2 Language

The formal language of HLE is a higher-order generalization of the first-order language developed in Fine’s LE [Reference Fine21]. In LE, essentialist statements are regimented by means of operators subscripted with a one-place predicate, written $\square _{F}$ . On its intended meaning, a sentence of the form $\square _{F}\phi $ says that it lies in the nature of the things which F that $\phi $ .Footnote ⁸ Thus the subject of the essentialist attribution is the collection of objects which F. However, Fine’s formalization only allows us to pick out (collections of) objects as the subjects of essentialist attributions: LE uses a first-order language in which predicates can only combine with terms in name position to form sentences. But as pointed out in §1, some essentialist statements, such as ‘It lies in the collective nature of identity and universality that everything is self-identical’ cannot be straightforwardly expressed in such a language. We therefore need to extend our expressive resources. The basic idea is to formalize such sentences by picking out what is expressed by words and phrases of different grammatical categories—such as the categories of predicates, sentences or operators—as the subjects of essentialist attributions. The most straightforward way to achieve this is to use a higher-order language in which predicates can combine with terms of different grammatical categories to form sentences. For example, if we have at our disposal a predicate that is satisfied by conjunction in the same way in which ‘is identical to Socrates’ is satisfied by Socrates, we could state what lies in the nature of conjunction in the same way as we can state what lies in the nature of Socrates.

Let me make this a bit more precise. The language we will be working with is a relationally typed language with lambda abstraction.

Clauses (i) and (ii) are the usual clauses for relational types: e is the type of objects (individuals), $\langle \tau _{1},\ldots ,\tau _{n}\rangle $ is the type of ordinary relations between entities of type $\tau _{1},\ldots ,\tau _{n}$ with the special case $\langle \rangle $ being the type of propositions, and clause (iii) concerns the type of rigid properties. Intuitively, a rigid property is a property of being one of $x_{1},x_{2},\ldots $ , for certain specific entities $x_{1},x_{2},\ldots $ . We can thus think of rigid properties as predicative analogs of pluralities.

We assume given, for each $\tau \in \mathcal {T}$ , a denumerably infinite set $\mathcal {V}_{\tau }$ of variables with unique type $\tau $ . The set of all variables is $\mathcal {V}=\bigcup _{\tau \in \mathcal {T}}\mathcal {V}_{\tau }$ . A signature is a set of uniquely typed constants. If $\Sigma $ is a signature, we write $\Sigma _{\tau }$ for the set of constants from $\Sigma $ that have type $\tau $ . We will generally assume that $\Sigma $ is disjoint from $\mathcal {V}$ . Note that $\lbrace \mathcal {V}_{\tau }:\tau \in \mathcal {T}\rbrace $ is a typed family. To define our language, we first define a more general kind of language, a $\lambda K$ -language, in which vacuous $\lambda $ -abstracts are allowed.

The notions of free and bound occurrences of a variable are defined in the standard way. The definition of the sets of $\Sigma $ -terms $\mathcal {L}^{\Sigma }_{\tau }$ of type $\tau $ , for each type $\tau $ , is the same as the definition of $\mathcal {L}K^{\Sigma }_{\tau }$ , except that clause (iv) is replaced by the following clause:

1. (iv′) if $\phi \in \mathcal {L}^{\Sigma }_{\langle \rangle }$ and $x_{1}\in \mathcal {V}_{\tau _{1}},\ldots ,x_{n}\in \mathcal {V}_{\tau _{n}}$ are pairwise distinct variables all of which are free in $\phi $ , $n\geq 1$ , then $(\lambda x_{1}\ldots x_{n}.\phi )\in \mathcal {L}^{\Sigma }_{\langle \tau _{1},\ldots ,\tau _{n}\rangle }$ .

The set of all ${\Sigma }$ -terms $\bigcup _{\tau \in \mathcal {T}}\mathcal {L}^{\Sigma }_{\tau }$ is denoted by $\mathcal {L}^{\Sigma }$ and constitutes the $\lambda I$ -language of $\Sigma $ . Note that $\lbrace \mathcal {L}_{\tau }^{\Sigma }:\tau \in \mathcal {T}\rbrace $ is a typed family. In what follows, we will be exclusively concerned with such $\lambda $ I-languages in which vacuous $\lambda $ -abstracts are not well-formed. A justification for this choice of language is provided in §4. We call terms of type $\langle \rangle $ formulas (or sentences if they don’t contain free variables), terms of type $\langle \tau _{1},\ldots ,\tau _{n}\rangle $ (ordinary) predicates, and terms of type $[\tau ]$ rigid predicates. Our use of rigid predicates is a higher-order generalization of the category of rigid predicates in LE.Footnote ⁹ We call types of the form $\langle \sigma \rangle $ or $[\sigma ]$ one-place predicate types and terms of this type one-place predicate terms. A combinator is a term without free variables or constants, such as $\lambda p.p$ . If not otherwise specified, we assume that each signature $\Sigma $ contains the following logical constants:

1. $\neg \in \Sigma _{\langle \langle \rangle \rangle }$ ;

2. $\wedge ,\vee $ , $\rightarrow ,\leftrightarrow \;\in \Sigma _{\langle \langle \rangle ,\langle \rangle \rangle }$ ;

3. $\forall _{\sigma },\exists _{\sigma }\in \Sigma _{\langle \langle \sigma \rangle \rangle }$ for all $\sigma \in \mathcal {T}$ ;

4. $=_{\sigma }\in \Sigma _{\langle \sigma ,\sigma \rangle }$ for all $\sigma \in \Sigma $ ;

5. $\square _{\tau _{1},\tau _{2},\ldots ,\tau _{k}}\in \Sigma _{\langle \tau _{1},\ldots ,\tau _{k},\langle \rangle \rangle }$ for any $k\geq 0$ and any one-place predicate types $\tau _{i}\in \mathcal {T.}$

Constants of this last kind are essentialist operators. The case where $k=0$ is written $\square _{\varnothing }$ . When $F_{1}, F_{2},\ldots ,F_{n}$ are one-place predicate terms and $\phi $ is a formula, we write $\square _{F_{1}, F_{2},\ldots ,F_{n}}\phi $ for $\square _{\tau _{1},\ldots ,\tau _{n}}(F_{1},\ldots ,F_{n},\phi )$ . We also write $\forall x^{\tau }(\phi )$ for $\forall _{\tau }((\lambda x^{\tau }.\phi ))$ , and $\phi \wedge \psi $ for $\wedge (\phi ,\psi )$ , and similarly for other logical constants. Note that in this setting, quantifiers don’t bind variables, and they combine with predicates to make sentences. For instance, if F is a predicate of type $\langle \sigma \rangle $ , then $\forall _{\sigma }(F)$ states that F applies to every entity of type $\sigma $ (or, equivalently, that F is universal). Thus, to express that everything of type e is self-identical, we can write $\forall _{e}((\lambda x.x=_{e}x))$ , which our convention above allows us to abbreviate to the more usual $\forall x^{e}(x=_{e}x)$ . I will generally omit parentheses and type annotations whenever no confusion is likely to arise and follow other standard conventions. For example, we may write $\forall x(x= x)$ instead of $\forall x^{e}(x=_{e}x)$ whenever it is clear from the context or it doesn’t matter what the appropriate types are.

When A is a term, $x_{1},\ldots ,x_{n}$ are pairwise distinct variables, and $B_{1},\ldots ,B_{n}$ are terms, we write $A[B_{i}/x_{i}]$ for the term that results from replacing each free occurrence of $x_{1}$ in A with $B_{1}$ , and each free occurrence of $x_{2}$ with $B_{2}$ , and so on, replacing bound variables in such a way that no free variables in any $B_{i}$ become bound. We also adopt the following abbreviations:

1. (i) $F\subseteq G$ for $\forall x(F(x)\rightarrow G(x))$ , where x is the first variable of the relevant type not free in either F or $G;$

2. (ii) $F\approx G$ for $(F\subseteq G\wedge G\subseteq F);$

3. (iii) $[A]$ for $\lambda x.x= A$ , where x is the first variable of the same type as A not free in $A;$

4. (iv) $(F_{1},\ldots ,F_{n})\succeq _{\sigma _{1},\ldots ,\sigma _{n},\tau } A$ for $\exists X^{\langle \tau \rangle }\square _{F_{1},\ldots ,F_{n}}X(A)$ , where $X^{\langle \tau \rangle }$ is the first variable not free in $F_{1},\ldots ,F_{n},A;$

5. (v) $c_{\tau }(F_{1},\ldots ,F_{n})$ for $\lambda x^{\tau }.(F_{1},\ldots ,F_{n})\succeq _{\sigma _{1},\ldots ,\sigma _{n},\tau } x^{\tau }$ , where $x^{\tau }$ is the first variable not free in $F_{1},\ldots ,F_{n}$ ;

6. (vi) $A\geq _{\sigma ,\tau } B$ for $([A])\succeq _{\langle \sigma \rangle ,\tau }B.$

The predicate $\succeq _{\sigma _{1},\ldots ,\sigma _{n},\tau }$ expresses the relation of essential involvement between a collection of entities and a single entity. We can pronounce a statement of the form $(F_{1},\ldots ,F_{n})\succeq _{\sigma _{1},\ldots ,\sigma _{n},\tau } A$ as ‘the nature of the collection of entities picked out by $F_{1},\ldots ,F_{n}$ involves A’, where the collection of entities picked out by $F_{1},\ldots ,F_{n}$ consists of all the entities in the extensions of $F_{1},\ldots ,F_{n}$ . By (iv) this means that there is some property $X^{\langle \tau \rangle }$ such that it lies in the nature of the collection picked out by $F_{1},\ldots ,F_{n}$ that A has X. We will sometimes suppress the types of the predicates on the left-hand side and simply write $\succeq _{\tau }$ for $\succeq _{\tau _{1},\ldots ,\tau _{n},\tau }$ . The predicate $c_{\tau }(F_{1},\ldots ,F_{n})$ applies to all and only those entities of type $\tau $ that are involved in the nature of the collection of entities picked out by $F_{1},\ldots ,F_{n}$ . For example, $c_{e}([s])$ applies to all and only the objects involved in Socrates’ nature and $c_{\langle \rangle }([\neg ])$ applies to all and only the propositions involved in the nature of negation. The predicate $\geq _{\sigma ,\tau }$ expresses the relation of essential involvement between a single entity of type $\sigma $ and a single entity of type $\tau $ . Thus a statement of the form $A\geq _{\sigma ,\tau }B$ expresses that the nature of A involves B. We will sometimes omit type subscripts and write $\geq $ instead of $\geq _{\sigma ,\tau }$ .

