There are two positions in this debate: necessitism and contingentism
Necessitism is Williamson’s view. Everything exists necessarily; domain is fixed across worlds.
Contingentism: certain objects exist at some worlds but not others.
BF: ◇∃xFx → ∃x◇Fx
If there could have been something that was F, then there is something that could have been F.
The dual is ∀x□Fx → □∀xFx: if everything is necessarily F, then necessarily everything is F.
CBF: ∃x◇Fx → ◇∃xFx
If there is something that could have been F, then there could have been something that was F.
The dual is □∀xFx → ∀x□Fx: if necessarily everything is F, then everything is necessarily F.
Necessity of existence: ∀x □∃y (x=y)
For any object x, there’s necessarily something identical to it – i.e., x couldn’t have failed to exist.
This follows from CBF and the uncontroversial claim that necessarily everything is self-identical, □ ∀x∃y(x=y).
These formulae each require that the quantifier ∀ and modal operator □ commute, which semantically corresponds to fixing a single domain across all possible worlds.
BF/CBF bridge from de dicto to de re modality; the counterexamples exploit the gap between the two.
The methodological case for SQML
Williamson’s main argument is that if you take the simplest quantified modal logic (our SQML), then BF and CBF are axiomatically provable and semantically valid.
In particular, take the standard axioms of predicate logic and combine them with the S5 axioms.
You don’t need any special axioms about the interaction of quantifiers and modals, but BF and CBF fall out as theorems.
To reject BF/CBF, you need to relativise domains to worlds (VDQML; “Kripke semantics”).
To do this, you need to reject universal instantiation [PC1]: ∀x φ(x) → φ(y/x) can fail when y is assigned something outside the domain of the evaluation world.
Williamson’s objection is that this creates a deeper problem, about the coherence of your metalanguage.
To state the variable domain semantics, you need to say things like “there exists an object d in D(w₁) but not in D(w₂)”.
But the quantifier “there exists” in the metalinguistic sentence is ranging over all domains – i.e., is unrestricted.
So the VDQML theorist is using unrestricted quantification in the metalanguage, while claiming that object-language quantifiers are restricted.
But this seems unprincipled. If you’re going to accept unrestricted quantification in the metalanguage, why not in the object language?
Also, Williamson thinks that it is hard to explain how CBF could fail.
We can make a semantic proof by contradiction of its validity quite easily.
Suppose that necessarily everything is F, but something d could have failed to be F (for a putative counterexample to □∀xFx → ∀x□Fx).
Then there’s a possible situation where everything is F, yet d is not F.
But then d would be a counterexample to the universal generalisation – so not everything is F in that situation.
So it’s false that necessarily everything is F. Contradiction.
The opponent would say that d lacks some ontological status in that world where it isn’t F, so it’s not actually a counterexample.
But Williamson’s point is that the universal generalisation ∀ in our premiss says everything is F, without restriction – not “everything with some special ontological status is F” or “everything that exists in this world is F”.
So, you have to restrict the quantifier ∀ if you’re going to make this move, which Williamson thinks is unmotivated.
Complications with axiomatisation and completeness
To make an axiomatic system for VDQML, we’d need to abandon PC1 and replace it with something else. This makes the axiomatic system more complex.
And completeness proofs also become more involved.
Williamson takes this as evidence that something has gone wrong philosophically, because we’ve had to weaken universal instantiation (PC1).
Natural-language counterexamples and Williamson’s response
Natural-language counterexamples.
To BF: Wittgenstein.
Wittgenstein died childless but could have fathered someone: ◇∃x(Wittgenstein fathered x).
BF then says that there actually is something Wittgenstein could have fathered: ∃x◇(Wittgenstein fathered x).
But given necessity of origin, no actual person could have been fathered by him.
To CBF: the river Inn.
The Inn could have failed to exist (if no water had covered that terrain), i.e. ◇¬∃y(Inn=y). So ∃x◇¬∃y(x=y).
CBF then gives ◇∃x¬∃y(x=y); possibly there exists something that’s not identical with anything.
But necessarily everything is self-identical [□∀x∃y(x=y)], so we have a contradiction.
Williamson tries to explain how it’s palatable that everything that could exist does exist, and everything exists necessarily.
The natural-language counterexamples only work if you confuse two claims:
There is something in space-time that Wittgenstein could have fathered; there is something in space-time that could have been nothing – both false, intuitively and according to Williamson.
There is something, unrestricted, that Wittgenstein could have fathered; there is something which could have been nothing – both true, according to Williamson.
Williamson’s solution is that the unrestricted something we’re looking for exists as a “bare possibilium”.
It has the modal property of, say, possibly being a person fathered by Wittgenstein, but beyond that doesn’t have many interesting non-modal properties in the actual world.
It’s not in space-time, it’s not a person, it’s not abstract, it’s just something, individuated by what it could be (see below).
An analogy with tense helps motivate Williamson’s view.
Consider the river Inn, which has dried up completely. Is it still something?
Williamson says yes – it’s a past river.
You can refer to it; it’s counted when you ask “how many rivers have there ever been”; it’s not a ghost or an abstract object.
Applying this modally: if the Inn had never existed, it would still be something – a merely possible river.
Temporal displacement (being past) doesn’t strip an object of its status as being something; similarly modal displacement (being merely possible) doesn’t.
(Williamson doesn’t think it proves the modal case, but it makes the ontological picture seem less bizarre.)
