(p.200) Appendix C The Canonical Space of Worlds
(p.200) Appendix C The Canonical Space of Worlds
(p.200) Appendix C
The Canonical Space of Worlds
This appendix contains proofs of the technical results described in Chapter 5, and presupposes the machinery described in Chapter 6.
Preliminaries Let L be a first‐order language, L ^{◊} be the result of adding ‘◊’ to L, and A be an assignment of truth‐values to sentences in L and basic ‘just is’‐statements built up from vocabulary in L. I shall assume that L has a set‐sized vocabulary and a set‐sized domain.
We will construct the canonical space of worlds in stages. At each stage, we do two things: (1) we introduce a set of objects and assign each of them an ‘essence’, and (2) we introduce a set of a‐worlds for L ^{◊}.
I will begin by explaining what an essence is, and then describe the construction in further detail.
Essences We start with the notion of a constitutive predicate. Where $\overrightarrow{v}$ is a sequence of variables v _{1}, …, v_{k}, which we will think of as parameters, we will refer to $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ as an m‐place predicate, and say that $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ is constitutive by the lights of A (or A‐constitutive) just in case the basic statement
is true according to A.
Informally speaking, an ‘essence’ is a set of m‐place constitutive predicates in which any parameters have been filled by objects from a given domain.
Formally speaking, essences are defined as follows. Let D be a domain of objects, and let o be an arbitrary object not in D, which we shall refer to as the ‘outside object’. (The ‘outside object’ will be used to keep track of (p.201) the behavior of constitutive predicates whose parameters have been filled with objects which lie outside the domain of the world with respect to which the predicate is being evaluated.) If $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ is an (m‐place) predicate of L, we say that an (m‐place) D‐completion of $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ is a pair $\u3008\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v}),f\u3009$ where f is a function that assigns an element of D ⋃ {o} to each variable in $\overrightarrow{v}$. A set E of (m‐place) D‐completions of A‐constitutive predicates is an (m‐place) D‐essence just in case a sequence of objects a _{1}, …, a_{m} could consistently instantiate every D‐completion in E, in the following sense:
There is an a‐world w with domain D such that: (1) w verifies (the modal analogue of) every unconditional ‘just is’‐statement that is counted as true by A, and (2) some sequence of objects a _{1}, …, a_{m} ∈ D is such that for any $\u3008\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v}),f\u3009$ in E, $\varphi (z,\overrightarrow{v})$ is true in w relative to a variable assignment that assigns a_{j} to z_{j} and f (w_{i}) to each w_{i} in $\overrightarrow{v}$.
(Since the ‘outside object’ o is not in the domain D of w, constitutive predicates whose parameters have been filled with o will behave in w like predicates containing empty names.)
The Construction The intuitive picture is very simple. Our construction proceeds in stages. At stage 0 we introduce the ‘actualized’ a‐world. At each later stage we introduce as varied a set of a‐worlds as we can, while respecting the essences assigned to sequences of objects introduced earlier in the process.
Here is how the construction works in greater detail. At stage 0, we introduce the ‘actualized’ a‐world for L ^{◊}, that is, the a‐world whose domain D _{0} consists of pairs 〈x, ‘actual’〉 for x in the domain of L, and in which every predicate receives its intended interpretation (corrected for the fact that the domain consists of ordered pairs 〈x, ‘actual’〉 instead of their first components). For each z _{1}, …, z_{m} ∈ D _{0}, we let the (m‐place) D _{0}‐essence of z _{1}, …, z_{m} be the set of D _{0}‐completions of (m‐place) A‐constitutive predicates that are true of z _{1}, …, z_{m} in the actualized a‐world. (I shall assume that the ‘outside object’ o is not an ordered pair, and therefore not in D _{0}.)
For k a natural number, stage k + 1 is constructed as follows. We let μ be the maximum of: (a) ω, (b) the cardinality of D _{0}, and (c) the cardinality of L. Next, we let D_{k} _{+1} be the union of D_{k} and of the set of objects 〈α_{k} _{+1}, ‘non‐actual’〉 for α ∈ μ. (Since the ‘outside object’ is not an ordered pair, it is not in D_{k} _{+1}.) Each sequence d _{1}, …, d_{m} ∈ D_{k} _{+1} is then assigned an (m‐place) D _{k+1}‐essence in such a way that:

(p.202) 1. D_{k}‐essences are preserved (that is, if d _{1}, …, d_{m} ∈ D_{k}, then the m‐place D_{k} _{+1}‐essence of d _{1}, …, d_{m} is just the m‐place D_{k} essence of d _{1}, …, d_{m}).

