, because neither sets nor the membership relation are.Footnote ³⁰ If being red is being a member of a set, it is not an observable property. This is not just because we do not see all the red things. When we see six glasses, the six exhaust a set's membership, but we do not see that the glasses are a member of their set. We observe the six, and infer the membership. Though we see all six members, we do not see the set: abstract entities can't reflect light. If we see a group, that is because a group is not abstract. Seeing the group is seeing them, i.e. the glasses.

It won't do to say that when we see the apple, we see what it looks like to belong to the red-set. There is nothing it looks like to belong to a set, because membership has no effects, and so no observable effects. It won't do to appeal to the content/constitution distinction either. For a way of putting my point is that the class nominalist's constitution story isn't compatible with the property's content: nothing which consists in set membership is observable. The theist's constitution story is compatible with the theist content story: the apple's reflecting and absorbing light is the tip of a causal iceberg whose submerged depth is divine. If things' being red is their being disposed to interact with light a certain way, at a deeper level it is their being constantly caused to have this disposition – and of course it can be both. Belonging to a set can't confer a reflectancy property, nor is having the reflectancy necessary for belonging to the set,Footnote ³¹ nor does belonging to the set depend on having the reflectancy,Footnote ³² nor does an item's being thus-reflective suffice for the set's being the red-set.Footnote ³³ Thus there is no clear way for the content of belonging to a set to be a reflectancy,Footnote ³⁴ and in fact, as noted earlier, a disposition to appear red isn't part of the content of belonging to the set, because belonging to the set presupposes having it. We might next ask whether when we see an apple's redness, we see a sign of its being a member of the red-set. We certainly see it looking as it will look just if it belongs. But we believe we can observe the apple's being red, not just signs of it.

Theism has an edge here. Red-making divine mental causal relations to apples have observable effects. When we see a red thing, the mental causal relations are among the (distal) causal conditions of our perception. We see effects of being rightly related to them. So in one sense of the phrase, we see what it looks like to be rightly related to them. So we see what it looks like to be red. What else would we expect to see? One might reply that a thing's being red should be directly, not indirectly observable. But perhaps it is, on the theist account. If we let causal mediation suffice for rendering states of affairs only indirectly observable, no state of affairs whose effects are not causally immediate to the eye (or the brain?) will count as directly observable. It's not clear that a parsing which lets something as causally removed from us as a reflection-event count as directly observable will not let the thing's being red count as directly observable, even on the theist account. One might reply that seeing the thing should suffice for seeing it to be red (given normal perceptual capacities and conditions), and this isn't so on the theist story. But it is: it's just that on the theist account there is a further, distal causal story behind our so seeing the thing.

The coextension problem

Again, class nominalism has a well-known problem with contingently co-extensive attributes:

1. 10. the set of unicorns = the set of centaurs = ø. So  (premise)

2. 11. unicornhood = centaurhood.  (10, class nominalism)

3. 12. All identities are necessary. So  (premise)

4. 13. Necessarily unicornhood = centaurhood.  (11, 12)

5. 14. Possibly some unicorns are not centaurs. So (premise)

6. 15. possibly the set of unicorns ≠ the set of centaurs. So  (14, set identity-conditions)

7. 16. possibly unicornhood ≠ centaurhood.  (15, class nominalism)

8. 17. ¬(13).  (16, modal rule)

Since a contradiction has emerged, something has to go. But (10), (14) and the set identity-conditions are unassailable, and rejecting (12) is unappealing. One move is to say that there is a role, collecting all and only unicorns, playing which is sufficient to make something a world's unicorn-set and so its unicornhood, and this role is played by different sets in different worlds. This lets the class nominalist deny that class nominalism gets us (11). It suggests that instead, unicornhood is e.g. a pair of the unicorn-set and the unicornhood-role, centaurhood a pair of the centaur-set and the centaurhood-role, and so the attributes are distinct even if there are neither unicorns nor centaurs.

But a role is an attribute, and so on class nominalism a set. Perhaps the unicorn-role in a world W is the unit set of W's unicornhood. Or perhaps it is a set whose members are W's unicornhood and all W's unicorns. I see no other plausible proposals if every world has its own unicorn-role. Either way, if there are neither unicorns nor centaurs, the centaurhood and unicornhood roles wind up identical, and so we still get (11). What's required, then, is that the unicorn role be a trans-world set, one containing every world's unicorns coupled with that world's unicornhood. This generates a different set to conjoin with W's unicorns than is conjoined with W's centaurs, even if W lacks both, and so keeps unicornhood and centaurhood distinct. But this generates a commitment to possibilia, merely possible unicorns.

Once this commitment rears its head, rather than talking about roles, the class nominalist can deny (10), by bringing non-actual animals into the sets to distinguish them. But if we do this in one case, we have no principled reason not to do it everywhere else: and so class nominalism winds up committed to possibilia to instance all possible attributes. This is a far larger ontological cost than positing one God, in terms of both individuals and kinds posited. It is also a far more radical revision of our general ontology: theism posits one unusual actually existing individual, while possibilism requires us to believe in things that either do not exist or are not actual but are unicorns- and gods. If possibly God exists, then even if He does not actually exist, the possibilist must accept that there ‘is’ a God- just one that doesn't happen to exist, or be actual. Even if there are irredeemable defects in the standard concept of God, there are surely other concepts nearby without these. If the problem is (say) that omnipotence and moral perfection are incompatible, there are the consistent concepts of someone just like God save for being not quite omnipotent and someone just like God save for being not quite morally perfect. Even if both are concepts of necessary beings and there are reasons to deny that there could be more than one near-divine necessary being per possible world, there are also consistent concepts of contingent near-Gods, and so surely there are possible worlds in which such things exist. So the possibilist must at a minimum accept that many possible worlds contain near-Gods – just (again) ones that doesn't happen to exist, or be actual. Some might think this fortunate, but the ontologist must judge that a non-existent or non-actual near-God is a stranger beast than theism believes in. So class nominalism avoids the contingent coextension problem only by taking on ontological commitments due to which a theist theory wins against it on economy grounds if it can do at least the ontological work class nominalism does and has no insuperable internal difficulties.

