Truthmakers

Truth depends on the world. That much is common ground: what we say is true or false according to how things are, not according to what we think. Truthmaker theory is the attempt to make the dependence precise, and its central principle is that for every truth there is some entity whose existence accounts for it.

The principle looks modest and has consequences throughout ontology, because it functions as a rival to Quine's criterion of commitment — and a far more demanding one.

The principle

The truthmaker principle: for every true proposition $p$, there is some entity $T$ such that $T$ is a truthmaker for $p$. Standardly this is spelled out as necessitation:

$$T\text{ is a truthmaker for }p{=}_{\mathrm{df}}\square (T\text{ exists}\to p\text{ is true})$$

The existence of the truthmaker is enough for the truth; there is no possible situation in which $T$ exists and $p$ is false.

Two strengths of the doctrine:

• Truthmaker maximalism: every truth has a truthmaker.
• Non-maximalism: some truths have truthmakers and others — typically negative and general ones — do not, being instead true in virtue of what fails to exist.

And a weaker principle often preferred to both, Lewis's truth supervenes on being: there can be no difference in what is true without a difference in what exists and what properties and relations it instantiates. This captures the anti-idealist intuition while requiring no particular entity per truth, and it is much easier to defend. Whether it captures enough is the question — it says truth covaries with being, not that it is grounded in it, and the truthmaker theorist wants the stronger claim.

Why it matters: cheater-catching

The principle's main use is as a discipline. Armstrong's charge is that some theories help themselves to truths they have no resources to make true — a practice he calls ontological cheating. Three standing accusations:

Ostrich nominalism. "$a$ is red" is true, and the ostrich says $a$ is the only entity involved. But $a$ can exist without being red, so $a$ does not necessitate the truth. Either something more exists — a universal, a trope, a state of affairs — or the truth floats free of being.

Presentism. The presentist holds that only present things exist, yet asserts that there were dinosaurs. What makes that true? Nothing present is such that it could not exist unless dinosaurs had. The presentist must find a present truthmaker (world-properties, "Lucretian" past-tensed properties of the present world) or restrict maximalism. This is the sharpest application of the principle, and it is developed in the space-and-time treatment of presentism and eternalism.

Phenomenalism and behaviourism. Counterfactual truths about what would be perceived, or how someone would behave, need grounds in what actually exists.

Notice the shape of the move. Quine's criterion asks what a theory quantifies over; truthmaker theory asks what must exist for the theory's claims to be true. The second catches theories that avoid quantifying over what they nonetheless need — which is precisely the manoeuvre the paraphrase method makes available.

The hard cases

The principle's difficulties are concentrated in a few classes of truth, and how a theory handles them largely determines its shape.

Negative existentials. What makes "there are no unicorns" true? The world contains everything it contains and no unicorn; but no entity seems to necessitate the absence, since any collection of things could have coexisted with a unicorn elsewhere. Options:

• Totality facts (Armstrong): a higher-order state of affairs to the effect that these are all the things there are. This necessitates the negative truth. The cost is an entity of an unusual kind, whose constituents are everything and whose existence excludes further things.
• Absences: admit negative entities. Few take this route.
• Incompatibility (Demos): "there are no unicorns" is made true by positive facts incompatible with there being unicorns. Attractive until one asks what makes the incompatibility true.
• Non-maximalism: negative truths simply lack truthmakers and are true because of what does not exist. Clean and principled, and it concedes that being is not the only determinant of truth.

Universal generalisations. "All ravens are black" is not made true by the black ravens, since they could exist alongside a white one. Same options, same trade-offs.

Counterfactuals, modal, and mathematical truths. What makes "the vase would have broken" true, or "there could have been talking donkeys," or "$7+5=12$"? Modal realism supplies concrete worlds as truthmakers; the essentialist supplies natures; the mathematical platonist supplies numbers. Each is an ontological commitment incurred by the demand for truthmakers, which shows how the principle drives inventory.

Necessary truths. If a truthmaker need only necessitate, then anything whatever is a truthmaker for every necessary truth — since necessarily, if anything exists, $2+2=4$. This is technically correct and evidently wrong in spirit, and it is the standard reason for thinking that necessitation is not sufficient for truthmaking.

Relevance and grounding

That last problem generalises. Necessitation is too coarse: it makes irrelevant entities into truthmakers, and it also entails that any truthmaker for $p$ is a truthmaker for every necessary consequence of $p$. What is wanted is that the truthmaker be relevant — that the truth hold in virtue of it.

But "in virtue of" is the language of grounding, and the natural conclusion is that truthmaking is a species of grounding rather than a modal relation at all. On this view $p$ is true in virtue of $T$, where the "in virtue of" is a primitive explanatory relation, and the modal necessitation is a symptom rather than the analysis.

This is the direction most current work takes, and it relocates the whole discussion. The demand becomes: for every truth, an explanation of why it is true, in terms of what there is. That is stronger than covariation and weaker than necessitation, and it is the notion the section's later treatment of fundamentality and dependence develops.

Where truthmakers sit

Truthmaker theory belongs to the status question as much as to the inventory. It is a proposal about what ontology is answerable to — a rule for admitting entities — and it competes directly with the Quinean rule the section has used throughout.

The competition is uneven in an instructive way. Quine's criterion is precise and permissive: it reads commitment off logical form and lets any paraphrase discharge it. Truthmaker theory is imprecise and demanding: it cannot be stated without controversy, but it catches exactly the theories that use paraphrase to avoid commitments they still rely on. Which is the better instrument depends on whether one thinks the point of ontology is to be economical or to be honest — and that is not a question either criterion can settle.