1.

A strict order is a relation that is transitive, irreflexive, and asymmetric. My aim in this paper is to argue that grounding is not a strict order. More precisely, I shall argue that grounding is neither transitive, nor irreflexive, nor asymmetric.Footnote ¹ This is controversial and original. It is controversial because some philosophers maintain that grounding is a strict order, or that it has some of the three features necessary for being a strict order; for instance, Correia (Reference Correia2010) and Raven (Reference Raven2013: 193–94) think it has all three; Fine thinks it is irreflexive and transitive (2010: 100), which entails it is also asymmetric (see also Fine Reference Fine, Correia and Schnieder2012: 56). Audi, Bennett, Rosen, and Schaffer think that it is irreflexive and asymmetric (Audi Reference Audi, Correia and Schnieder2012a: 102, Audi Reference Audi2012b: 691–92, Bennett, forthcoming, Schaffer Reference Schaffer, Chalmers, Manley and Wasserman2009: 364, Rosen Reference Rosen, Hale and Hoffman2010: 115–16). Moreover, my argument is original because although some philosophers have maintained that grounding is not a strict order because it lacks one of the three features, as far as I know no one has argued that it lacks all three of them (Schaffer (Reference Schaffer, Correia and Schnieder2012) has argued that grounding is not transitive; Thompson (Reference Thompson2014) has argued that it is not asymmetric, and Correia (Reference Correia2014) has rejected its irreflexivity. In addition, Jenkins (Reference Jenkins2011) has argued that one should not assume that grounding is irreflexive, and Barnes (Reference Barnes2014) has argued that dependence is not asymmetric, but she distinguishes dependence from grounding). Furthermore, the way in which I shall argue for my thesis is original, since some of my examples that grounding is neither irreflexive nor asymmetric nor transitive will be examples of truthmaking where irreflexivity, asymmetry, and transitivity fail.Footnote ²

The structure of the paper is as follows. In section 2 I shall characterize grounding in a general way. From this characterization it will follow that truthmaking and another relation that I shall call alethic-fact grounding are species of grounding. Thus, if either of these relations lacks one of the features necessary for being a strict order, then grounding is not a strict order either. In section 3 I shall discuss and reject Schaffer's counterexamples to the transitivity of grounding. In section 4 I shall present my case against the transitivity of truthmaking. In section 5 I shall present my case against the irreflexivity of truthmaking and alethic-fact grounding. In section 6 I shall present my case against the asymmetry of truthmaking and alethic-fact grounding. In section 7 I shall argue that a contrastive quaternary relation of grounding, like the one proposed by Schaffer (Reference Schaffer, Correia and Schnieder2012: 130) is neither irreflexive nor asymmetric. Section 8 is a brief conclusion.

2.

Truthmaking is the relation that obtains between a true proposition and that in virtue of which it is true, its truthmaker. For instance, according to many, the fact that Socrates is white is the truthmaker of the proposition <Socrates is white>.Footnote ³ That is, the fact that Socrates is white and the proposition <Socrates is white> are related by the truthmaking relation. Similarly, according to many, Socrates is the truthmaker of the proposition <Socrates exists>. Thus, Socrates and the proposition <Socrates exists> are related by the truthmaking relation.

Is truthmaking a case of grounding? I think it is, and others think so as well (Schaffer Reference Schaffer, Chalmers, Manley and Wasserman2009: 375, Liggins Reference Liggins, Correia and Schnieder2012: 269; cf. Schnieder Reference Schnieder2006: 30). The relation between truthmaker and truth seems analogous to the relation between member and set, between part and whole, and between being a cruel action and being a wrong action, and these are all initially plausible cases of ground and grounded. It is thus initially plausible that truthmaking is a case of grounding.

Sometimes grounding is characterized as a non-causal relation of dependence (Audi Reference Audi, Correia and Schnieder2012a: 101; Reference Audi2012b: 690). But then truthmaking is not a case of grounding, since the proposition <Socrates is white> does not depend on the fact that Socrates is white. What depends on the fact that Socrates is white is not the proposition itself, but its truth or its having the property of being true. In other words, the fact that the proposition exists does not depend on the fact that Socrates is white, but the fact that the proposition is true does.

As I said, truthmaking seems to be a case of grounding. Thus, the considerations of the previous paragraph suggest that grounding should not be characterized as a non-causal relation of dependence. And yet there seems to be something right in the idea of grounding as a non-causal relation of dependence.

I think the way out of the dilemma is the following. Grounding is the non-causal generic relation of being F in virtue of (or, equivalently, the generic relation of being F non-causally in virtue of). Thus a relation is a case or species of grounding if it is a specification of the non-causal generic relation of being F in virtue of. Therefore, truthmaking is a case or species of grounding since it is the non-causal relation of being true in virtue of: the proposition <Socrates is white> is true in virtue of the fact that Socrates is white. Similarly, being right in virtue of, being blue in virtue of, existing in virtue of, and many other such relations are also cases or species of grounding.