Here are some examples of formalized essentialist statements:

1. 1.

   1. (a) $\square _{[s]}H(s).$

      (It is essential to Socrates that he is human.)

   2. (b) $\neg \square _{[H]}H(s).$

      (It is not in the nature of being human that Socrates is human.)

   3. (c) $\square _{N} \forall x(N(x)\rightarrow (E(x)\vee O(x)).$

      (It lies in the collective nature of the natural numbers that every natural number is either even or odd.)

   4. (d) $\square _{[\forall ],[=]}\forall x(x= x).$

      (It lies in the nature of identity and universality taken together that everything is self-identical.)

   5. (e) $\square _{\lambda p^{\langle \rangle }.p= p}\exists p\;p.$

      (It is true in virtue of the collective nature of all propositions that there is a true proposition.)

Note the difference between $\square _{[F]}$ and $\square _{F}$ : The former expresses truth in virtue of the nature of being F, so the property of being F is the subject of the essentialist attribution, whereas the latter expresses truth in virtue of the nature of the entities which instantiate F, so the entities in the extension of F are the (collective) subject of the essentialist attribution.

3 The system

In this section, we present the proof system of HLE. The system HLE is constituted by the following axioms and rules.

1. (i) Background logic:

   1. (PC) $\phi $ , whenever $\phi $ is a tautology.

   2. (UI) $\forall _{\tau }(F)\rightarrow F(A).$

   3. (EG) $F(A)\rightarrow \exists _{\tau }(F).$

   4. (MP) If $\vdash \phi $ and $\vdash \phi \rightarrow \psi $ , then $\vdash \psi .$

   5. (GEN) If $\vdash \phi \rightarrow \psi $ and x is free in $\psi $ but not $\phi $ , then $\vdash \phi \rightarrow \forall _{\tau }x(\psi ).$

   6. (INST) If $\vdash \phi \rightarrow \psi $ and x is free in $\phi $ but not $\psi $ , then $\vdash \exists _{\tau }x(\phi )\rightarrow \psi .$

   7. (Ref) $F=_{\sigma }F.$

   8. (LL) $F=_{\sigma }G\rightarrow (\phi [G/x]\rightarrow \phi [F/x]).$

   9. (β-conversion) $\phi \leftrightarrow \phi ^{*}$ , where $\phi ^{*}$ results from $\phi $ by replacing some constituent of the form $(\lambda x_{1}\ldots x_{n}.\psi )(t_{1},\ldots ,t_{n})$ with $\psi [t_{i}/x_{i}].$

   10. (η-conversion) $\phi \leftrightarrow \phi ^{*}$ , where $\phi ^{*}$ results from $\phi $ by replacing some constituent of the form $(\lambda x_{1}\ldots x_{k}.F(x_{1},\ldots ,x_{k}))$ , where none of $x_{1},\ldots ,x_{k}$ is free in F, with $F.$

2. (ii) Background essentialist axioms

   1. (Permutation) $\square _{F_{1},\ldots ,F_{n}}\phi \rightarrow \square _{F_{\pi (1)},\ldots ,F_{\pi (n)}}\phi $ , where $\pi $ is a permutation of $1,\ldots ,n.$

   2. (Idempotence) $\square _{F_{1},\ldots ,F_{n}}\phi \rightarrow \square _{F_{1},\ldots ,F_{n-1}}\phi $ , if $F_{n}=F_{i}$ , for some $i=1,\ldots ,n-1.$

   3. (Separation) $\square _{F_{1},\ldots ,F_{n},\lambda x.(G(x)\vee H(x))}\phi \rightarrow \square _{F_{1},\ldots ,F_{n},G,H}\phi .$

   4. (Subtraction) $\forall x^{\sigma }\neg F_{n+1}(x)\rightarrow (\square _{F_{1},\ldots ,F_{n+1}}\phi \leftrightarrow \square _{F_{1},\ldots ,F_{n}}\phi ).$

   5. (MON) $\forall x^{\sigma }(F_{1}(x)\rightarrow G(x))\rightarrow (\square _{F_{1},\ldots F_{k}}\phi \rightarrow \square _{G,F_{2},\ldots ,F_{k}}\phi )$ , $1\leq k.$

   6. (Decomposition) $\square _{F_{1},\ldots ,F_{k},[B]}\phi \rightarrow \square _{F_{1},\ldots ,F_{k},[A_{1}],\ldots ,[A_{n}]}\phi $ , where $A_{1},\ldots ,A_{n}$ are all the constants and free variables of $B.$

3. (iii) Axioms for rigidity

   1. (R-Comp) $\forall X^{\langle \sigma \rangle }\exists Y^{[\sigma ]}\forall x^{\sigma }(X(x)\leftrightarrow Y(x)).$

   2. (R-Ext) $\forall X^{[\sigma ]}\forall Y^{[\sigma ]}(\forall x^{\sigma }(X(x)\leftrightarrow Y(x))\rightarrow X=_{[\sigma ]}Y).$

   3. (Rigidity) $R^{[\sigma ]}(x)\rightarrow \square _{R}R(x).$

   4. (R-Equiv) $\square _{R,F_{1},\ldots ,F_{n}}\phi \leftrightarrow \square _{[R],F_{1},\ldots ,F_{n}}\phi $ , where R is a rigid predicate.

4. (iv) Core essentialist axioms and rules

   1. (CH) $\square _{G_{1},\ldots ,G_{n},c_{\sigma }(F_{1},\ldots ,F_{k})}\phi \rightarrow \square _{G_{1},\ldots ,G_{n},F_{1},\ldots ,F_{k}}\phi .$

   2. (RC) If $\vdash \phi _{1}\wedge \cdots \wedge \phi _{n}\rightarrow \psi $ , then $\vdash \square _{F_{1},\ldots ,F_{k}}\phi _{1}\wedge \cdots \wedge \square _{F_{1},\ldots ,F_{k}}\phi _{n}\rightarrow \square _{F_{1},\ldots ,F_{k},[A_{1}],\ldots ,[A_{m}]}\psi $ , where $A_{1},\ldots ,A_{m}$ are all the constants and free variables occurring in $\psi $ but not any of $\phi _{1},\ldots ,\phi _{n.}$

   3. (T) $\square _{F_{1},\ldots ,F_{n}}\phi \rightarrow \phi .$

   4. (4) $\square _{F_{1},\ldots ,F_{n}}\phi \rightarrow \square _{F_{1},\ldots ,F_{n},[\square _{\sigma _{1},\ldots ,\sigma _{n}}]}\square _{F_{1},\ldots ,F_{n}}\phi $ , whenever $F^{\sigma _{1}}_{1},\ldots ,F^{\sigma _{n}}_{n}$ are rigid predicates.

   5. (5) $\neg \square _{F_{1},\ldots ,F_{n}}\phi \rightarrow \square _{F_{1},\ldots ,F_{n},[\neg ],[\square _{\sigma _{1},\ldots ,\sigma _{n}}],[\phi ]}\neg \square _{F_{1},\ldots ,F_{n}}\phi $ , whenever $F^{\sigma _{1}}_{1},\ldots ,F^{\sigma _{n}}_{n}$ are rigid predicates.

I will only briefly comment on the rationale for the different axioms and rules. For a more detailed justification, the reader is referred to Ditter [Reference Ditter15]. The background logic consists of the principles of classical higher-order logic with identity together with the standard principles of $\beta $ - and $\eta $ -conversion. The first five axioms in group (ii) capture the principle of Monotonicity: If X is a subcollection of Y, then if it lies in the nature of X that $\phi $ then it lies in the nature of Y that $\phi $ . They do so by making sure that if the predicates $F_{1},\ldots ,F_{n}$ pick out some entities that form a (proper or improper) subcollection of the entities picked out by $G_{1},\ldots , G_{m}$ , then $\square _{F_{1},\ldots ,F_{n}}\phi $ materially implies $\square _{G_{1},\ldots ,G_{m}}\phi $ .Footnote ¹⁰ In writing formal proofs, I will often refer to one or more of these axioms simply by ‘Monotonicity’.