The membership relation

A last problem for class nominalism is that it creates difficulties for the metaphysics of sets. ZF is the generally accepted theory of sets. At least within the confine of ZF set theory, on class nominalism, there is no such thing as the membership relation, ∈. For ∈ is a relation, an attribute. It links every member to every set of which it is a member. So within class nominalism, ∈ is a set of pairs, each containing a member and a set. ∈ has members: the pairs. So there are also pairs consisting of a pair and ∈, and these pair-sets are in ∈. So ∈ is a member of a member of itself. This conflicts with ZF's axiom of well-founding. So ZF rules such a set out. Mathematicians do consort occasionally with non-well-founded sets, but philosophers prefer more respectable company. The class nominalist, then, must deny the existence of a membership relation. But without a membership relation to relate members to sets containing them, how can we understand the difference between sets containing an item a and sets not containing a? I see seven options.

The identity option

Quine suggests identifying singletons with their members. There is then no difficulty understanding what makes {a} different from {b}: it is just whatever makes a differ from b. We might consider extending this to other sets: saying that ‘{a, b}’ is just a plural referring device that picks out a and b, that ‘{{a}}’ differs in sense from ‘{a}’ but has the same referent, and so on. This would turn set theory into an elaborate notational device whose sole ontology is ur-elements. Whatever the charms of this, the class nominalist can't accept it. The class nominalist believes in sets to be attributes. This device eliminates sets. One could instead combine it with some ideas of David Lewis to eliminate non-sets, but that is surely a non-starter.Footnote ³⁵

An Intrinsic property?

If identity won't take the place of a membership relation, we might try to take containing a as an intrinsic property of {a}, not a relation between {a} and a, and so the difference between {a} and {b} vis-à-vis a as a matter of having this property. But this is mysterious. And it makes it equally mysterious why {a} cannot exist if a does not. If containing a is an intrinsic property of {a}, it should be possible for {a} to exist and have the property though a does not exist: if a's being F entails that b exists or has some attribute, Fness is ipso facto an extrinsic property. Yet of course, {a} can't exist without a. Further, if {a} contains a, {a} contains something: if there is such a property as containing a, there is presumably a property of containing something, which appears just where we thought we had a membership relation. On class nominalism, this property is a set, whose members are all the sets which contain something- all the non-empty sets. But then it contains itself, and so is non-well-founded. So ZF won't permit it.

Constituency work

We might next consider taking a as a constituent of sets containing it. But if a is a constituent of {a}, is there any more to {a} than a? There isn't some further entity to which the brackets correspond, which when added to a gives us a singleton. A singleton doesn't consist of a collecting lasso and the thing collected. So if a is a constituent of {a}, there is nothing more to {a} than a. Yet that can't be right either, because it seems to imply that a = {a}. Again, Quine has suggested this identity, but if we limit it to singletons it does not fully explain the difference between sets containing and sets not containing a (non-singletons also contain a), and if we generalize it we eliminate sets or non-sets. It can't be that {a} consists of constituents, {a}'s only constituent is a, there is nothing ‘out there’ to account for the non-identity of a and {a}, and yet a ≠ {a}. There is nothing to account for their non-identity if a is a constituent of {a}; in fact, a ≠ {a}; and if a is a constituent of {a}, {a} consists of constituents and {a}'s only constituent is a. So a is not a constituent of {a}.

Constitution

We might next appeal to constitution, a relation which (say some) obtains between statues and the blocks of which they consist. This relation can be 1:1. We might say that a constitutes {a}: that a's existence generates {a}'s, and a is not a constituent of {a}, but is instead related to it somewhat as a block of marble is to a statue. The problem here is what to make of the constitution-relation. Some treat it as contingent identity.Footnote ³⁶ Such proposals just fall under my treatment of identity above. Some treat it in terms of shared temporal parts.Footnote ³⁷ This is a non-starter if sets are not in time, as many think. If they are in time, still this amounts to the claim that for some period of time, an item is both a and {a}. But this would have to be the whole period of their existence – a cannot outlive {a}, nor vice-versa – and necessarily so if set theory is necessarily true. So in this particular case, we would have it that one item is both a and {a} and the two are not even possibly distinct: which boils down to the identity proposal already discussed. Some proposals treat constitution as relative identity or its close kin.Footnote ³⁸ In the case at hand this would require finding a sortal under which a non-set and a set can coincide – probably a non-starter. Finally, some treat constitution as a relation between distinct but coinciding entities. The best-developed account of this is Lynne Rudder Baker's. Her official definition is this:

If a thing is necessarily accompanied by its singleton, any circumstance is singleton-favorable; more generally any circumstance containing a and b is {a, b}-favorable. Sets don't consist of stuff, and as usually conceived don't spatially coincide with anything. There is also a difficulty about primary-kind properties: in the context of class nominalism, these can only be sets, but it's not clearly licit to appeal to sets in trying to give an account of a relation basic to making sense of what a set is. So the account seems simply not to apply to this case. If we say that sets are where their members are, the account applies if we ignore the worry about sets and eliminate the clause about possible non-coincidence and stuff-clauses. We must also eliminate the sole-coincidence requirement. If a is a member of many sets, and sets are where their members are, presumably they all spatially overlap a; if we retain sole coincidence as a requirement by requiring not just overlap but complete coincidence, the account will only apply to singletons. With these moves made, we'd be saying that constitution for sets reduces to spatial coincidence with their elements. But this seems inadequate. Coincidence is a symmetrical relation. Its presence can't explain the asymmetry of membership: {a} and {{a}} coincide, so why does only one constitute the other? Whatever moves we make, further, we're left with a more basic difficulty, which applies too if we leave the constitution-relation primitive, as in Wiggins. On set-Platonism, the constitution relation is a set. If constitution replaces the member-set relation, constitution simply becomes our new relation that isn't there.

