2 Henkin models of second-order logic and its subsystems

We work with Henkin models of second-order logic and its subsystems. These are an important class of models since the Completeness Theorem holds for them and their validity relation is recursively enumerable.Footnote ⁸ In this section, we introduce notation for Henkin models.Footnote ⁹ Readers familiar with second-order logic may safely skip this section and come back to it as needed.

Henkin models have the form $\mathcal {A}=(A, S_1[A], S_2[A], \ldots )$ where A is a non-empty set and $S_n[A]$ is a non-empty collection of subsets of the n-th Cartesian product $A^n$ of A. The idea is that A is the first-order domain and that $S_n[A]$ is the domain reserved for the n-ary second-order entities. The membership relation is stipulated to be interpreted absolutely, i.e., $\mathcal {A}\models R\overline {a}$ iff $\overline {a}$ is in R, for an element R of $S_n[A]$ and an n-tuple $\overline {a}=(a_1, \ldots , a_n)$ from $A^n$ . A Henkin model $\mathcal {A}=(A, S_1[A], S_2[A], \ldots )$ is standard if for all $n\geq 1$ one has $S_n[A]=P(A^n)$ , i.e., the interpretation $S_n[A]$ of the range of quantifiers over n-ary relation is the full powerset of the n-th Cartesian product $A^n$ of the underlying first-order domain A, as judged by the ambient metatheory. The standard models satisfy all instances of the following comprehension schema:

(2.1) $$ \begin{align} \exists \; R \; \forall \; \overline{x} \; (R\overline{x} \leftrightarrow \varphi(\overline{x})), \end{align} $$

wherein $\varphi (x_1, \ldots , x_n)$ has free first-order variables $x_1, \ldots , x_n$ , along with perhaps other free variables (called parameters), and wherein the n-ary second-order variable R does not appear free in $\varphi (x_1, \ldots , x_n)$ .

Some minimal set-theoretic apparatus is available in any Henkin model $\mathcal {A}$ which satisfies the comprehension schema (2.1) for formulas $\varphi $ which do not contain second-order quantifiers. First, $S_n[A]$ is a Boolean algebra using the obvious formulas: e.g., treating $X,Y$ in $S_1[A]$ as parameters and using the formula $Xx\wedge Yx$ , we have a Z such that: for all x in A, one has $Zx$ iff $Xx\wedge Yx$ ; and we write this as $Z=X\cap Y$ . We likewise freely use the operations $X\cup Y, X\setminus Y, X\times Y, X^n$ with their usual meaning in the object language, and for each $n\geq 1$ we use $\emptyset $ for the empty n-ary relation (and let context determine which $n\geq 1$ is intended). To sometimes avoid having to display arities, if R is n-ary and X is unary, we stipulate

(2.2) $$ \begin{align} R\cap X := R\cap X^n. \end{align} $$

To this we add the projection functions: for n-ary R and $1\leq i\leq n$ we have

(2.3) $$ \begin{align} \pi_i R =\{ x_i : \exists \; x_1, \ldots, x_{i-1}, x_{i+1}, \ldots, x_n \; R(x_1, \ldots, x_{i-1}, x_i, x_{i+1}, \ldots, x_n) \} \end{align} $$

as well as the field $\mathscr {U} R = \bigcup _{i=1}^n \pi _i R$ of the n-ary relation R, where the scripted $\mathscr {U}$ is a mnemonic for ‘union’. We similarly define, for a tuple $\overline {R}=(R_1, \ldots , R_m)$ , the union of its fields $\mathscr {U} \overline {R}=\bigcup _{j=1}^m \mathscr {U} R_j$ . Further, for each $(n+m)$ -ary R and tuple $\overline {x}$ of length n, we have the function $(R,\overline {x})\mapsto R[\overline {x}]$ given by $R[\overline {x}]:=\{\overline {y}: R\overline {x}\,\overline {y}\}$ .Footnote ¹⁰ In addition to these operations, we use the subset relation $R\subseteq R^{\prime }$ with its usual meaning.

A further aspect of this minimal set-theoretic apparatus is that it is absolute between Henkin models and the metatheory and hence also between two Henkin models, a theme to which we will return to in §7.

Due to their absoluteness and their definability in systems which have comprehension for formulas without second-order quantifiers, we may regard ourselves as working in a definitional expansion of second-order logic which has constant, relation, and function symbols for all this minimal set-theoretic apparatus, namely: $\emptyset $ , $X\subseteq Y$ , $X\cap Y$ , $X\cup Y$ , $X\setminus Y$ , $X\times Y$ , $X^n$ , $\pi _i R$ , $\mathscr {U}\overline {R}$ , and $R[\overline {x}]$ .

However, occasionally we avail ourselves of non-absolute definitions. For instance, we occasionally use $\mathbb {V}:=\{x: x=x\}$ as an abbreviation for the universal unary relation. Likewise, we sometimes use $Y^c:=\mathbb {V}\setminus Y$ as an abbreviation for the relative complement of Y. But since these are not absolute, we do not add constants and function symbols for these and regard them as mere abbreviations.

As in other areas, it is convenient to allow constant symbols, both for individuals and for n-ary relations. As usual, what constant symbols are available in a given context is specified by the signature. For instance, in a setting where one wants names for individuals, one works in a signature with constants $c,d,\ldots $ , and any model $\mathcal {A}$ in this signature comes equipped with interpretations $c^{\mathcal {A}}, d^{\mathcal {A}}, \ldots $ from A. Likewise, in a setting where one wants named binary relations for, e.g., orders or graphs of functions, one simply works in a signature with binary relation symbols $\preceq , g, \ldots $ on individuals, and any model $\mathcal {A}$ in this signature comes equipped with interpretations $\preceq ^{\mathcal {A}}, g^{\mathcal {A}}$ which are subsets of $A^2$ . And similarly for n-ary relations for values besides $n=2$ . Since we always work with at least comprehension for formulas with no second-order quantifiers, the interpretation $C^{\mathcal {A}}$ of an n-ary relation symbol C in a Henkin model $\mathcal {A}$ is identical to an element of $S_n[A]$ , and in particular $C^{\mathcal {A}}$ is identical to the set of n-tuples $\overline {x}$ from $A^n$ which satisfy the n-ary relation C on $\mathcal {A}$ . Hence even though we allow extra-logical relation symbols beyond membership, all atomics are equivalent to those formed from membership and identity.

As a deductive theory, second-order logic $\mathsf {SOL}$ just consists of the comprehension schema (2.1) and the extensionality axioms, one for each $n\geq 1$ , where $R,S$ are n-ary:

(2.4) $$ \begin{align} \forall \; R \; \forall \; S \; \bigg(R=S\leftrightarrow \forall \; \overline{x} \; (R\overline{x}\leftrightarrow S\overline{x})\bigg) \end{align} $$

together with the explicit definitions of the minimal set-theoretic apparatus described above. One generates subsystems of second-order logic by keeping extensionality and restricting the comprehension schema. A $\Sigma ^1_1$ -formula is one of the form $\exists \; R_1, \ldots , R_{\ell } \; \varphi $ , where $\varphi $ does not contain any second-order quantifiers. A $\Pi ^1_1$ -formula is one of the form $\forall \; R_1, \ldots , R_{\ell } \; \varphi $ , where $\varphi $ does not contain any second-order quantifiers. A formula $\varphi $ of second-order logic is first-order if it does not contain any second-order quantifiers. Since the minimal set-theoretic apparatus is all first-order definable, its inclusion does not impact what is $\Sigma ^1_1$ -definable, $\Pi ^1_1$ -definable, and first-order definable.

A traditional predicative subsystem is $\Delta ^1_1$ -comprehension, abbreviated $\mathsf {\Delta ^1_1{-}CA_0^{-}}$ , which consists of the extensionality axioms and the following $\Delta ^1_1$ -comprehension schema:

(2.5) $$ \begin{align} \forall \; \overline{x} \; (\varphi(\overline{x}) \leftrightarrow \psi(\overline{x}))\rightarrow \exists \; R \; \forall \; \overline{x} \; (R\overline{x} \leftrightarrow \varphi(\overline{x})), \end{align} $$

wherein $\varphi $ is $\Sigma ^1_1$ and $\psi $ is $\Pi ^1_1$ . Another traditional predicative system is $\Sigma ^1_1$ -choice, abbreviated $\mathsf {\Sigma ^1_1{-}AC_0^{-}}$ which consists of the extensionality axioms, the comprehension schema (2.1) restricted to first-order formulas, and the following $\Sigma ^1_1$ -choice schema:

(2.6) $$ \begin{align} \forall \; \overline{x} \; \exists \; R \; \varphi(\overline{x}, R)\rightarrow \exists \; R^{\prime} \; \forall \; \overline{x} \; \varphi(\overline{x}, R^{\prime}[\overline{x}]), \end{align} $$

wherein $\varphi (\overline {x}, R)$ is $\Sigma ^1_1$ and $R^{\prime }$ is $(n+m)$ -ary and $\overline {x}$ is n-ary and R is m-ary. Finally, the system $\mathsf {ACA_0^{-}}$ is simply the extensionality axioms and all instances of the comprehension schema (2.1) such that the formula $\varphi $ does not contain second-order quantifiers. In addition to these three predicative theories, a traditional intermediary impredicative system is $\mathsf {\Sigma ^1_1{-}CA_0^{-}}$ , which consists of the extensionality axioms and the comprehension schema (2.1) for $\Sigma ^1_1$ -formulas. The names of these subsystems of $\mathsf {SOL}$ are taken from the corresponding well-studied theories of second-order arithmetic [Reference Simpson55], but we have added a minus superscript to them to disambiguate and to indicate that they are logical rather than arithmetical in nature.

In any of these systems of second-order logic, one can readily regiment statements about cardinality. Whereas in ordinary set-theoretic mathematics, the cardinality operator $\left |\cdot \right |$ is function from sets to cardinals, in second-order logic one defines the comparative notions $\left |X\right |\leq \left |Y\right |$ and $\left |X\right |< \left |Y\right |$ explicitly as follows.