2. Essences are distributed as diversely as possible.
Let me be more specific. For each n ≥ 1, let an n‐distribution of essences be a function f such that, for each i ≤ n, f assigns an i‐place essence to each i‐membered sequence built from numbers less than n. Then we shall require our assignment of essences to be such that each n‐distribution of essences applies to μ sequences d _{1}, …, d_{n} ∈ D_{k} _{+1} (where an n‐distribution ‘applies’ to a sequence of objects d _{1}, …, d_{n} just in case each i‐membered sequence built from d _{1}, …, d_{n} is assigned the essence that the n‐distribution assigns to the corresponding sequence of numbers).
This procedure guarantees that our assignment of essences is unique, up to isomorphism.
Finally, we introduce an a‐world w just in case it meets the following conditions:

C0 w respects the intended interpretations of singular terms (all of which must refer to objects in the domain of the actualized a‐world, D _{0}.)

C1 The domain of w is a subset of D_{k} _{+1}.

C2 If ⌜ø(x _{1}, …, x_{n}) ≡_{x1}, …_{,xn} ψ (x _{1}, …, x_{n})⌝ is counted as true by A, 𢌜∀x _{1} … ∀x_{n}(ø(x _{1}, …, x _{n}) ↔ ψ(x _{1}, …, x_{n}))⌝ is true at w.

C3 Let $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ be an (m‐place) A‐constitutive predicate, let x _{1}, …, x_{m} be objects in the domain of w, and let $\u3008\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v}),f\u3009$ be a D_{k} _{+1} ‐completion of $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$. Let s ≤ k + 1 be the first stage at which x _{1}, …, x_{m} were all introduced, and let f* be just like f except that whenever f (v_{i}) is an object outside D_{s}, f*(v_{i}) is the ‘outside object’ o.
Then x _{1}, …, x_{m} satisfies $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ in w relative to the parameters assigned to $\overrightarrow{v}$ by f just in case $\u3008\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v}),{f}^{*}\u3009$ is in the (m‐place) D_{k} _{+1}‐essence assigned to x _{1}, …, x_{m}.
We let the canonical space of worlds, M_{A}, consist of all and only a‐worlds w such that w is introduced at some finite stage of this process, and designate the actualized a‐world as M_{A}'s center.
It is clear from the construction that M_{A} has been specified uniquely (up to isomorphism). We can also prove:
Theorem C.1 M_{A} satisfies the following two conditions:

(p.203) 1. If $\varphi \left(\overrightarrow{x}\right){\u261e}_{\overrightarrow{x}}\psi \left(\overrightarrow{x}\right)$ is true according to A, then ${M}_{A}=\square (\forall \overrightarrow{x}(\varphi \left(\overrightarrow{x}\right)\leftrightarrow \psi \left(\overrightarrow{x}\right)\left)\right)$.