Armstrong

Our fifth alternative grins and bears it. Armstrong embraces the non-existence of ∈, rather than seeing it as a problem. He sees ∈ as unneeded: he holds that the existence of a and {a} is all we need to make it true that a is a member of {a}. The truth of the relational proposition ‘supervenes’ on these though the relation does not exist.Footnote ⁴⁰ But this won't do. It might just be a way to say that it is primitive, non-relational fact about {a} that a is its member – i.e. that this is a non-relational property of {a}. If it is, I've already dealt with it. If not, it is just ‘solving’ the problem by magic if we can point to nothing about the singleton that makes a its member. And if a is not in an intrinsic membership-relation to {a}, not a constituent of {a}, nor ‘in’ it as a block of marble is ‘in’ a statue it constitutes, there is no such intrinsic thing to be had. We need the relation as something given in reality to explain what it is about the set that lets it help make the relational proposition true.

Whatever {a} is, it is the unique entity to which only a bears the membership relation. If the member-set relation is not intrinsic- and it is hard to see how a relation to something spatial could be intrinsic to something non-spatial – we need an extrinsic membership relation. If so, being a set is an extrinsic property- what makes something a set is that something has to it a relation of membership. And something extrinsic, a relation between a and {a}, makes it the case that a is a member of {a}. Jubien contends that if sethood is extrinsic, it turns contingent that a given entity is a set, and its members are its members:

But there is nothing incoherent about the idea of an extrinsic essential property. Perhaps I am essentially the result of the meeting of sperm S and egg E to form a zygote, and that zygote's splitting in two. If so, I essentially share a possible world with the other bit of that zygote. A set's extrinsic essential attributes may include its relations to its members.

Identity and existence

Rodriguez-Pereyra suggests that being a member of {a} is the property of being identical to a and such that {a} exists.Footnote ⁴² On class nominalism, this conjunctive property is a set, and so are its conjunct properties. As class nominalism identifies properties with the sets which are their extensions and having properties with belonging to those sets, identity with a is the set, {a}, having this property is being a member of {a}, and being such that {a} exists is the set of things such that {a} exists. Both conjunct properties are problematic on class nominalism. To see the problem with the first, recall that on Rodriguez-Pereyra's account, a ∈ {a} =df. a has the property of identity with a and is such that {a} exists. But on class nominalism, the first really asserts that a ∈ {a}: Rodriguez-Pereyra's definition turns circular. The problem with the second property is that everything is such that {a} exists. So there is a set of things such that {a} exists only if there is a universal set – and there is not. If one appeals instead to a universal proper class, then the same proper class will be being such that {b} exists, being such that {c} exists, etc. So these all wind up as the same attribute. But they are not. They are not even necessarily co-extensive; in worlds in which a exists but b does not, everything is such that {a} exists but nothing is such that {b} exists. Leaving this aside, their identity means that if we conjoin class nominalism, a universal proper class and Rodriguez-Pereyra's suggestion, it follows that being a member of a is the property of being identical with a and such that {b} exists. Finally, consider the conjunction as a whole. If properties are just sets of their extensions, this property is just {a}. So if we conjoin class nominalism with Rodriguez-Pereyra's suggestion, the relation of membership in {a} is just {a} itself. I can make no sense of this – certainly if sets presuppose membership relations, but also even if they do not. So whatever the merits of Rodriguez-Pereyra's suggestion, it does not sit well with class nominalism.

Type theory

We might last consider moving to a typed class nominalism, saying that there are many membership relations, not one: that if the highest type-level of any member of set S is level n, S is of level n + 1, and so the relation linking S to its members is of level n + 2. On such an account, we never get a non-well-founded membership relation: if a membership relation R is a set of level n, then R's membership-relation to any set must be of at least level n + 2. There is no ontological bloat here; if there are sets at all, these sets all exist anyway. But this puts class nominalism at a disadvantage in simplicity to any theory that can get by with a single relation between properties and their instances. Moreover, it sits ill with the iterative conception of set. The guiding thought of the iterative conception is that a set's existence presupposes only lower-level items: thus every set has as members only ur-elements or sets formed at lower points in the set hierarchy.Footnote ⁴³ A non-empty set presupposes the existence of its members- and exists just if they are its members. If that's right, it presupposes a membership relation as well as the existence of its members. If sets cannot presuppose items from higher levels of the hierarchy, and their existence presupposes a membership relation's existence, relations between sets and members cannot be sets of higher levels of the hierarchy.

On class nominalism, each membership relation is itself a set. And on the iterative conception, sets presuppose the existence only of sets from earlier or lower in the hierarchy. The iterative conception, then, seems inconsistent with ZF and class nominalism. So it seems that class nominalism is false if ZF and the iterative conception are true. Theism, however, is compatible with both – and a structuralist account of both would allow for items involving God to take sets' place as the items bearing the structure.