Of course, functions $f:X\rightarrow Y$ are identified with their graph as a subrelation of $X\times Y$ . Being able to define such notions is one of the chief advantages of working with polyadic as opposed to monadic second-order logic: namely, quantification over binary relations allows one to quantify over functions between different unary second-order entities in addition to being able to quantify over the second-order entities themselves. That said, there are other perspectives from which polyadic second-order quantification is a disadvantage. For instance, plurals are thought to be monadic and not polyadic,Footnote ¹¹ and there are many decidability methods for monadic second-order theories which have no polyadic analogue.Footnote ¹²

3 Generalized quantifiers: Definition and examples

Second-order logic is an extremely expressive system, even when comprehension is restricted. It is thus a good framework to study generalize quantifiers, since the truth conditions for these quantifiers are often given by a set-theoretic formula which can be readily transcribed into the object language of second-order logic. Thus, in this paper, we define:

This is abbreviated by saying that the generalized quantifier $Q(R_1, \ldots , R_{\ell })$ is of type $( n_1, \ldots , n_{\ell })$ . If the particular values of $n_1, \ldots , n_{\ell }$ are not important, such as when one is just stating general claims about all generalized quantifiers, one further abbreviates this by saying that $Q(\overline {R})$ is of type $( \overline {n})$ , or for simplicity that $Q(\overline {R})$ is of type $\overline {n}$ . Note that the type $\overline {n}$ specifies both the length of the sequence $\overline {R}$ as well as the arities of its individual entries. In a Henkin model $\mathcal {A}$ , one correspondingly works with elements $(R_1, \ldots , R_{\ell })$ of $S_{n_1}[A]\times \cdots \times S_{n_{\ell }}[A]$ , and we similarly just abbreviate this as the element $\overline {R}$ of $S_{\,\overline {n}\,}[A]$ .

The tradition of generalized quantifiers typically works with standard models. If $Q(R_1, \ldots , R_{\ell })$ is a generalized quantifier in the sense of Definition 3.1 and $\mathcal {M}$ is a standard model, then the interpretation of $Q(R_1, \ldots , R_{\ell })$ on $\mathcal {M}$ is simply given by a subset of $P(M^{n_1})\times \cdots \times P(M^{n_{\ell }})$ . Indeed, this is the traditional understanding of the semantic value of a generalized quantifier within linguistics, namely, it is a collection of tuples of subsets of Cartesian products of the underlying domain. In restricting to collections which are defined by second-order formulas, Definition 3.1 is more restrictive than this tradition; although, as the subsequent list of examples suggests, nothing is lost by this restriction. In being applicable to Henkin models which are not standard models, Definition 3.1 is more expansive than this tradition. It is a chief burden of this paper to make a case that this is a fruitful expansion.

Here are some famous examples of generalized quantifiers of type $(1,1)$ , where we list their truth conditions followed by an example.

Of course, in the tradition of Aristotelian logic, one defines the universal so that it has existential import.Footnote ¹³

Other familiar type $(1,1)$ generalized quantifiers are the following.Footnote ¹⁴

In this, we use the regimentation of $\left |X\right |>\left |Y\right |$ from Definition 2.2.

But there are many examples of natural quantifiers of different type than $(1,1)$ . Here is an example of a type (1) quantifier associated with Montagovian individuals.Footnote ¹⁵

Here is an example of type $(1,1,1)$ .Footnote ¹⁶

Here is an example of type $(2,2)$ .Footnote ¹⁷

Lastly, consider the following generalized quantifier of type $(1,1,2)$ .Footnote ¹⁸

This is Barwise’s template for Hintikka’s branching quantifiers, where the generalization emerges by replacing one or both instances of $\mathsf {Most}$ by other type $(1,1)$ generalized quantifiers and perhaps adding more than two of them or just one of them (cf. (7.4)). Barwise’s example is intended to have the stated truth conditions, and hence it is distinct from $\mathsf {Usually}(F\times G,R)$ . However, it seems plausible that the truth conditions of Barwise’s example derive from the reciprocal “one another” or the “all” in the verb phrase,Footnote ¹⁹ and hence for reasons ultimately extrinsic to the generalized quantifier in the noun phrase. There is then some reason to regard Example 3.10 as slightly set apart from the previous examples.Footnote ²⁰

4 Substructures and Henkin domain independence

In formal semantics, one of the main conditions on natural language quantifiers is the domain independence condition,Footnote ²¹ which says that the quantifier is both upwards and downwards invariant under expansions of the first-order domain. The natural notion of expansion in Henkin models is the model-theoretic notion of substructure.

If the two models are standard and one is working in signature with no extra-logical constants, then this happens iff $A\subseteq B$ . But note that if $\mathcal {A}$ is merely a Henkin model, then specifying an extension requires not only specifying a superset B of A but also an associated sequence of supersets $S_n[B]$ of $S_n[A]$ which satisfy $S_n[B]\subseteq P(B^n)$ .

In this case, the superstructure is formed by keeping the first-order domain the same and by expanding the domains associated with the higher-order objects by recourse to a metatheoretic operation. Another kind of example is where we shrink the first-order domain and correspondingly take the second-order objects to be their intersection with the new first-order domain.

Note that in this case, if $\mathcal {B}$ is standard then $\mathcal {A}$ is standard.

For some concrete examples of pairs of Henkin models $\mathcal {A},\mathcal {B}$ where $\mathcal {A}$ is a substructure of $\mathcal {B}$ and $A=B$ while $S_n[A]\subsetneq S_n[B]$ for $n\geq 1$ , see Examples 6.2 and 6.3 in §6. One can also produce such examples by using the Downward Löwenheim–Skolem Theorem.Footnote ²²

We draw attention to a feature of many theories which is related to internal substructures.

Many logical theories, including $\mathsf {ACA}_0^{-}$ , $\mathsf {\Delta ^1_1{-}CA}_0^{-}$ , $\mathsf {\Sigma ^1_1{-}AC_0^{-}}$ , $\mathsf {\Sigma ^1_1{-}CA_0^{-}}$ and $\mathsf {SOL}$ , are closed downward under internal substructures. This is because the formula $\varphi $ in the Comprehension Schema (2.1) is allowed to use A as a parameter and hence one can bind all the quantifiers in $\varphi $ to A without increasing the relevant kinds of complexity (cf. Definition 7.2 for the definition of binding).

We now define the generalization of domain independence to Henkin models.

Definition 4.5 (Henkin independence notions).

Suppose that T is a theory and $Q(\overline {R})$ is a generalized quantifier. Then we say that Q is Henkin upwards independent relative to T if for all pairs of Henkin models $\mathcal {A}, \mathcal {B}$ of T, if $\mathcal {A}$ is a substructure of $\mathcal {B}$ , then for all $\overline {R}$ from $S_{\,\overline {n}\,}[A]$ :

(4.1) $$ \begin{align} \mathcal{A}\models Q(\overline{R}) \Longrightarrow \mathcal{B}\models Q(\overline{R}). \end{align} $$

If the arrow is reversed, we say that Q is Henkin downwards independent relative to T. If the biconditional holds, then we say that Q is Henkin domain independent relative to T. If T is clear from context, then we omit “relative to T” for brevity.

We look at simple examples which violate Henkin domain independence in §6. We discuss the motivations for Henkin domain independence in §10.

5 Henkin conservativity

In order to define conservativity,Footnote ²³ one has to work with generalized quantifiers $Q(\overline {R},\overline {R}^{\prime })$ where the free variables are split into two non-empty parts $\overline {R},\overline {R}^{\prime }$ . To avoid a great deal of notational clutter, we are going to suppose that the tuple $\overline {R}$ comes before the tuple $\overline {R}^{\prime }$ in the enumeration of the free variables in $Q(\overline {R},\overline {R}^{\prime })$ , although as one will see that nothing in what follows depends on this.Footnote ²⁴ We define:

Definition 5.1 (Henkin conservative).

Suppose that T is a theory and $Q(\overline {R},\overline {R}^{\prime })$ is a generalized quantifier. Then we say that Q is Henkin conservative relative to T if T proves

(5.1) $$ \begin{align} \forall \; \overline{R} \; \forall \; \overline{R}^{\prime} \; \big(Q(\overline{R}, \overline{R}^{\prime})\leftrightarrow Q(\overline{R}, \mathscr{U}\overline{R}\cap \overline{R}^{\prime})\big). \end{align} $$

If T is clear from context, then we omit “relative to T” for brevity.

In equation (5.1), the notation $\mathscr {U}\overline {R}\cap \overline {R}^{\prime }$ means the tuple of the same length as $\overline {R}^{\prime } = (R_1^{\prime }, \ldots , R_{\ell ^{\prime }}^{\prime })$ whose i-th component is $\mathscr {U}\overline {R}\cap R_i^{\prime }$ , where we use the convention from (2.2).

The content of Henkin conservation is best understood by way of example.

To spell this out with the sentence from Example 3.6, the biconditional $Q(X,Y)\leftrightarrow Q(X,X\cap Y)$ is: most French majors go to study abroad iff most French majors are French majors who go to study abroad. The intuitiveness of biconditionals like this is some of the primary evidence for conservativity.Footnote ²⁵

We discuss further the motivations for Henkin conservativity in §10.

Henkin conservativity has the following consequence for the empty relation.

Remark 5.5 (Henkin conservativity and the empty relation).

Suppose that $Q(\overline {R},\overline {R}^{\prime })$ is Henkin conservative relative to T. Then T proves all three of the following:

$$ \begin{align*} \forall \;\overline{R}^{\prime} \; \big(Q(\emptyset, \overline{R}^{\prime})\leftrightarrow Q(\emptyset,\emptyset)\big), \hspace{5mm} Q(\emptyset,\emptyset)\leftrightarrow \forall \;\overline{R}^{\prime} \; Q(\emptyset, \overline{R}^{\prime}), \hspace{5mm} Q(\emptyset,\emptyset)\leftrightarrow \exists \;\overline{R}^{\prime} \; Q(\emptyset, \overline{R}^{\prime}). \end{align*} $$

The first of these follows because $\overline {R}$ is empty iff its field $\mathscr {U}\overline {R}$ is empty, and the intersection of anything with the empty relation is the empty relation. The other two follow from the first by elementary quantifier manipulations.

6 The disjunctive examples

Below are three examples of quantifiers that are not Henkin domain independent but are Henkin conservative. This kind of example is well-known.Footnote ²⁶ We elaborate on it by producing examples where the disjunctive behavior pertains to both the first- and second-order part.

Example 6.3 (Disjunctive Example 3).

