Truthmaker Theory

The last article was about entities depending on entities: a smile on a face, a set on its members, for their existence or for their nature, on a particular thing or merely on something of a kind. This article takes up a dependence of a different shape, one that runs across categories. A true proposition is not an object in the world, and the world is not a proposition. Yet the truth of "Socrates is pale" seems to depend on Socrates and how he is, and not the other way round. Truthmaker theory is the attempt to say what that dependence is, how far it reaches, and what the world must contain if it reaches everywhere.

The asymmetry was noticed early. In a passage of the Categories, Aristotle observes that if there is a man, then the statement that there is a man is true, and that the implication also runs in reverse. Yet the true statement is in no way the cause of the man's existence, whereas the man is in some way the cause of the statement's being true. Mutual implication with one-way explanation: that pattern runs through everything below.

Truths and what makes them true

The intermediate article on facts, events and states of affairs described a fact as whatever makes a true sentence or proposition true. Truthmaker theory takes that job description and asks, in general, what does the job. A truthmaker for a truth is an entity in virtue of whose existence the truth is true: Socrates, perhaps, for "Socrates exists". The truthmaker principle is the claim that truths need truthmakers, that for a proposition to be true something must exist that makes it so. Both ideas also have a place in ontology, where they offer a rival to Quine's test for what a theory is committed to; here they are needed only in outline.

The modern vocabulary comes from a 1984 paper by Kevin Mulligan, Peter Simons and Barry Smith, called simply "Truth-Makers", though the thought is much older. D. M. Armstrong, who did more than anyone to build it into a systematic metaphysics, credited it to his colleague C. B. Martin, who pressed it against phenomenalism. The phenomenalist says that the table in the empty room exists in the sense that, were anyone to look, they would have table-like experiences. Martin's complaint was that such counterfactuals cannot simply be true. If nobody is looking, something must be there in virtue of which a looker would have those experiences, and the obvious candidate is the table. The principle works as a check: a theory that helps itself to truths must say what in the world they answer to.

Truthmaking is not one to one. "There are dogs" is made true by Fido, and by Rex, and by any other dog, and Fido makes true "there are dogs", "there are animals", "something exists" and much else. Armstrong called the smallest entities that do the job minimal truthmakers: Fido and Rex together make "there are dogs" true, but not minimally, since Fido alone suffices.

How far does the principle reach? A modest truthmaker theorist asks only that some truths have truthmakers, perhaps the contingent positive ones. The ambitious version asks it of all.

Maximalism is attractive because it is simple and because restricted versions look arbitrary. If some truths need no ground in being, a theory owes an account of which ones, and why those. Its difficulty is that it has to find a truthmaker for everything, including truths that say what is not there.

Necessitation and states of affairs

The first question is what the "makes" in truthmaking amounts to. The standard answer, which Armstrong made central in Truth and Truthmakers (2004), is necessitation. If an entity makes a proposition true, then it is impossible for that entity to exist and the proposition to be false. The relation is cross-categorial, running from something in the world to something that bears truth, and it is not entailment, since entailment holds between propositions.

Necessitation is enough to show that substances alone cannot do all the work. Socrates necessitates "Socrates exists", and if being human is essential to him, he necessitates "Socrates is human". He does not necessitate "Socrates is pale", since he could have existed and been tanned. Adding the property of paleness does not help, and nor does taking Socrates and paleness together.

This is the reasoning behind Armstrong's A World of States of Affairs (1997), and it is the truthmaking side of the unity problem met in the intermediate tier. A state of affairs is needed precisely because its constituents can exist without it, which is why Armstrong treats it as composed in a non-mereological way. The argument also shows the shape of the debate to come: each kind of truth that substances and properties fail to necessitate seems to call for a new kind of entity.

There is a rival at this point. Mulligan, Simons and Smith proposed that the truthmakers for contingent predications are particular properties, which they called moments and which the ladder has met as tropes: the particular paleness of Socrates, which could not exist without him and could not have belonged to anything else. A trope that is essentially Socrates' necessitates "Socrates is pale" without any further item to tie it to its bearer. Whether that is an economy, or the same commitment under another name, depends on what you think the tie in a state of affairs was doing.

Too many truthmakers

Necessitation is a modal notion, and it inherits the coarseness of modal notions. Three problems follow.

The first concerns necessary truths. A necessary truth cannot be false, so it is impossible for anything at all to exist while it is false. On the necessitation account, then, every entity is a truthmaker for every necessary truth: the Eiffel Tower makes it true that two plus two is four, and so does your left shoe. Necessitation was meant to capture the idea that a truth holds because of its truthmaker, and here the truth holds whatever the supposed truthmaker is like.

