The Broadest Necessity

Abstract

In this paper the logic of broad necessity is explored. Definitions of what it means for one modality to be broader than another are formulated, and it is proven, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. It is shown, moreover, that it is possible to give a reductive analysis of this necessity in extensional language (using truth functional connectives and quantifiers). This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. This conjecture is formulated precisely in higher-order logic, and concrete cases in which it fails are examined. The paper ends with a discussion of the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. Some philosophical reasons to think that the logic of broad necessity does not include the S5 principle are given.

Appendix

1.1 A.1 Models of Higher-Order Logic

We work within the simply typed λ-calculus with one base type, t. All other types may be obtained as follows: t is a type, and if σ and τ are types, so is σ → τ. ⁵¹

Type signatures such as the one described above can in general be modeled by applicative structures (see Mitchell [34]). Here we shall focus on a particular kind of applicative structure:

Definition A.1 (Henkin structure)

A Henkin structure is a collection of sets A ^σindexed by the types σ with the property that

• $A^{\sigma \to \tau } \subseteq {A^{\tau }}^{A^{\sigma }}$ for each σ and τ

A Henkin structure is moreover full if:

• $A^{\sigma \to \tau } = {A^{\tau }}^{A^{\sigma }}$

Henkin structures then assign meanings to all type expressions. In general Henkin structures are too impoverished too interpret higher-order logic: we need to ensure that they contain enough functions to interpret the typed λ-calculus.

Definition A.2 (Rich Henkin structure)

A Henkin structure is rich iff, for each types σ, τ, υ there are elements K _{σ τ} ∈ A ^σ→τ→σand S _{σ τ υ} ∈ A ^{(σ → τ → υ)→(σ → τ)→σ → υ} satisfying the following properties:

• K _{σ τ} x y = x whenever x ∈ A ^σand y ∈ A ^τ

• S _{σ τ υ} x y z = x z(y z)whenever x ∈ A ^{σ → τ → υ}, y ∈ A ^σ→τand z ∈ A ^σ

Note that any model of the simply typed λ-calculus must contain S and K in each relevant type, because we can define functions with their behavior just using λ-terms: λ x λ y x and λ x λ y λ z x z(y z). Of more note is the fact that if a Henkin structure contains S and K then it contains every λ-definable function (see Mitchell [34] chapter 4).

Definition A.3 (Signature)

A signature Σ for a simply typed λ-calculusis a set of constants c and a type assignment function T y p mapping each constant to a type.

Given an infinite set of variables V a r, the type assignment function can be expanded so that it surjectively assigns types to every member of V a r in such a way that the preimage of each type is infinite. We then expand T y p to assign types to strings of symbols from our signature as follows (below and elsewhere we write ‘ α has type σ’ when T y p(α) = σ):

• c has type T y p(c)

• x has type T y p(x)

• α β has type τ when α has type σ → τ and β has type σ

• λ x α has type σ → τ when x has type σ and α type τ

A term of the simply typed λ-calculus of signature Σ, $\mathcal {L}_{\Sigma }$ , is any string in the domain of T y p.

A variable assignment is a function g on V a r such that g(x) ∈ A ^Typ(x) for each x ∈ V a r. We write g[x↦d] for the assignment that is exactly like g except it maps x to d. If a Henkin structure is rich it is possible to interpret the simply typed λ-calculus over a given signature.

Definition A.4 (Henkin model)

A Henkin model of a signature Σ is a pair (A, [[ ]]) where A is a rich Henkin structure and [[⋅]]^⋅ a function taking each term of $\mathcal {L}_{\Sigma }$ of type σ and variable assignment to an element of A ^σ, satisfying the following properties:

• 1. [[c _σ ]]^g ∈ A ^σ

• 2. [[x]]^g = g(x)

• 3. [[α β]]^g = [[α]]^g([[β]]^g)

• 4. [[λ x α]]^g = the unique function f ∈ A ^σ→τsuch that f(d) = [[α]]^g[x↦d] for every d ∈ A ^σ