The principle of Decomposition says that whatever lies in the nature of an entity expressed by a term B also lies in the collective nature of the entities denoted by the free variables and constants occurring in B. For instance, Decomposition allows us to infer $\square _{[\forall ],[=]}\forall x(x= x)$ from $\square _{[\forall x(x= x)]}\forall x(x= x)$ .Footnote ¹¹ Model-theoretically, Decomposition corresponds to the condition that the denotation of a term exists at a world if the denotations of all of its constants and free variables exist at that world. This condition is crucial not only for the validity of Decomposition but for virtually every principle of the system, as the proof of the Soundness Theorem (Theorem 1) will illustrate.

The axioms in group (iii) state characteristic principles for rigid properties. R-Comp says that for any property of any type there is a rigid property coextensive with it.Footnote ¹² R-Ext is an extensionality principle for rigid properties. It says that coextensiveness suffices for the identity of rigid properties. This is in line with our thinking of rigid properties as predicative analogs of pluralities. The axiom of Rigidity says that if x is one of some entities, then it lies in the nature of those entities that x is one of them. This axiom is a higher-order generalization of Fine’s axiom of Rigidity in LE. R-Equiv concerns the relation between a rigid property and its extension. It says that whatever lies in the nature of a rigid property lies in the collective nature of the entities in its extension, and vice versa. The nature of a rigid property is thus exhausted by the collective nature of the entities that it applies to. R-Equiv and Rigidity together entail that if x is one of some entities, then it lies in the nature of the property of being one of those entities that x is one of them. This is analogous to the essentialist principle about sets and pluralities that says that sets and pluralities have their members essentially.

The last three axiom schemas in group (iv) are essentialist analogs of the modal T-, 4- and 5-schemas. The T-schema says that whatever is true in virtue of the nature of a collection of entities is true. The 4-principle says that if $\phi $ is true in virtue of the nature of a collection X, then it is true in virtue of the nature of X together with the nature of essentiality (i.e., what is expressed by the essentialist operator) that it is true in virtue of the nature of X that $\phi $ . In conjunction with the widely accepted principle that essence implies necessity (i.e., if $\phi $ is true in virtue of the nature of some entities then $\phi $ is necessary), the 4-principle entails the plausible claim that whatever is true in virtue of the nature of a collection X is necessarily true in virtue of the nature of X—entities have their essence necessarily.

In stating the essentialist 4-schema in our formal language it is crucial that the predicates $F_{1},\ldots ,F_{n}$ be rigid, since the inner essentialist operator in the consequent should have the same subject as the operator in the antecedent. The schema does not in general hold for non-rigid predicates. For example, suppose that Mary is wearing a hat and that it is essential to the hat wearers that Mary is human. Then it may be false that it is essential to the hat wearers, taken together with essentiality, that it is essential to the hat wearers that Mary is human, since it may not be the case that it is essential to Mary to be wearing a hat, and so it may not be essential to whoever is wearing a hat that Mary is human. Such a case is averted if the predicates in (4) rigidly pick out Mary.

The same restriction to rigid predicates is required in the statement of the essentialist 5-principle. This principle says that if it is not true in virtue of the nature of a collection X that $\phi $ , then it is true in virtue of the nature of X together with negation, essentiality and $\phi $ that it is not true in virtue of the nature of X that $\phi $ . Together with the principle that essence implies necessity, (5) entails that if $\phi $ is not true in virtue of the nature of X, then this is necessarily so—the nature of X could not have been richer.Footnote ¹³

The first two principles in group (iv) are the key components of the account of logical closure provided by HLE. The principle CH is a generalization of the ‘chaining axiom’ from LE to our higher-order framework. Recall that $c_{\sigma }(F_{1},\ldots ,F_{k})$ picks out the collection of entities of type $\sigma $ involved in the nature of the entities picked out by $F_{1},\ldots ,F_{k}$ . Let p be some proposition true in virtue of the collective nature of the entities in $c_{\sigma }(F_{1},\ldots ,F_{k})$ . Then CH says that p is also true in virtue of the collective nature of the entities picked out by $F_{1},\ldots ,F_{k}$ . Thus every collection of entities inherits the collective natures of the entities involved in its nature. RC is a closure principle for essences. In HLE, any type of logical closure manifested by the essence of a collection of entities derives from the logical operations involved in the essence of the collection. So, for example, the essence of a collection of entities will be closed under conjunction elimination and introduction just in case the essence of the collection involves conjunction, and it will be closed under modus ponens just in case it involves the conditional. For a concrete example, suppose it lies in the nature of the number two that it is a number and even ( $\square _{[2]}(N(2)\wedge E(2))$ ). RC then immediately implies that it lies in the nature of the number two that it is a number ( $\square _{[2]}N(2)$ ). The rationale for closure under conjunction elimination is that the hypothesis implies that the nature of the number two involves conjunction (in the sense of involvement defined above), since $\square _{[2]}(N(2)\wedge E(2))$ implies $\square _{[2]}\lambda X.(X(N(2),E(2))(\wedge )$ by $\beta $ -conversion, which implies $2\geq \wedge $ by $\beta $ -conversion and EG. The converse implication is only valid under the additional assumption that the number two involves conjunction. By RC alone we have that $\square _{[2]}N(2)$ and $\square _{[2]}E(2)$ together imply (*) $\square _{[2],[\wedge ]}(N(2)\wedge E(2))$ ; if we further assume $2\geq \wedge $ , then $c_{\langle \langle \rangle ,\langle \rangle \rangle }([2])(\wedge )$ and thus $\square _{[2],c_{\langle \langle \rangle ,\langle \rangle \rangle }([2])}(N(2)\wedge E(2))$ by Monotonicity and (*), whence $\square _{[2]}(N(2)\wedge E(2))$ follows by CH and Monotonicity. RC and CH together imply that the nature of a collection of entities is closed under precisely those consequences that are stated solely in terms of vocabulary expressing entities that are involved in the nature of the collection.

The special case of RC where $n=0$ gives us the following essentialist “necessitation-rule”:

1. (RC⁰) If $\vdash \psi $ , then $\vdash \square _{[A_{1}],\ldots ,[A_{m}]}\psi $ , where $A_{1},\ldots ,A_{m}$ are all the constants and free variables occurring in $\psi .$

RC $^{0}$ implies that every logical truth whose only constants are logical constants is true in virtue of the nature of the logical operations expressed by these constants. So, for example, since $\forall x(x= x)$ is a logical truth whose only constants are $\forall $ and $=$ , RC $^{0}$ implies $\square _{[\forall ],[=]}\forall x(x= x)$ . Note, however, that RC $^{0}$ does not imply that every logical truth is true solely in virtue of some logical operations. This would have the implausible consequence that the collective nature of the logical operations involves absolutely every entity of any type, but plausibly the nature of identity does not involve Socrates, for example.

Correia [Reference Correia8, Reference Correia and Dumitru9] has proposed an account of the way in which essences are closed under logical consequence which differs in important ways from the account proposed here. A brief comparison between HLE and Correia’s account will be instructive. Correia’s target notion of essence, which he calls the ‘Finean notion of essence’, is characterized in terms of a primitive notion of a proposition being basically essential to a plurality (the logic of which is not specified), and a relation of relative logical consequence, defined as follows:Footnote ¹⁴

After providing some general structural principles guiding the relation of relative logical consequence (the details of which don’t matter for present purposes), Correia goes on to define what he calls the ‘Finean notion of essence’, thus:

I will refer to the propositions that are true in virtue of the nature of X so defined as just the essence or nature of X. According to Correia, this notion of essence differs importantly from Fine’s consequential notion, since, e.g., ‘the propositions which are true in virtue of the nature of Socrates are, on the proposed account, just the propositions which belong to his basic nature.’ (ibid.) More generally, he maintains, ‘in case X comprises no logical concepts, the propositions which are true in virtue of the nature of X are just those which belong to the basic nature of X.’ (ibid.) Thus, in contrast to HLE, the logical closure of the essence of X does not depend on which logical concepts are involved in the nature of X, but only on whether X comprises any logical concepts. The notion of essential involvement plays no role in Correia’s account.

This leads to some interestingly different predictions about the logical closure properties of essences. It is consistent with both accounts that, say, Socrates’ nature is “logically simple” in the sense of not being closed under logical consequence at all. However, if it is essential to Socrates that he is human and mortal ( $\square _{[s]}(H(s)\wedge M(s))$ ), then HLE entails that he is essentially human ( $\square _{[s}H(s)$ ) and essentially mortal ( $\square _{[s]}M(s)$ ), whereas this doesn’t follow on Correia’s account. Furthermore, Correia’s account predicts that it lies in the nature of identity alone that Socrates is self-identical, because the proposition that Socrates is self-identical can be proved using only an instance of (Ref), whereas in HLE this can be consistently denied, because we can consistently deny that the nature of identity involves Socrates.