Thus, truthmaking is a species of grounding. But Kit Fine has argued that truthmaking is not a case of grounding even if the two are close cousins. Among other points, he maintains that grounding, if a relation, should hold between entities of the same type, and insofar as a choice needs to be made, such entities should probably be facts rather than propositions (2012: 43; Fine's own view is that grounding is an operation rather than a relation). This excludes truthmaking as a case of grounding since the truthmaking relation links entities of different kinds to propositions. But no reason has been given why ground should only link entities of the same type. And, indeed, other grounding theorists take ground to link entities of different kinds (Schaffer Reference Schaffer, Chalmers, Manley and Wasserman2009: 375).

Fine objects to truthmaking in other ways too. For instance, he argues that truthmaking theory has an unduly restricted conception of what grounds and of what is grounded and that it is ill-suited to the step-by-step procedure of determining what grounds what (2012: 43–44). But these are objections to the project of letting truthmaking theory play all the roles associated with a theory of grounding. These objections do not counter the more limited claim, which is my claim here, that truthmaking is a species of grounding. Fine also objects that the relation of truthmaking is defined in modal terms, namely, as the relation of necessitation, and that this has, among others, the problem that any necessary truth will be grounded by anything (2012: 45). But there is no need to define truthmaking as necessitation; in fact, truthmaker theorists often define the truthmaking relation in terms of being true in virtue of (see, for example, Armstrong Reference Armstrong2004: 5, 7, 16–17; Rodriguez-Pereyra Reference Rodriguez-Pereyra2002: 34; Reference Rodriguez-Pereyra, Dodd and Beebee2005: 17–18). Thus, although Fine is right that there is more to grounding than truthmaking, nothing he has said seems to undermine the idea that truthmaking is a species of grounding.

Of course, some theorists, Fine included, may reject my claim that truthmaking is a species of grounding by rejecting my general characterization of grounding as the non-causal generic relation of being F in virtue of. They may think that grounding should be characterized as the relation of in virtue of, or as being F in virtue of being G, or in some other way. But given the scope and plan of this article, I cannot engage in defending my general characterization of grounding. It is initially plausible that truthmaking is a species of grounding, and all I can do here is give reasons why this is the case, but I cannot defend those reasons. My aim, after all, is to argue that grounding is not a strict order, and I will do so by arguing that two species of grounding relations, one of which is truthmaking, fail to satisfy the definition of a strict order. Even those who think that truthmaking is not a species of grounding could still be persuaded by my argument that the other grounding relation is not a strict order, and that therefore, in general, grounding is not a strict order (though I need the claim that truthmaking is a species of grounding to reach my conclusion that grounding is not transitive).

What is the other relation I shall be discussing here? It is a close relative of truthmaking. Think of a truthmaker and the proposition it makes true. Because of the truthmaker, there is a fact that the proposition is true. And so the fact that the proposition is true exists, or obtains, in virtue of the truthmaker. The fact that the proposition is true and the truthmaker stand in the non-causal exists (or obtains) in virtue of relation. Thus the truthmaker grounds the fact that the proposition is true. For instance, Socrates grounds the fact that <Socrates exists> is true, since the latter fact exists or obtains in virtue of Socrates. Similarly, the fact that Socrates is white grounds the fact that <Socrates is white> is true, since the latter fact exists or obtains in virtue of the fact that Socrates is white. I shall call this relation obtaining between a truthmaker and the fact that the proposition it truthmakes is true the relation of alethic-fact grounding. It must be noted here that taking these two relations as cases of grounding commits one to grounding not being a relation that obtains only between facts. This is controversial in the theory of grounding. There are those who maintain that grounding can obtain only between facts (e.g., Audi Reference Audi, Correia and Schnieder2012a and Rosen Reference Rosen, Hale and Hoffman2010; cf. Fine Reference Fine, Correia and Schnieder2012: 43) and those who maintain it can obtain between other entities too (e.g., Schaffer Reference Schaffer, Chalmers, Manley and Wasserman2009).

Thus, both truthmaking and alethic-fact grounding are species of grounding. But for the genus to be a strict order means that all its species must be so. Thus, by showing that neither truthmaking nor alethic-fact grounding are strict orders, I shall have shown that grounding is not a strict order either.

When an entity is a truthmaker of a certain proposition, that entity is the alethic-fact ground of the fact that that proposition is true. The relations of truthmaking and alethic-fact grounding are thus closely linked. But this close link does not mean that they must have the same formal properties. That is, the fact that truthmaking is neither transitive nor irreflexive nor asymmetric, does not mean that alethic-fact grounding has none of those three formal properties. Thus, my examples showing that truthmaking is neither irreflexive nor asymmetric and my examples showing that alethic-fact grounding is neither irreflexive nor asymmetric will be independent of each other, albeit similar. Furthermore, although I shall argue that truthmaking is not transitive, I am not in a position to argue that alethic-fact grounding is not transitive. But I shall first argue that Schaffer has not established that grounding is not a strict order since he has not established that grounding is not transitive.

3.

Schaffer has presented the following counterexample to transitivity. Consider the set S = {a, b, c}. Schaffer argues that (a) the fact that c is a member of S grounds the fact that S has exactly three members and (b) the fact that S has exactly three members grounds the fact that S has finitely many members, but (c) it is not the case that the fact that c is a member of S grounds the fact that S has finitely many members (Schaffer Reference Schaffer, Correia and Schnieder2012: 127–28).