Notice that if there is a function satisfying the condition in the last clause, it is unique by the functionality of Henkin models. The fact that we have required the Henkin model to be rich guarantees that there is always at least one function that satisfies the condition; this follows from the point, noted above, that a rich Henkin structure contains every λ-definable function.⁵²

To interpret higher-order logic we focus on the signature H = {→}∪{∀ _σ ∣σ ∈ T y p e} where → has type t → t → t and ∀ _σ type (σ → t) → t. From these the other logical operations may be defined in the usual way, e.g. ⊥ := ∀_{(t → t)→t} ∀ _t , ¬ = λ p (p →⊥), and so on.

Definition A.5 (Logical Henkin model)

A logical Henkin model is a a triple (A, [[ ]], T) where (A, [[ ]]) is a Henkin model, ∅ ⊂ T ⊂ A ^t, and moreover,

• [[ → ]](a)(b) ∈ T iff a ∉ T or b ∈ T

• [[∀ _σ ]](f) ∈ T iff f(a) ∈ T for every a ∈ A ^σ.

Term ϕ of type t is true in a logical model M if and only if [[ϕ]]^g ∈ T for every assignment g.

Logical Henkin models are models of higher-order logic: It is easily verified that every theorem of H is true in a logical Henkin model. Note that logical Henkin models do not encode any assumptions about how fine-grained propositions are. For example, our constraints ensure that [[¬¬ϕ]] and [[ϕ]] have the same truth value, but not that they are identical.

Example A.1 (Boolean models)

Suppose A is a rich Henkin structure, and that A ^tis a Boolean algebra such that every subset of A ^twhich is the range of some function f ∈ A ^{σ → t}has a conjunction in A ^t. (This is satisfied, in particular, if A ^tis a complete Boolean algebra.) We may then define a logical Henkin model as follows. Let $\llbracket \forall _{\sigma }\rrbracket (f) = \bigwedge _{a\in A^{\sigma }} f(a)$ and [[ → ]](a)(b) = ¬a ∨ b(where ¬, ∨ and $\bigwedge$ express the Boolean operations), and let T be an ultrafilter on A ^t with the property that whenever the range of some f ∈ A ^σ→τ is a subset of T, then its conjunction is also in T. It is easily verified that (A, [[ ]], T) is a logical Henkin model, that moreover makes all of the theorems of HE true.

The status of the functionality principle is more subtle. In a Henkin model, but not in an arbitrary applicative structure, if two elements f, g ∈ A ^σ→τ output the same thing for every argument, they are identical. This ensures that we can close our theory under a weak functionality rule: if ⊩ ϕ x = ψ x then ⊩ ϕ = ψ. However there are logical Henkin models in which the functionality axiom of Section 4 is false; we will attend to this matter later in Corollary A.1.

1.2 A.2 Kripke Logical Relations

We now introduce an important definition from Plotkin [36].

Definition A.6 (Kripke logical relation)

Let A be a Henkin structure, and let (W, R) be a reflexive transitive Kripke frame. A binary Kripke logical relation over (W, R)is, for each x ∈ W, a typed family ofbinary relations $\sim _{\sigma }^{x}$ over A ^σwith the following properties.

• For every a, b ∈ A ^σ, if $a\sim ^{\sigma }_{x} b$ and R x y then $a\sim ^{\sigma }_{y}b$.

• For f, g ∈ A ^σ→τ, $f\sim ^{\sigma \to \tau }_{x}g$ if and only if $fa\sim ^{\tau }_{y}gb$ for every y such that R x y and a, b ∈ A ^σsuch that $a\sim ^{\sigma }_{y}b$.

