18 The Being Constraint

One important feature of the variable domain model theory for free quantified modal logic is that we made allowance for an atomic formula of the form $F x$ to be true at a world $w$ relative to an assignment of an object $d$ outside the inner domain of $w$ . That is, if $α$ is an assignment such that $α (x) \notin Q (w)$ but $α (x) \in I_{w} (F)$ , then we evaluate the formula $F x$ as true in $w$ . That feature of the model theory appears to clash with the inchoate thought that one has to be something in order for one to be able to possess an attribute; one is not human or a philosopher or a US citizen unless one is something. This is what we will call the being constraint, which we now set out to study.

One initial articulation of the constraint for each $n$ -place predicate in the language of quantified modal logic amounts to the requirement to make true a variety of formulas: $\begin{array}{lllll} B C_{F} & ◻ \forall x ◻ (F x \to \exists y x = y) \\ B C_{R} & ◻ \forall x ◻ \forall y ◻ (R x y \to \exists z x = z \land \exists z y = z) \\ \dots & \dots \\ B C_{R^{n}} & ◻ \forall x_{1} \dots ◻ \forall x_{n} ◻ (R x_{1} \dots x_{n} \to \exists z x_{1} = z \dots \exists z x_{n} = z) \\ \dots & \dots \end{array}$ It is important to note at this point that these are all theorems of the simplest quantified modal logic as given the combination of $K$ and necessitation, and thus the Converse Barcan Formula, they become simple consequences of the following theorems of quantificational logic: $\begin{array}{lllll} \forall x (F x \to \exists y x = y) \\ \forall x \forall y (R x y \to (\exists z x = z \land \exists z y = z)) \\ \dots \\ \forall x_{1} \dots \forall x_{n} (R x_{1} \dots x_{n} \to (\exists z x_{1} = z \dots \exists z x_{n} = z)) \\ \dots \end{array}$ On the other hand, none of the above are theorems of free quantificational logic. Nor are the earlier formulations of the being constraints provable in some of the formulations of free quantified modal logic we have considered thus far. One way to verify this is to note that the axioms of free quantified modal logic are sound with respect to the class of variable domain models, yet none of the above formulas turn out to be valid with respect to that class of models. One may now be tempted to alter with the model theory to make sure we validate the formulas that give voice to the being constraint for each predicate of the language.

It will be helpful to distinguish three approaches to the semantics for free quantified modal logic. One important choice point for these approaches concerns the satisfaction of atomic formulas of the form $R^{n} x_{1} \dots x_{n}$ at a world $w$ relative to assignments of objects outside the inner domain of $w$ . Let us now distinguish three routes one may take at this point.

A Problem for the Being Constraint

It is clear now that in order to enforce the being constraint as initially formulated, one should adopt a negative semantics for free quantified modal logic, which we would presumably expand to free quantified modal logic with identity. We would embrace the following clauses for the satisfaction of identity and other atomic formulas:

An atomic formula of the form $x = y$ is true at a world $w$ relative to an assignment $α$ if, and only if, $α (x) \in Q (w)$ , and $α (x) = α (y)$ .
An atomic formula of the form $R^{n} x_{1} \dots x_{n}$ is true at a world $w$ relative to an assignment $α$ if, and only if, for all $x_{n}$ , $α (x_{n}) \in Q (w)$ , and $(α (x_{1}), \dots, α (x_{n})) \in I (R^{n})$ .

Here is an important observation to make at this point.

Proposition 18.1 $⊨ ◻ \forall x ◻ (F x \to F x)$

Proof. Given a variable domain model $M = ⟨ W, R, D, I ⟩$ , a world $w \in W$ , and an assignment $α$ for $M$ , note that if $u \in W$ is such that $R w u$ , then $M, u, α ⊩ \forall x ◻ (F x \to F x)$ . For given $d \in Q (u)$ , $M, v, α [d / x] ⊩ ◻ (F x \to F x)$ . For suppose $v \in W$ is such that $R u v$ . There are two cases to consider. If $d \notin Q (v)$ , then $M, v, α [d / x] ⊮ F x$ and $M, v, α [d / x] ⊩ F x \to F x$ . Otherwise, regardless whether $d \in I_{v} (F)$ , $M, v, α [d / x] ⊩ F x \to F x$ .

But this is problematic for those philosophers tempted by the following generalization of the being constraint: $\begin{array}{lllll} B C_{φ (x)} & ◻ \forall x ◻ (φ (x) \to \exists y x = y) \end{array}$ They would be inclined to accept the instance corresponding to $F x \to F x$ : $\begin{array}{lllll} B C_{F x \to F x} & ◻ \forall x ◻ ((F x \to F x) \to \exists y x = y) \end{array}$ But the combination of $◻ \forall x ◻ (F x \to F x)$ and $◻ \forall x ◻ ((F x \to F x) \to \exists y x = y)$ would seem to deliver the necessity of existence as an immediate consequence: $\begin{matrix} (NE) & ◻ \forall x ◻ \exists y x = y \end{matrix}$

One consequence of all this is that $B C_{F x \to F x}$ is not valid in every variable domain model even after we are careful to adopt a negative semantics for the language.

Proposition 18.2 $⊭ ◻ \forall x ◻ ((F x \to F x) \to \exists y x = y)$

Advocates of the being constraint face a dilemma. They should either

reject $◻ \forall x ◻ ((F x \to F x) \to \exists y x = y)$ as giving partial voice to the being constraint, or else

That means that there is an important difference between atomic and complex formulas when it comes to the being constraint. One has to be something in order to satisfy an atomic formula, but one may satisfy a complex formula even if one is nothing. Indeed, there is no need for an object to be something in order to satisfy the negation of an atomic formula, e.g., $\neg F x$ .
reject the negative semantics for free quantified modal logic and find some other approach to use in its place.

One option at this point might be to opt for a neutral semantics on which a formula $φ (x)$ may generally be considered neither true nor false at a world $w$ when the free variable is assigned something outside the inner domain of that world. The problem in that case is that we will in effect give up classical logic in favor of some non-classical alternative designed to accommodate failures of Excluded Middle.

18 The Being Constraint

Positive Semantics for Quantified Modal Logic

Negative Semantics for Quantified Modal Logic

Neutral Semantics for Quantified Modal Logic

A Problem for the Being Constraint