Schaffer supports (a) by saying that if c were not a member of S then S would have had two members and not three. Given that sets have their members necessarily, it is controversial that if S had not had c as a member, it would have had two members. But it is still plausible that if it had not had c as a member, it would not have had three members, since it would not have existed. Still, Schaffer's reasoning treats the fact that something is counterfactually necessary for another fact as a sufficient condition of the former fact being a ground of the latter, where a fact [P] is counterfactually necessary for a fact [Q] if and only if had [P] not obtained, [Q] would not have obtained.

This differs from Schaffer's support for (c). For he says that (c) is true because S would have been finite even without c as a member. But, it might be objected, S would not have been finite without c as a member since S would not have existed. This problem can be circumvented by changing the example and using a property that contingently has three instances (as noted by Litland Reference Litland2013: 25–26). So I shall ignore this problem and continue using Schaffer's example of set S.

Thus, according to Schaffer, the fact that c is a member of S does not ground the fact that S has finitely many members because if c had not been a member of S, S would still have been finite. But this means that a fact being counterfactually necessary for another fact is a necessary condition of the former grounding the latter. And given what he says about (a), that a fact is counterfactually necessary for another fact is necessary and sufficient for the former grounding the latter.

But that a fact is counterfactually necessary for another fact is a necessary condition for the former grounding the latter conflicts with (b), since the fact that S has exactly three members is not counterfactually necessary for the fact that S has finitely many members. But all Schaffer says about (b) is that it is plausible because having finitely many members is a determinable, of which having exactly three members is a determinate (2012: 127). So perhaps the idea is that, provided property F is not a determinate of property G, if the fact that something is F grounds the fact that it is G, then that it is F must be counterfactually necessary for its being G. But it is not clear what the rationale for exempting the case of determinates and determinables would be. Furthermore, it seems that other exceptions will be needed since it is a standard idea that disjunctive facts are grounded in each of their true disjuncts, despite the latter often not being counterfactually necessary for the former. But then what is difficult to see is why in the case of the facts that c is a member of S and that S has finitely many members, for the former to ground the latter, the former must be counterfactually necessary for the latter.

Now, in private correspondence Schaffer has said that his intention was to use the counterfactual test not to establish his claims (a) and (c) but as a heuristic to buttress the intuitions that (a) and (c) are correct. But since no reason has been given for why the counterfactual test is a good heuristic in the cases in which it is supposed to be a good heuristic, it is not clear that it buttresses any intuitions. Furthermore, this methodology seems impotent to convince anyone who does not share the initial intuitions. This is my own case. Indeed, I reject (a). For the identity of its members has nothing to do with the fact that S has exactly three of them—what grounds the fact that S has exactly three members is the fact that there is something, something else, and a third thing such that none of them is identical with either of the others, all three are members of S, and nothing else is.Footnote ⁴

Schaffer produces two other counterexamples to the transitivity of grounding. One is the dented sphere case. But this case has exactly the same structure as the case of the three-membered set S, and so what I said about the latter applies to the former. The other case is that of Cadmus's meow. According to Schaffer, (d) the fact that the creature was produced from the meeting of this sperm and that ovum grounds the fact that Cadmus is meowing, and (e) the fact that Cadmus is meowing grounds the fact that something is meowing, but (f) the fact that the creature was produced from the meeting of this sperm and that ovum does not ground the fact that something is meowing. Given what Schaffer says about (d), (e), and (f), it is not clear that this case is intended to have the same structure as the other two.

I think this example is not a failure of transitivity. In defense of (d) Schaffer says that, given origin essentialism, (d) is plausible, since being produced from the meeting of this sperm and that ovum helps make the creature Cadmus as opposed to some other cat. If one wonders why Cadmus is meowing, Schaffer says, the facts that make the meowing creature Cadmus are part of the reason (2012: 128). So the grounded fact is the fact that Cadmus is meowing. This is not the fact that Cadmus is meowing, since this fact is not grounded by what makes the creature Cadmus—what makes the creature Cadmus has nothing to do with the fact that it is meowing.

And Schaffer says that (e) is plausible since the fact that Cadmus is meowing provides a factual witness for the existential generalization that something is meowing, and factual witnesses ground existential generalizations. If one wonders why something is meowing, Schaffer says, the fact that Cadmus is meowing provides a sufficient reason (2012: 128).

Note that given his commitments in the case of set S, Schaffer will have some explaining to do concerning why (e) is acceptable since the fact that Cadmus is meowing is not counterfactually necessary for the fact that something is meowing. But, independently of this, what grounds the fact that something is meowing is not that Cadmus is meowing but that Cadmus is meowing. That Cadmus is meowing grounds the fact that Cadmus is doing something, not the fact that something is meowing. That in virtue of which something is meowing is that Cadmus is meowing, not that Cadmus is meowing. And so there is no failure of transitivity in this case since the fact grounded in (d) is not the grounding in (e).