However, there are some important details about Correia’s account that need to be spelled out before we can offer a more systematic comparison between it and HLE. As we have seen above, and as Correia himself stresses, his proposal crucially rests on taking the relation of relative logical consequence, as well as the notions of proof and rule of inference in terms of which it is defined, not to be relations between sentences (as logical consequence is usually defined) but to be relations between propositions (ibid. p. 647 fn. 10). (Otherwise the account would be problematically meta-linguistic.) But the usual characterizations of the notions of proof and rule of inference crucially presuppose various syntactic notions which do not obviously have propositional analogs. For example, in order to state an inference rule governing a given connective, we need to be able to identify the occurrence of the connective in a certain syntactic position of a formula. But unless propositions are structured in something like the way sentences are structured—an assumption that we have reason to question, given the Russell–Myhill paradoxFootnote ¹⁶ —it is not obvious that there are propositional analogs of these notions which work in the intended way. Without further details, it thus remains an open question how Correia’s central notion of relative logical consequence as well as the corresponding notion of essence are to be understood, and whether they can ultimately be developed in a consistent way.

4 Some theorems

This section establishes a number of theorems of HLE and discusses their philosophical ramifications. Principles of classical logic will be referred to by the label ‘CL’. Theoremhood is defined in the usual way. We begin with a useful consequence of the Monotonicity axioms.

From now on, I will normally use the label ‘Monotonicity’ to refer to either Proposition 1 or one or more of the Monotonicity axioms. The following result concerns some important properties of the relation $\geq $ of essential involvement between particular entities.

Proposition 2(i) and (ii) concern the reflexivity and transitivity of $\geq $ , respectively. Part (iv) shows that the relation $\geq $ is not asymmetric across types: For every entity of any type, there is a proposition such that the nature of the entity involves the proposition and the nature of the proposition involves the entity. Let us call the rigid property that applies to x and only x the haecceity of x. (Uniqueness is justified by R-Ext.) When F is the haecceity of x, we say that $F(x)$ is the haecceity proposition of x. For any x, the haecceity proposition of x is a canonical example of a proposition whose nature involves x and which is involved in the nature of x. Although Proposition 2(iv) crucially relies on Rigidity, it is worth noting that the same result would follow without Rigidity if we posited that for any entity x, there is a proposition true in virtue of the nature of x that predicates something of x.

We now turn to some important consequences of CH.

Note that Proposition 3(iv) is the converse of Decomposition. Proposition 3(iii) bears resemblance to, but is importantly different from, the axiom of Localization in LE. The latter says that if $\phi $ is true in virtue of the nature of a collection of objects, then any object x which ‘occurs in’ $\phi $ is such that the nature of one of the members of the collection must involve x. Formally (in the language of HLE):Footnote ¹⁷

1. (Localization) $\square _{F}\phi \wedge x\eta \phi \rightarrow \exists y^{e}(F(y)\wedge y\geq _{e,e} x).$

We can state a higher-order generalization of Localization as follows, replacing $x\eta \phi $ with $\phi \geq _{\langle \rangle ,\sigma }x$ :Footnote ¹⁸

1. (Localization^*) $\square _{F_{1},F_{2},\ldots ,F_{n}}\phi \wedge \phi \geq x^{\sigma }\rightarrow (\exists x_{1}^{\sigma _{1}}(F_{1}(x_{1})\wedge x_{1}\geq x)\vee \cdots \vee \exists x_{n}^{\sigma _{n}} (F_{n}(x_{n})\wedge x_{n}\geq x)).$

Localization $^{*}$ says that if $\phi $ is true in virtue of the nature of some collection of entities and the nature of $\phi $ involves some entity x, then the nature of some member of the collection involves x.Footnote ¹⁹ The difference between Localization $^{*}$ and Proposition 3(iii) is that the former requires the nature of a particular member of the collection to involve x, whereas the latter only requires the nature of the collection to involve x. Localization $^{*}$ immediately yields problematic results in HLE. For example, since both $\square _{[\forall ],[\neg ],[\vee ]}\forall p(p\vee \neg p)$ and $\forall p(p\vee \neg p)\geq \forall p(p\vee \neg p)$ are theorems of HLE (by RC $^{0}$ and Proposition 2(i), respectively), Localization $^{*}$ implies that the nature of one of $\forall $ , $\neg $ or $\vee $ involves $\forall p(p\vee \neg p)$ , and therefore each of $\forall $ , $\neg $ and $\vee $ since the nature of $\forall p(p\vee \neg p)$ involves each of them (by Proposition 2(ii) and (iii). Similarly, if it is true in virtue of the nature of x and y that they are distinct, then it follows from Localization $^{*}$ and the theoremhood of $\neg (x= y)\geq \neg (x= y)$ that either x or y involves the proposition $\neg (x= y)$ ; hence, it follows that either x involves y or vice versa, since the nature of the proposition $\neg (x= y)$ involves both x and y.Footnote ²⁰

Although the failure of the higher-order generalization already presents at least a prima facie reason to reject Localization, the above problems for Localization $^{*}$ make crucial use of higher-order resources and therefore do not directly challenge Fine’s first-order principle of Localization. However, potential counterexamples to Localization are not hard to come by. Fine [Reference Fine21, p. 249 f.] himself notes that potential counterexamples to Localization arise from cases ‘in which the nature of several objects is simultaneously understood in terms of one another.’ He discusses the following kind of case.Footnote ²¹ Suppose it lies in the nature of a mind and body taken together to comprise a given person. Localization then entails that either the nature of the mind or the nature of the body involves the person. But it is natural to think that the person is not involved in the nature of either of them, taken individually.Footnote ²² However, Fine does not take this case to be a counterexample to Localization. But the reason is not that he argues that such a case is impossible; rather, he holds that

Fine resolves the potential counterexample above in accordance with this decision: just as each point of Euclidean space is involved in the nature of every other point, so too do the nature of the mind and that of the body individually involve the person as well as each other (ibid., p. 249 f.).Footnote ²³

In my view, the question of whether Localization is true is a substantive question that cannot be resolved in the way Fine suggests. I also think that the potential counterexample to Localization discussed above is an actual counterexample. Indeed, the case of a body and a mind essentially comprising a person is in an important way analogous to the problems I have pointed out for Localization $^{*}$ but disanalogous to the case of points of Euclidean space. Both the problems for Localization $^{*}$ and the case of the body and the mind essentially comprising a person represent cases where some entity that is not involved in the nature of any individual member of a given collection is involved in the nature of the collection. By contrast, the case of the points of Euclidean space Fine describes is not of this sort: since each of the points is involved in its own nature, there will be, for any point, a point whose nature involves it. Therefore, the case of the points of Euclidean space essentially entering into certain geometric relationships with one another does not pose a threat to Localization.Footnote ²⁴ But this disanalogy between the case of the mind and the body on the one hand and the points of Euclidean space on the other further calls into question Fine’s response to the potential counterexample concerning a person being comprised by a mind and body.

As mentioned above, counterexamples to Localization $^{*}$ (and Localization) arise whenever the nature of a collection involves new entities, in the sense that those entities are not involved in the nature of any individual members of the collection. The proposition $\forall p(p\vee \neg p)$ , for example, is involved in the collective nature of $\forall $ , $\vee $ and $\neg $ , but arguably not in the nature of any of them individually. The following consequence of Localization $^{*}$ is therefore subject to counterexamples:

1. (F-Collective) $(F_{1},\ldots ,F_{n})\succeq _\sigma x\rightarrow (\exists x_{1}^{\sigma _{1}}(F_{1}(x_{1})\wedge x_{1}\geq x)\vee \cdots \vee \exists x_{n}^{\sigma _{n}}(F_{n}(x_{n})\wedge x_{n}\geq x)).$

F-Collective says that whenever the nature of some collection involves some x, then the nature of some member of the collection involves x. The failure of F-Collective entails that we cannot define collective essential involvement in terms of individual essential involvement, as Fine does in LE. Individual essential involvement ( $\geq $ ) is a special case of collective essential involvement $(\succeq )$ , and not vice versa. As we will see later, the failure of Localization $^{*}$ requires a modification (which turns out to be a simplification) of Fine’s semantic clause for the essentialist operator.

Our next result concerns some consequences of RC.

By Decomposition, part (i) of Proposition 4 is in fact equivalent to RC. Note that part (ii) immediately implies that the essence of any collection of entities whose nature involves every proposition is closed under unrestricted consequence.

Next we state some consequences of Decomposition.

Part (iii) of the above result illustrates the special status of the combinators in HLE: The nature of the operations expressed by the combinators is involved in the nature of any collection of entities. The combinators are naturally regarded as expressing broadly “grammatical” operations presupposed by the nature of any entity. Part (i) is equivalent to Decomposition given the other principles of HLE. It is a kind of non-linguistic counterpart of the idea that for any linguistic expressions of a given grammar, the complex expressions that can be formed from them by the grammatical operations are part of the collective nature of the linguistic expressions. For example, it is natural to think that it lies in the nature of the name ‘Socrates’ and the predicate ‘is human’ to form such sentences as ‘Socrates is human’. And just as it lies in the collective nature of the name ‘Socrates’ and the predicate ‘is human’ to form the sentence ‘Socrates is human’, so too is it natural to think that it lies in the nature of Socrates and being human to form the proposition that Socrates is human. More generally, the entities that are formed from Socrates and being human by broadly “grammatical” operations like predication and (the metaphysical counterpart of) predicate abstraction are involved in the collective nature of Socrates and being human.Footnote ²⁵

Decomposition also implies that whatever is true in virtue of the nature of the converse of a relation is true in virtue of the relation itself. For let L stand for the relation loves, for example. Then its converse $\lambda xy.L(y,x)$ stands for being loved by. By Decomposition, we can infer $\square _{[L]}\phi $ from $\square _{[\lambda xy.L(y,x)]}\phi $ . The same is true for the “reflexivization” of L, $\lambda x.L(x,x)$ , which represents the relation of loving oneself.