Clearly once you have fixed the behavior of a Kripke logical relation on the base domain A ^t its behavior is determined on all higher types. There is a close connection between Kripke logical relations and the Kripke semantics for intuitionistic logic, in which we talk of types being true at worlds, and in which a functional type σ → τ is true at a world x only if τ is true at every σ world that is R-accessible from x. For convenience we shall continue to suppress the type superscripts when the types can be inferred from the context. The notion of Kripke logical relation we employ in the following is less general than Plotkin’s in a couple of respects. Firstly, we will restrict attention to Kripke logical relations generated by a family of equivalence relations ${\sim _{x}^{t}}$ on the base type. (It’s worth noting that, even if a Kripke logical relation is generated from an equivalence relation on the base types, it does not follow that it is an equivalence relation on the higher types.) Secondly, Plotkin’s Kripke logical relations can have any arity, whereas we restrict attention to binary relations.

Kripke logical relations where originally introduced by Plotkin in order to give a characterization of the λ-definable functions in a model of type theory. For our purposes, the most important result concerning Kripke logical relations is that every λ-definable function is invariant under every Kripke logical relation.

Theorem A.1 (Plotkin)

Let (A, [[ ]]) be a Henkin model of the empty signature, (W, R)a transitive Kripke frame, and ∼a Kripke logical relation on A with respect to (W, R). Then for every closed term α, and world x ∈ W, [[α]]∼ _x [[α]].

Proof

Given two variable assignments g and h we say that g ∼ _x h if g(v) ∼ _x h(v)for every variable v ∈ V a r.

We will prove, by induction on term complexity, a stronger hypothesis that if g ∼ _x h then [[α]]^g ∼ _x [[α]]^hfor arbitrary terms α. The hypothesis is clearly true for variables.

Suppose that it is true for α and β of types σ → τ and σ respectively, and suppose that g ∼ _x h. Then [[α]]^g ∼ _x [[α]]^h by inductive hypothesis. By the definition of a Kripke logical relation, that means that if R x y and a ∼ _y b, [[α]]^g (a) ∼ _y [[α]]^h (b).In particular, since [[β]]^g ∼ _x [[β]]^hby inductive hypothesis, and R x x, if follows that [[α]]^g([[β]]^g) ∼ _x [[α]]^h([[β]]^h). Thus [[α β]]^g ∼ _x [[α β]]^has required.

Now suppose that the hypothesis is true for α of type τ, let v be a variable of type σ and suppose g ∼ _x h. We wish to show that [[λ v α]]^g ∼ _x [[λ v α]]^h, so suppose that R x y and a ∼ _y b. Then g[v↦a] ∼ _y h[v↦b], and thus by inductive hypothesis [[α]]^g[v↦a] ∼ _y [[α]]^h[v↦b].This is just to say that [[λ v α]]^g (a) ∼ _y [[λ v α]]^h (b), as required for the equivalence of functions. □

The restriction to the empty signature can be lifted if we additionally impose that $\llbracket c_{\sigma }\rrbracket \sim _{x}^{\sigma } \llbracket c_{\sigma }\rrbracket$ for each constant c _σ in the language.

1.3 A.3 Modalized Domains and Models of HFE

Here we define a general class of models for HFE. In these models propositions will be represented by sets of worlds. As we noted in Section 5.3, if there is contingent identity then there are principled reasons why the interpretations of functional types are not full (i.e. do not contain all functions between the source and target types). The models that follow are, in a natural sense, as full as they can be once you’ve laid down the structure of the broadest necessity — i.e. once you’ve specified the set of worlds from which propositions are constructed, and the accessibility relation representing the broadest necessity. The logic validated by this class of models thus has many of the properties of the logic of standard models: it will not, for example, be compact, or complete for any recursive axiomatic system.

To define these models we introduce an extension of Henkin structures that carry with them information about modal structure. Recall that a Henkin structure consisted of a type-indexed collection of domains A ^σ, where a domain is just an ordinary set. Instead of using sets, our domains will simply be certain Kripke logical relations. Let $\mathcal {F}= (W,R)$ be a Kripke frame.