This presupposes that the facts that Cadmus is meowing and that Cadmus is meowing are different facts, and it is not clear exactly what the difference between them consists in. But that these facts are different is presupposed by what Schaffer says. For instance, in defense of (f) Schaffer says ‘the fact that the creature was produced from the meeting of this sperm and that ovum helps make it be Cadmus meowing, but doesn't help make it be Cadmus meowing’ (2012: 129). And for my purposes here I do not need to clarify the difference between these facts. For, whatever the difference between these facts is, what Schaffer says suggests that the two facts are different. Furthermore, for (f) to be plausible, the fact grounded in (d) must be the fact that Cadmus is meowing, while the grounding in (e) must be the fact that Cadmus is meowing. And if this is so, then the present case is not a counterexample to the transitivity of grounding.

Thus, Schaffer's case does not establish that grounding is not transitive.

4.

In this section I shall argue that truthmaking is not transitive. Consider (1) and (2):

1. (1) The fact that Socrates is white makes <Socrates is white> true.

2. (2) <Socrates is white> makes <There are propositions> true.

For truthmaker theorists, (1) is plausible and so is (2). For it is plausible that the truthmakers for an existential generalization are its instances if those instances are necessarily its instances. But it is also plausible that propositions are necessarily propositions. Thus, any proposition is a truthmaker for <There are propositions>. Now, transitivity would give us (3):

1. (3) The fact that Socrates is white makes <There are propositions> true.

But (3) is false. The proposition <There are propositions> is not true in virtue of the fact that Socrates is white. Although the fact that Socrates is white makes true a truthmaker of <There are propositions>, the fact that Socrates is white is not that in virtue of which <There are propositions> is true. Indeed, the fact that Socrates is white is not even that in virtue of which <Socrates is white> exists, and so one should not expect it to be that in virtue of which <There are propositions> is true. Only that in virtue of which propositions exist can be the truthmaker of <There are propositions>. Thus (3) is false, and transitivity fails.Footnote ⁵

Raven says that truthmaking cannot be a species of grounding because the relata of truthmaking are of different types and this precludes chaining (Raven Reference Raven2013: 194; cf. Fine Reference Fine, Correia and Schnieder2012: 44–45). But not all the relata of a truthmaking relation are of different types, as this example illustrates. A proposition can both be made true and make true another proposition. This is the case with <Socrates is white> in the present example.

Is the alethic-fact relation non-transitive? I am not in a position to argue for such a claim. Consider, for instance, the following three claims:

1. (4) The fact that Socrates is white grounds the fact that <Socrates is white> is true.

2. (5) The fact that <Socrates is white> is true grounds the fact that <There are propositions> is true.

3. (6) It is not the case that the fact that Socrates is white grounds the fact that <There are propositions> is true.

These three claims do not show that alethic-fact grounding is non-transitive, for although (4) and (6) are true, (5) is false. What <There are propositions> is true in virtue of is not the fact that <Socrates is white> is true, but rather the fact that <Socrates is white> exists.

One might then think that the following three claims show that alethic-fact grounding is non-transitive:

1. (4) The fact that Socrates is white grounds the fact that <Socrates is white> is true.

2. (7) The fact that <Socrates is white> is true grounds the fact that <There are true propositions> is true.

3. (8) It is not the case that the fact that Socrates is white grounds the fact that <There are true propositions> is true.

But these three claims do not show that alethic-fact grounding is non-transitive. For although both (4) and (7) are true, I see no reason to accept (8) that does not assume that alethic-fact grounding is non-transitive. Points made by Nick Jones and Ezequiel Zerbudis made me see that I am not in a position to argue that alethic-fact grounding is non-transitive on the basis of the example based on claims (4), (7), and (8).

Tuomas Tahko (Reference Tahko2013) argues that what he calls truth-grounding is not transitive, and what he calls truth-grounding seems to be what I am calling here alethic-fact grounding. Simplifying a little bit, he argues that while the fact that bottle b exists is grounded in the fact that b has a stable macrophysical structure, and the fact that b has a stable macrophysical structure is grounded in the fact that the Pauli exclusion principle holds, it is not the case that the fact that b exists is grounded in the fact that the Pauli exclusion principle holds (2013: 336–37). The problem with this is that it does not seem to counterexemplify the transitivity of truth-grounding in particular since no fact involved in this case is a fact about something's being true. But I do not think the example even works against the transitivity of grounding in general. Tahko thinks that general laws like the Pauli exclusion principle in no way contribute to answering the question about the existence of bottle b (2013: 337). But if the fact that the Pauli exclusion principle holds grounds the fact that b has a stable macrophysical structure, and this latter fact grounds the fact that b exists, then I see no reason, other than assuming that transitivity must fail, why the fact that the Pauli exclusion principle holds should not ground the fact that bottle b exists or even the fact that <bottle b exists> is true. One might try reformulating the example as follows: the fact that <b has a stable macrophysical structure> is true grounds the fact that <b exists> is true, the fact that the Pauli exclusion principle holds grounds the fact that <b has a stable macrophysical structure> is true, but it is not the case that the fact that the Pauli exclusion principle holds grounds the fact that <b exists> is true. But this reformulation has the additional problem that the first claim, that the fact that <b has a stable macrophysical structure> is true grounds the fact that <b exists> is true, seems incorrect: what grounds the truth of <b exists> is, in any case, something about b, say that it has a stable macrophysical structure, but not something about the truth of another proposition.