Let $\varphi _{PA^2}$ say that there is a model of second-order Peano arithmetic:

$$ \begin{align*} \exists \; z \; & \exists \; N \; \exists \; S \; \\ \&\; &\forall \; x \; (Nx \rightarrow \exists \; ! \; y \; (Ny \wedge Sxy)) \\ \&\; & \forall \; x \; \forall \; y \; \forall \; w \; \big((Nx\wedge Ny\wedge Nw \wedge Sxw\wedge Syw)\rightarrow x=y\big) \\ \&\; & \forall \; x \; (Nx\rightarrow \neg Sxz) \\ \&\; & \forall \; X \; \bigg(\big(X\subseteq N \; \& \; Xz \; \& \; \big(\forall \; x \; \forall \; y \; ((Nx \; \& \; Ny \; \& \; Sxy \; \& \; Xx)\rightarrow Xy)\big)\big)\rightarrow N\subseteq X\bigg). \end{align*} $$

Let $Q(X,Y)$ be $(\neg \varphi _{PA^2} \rightarrow X\cap Y\neq \emptyset )\wedge ( \varphi _{PA^2} \rightarrow X\subseteq Y)$ . That is, $Q(X,Y)$ is the type $(1,1)$ quantifier which is the existential quantifier if the domain does not contain a model of second-order Peano arithmetic, and which is the universal quantifier if the domain does contain a model of second-order Peano arithmetic.

Consider the Henkin model $\mathcal {A}$ such that A is uncountable and $S_n[A]$ is the collection of first-order definable subsets of A as a first-order structure in empty signature. Consider the standard model $\mathcal {B}$ such that B is A. Then $\mathcal {A}$ is a model of $\mathsf {\Delta ^1_1{-}CA}_0^{-} + \neg \varphi _{PA^2}$ ,Footnote ²⁸ while $\mathcal {B}$ is a model of $\mathsf {\Delta ^1_1{-}CA}_0^{-}+\varphi _{PA^2}$ .

The model $\mathcal {A}$ is the setting where the only properties are finite and cofinite collections of objects, and $\mathcal {B}$ is the setting where a richer array of properties exist.

As in the previous example, $Q(X,Y)$ is neither upwards nor downwards Henkin independent relative to $\mathsf {\Delta ^1_1{-}CA}_0^{-}$ , but it is Henkin conservative relative to any theory extending $\mathsf {ACA}_0^{-}$ .

7 Definability, bounded definability, and absoluteness

In this short section, we say what it means for a generalized quantifier itself to be definable and bounded definable, and we discuss the absoluteness of certain definitions.

Definition 7.1 (Definability relative to T).

Suppose that T is a theory and $Q(\overline {R})$ is a generalized quantifier. Then Q is $\Sigma ^1_1$ -definable relative to T if there is a $\Sigma ^1_1$ -formula $\Phi (\overline {R})$ with exactly the free variables displayed such that T proves

(7.1) $$ \begin{align} \forall \; \overline{R} \; (Q(\overline{R})\leftrightarrow \Phi(\overline{R})). \end{align} $$

Further, $\Pi ^1_1$ -definable relative to T is defined exactly the same, but with the $\Sigma ^1_1$ -formula replaced by a $\Pi ^1_1$ -formula. And first-order definable relative to T is defined exactly the same but with $\Sigma ^1_1$ -formula replaced by a first-order formula. Finally Q is $\Delta ^1_1$ -definable relative to T if it is both $\Sigma ^1_1$ - and $\Pi ^1_1$ -definable relative to T. If T is clear from context, then we omit “relative to T” for brevity.

We need a similar notion of bounded quantifiers and bounded definability.

Obviously, $\varphi ^X$ as in (3) is bound in the sense of (2); and any formula bound in the sense of (2) is of the form $\varphi ^X$ .

Here are three examples.

However, we note the following, which is the familiar problem of predicate logic on empty domains.Footnote ²⁹ $^{{,\;}}$ Footnote ³⁰

Despite this poor behavior in this limiting case, binding notions are important for their absoluteness properties. We encountered such properties already in Proposition 2.1. It is a recurring theme in mathematical logic: for instance, bounded formulas are absolute between transitive models of set theory, and recursive formulas are absolute between models of first-order Peano arithmetic.Footnote ³¹ The following is the version of absoluteness which is useful in the setting of second-order logic.

Theorem 7.7 (Grades of absoluteness and bounded definability).

Suppose that $\mathcal {A},\mathcal {B}$ are two Henkin models of $\mathsf {ACA}_0^{-}$ with $\mathcal {A}$ a substructure of $\mathcal {B}$ . Suppose that $\varphi (\overline {x},\overline {R})$ is a formula with free variables among $\overline {x},\overline {R}$ , and suppose X is a unary term. Suppose that $\overline {x},\overline {R}$ are from $\mathcal {A}$ and the free variables of X are from $\mathcal {A}$ and are not bound in $\varphi (\overline {x},\overline {R})$ .

If $\varphi (\overline {x},\overline {R})$ is $\Sigma ^1_1$ , then

(7.2) $$ \begin{align} \mathcal{A}\models \varphi^X(\overline{x},\overline{R}) \Longrightarrow \mathcal{B}\models \varphi^X(\overline{x},\overline{R}). \end{align} $$

If $\varphi (\overline {x},\overline {R})$ is $\Pi ^1_1$ then the arrow is reversed. If $\varphi (\overline {x},\overline {R})$ is equivalent to a single $\Sigma ^1_1$ -formula on both $\mathcal {A},\mathcal {B}$ and a single $\Pi ^1_1$ -formula on both $\mathcal {A},\mathcal {B}$ , then the biconditional holds. In particular, if $\varphi (\overline {x},\overline {R})$ is first-order, then the biconditional holds.

We prove this in §14, and it builds off of Proposition 2.1. Note that Example 6.1 shows that one cannot remove the binding to X and maintain the conditional in (7.2) for first-order formulas $\varphi (\overline {x},\overline {R})$ .

In the case of internal substructures (cf. Definition 4.3) one can improve Theorem 7.7 as follows.

We prove this in §14. It has the following immediate corollary.

Having defined bounded quantifiers, we can now define bounded definability.Footnote ³²

Definition 7.10 (Bounded definability of generalized quantifiers).

Suppose that T is a theory and $Q(\overline {R},\overline {R}^{\prime })$ is a generalized quantifier. Then Q is bounded $\Sigma ^1_1$ -definable relative to T if there is a $\Sigma ^1_1$ -formula $\Phi (\overline {R}, \overline {R}^{\prime })$ with all free variables displayed such that T proves

(7.3) $$ \begin{align} \forall \; \overline{R} \;\forall \; \overline{R^{\prime}} \; (Q(\overline{R}, \overline{R^{\prime}})\leftrightarrow \Phi^{\mathscr{U}\overline{R}}(\overline{R},\overline{R^{\prime}})). \end{align} $$

Further, bounded $\Pi ^1_1$ -definable relative to T is defined exactly the same, but with the $\Sigma ^1_1$ -formula replaced by a $\Pi ^1_1$ -formula. And bounded first-order definable relative to T is defined exactly the same but with $\Sigma ^1_1$ -formula replaced by a first-order formula. Moreover, Q is bounded $\Delta ^1_1$ -definable relative to T if it is both bounded $\Sigma ^1_1$ -definable relative to T and bounded- $\Pi ^1_1$ -definable relative to T. Finally, Q is bounded definable relative to T if T proves (7.3) for some formula $\Phi (\overline {R},\overline {R}^{\prime })$ . If T is clear from context, then we omit “relative to T” for brevity.

One then has the following immediate corollary.

If one restricts to standard models, one gets something more, namely that all bounded definable generalized quantifiers, regardless of the complexity of the definition, are domain invariant across standard models.

The previous corollary indicates that domain invariance across standard models is compatible with arbitrarily complex bounded definable formulas. As we will see in the next section, things turn out quite differently with Henkin models.

We close by noting that branching quantifiers like Example 3.10 need not be $\Delta ^1_1$ -definable relative to $\mathsf {\Delta ^1_1{-}CA}_0^-$ .Footnote ³³

Proposition 7.13 (Branching quantifiers are not necessarily bounded $\Delta ^1_1$ -definable).

Suppose that $Q_1, \ldots , Q_k$ are k-many type $(1,1)$ generalized quantifiers. Consider the associated branching quantifier of type $(1, \ldots , k)$ :

(7.4) $$ \begin{align} \mathsf{Br}_{\overline{Q}}(\overline{A},R) { iff } \exists \; B_1\subseteq A_1, \ldots, B_k\subseteq A_k \; \bigwedge_{i=1}^k Q_i(A_i,B_i) \wedge B_1\times \cdots \times B_k\subseteq R. \end{align} $$

If $Q_1, \ldots , Q_k$ are bounded $\Sigma ^1_1$ -definable relative to theory T, then so is $\mathsf {Br}_{\overline {Q}}$ . But there are examples of $k\geq 1$ and $Q_1, \ldots , Q_k$ which are bounded $\Sigma ^1_1$ -definable relative to $\mathsf {\Delta ^1_1{-}CA}_0^-$ such that $\mathsf {Br}_{\overline {Q}}$ is not bounded $\Delta ^1_1$ -definable relative to $\mathsf {\Delta ^1_1{-}CA}_0^-$ .

8 Statement of main results

The first main result says that Henkin domain independent generalized quantifiers are low-level definable.

We prove Theorem 8.1 in §16, and the proof follows Marker’s proof of [Reference Marker46] of Feferman’s Preservation Theorem [Reference Feferman15]. See the outset of §16 for more discussion of Marker’s proof.

Note that the converse of Theorem 8.1 is false, as one can see from Example 6.1.

In his later work on the Tarski–Sher logicality thesis, Feferman himself mentions his earlier Preservation Theorem. First, in Feferman [Reference Feferman16, §5], he mentions it in passing in motivating a discussion of Manders’ result that absoluteness conditions will force first-order definability in certain circumstances. Second, in Feferman [Reference Feferman and Torza17, p. 24], he appeals to his Preservation Theorem in order to prove that if a generalized quantifier can be implicitly defined as the unique solution to a sentence, then the generalized quantifier is first-order definable. This seems interesting and worthy of more discussion than I am able to give it here. But nothing in domain independence requires logicality, such as the type (1) quantifier in Example 3.7 involving a named individual (but see footnote 15).

We can improve Theorem 8.1 by obtaining bounded predicative definability in certain situations. In order to describe these situations, we need one last definition.

Definition 8.2 (Strong existential import).