Putting them to work

Attributes are invoked to do a variety of philosophical work. I have already suggested that God can do at least as much of this work as sets. That is enough for the purposes of my argument. But in fact, God can do more than sets, and showing this will strengthen my case. Immanent universals are the gold standard here. So I frame my discussion in terms of jobs they are invoked to do. I've shown elsewhere that God can do all the work immanent universals can do.Footnote ⁴⁴ I do not claim that a theist theory of attributes does all this work. Rather, my theist theory commits ontologically only to God plus created concreta, and I remain within the bounds of that theory as long as I do all the needed work with only the ontology it permits. I now argue that sets (with or without concreta) can't do some of this work.

Universals provide referents for abstract singular terms and so help make statements involving abstract terms true: that courage is a virtue is true if the universal courage has a particular relation to the universal virtue. ‘Courage’ could refer equally well to the realities behind talk of a divine concept. But there are well-known difficulties in pressing sets into this role.Footnote ⁴⁵ And they are not helped even by expanding the sets to include possibilia.

Universals help ground modal truths: there can be more cats than there ever have been because universals are always further-exemplifiable. So too, there can be more cats than there ever have been because God's cat-concept is always further-satisfiable – God can always make more. Some necessities can be traced to universals' contents and relations (e.g. of incompatibility, grounded on their contents) – and so too, these necessities can be traced to the contents and relations of the realities behind divine-concept-talk. Sets have their members essentially. So it is not the case that there can be more cats because the actual cat-set can have more members. If there can be more cats, another set can be the cat-set (though not another actual set, if there are no possibilia), but the actual cat-set contributes nothing to this being so. Actual concrete non-divine particulars don't always provide the possibility of further cats, either. For more cats will be possible even if the world comes to contain only non-divine concreta that essentially could neither cause cats to be nor turn into cats. Granted, it would still be true in that case that past things could have turned into or caused more cats – but this gives us only that there could have been more cats, not that there can be more cats. So it seems that sets and actual non-divine elements can't do all of immanent universals' modal work. If the cats include possibilia, there can be more cats because there are merely possible cats, not because there are merely possible sets of cats. So even if we move to a possibilist version of class nominalism, the sets won't do the work here. Only the possibilia do. Further, sets don't have incompatibility relations, as universals or divine concepts may. What are incompatible are (say) belonging to the cat-set and at the same time belonging to the dog-set- but nothing about the sets explains this. The class nominalist needs (I think) again to appeal to possibilia- the incompatibility can be the fact that nothing is a possible cat-and-dog. Once the class nominalist has brought in possibilia, sets are in many cases idle wheels. And I have already suggested that possibilia are a greater ontological cost than God is.

Again, causes (we think) effect what they do in virtue of their properties: if water's coming into contact with salt dissolves the salt, this is because of their molecular structures. The theist can say the same, only adding that having that molecular structure consists in depending on the realities behind talk of a divine concept of such a molecule for being such as to satisfy it. This amounts to saying that the molecules are always being caused in a particular way to have the structures they do – which does not deny the structures their causal role, any more than being caused to speak denies the speaking its causal role. On class nominalism, our target claim becomes that causes effect what they do in virtue of belonging to (some) sets they belong to. But membership doesn't seem a causally relevant relation. Perhaps the class nominalist is best advised to reject the claim that causation is due to properties, saying instead that particulars just do cause what they do. But the loss of an intuitive thesis about causality is legitimately counted as a cost of the position, a cost theism needn't pay. Again events seem to lose their role as causal relata on class nominalism- at least on any theory that takes events to involve exemplifying properties at times. For on class nominalism, this becomes things' belonging to sets at times. And again, set-membership isn't causally relevant. So how could an event be a causal relatum, on this sort of account plus class nominalism?

Universals would ground resemblance. Intuitively, cats are alike qua cats. Being cats makes them alike. It explains their resemblance. Realists about universals can accept this: their all being cats is their having the same universal property, cathood. I accept this too, and say that what explains their resemblance is their all depending on the same divine mental events. Cats are termini of causal paths with one origin. One could object here: that cats are cats is a fact about them. On my account, it is a fact about their relation to something outside them, something extrinsic – and that's odd. But it's not clear that relations to God are extrinsic. God is omnipresent. He is in some sense everywhere. So He is within the boundaries of every cat. A cat's relation to items within it are intrinsic. Again, relations to a necessary being come out intrinsic on many accounts of being intrinsic, e.g. Yablo's.Footnote ⁴⁶

In the order of explanation, first cats are cats, which suffices to make them all alike, then they are for this reason members of the cat-set. What explains cats' resemblance must precede it in the order of explanation. So membership in the cat-set can't ground resemblance. So the class nominalist must argue that while cats do resemble, it is not because they are cats. Rather, they just resemble as a whole; respects of resemblance don't enter the account, save as defined in terms of overall resemblance as wholes. This gives up on doing the work of explaining resemblance, and so constitutes another respect in which the realities behind divine concept talk can do more work. Further, showing the overall-resemblance account to be tenable would take one a long way toward showing the tenability of resemblance nominalism, of which it is a central thesis. If resemblance nominalism works at all, it arguably works without identifying attributes with sets.Footnote ⁴⁷ So I suggest that class nominalists face a dilemma here. They can say that resemblance needs to be grounded in property-identity, and then admit that they can't do needed ontological work. Or they can defend the claim that resemblance needn't be grounded in identity of property – and then try to show that this doesn't undermine their position.

Overall, I submit, a theist theory of attributes does more work than a class nominalist theory, and does it at less overall ontological cost. So we have reason to prefer a theist theory to class nominalism- unless of course the theist view has still worse problems of its own.Footnote ⁴⁸ I close by considering three objections it is bound to raise.