There’s a three-category ontology, on the SQML view:
Concrete objects (spatiotemporally located), e.g. Bush in the actual world
Contingently non-concrete, e.g. Bush in worlds where he intuitively doesn’t exist [but is still in the domain, just not space-time]
Abstract (not concrete at any world), e.g. numbers, pure sets.
The Linsky/Zalta view doesn’t include contingent abstracta. But see Hayaki objection below.
One worry with these bare possibilia is how you individuate them, given that they have almost no non-modal properties.
For example, consider all the merely possible children Wittgenstein could have fathered. At the actual world, they’re non-concrete, non-spatiotemporal, non-human. So how do you distinguish them?
Williamson appeals to two laws for this, both provable in SQML using II:
Necessity of identity. x = y → □(x = y)
Necessity of non-identity (distinctness). x ≠ y → □(x ≠ y)
Since identity and distinctness are necessary, they don’t need to be grounded in the objects’ actual-world, non-modal properties.
Two possible children of Wittgenstein are necessarily distinct, as a brute modal fact.
The way you in-practice individuate them is by looking for any (counterfactual) worlds where they are both concrete, and then checking if they meet the ordinary criteria for identity between people
Ontological costs of fixed domains
Hayaki’s essence objection.
To symbolise “x is essentially F”, you’d normally write □(x exists → x is F).
The point of the antecedent is to restrict our attention to worlds where x is [roughly speaking] around.
But with a fixed domain, everything exists necessarily. So then if x exists, it exists in every world, and our conditional collapses to just □(x is F). The problem is that this is too strong for most essentialist properties: Bush is essentially human, but he clearly isn’t human in the worlds where he’s non-concrete!
So instead you symbolise it as □(x is concrete → x is F). (From Linsky/Zalta)
Now, if you try to symbolise “x is essentially concrete”, we’d have as □(x is concrete → x is concrete)
But this is a tautology satisfied by everything, including things that are never concrete.
Two solutions:
Deny essential concreteness. E.g., Bush isn’t essentially concrete because he’s non-concrete in some worlds.
But by this logic he isn’t essentially human either.
So we have a general anti-essentialism – nothing is essentially anything, because everything is non-concrete somewhere.
Redefine “essentially concrete” as “possibly concrete”: ◇Cx.
Then we preserve the intuition that Bush is essentially concrete
But we’ve collapsed a genuine distinction: a ring is essentially circular, while the gold it’s made from is only contingently circular. Under the redefinition, though, both are essentially possibly circular.
Contingent abstracta problem. Consider e.g. Sherlock Holmes.
He is both abstract (not spatiotemporal) and contingent (if Doyle had had more work as a doctor, then he’d never have written the books).
On the fixed-domain view, Holmes exists in all worlds. But what is his status in worlds where Doyle never writes?
It doesn’t make sense to call him contingently non-concrete, since he’s non-concrete in every world.
And he’s not contingently non-abstract either, since whenever he does exist he is abstract.
So we need a new predicate, like “contingently unrealised”.
The problem is that now the antecedent of essentialist conditionals varies by ontological category.
E.g., “x is essentially human”: □(concrete(x) → human(x))
But “x is essentially fictional”: □(realised(x) → fictional(x))
Some moves for Williamson – Hayaki’s objection is only as strong as the claim that there really are contingent abstracta.
Deny that fictional characters are genuine entities (eliminativism)
Deny that they’re contingent (maybe abstract structures like “the Holmes-role” exist necessarily, and the contingency is just whether or not they’re authored)
Or bite the bullet and accept that we need multiple predicates but this is worth the price for logical simplicity.
Tying together the above objections, a fixed-domain view is just much less ontologically parsimonious than a variable-domain one.
Regarding predicates: we need “concrete”, “abstract”, “realised” etc where previously just “exists” was sufficient.
Translations: we need multiple formats for essentialist claims, rather than just one.
Populations: every possible world has the same, maximal domain.
Williamson tries to make one pre-theoretic argument in favour of the fixed-domain view: his counting argument.
Suppose we have two blades, and two handles. How many knives could be made?
In one sense, two (you can only actually make two at once).
But in another sense, four.
Williamson thinks that when we answer four, we’re counting the four possible knives – i.e., quantifying over them.
This is meant to show that ordinary thought commits us to quantifying over possibilia. That’s a pre-theoretic argument for a fixed domain.
But Hayaki replies: we’re counting ways of assembling parts, not objects.
We often quantify over ways of doing things – e.g. “more than one way to skin a cat”.
Also, even granting that we’re quantifying over possible knives, that doesn’t show that they are actual.
Lewis has “counterpart theory” which allows you to quantify over possible objects without saying that they exist in this world.
So the counting argument doesn’t demonstrate that possibilia are actual.
Conclusions
The strongest argument is not so much the counterexamples, which Williamson can handle with possibilia, but the wider costs Hayaki identifies.
The debate is really about what you think is more costly:
Necessitism / SQML has ontological profligacy (necessary existence of everything)
Contingentism / VDQML has logical complexity (variable domains, complications in axiomatisation and completeness proofs)
Note that free logicians reject PC1, which means that they deny CBF and BF, but are still committed to NI and ND.
We might have instead PC1’: ∀αφ → (∃y y=β → φ(β/α))
Free logic allows for:
Empty names
Empty domains
It does this by creating an “outer domain” with nonexistent objects, and then an inner domain with existent objects. The union must be nonempty, but the inner domain can be empty.
Quantifiers range over the inner domain.
Also, denotation of a constant isn’t required to be in the inner domain, so ∃x x=a fails.