2. if
$$\frac{\varphi ({z}_{1},\mathrm{\dots}{z}_{m},\overrightarrow{v})}{{x}_{1}={z}_{1}\wedge \mathrm{\dots}{x}_{m}{\u3009}_{{x}_{1,\mathrm{\dots},{x}_{m}}}\varphi ({x}_{1},\mathrm{\dots},{x}_{m},\overrightarrow{v})}$$
is true according to A, then
Proof: It is straightforward to check that the first condition is verified. To see that the second condition is verified, assume that $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ is A‐constitutive. Let w be an a‐world in M_{A} and suppose that a _{1}, …, a_{m}, b _{1}, …, b_{k} is a sequence of objects in the domain of w such that $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ is true at w relative to a _{1}, …, a_{m}, b _{1}, …, b_{k}. We show that, if w^{′} ∈ M_{A} has a _{1}, …, a_{m} in its domain, then $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ is true at w^{′} relative to a _{1}, …, a_{m}, b _{1}, …, b_{k}. Let the sequence b _{1}*, …, b_{k}* be just like the sequence b _{1}, …, b_{k} except that b _{1}* is the ‘outside object’ o whenever the a _{1}, …, a_{m} were all introduced before b_{i} in our construction. Since $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ is A‐constitutive, it follows from condition C3 above that $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ is true at w relative to a _{1}, …, a_{m}, b _{1}, …, b_{k} just in case $\u3008\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v}),{f}^{*}\u3009$ is in the essence of a _{1}, …, a_{m}, where f* (v_{i}) = b_{i}*. So $\u3008\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v}),{f}^{*}\u3009$ is in the essence of a _{1}, …, a_{m}. But it also follows from C3 that $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ is true at w^{′} relative to a _{1}, …, a_{m}, b _{1}, …, b_{k} just in case $\u3008\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v}),{f}^{*}\u3009$ is in the essence of the a _{1}, …, a_{m}. So $\varphi ({z}_{1},\mathrm{\dots},{z}_{m},\overrightarrow{v})$ is true at w^{′} relative to a _{1}, …, a_{m}, b _{1}, …, b_{k}.
Possibility Let a possibility statement be a sentence of L ^{◊} of the following form:
where none of the ø_{i} contain boxes or diamonds.
Intuitively speaking, a possibility statement will be said to be ‘good’ just in case it meets the following three constraints: (1) ‘${\varphi}_{1}\left({\overrightarrow{x}}_{1}\right)$’ is satisfied by the actualized a‐world of M_{A}, (2) each of the ‘${\varphi}_{i}({\overrightarrow{x}}_{1},\mathrm{\dots},{\overrightarrow{x}}_{i})$ is consistent with the unconditional basic statements that are true by the lights of A, and (3) there are no clashes amongst the ø_{i} regarding the constitutive properties demanded of the referents of a given sequence of variables.
We wish to prove the following result:
(p.204) Theorem C.2 Every good possibility statement is true in M_{A}.
Before turning to the proof, we need to give a proper definition of ‘good’. Say that a possibility statement ø is good just in case there is:

(a) an essence‐assignment η such that, for any variables x _{1}, …, x_{m} occurring in ø, there is a k such that η assigns x _{1}, …, x_{m} some D_{k}‐essence; and

(b) a decision δ as to which variables to treat as coreferential;
such that η and δ satisfy the following condition:
Start by expanding the language with a new individual constant for each of the objects filling parameters in the essences that η assigns to sequences of variables in ø (including an empty individual constant ‘o’ corresponding to the ‘outside object’ o).
Next modify the possibility statement so as to get the following:
(Here ${E}_{i}({\overrightarrow{x}}_{1},\mathrm{\dots},{\overrightarrow{x}}_{\mathrm{i}})$ is the (possibly infinite) conjunction of formulas θ whose D_{k}‐completions are in the D_{k}‐essence that η assigns to subsets of ${\overrightarrow{x}}_{1},\mathrm{\dots},{\overrightarrow{x}}_{i}$, where each free variable in θ to which the D_{k}‐completion assigns an object has been replaced with the new constant that refers to that object (and where each variable ‘z_{i}’ has been replaced by an appropriate variable x_{j}). And I_{i}(…) is the conjunction consisting of a conjunct ‘∃y∃z(x _{i} = y∧x_{j}=z) → x_{i} = x_{j}’ whenever x_{i} and x_{j} are treated as coreferential by χ and a conjunct ‘∃y∃z(x_{i} = y∧x_{j} = z) → x_{i} ≈ x_{j}’ whenever x_{i} and x_{j} are treated by χ as referring to different objects.
then each of the following conditions is satisfied:

1. $\exists {\overrightarrow{x}}_{1}({E}_{1}({\overrightarrow{x}}_{1})\wedge {I}_{1}({\overrightarrow{x}}_{1})\wedge {\varphi}_{1}({\overrightarrow{x}}_{1}))$ is true in the actualized a‐world (when all constants are assigned their intended interpretations, whether or not they refer to objects in the domain of the actualized a‐world).