Three questions for the theist view

The theist view set out above raises at least three objections. One is that it seems automatically to lose out in comparison with any non-theist view. Universals, for instance, do not raise the problem of evil – one need not explain the presence of evil in the world to assert realism. So there is a large problem for theist views which seems likely to outweigh difficulties in their non-theist rivals. But actually one deals with evil partly just by arguing that God exists. If God exists, there are sufficient answers to all arguments from evil. A reason to believe that God exists, then, is reason to believe that there are such answers. So any good argument for God's existence is just as such a minimal response to the problem of evil. Now as long as one hasn't seen more than this sort of minimal response, one may reasonably give theism less credence than one might have: if (say) an argument might on its own lead one to assign theism a credence of .7, lingering doubt that a viable account of evil is really to be had could reasonably lower that credence, perhaps to below .5 despite the argument. But despite this, it could be that if we consider only the problem of attributes, a theist theory provides the best answer. Even if that answer could not convince fully save as part of an overall package of arguments including arguments about evil, it would be worth giving, for a viable treatment of evil may well be available. Further, whether the project of explaining evil is worth looking into depends in part on how much better theist answers are than their rivals; in advance even of considering a theist theory, there is no way to show that its advantages do not justify the extra work, nor that they are not big enough to incline one more toward theism than toward its rivals even in advance of that work. So while evil is a problem for theism and not (say) realism, there is no reason to think it must outweigh theism's advantages over (say) realism in the theory of attributes.

Conservation

My theist theory requires that divine conservation be a possible sort of causality. The causal role of conservation is to explain non-probabilistically why the universe begins or continues to exist. No natural cause can do either. This is obvious in the case of beginning: the universe's beginning to exist is the appearance of the first natural items able to cause.Footnote ⁴⁹ Again, no matter what natural causes we appeal to, it is compatible with their operation that history cease. It is possible, even physically possible, that time stop at any point: General Relativity could be satisfied in a spacetime of any size, and so one which stopped just then. For it to be a non-probabilistic natural law that E-type events follow C-type events is for it to be a law that if history continues long enough after a C-type event, there will be an E-type event. Even the conservation laws, crudely put as ‘mass-energy is neither created nor destroyed’, do not entail that history will continue, for they are more precisely put as claims that the amount of mass-energy in the universe remains constant in every state of the universe, and this fact is compatible with there being a last state.

The laws do not guarantee that history will continue. They at most make it probable that it will. So a non-probabilistic explanation of the universe's persistence would require a different, non-natural sort of causation, involving a cause whose action eliminated the possibility that history cease – one whose action is incompatible with its effect's not occurring. Now it is not possible that an omnipotent will be thwarted. So while no natural cause's operation can entail that time will last long enough for it to have a given effect, if an omnipotent will decides that time shall last long enough, it cannot fail to do so. Divine conservation, then, is just a causation fit to do this particular explanatory job. Divine conservation involves a primitive causal connection: it just is a fact that if God wills that x continue to exist, x continues to exist, because God so wills. But we find good reason in the explanatory work they would do to posit other primitive causal connections. It is not clear why we should treat this case differently.

Again, competitor theories of attributes that introduce new ontology also introduce new primitive relations between their posits and the particulars which bear properties. Realism posits universals and also exemplification to tie them to particulars; trope theories posit tropes and also compresence to bind them to one another in particulars or exemplification to bind them to particulars which are not just trope-bundles; set theories introduce membership. In each case, the theories need the relations, and posit in them just the properties needed to perform their theoretical work. The theist theory's use of conservation is no different – though it has perhaps a slight epistemic edge in that the theist can rightly claim that the relation was not invented to solve this problem, but instead has its home outside the theory of attributes. Competitor theories that introduce no new ontology (concept – and resemblance-nominalism) also do without special relations: concepts bear satisfaction-relations to concreta and the important relation in resemblance nominalism is just ordinary likeness. But I hope to show elsewhere that concept nominalism is simply inadequate and resemblance nominalism solves its problems only at an ontological cost much greater than theism's. If I'm right, the theist need worry only about competitor theories of attributes that also introduce special relations, and it is not clear that conservation is the worst posit to make.

Conservation is apt to sound like magic: God wills it, and just for this reason it is so. At one level, this resemblance is a good thing for theists. We have all read stories of wizards who say words and by so doing cause effects. We understood those stories. We had no idea how such a world could work, but we could vividly imagine the world working that way. In so doing, it is unlikely that we were imagining etc. anything metaphysically impossible. The very vividness and completeness of the imagination involved is evidence against that. We were imagining a world with radically different natural laws than ours, where laws link the occurrence of words with the occurrence of effects. It is part of the idea of conservation that there is a set of natural laws which link certain sorts of thought words with events outside the mind.Footnote ⁵⁰ The more strongly something we imagine is like something whose possibility we're trying to assess, the more evidence the ability to imagine is in favor of the possibility. So the imaginability of a magic world is considerable evidence in favor of the possibility of conservation.

This isn't to say that believing in conservation is like believing in magic. Believing in magic is believing in something we know doesn't work, that purports to explain things for which better explanations are available and even known. So believing in magic is either irrational or a function of defective information, or both. Belief in conservation is neither. We don't know that conservation doesn't work or doesn't occur. And conservation, if it occurred, would explain something for which no other explanation seems possible. The concept of a world in which magic works is like the concept of a world in which conservation occurs. It doesn't follow that belief in conservation and belief in magic have the same epistemic status. The epistemic status of belief in conservation is more like that of belief in a fundamental, brute-fact physical causal connection. Conservation, if it occurred, would be a brute-fact though lawful causal connection between God's volitions and the existence of creatures. It could not be otherwise: there is nothing in virtue of which conservation works except the nature of God's power, i.e. the nature of God, and it is a brute, not further explained fact that this is as it is.