The previous results allow us to illustrate why we are working in a $\lambda $ I-language in which vacuous abstraction is not allowed. Suppose that we allowed for vacuous abstraction, so that we can form lambda-abstracts like $\lambda x.\phi $ , where x is not free in $\phi $ . Let A and B be any terms of arbitrary type. By Proposition 2(i), $\vdash \exists X\square _{[A]}X(A)$ . Now let z be a variable not free in $X(A)$ . Then $\vdash X(A)=(\lambda z.X(A))(B)$ by a vacuous instance of $\beta $ -conversion. By CL and LL, this implies $\vdash \exists Y\square _{[A]}Y(B)$ , and thus $\vdash A\geq B$ for arbitrary A and B. It follows that all subscripted essentialist operators are logically coextensive: $\vdash \square _{F_{1},\ldots ,F_{n}}\phi \leftrightarrow \square _{G_{1},\ldots ,G_{k}}\phi $ . Instead of avoiding this result as we do here by working with a language in which vacuous instances of $\beta $ -conversion are not well-formed we could instead work with a $\lambda $ K-language and restrict the principle of $\beta $ -conversion to its non-vacuous instances. The present choice of language is primarily a matter of convenience, although there is something to be said for taking the $\lambda $ I-language to be metaphysically more perspicuous than the $\lambda $ K-language in a context in which vacuous $\beta $ -conversion is rejected.Footnote ²⁶

We now turn to some important results about identity and distinctness.

Proof. (i) By RC $^{0}$ , $\vdash \square _{[x],[=]}x= x$ . Hence, by CL and LL, $\vdash x= y\rightarrow \square _{[x],[=]}x= y$ .

(ii) Let $X,Y,Z$ be variables of types $[\sigma ]$ , $[\sigma ]$ and $[\langle \sigma ,\sigma \rangle ]$ , respectively. By part (i) and Monotonicity, we have

(1) $$ \begin{align} \vdash X(x)\wedge Y(y)\wedge Z(=)\rightarrow (x= y\rightarrow\square_{X,Y,Z}x= y). \end{align} $$

Hence, by CL,

(2) $$ \begin{align} \vdash X(x)\wedge Y(y)\wedge Z(=)\rightarrow (\neg\square_{X,Y,Z}x= y\rightarrow\neg(x= y)). \end{align} $$

In what follows, we sometimes abbreviate $\square _{X,Y,Z,[\neg ],[=],[\square _{[\sigma ],[\sigma ],[\langle \sigma ,\sigma \rangle ]}]}$ by $\square _{*}$ and use $\square $ to abbreviate $\square _{[\sigma ],[\sigma ],[\langle \sigma ,\sigma \rangle ]}$ in subscript position. From (2), it follows by RC that

(3) $$ \begin{align} \vdash\square_{*}X(x)\wedge\square_{*}Y(y)\wedge\square_{*}Z(=)\rightarrow(\square_{*}\neg\square_{X,Y,Z}x= y\rightarrow\square_{*}\neg(x= y)). \end{align} $$

From (3) by CL,

(4) $$ \begin{align} \vdash\square_{*}X(x)\wedge\square_{*}Y(y)\wedge\square_{*}Z(=)\rightarrow(\neg\square_{*}\neg(x= y)\rightarrow\neg\square_{*}\neg\square_{X,Y,Z}x= y). \end{align} $$

By Monotonicity, the essentialist 5-schema, and Decomposition:

(5) $$ \begin{align} \vdash [x]\approx X\wedge [y]\approx Y\rightarrow(\neg\square_{X,Y,Z,[\neg],[=],[\square]}\neg\square_{X,Y,Z}x= y\rightarrow\square_{X,Y,Z}x= y). \end{align} $$

By Rigidity and Monotonicity:

(6) $$ \begin{align} \vdash [x]\approx X\wedge [y]\approx Y\wedge [=]\approx Z\rightarrow( \square_{*}X(x)\wedge\square_{*}Y(y)\wedge\square_{*}Z(=)). \end{align} $$

So by (4)–(6), (T), and CL:

(7) $$ \begin{align} \vdash [x]\approx X\wedge [y]\approx Y\wedge [=]\approx Z\rightarrow(\neg\square_{X,Y,Z,[\neg],[=],[\square]}\neg(x= y)\rightarrow x= y)). \end{align} $$

Monotonicity then yields:

(8) $$ \begin{align} \vdash [x]\approx X\wedge [y]\approx Y\wedge [=]\approx Z\rightarrow(\neg\square_{[x],[y],[=],[\neg],[\square]}\neg(x= y)\rightarrow x= y)). \end{align} $$

By R-Comp and CL, we then obtain

(9) $$ \begin{align} \vdash\neg (x= y)\rightarrow\square_{[x],[y],[=],[\neg],[\square]}\neg(x= y). \end{align} $$

(iii) Similar to the proof of part (ii), using Rigidity in place of part (i) in the first step.

Parts (i) and (ii) are essentialist analogs of the necessity of identity and distinctness. A stronger form of (i) to the effect that if x and y are identical, then it lies in the nature of x (or y) that they are identical is not derivable in HLE, as can be shown by the semantic methods developed in §6. The strengthenings of parts (ii) and (iii) where the essentialist operator is deleted from the subject of the essentialist operator in the consequent do not seem to be provable in HLE, although they are valid in the models developed in later sections. These strengthenings do strike me as very plausible, at least if the essentialist 5-schema is adopted. However, I have preferred not to take them as axioms here, in part because it is instructive to see that the weaker principles are already theorems of HLE. We could in principle have adopted a similar attitude with regard to the essentialist 4-schema, a weaker version of which is also derivable from the other principles of the system.Footnote ²⁷ But it seems preferable to adopt (4) as an axiom, both because there is a direct and compelling justification for it and because it is more evident than the 5-schema which is needed for the derivation of the weaker version.

Our next result states that every essentialist statement is equivalent to an essentialist statement whose subject is a collection of propositions. To state the result, we introduce the following abbreviation. Let $F_{1},\ldots ,F_{n}$ be one-place predicates. Then,

$$ \begin{align*} \mathfrak{h}(F_{1},\ldots,F_{n}) &=_{\text{df}}\lambda p^{\langle\rangle}.(\exists x(F_{1}(x)\wedge\exists X^{[\sigma_{1}]}(\forall z(X(z)\leftrightarrow x=_{\sigma_{1}}z)\wedge p= X(x))) \vee\dots\vee\\ &\quad (\exists x(F_{n}(x)\wedge\exists X^{[\sigma_{n}]}(\forall z(X(z)\leftrightarrow x=_{\sigma_{n}}z)\wedge p= X(x))). \end{align*} $$

The predicate $\mathfrak {h}(F_{1},\ldots ,F_{n})$ picks out the haecceity propositions of the entities picked out by $F_{1},\ldots ,F_{n}$ .

The following are immediate consequences of Proposition 7.

Proposition 8(i) says that for the essences of two collections of entities to be the same, it suffices that their collective natures involve the same propositions. Proposition 8(ii) says that if the natures of two collections involve the same propositions, then for any type $\sigma $ , they involve the same entities of that type; or more briefly, for the natures of two collections to involve exactly the same entities (of any type), it is sufficient that they involve the same propositions. It is worth noting that the converse of Proposition 8(ii) is not provable for every type $\sigma $ . For example, the nature of the collection of all objects and the nature of the collection of all propositions each involve all objects but may fail to involve the same propositions, because there may be no objects whose nature involves negation, for instance. (The consistency of this can be easily demonstrated using the semantics developed in §6.) This counterexample to the converse of Proposition 8(ii) is specific to the case where $\sigma $ is e, however. It is less clear whether there are counterexamples to the instances of the converse of Proposition 8(ii) in which $\sigma $ is not e or $\langle \rangle $ . These instances do not seem to be provable either, although they are valid on the semantics developed in §6.

5 The logic of necessity—Part I

This section shows how we can develop the logic of metaphysical necessity within HLE. Fine [Reference Fine20] proposed that metaphysical necessity is a special case of essence. His proposal is that for a proposition to be metaphysically necessary is for it to be true in virtue of the nature of all objects. We can use the operator $\square _{\lambda x^{e}.x=_{e}x}$ to formally express truth in virtue of the nature of all objects. In the system E5 of Fine’s LE, the propositional logic of this operator (more precisely, its analog in the language of LE) is exactly S4.Footnote ²⁸ Fine also shows how to extend the system E5 in such a way that the logic of $\square _{\lambda x^{e}.x=_{e}x}$ is S5.Footnote ²⁹ By contrast, in HLE the operator does not even obey S4 without adding further principles. The reason is that HLE does not require the collective nature of all objects to involve any of the logical operations, or any other entities expressed by constants of type $\neq e$ . This implies that in HLE, $\square _{\lambda x^{e}.x=_{e}x}$ is not even closed under unrestricted consequence. However, there is another way of reducing metaphysical necessity to essence that suggests itself in our higher-order setting. Instead of taking metaphysical necessity to be truth in virtue of the nature of all objects, we may identify it with truth in virtue of the nature of all propositions. We can use the operator $\square _{\lambda p^{\langle \rangle }.p=_{\langle \rangle }p}$ to formally express truth in virtue of the nature of all propositions. It turns out that this operator does obey exactly the principles of S4 already in HLE. This section first demonstrates this fact and then goes on to show how the system can be extended so that $\square _{\lambda p^{\langle \rangle }.p=_{\langle \rangle }p}$ obeys exactly S5. After that, we will consider ways of extending HLE so as to make $\square _{\lambda x^{e}.x=_{e}x}$ obey S4 and S5, respectively. A detailed discussion of the philosophical ramifications of the results in this section can be found in Ditter [Reference Ditter15].