2. For each i 〉 1, there is an a‐world w and a way of further expanding the language with a family of interpreted constants c_{α} (which may or may not refer to objects in the domain of w such that: (1) w satisfies C _{2}, and (2) $\exists {\overrightarrow{x}}_{i}({E}_{i}({\overrightarrow{c}}_{1},\mathrm{\dots},{\overrightarrow{c}}_{i1},{\overrightarrow{x}}_{i})\wedge {I}_{1}({\overrightarrow{c}}_{1},\mathrm{\dots},{\overrightarrow{c}}_{i1},{\overrightarrow{x}}_{i})\wedge {\varphi}_{1}({\overrightarrow{c}}_{1},\mathrm{\dots},{\overrightarrow{c}}_{i1},{\overrightarrow{x}}_{i}))$ is true at w (when all constants are assigned their (p.205) intended interpretations, whether or not they refer to objects in the domain of the actualized a‐world).
We now turn to the proof of the theorem.
Let ψ be a good possibility statement containing n diamonds. We know that there are assignments η and χ of essences and coreferentiality respectively, such that a version of ψ modified as above satisfies conditions 1 and 2. Since modified‐ψ entails ψ, it will suffice to show that modified‐ψ is true in M_{A}.
We proceed by reductio. For k ≤ n +1, let the k‐truncation of modified‐ψ be the result of eliminating from modified‐ψ the subformula beginning with the kth diamond and the conjunction sign that precedes it (or doing nothing, if k = n + 1). Suppose that modified‐ψ is false in M_{A}. It follows from condition 1 of the definition of goodness, and the fact that the M_{A}'s center is the actualized a‐world, that the 1‐truncation of modified‐ψ is true in M_{A}. So there must be some k ≤ n such that the kth‐truncation of modified‐ ψ is true in M_{A} and the k + 1th truncation is false in M_{A}.
Since the kth‐truncation of modified‐ψ is true, there must be a sequence of a‐worlds w _{1}, … w_{k} in M_{A} and a sequence of objects ${\overrightarrow{a}}_{1},\mathrm{\dots},{\overrightarrow{a}}_{k}$ such that: (1) w _{1} is the actualized a‐world, (2) each of the ${\overrightarrow{a}}_{i}$ is in the domain of w_{i}, and (3) w_{i} verifies the subformula of the kth‐truncation of modified‐ψ that follows the string of existential quantifiers that binds ${\overrightarrow{x}}_{i}$ when each variable in ${\overrightarrow{x}}_{j}$ is assigned the corresponding ${\overrightarrow{a}}_{j}$ as a value (j ≤ i). By condition 2 of the definition of goodness, there is an a‐world w verifying every ‘just is’‐statement counted as true by A and such that the subformula $\exists {\overrightarrow{x}}_{k+1}({E}_{k+1}({\overrightarrow{x}}_{1}\mathrm{\dots},{\overrightarrow{x}}_{k+1})\wedge {I}_{k+1}({\overrightarrow{x}}_{1},\mathrm{\dots},{\overrightarrow{x}}_{k+1})\wedge {\varphi}_{k+1}({\overrightarrow{x}}_{1},\mathrm{\dots},{\overrightarrow{x}}_{k+1}))$ of ψ is true when the $\overrightarrow{{x}_{j}}$ (j 〈 k + 1) are replaced by suitable new constants. Where m is the maximum of the stages at which w _{1}, … w_{k} were introduced to M_{A} and stages k such that η assigns to some sequence of variables in ψ a D_{k}‐essence, one can verify that there is a world w* which is isomorphic to w and is introduced to M_{A} at stage m + 1.
To see this, we assume with no loss of generality that w has a countable domain, and replace each object z in the domain of w by z*, where (…)* is defined as follows:

1. If c_{j} ^{i} is the new constant introduced to take the place of the ith member of $\overrightarrow{{x}_{j}}$, and if the referent z of c_{j} ^{i} is in the domain of w, then z* is ith member of $\overrightarrow{{a}_{j}}$.
(p.206) (This assignment is guaranteed to be one‐one because we know that the result of substituting new constants for variables in ${I}_{k+1}({\overrightarrow{x}}_{1},\xb7\xb7\xb7{\overrightarrow{x}}_{k+1})$ is true at w.)