Do we really need God?

One might also ask whether God is a more complex piece of equipment than we really need to handle attributes. God has (we may think) many attributes not needed to make my theory go. If so, the most my argument really warrants is a stripped-down semi-deity: even if it succeeds, it is best taken not as an argument for God's existence, but as one for the instancing of a description which falls short of full deity. This problem of course afflicts many sorts of natural theology: a first cause or an intelligent designer need not (one can argue) have all the other divine attributes.

I reply that the best version of a theory broadly like mine includes more standard divine attributes than one might think. To begin: on the theory, whatever is F is the paradigm F, or conserved by it, or conserved by a being with appropriate mental events. It's conceivable that there be many paradigms or conservers, that many beings split the appropriate mental events between them, that nothing both is a paradigm and has any of the right mental events, etc. But it's obviously more parsimonious to combine all the roles in one being, and there is no countervailing reason not to. So if ontological parsimony is a guide to truth or at least reasonability, the most likely true or at least most reasonable version of my sort of theory will be that at any time, at most one being plays all these roles for all attributes. The one-being version ties the others in its ability to explain the facts; the tie-breaker must come from elsewhere, and rules against the others.

Again, on the theory, there must be at least one conserver at all times at which anything other than a paradigm has an attribute. It's conceivable that there be different ones at different times, and none at all when nothing which is not a paradigm has an attribute. But it is more parsimonious to have one being play all the roles at all times at which anything plays them. This being will either persist at all times, even when there is nothing to conserve, or disappear when there is nothing to conserve and reappear when there is. If we hold that this being persists even when there is nothing to conserve, we must say why and how it persists. But we need not say why it ceases, why it does so when it does, why it pops back into existence when it does, what brings this about, or what secures its identity over the gap of non-existence, and intuitively all this is more (and harder) to explain. If we do not make the parsimonious move, we have different beings playing the role at various times. Then we have to explain for each that ceases why it does, and why it does so just then. For any that re-appear after an interval we must answer all questions above. For any that do not but have successors which have not previously appeared, we must explain why the successors appear, why just when they do, and what brings this about. All this is a complicated story. The parsimonious move, then, gets added support from the fact that it has (probably) the simplest, easiest further explanation to supply. In the abstract, it seems likely that the conjunction of answers to the questions about less parsimonious versions will be less plausible overall than the conjunction for the parsimonious version – if only because there seem likely to be more independent conjuncts to the answer, and where the conjuncts of a theory are independent and each has an epistemic probability of less than 1, the more the conjuncts, the lower the epistemic probability of the conjunction. So the most plausible version of the theory will have a single eternal being always available to play the metaphysical role described for all attributes.

The theory is a metaphysically necessary truth if true at all (just as it is necessary if true at all that actual attributes are universals, or tropes). It's conceivable that different beings play the metaphysical role in different possible worlds, but if we have just one being do it in all worlds, we need not explain why there are many across worlds, and why there are the ones there are in different worlds. There is no good argument against the necessary-being version of the theory. This being is concrete, since it is a conserving cause, and there have been arguments that there cannot be concrete necessary beings: but I dispose of these elsewhere.Footnote ⁵¹ If we say that one being is in all worlds, questions arise about why it is necessary and why there are not also others. But it's plausible that necessary beings are so just because of the kind of thing they are, and to the second we may just say that Ockham's Razor bids us not believe in others unless one being of this sort can't do all we want of it: and if it is necessary, it can do all we want. If there is just the one necessary being, then its being there is just one brute fact across all possible worlds; if there are many beings across all possible worlds, either there are many brute facts (which as brute are minimally plausible additions to the theory) or there are many explanations of why there is this being, not that, in a given world, and their sheer number is reason to think that the theory will be much less plausible for including them. So overall, the necessary-being version of the theory seems likely to be the most plausible.

This being would conserve all other concreta at all times in all possible worlds. Now we believe that the universe had a beginning. A being that conserves all things at all times conserves them even at their origins: and on the part of the cause, conservation at the universe's origin is just creation ex nihilo. That is, what the conserving cause does is fundamentally the same in each case, namely make the difference between there being and there not being a segment of universe-history. So this being will also be a creator ex nihilo in worlds where a universe begins to exist. I would argue that nothing could be both necessarily existent and composed of matter: hence I would argue that this being would be immaterial in any world in which it was necessary. On the theory, this thing has per se any properties it both has and conserves in others. As these include existence, it is not itself conserved in existence by anything else.

So far, I've suggested that the most plausible version of the theory involves a single eternal, necessary, immaterial, uncreated creator ex nihilo and conserver. This being is divine, though, only if it knows and wills. So the next question is whether it does: why not a conserving cause whose inner states aren't mental?Footnote ⁵² Well, one either does or does not believe in immaterial items with mental events. Those who do have been happy to hold that any concrete immaterial thing has mental events.Footnote ⁵³ Some who don't may be happy to grant that if there were such things, they would have mental events. They would have non-transcendental properties. Ex hypothesi they would have neither physical nor distinctive abstract-entity properties. Other than some natural kinds, we have little idea what other sort of non-transcendental property they could have. So the need to fill in the blanks on them somehow will strike some as weak reason to think they would have mental properties. Again, if some immaterial concreta would have mental properties and some not, there would presumably have to be something else different about the two sorts to explain this: there certainly is where the things lacking and possessing mental events are physical. Difference in kind – ‘the one with mental events is a human soul, and the one without isn't’– doesn't provide any explanation we can grasp. So with no idea what other sorts of non-transcendental intrinsic states they might have, we must choose between an unexplained difference with regard to mental events or the claim that all immaterial concreta, as such, would have mental events. The latter seems more plausible. So the question ‘why not one without mental events?’ is likely to be put only by some who think that whatever has mental events is physical. Among these, functionalism is the leading theory of the mind.Footnote ⁵⁴ So I point my further reply at a functionalist, by considering how a conserving machine (henceforth Con) might work, given the actual character of the world. I argue that Con most likely turns itself on and off in non-random ways, and the ways it does so are grounds for ascribing mental events to it.