In what follows, we will use the following abbreviations: $\Omega :=\lambda x^{e}.x=_{e} x$ ; $\Pi :=\lambda p^{\langle \rangle }.p=_{\langle \rangle } p$ . We start by observing that the following essentialist analog of the K-principle from modal logic is an immediate consequence of RC.

Next we show that the propositional logic of $\square _{\Pi }$ is at least S4 in $\mathsf {HLE}$ . We first note that if $\square _{\Pi }$ is taken to express metaphysical necessity, then the principle that essence implies necessity is a theorem of HLE.

The principles of S5 become provable once we add an axiom to the effect that it is true in virtue of the nature of all propositions that they are all the propositions:

1. (DOM^⟨⟩) $\forall pX^{[\langle \rangle ]}(p)\rightarrow \square _{X}\forall pX(p).$

The same principle restricted to objects (entities of type e) is invoked by Fine [Reference Fine21] in LE to show that the operator that expresses truth in virtue of the nature of all objects satisfies all the principles of S5 in his system E5⁺ (see Theorem 4 in Fine [Reference Fine21]). We write $\vdash _{+}$ to indicate theoremhood in the system $\mathsf {HLE+DOM^{\langle \rangle }}$ .

In §9.1 we will show by semantic techniques that the propositional logic of $\square _{\Pi }$ is exactly S4 (S5) in $\mathsf {HLE}$ ( $\mathsf {HLE+DOM^{\langle \rangle }}$ ).

We now turn to the logic of $\square _{\Omega }$ . As mentioned above, $\square _{\Omega }$ does not obey S4 in HLE, since it is not even closed under unrestricted logical consequence. One way to ensure that $\square _{\Omega }$ obeys S4 is to add a principle to HLE to the effect that every entity of type $\neq e$ is involved in the nature of some objects (entities of type e). There are several ways to motivate such a principle, though perhaps the most obvious option is to endorse some form of Platonism, positing for every entity of type $\neq e$ some objectual surrogate or Form of type e that stands in a specific relation to the higher-order entity. Here I will focus on a fairly minimalist, “Propositional Platonist” theory that posits objectual surrogates for propositions:

1. (P-Truth) $\forall p^{\langle \rangle }\exists x^{e}\square _{[x]}(true(x)=_{\langle \rangle }p).$

It turns out that this suffices for the purpose at hand. Supplementing HLE with P-Truth allows us to infer EN $^{\square _{\Omega }}$ . Once we have EN $^{\square _{\Omega }}$ , the analogs for $\square _{\Omega }$ of the preceeding results are proved exactly like those for $\square _{\Pi }$ , except that we use EN $^{\square _{\Omega }}$ and RN $^{\square _{\Omega }}$ instead of EN $^{\square _{\Pi }}$ and RN $^{\square _{\Pi }}$ in the relevant places. We write $\vdash _{*}$ for provability in HLE+P-Truth.

As in the case of $\square _{\Pi }$ , we obtain S5 for $\square _{\Omega }$ if we add an axiom to the effect that it is true in virtue of the nature of all objects that they are all the objects:

1. (DOM) $\forall xX^{[e]}(x)\rightarrow \square _{X}\forall xX(x).$

We write $\vdash _{*+}$ for provability in HLE+P-Truth+DOM.

We will show in §9.2 that the propositional logic of $\square _{\Omega }$ is exactly S4 (S5) in HLE+P-Truth (HLE+P-Truth+DOM).

6 Semantics

We now proceed to the development of a sound model-theoretic semantics for HLE. The basic structure of the model theory bears some similarity to the structure of the model theory of intensional type theory with varying domains developed in Fine [Reference Fine19] and Fritz and Goodman [Reference Fritz and Goodman24]. We start with a frame, an ordered pair $\langle W,D\rangle $ consisting of a non-empty set W and a domain function D taking each member w of W and $\tau $ of the set of types $\mathcal {T}$ into some non-empty set $D^{w}_{\tau }$ . The set W can informally be thought of as the set of worlds, and the sets $D^{w}_{\tau }$ as the domains of entities of type $\tau $ in w. A characteristic feature of the model theory developed here concerns the structure of the elements of the domains $D^{w}_{\tau }$ . The frames we are interested in are ones in which every element of a domain is an ordered pair whose first component can be informally thought of as the “worldly” content and whose second component as the “intensional” content of the entity. In the case of entities of type $\langle \rangle $ the second component is just a subset of W, whereas for entities of type $\langle \tau \rangle $ it is a function from entities of type $\tau $ to subsets of W; there are no constraints on what the second component is for entities of type e. The worldly content of each ordered pair in the domain is provided by a complete lattice, allowing the worldly content of each entity of any type to freely combine with the worldly content of any entity of arbitrary type. For example, the worldly content of the denotation of the sentence ‘Socrates is human’ will consist of the (lattice-theoretic) join of the worldly contents of the denotations of ‘Socrates’ and ‘human’.

An essentialist statement $\square _{F_{1},\ldots ,F_{n}}\phi $ is true at a world w just in case $\phi $ is true at every world whose domain contains the union of the extensions of $F_{1},\ldots ,F_{n}$ at w. For example, the sentence $\square _{[s], [\wedge ]}\forall x(x=_{e}x)$ (‘It is essential to Socrates and conjunction that everything is self-identical’) is true at w just in case $\forall x(x=_{e}x)$ is true at every world whose domain contains the denotations of s and $\wedge $ . Crucially, the models allow that the denotation of a constant of any type can in principle fail to exist at any world. Thus, for example, the denotations of s and $\wedge $ need not exist in the same worlds. The denotation of a syntactically complex term A is in the domain of a world just in case all of the denotations of its constants and free variables (relative to an assignment) are in the domain of that world. Thus, the denotation of $\forall x(x=_{e}x)$ exists in exactly those worlds in which both the denotations of $\forall _{e}$ and $=_{e}$ exist, though it fails to exist in a world in which only one of them exists.

An important consequence of the semantics is that a collection of entities $\mathbf {x}_1,\mathbf {x}_2,\ldots $ essentially involves an entity $\mathbf {y}$ if and only if $\mathbf {y}$ exists in every world in which all of $\mathbf {x}_1,\mathbf {x}_2,\ldots $ exist. Thus the relation of essential involvement expressed by the defined predicates $\succeq _{\sigma _{1},\ldots ,\sigma _{n},\tau }$ is modeled by a relation between the existence sets of entities: $(F_1,\ldots ,F_n)\succeq _{\sigma _1,\ldots ,\sigma _n,\tau }x$ is true at w just in case the denotation of x exists in every world whose domain contains the union of the extensions of $F_{1},\ldots ,F_{n}$ at w. So, for example, $([s])\succeq \neg $ (or, equivalently, $s\geq \neg $ ) is true at w just in case the denotation of $\neg $ exists in every world in which the denotation of s exists.

Another distinctive feature of the semantics is that formulas are only evaluated at those worlds at which their denotations exist. This is compatible with classical logic given the notion of validity adopted here, according to which a formula $\phi $ is valid if and only if for every model, $\phi $ is true at every world at which its denotation exists.Footnote ³⁰ The present semantics is similar in this respect to the first-order semantics for LE in Fine [Reference Fine22], in which a formula is evaluated only at those worlds at which the objectual content of the formula exists, where the objectual content of a formula consists, intuitively, of the objects that the formula is about. Our notion of worldly content is a generalization of the notion of objectual content within a higher-order setting. One consequence of Fine’s semantics is that a sentence like $\forall x(\text {set}(x)\vee \neg \text {set}(x))$ , which has no objectual content on Fine’s semantics, is true at every world in every model, which, given Fine’s interpretation of the essentialist operator, entails that $\forall x(\text {set}(x)\vee \neg \text {set}(x))$ is true in virtue of the nature of any particular object. The present semantics, by contrast, allows us to avoid this, because the denotation of $\forall x(\text {set}(x)\vee \neg \text {set}(x))$ may fail to be in the domain of some world, since all of the denotations of $\forall _{e}$ , $\text {set}$ , $\vee $ and $\neg $ may fail to be in the domain of some world, as explained above. I will now turn to the details.

When $\mathcal {N}=\langle N,\sqsubseteq \rangle $ is a complete lattice, we usually indicate membership in the lattice by simply writing $x\in \mathcal {N}$ instead of $x\in N$ . A canonical example of a complete lattice is the power set of a given set, ordered by inclusion, with the join and meet given by union and intersection of subsets, respectively. We will make use of such power set lattices in the construction of our example models in §7. We now define the “worldly content structures” that provide the basis of the worldly contents of the elements of the frames we are primarily interested in.