2. To each remaining object z in the domain of w, (…)* assigns a distinct m + 1‐stage object whose essence matches the distribution of constitutive predicates that are satisfied by z in w.
The construction of M_{A} guarantees that some a‐world introduced to M_{A} at stage m + 1 is isomorphic to w under (…)*.
This can be used to show that the k + 1th truncation of modified‐ψ is true in M_{A}, and therefore to complete our reductio. It suffices to check that the subformula $\exists {\overrightarrow{x}}_{k+1},({E}_{k+1},({\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1})\text{}\wedge {I}_{k+1}({\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1})\text{}\wedge \text{}{\varphi}_{k+1}\text{}({\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1}))$ of modified‐ψ is true in w* when the values of the ${\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k}$ are taken to be ${\overrightarrow{a}}_{1},\xb7\xb7\xb7,{\overrightarrow{a}}_{k}$. Start by fixing referents ${\overrightarrow{b}}_{k+1}$ for ${\overrightarrow{x}}_{k+1}$ in the domain of w that witness the truth of $\exists {\overrightarrow{x}}_{k+1}({E}_{k+1}({\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1})\text{}\wedge \text{}{I}_{k+1}({\overrightarrow{c}}_{1},\xb7\xb7\xb7,{\overrightarrow{c}}_{k},{\overrightarrow{x}}_{k+1}\text{})\text{}\wedge {\varphi}_{k+1}({\overrightarrow{c}}_{1},\xb7\xb7\xb7,{\overrightarrow{c}}_{k},{\overrightarrow{x}}_{k+1}\text{}))$ in the domain of w that witness the truth of $\exists {\overrightarrow{x}}_{k+1}({E}_{k+1}({\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1})\text{}\wedge \text{}{I}_{k+1}({\overrightarrow{c}}_{1},\xb7\xb7\xb7,{\overrightarrow{c}}_{k},{\overrightarrow{x}}_{k+1}\text{})\text{}\wedge {\varphi}_{k+1}({\overrightarrow{c}}_{1},\xb7\xb7\xb7,{\overrightarrow{c}}_{k},{\overrightarrow{x}}_{k+1}\text{}))$ in w. Let σ be a variable‐assignment that assigns the referent of a new constant replacing a given variable to that variable and assigns ${\overrightarrow{b}}_{k+1}$ to ${\overrightarrow{x}}_{k+1}$ let σ* be a variable‐assignment that assigns ${\overrightarrow{a}}_{1},\xb7\xb7\xb7,{\overrightarrow{a}}_{k}$ to ${\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k}$, assigns z* to a variable v whenever σ(v) = z and z is in the domain of w, and assigns an object outside the domain of w* to v whenever σ(v) is an object outside the domain of w.
To prove the result, it suffices to show that ${\varphi}_{k+1}{\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1}$ is true in w relative to σ just in case it is true in w* relative to σ*. We proceed by induction on the complexity of ø_{k} _{+1}:

• ${\varphi}_{k+1}({\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1})\text{}\text{}\text{is}P({\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1})\text{}$ for P atomic.
I am assuming that atomic formulas (including formulas of the form ‘y = z’) are false at an a‐world whenever they involve empty terms. So the empty‐term case follows from the observation that the result of applying σ to ${\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1}\text{}\text{}\text{}$ is outside the domain of w just in case the result of applying σ* to ${\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1}\text{}\text{}\text{}$ is outside the domain of w*. When there are no empty terms, the result follows from the observation that w and w* are isomorphic under (…)*.

• ${\varphi}_{k+1}({\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1})\text{}\text{}\text{}$ is ${\exists}_{k+1}\left(\theta \right({\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1},z\left)\text{}\right)\text{}\text{}$. Suppose ${\exists}_{z}\left(\theta \right({\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1},z\left)\text{}\right)\text{}\text{}$ is true in w relative to σ, then there is some y in the domain of w such that $\theta ({\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1},z)\text{}\text{}\text{}$ is true in w relative to σ [y/z]. By inductive hypothesis, $\theta ({\overrightarrow{x}}_{1},\xb7\xb7\xb7,{\overrightarrow{x}}_{k+1},z)\text{}\text{}\text{}$ is true in w* relative to σ[y* / z]. So $\exists z\left(\theta \right({\overrightarrow{x}}_{1},\xb7\xb7\xb7{\overrightarrow{x}}_{k+1},z\left)\right)\text{}\text{}\text{}$ is true in w* relative to σ*. The converse is analogous.

• The remaining cases are trivial.
(p.207) Reduction Assume that every object a in the domain of the actualized a‐world of M_{A} has a unique name c_{a}. If w is an a‐world in M_{A}, let S_{w} be the (possibly infinite) sentence that is constructed as follows:

1. Assign a variable x_{a} to each member a of the domain of w.

2. Form the (possibly infinite) conjunction whose conjuncts are: (i) the formulas P(x_{a} _{1}, …, x_{an}) such that 〈a _{1}, …, a_{n}〉 satisfies the atomic predicate P in w, (ii) the formulas x_{ai} ≠ x_{aj} for a_{i} ≠ a_{j}, (iii) the formulas c_{ai} ≠ x_{aj} for a_{i} ≠ a_{j}, (iv) the formulas c_{ai} = x_{ai}, and (v) the (possibly infinite) formula ∀z(z = x_{a} _{1} ∨ z = x_{a} _{2} ∨ …).

