Easy Ontology and Neo-Fregeanism
A second route to deflationism reaches the conclusion by the opposite path from quantifier variance. Where the variantist says ontological questions have no determinate answers, Amie Thomasson's easy ontology says they have answers that are too easy — obtainable by anyone with the relevant concepts plus some uncontroversial facts, and requiring no distinctively metaphysical inquiry at all.
If she is right, the conclusion for the discipline is the same: there is no serious subject here. But the diagnosis differs, and so do the objections.
The easy arguments
The pattern is a trivial inference from an uncontested premise to an existence claim:
There are five books on the table. ⟹ The number of books on the table is five. ⟹ There is a number, namely five.
The shirt is red. ⟹ The shirt has the property of being red. ⟹ There is a property.
There are particles arranged tablewise, in contact and functioning as a support. ⟹ There is a table.
Each step is one a competent speaker is entitled to make. The move from "there are five books" to "the number of books is five" is not a substantive discovery; it is a transformation the rules of the language license, and refusing it betrays a misunderstanding of number talk rather than a rival metaphysics.
Thomasson's underlying account is that existence questions are settled by application conditions: rules, mastered in learning a term, that specify what it takes for the term to apply. To ask whether numbers exist is to ask whether the application conditions for "number" are satisfied, and those conditions are met by facts of the ordinary sort. There is no further, framework-independent question — which is where the position shows its Carnapian lineage, though Thomasson does not need the analytic/synthetic distinction to state it.
The corollary is that the ontologist's characteristic activity is misconceived. Debating whether the trivial inference is really valid, or whether the resulting entities really exist, presupposes an external standpoint that easy ontology denies is available.
Neo-Fregean abstraction
The formal counterpart is the neo-Fregean programme of Wright and Hale, treated in more detail on the nominalist programs page. Its engine is Hume's Principle:
The right-hand side quantifies only over the s and s and says they are equinumerous; the left introduces numbers as objects. Frege's Theorem shows that second-order Peano arithmetic follows from Hume's Principle in second-order logic alone. So arithmetic — a substantial body of truth about a domain of objects — is derivable from a principle claimed to be available a priori as an implicit definition.
The philosophical claim is that abstraction principles stipulate the meaning of a new term-forming operator, and that in doing so they entitle us to the objects. Knowledge of numbers then needs no causal contact and no faculty of intuition, which dissolves the access problem that drives the mathematical objects debate.
Thomasson's programme generalises this beyond mathematics: abstraction and trivial inference are the same phenomenon, and both show that entities can be introduced by rules rather than discovered by investigation.
The objections
It proves too much. The most immediate worry. If trivial inferences deliver numbers and properties, why not everything? From "the average family has 2.3 children" we should get the average family; from "he did it for her sake" we should get sakes. Thomasson's reply is that the inferences are licensed only where genuine application conditions exist, and that "average family" and "sake" are precisely cases where the surface form does not track a term with such conditions. The reply is principled, but it means a great deal turns on which terms have application conditions — and that is a question the easy method does not itself answer easily.
The Bad Company problem. Abstraction principles of exactly the form of Hume's Principle can be inconsistent. Frege's own Basic Law V — the extension of equals the extension of iff the s and s are coextensive — has the same shape and yields Russell's paradox. Others are individually consistent but jointly unsatisfiable, or satisfiable only in domains of some particular size. Since syntactic form does not separate the good from the bad, a criterion of acceptability is needed; and any criterion adequate to the task seems to presuppose knowledge about what there is, which was supposed to be the output of the method.
The Caesar problem. Hume's Principle fixes when the number of s is the number of s. It says nothing about whether the number of s is Julius Caesar. An implicit definition that leaves the identity conditions of its objects incomplete has not fully specified them, and cannot claim to have introduced determinate objects.
Existence cannot be cheap if the entities are substantial. The realist's complaint is that something has gone wrong when a stipulation about language yields a commitment about the world. Either the entities introduced are metaphysically weightless — in which case the platonist has not been vindicated but changed the subject — or a stipulation has produced substantial knowledge, which looks impossible. Thomasson's answer is to deny the dilemma: the entities are perfectly real, and their reality is not the kind of thing that could have been in doubt, because doubting it requires a standpoint outside all frameworks.
Deflationism, and what it deflates
It is worth distinguishing what the position denies. Easy ontology is not anti-realism about numbers or tables: it affirms that they exist. What it denies is that their existence is a deep fact, discoverable by metaphysical inquiry and contestable by argument. It is deflationism about the questions, not about the entities.
This makes it an awkward opponent. The nominalist and the platonist both take themselves to be engaged in something difficult; easy ontology agrees with the platonist's conclusion and undermines the platonist's method. That is why the neo-Fregean position is best described as a deflationary platonism — maximally permissive about what exists, maximally dismissive of the discipline that argues about it.
Where this sits
Like quantifier variance, this is the status question answered against the substantiality of ontology, but it locates the defect differently. The variantist says the disputants are talking past each other; the easy ontologist says they are labouring over something already settled. A dispute could be verbal without being easy, or easy without being verbal, so the two deflationisms are independent — and an opponent must answer both.
The realist reply to both, and the strongest defence of ontology as a substantive discipline, is the subject of the next page: ontological realism.