If Con exists, it is not always switched off, for if it were, nothing other than itself would ever exist. So Con must either switch on and off, or be always on. But it is unlikely that it always switches off and on randomly. If it did, then whether Con provides a sufficient or merely a necessary condition of conservation, it very likely would sometimes so switch off as to cause things to lapse from existence randomly. This does not (we believe) happen on the macro-level. So either Con does not always act randomly or there has been a remarkable run of luck, with random off-switchings for various items always just happening to occur when they had already ceased to exist (or were just doing so) for other reasons, and perhaps Con's ‘going random’ so timed as to produce this result. It is also likely (though less so) that Con does not ever for any extended time act randomly. It thus seems likely that if Con exists, it switches on and off non-randomly to conserve things in the patterns in which we see them appear and disappear, or else is always switched on to provide a standing necessary causal condition for conservation of whatever it conserves.

If Con is always ‘on’, though, then in non-probabilistic conserving causation – and so (I suspect) all conservation at the macro-level – conditions among conserved entities as time moves toward t constitute the rest of a causally sufficient condition for conservation at t, and this ‘triggers’ the conservation. If this is how things work, the presence of appropriate conditions just before t entails the occurrence of conservation, and so history's continuance up to t. I have already argued that this is not the case: that it is possible, no matter what has occurred to things other than Con just before t, that such things' history end without ever reaching t. If this is possible, then nothing just before t infallibly triggers or entails Con's action. Instead, there are conditions before t which provide an opportunity for Con to turn itself on for t, and it does or does not do so, with the possibility of doing otherwise.

If Con exists, then, it is most likely that Con switches on and off non-randomly to conserve things in the non-random patterns in which we see them appear and disappear. Given that Con does turn on and off, that it does so non-randomly might seem a trivial claim, for one might doubt that Con's action could count as random. Con's contribution to the world is part of what determines the patterns we see. So whatever the world's regularities are, Con's output is one factor determining them, and so if there are any, one might think, Con's activity will appear lawful, not random. I suggest, though, that if Con's activity is truly random, then it is likely that eventually its pattern changes. It is likely, in other words, that some lawlike regularities involving things' beginning and ceasing to exist eventually alter, and perhaps be replaced by no discernible regularity at all. Con's activity, if it is random, can also appear random, by contrast with its previous pattern and with patterns in events other than beginnings and ceasings to exist. Because this could happen, it has content to say that Con does not act randomly. Con acts regularly, and some of these regularities do not change over time. If a ball moves toward a wall, the ball does not disappear, and reaches the wall if it has enough momentum: this is inter alia because Con continues to contribute to the ball's existence. Thus Con's non-random activity is a major part of the reason we live in a world of regularities.

Things are Fs inter alia because Con conserves them as Fs, on my view. Further, on my view, whether or not Con is personal, different internal states operate in conserving Fs as Fs than operate in conserving Gs as Gs. If Con has no mental events, (9)'s talk of the states behind F-conservation fictionally involving concept-possession is very fictional indeed, but the theory demands that something in what Con uses to conserve Fs as Fs differ from what Con uses to conserve Gs as Gs, else there will be nothing in reality to distinguish being F from being G. If something does differ, then Con logs in the current state of the world it conserves, in the form of the current pattern of activation of these states. If Con conserves a frog, Con's frog-conserving states are activated. So there is information about the state of the world encoded in Con's inner states. They represent the state of the world – that it contains a frog – if only as rings in a tree represent the tree's age, and their changes represent changes in the world.

We can use the regularity of Con's activity to argue that while earlier states of the physical world do not determine Con's current activity, Con in fact logs in the content of the world's earlier states somehow – minimally, by retaining some record of the sequence of activations of its inner states – and therefore produces output apt to maintain earlier regularities. For if this is not the case, Con's maintenance of these is an improbable cosmic coincidence. The coincidence could be that though Con's output is random and not based on the past, it just happens to be what's needed to continue them. If Con's output is not random but not past-based, the coincidence runs this way. As non-random, Con operates entirely out of an inner program, switching on and off its conservation of various things. As it is not past-based, the program is not responsive to conditions before these things begin or cease to exist (perhaps Con keeps no records). Further, earlier parts of the program in no way causally condition the content of later parts; if anyone wrote it, the author of each instant's worth had no knowledge of the content of other instants. (Otherwise the program's later output would be in a way at least partly past-based.) If conservation involves co-operation between Con and conserved things, the program is not responsive to the availability of things to co-operate or keyed to providing them earlier to have them for use later. But it just so happens that this program so regulates conservation that a lawful physical world results, and if co-operation is needed, it is available enough of the time for there to be regularities about beginning and ceasing to exist.

Lawful regularities are much more likely on the assumption that Con currently has information about the world's earlier state and bases its current output on this, whether it has that information from without or by keeping records within. So the fact that there are natural laws strongly supports this thesis. But if Con logs in the earlier world's state and then produces appropriate current output, it has internal states somehow encoding this information, internal states which somehow process this information, and internal states which produce outputs at least partly determined by this information. Trees do not do internal processing on or produce responses to the content of their rings. It thus seems plausible to view Con as inter alia an information-processing machine, and Con's inputs as providing it information.