But it is clear that statements (1)–(3) show that truthmaking is not transitive, and so truthmaking is not a strict order. And this is enough to show that grounding is not transitive and therefore also not a strict order either. But there are other reasons why truthmaking is not a strict order: it is neither irreflexive nor asymmetric, as we shall now see. Thus, grounding is also neither irreflexive nor asymmetric, as we shall now see.

5.

In this section I shall argue that grounding is not irreflexive. Consider the proposition A: <A exists>. This proposition says of itself that it exists. Let us suppose that A exists. If so, A is true. What makes it true? According to standard truthmaking ideas, a proposition of the form <e exists> is made true by e itself. Thus, if it exists, A is made true by itself. Thus, truthmaking is not irreflexive and, therefore, neither is grounding.

But there is another reason why grounding is not irreflexive, since alethic-fact grounding also fails irreflexivity. To see this, consider the truth-teller, B: <B is true>. This proposition says of itself that it is true. Let us suppose that B is indeed true. What grounds the fact that it is true? The fact stated by B itself. But what B states is that it itself is true. Thus, if B is true, the fact that B is true grounds the fact that B is true. Therefore, alethic-fact grounding is not irreflexive. And since alethic-fact grounding is a species of grounding, grounding is not irreflexive.Footnote ⁶ (Note that if A is its own truthmaker, it is its only truthmaker, and if the fact that B is true is grounded by the fact that B is true, the fact that B is true is its sole ground. This means that A is its own full ground, and the fact that B is true is its own full ground).Footnote ⁷

Kripke has proposed a theory of truth in which the concept of groundedness plays a central role. The truth-bearers in Kripke's theory are sentences. On Kripke's theory, the truth-teller, the sentence that says of itself that it is true, is ungrounded (Kripke Reference Kripke and Kripke2011: 85, 91). Thus, what I am proposing here seems to conflict with Kripke's theory of truth.

But it is important to appreciate the real point of conflict. In Kripke's theory, that a sentence is ungrounded basically means that its truth-value is not ultimately determined by non-semantic facts (Kripke Reference Kripke and Kripke2011: 88). The motivation of Kripke's theory is to give an explanation of truth that avoids the semantic paradoxes, and he does this in a way that accounts for how someone who initially does not understand the truth predicate, but who understands other non-semantic predicates, would learn to use the truth predicate (2011: 85). At the base of Kripke's construction there are sentences featuring only non-semantic predicates. The assignment of truth-values to sentences featuring non-semantic predicates determines a minimal fixed point in which the truth value of sentences featuring semantic predicates is ultimately determined by the truth value of sentences featuring only non-semantic predicates. The sentences that do not have a truth-value in the minimal fixed points are ungrounded. Thus, for Kripke ‘ungrounded sentence’ means ‘sentence not ultimately grounded in non-semantic facts’.

But, as Kripke is of course aware, the truth-teller is not a paradoxical sentence. The truth-teller can consistently be assigned the truth-value true, and Kripke's theory allows for this—the truth-teller has a truth-value in every maximal fixed point. But since in those cases the truth of the truth-teller is not determined by non-semantic facts at all, the truth-teller counts as a truth but an ungrounded truth (2011: 91–92).

Translated into my schema of propositions and facts about their being true, this becomes the idea that a proposition is ungrounded only if the fact that it is true is not ultimately determined by non-semantic facts. Now, insofar as Kripke would maintain that the fact that B is true is not determined by non-semantic facts at all, I am not in conflict with him since I maintain that the fact that B is true is not determined by non-semantic facts at all. But insofar as Kripke seems to maintain that the semantic can only be grounded in the non-semantic, I disagree. I can see no reason why semantic facts—that is, facts that a certain proposition is true—must ultimately be grounded in non-semantic facts. And, indeed, the example of proposition B convinces me that there are semantic facts ultimately grounded in semantic facts. Furthermore, the idea that the semantic can only be grounded in the non-semantic does not seem to be essential to Kripke's solution of the paradoxes. What is essential for that is simply that the truth value of certain sentences is not determined by non-semantic facts at all. And this is independent of the idea that the semantic can only be grounded in the non-semantic.

Note, by the way, that my case that proposition A is made true by itself does not presuppose that semantic facts can be ultimate grounds. For proposition A is a proposition, not a fact, and therefore not a semantic fact. Nor is the fact that proposition A exists a semantic fact, since a semantic fact is a fact about whether a certain proposition is true or not. Thus, even those who believe that semantic facts cannot be ultimate grounds will have reason to reject the irreflexivity of truthmaking and therefore that of grounding too.