The second concerns direction. Socrates necessitates the truth of "Socrates exists", but the truth of "Socrates exists" equally necessitates Socrates, since it cannot be true unless he exists. Necessitation runs both ways, and Aristotle's asymmetry falls through it. Worse, Socrates' singleton set necessitates the truth too, since the set cannot exist without its member, and so the singleton passes the necessitation test exactly as well as Socrates does. This is the pair that showed, in the article on essence and modality, that necessary connections do not track natures, and it makes the same point here.

The third is formal. Greg Restall showed in 1996 that necessitation sits badly with two principles that truthmaker theorists had found natural: that whatever makes a proposition true makes true everything it entails, and that whatever makes a disjunction true makes one of its disjuncts true. Take any truthmaker for any truth, and any proposition $q$. Every proposition entails "$q$ or not $q$", so by the first principle the truthmaker makes that disjunction true. By the second it makes $q$ true or makes "not $q$" true, and since a truthmaker only makes truths true, it makes true whichever of them is true. Every truthmaker then makes every truth true, and the theory collapses. The usual moral is that the entailment principle must use a stricter notion of consequence than classical entailment, which is another way of saying that necessitation was too blunt.

Truth grounded in being

The three problems share a diagnosis. Necessitation records that two things go together across all possible worlds, and truthmaking is supposed to record that one holds because of the other. That is the difference between a modal relation and an explanatory one, and the ladder already has a name for the explanatory relation. Grounding, the in-virtue-of relation from the opening rung of this tier, is hyperintensional, so it can tell apart facts that hold at exactly the same worlds, and it is asymmetric, so it never runs in both directions at once.

Several writers, Gideon Rosen and Jonathan Schaffer among them, have proposed that truthmaking is a special case of grounding. On a typical version, an entity makes a proposition true when the fact that the proposition is true is grounded in the existence of that entity, or in some fact about it. Each problem then dissolves. The Eiffel Tower necessitates the truths of arithmetic but does not ground them, since nothing about the tower is why two plus two is four. Socrates grounds the truth of "Socrates exists", and because grounding is asymmetric, the truth does not ground Socrates. The singleton drops out as well: its existence is itself grounded in Socrates', so it sits downstream of the truthmaker and is not a truthmaker in its own right.

An essence-based version runs through the previous article. E. J. Lowe, among others, defended an account on which truthmaking is fixed by the nature of the truth-bearer: roughly, an entity makes a proposition true when it lies in the essence of the proposition that it is true if that entity exists. It belongs to the essence of the proposition that Socrates exists that it is true if Socrates exists, and nothing in that essence mentions his singleton or the Eiffel Tower. On this view a truth bears a kind of essential dependence on its truthmaker, and the asymmetry comes from the asymmetry of real definition: the proposition is defined partly by reference to Socrates, and Socrates is not defined by reference to the proposition.

The slogan that results is that truth is grounded in being. It keeps what necessitation got right, since if grounding necessitarianism is true, every full ground of a truth still necessitates it, and it drops what necessitation got wrong. What it does not do, by itself, is say what the grounds of every truth are. Necessitation gave maximalism a test that any candidate truthmaker had to pass. Grounding gives a stricter test, and the hardest cases fail both.

Negative existentials

Not every negative truth is so hard. Plato's Sophist uses "Theaetetus flies" as its example of a false statement. Its negation, "Theaetetus is not flying", has a promising truthmaker in Theaetetus' sitting, since sitting excludes flying. This incompatibility approach, which Raphael Demos defended in 1917 and Russell discussed in his lectures on logical atomism, finds positive truthmakers for many negative predications. Negative existentials resist it. No positive thing seems to rule out unicorns, since any ordinary thing, a horse, a mountain, the Atlantic, could have existed alongside a unicorn. What would rule out unicorns is something that rules out everything not already there.

George Molnar gave the difficulty its standard form in "Truthmakers for Negative Truths" (2000). He set out four theses, each plausible taken alone, which in paraphrase are these:

1. The world is the totality of what exists.
2. Whatever exists is positive.
3. Some negative claims about the world are true.
4. Every truth about the world is made true by something that exists.

Together they say that some negative truths are made true by positive things. If nothing positive can necessitate an absence, as the case of the unicorns suggests, at least one thesis has to go, and the main responses to negative existentials can be sorted by which one they give up. Accepting negative facts as basic entities gives up the second. Lewis, as will be seen, gives up the fourth. Denying the third, by treating negative claims as not really about the world, is a minority option. Armstrong's answer is subtler, since it claims to keep all four by finding one entity that is not obviously negative and that does necessitate an absence.