Suppose that T is a theory extending $\mathsf {ACA}_0^{-}$ and suppose that $Q(\overline {R},\overline {R}^{\prime })$ is a generalized quantifier. Then Q has strong existential import relative to T if T proves

(8.1) $$ \begin{align} \forall \; \overline{R} \; \forall \; \overline{R}^{\prime} \; \big(Q(\overline{R}, \overline{R}^{\prime})\rightarrow \mathscr{U}\overline{R}\neq \emptyset\big) \end{align} $$

and if for every individual constant c in the signature, T proves

(8.2) $$ \begin{align} \forall \; \overline{R} \; \forall \; \overline{R}^{\prime} \; \big(Q(\overline{R}, \overline{R}^{\prime})\rightarrow (\mathscr{U}\overline{R})c \big). \end{align} $$

To secure (8.1) one can of course define $Q^{\prime }(\overline {R}, \overline {R}^{\prime })$ to be $Q(\overline {R}, \overline {R}^{\prime })\wedge \mathscr {U}\overline {R}\neq \emptyset $ . This is how the Aristotelian universal (Example 3.4) is defined from the ordinary universal (Example 3.2). Likewise, note that (8.2) is satisfied by the paradigmatic example of a generalized quantifier defined with a constant symbol, namely the example of Montagovian individuals (Example 3.7). The general reason why we need to restrict to generalized quantifiers with strong existential import is that the empty set interacts poorly with binding (cf. Remark 7.6) and because substructures need to contain the denotations of the individual constant symbols (cf. Definition 4.1).

Restricting to generalized quantifiers with strong existential import does not seem like a deep restriction given the way in which $Q(\emptyset , \overline {R}^{\prime })$ is controlled entirely by $Q(\emptyset , \emptyset )$ for Henkin conservative quantifiers (cf. Remark 5.5). In most cases, one anticipates that either (i) $Q(\emptyset , \emptyset )$ will be derivable from the background theory T or (ii) $\neg Q(\emptyset , \emptyset )$ will be derivable from the background theory T. In case (i), we simply define $Q^{\prime }(\overline {R}, \overline {R}^{\prime })$ by $Q(\overline {R}, \overline {R}^{\prime })\wedge \mathscr {U}\overline {R}\neq \emptyset $ and work with it instead. In case (ii), we get an entailment to (8.1) when Q is Henkin conservative with respect to the background theory T (cf. Remark 5.5). Note that in Examples 3.3–3.9 we are in case (ii) for the minimal background theory $\mathsf {ACA}_0^{-}$ .

Here is then the theorem on bounded definability.

The proof of Theorem 8.3 is given at the end of §16. Note that a partial converse to this theorem is in Corollary 7.11. However, there are many bounded formulas which are not Henkin conservative relative to natural theories, and so a full converse which goes from bounded definability to Henkin conservativity is not possible.

It is no accident that domain independence and conservativity result in predicative definitions. For, historically a motivating idea of predicative constraints was that these definitions were stable across expansions,Footnote ³⁴ and Theorems 8.1(3) and 8.3 give vivid displays of how certain kinds of stability entail varieties of $\Delta ^1_1$ -definability, which is the most traditional formal precisification of predicative constraints. By Theorem 7.7, one can of course see that certain kinds of predicative definability entail a kind of stability or absoluteness. The surprise of course in Theorems 8.1 and 8.3 is that this absoluteness requires definability of any kind.

I have not seen anything like Theorem 8.3 in the literature. However, in [Reference van Benthem8], van Benthem posed the problem of “find[ing] a preservation result characterizing conservativity of first-order sentences” ([Reference van Benthem8, p. 462], cf. [Reference van Benthem9, p. 38]). He then says: “The obvious conjecture is that $Q(X,Y)$ is conservative in X iff Q is logically equivalent to some sentence with all quantifiers X-restricted. Kit Fine observed that this follows, indeed, from the work of Feferman” where Feferman’s [Reference Feferman15] is cited (notation changed in quotation to match my own notation). But this discussion seems to assume at the outset that Q is first-order definable, whereas Theorem 8.3 is compatible with Q merely being $\Delta ^1_1$ -definable, as will be the case with the cardinality quantifiers (cf. Theorem 12.1). I do not know anywhere that van Benthem or Fine say how they take Feferman [Reference Feferman15] to help establish the stated results for first-order definable Q. That said, the proof of Theorem 8.3, given at the close of §16, is comparatively direct once the definitions are in place.

9 Applications to explanatory projects in formal semantics

The semantics of generalized quantifiers are again given by a collection of tuples of subsets of Cartesian products of the underlying domain.Footnote ³⁵ A chief challenge is to explain why so few of these are realized in natural language. For instance, Sternefield writes in a recent survey: “the number of possible quantifiers left as a result of such axioms as (17) [conservativity] is still huge. […] there is nothing in the theory that would allow for isolating exactly the ‘primitive’ or even the ‘lexical’ quantifiers of a natural language” [Reference Sternefeld58, p. 11]. Similarly, von Fintel–Matthewson say that “It is clear that natural languages do not nearly make full use of the logically possible space of determiner meanings. The conservativity universal may be fraying at the edges, but the investigation of the limits on possible determiner meanings is still a promising avenue of research” [Reference von Fintel and Matthewson20, pp. 163–164].

The main reason why Theorems 8.1 and 8.3 are significant is that they explain why so few options for generalized quantifiers are realized in natural language. Namely, if one extends the domain independence conditions and the conservativity conditions to Henkin models, then these theorems directly entail the comparative low-level definability of the generalized quantifiers. Or, if one thought that domain independence and conservativity were just as much in need of explanation as the limited options for generalized quantifier meanings, then these theorems say that any explanation of domain independence and conservativity that also works for Henkin models itself suffices to explain the limited options for generalized quantifier meanings; and hence these explanatory tasks are not independent of one another.

Before turning to whether the motivations for the semantic constraints extend to Henkin models, it is worth being a little more explicit about what it means to say that so few of the theoretically possible options for generalized quantifiers are realized in natural language. This should not be understood as a claim about what generalized quantifiers appear in the lexicon of natural languages. For the lexicon will always be limited and will face pressure to be sparse, and what a sparse lexicon can generate can be vast: for instance, generalized quantifiers are closed under Boolean operations.Footnote ³⁶ I take it rather that the following is intended: there are some determiners which we recognize as generalized quantifiers, and of this subclass of determiners, their truth conditions seem to be a comparatively small slice of those which are theoretically possible, given that generalized quantifiers have as their meanings simply arbitrary collections of tuples of subsets of Cartesian products of the underlying domain.

It is however important to be precise about the meaning of “comparatively small.” On the one hand, it might be understood in terms of numerical constraints. For instance, if a standard model has first-order domain with n elements, then there are $2^{2^n\times 2^n}$ many theoretically possible denotations for type $(1,1)$ quantifiers, but only $2^{3^n}$ many theoretically possible denotations for type $(1,1)$ quantifiers which are conservative.Footnote ³⁷ Alternatively, it might be understood in terms of definability constraints on the truth conditions for these quantifiers. That is, in inspecting the list of examples of generalized quantifiers in §3, one notes that they all seem to be easy or simple to define, and definitions featuring, e.g., 27 second-order quantifiers appear to be absent. Obviously Theorems 8.1 and 8.3 fare better in contributing to the explanatory task at hand when one works with definability constraints rather than numerical constraints.

But one’s applications of Theorems 8.1 and 8.3 will vary depending on what one thinks of Example 3.10, our paradigmatic example of a branching quantifier (cf. (7.4)). Examples like these are often bounded definable, and by boundedness, they are domain invariant across standard models by Corollary 7.12, and they are often Henkin conservative and have strong existential import relative to minimal and natural theories. But Theorems 8.1 and 8.3 indicate that they are Henkin domain invariant relative to a theory only if they are bounded $\Delta ^1_1$ -definable relative to the theory. Some of them are demonstrably not so definable (cf. Proposition 7.13), and for others one may suspect that are not so definable. For someone who took Example 3.10 as a core rather than periphery case of a generalized quantifier, this would be evidence that not all generalized quantifiers are downwards Henkin invariant. This is compatible with still holding that generalized quantifiers of natural languages are Henkin upwards invariant as well as Henkin conservative. From the perspective of someone who held this, Theorem 8.1(1) could still well serve to explain why so few theoretically possible generalized quantifiers appear in natural language, but with the definability constraint set at $\Sigma ^1_1$ -definable rather than bounded $\Delta ^1_1$ -definable.

This discussion also gestures towards a possible method, which is simultaneously empirical and mathematical, of refuting the contention that the generalized quantifiers of natural language are Henkin domain invariant as well as Henkin conservative. Namely, one could try to produce an example of something which was recognizably a generalized quantifier of natural language and yet which was demonstrably not bounded $\Delta ^1_1$ -definable relative to some natural theory. Likewise, apropos the previous paragraph, one could attempt to produce an example, like Example 3.10, at the periphery of the generalized quantifiers of natural language, but which was yet demonstrably not $\Sigma ^1_1$ -definable relative to some natural theory. Such examples would of course be of further interest since they do not occur in the lists of the generalized quantifiers of natural language in the standard surveys.Footnote ³⁸ $^{{,\;}}$ Footnote ³⁹

10 Motivations for semantic constraints extend to Henkin models

Given the contributions to the explanatory projects discussed in the previous section, a real question then is whether the motivations for insisting on domain independence and conservativity extend from standard models to Henkin models. This seems very clear in the case of conservativity. For, conservativity is just a biconditional, namely in the case of type $(1,1)$ quantifiers, the biconditional $Q(X,Y)\leftrightarrow Q(X,X\cap Y)$ .Footnote ⁴⁰ Since this is expressible in the object language of second-order logic and hence does not distinguish between Henkin models and standard models, it seems that whatever motivates it or explains it will not distinguish between these two classes of models. For instance, conservativity is often explained by natural language needing a mechanism by which to temporarily introduce a domain of discourse. Keenan calls conservativity the “domain setting” property of a generalized quantifier.Footnote ⁴¹ If this domain setting role is accomplished by the aforementioned biconditional, then it can be done in both Henkin and standard models. However, conservativity is sometimes explained by appealing to structural features of syntactic processing.Footnote ⁴² In this case, to the extent that the syntactic processing is constrained to act primarily on the logical forms correlated with $Q(X,Y)$ and $Q(X,X\cap Y)$ , it seems this will not distinguish between Henkin and standard models.

As for domain independence, Peters and Westerståhl describe it as follows:

The lack of change here of course has to be understood to mean that subsets which are in the denotation of the quantifier prior to the extension (or contraction) of the universe continue to be in the denotation of the quantifier after the extension (or contraction). Hence, the question is whether any of the reasons we have for insisting on this lack of change in standard models carries over to Henkin models. All finite models which satisfy first-order comprehension are standard models, and so looking only at finite models does not distinguish between the Henkin and standard semantics. But if one wants a semantics with a Soundness and Completeness Theorem and with the recursive enumerability of the validity relation, then the Henkin semantics are the clear choice.Footnote ⁴³ Further, experience suggests that many arguments involving standard models can be recast with minimal modification as derivations in second-order logic which are hence applicable to Henkin models. For, one simply collects together the hypotheses about models to which one is appealing, and if they simply speak about elements of the underlying domain and relations on the underlying domain, then they can be regimented into sentences in the object language of second-order logic.Footnote ⁴⁴

Peters and Westerståhl also describe domain independence as follows:

If ‘given by the same rule’ simply means ‘given by a single rule definable in second-order logic,’ as in Definition 3.1, then this is very undemanding requirement, and will not get one to domain independence, in either the standard or Henkin semantics. But the latter part of the quotation seems to get closer to the core idea: namely a quantifier is domain independent if it does not query the underlying domain in the wrong way.