Now, proposition B can consistently be thought of as true, it can consistently be thought of as false, and it can consistently be thought of as indeterminate; there is no evidence about which it really is. Therefore, it might be objected, there is no evidence that proposition B is true, and therefore no evidence has been presented that grounding is not irreflexive. But this objection wrongly assumes that the relevant target is the thesis that there are no actual instances of reflexive grounding, that is, no actual instances of entities grounding themselves. But this is not the thesis I am trying to refute. The thesis I am trying to refute is that it is conceptually necessary, or at least metaphysically necessary, that grounding is irreflexive. That is what those who maintain the irreflexivity of grounding maintain. And, in any case, that it is conceptually necessary, or at least metaphysically necessary, that grounding is irreflexive is what defenders of the irreflexivity of grounding should maintain. For when postulating that grounding is irreflexive (or asymmetric or transitive) they are concerned with the most general and abstract formal features of grounding, not with aspects that might be true but need not be true of it. But if, under the non-contradictory assumption that B is true, it is shown that the fact that B is true would ground itself, then it is not conceptually true that alethic-fact grounding is irreflexive. And, therefore, the thesis that grounding is irreflexive, in the intended sense, has been refuted.

A similar point can be made about proposition A. For given that under the non-contradictory assumption that A exists I have shown that A would be its own truthmaker and so its own ground, I have shown that it is not conceptually true that grounding is irreflexive.

It might be claimed that although it is not conceptually necessary that alethic-fact grounding and truthmaking are irreflexive, it is metaphysically necessary that this is so. One way for this to happen is for it to be metaphysically necessary that if A exists, it is not its own truthmaker. Likewise, it would have to be metaphysically necessary that if B is true, the fact that it is true is not grounded in itself.

Now, given that propositions of the form <e exists> are made true by e itself, it is very difficult to see how A could exist without being made true by itself. Similarly, given that propositions are grounded in the facts stated by them, it is very difficult to see how the fact that B is true could fail to be grounded in itself.

Perhaps, then, it is necessary that A does not exist? But why should this be the case? There is certainly no contradiction in its existence. And there are plenty of other self-referring objects. Indeed, there are some objects very similar to A. Consider this token sentence S: S does not exist. S does exist—it is located on this page. So why couldn't proposition A exist? What does not exist is the case for the necessary non-existence of A.

Depending on one's metaphysics of propositions, one might have doubts about the existence of propositions like A. For instance, if propositions are broadly Russellian in the sense of being complex objects containing the objects they are about, proposition A would contain itself as a proper constituent. This need not be a reason for rejecting self-referential propositions though, since the notion of something being a proper constituent of itself need not be absurd. But even if it is, there are other conceptions of propositions in which proposition A would not be a proper constituent of itself. Furthermore, truthmaker theorists are unlikely to adopt any Russellian view of propositions, because these are too similar to facts, and they need to maintain a clear distinction between facts and propositions.

How about the fact that B is true? Perhaps it is necessary that B is untrue? It may be argued that this is so because it is metaphysically necessary that truths are ultimately determined by non-semantic facts. Since the truth of B cannot be determined by non-semantic facts, then B cannot be true, and so the fact that B is true cannot exist.

But there is a problem with this line of thinking, namely, if it is correct, there must be truths that are not ultimately grounded in non-semantic facts. For suppose that B is untrue because it is not ultimately grounded in non-semantic facts. Consider then the proposition B*: <B is untrue>. B* is true, since, we are assuming, B is untrue. But the truth of B* is not ultimately determined by non-semantic facts: it is, on the contrary, ultimately determined by the semantic fact that B is untrue.

Thus, there is not much prospect in arguing that although it is not conceptually necessary that grounding is irreflexive, it is metaphysically necessary that this is so.

It is interesting to note that my counterexamples to the irreflexivity of grounding and truthmaking share an important feature that not all other potential counterexamples share. The potential counterexample to the irreflexivity of grounding Jenkins considers is that of a pain being grounded in itself, by virtue of its being grounded in a brain state to which it is identical (Jenkins Reference Jenkins2011: 271). This depends on a controversial metaphysics of mind. But there is another reason why this example is controversial. The idea that mental states are grounded in brain states is motivated by an interest in relating two distinct sets of states. It is difficult to see why an identity theorist should feel inclined to maintain that mental states are grounded in brain states. But truthmaker theory is committed to the claim that a proposition of the form <e exists> is made true by e, and so it entails that if proposition A exists, it is made true by itself. Similarly, the theory of grounding maintains that a fact that a certain proposition is true is grounded in the fact stated by that proposition, and so it entails that if proposition B is true, the fact that it is true is grounded in itself.

6.

If truthmaking and alethic-fact grounding are asymmetric, they must be irreflexive. Their non-irreflexivity, however, guarantees their non-asymmetry only by guaranteeing that cases of symmetric truthmaking and alethic-fact grounding are cases of reflexive truthmaking and alethic-fact grounding. But if what I have said above is accepted, it will not be difficult to see that truthmaking and grounding are non-asymmetric in the full sense of there being distinct entities that ground each other.

Consider the propositions C = <D exists> and D = <C exists> and assume that both of them exist. The truthmaker of C is D, and the truthmaker of D is C. Thus, truthmaking is not asymmetric and therefore neither is grounding.

Now consider the propositions E = <F is true> and F = <E is true>, and assume they are both true. If so, given what they say, the fact that E is true grounds the fact that F is true, and the fact that F is true grounds the fact that E is true (Thompson [Reference Thompson2014] has given a similar example). Therefore, alethic-fact grounding is not asymmetric and neither is grounding.