Definition A.7 (Modalized domains)

Let $\mathcal {F} = (W,R)$ be a transitive reflexive Kripke frame. A modalized domain based on $\mathcal {F}$ is an ordered pair $A = (|A|, \sim _{\cdot }^{A})$ such that:

• A is a set

• For each x ∈ W, ${\sim _{x}^{A}}$ is an equivalence relation on W.

• Whenever a, b ∈|A|, R x y and a ∼ _x b, a ∼ _y b.

We shall drop the superscript from ${\sim _{x}^{A}}$ when it is clear from context.

Roughly, a modalized domain is a set of elements, along with a notion of identity, ∼ _x , telling us which elements of A are identical at the world x. Given two modalized domains on $\mathcal {F}$, A = (|A|, ∼^A),B = (|B|, ∼^B), we understand a function between them to be a mapping which preserves identity at each world:

Definition A.8 (Mapping between modalized domains)

If A and B are modalized domains, we write f : A → B to mean:

• f : |A| → |B|

• For all w ∈ W, if a ∼ _w b then f a ∼ _w f b

Given this one can define the ‘full’ function space between two modalized domains as follows:

Definition A.9 (Full function space for modalized domains)

If (|A|, ∼^A)and (|B|, ∼^B)are modalized domainson $\mathcal {F}$ we define thefull function space, (|A ⇒ B|, ∼^{A ⇒ B}) as follows:

• |A ⇒ B| = {f∣f : A → B}

• $f\sim _{x}^{A\Rightarrow B} g$ if and only if, for each a, b ∈|A|and each y such that R x y, if $a{\sim _{x}^{A}} b$ then $fa {\sim _{x}^{B}} gb$

Note that f : A → B if and onlyif $f \sim _{x}^{A\Rightarrow B} f$ , meaning that$\sim _{x}^{A\Rightarrow B}$ is an equivalencerelation on |A ⇒ B|for each x ∈ W. It is similarlystraightforward to show that $\sim _{x}^{A\Rightarrow B} \subseteq \sim _{y}^{A\Rightarrow B}$ whenever R x y, showing that A ⇒ B is indeed a modalized domain.

For the function space, the notion of identity on A and B respectively is lifted to functions between A and B using the rule for Kripke logical relations. The condition for functions to belong to |A ⇒ B| is just the condition that functions preserve identity at each world x (a condition that, of course, must obtain if we are to validate Leibniz’s law). Note that Plotkin’s theorem says that each closed term built only out of λ s and variables satisfies: [[α]]∼ _x [[α]]. This corresponds to the idea that λ definable functions will ‘necessarily preserve identity’: for by the definition of ∼ _x between functions, the above means that whenever R x y and a ∼ _y b, [[α]](a) ∼ _y [[α]](b).

Definition A.10 (Full modalized structure)

A full modalized structure is a type indexed collection of modalized domains, A ^σ, withthe property that:

• A ^σ→τ = A ^σ ⇒ A ^τ

It should be stressed that when you throw away the W-indexed equivalence relations, a full modalized structure is just a special kind of (non-full) Henkin structure, B, given by setting B ^σ := |A ^σ|, and that the relations associated with each modalized domain form a Kripke logical relation on B. Indeed, we could have defined the class of Henkin models we were interested in directly, without the detor through modalized domains. However, it is often helpful to carry the information encoded by the equivalence relations around with the domains, and doing so fixes the behavior of the function space constructor uniquely. In the following, we shall move between the modalized structure and the ordinary Henkin structure it corresponds to without comment.⁵³

The definition of a rich structure, a model and a logical model over modalized structures follows the same sequence of definitions as in Section Appendix. In particular, a full modalized structure A ^σ is rich, a model, or a logical model iff the Henkin structure |A ^σ| is rich, a model, or a logical model. As before, a logical model on a modalized structure does not include any assumptions about how fine-grained the elements of the propositional type are. However, in the intended models, to be described shortly, there is a tight connection between the frame (W, R) and the interpretation of the propositional type (A ^t, ∼^t).

Proposition A.2

Every full modalized structure is rich.