This can be borne out by reflecting back on the type $(1,1)$ generalized quantifier $Q(X,Y)$ from Example 6.1, which is the existential quantifier if there are fewer than three objects in the underlying domain but is the universal quantifier if there are at least three objects in the underlying domain. This disjunctive example acts differently on $X,Y$ depending on how things stand in other first-order parts of the universe; and as we saw in Example 6.1, it is not domain independent. This motivation then extends clearly to Henkin models, and just as one wants to restrict attention to type $(1,1)$ quantifier $Q(X,Y)$ which leave the focus on $X,Y$ and do not query other first-order parts of the universe, so one would want them to not query other second-order parts of the universe. This was illustrated by Examples 6.2 and 6.3, where common idea was that the quantifier was the existential quantifier if the second-order part of the universe was expressively impoverished and it was the universal quantifier if the second-order part of the universe was expressively rich.

Hence the suggestion is that one should accept both domain independence and Henkin independence of generalized quantifiers because one wants to restrict attention to type $(1,1)$ generalized quantifiers $Q(X,Y)$ whose satisfaction conditions pertain to $X,Y$ and not to other regions of the first- or second-order domain. (For notational ease, we continue to focus the discussion on type $(1,1)$ generalized quantifiers; but this applies mutatis mutandis for generalized quantifiers of other types.) An advantage of this proposal is that brings the motivation for domain independence closer to the motivation for conservativity.Footnote ⁴⁵ Recall that the latter was a “domain setting” feature, where the idea is that $Q(X,Y)$ results in making X the temporary domain of discourse to which Y is restricted. Our discussion of the motivation of domain independence suggests the further thought: in addition to considering only Y’s intersection with X, one must not query either first-order or second-order facts about the model which are outside of X.

However, the proposal does come with disadvantages and challenges. First, the proposal introduces more explicitly notions of “aboutness” or “topicality” into this understanding of the motivations for domain independence and conservativity.Footnote ^{46 ,} Footnote ⁴⁷ It is well-known that this kind of intentionality is hard to explicate and model, at least when one restricts oneself to the tools of formal semantics and logic. Second, the proposal conceives of a core task of domain independence as getting rid of the disjunctive examples in §6. But the history of Nelson’s grue should make us cautious in claiming anything more than provisional success in weeding out disjunctive laws. Third, a presupposition of the above discussion is that linguistic agents are moving between different Henkin models in the same way that they are moving between contexts in a more quotidian sense, and that the semantic constraints on quantifiers allows agents to hone in and focus on a topic that is fixed amid this wider flux. It does seem that some variation in the second-order domain is quotidian: in some discussions a limited conception of properties is at issue and in other discussions a more fine-grained conception of properties is at issue. That said, one would want a better understanding of the circumstances in which the second-order domain becomes subject to substantial variation, and whether the variation could be understood like Examples 6.2 and 6.3 in which one goes back and forth between a mathematically impoverished and a mathematically enriched setting.

11 Lowering to first-order definability

The theorems discussed thus far, namely Theorems 8.1 and 8.3, get one to bounded $\Delta ^1_1$ -definability. But the standard list of examples, such as we sampled in §3, consists in large part of generalized quantifiers which are first-order definable. In this section we show that by switching to a slightly richer background theory we can improve the definability to first-order definability.

We expand from pure second-order logic to a well-order.Footnote ⁴⁸

Note that if $T^{-}$ be one of $\mathsf {ACA_0^{-}}$ , $\mathsf {\Delta ^1_1{-}CA_0^{-}}$ , $\mathsf {\Sigma ^1_1{-}AC_0^{-}}$ , $\mathsf {\Sigma ^1_1{-}CA_0^{-}}$ , or $\mathsf {SOL}$ , then both $T^{\mathsf {wo}}$ and $T^{\mathsf {fin}}$ are closed downwards under internal substructures (cf. Definition 4.4), since initial segments of well-orders are still well-orders, and since Dedekind infinitude is bounded $\Sigma ^1_1$ -definable.

Then we have the following results.

We prove these two theorems in §17. However, note that Theorems 11.2 and 11.3 pertain to a specific theory, namely $\mathsf {\Sigma ^1_1{-}AC_0^{wo}}$ , whereas Theorems 8.1 and 8.3 are about general background theories T of second-order logic.

12 Quantifiers defined in terms of cardinality

Finally, we make good on an obligation to show that Examples 3.6 and 3.8 of $\mathsf {Most}$ and $\mathsf {More{-}than}$ fit into the framework described in this paper. We have the following, where $\mathsf {\Sigma ^1_1{-}CA_0^{fin}}$ was defined at the end of Definition 11.1.

We prove this in §18. Obviously (1) follows from (2) and Theorems 8.1 and 8.3. But to our knowledge, there is no easy way to verify the Henkin domain independence in (2) without going through the specific definitions produced in the proof of (1) and then appealing to Theorem 7.7.

It is natural to ask whether the restriction to Dedekind finite universes in Theorem 12.1 is necessary. Let us consider the case of $\mathsf {More{-}than}$ (cf. Example 3.8). Note that one has $\left |X\right |>\left |Y\right |$ iff $\mathsf {More{-}than}(X,Y,\mathbb {V})$ . Hence if $\mathsf {More{-}than}$ is bounded $\Delta ^1_1$ -definable relative to a theory T, then $\left |X\right |>\left |Y\right |$ is $\Delta ^1_1$ -definable relative to T, by quantifiers which are bound to $X\cup Y$ (in the sense of Definition 7.2(2)).

But, one can show using the method of forcing that this is false for $\mathsf {ACA}_0^-$ .Footnote ⁴⁹ For, using this method, one can produce two Henkin models $\mathcal {A}, \mathcal {B}$ of $\mathsf {ACA}_0^-$ such that $\mathcal {A}$ is a substructure of $\mathcal {B}$ and one can produce two $X,Y$ in $S_1[A]$ such that $\mathcal {A}\models \left |X\right |< \left |Y\right |$ but $\mathcal {B}\models \neg ( \left |X\right |< \left |Y\right |)$ . Namely, $\mathcal {A}$ is the countable transitive ground model together with its first-order definable subsets, and $\mathcal {B}$ is a generic extension together with its first-order definable subsets; one further chooses $X=\omega $ and $Y=\kappa $ for some uncountable cardinal $\kappa $ in $\mathcal {A}$ which is collapsed in $\mathcal {B}$ . If $\left |X\right |<\left |Y\right |$ was $\Delta ^1_1$ -definable relative to $\mathsf {ACA}_0^-$ with quantifiers bound to $X\cup Y$ , then we would contradict Theorem 7.7.

The restriction to the finite in Theorem 12.1 does not seem problematic, as far as empirical coverage goes. For, our judgments about $\mathsf {Most}$ and $\mathsf {More{-}than}$ are primarily informed by our experience with finite collections of, e.g., people and other middle-sized objects. But the restriction to the finite does suggest an interesting test case for someone who wanted to deny that generalized quantifiers satisfy Henkin domain independence. For, by the above, $\mathsf {More{-}than}$ is not Henkin domain independent relative to $\mathsf {ACA}_0^-$ . But it is domain independent on standard models by Corollary 7.12. If one thought that $\mathsf {More{-}than}$ was a generalized quantifier regardless of the size of the ambient universe, then this would be a reason to reject that all generalized quantifiers satisfy Henkin domain independence. Pursued in this direction, the discussion would start blending with discussions in philosophy of mathematics about whether and to what extent models produced by the method of forcing are or are not good candidates for intended models of set theory.Footnote ⁵⁰ But those who outright reject the Henkin domain independence of generalized quantifiers would be in need of another explanation of why natural language generalized quantifiers realize so few of the theoretically possible options for collections of subsets of the underlying domain.

13 Applications to upper bounds on complexity

Finally, we mention an immediate corollary of Theorem 8.1(3) (there are similar corollaries for (1) and (2)). (Recall the axioms for well-orders in Definition 11.1).

This corollary leverages how predicatively definable sets occur naturally throughout mathematics and computation and are tied to core notions like the Borel sets and the traditional complexity classes. Let us discuss each of these in turn.

Borel sets are the events most commonly used in probability, measure-theoretic mathematics, and their many applications. The subject matter of descriptive set theory is definability in Polish spaces, where the Borel sets form the lower end of a hierarchy of definability.Footnote ⁵³ Corollary 13.1(1) says that the quantifiers at issue satisfy the usual condition for the theorems of probability theory and measure-theoretic mathematics to apply to them, and that they are not a source of set-theoretic pathology. However, the applicability of this is limited. As we saw in §12, the theory $\mathsf {ACA}_0^-$ , which is true on all standard models with first-order domain which is countably infinite, is such that the quantifier $\mathsf {More{-}than}$ is not Henkin domain independent relative to $\mathsf {ACA}_0^{-}$ .

Computational complexity theory studies feasible models of computation, which admit computations that are a strict subclass of those allowed under the more traditional Turing models of computation.Footnote ⁵⁴ Corollary 13.1(2) identifies an upper bound on the computational complexity of the quantifier. If the motivations for domain independence extend to Henkin domain independence, then Corollary 13.1(2) would predict that natural language generalized quantifiers are not excessively infeasible to compute. Finally, in contrast to the conclusion of the previous paragraph, in §12 we identified the theory $\mathsf {\Sigma ^1_1{-}CA_0^{fin}}$ which satisfies the hypotheses of Corollary 13.1(2) and relative to which $\mathsf {Most}$ and $\mathsf {More{-}than}$ are Henkin domain invariant and Henkin conservative. Hence $\mathsf {Most}$ and $\mathsf {More{-}than}$ are $\mathsf {NP}\cap \mathsf {co{-}NP}$ on the finite models described in Corollary 13.1(2).

Corollary 13.1(2) is rather complementary to Szymanik’s work, which puts the upper bound at $\mathsf {NP}$ rather than $\mathsf {NP}\cap \mathsf {co{-}NP}$ .Footnote ⁵⁵ This is not lowered due to branching quantifiers and Ramsey quantifiers (cf. Example 3.10).Footnote ⁵⁶ For someone who thought that branching quantifiers and Ramsey quantifiers were core rather than periphery generalized quantifiers, one would rather focus on the analogue of Corollary 13.1(2) which is centered around Fagin’s Theorem and Theorem 8.1(1) rather than Theorem 8.1(3).