Totality states of affairs

Russell had faced the problem in the lectures on logical atomism he gave in 1918, and he accepted both negative facts and general facts as part of the world. He reported that arguing for negative facts at Harvard had nearly caused a riot. Armstrong rejected negative facts but kept a descendant of Russell's general facts, and the result is his answer to negative existentials.

A totality state of affairs, in Armstrong's sense, is one in which some aggregate of things stands in a relation of totality to a property: these things are all the things with that property. Take the aggregate of all the first-order states of affairs, everything there is at the base level, together with the property of being a first-order state of affairs. That the aggregate is all of them is a further state of affairs, and its existence necessitates that there are no unicorns. If there were a unicorn, there would be a first-order state of affairs outside the aggregate, and then the aggregate would not be all of them. One entity thus serves as truthmaker for every negative existential at once, and for general truths such as "every electron has the same charge" too. The opening article of this tier met the same device on a small scale, when a totality fact was added to the redness of three apples so that their being red would necessitate that all the apples on the table are red.

The objection that Lewis and others pressed is Humean in spirit.

Truth supervenes on being

John Bigelow had reached the same impasse earlier. In The Reality of Numbers (1988) he began from a truthmaker principle and, finding that negative truths would force entities on him that he did not want, retreated to a weaker principle summed up in a slogan: truth supervenes on being. Lewis took the slogan over in "Truthmaking and Difference-Making" (2001) and argued that it captures what is right in the truthmaker idea.

Apply it to the unicorns. Compare our world with one exactly like it except that it also contains a unicorn. "There are no unicorns" is true here and false there, and the two worlds differ in being, since one contains a thing the other lacks. The principle is satisfied. Yet no particular thing in our world makes the negative existential true. On Lewis's account a positive truth is true because of something that exists, and a negative existential is true because something does not: it has no truthmaker, and it is true for lack of anything that would make it false. Both kinds answer to being, but only one has an entity to point to.

This gives up Molnar's fourth thesis and keeps the other three, and its costs fall in two places. The first is that maximalism goes, and with it the simple test that made the truthmaker principle useful against theories that take truths for free. A presentist or a phenomenalist can no longer be asked, for every truth, which entity makes it true; they can be asked only whether their truths vary with being. That remains a real constraint, as the next section shows, but it is a weaker one.

The second cost is that the principle is only a supervenience thesis, and supervenience says that two things vary together, not that one varies because of the other. The opening article of this tier argued that supervenience was never going to do the work of grounding, and the point applies here directly. Necessary truths do not vary across worlds, so they supervene trivially on anything and the principle says nothing about them. Aristotle's asymmetry is lost as well. Being supervenes on truth just as truth supervenes on being, since two worlds alike in every truth, including every truth about what exists, are alike in being. So the weaker principle, though it handles negative existentials, gives up the explanatory ambition that drove the move from necessitation to grounding. A grounding theorist can try for both: negative existentials are grounded in being, though not in any one being, perhaps in all the positive facts together with the fact that they are all. That is close to Armstrong's totality state of affairs under a new description, and whether it is a cost depends on which of Molnar's theses you would rather lose.

Truths about the past

The intermediate article on theories of time met the truthmaker objection to presentism. Truth supervenes on being sharpens it. Russell once entertained the hypothesis that the world sprang into existence five minutes ago, complete with memories and fossils. Take two presentist worlds exactly alike in everything that now exists and in how those things now are, one with dinosaurs in its past and one of Russell's five-minute worlds without them. "There were dinosaurs" is true in the first and false in the second, and since for the presentist only present things exist, nothing in being differs between them. The presentist breaks even the weak principle, not just maximalism.

The natural response is to put more into present being. John Bigelow, whose slogan this article has been using, defended such a view in "Presentism and Properties" (1996): the world as a whole now has properties such as having contained dinosaurs. With properties like that in the base, the two worlds differ in being after all. Theodore Sider pressed the reply in Four-Dimensionalism (2001) that such properties are suspect, because their whole content concerns what lies outside the present, so that positing them pays for the past truths with an entity whose only credential is that it makes them true. The presentist answers that every theory takes some features as primitive, and that a primitive tense is no worse than the eternalist's primitive spread of times. The dispute turns on what may count as being, and truthmaker theory, in any of its versions, leaves that to be settled elsewhere.

Next

Truthmaker theory treats being as what truths answer to, and asks that differences in truth be differences in being. The next article tests that demand on a case where the being seems to be shared. A statue and the lump of clay it is made of occupy the same place, have the same parts and weigh the same, yet the lump could survive being squashed into a ball and the statue could not. If they share every part, something has to account for the difference in what is true of them, and the problem of material constitution is the question of what that is, or whether the statue and the clay are one thing after all.