Proof

To show that a modalized structure A is rich we must show that the S _{σ τ υ} and K _{σ τ} combinators belong to the corresponding domains in A. To show that K _{σ τ} ∈ A ^σ→τ→σ, for example, we must show that K _{σ τ} preserves the Kripke logical relation associated with A ^σand A ^{τ → σ}respectively. Indeed, this preservation holds for all closed terms, by the samereasoning employed in Plotkin’s Theorem A.1 (we shall not reproduce it here). □

In what follows we write R(x) to abbreviate {y ∈ W∣R x y}, for a relation R ⊆ W × W. We are now in a position to describe, for each transitive reflexive Kripke frame (W, R), the intended model for HFE for that frame (we may think of this as the intended model, on the assumption that modal reality is accurately represented by (W, R)).

Definition A.11 (Intended models)

Let $\mathcal {F}$ be a pointed Kripke frame (W, R, @)where (W, R)is a transitive reflexive frame, and @ ∈ W the designated world. Then (W, R, @)determines a unique logical model, $M_{\mathcal {F}} = (A,\llbracket \cdot \rrbracket , T)$ ,based on a full modalized structure A. It is constructed as follows. It sufficesto: (i) say which modalized domain we use to interpret the base type, A ^t, (ii) specify T, and (iii) specify the interpretations of ∀and →.

For (i) we get an modalized domain (A ^t, ∼^t)as follows:

• $A^{t} = \mathcal {P}(W)$

• $p {\sim _{x}^{t}} q$ iff p ∩ R(x) = q ∩ R(x).

In other words, p and q are identical at x iff they are necessarily equivalent at x. For (ii) and (iii) wefollow Example A.1:

• p ∈ T if and only if @ ∈ p.

• $\llbracket \forall _{\sigma }\rrbracket (f) = \bigcap _{a\in A^{\sigma }} f(a)$

• [[ → ]](p, q) = (W ∖ p) ∪ q

For each transitive reflexive pointed Kripke frame $\mathcal {F}$ denote the corresponding model of higher-order logic $M_{\mathcal {F}}$. We let $\mathcal {C} := \{M_{\mathcal {F}}\mid \mathcal {F}$ a transitive reflexive pointed Kripke frame}.

To show that the above really is a model we must show that $M_{\mathcal {F}}$ is rich, and moreover contains the interpretations of ∀ _σ and → above. (In what follows we assume a fixed frame $\mathcal {F}$ , and we omit the subscript from $M_{\mathcal {F}}$ accordingly.)

Proposition A.3

M is a model of HE.

Proof

That M is rich follows from Proposition A.2. It remains to show that the interpretations of ∀ _σ and →given by Definition A.11 really are mappings between modalized domains: [[∀ _σ ]] : A ^{σ → t} → A ^t and [[ → ]] : A ^t → A ^{t → t}.That is to say, we must show that these functions preserve ∼ _w at each world.

The case of →is straightforward. For ∀ _σ ,suppose f, g ∈ A ^{σ → t}and f ∼ _x g. For each a ∈ A ^σ, a ∼ _x a so f a ∼ _x g a. Expanding the definition of ∼ _x ,this means that R(x) ∩ f a = R(x) ∩ g a for each a, and so $R(x)\cap \bigcap _{a\in M_{\sigma }} fa = R(x) \cap \bigcap _{a\in A^{\sigma }} ga$. That is to say [[∀ _σ ]](f) ∼ _x [[∀ _σ ]](g).

M is thus a logical Henkin model, in which A ^tis a complete Boolean algebra, and T an ultrafilter on A ^t, as described earlier. Thus M is a model of HE. □

The next proposition shows that propositional identity in our model, which is defined by Leibniz equivalence — ∀X(X p ⇔ X q) — amounts to the same thing as necessary equivalence relative to the modality defined by the accessibility relation R. In particular x ∈ [[∀X(X p ⇔ X q)]] if and only if every world accessible to x belongs to [[p ⇔ q]]. This also has the consequence that x ∈ [[L A]] if and only if y ∈ [[A]] for every y such that R x y — that is, L is governed by a standard Kripke semantics in terms of the accessibility relation R.