3. Bind the free variables with a (possibly infinite) initial string of existential quantifiers.
If ø is a sentence of L ^{◊}, let ø^{L} be the (possibly infinite) disjunction of (possibly infinite) sentences S_{w} such that ø is true at w.
Theorem C.3 For every w ∈ M_{A}, w ⊨ ø if and only if w ⊨ ø^{L}.
The left‐to‐right direction is trivial, in light of the definition of ø^{L}. To verify the right‐to‐left direction, we begin by proving a lemma:
Let v and v ^{′} be a‐worlds in M_{A} and assume that there is an isomorphism g from v onto v ^{𠈲}. (An ‘isomorphism’ is a function from the domain of v onto the domain of v ^{′} that respects the interpretation of every piece of vocabulary in L.) Let s be a variable assignment with the special feature that (1) whenever s assigns to some variable a value x outside the domain of v, x is also outside the domain of v ^{′}, and (2) there is some k such that every object in the range of s was introduced by the kth stage of the construction of M_{A}. Then a formula ø of L ^{◊} is true at v relative to s just in case it is true at v ^{′} relative to the variable assignment s^{g} (which is just like s except that if the value that s assigns to a variable is an object x in the domain of v, then the value that s^{g} assigns to that variable is g(x)).
We prove the lemma by induction on the number of occurrences of ◊ in ø. Assume, first, that ø has no occurrences of ◊. We proceed by induction on the complexity of ø. Every case is straightforward except when ø is atomic. So all we need to show is that, for ø atomic, ø is true at v relative to s just in case ø is true at v ^{′} relative to s^{g}. We verify the left‐to‐right direction. (The converse is analogous.) Assume that ø is true at v relative to s. If every free variable in ø is assigned by s a value in the domain of v, it follows from the fact that g is an isomorphism from v onto v ^{′} that ø is true at v ^{′} relative to s^{g}. If, on the other hand, there are free variables in ø to which s assigns values outside the domain of v, the truth‐value of ø (p.208) at v relative to s doesn't depend on the particular choice of those values (as long as they are chosen to be outside the domain of v). Likewise, the truth‐value of ø at v ^{′} relative to s^{g} doesn't depend on the particular choice of values for those variables (as long as they are chosen to be outside the domain of v). But we are working on the assumption that any object in the range of s that is outside the domain of v is also outside the domain of v ^{′}. So if s assigns to a free variable in ø an object outside the domain of v, s (and therefore s^{g}) assigns to that variable an object outside the domain of v ^{′}. Since ø is true at v relative to s it must therefore be the case that ø is true at v ^{′} relative to s^{g}.
Let us now assume that the lemma holds for the case in which ø has k or less occurrences of ◊, and show that it holds when ø has k + 1 or less occurrences of ◊.
We proceed by induction on the complexity of ø. Every case is straightforward except when ø is of the form ◊ψ. So all we need to show is that ◊ψ is true at v relative to sjust in case ◊ψ is true at v ^{′} relative to s^{g}.
We verify the left‐to‐right direction. (The converse is analogous.) Assume that ◊ψ is true at v relative to s. Then there is an a‐world u in M_{A} such that ψ is true at u relative to s. The construction of M_{A}—together with the fact that every object in the range of s was introduced by stage k of the construction—guarantees that there is a world u ^{′} in M_{A} which is isomorphic to u and which is such that the relevant isomorphism, h, has three properties: (1) h(x) = g(x) for any x in the domain of v, (2) h(x) = x for any x such that s assigns x to some variable occurring free in ψ and x is outside the domain of v, and (3) if s assigns to some variable an object x outside the domain of u, x is also outside the domain of u ^{′}. Since ψ is true at u relative to s, it follows from our (outer) inductive hypothesis (and from property (3) of h) that ψ is true at u ^{′} relative to s^{h}. But—thanks to properties (1) and (2) of h—s^{g} agrees with s on the values of any variables occurring free in ψ. So the fact that ψ is true at u ^{′} relative to s^{h} entails that ◊ψ is true at v ^{′} relative to s^{g}.
This concludes the proof of our lemma. To prove the theorem, suppose that ø is true at w. Then some disjunct S_{t} of ø^{L} is true at w, which entails that t and w are isomorphic. But the definition of ø guarantees that ø is true at t. So our lemma guarantees that ø is true at w.