We now turn to the more technical part of the paper, in which we prove the various theorems in turn.

14 Proofs of Theorems 7.7 and 7.8

In this theorem, recall that the binding concepts and notation was defined in Definition 7.2.

Theorem 7.7 (Grades of absoluteness and bounded definability).

Suppose that $\mathcal {A},\mathcal {B}$ are two Henkin models of $\mathsf {ACA}_0^{-}$ with $\mathcal {A}$ a substructure of $\mathcal {B}$ . Suppose that $\varphi (\overline {x},\overline {R})$ is a formula with free variables among $\overline {x},\overline {R}$ , and suppose X is a unary term. Suppose that $\overline {x},\overline {R}$ are from $\mathcal {A}$ and the free variables of X are from $\mathcal {A}$ and are not bound in $\varphi (\overline {x},\overline {R})$ .

If $\varphi (\overline {x},\overline {R})$ is $\Sigma ^1_1$ , then

(7.2) $$ \begin{align} \mathcal{A}\models \varphi^X(\overline{x},\overline{R}) \Longrightarrow \mathcal{B}\models \varphi^X(\overline{x},\overline{R}). \end{align} $$

If $\varphi (\overline {x},\overline {R})$ is $\Pi ^1_1$ then the arrow is reversed. If $\varphi (\overline {x},\overline {R})$ is equivalent to a single $\Sigma ^1_1$ -formula on both $\mathcal {A},\mathcal {B}$ and a single $\Pi ^1_1$ -formula on both $\mathcal {A},\mathcal {B}$ , then the biconditional holds. In particular, if $\varphi (\overline {x},\overline {R})$ is first-order, then the biconditional holds.

15 Henkin models and faithfulness

Henkin models as defined in §1 are required to interpret membership absolutely. It is natural to view them as models of many-sorted first-order logic, and hence to view A as the interpretation of the sort associated with objects and $S_n[A]$ as the interpretation of the sort associated with n-ary second-order relations. In order to be able to apply the full metatheory of many-sorted first-order logic,Footnote ⁵⁷ one needs (i) to have the freedom to temporarily relax the assumption that membership is interpreted absolutely, since, e.g., the models coming from compactness will not in general have this property; and one needs simultaneously (ii) to know how to reinstate it by passing to a related model, since we want eventually to restate things in terms of Henkin models.

Hence we define the following definition.

Definition 15.1 (Pre-Henkin model).

A pre-Henkin model in a signature is a many-sorted structure of the form:

(15.1) $$ \begin{align} \mathcal{A}=(A,S_1[A], S_2[A], \ldots; E_1^{\mathcal{A}}, E_2^{\mathcal{A}}, \ldots), \end{align} $$

where A is a non-empty set, and where

1. (1) For all $n\geq 1$ , the interpretation $E_n^{\mathcal {A}}$ of the membership symbol $E_n$ has sort $S_n[A]\times A^n$ and in the object language we suppress $E_n$ and write $R\overline {a}$ instead of $E_n(R,a_1, \ldots , a_n)$ .

2. (2) For all $n\geq 1$ the structure $\mathcal {A}$ satisfies extensionality:

   (15.2) $$ \begin{align} \forall \; R \; \forall \; S \; \bigg(R=S\leftrightarrow \forall \; \overline{x} \; (R\overline{x}\leftrightarrow S\overline{x})\bigg), \end{align} $$
   where $R, S$ are n-ary variables and the first-order tuple $\overline {x}$ has arity n.

3. (3) The model $\mathcal {A}$ satisfies the comprehension schema (2.1) for first-order formulas in the signature.

4. (4) For the interpretation of the signature, we have:

   1. (a) For all individual constant symbols c in the signature, it is interpreted as an element $c^{\mathcal {A}}$ of A.

   2. (b) For all $n\geq 1$ and all n-ary relation constant symbols C in the signature, it is interpreted as an element $C^{\mathcal {A}}$ of $S_n[A]$ .

   3. (c) For all $n\geq 1$ and all n-ary relation symbols C in the signature, it is interpreted as a subset $C^{\mathcal {A}}$ of $A^n$ .

In (1), the idea is that $E_n$ interprets the membership relation between the domain $S_n[A]$ reserved for the n-ary second-order objects and n-tuples of elements from the first-order domain A. Since we only focus on models which satisfy first-order comprehension, and since the minimal set-theoretic apparatus described in §2 is all first-order definable, we may regard ourselves as working in the signature consisting just of membership symbols from (1) and the symbols from (4).

In §2 we only had symbols corresponding to (4a) and (4c). It is useful to include (4b) since in the setting of many-sorted first-order logic, they will act differently under morphisms. For instance, if $f:\mathcal {A}\rightarrow \mathcal {B}$ is an embedding of pre-Henkin structures, then for a binary relation constant symbol $C_1$ in the sense of (4b), the standard definition of an embedding requires that (i) $f(C^{\mathcal {A}}_1)=C^{\mathcal {B}}_1$ , while for a binary relation symbol $C_2$ in the sense of (4c), the standard definition of an embedding requires that (ii) $\mathcal {A}\models C_2 xy $ iff $\mathcal {B}\models C_2 f(x) f(y)$ for all $x,y$ in A. In the case where $\mathcal {A},\mathcal {B}$ are Henkin models and $f:\mathcal {A}\rightarrow \mathcal {B}$ is the inclusion map of a smaller Henkin model $\mathcal {A}$ into a larger Henkin model $\mathcal {B}$ , this is the difference between (i) $C^{\mathcal {A}}_1=C^{\mathcal {B}}_1$ and (ii) $C_2^{\mathcal {A}}\cap A^2= C_2^{\mathcal {B}}$ . Case (ii) is what one wants when one considers, e.g., a larger linear order $\mathcal {B}$ and its restriction to a smaller linear order $\mathcal {A}$ . But in model theory it is of course commonplace in proofs to name elements of the domains with fresh constant symbols, and for this one wants to be in case (i). Since we are doing the proofs in these sections, we need to include (4b).

Note that in a signature with n-ary relation symbols C as in (4c), there are three kinds of atomics: (I) equalities, (II) atomics coming from the membership symbols from (1), and (III) atomics coming from the n-ary constant relation symbols from (4). By extensionality (2), all equalities on second-order entities are definable in terms of membership. By (3), every n-ary constant relation symbol determines an element of $S_n[A]$ , and so anything from (III) is equivalent to something from (II). Hence, in a fixed pre-Henkin model, as well as a fixed Henkin model, we can assume without loss of generality that the atomics are: (I $^{\prime }$ ) equalities between individuals and (II $^{\prime }$ ) atomics coming from the membership symbol.

Note that being a pre-Henkin model is preserved under isomorphism.

Having defined this more general notion, we can then specify the notion defined in §1 as a special case.

One has the following elementary observation.

In the above proof, we use $f(\overline {x})$ as an abbreviation for the tuple $(f(x_1), \ldots , f(x_n))$ . Similarly, often in what follows we write $f(\overline {x})=\overline {y}$ , which is an abbreviation for $f(x_1)=y_1, \ldots , f(x_n)=y_n$ .

However, not every substructure of a Henkin model is Henkin model.Footnote ⁵⁸

Example 15.4 (Among pre-Henkin models, being a Henkin model is not preserved under substructure).

Consider a standard model $\mathcal {B}$ in a countable signature where B is uncountable. By Downward Löwenheim–Skolem, take an elementary substructure

(15.3) $$ \begin{align} \mathcal{A}=(A,S_1[A], S_2[A], \ldots; E_1^{\mathcal{A}}, E_2^{\mathcal{A}}, \ldots), \end{align} $$

where A is countably infinite and each $S_n[A]$ is countably infinite. Then $\mathcal {A}$ is a pre-Henkin model. Let X be the unique element of $S_1[A]$ such that $\mathcal {A}\models \forall x \; Xx$ . Then since it is an elementary substructure one has that $\mathcal {B}\models \forall \; x \; Xx$ . Then since $\mathcal {B}$ is a Henkin model, X is in $S_1[B]$ and by extensionality $X=B$ . Since A is countable and B is uncountable, we have that X is in $S_1[A]\setminus P(A)$ , and thus $S_1[A]\nsubseteq P(A)$ . Thus $\mathcal {A}$ is not a Henkin model.

We can also describe this example in terms of embeddings.

This example motivates the following definition, which is the natural analogue of Marker [Reference Marker46, definition 1.2] in the setting of second-order logic.

These are related by the following.

16 Proof of Theorems 8.1 and 8.3

As mentioned in the abstract and the introduction, Theorem 8.1 basically follows from Feferman’s Preservation Theorem [Reference Feferman15], which Marker [Reference Marker46] proved model-theoretically. Since readers interested in generalized quantifiers may not be interested in the model-theoretic details of Marker’s presentation, we specialize Marker’s proof to the setting of second-order logic. Further, since the proof is short, this requires about the same amount of space needed to verify that Theorem 8.1 follows from from Marker’s result.Footnote ⁵⁹

Proof. For (1), suppose that T is a theory extending $\mathsf {\Sigma ^1_1{-}{AC}_0}$ , and suppose that Q is Henkin upward independent relative to T. We show that Q is $\Sigma ^1_1$ -definable relative to T. Without loss of generality, Q is instantiated in a model of T (otherwise choose an inconsistent $\Sigma ^1_1$ -definition).

Let $R_1, \ldots , R_{\ell }$ be new relation constant symbols, in the sense of Definition 15.1(4b), corresponding to the type of Q. We abbreviate $R_1, \ldots , R_{\ell }$ as $\overline {R}$ . It suffices to show: there is a $\Sigma ^1_1$ -formula $\Phi $ such that the theory T proves: $\Phi (\overline {R})\leftrightarrow Q(\overline {R})$ .

Note that since Q is instantiated in a model of T, we have that there is a model $\mathcal {A}$ of $T+Q(\overline {R})$ . For any model $\mathcal {A}$ in the expanded signature, let $\Sigma ^1_1(\mathcal {A})$ be the set of all $\Sigma ^1_1$ -sentences in the expanded signature which are true in $\mathcal {A}$ . Consider the following.

Case 16.1 There is a pair of models $\mathcal {A}_0$ and $\mathcal {B}_0$ such that $\mathcal {A}_0\models T+Q(\overline {R})$ and $\mathcal {B}_0\models T+\neg Q(\overline {R}) +\Sigma ^1_1(\mathcal {A}_0)$ .