Elements of the set $\bigcup _{w\in W} I^{w}$ will serve as the second components of the members of the set of individuals in the frames defined below. The function $\overline {E}$ assigns every element of the lattice to a set of worlds which can be thought of as the “existence set” of the element. By the last part of clause (v), we have for every $o\in \bigcup _{w\in W} I^{w}: o\in I^{w}$ if and only if $w\in \overline {E}(\iota (o))$ . In the frames defined below, this means that the existence set of an individual $\langle \iota (o),o\rangle $ is just $\{w\in W: o\in I^w\}$ . If a lattice element n is below a lattice element m in the lattice ordering, then the existence set of n is a superset of the existence set of m. We record this useful fact in the following lemma.

Here are two simple examples of worldly content structures. We will use variants of these structures in our model constructions in §7.

As mentioned above, the frames we are primarily interested in are ones in which the elements of domains of all types are ordered pairs. Where x is an ordered pair, we denote its first component by $\pi _{0}(x)$ and its second component by $\pi _{1}(x)$ . In the frames we define below, every element of the outer domain is an ordered pair whose first component is an element of a complete lattice drawn from a worldly content structure underlying the frame. We will later show that the existence set of an element of the outer domain of a frame is just the value of $\overline {E}$ applied to its first component.

Conditions (iii) and (v) of Definition 8 jointly ensure that for any world w in a frame, the domain of individuals $D_{e}^{w}$ is non-empty. For entities of type $\neq e$ this is guaranteed by the condition that $\overline {E}(\bot )=W$ . As the next lemma shows, the existence set $E(\{\mathbf {x}\})$ of an element of the outer domain of a frame is identical to the existence set $\overline {E}(\pi _{0}(\mathbf {x}))$ of its first component; more generally, we have $E(\{\mathbf {x}_{1},\ldots ,\mathbf {x}_{n}\}=\overline {E}(\bigsqcup \{\pi _{0}(\mathbf {x}_{1}),\ldots ,\pi _{0}(\mathbf {x}_{n})\})$ . Thus the existence set of an element of the outer domain of a frame is uniquely determined by the worldly content structure underlying the frame.

An important feature of clause (ii) is that the intensional content $\pi _{1}(\mathbf {x})$ of an element of the domain of propositions $D_{\langle \rangle }$ is a subset of its existence set $E(\{\mathbf {x}\})$ . We can think of the intensional content of a proposition as the set of worlds at which the proposition is true. The condition that the intensional content of a proposition should be a subset of its existence set therefore entails that a proposition can only be true at those worlds at which it exists. Similarly, we can think of the intensional content $\pi _{1}(\mathbf {X})$ of a non-rigid property or relation (i.e., an element of some $D_{\langle \tau _{1},\ldots ,\tau _{k}\rangle }$ ) specified in clause (iii) as the function that takes entities of appropriate type to the set of worlds at which the entities have the property or relation in question. For example, if $\mathbf {X}\in D_{\langle \tau \rangle }$ and $\mathbf {x}\in D_{\tau }$ , then $\pi _{1}(\mathbf {X})(\mathbf {x})$ represents the set of worlds at which $\mathbf {x}$ has property $\mathbf {X}$ , or in other words, the intensional content of the proposition that $\mathbf {x}$ has $\mathbf {X}$ . Clause (iii) guarantees that the set of worlds $\pi _{1}(\mathbf {X})(\mathbf {x})$ is a subset of the set $E(\{\mathbf {X},\mathbf {x}\})$ , the set of worlds at which both $\mathbf {X}$ and $\mathbf {x}$ exist. Thus an entity can only instantiate a property at those worlds at which both the entity and the property exist.

Unlike the intensional content $\pi _{1}(\mathbf {X})$ of a non-rigid property, the intensional content of a rigid property (i.e., an element of some $D_{[\tau ]}$ ) is simply its extension, i.e., a subset of the domain of entities of appropriate type. For example, if $\mathbf {X}\in D_{[\tau ]}$ , then $\pi _{1}(\mathbf {X})$ is a subset of $D_{\tau }$ , and an entity $\mathbf {x}\in D_{\tau }$ has $\mathbf {X}$ if and only if $\mathbf {x}\in \pi _{1}(\mathbf {X})$ . One difference between rigid and non-rigid properties is that the extension of a rigid property always remains fixed across worlds (see Lemma 4 (iv)), whereas the extension of a non-rigid property may vary across worlds. Another difference is that the worldly content $\pi _{0}(\mathbf {X})$ of a rigid property—but not that of a non-rigid property—is determined by its intensional content $\pi _{1}(\mathbf {X})$ , and thus by the entities in its extension. In fact, the worldly content $\pi _{0}(\mathbf {X})$ of a rigid property $\mathbf {X}$ is just the lattice join of the set of worldly contents of the entities in its extension. So a rigid property exists in exactly those worlds at which the entities in its extension exist (see Lemma 4 (i)).

Next we need to check that the rigid properties behave in the intended way.

We now specify how terms of our formal language are interpreted in a dual-content frame.

An easy induction on the complexity of terms shows that the function is well-defined, such that whenever defined, is indeed in $\bigcup D$ . To construct a dual content model based on a given worldly content model we only have to assign appropriate worldly and intensional contents to every non-logical constant of our signature; in the case of logical constants we only have to assign a worldly content, because the intensional contents of the logical constants are fixed by the clauses for the logical constants. Example models will be given in §7. Note that by compositional clause 3, the denotation of a term A is defined if and only if the denotations of its free variables and constants all exist together at some world. Formulas are only evaluated at worlds at which their denotations exist. A formula $\phi $ is said to be true at relative to a model $\mathfrak {M}$ and assignment g just in case ; otherwise it is false at w. We sometimes abbreviate this by $\mathfrak {M},w,g\vDash \phi $ and $\mathfrak {M},w,g\nvDash \phi $ , respectively. A formula is neither true nor false at a world at which its denotation does not exist. In what follows, we generally omit the assignment and write instead of when A is a term without free variables. Using Definition 11, we can derive the familiar truth-clauses for the logical constants. For example, the clauses for negation, conjunction, existential quantification and the essentialist operator are as follows. Let be a dual-content model. Then:

1. For iff $\mathfrak {M},w,g\nvDash \phi .$

2. For iff $\mathfrak {M},w,g\vDash \phi $ and $\mathfrak {M},w,g\vDash \psi .$

3. For iff for some $\mathbf {x}\in D^{w}_{\tau } :\mathbf {x}\in Ext^{w}$

4. For iff for all $v\in W$ : if , then $\mathfrak {M},v,g\vDash \phi .$

In words: if is in the domain of w, then $\square _{F_{1},\ldots ,F_{n}}\phi $ is true at w (relative to g) just in case $\phi $ ist true at every world v (relative to g) whose domain contains the union of the extensions of $F_{1},\ldots ,F_{n}$ at w. For example, $\square _{\lambda p.p=_{\langle \rangle }p}H(s)$ (‘It is true in virtue of the collective nature of all propositions that Socrates is human’) is true at w just in case $H(s)$ is true at every world that contains all the propositions in w, or in other words, at every world that contains . For another example, $\square _{[s]}H(s)$ (‘It is true in virtue of the nature of Socrates that Socrates is human’) is true at w just in case $H(s)$ is true at every world at which exists.Footnote ³³ So $\square _{[s]}H(s)$ is false at w just in case either is not in the domain of every world whose domain contains , or $H(s)$ is false at some world whose domain contains . It follows from Lemma 6 below that the denotation of a term (relative to an assignment g) is in the domain of a world w just in case the denotations (relative to g) of the constants and free variables occurring in it are in the domain of w. So, for example, is in the domain of w just in case both and are in the domain of w. As we will see below, it is straightforward to construct models where the denotations of different constants have different existence sets, so that even, e.g., and may be in the domains of some, none, or all of the same worlds.

The following derived clause for $\square _{\Pi }$ deserves special mention:

1. For iff for all $v\in W$ : if $D^{w}\subseteq D^{v}$ , then $\mathfrak {M},v,g\vDash \phi $

If, as discussed in §5, $\square _{\Pi }$ expresses metaphysical necessity, then the clause says that $\phi $ is metaphysically necessary at w if and only if it is true at every world that contains $D^{w}$ , i.e., the domain of all entities that exist at w. A world v is thus metaphysically accessible from a world w if and only if $D^w\subseteq D^v$ . But even if $\square _{\Pi }$ does not express metaphysical necessity, any candidate metaphysical necessity operator would have to obey the same semantic clause, provided that essence implies necessity and the logic of metaphysical necessity is at least S4.

Our semantic clause for the essentialist operator is a higher-order generalization of the clause for the essentialist operator in Fine [Reference Fine22] except for one small but crucial difference. On Fine’s semantics, $\square _{F}\phi $ is true at a world at which the formula is defined just in case (a) for every object x in the objectual content of $\phi $ , there is some element y in the extension of F that depends on x,Footnote ³⁴ and (b) $\phi $ is true at every world that contains the extension of F.Footnote ³⁵ Part (a) of Fine’s clause ensures the validity of Localization, a principle that we have seen reason to reject. Dropping part (a) constitutes a welcome simplification of the semantic clause for the essentialist operator.Footnote ³⁶

As mentioned previously, we also have the following derived clause for the relation of essential involvement, the proof of which is supplied in Lemma 11:

1. For iff depends on .

The relation of essential involvement can thus be modeled as a relation between existence sets of entities. This is another important difference to Fine’s first-order semantics, in which essential involvement (Fine’s dependence relation) is interpreted by a primitive dependence relation in the structures which cannot be defined in terms of existence sets.Footnote ³⁷

Before we move on to some example models, we supply the definition of validity.