But there is a natural objection to this. This objection is that grounding must be asymmetric because the ground is more fundamental than the grounded, and being more fundamental than is an asymmetric relation (cf. Rosen Reference Rosen, Hale and Hoffman2010: 116, Barnes Reference Barnes2014, Bennett, forthcoming). The same kind of consideration can be used to argue that grounding must be irreflexive, since being more fundamental than is an irreflexive relation.

Now, the relations of grounding and being more fundamental than are not the same relation. For something might be more fundamental than another thing without grounding it. This will be the case, for instance, if the former thing is fundamental and the latter one is grounded, but grounded in something other than the former thing. But the point that grounding must be irreflexive and asymmetric need not rely on an identification of the relations of grounding and being more fundamental than but only on the fact that when x grounds y, x must be more fundamental than y. But I shall argue that either the fact that x grounds y does not entail that x is more fundamental than y or there is no reason to think that it does.

Let us assume that the more fundamental than relation is asymmetric and irreflexive. But what is it to be more fundamental than something else? One answer is that it is to be closer to a fundamental (that is, ungrounded) entity than another thing (where it is understood that a fundamental entity is as close to a fundamental entity—itself—as anything can be). And the obvious way to understand this idea is that x is closer to a fundamental entity than y if and only if, in their chain of grounding, there are fewer elements between x and a fundamental entity than between y and a fundamental entity.

Now the relation of grounding is not restricted to finite chains. Nothing in the idea of grounding excludes the possibility of chains of grounding containing infinitely many distinct elements and such that there is no first element, no ungrounded ground. For instance, if wholes are grounded in their parts, a gunky whole—that is, a whole all of whose parts have proper parts—would be at the top of an infinitely long chain of grounding with no first element. Now take two members of such a chain, x and y, such that x grounds y. Is x more fundamental than y? No, for there is no fundamental entity in their chain of grounding, and therefore it is not the case that there are fewer elements between x and a fundamental entity than between y and a fundamental entity.

If so, then that x grounds y does not entail that x is more fundamental than y, and so the fact that reflexive and symmetric cases of grounding are cases in which the more fundamental than relation does not obtain between the ground and the grounded is no objection to them. Reflexive and symmetric cases of grounding are like bottomless chains of grounding: cases of grounding without different degrees of fundamentality. And in all three cases the reason why there are no different degrees of fundamentality is that there are no fundamental entities in the chains of grounding: neither bottomless chains of grounding, nor ‘reflexive’ chains of grounding, nor ‘symmetric’ chains of grounding, contain fundamental, ungrounded entities.Footnote ⁸

It might be objected that there cannot be bottomless grounding chains, that is, that every chain must end in an ungrounded ground. I do not see why there cannot be such ‘bottomless’ chains. But I do not need their possibility to make my point. For it is sufficient that there can be dense chains of grounding, that is, chains of grounding such that between any two elements there is a further element. In this case it will also be true that no element of such a chain will be closer to a fundamental entity than any other element of it (except, of course, when the member in question is the ungrounded ground of the chain).

It might be objected that the real number line is dense but one can make sense of one positive real number being closer to zero than another positive real number. But this is because we can independently make sense of the notion of one positive real being bigger or smaller than another one. In this case we are working with a notion of relative fundamentality according to which x is closer to a fundamental entity than y if and only if, in their respective chains of grounding, there are fewer elements between x and a fundamental entity than between y and a fundamental entity.

A better objection would be against the characterization I have given of what it is to be more fundamental than something else. Perhaps x is more fundamental than y if and only if every ground of x is a ground of y but not vice versa? This definition is satisfied by suitable pairs of elements in grounding chains with infinitely many elements, whether such chains have an ungrounded first element or not. But if this is what being more fundamental than is, what reasons are there for thinking that if x grounds y, x must be more fundamental than y? There do not seem to be any. Indeed, there seem to be no reasons why if x grounds y, then everything in virtue of which x is F is something in virtue of which y is F, but not vice versa. Furthermore, my counterexamples to the irreflexivity and the asymmetry of grounding are counterexamples to the idea that if x grounds y, every ground of x must be a ground of y but not vice versa. Thus, not only do there seem to be no reasons to think that if x grounds y, x must be more fundamental than y in this sense, but there are actually reasons against it.

(Three comments on the definition of being more fundamental than. (1) According to this definition a fundamental thing is more fundamental than what it grounds since they vacuously satisfy the condition that every ground of the fundamental thing is a ground of what it grounds and non-vacuously satisfy the condition that not every ground of what the fundamental thing grounds is a ground of it. (2) As formulated, this definition restricts the relation of being more fundamental than to pairs of things that are elements of the same grounding chain. If something like this definition is correct, then whether it can be extended to apply to pairs of things from different chains should be investigated. But this is not the topic of this paper, and the objection I just made does not depend on whether or not such a definition can be thus extended. (3) As formulated, this definition assumes the transitivity of grounding. A more careful formulation of the definition, which does not assume the transitivity of grounding, says that x is more fundamental than y if and only if the transitive closure of the grounds of x is a subset of the transitive closure of the grounds of y, but not vice versa. But it is only for simplicity that I gave the less careful formulation and, anyway, the point I made in the preceding paragraph against the less careful version of the definition would also apply to the more careful version.)