Proposition A.4

Let a, b ∈|A ^σ|.Then for each x ∈ W, a ∼ _x b if and only if, for every f ∈|A ^{σ → t}|, x ∈ f(a) ⇔ x ∈ f(b). In particular, when σ = t, Leibniz equivalence corresponds to being necessarily equivalent (relative to R) in our model.

Proof

Suppose that a ∼ _x b and let f ∈|A ^{σ → t}|. Since f preserves ∼ _x it follows that f(a) ∼ _x f(b)— i.e. f(a) ∩ R(x) = f(b) ∩ R(x). Since R is reflexive, x ∈ R(x) and so x ∈ f(a) if and only if x ∈ f(b).

Conversely suppose that a≁ _x b. Define a function f as follows: f(X) := {y∣X ∼ _y a}. Clearly x ∈ f(a) and x ∉ f(b). It remains to show that f ∈|A ^{σ → t}|. That amounts to showing f : A ^σ → A ^t, or, more explicitly, that f preserves ∼ _z for each world z.

Suppose, then, that X ∼ _z Y. We want to show that f(X) ∼ _z f(Y): that every f(X) world accessible to z is an f(Y) world and conversely. Let R z w, and suppose that w ∈ f(X). By the definition of f that means X ∼ _w a. Since X ∼ _z Y, X ∼ _w Y and since ∼ _w is an equivalence relation Y ∼ _w a. So w ∈ f(Y). The converse direction proceeds in exactly the same way, so f(X) ∼ _z f(Y). □

Corollary A.5

The functionality principle is true in M.

Proof

For any given world, x we want to show that x ∈ [[∀x(F x = G x) → F = G]], recalling again that = is short for Leibniz equivalence.

By Proposition A.4 it suffices to show that if f, g ∈ A ^σ→τand f a ∼ _x g a for every a ∈ A ^σthen f ∼ _x g, since we have ∼ _x corresponds to Leibniz equivalence in our model. So suppose the hypothesis, and let a ∼ _x b. Since f and g are in M they preserve ∼ _x ,and so f a ∼ _x f b and g a ∼ _x g b. Since ∼ _x is an equivalence relation, f a ∼ _x g b, and since this holds for every such a and b, f ∼ _x g as required. □

1.4 A.4 The Completeness of S4

We can now prove Theorem 5.1 as a corollary. Let $\mathcal {F}=(W,R,@)$ be a pointed Kripke frame.

Corollary A.6

A sentence of $\mathcal {L}_{L}$ is true in a pointed Kripke model $(\mathcal {F}, \llbracket \cdot \rrbracket )$ iff it is true in a corresponding model of higher-order logic $M_{\mathcal {F}}$.

Proof

Here we construct $M=M_{\mathcal {F}}$ as above, except we also need to provide interpretations, [[P]]^M for the propositional letters; these interpretations may simply betransferred from the Kripke model. By Proposition A.4 we know that w ∈ [[L A]]^Miff x ∈ [[A]] for every x such that R w x, provided [[A]]^M = [[A]], and so by a simple induction we can show that for any formula 𝜃 of the modal language $\mathcal {L}_{L}$ (defined in Section 4), w ∈ [[𝜃]]^M if and only if w ∈ [[𝜃]]. □

Now if ϕ is not provable in S4, it is false in some pointed Kripke model $\mathcal {F}$ with a transitive and reflexive accessibility relation (see Hughes and Cresswell [25]). So ϕ is false in $M_{\mathcal {F}}$. Since $M_{\mathcal {F}}$ satisfies HFE, it follows that one cannot prove ϕ from HFE. Conversely, we have shown in Section 4 that every theorem of S4 is provable in HFE, establishing Theorem 5.1.