Note that since we always have , because a proposition can only be true at those worlds at which it exists, entails .

7 Example models

We will now construct some examples of models that show the consistency of some simple HLE-theories. The models constructed here illustrate some general methods for constructing models of more complicated theories.

In the model constructed in Example 3, neither world is metaphysically accessible from the other, because the domain of neither is a subset of the other (no element whose first component is $\{v\}$ is in $D^w$ and no element whose first component is $\{w\}$ is in $D^v$ ). We may sometimes want to construct models based on a particular metaphysical accessibility relation on the set of worlds. The next example provides a template for constructing dual-content frames given a particular metaphysical accessibility relation.

Our next example shows the consistency of the schema $\neg (A\geq _{\sigma ,\tau }B)$ , where A and B are distinct constants of a given signature. The consistency of this schema implies in particular that it is consistent to assume that the nature of no logical operation involves any other logical operation, where by ‘logical operation’ we mean ‘denotation of a logical constant’ of a given signature.

The next examples illustrate that the denotations of logically equivalent formulas may fail to be identical in a model.

The model in the previous example is one in which the domain function D is such that for all $\sigma \in \mathcal {T}$ , $D_{\sigma }^{w}=D_{\sigma }^{v}$ for all $w,v\in W$ ; the typed domains are constant across worlds. Let us call models of this kind constant domain models. Constant domain models validate the schema $A\geq _{\sigma ,\tau }B$ , which, as discussed in §4, leads to the consequence that all subscripted essentialist operators become logically coextensive. The example also shows that two formulas $\phi $ and $\psi $ may be equivalent in the class of constant domain models without $\phi =\psi $ being valid in that class. However, any constant domain model in which $\mathcal {N}$ is a trivial one-element lattice is such that if , then .

There is also a natural class of models without constant domain in which sameness of intensional content and existing at the same worlds does entail sameness of denotation. Call a dual-content model injective if it is based on a worldly content model in which the function $\overline {E}$ is injective. The models constructed in Examples 3 and 5 are examples of injective models. In an injective model, any elements of the domain that exist in the same worlds have the same worldly content.

Injective models validate some interesting interaction principles between essence and identity that are not valid in the class of all dual-content models. Among the injective dual-content models, those in which the denotations of $=$ and $\wedge $ exist at all worlds are of special interest because they validate an especially tight connection between essence and identity. For example, suppose $\square _{[x]}p$ is true at some world w for some assignment g in such a model. Then , and so $(x= x)= ((x= x)\wedge p)$ is true on g at w as well. Conversely, if $(x= x)= ((x= x)\wedge p)$ is true at w on some assignment g in such a model, then $\square _{[x]}p$ is true on g at w too, if the denotation of $\square _{\sigma }$ exists at w. Thus, the formula $\square _{[x]}p\leftrightarrow ((x= x)= (x= x)\wedge p)$ is valid in this class of models. The argument above can be easily generalized to show that $\square _{[x_{1}],\ldots ,[x_{n}]}p\leftrightarrow ((x_{1}= x_{1})\wedge \cdots \wedge (x_{n}= x_{n})= ((x_{1}= x_{1})\wedge \cdots \wedge (x_{n}= x_{n})\wedge p)$ is also valid in this class of models. Whether or not these connections are plausible is a question that deserves more careful investigation elsewhere.

8 Soundness

In this section, we prove the soundness of HLE with respect to the semantics developed in §6. We start with a few auxiliary lemmas.

Note that, by Lemmas 2, 5 and 6, a formula $\phi $ is defined in $\mathfrak {M}$ just in case exists at some world in $\mathfrak {M}$ , for some assignment g.

We will now prove that any two $\beta \eta $ -equivalent terms have the same denotation in a dual-content model.

Proof. It suffices to show that whenever A one-step $\beta \eta $ -reduces to B. The proof is by induction using the definition of one-step $\beta \eta $ -reduction. We only show , the case of being an immediate consequence of Definition 114. Base case: A immediately $\beta \eta $ -reduces to B. (i) A is of the form $\lambda x_{1}\ldots x_{n}.C(x_{1},\ldots ,x_{n})$ , where $x_{1},\ldots ,x_{n}$ are not free in C, and B is C. Let g be an arbitrary assignment. Then,

The last equation holds by Lemma 7 because none of $x_{1},\ldots ,x_{n}$ is free in C. (ii) A is of the form $(\lambda x_{1}\ldots x_{n}.\phi )(A_{1},\ldots ,A_{n})$ and B is $\phi [A_{i}/x_{i}]$ . Let g be an arbitrary assignment. Then,

(By Lemma 9)

Induction step: Suppose that $A,B\in \mathcal {L}^{\Sigma }_{\sigma }$ are such that for all assignments g. Then if $\sigma $ is of the form $\langle \sigma _{0},\ldots ,\sigma _{m}\rangle $ (and similarly if $\sigma $ is a rigid type) and $C_{0},\ldots ,C_{m}$ are of appropriate types, , for any assignment h, by Lemma 8. Moreover, if $C_{0},\ldots ,C_{m}$ are of appropriate types, then , $0\leq k\leq m$ by Lemma 8. Finally, if $\sigma $ is $\langle \rangle $ and A and B contain all of $x_{1},\ldots ,x_{n}$ free, then for all assignments g,

It is evident that the formulas that are valid in the class of dual-content models do not coincide with the provable formulas of HLE, because among the validities are all validities of full second-order logic. So there cannot be a completeness result with respect to this class of models. For present purposes, the model theory primarily serves as a tool for checking the consistency of various essentialist theories. Still, it would be interesting to develop a model theory for HLE (or some extension of it) for which the theory is complete. It is quite clear that, at least if the model theory is to take a similar form as the one proposed here, the theory HLE would have to be strengthened in order to prove completeness. For example, we would plausibly have to adopt the strengthened version of Proposition 6(ii) and (iii) discussed above as axioms, since any class of models that validates the weaker versions would plausibly also validate the stronger versions of these principles.

Apart from the question of completeness, the class of dual-content models validates some principles that are not only plausibly unprovable in HLE but are also intuitively false on their intended interpretation. For example, the models validate the claim that the nature of any entity whatsoever involves, for any given property, some property that is necessarily coextensive with it. For instance, let $\langle n,f\rangle $ be the representation of the property of being human in some model. Then for any entity, the property $\langle \bot ,f\rangle $ exists in every world in which the entity exists, because $\overline {E}(\bot )=W$ . This seems undesirable. Since it is the “fullness” of the domains that is responsible for these validities, it would seem that they would be invalidated in appropriately generalized Henkin models, in which the typed domains of a model are subsets of the domains $D^{w}_{\sigma }$ , as defined in Definition 9, rich enough to validate R-Comp and to provide an interpretation , relative to an assignment g, for every term A for which , where $A_{1},\ldots , A_{n}$ are all the free variables and constants occurring in A. Such generalized Henkin models remain to be further explored.

Among the most important extensions of HLE that remain to be explored are those which add further principles concerning the interaction between identity and other the logical constants. In HLE the only axioms that specifically concern identity ( $=$ ) are Ref, LL and R-Ext. The system is deliberately neutral about more substantive principles concerning identity, such as, for example, the principle that conjunction is commutative ( $(p\wedge q)= (q\wedge p)$ ) or more substantive interaction principles between essence and identity. While a fuller investigation of this matter is left for future work, it is interesting to observe that the following rule, which generates a number of interesting identities, preserves validity in the class of dual-content models:

1. (RI) If $\vdash \forall x_{1}\ldots \forall x_{n}(R^{\langle \sigma _{1},\ldots ,\sigma _{n}\rangle }(x_{1},\ldots ,x_{n})\leftrightarrow S^{\langle \sigma _{1},\ldots ,\sigma _{n}\rangle }(x_{1},\ldots ,x_{n}) )$ and R and S contain exactly the same free variables and constants, then $\vdash R= S.$

RI immediately delivers such principles as the commutativity of conjunction and disjunction. It is interesting to observe, however, that it does not imply idempotence ( $\phi =(\phi \wedge \phi )$ ) for conjunction or disjunction, and it does not imply involution ( $\phi =\neg \neg \phi $ ). However, we do get weakened versions of these: each of $(\phi \wedge \phi )=(\phi \wedge (\phi \wedge \phi ))$ , $(\phi \vee \phi )=(\phi \vee (\phi \vee \phi ))$ and $\neg \phi =\neg \neg \neg \phi $ are implied by RI given the other principles of HLE. The weakened versions of involution and the commutativity of conjunction and disjunction seem rather natural from an essentialist standpoint.

9 The logic of necessity—Part II

This section provides model-theoretic proofs of some results announced in §5 regarding the logics of the operators $\square _{\Pi }$ and $\square _{\Omega }$ .