It seems, then, that if there actually is a conserver, it turns itself on and off, initiates its own activity whenever it acts, does so with full possibility of not doing so, has internal states which somehow represent external information, and produces activity based on processing of these states' content, activity appropriate to maintaining a law-governed universe. Con's representation of external information is mental (as vs. the tree-ring sort) if the states encoding the information are mental.

The relations between Con's input, inner representational states, internal processing and output look in outline like those functionalists claim to constitute mental systems. For functionalism, to be a mental state is to be a state with a particular functional role in a causal system of such states, mediating between environmental input and (at least possible) behavioral output. If any of Con's states is somehow caused from without to represent states of the universe, it plays the role of a perceptual belief. If Con works entirely from internal records, the states providing these play the role of memory beliefs. Con is not controlled from without and is self-activating; those of its states which trigger output in response to information from the environment or remembered information seem to play the role of intentions or volitions. But whether any of these really is a mental state depends on the overall character of the system in which it is embedded, for functionalists.Footnote ⁵⁵ This in turn depends on whether what takes us from input to output is of the right sort to count as mental.

Suppose for instance that Con is simply equipped ab initio with a huge program which pairs one response with every possible set of inputs.Footnote ⁵⁶ In this case it is not plausible that Con's inner setup constitutes a functionalist mind. If this is not how Con works internally, then Con has its inner representations, its possible responses, and taking it from one to the other something which falls short of pairing one response with every possible set of inputs. It has instead some sort of programming which selects an output based on the inputs and some inbuilt algorithm(s). If the algorithms uniquely determine the selection, though, we have again a setup which simply pairs one response with every possible set of inputs- only the pairing isn't (so to speak) written out in advance. Given what we've seen so far about Con, though, it is unlikely that this is how Con works within. For if Con works this way within, external conditions determine (and together with Con's condition just before t entail) that conservation shall occur and history shall continue. So Con must have a different internal setup. To have one different in the required respect, Con needs algorithms which work probabilistically: which might given the same inputs at least sometimes lead to different responses. If Con has these, Con is a probabilistic automaton: which is what the early ‘machine state’ version of functionalism took us to be.Footnote ⁵⁷ Functionalism has evolved beyond the ‘machine state’ version, but its core idea remains near the center of later functionalist views. So I submit that functionalists are ill-placed to push the question ‘why not a conserver without mental events?’ For when we look at how Con must act, it appears that Con has mental states as functionalists understand such states, or at least states enough like functionalist mental states that a functionalist's denial that they are mental looks strained. Thus to both dualists and functionalist materialists it should seem likely, based on how Con acts if Con exists, that Con has both beliefs and intentions.

It is a further question whether Con's beliefs are true. As we do not know how Con logs information in, we have no reason to suppose that a medium or its perceptual faculties will somehow distort incoming data and lead to inaccurate representation of the world's state, or that Con has immaterial memory traces which somehow decay over time. Now Con operates by way of algorithm. Algorithms may encode goals, if only those of their programmers: a thermostat algorithm which is biased toward selecting a lower temperature encodes a goal of cooling rooms. Whether we can infer accuracy from Con's output activity depends on what we suppose about the goals encoded in Con's algorithms. If Con's goals include maintaining certain natural regularities, Con's output suggests an accurate encoding of the world's state. If Con's goals include radically altering the regularities, Con's output suggests a wildly inaccurate encoding so interacting with the goal that acting (so to speak) with the intention of altering regularities but based on wildly false beliefs leads to just the output needed to maintain the regularities. However, if Con has a mind, principles of charity apply in interpreting its activity as they do in reading the activities of other humans: which suggests that we should favor accuracy and fulfilled goals over inaccuracy and thwarted goals. Further, Pr (Con maintains the regularities/Con has mostly true beliefs and the power to maintain regularities and has maintaining the regularities as a goal) is high, while Pr (Con maintains the regularities/Con has mostly false beliefs and the power to maintain regularities, and has altering the regularities as a goal) is not high: for there are many ways Con's beliefs could be false which would not lead to Con's maintaining the regularities, given its powers and goal. So we should hold that Con has mostly true beliefs and an intention to maintain regularities.

To do its job, further, Con must log in the entire state of the universe: and so if it has a mind, it is at every time at least well on the way to omniscience about the universe's state just before then and just then. If Con is to have enough concepts (speaking fictionally) to provide all possible non-divine attributes, it will have to possess (speaking fictionally) all possible concepts of non-divine attributes. What take the place of an F-concept on my account, again, are events with a causal role such that when appropriately tokened, Fs result. So for every possible non-divine-attribute-concept F, Con can bring it about that there is an F. This gives it at least a large chunk of the traditional range of omnipotence.

Nothing can be more perfectly one meter long than the standard meter, at least on the reading of ‘standard meter’ which treats the meter as whatever length that particular metal bar in Paris happens to have. So suppose that Con is actually the paradigm – standard- F. If nothing at a time can be more F than the standard F, then however great an F could appear at t, Con will have also to be at least that F by t. So at all times, Con's level of Fness, where it is the paradigm F, must be at least as great as any other F that could appear just then.

In sum, then: while it would be possible to formulate complex versions of my sort of theory in which no single being has anything approaching the divine nature, the best version of the theory involves a being which is eternal, necessary, uncreated, immaterial, creator/conserver, very knowledgeable and powerful and as perfect in relevant respects as any other thing that might co-exist with it.Footnote ⁵⁸ Were Richard Dawkins to learn that something had just these properties and no more, I doubt he'd say ‘well that's that, then – atheism is true’. Western monotheists would happily call it God even if it falls short of the standard perfect-being package. And of course, nothing rules it out that this being does have the rest of the perfect-being package: a being could well have the requisites for performing a role in the best version of my theory and much more besides.