1.5 A.5 Modalized Functionality

Although we found the functionality principle to be attractive, and consequently adopted it as a working hypothesis, many of the results in this paper do not rest on it. Here is an extremely natural way to weaken the functionality principle:

Modalized Functionality :

L∀ _σ x(F x= _τ G x) → F= _{σ → τ} G

Modalized Functionality is sufficient to prove the uniqueness of identity and the broadest necessity, and so is sufficient to prove, with the exception of the Barcan formula, the results in Section 4 given HE. ⁵⁴

It is relatively simple to tweak our models to generate a more general class of models that invalidate Functionality and the Barcan formula, but validate Modalized Functionality. Here we just outline the basic theory, leaving a more thorough treatment to future work. A partial equivalence relation, or a PER, on a set D is a transitive symmetric relation on D. A partial equivalence relation on a domain D can be equivalently thought of as an equivalence relation on some subset of D (namely, the set {x ∈ W∣x ∼ x}, where ∼ is a PER in the first sense).⁵⁵

Definition A.12 (Expanding modalized domain)

An expanding modalized domain for a frame (W, R) is apair A = (|A|, ∼_⋅) where:

• |A| is a set

• ∼_⋅is a W-indexed set of PERs on |A|such that every a ∈|A| is in the field of some ∼ _w .

• For all a, b ∈|A|, and x, y such that R x y, if a ∼ _x b, a ∼ _y b

Our choice of name for our domains is justified as follows:

Definition A.13

The inner domain, D(w), of an expanding modalized domain A at a world w is D(w) := {a ∈|A| : a ∼ _w a}.

It follows straightforwardly from Definition A.12 that the inner domains are expanding in the sense that D(x) ⊆ D(y) whenever R x y.

The full function space between expanding modalized domains is defined as follows

Definition A.14

Given expanding modalized domains, A and B, define A ⇒ B by:

• |A ⇒ B| = {f : |A| → |B|∣∃x ∈ W such that ∀y ∈ W with R x y and ∀a, b ∈|A|, if a ∼ _y b, f a ∼ _y f b},

• $f\sim _{x}^{A\Rightarrow B} g$ iff, for each a, b ∈|A|and each y such that R x y, if $a{\sim _{y}^{A}} b$ then $fa {\sim _{y}^{B}} gb$.

Note the quantifiers in the definition of |A ⇒ B|: functions don’t need to preserve the PER at every world, they merely need to preserve it from some world onwards (it is worth thinking about why this must be.⁵⁶)

The notions of richness, model, and logical model carry over as before. Following Definition A.11, one can define a class of intended models for each pointed frame as follows. A model based on a frame (W, R, @) consists of an expanding modalized Henkin structure, A, a truth set T, and interpretations of the logical operations, [[∀ _σ ]]∈ A ^{(σ → t)→t}, [[ → ]]∈ A ^{t → t → t}, subject to the following constraints:

• $A^{t} = \mathcal {P}(W)$

• If p, q ∈ D(w) then: $p{\sim _{w}^{t}} q$ iff R(w) ∩ p = R(w) ∩ q.

• ${\sim ^{t}_{x}}$ is a congruence with respect to arbitrary Boolean operations:

  • W ∖ p ∼ _x W ∖ q whenever p ∼ _x q

  • $\bigcap _{i\in I} p_{i} \sim _{x} \bigcap _{i\in I} q_{i}$ whenever p _i ∼ _x q _i for every i ∈ I

• p ∈ T iff @ ∈ p.