Suppose that Case 16.1 holds; we show that this eventuates in a violation of Henkin upward independence. Since $\mathcal {B}_0\models \Sigma ^1_1(\mathcal {A}_0)$ , one has that $\mathcal {B}_0$ is infinite iff $\mathcal {A}_0$ is infinite; and likewise $\mathcal {B}_0$ is finite and of size n iff $\mathcal {A}_0$ is finite and of size n. Hence, suppose that both $\mathcal {A}_0, \mathcal {B}_0$ are finite and of size n. Finite models of the same size which satisfy the same first-order sentences in a signature are isomorphic and hence satisfy the same second-order sentences, which is impossible since $\mathcal {A}_0\models Q(\overline {R})$ and $\mathcal {B}_0\models \neg Q(\overline {R})$ . Henceforth suppose that both $\mathcal {A}_0, \mathcal {B}_0$ are infinite, and without loss of generality countably infinite. Let $(\mathcal {A}_0, \mathcal {B}_0)$ be the disjoint sum of the two many-sorted structures. Let $(\mathcal {A}, \mathcal {B})$ be a countable recursively saturated elementary extension of $(\mathcal {A}_0, \mathcal {B}_0)$ .Footnote ⁶⁰ By taking an isomorphic copy à la Proposition 15.3, we may assume that both $\mathcal {A},\mathcal {B}$ are Henkin models. Enumerate both the first- and second-order parts of $\mathcal {A}$ simultaneously as $\alpha _1, \alpha _2, \ldots $ , and similarly enumerate $\mathcal {B}$ as $\beta _1, \beta _2, \ldots $ . We build a faithful embedding $f=\bigcup _n f_n:\mathcal {A}\rightarrow \mathcal {B}$ in stages $f_0,f_1, f_2, \ldots $ such that $f_n$ satisfies:

1. (i) The domain of $f_n$ is a finite subset, including $\alpha _1, \ldots , \alpha _n$ .

2. (ii) For any $\Sigma ^1_1$ -formula $\varphi (\upsilon _1, \ldots , \upsilon _j)$ with free variables among $\upsilon _1, \ldots , \upsilon _j$ , if $\mathcal {A}\models \varphi (\gamma _1, \ldots , \gamma _j)$ where $\gamma _1, \ldots , \gamma _j$ enumerates the domain of $f_n$ , then $\mathcal {B}\models \varphi (f_n(\gamma _1), \ldots , f_n(\gamma _j))$ .

3. (iii) If $\beta _i$ is the least first-order object with $i\leq n$ which is in some tuple in some second-order object $\beta $ in the range of $f_n$ , then $\beta _i$ is in the range of $f_n$ .

Note that while each of the variables $\upsilon _1, \ldots , \upsilon _j$ in (ii) is either first-order or second-order (and if second-order is unary, binary, etc.), we are enumerating them all simultaneously but not marking their status in the notation: one could easily mark them as first-order or second-order (and if second order one could mark whether they are unary, binary, etc.), but this would terribly clutter the notation. By way of motivation for these conditions, note that:

• – Condition (ii) ensures that $f:\mathcal {A}\rightarrow \mathcal {B}$ is an embedding. In particular it ensures the injectivity of f and and that f preserves the individual constant symbols and n-ary relation constant symbols. For, consider the formula $\varphi _c(\upsilon _k)\equiv c=\upsilon _k$ for an individual constant symbol c and individual variable $\upsilon _k$ , and consider the formula $\varphi _i(\upsilon _j)\equiv R_i = \upsilon _j$ for one of the $R_i$ from our sequence of $R_1, \ldots , R_{\ell }$ of relation constants symbols and $\upsilon _j$ an n-ary variable. Since $\mathcal {A}\models \varphi _c(c^{\mathcal {A}})$ we have by condition (ii) that $\mathcal {B}\models \varphi _c(f(c^{\mathcal {A}}))$ , which is to say that $f(c^{\mathcal {A}})=c^{\mathcal {B}}$ . Since $\mathcal {A}\models \varphi _i(R_i^{\mathcal {A}})$ we have by condition (ii) that $\mathcal {B}\models \varphi _i(f(R^{\mathcal {A}}_i))$ , which is to say that $f(R^{\mathcal {A}}_i)=R^{\mathcal {B}}_i$ .

• – Condition (iii) ensures that $f:\mathcal {A}\rightarrow \mathcal {B}$ is a faithful embedding.

For the base case, one takes $f_0=\emptyset $ , and condition (ii) is satisfied because $\mathcal {B}\models \Sigma ^1_1(\mathcal {A})$ . For the inductive step, we consider the following recursive type $\Gamma (\upsilon )$ , where $\gamma _1, \ldots , \gamma _j$ enumerates the domain of $f_n$ . We insert an initial formula in $\Gamma (\upsilon )$ which says that $\upsilon $ comes from $\mathcal {B}$ , and in general we put a formula in $\Gamma (\upsilon )$ if it has the form

(16.1) $$ \begin{align} \varphi^{\mathcal{A}}(\gamma_1, \ldots, \gamma_j, \alpha_{n+1})\rightarrow \varphi^{\mathcal{B}}(f_n(\gamma_1), \ldots, f_n(\gamma_j), \upsilon), \end{align} $$

where $\varphi (\upsilon _1, \ldots , \upsilon _j, \upsilon )$ is a $\Sigma ^1_1$ -formula, and where as in (ii) the variables $\upsilon _1, \ldots , \upsilon _j, \upsilon $ may be first-order or second-order. This recursive type $\Gamma (\upsilon )$ is indeed finitely satisfied. For, suppose that we have m formulas:

(16.2) $$ \begin{align} \varphi_1(\upsilon_1, \ldots, \upsilon_j, \upsilon), \ldots, \varphi_m(\upsilon_1, \ldots, \upsilon_j, \upsilon). \end{align} $$

By reordering, we may suppose that $\mathcal {A}\models \varphi _i(\gamma _1, \ldots , \gamma _j, \alpha _{n+1})$ iff $i\leq m_0$ , for some $m_0\leq m$ . Then $\mathcal {A}\models \exists \; \upsilon \; \bigwedge _{i=1}^{m_0} \varphi _i(\gamma _1, \ldots , \gamma _j, \upsilon )$ . Since this is itself a $\Sigma ^1_1$ -formula, by (ii) one has that $\mathcal {B}\models \exists \; \upsilon \; \bigwedge _{i=1}^{m_0} \varphi _i(f_n(\gamma _1), \ldots , f_n(\gamma _j), \upsilon )$ . Thus $\Gamma (\upsilon )$ is indeed finitely satisfied and so satisfied in $\mathcal {B}$ by some $\beta $ , and then we set $f_{n+1}(\alpha _{n+1})=\beta $ . Thus we have satisfied (i) and (ii). To ensure (iii), suppose that $\beta _i$ is the least first-order object with $i\leq n+1$ which is in some tuple in some second-order object $\beta $ in the range of $f_{n+1}$ , say $\beta =f(\gamma _p)$ . Without loss of generality the tuple is, say, a 2-tuple, so that $(\beta _u,\beta _i)\in \beta $ . Consider the following recursive type $\Gamma ^{\ast }(v)$ , where v is a first-order variable. We put an initial formula in $\Gamma ^{\ast }(v)$ which says that v comes from $\mathcal {A}$ , and in general we put a formula in $\Gamma ^{\ast }(v)$ if it has the form

(16.3) $$ \begin{align} \varphi^{\mathcal{B}}(f_n(\gamma_1), \ldots, f_n(\gamma_j), f_{n+1}(\alpha_{n+1}), \beta_i)\rightarrow \varphi^{\mathcal{A}}(\gamma_1, \ldots, \gamma_j, \alpha_{n+1}, v), \end{align} $$

where $\varphi (\upsilon _1, \ldots , \upsilon _j, \upsilon _{j+1}, v)$ is a $\Pi ^1_1$ -formula. This recursive type is indeed finitely satisfied. For, suppose that we have m-such formulas

(16.4) $$ \begin{align} \varphi_1(\upsilon_1, \ldots, \upsilon_j, \upsilon_{j+1}, v), \ldots, \varphi_m(\upsilon_1, \ldots, \upsilon_j, \upsilon_{j+1}, v). \end{align} $$

By reordering, we again may suppose that $\mathcal {B}\models \varphi _i(f_n(\gamma _1), \ldots , f_n(\gamma _j), f_{n+1}(\alpha _{n+1}), \beta _i)$ iff $i\leq m_0$ , for some $m_0\leq m$ . Then one has

(16.5) $$ \begin{align} \mathcal{B}\models \exists \; v \; \exists \; u \; (f(\gamma_p))(u,v) \; \wedge\; \bigwedge_{i=1}^{m_0} \varphi_i(f_n(\gamma_1), \ldots, f_n(\gamma_j), f_{n+1}(\alpha_{n+1}), v). \end{align} $$

Since $\mathcal {B}$ is a model of $\mathsf {\Sigma ^1_1{-}AC_0^{-}}$ , this is equivalent to a $\Pi ^1_1$ -formula over $\mathcal {B}$ , and similarly over $\mathcal {A}$ , and thus by (ii) holding for $f_{n+1}$ as defined so far, we have that

(16.6) $$ \begin{align} \mathcal{A}\models \exists \; v \; \exists \; u \; \gamma_p(u,v)\;\wedge \; \bigwedge_{i=1}^{m_0} \varphi_i(\gamma_1, \ldots, \gamma_j, \alpha_{n+1}, v). \end{align} $$

Thus $\Gamma ^{\ast }(v)$ is indeed finitely satisfied and so satisfied in $\mathcal {A}$ by some $\gamma $ , and then we set $f_{n+1}(\gamma )=\beta _i$ . This completes the construction of the faithful embedding $f:\mathcal {A}\rightarrow \mathcal {B}$ . Let $\mathcal {D}$ be the image of $\mathcal {A}$ under the faithful embedding $f:\mathcal {A}\rightarrow \mathcal {B}$ . By Proposition 15.7, we have that $\mathcal {D}$ is a Henkin model which is a substructure of $\mathcal {B}$ . But the assumption that we are in Case 16.1 implies that $\mathcal {A}\models Q(\overline {R})$ , and since $f:\mathcal {A}\rightarrow \mathcal {D}$ is an isomorphism, we have $\mathcal {D}\models Q(\overline {R})$ . By Henkin upwards independence, we have $\mathcal {B}\models Q(\overline {R})$ , which contradicts that we are in Case 16.1, which says that $\mathcal {B}\models \neg Q(\overline {R})$ .