Finally, one might say that x being more fundamental than y means that x grounds y but is not grounded in it. But, again, there seem to be no reasons for thinking that if this is what being more fundamental than something means, then grounding something entails being more fundamental than that thing. For there is no reason to think that if x grounds y, then, although y is F in virtue of x, x is not F in virtue of y. And, again, my examples constitute reasons against the idea that if x grounds y, then x is not grounded in y.

Analogous replies could be made if someone objected that truthmaking must be asymmetric because if x is a truthmaker of y, x must be more fundamental than y.

Thus, these objections can be repelled. If my point that grounding is not asymmetric stands, it follows that grounding does not entail being more fundamental than. Therefore, it follows that grounding is not a relation of relative fundamentality, to use an expression of Bennett's (Bennett, forthcoming).

But there is another possible objection to my arguments that grounding is not asymmetric. This is that grounding is a kind of explanation, and explanations are asymmetric. For instance, Fine says that ‘there is a plausible demand on ground or explanation that we are unable to evade. For given a truth that stands in need of explanation, one naturally supposes that it should have a “completely satisfactory” explanation, one that does not involve cycles and terminates in truths that do not stand in need of an explanation’ (Fine Reference Fine2010: 105). This appeal to the asymmetry of explanation to defend the asymmetry of grounding is not uncommon (Audi Reference Audi, Correia and Schnieder2012a: 102, Rosen Reference Rosen, Hale and Hoffman2010: 116).

Now, it is true that explanations, in the sense of propositions put forward by an enquirer to advance our understanding, are asymmetric. Let us call explanations in this sense epistemic explanations. But there is no reason to extrapolate the formal features of epistemic explanations to grounding. I think that there is a close connection between epistemic explanations and grounding, namely, that explanations track determinative relations, and since grounding is a determinative relation, many explanations track grounding relations (cf. Rodriguez-Pereyra Reference Rodriguez-Pereyra, Dodd and Beebee2005: 28). But it does not follow that grounding must be asymmetric because epistemic explanations are. Indeed, it does not even follow that every relation of grounding must be susceptible of being tracked or represented by an epistemic explanation.

But the thought might be that there is a non-epistemic asymmetric sort of explanation and that grounding is such a non-epistemic sort of explanation or is at least a species of it.Footnote ⁹ Something like this is what Fine seems to be assuming in the passage I quoted above, when he speaks of ‘grounding or explanation’. But, whatever Fine was assuming, the idea that grounding is a non-epistemic asymmetric sort of explanation cannot be used as an objection against my points that grounding is not asymmetric. For in that case the objector would be saying: your thesis that grounding is not asymmetric must be wrong because grounding is asymmetric. Such an objection has no force.

But I have argued that grounding is not asymmetric. It follows that grounding is not a non-epistemic asymmetric explanatory relation.

7.

Schaffer has proposed a contrastive account of grounding according to which grounding is a quaternary relation of the form: the fact that F rather than the fact that F* grounds the fact that G rather than the fact that G*, where the facts that F and G are required to be obtaining facts, and the facts that F* and G* are required to be non-obtaining facts. Schaffer thinks that grounding respects the principles of transitivity, asymmetry, and reflexivity when these are formulated for differences. Thus:

Differential irreflexivity: It is not the case that the fact that F rather than F* grounds the fact that F rather than F*.

Differential asymmetry: If the fact that F rather than F* grounds the fact that G rather than G*, then it is not the case that the fact that G rather than G* grounds the fact that F rather than F*

Differential transitivity: If the fact that F rather than F* grounds the fact that G rather than G*, and the fact that G rather than G* grounds the fact that H rather than H*, then the fact that F rather than F* grounds the fact that H rather than H*.

Schaffer argues that his counterexamples to the transitivity of grounding are not counterexamples to the differential transitivity of grounding. And he suggests (2012: 133, fn. 17) that Jenkins’ counterexample to the irreflexivity of grounding does not threaten differential irreflexivity. This suggests that a quaternary relation of grounding may be a strict order.

But my examples are also examples against differential irreflexivity and differential asymmetry of grounding. The fact that B is true rather than B is not true grounds the fact that B is true rather than B is not true. So differential irreflexivity fails. And the fact that E is true rather than E is not true grounds the fact that F is true rather than F is not true, and the fact that F is true rather than F is not true grounds the fact that E is true rather than E is not true. So differential asymmetry fails.

Therefore, taking grounding to be a quaternary relation does not make it a strict order.

8.

To conclude, truthmaking is neither transitive nor irreflexive nor asymmetric, and alethic-fact grounding is neither irreflexive nor asymmetric. Since those two relations are species or cases of grounding, grounding is neither transitive nor irreflexive nor asymmetric, and so grounding is not a strict order. This has some consequences. One consequence of this is that if x grounds y, it does not follow that x is more fundamental than y. Thus, grounding is not a relation of relative fundamentality. Another consequence of this is that if non-epistemic explanation is an asymmetric relation, grounding is not non-epistemic explanation. The link between grounding and explanation consists in that instances in which grounding occurs irreflexively and asymmetrically back up epistemic explanations.