• [[ → ]](p)(q) = (W ∖ p) ∪ q

• [[∀ _σ ]](f) = {w ∈ W ∣ w ∈ f(a) for every a such that a ∈ D ^σ (w)}

• [[∀ _σ ]] ∈ |A ^σ→τ|

Note that the last condition amounts to the requirement that [[∀ _σ ]] preserve ∼ at all worlds. This condition was automatically satisfied in all the models $M_{\mathcal {F}}\in \mathcal {C}$ of Functionality (as proved in Proposition A.3). In this context, however, the constraint is not always satisfied. Consider, for example, the frame (3,≤, 0) where 3 = {0, 1, 2} and ≤ is the usual ordering of natural numbers. Then setting ${\sim ^{t}_{0}} = {\sim ^{t}_{2}} = \{(W,W), (\emptyset ,\emptyset )\}$ , and $p{\sim ^{t}_{1}} q$ iff p ∩ R(1) = q ∩ R(1), it can be verified that the sentence, ϕ := ∃p q r(p ≠ q ∧ q ≠ r ∧ p ≠ r), saying that there are at least three propositions, is true at world 1 only. Moreover, in order to have a model we must have [[ψ]] ∈ D ^t(0) for every closed sentence of the language, or else the principle of universal instantiation for the propositional quantifiers would not in general hold. This condition fails in this model, since [[ϕ]] = {1} and {1}≁₀{1}. The diagnosis in this case is that [[∀ _σ ]] does not preserve ∼ _x at every world. Doubtless more needs to be said here, however a proper investigation of these models would take us too far afield.

1.6 A.6 The Behavior of Bijections in Models of HFE

The sorts of models described in Section Appendix can be used to show other independence results. We end by briefly describing how to use these techniques to construct a model (mentioned in Section 5.2) in which there is a bijection of type e → t with no inverse in t → e. Here we use a Kripke frame (W, R) where $W=\mathbb {N}$ and R =≤.

• Let $A^{e} = A^{t} = P(\mathbb {N})$.

• For $x,y\in \mathbb {N}$ let R x y iff x ≤ y. (Any preorder that isn’t an equivalence relation would work here.)

Now we define modalized domains for the base types as follows:

• For a, b ∈ A ^e, a ∼ _w b iff a = b

• For a, b ∈ A ^t, a ∼ _w b iff R(w) ∩ a = R(w) ∩ b.

As before we obtain modalized domains for the higher types using the modalized function space construction. That is:

• For f, g ∈ A ^σ→τ, f ∼ _w g iff for every x such that R w x, and every a, b ∈ A ^σ such that a ∼ _x b, f a ∼ _x g b.

• A ^{a → b} = {f : A ^a → A ^τ∣ for every w, f ∼ _w f}

By setting @ := 0 we can define a logical modalized Henkin model as above which makes all of the theorems of HFE true.

It is immediate that any bijective function f : A ^e → A ^t preserves ∼ _x for each world x and thus that f ∼ _x f, because ${\sim _{x}^{e}}$ is just identity. So f ∈ A ^{e → t}. It should be noted that two propositions a, b ∈ A ^t are Leibniz equivalent at 0 (0 ∈ [[x = y]]^{g[x↦a, y↦b]}) if and only if a = b, since by Proposition A.4 a and b are Leibniz equivalent at 0 iff they are necessarily equivalent at 0. It is then easy to verify that a function f is satisfies the object language statement that f is a bijective concept of type e → t at 0, iff f is in fact a bijective function.

Note that bijections are at best contingently bijections. A bijection from A ^e to A ^t does not count as ‘bijective’ at any world > 0. Let $a=\mathbb {N}$ and $b=\mathbb {N}\setminus \{0\}$. Then a ∼ ₁ b, but since f is a bijection f(a)≠f(b) so f(a) and f(b) are not identical at 1 in A ^e. f thus fails to be injective at 1 because it takes distinct things at 1 of type e to identical things at 1 of type t.

This example also shows that no bijection can belong to A ^{t → e}, since to belong to this domain you must preserve ∼₁, yet $a{\sim _{1}^{t}} b$ but for any bijection f a ≠ f b (since a and b are distinct) and so $fa {\not \sim _{1}^{e}} fb$. Since no bijective functions belong to A ^{t → e}, the claim that there’s a bijective function from t → e is false at 0 (because, as noted above, a function counts as bijective at 0 iff it’s a bijection.) The object language claim ‘no bijection of type e → t has an inverse’ is also true in this model for similar reasons.