Hence, rather we are in the following case.

Case 16.2 There is no pair of models $\mathcal {A}_0$ and $\mathcal {B}_0$ such that $\mathcal {A}_0\models T+Q(\overline {R})$ and $\mathcal {B}_0\models T+\neg Q(\overline {R}) +\Sigma ^1_1(\mathcal {A}_0)$ .

Then for any model $\mathcal {A}\models T+Q(\overline {R})$ , one has that $T+\neg Q(\overline {R}) +\Sigma ^1_1(\mathcal {A})$ is inconsistent and so by compactness there is a $\Sigma ^1_1$ -sentence $\Phi _{\mathcal {A}}$ in $\Sigma ^1_1(\mathcal {A})$ such that

(16.7) $$ \begin{align} T\vdash \Phi_{\mathcal{A}} \rightarrow Q(\overline{R}). \end{align} $$

In this, we appeal to the fact that $\Sigma ^1_1$ -formulas are closed under finite conjunctions. Then consider the theory

(16.8) $$ \begin{align} \Delta = \{\neg \Phi_{\mathcal{A}}: \mathcal{A} \models T+ Q(\overline{R})\}. \end{align} $$

Then $T+Q(\overline {R})+\Delta $ is inconsistent. For, suppose that it had a model $\mathcal {A}$ . Then it would model $\neg \Phi _{\mathcal {A}}$ (since this sentence is in $\Delta $ ) and it would model $\Phi _{\mathcal {A}}$ (since this sentence is, by construction, in $\Sigma ^1_1(\mathcal {A})$ , and hence true on $\mathcal {A}$ ). Since $T+Q(\overline {R})+\Delta $ is inconsistent, there are a finite number of models $\mathcal {A}_1, \ldots , \mathcal {A}_m \models T+ Q(\overline {R})$ such that $T + Q(\overline {R})$ proves $\bigvee _{i=1}^m \Phi _{\mathcal {A}_i}$ . But then together with (16.7), we have that

(16.9) $$ \begin{align} T\vdash Q(\overline{R}) \leftrightarrow \bigvee_{i=1}^m \Phi_{\mathcal{A}_i}. \end{align} $$

This finishes the proof of (1)

The proof of (2) follows from (1) by replacing Q by $\neg Q$ ; and (3) follows directly from (1) and (2).

Now we turn to the proof of Theorem 8.3. We first prove a preliminary theorem.

Proof. Consider a Henkin model $\mathcal {B}$ of T, and consider the $\Sigma ^1_1$ -definition $\Psi (\overline {R},\overline {R}^{\prime })\equiv \exists \; \overline {S} \; \Psi _0(\overline {R},\overline {R}^{\prime }, \overline {S})$ of $Q(\overline {R},\overline {R}^{\prime })$ , where $\Psi _0(\overline {R},\overline {R}^{\prime }, \overline {S})$ contains no second-order quantifiers. Then the following $\Sigma ^1_1$ -formula $\Phi (\overline {R},\overline {R}^{\prime })$ also defines $Q(\overline {R},\overline {R}^{\prime })$ relative to T, where we have added the dummy existentials, where $c_1, \ldots , c_n$ are the individual constants in the signature:

(16.10) $$ \begin{align} \exists \; \overline{S} \; \bigg(\big((\exists \; x \; x=x) \wedge \bigwedge_{i=1}^n (\exists \; x \; x=c_i)\big) \wedge \Psi_0(\overline{R},\overline{R}^{\prime}, \overline{S})\bigg). \end{align} $$

Let $\Phi ^{\mathscr {U}\overline {R}}(\overline {R},\overline {R}^{\prime })$ be the result of bounding all quantifiers in $\Phi (\overline {R},\overline {R}^{\prime })$ to $\mathscr {U}\overline {R}$ , as in Definition 7.2(3), so that $\Phi ^{\mathscr {U}\overline {R}}(\overline {R},\overline {R}^{\prime })$ is equivalent to:

(16.11) $$ \begin{align} \exists \; \overline{S}\subseteq \mathscr{U}\overline{R} \; \bigg(\big((\exists \; x \; (\mathscr{U}\overline{R})x) \wedge \bigwedge_{i=1}^n (\mathscr{U}\overline{R})c_i \big) \wedge \Psi_0^{\mathscr{U}\overline{R}}(\overline{R},\overline{R}^{\prime}, \overline{S})\bigg). \end{align} $$

Let $\overline {R},\overline {R}^{\prime }$ be fixed in $S_{\overline {n}}[B]$ . We claim that $\mathcal {B}\models Q(\overline {R},\overline {R}^{\prime })\leftrightarrow \Phi ^{\mathscr {U}\overline {R}}(\overline {R},\overline {R}^{\prime }\cap \mathscr {U}\overline {R})$ .

First suppose that $\mathscr {U} \overline {R}$ is empty or does not include the interpretation of one of the individual constants $c_1, \ldots , c_n$ . By strong existential import (cf. Definition 8.2), one has that $\mathcal {B}\models \neg Q(\overline {R},\overline {R}^{\prime })$ . By (16.11) we have $\mathcal {B}\models \neg \Phi ^{\mathscr {U}\overline {R}}(\overline {R},\overline {R}^{\prime })$ . Hence we have the biconditional since both sides are false.

Second suppose that $\mathscr {U} \overline {R}$ is non-empty and contains all of the interpretations of the individual constants $c_1, \ldots , c_n$ . Then by choosing $A=\mathscr {U} \overline {R}$ we have a Henkin internal substructure $\mathcal {A}$ of $\mathcal {B}$ (cf. Definition 4.3). Since T is closed downwards under internal substructures, we have that $\mathcal {A}\models T$ . Further, one has $\mathcal {B}\models Q(\overline {R},\overline {R}^{\prime })$ iff $\mathcal {B}\models Q(\overline {R},\overline {R}^{\prime }\cap \mathscr {U}\overline {R})$ by Henkin conservativity. By Henkin domain invariance relative to T together with Proposition 2.1, this happens iff $\mathcal {A}\models Q(\overline {R},\overline {R}^{\prime }\cap \mathscr {U}\overline {R})$ . Using the $\Sigma ^1_1$ -definition in $\mathcal {A}$ this happens iff $\mathcal {A}\models \Phi (\overline {R},\overline {R}^{\prime }\cap \mathscr {U}\overline {R})$ . By Theorem 7.8, this happens iff $\mathcal {B}\models \Phi ^{\mathscr {U}\overline {R}}(\overline {R},\overline {R}^{\prime }\cap \mathscr {U}\overline {R})$ . Hence indeed $\mathcal {B}\models Q(\overline {R},\overline {R}^{\prime })\leftrightarrow \Phi ^{\mathscr {U}\overline {R}}(\overline {R},\overline {R}^{\prime }\cap \mathscr {U}\overline {R})$ .

Thus $Q(\overline {R},\overline {R}^{\prime })$ is bounded $\Sigma ^1_1$ -definable.

The same argument works for the $\Pi ^1_1$ -case and the first-order case.

Here is then the theorem on bounded definability.

17 Proof of Theorems 11.2 and 11.3

First we define a substitution notion.

Definition 17.1 (Substituting a first-order formula for a relation variable).

Suppose that the type of $\overline {R}$ is $\overline {n}=( n_1, \ldots , n_k)$ . Suppose that $\Phi (\overline {R})$ is a formula of second-order logic which has no second-order quantifiers and only second-order free variables, including the displayed $\overline {R}$ .

Suppose that we have specified an additional first-order signature, which has relation symbols corresponding to all the free second-order variables of $\Phi (\overline {R})$ excluding those of the displayed $\overline {R}$ (our substitution notion will remove these displayed $\overline {R}$ ).

Then a tuple of first-order formulas $\overline {\xi }$ in this signature is of type $\overline {n}$ if it has the form

(17.1) $$ \begin{align} (\xi_1(\overline{v_1},\overline{x}), \ldots, \xi_k(\overline{v_k}, \overline{x})), \end{align} $$

where the displayed $\overline {v_i}$ are of length $n_i$ , and all the variables in $\overline {v_1}, \ldots , \overline {v_n}, \overline {x}$ are pairwise distinct, and do not occur free or bound in $\Phi (\overline {R})$ .

If $\overline {\xi }$ is of type $\overline {n}$ , then we let $\Phi (\overline {\xi }(\cdot , \overline {x}))$ be the first-order formula in the signature which is the result of replacing each instance of $R_{i}\overline {u}$ in $\Phi (\overline {R})$ with $\xi _i(\overline {u}, \overline {x})$ .

Hence $\Phi (\overline {\xi }(\cdot ,\overline {x}))$ has only the free variables $\overline {x}$ coming from $\overline {\xi }$ . Note that when we are writing $\Phi (\overline {\xi }(\cdot , \overline {x}))$ , we are suppressing the $\overline {v}$ variables in (17.1). If one wanted a more explicit notation, one could write $\Phi (\overline {\xi }(\overline {v},\overline {x}))$ , but this has the disadvantage of creating the impression that $\overline {v}$ is free in this formula, which it is not. The following elementary example is illustrative.

Example 17.2 (Example of substitution).

Let $\overline {R}$ be $R_1,R_2$ , where $R_1$ is unary and $R_2$ is binary, and let $\Phi (\overline {R})$ be the following:

(17.2) $$ \begin{align} \forall \; x \; \forall \; y \; \big((R_1x\wedge R_1y \wedge x\neq y)\rightarrow R_2xy\big). \end{align} $$

Then the type of $\overline {R}$ is $( 1,2)$ . Suppose that we specify the first-order signature with unary relation symbol F and binary relation symbol S. Consider the following tuple of first-order formulas $\overline {\xi }$ of type $( 1,2)$ :

(17.3) $$ \begin{align} \big( Fv_{1,1} \wedge Sx_1 v_{1,1},\hspace{3mm} S v_{2,2} v_{2,1} \rightarrow S v_{2,1} x_2\big). \end{align} $$

Then one has that $\Phi (\overline {\xi }(\cdot , \overline {x}))$ is

(17.4) $$ \begin{align} \forall \; x \; \forall \; y \; \big((Fx \wedge Sx_1 x\wedge Fy \wedge Sx_1 y \wedge x\neq y)\rightarrow (Syx\rightarrow Sxx_2)\big), \end{align} $$

which has only free variables $x_1, x_2$ .

We begin the march to the proof of Theorems 11.2 and 11.3 by showing the following.

The same result holds for $\mathsf {ACA_0^{-}}$ . We prove it for $\mathsf {ACA_0^{wo}}$ because we need this for later results in this section.