Relations with Dependence Logic

Abstract

We saw how by bringing questions into play in logic we can capture dependency relations as cases of (contextual or logical) entailment, and we can analyze such relations using standard tools of logic.

We saw how by bringing questions into play in logic we can capture dependency relations as cases of (contextual or logical) entailment, and we can analyze such relations using standard tools of logic. The notion of dependency is central to another line of work in logic which has received much attention in recent years, with the rise of dependence logic (Väänänen [1]) and other related logics based on so-called team semantics (Hodges [2, 3]).

In this chapter, we discuss some of the similarities and differences between inquisitive logic and dependence logic, in particular with regard to the treatment of dependencies in these two frameworks. We will focus here on the standard system of dependence logic; for the connections between dependence logic and inquisitive logic in the propositional and modal setting, which are also significant, the reader is referred to Ciardelli [4, 5] and to Yang and Väänänen [6].

Dependence logic is similar to inquisitive semantics in many respects. First, like inquisitive logic, it aims to achieve a conservative extension of classical logic with a new kind of formulas; in the case of inquisitive logic, the new formulas are complex formulas expressing questions; in the case of dependence logic, they are atomic formulas expressing dependencies. Moreover, in both cases, the extension is made possible by revising the standard semantics of classical logic, replacing standard points of evaluations by sets of such points: in the case of inquisitive logic, the relevant points are possible worlds, modeling states of affairs; in the case of dependence logic, they are assignment functions fixing the values of variables. Thus, the semantics of dependence logic is given relative to sets of assignments, called teams. As in the case of inquisitive logic, this can (although it need not) be seen as a semantics where formulas are evaluated with respect to states of partial information (cf. Galliani [7, 8]), where this information concerns the values of variables rather than the state of affairs. Moreover, like inquisitive logic, dependence logic (though not its variants, such as independence logic [9] and inclusion logic [7]) satisfies persistency with respect to the information ordering. This means that the logical operators that can be naturally defined in these logics are essentially the same; and indeed, many of the operators we considered in inquisitive logic have also been explored independently in the dependence logic literature (see in particular Abramsky and Väänänen [10]); at the same time, sometimes the different motivations and history of the two traditions are reflected in different choices of logical repertoire. Nevertheless, the tight formal similarity between the two approaches allows for a fruitful transfer of results and insights between them. In fact, many logical systems can be legitimately regarded either as systems of dependence logic or as systems of inquisitive logic (this applies, e.g., to the system InqBT that we will discuss in Sect. 7.4); the difference between the two traditions is mostly one of aims and conceptual perspective, which is sometimes, but not always, reflected in different technical setup choices.

One significant difference between dependence logic and inquisitive logic is the conceptualization of the dependency relation. In dependence logic, dependency is viewed as a relation holding between variables, whereas in inquisitive logic, it is viewed as a relation between questions. In this chapter we will explore the connection in detail and we will argue that, while both perspectives are meaningful and natural, the question-based perspective has some important assets to it: it is more general, allowing us to capture a broader spectrum of dependence facts, and it allows us to connect dependency directly to the central notions of logic, including entailment, proofs, and the implication operator.

The chapter is structured as follows. We start in Sect. 7.1 by introducing the variable-based perspective on dependency, which long predates the rise of dependence logic and has received much attention in database theory. In Sect. 7.2 we present the standard version of dependence logic, which combines this conception of dependency with the idea of giving a team semantics for predicate logic. In Sect. 7.3 we show how the key ideas of inquisitive logic apply naturally in the team semantic setting, yielding a question-based perspective on dependency, and we discuss some attractions of this perspective. In Sect. 7.4 we illustrate this general point by showing that the inquisitive first-order logic of the previous chapter can be naturally adapted to the team semantic setting; we discuss the sort of questions and dependencies that can be captured in this system, and we mention some important open problems about the resulting logic. In Sect. 7.5 we show how to interpret inquisitive first-order logic in a more general semantic setup, from which both the standard semantics of the previous chapter and the team-based semantics discussed in the present chapter can be obtained as special cases. Finally, in Sect. 7.6 we conclude with a summary and some further considerations.

7.1 V-Dependency in a Team

The starting point to understand the analysis of dependence in dependence logic is the notion of a team. A team is a set of assignment functions.¹

Definition 7.1.1

(Teams) A team over a domain D is a set of assignments $g:\textsf {Var}\rightarrow D$.

We can visualize a team as a table, where the columns correspond to the variables, the rows to the assignments in the team, and the cell corresponding to assignment g and variable x contains the value g(x). For instance, Fig. 7.1 represents a team of six assignments over the domain of natural numbers; only the values of these assignments on the variables x, y, z are displayed.

Fig. 7.1

An example of a team, where only the values of assignments on the variables x, y, z are displayed

In the context of such a table, it makes sense to ask whether the value of a variable is or is not determined by the values of other variables. For instance, in the team of Fig. 7.1, the value of z is determined by the value of y: if we are told the value of y on a given row in the table, we can infer from it the corresponding value of z. Conversely, the value of y is not determined by the value of z: if we are given the information that the value of z is 1, for instance, we are unable to reconstruct from that the value of y. However, the value of y is jointly determined by the values of x and z: if we are given both the value of x and the value of z on a given row, we can infer the corresponding value of y.

Generalizing, we can view dependency as a relation that may or may not hold between variables in the context of a team. We will refer to this relation here as v-dependency, to contrast it with the q-dependency relation to be discussed below, which is a relation between questions.

Definition 7.1.2

(v-dependency) Let T be a team. A set X of variables determines a variable y in the context of T, denoted $\mathbb {D}_T(X;y)$, if for every $g,g'\in T$, if $g(x)=g'(x)$ for all $x\in X$, then $g(y)=g'(y)$. We write $\mathbb {D}_T(x_1,\dots ,x_n;y)$ as a short-hand for $\mathbb {D}_T(\{x_1,\dots ,x_n\};y)$. We refer to $\mathbb {D}_T$ as the relation of v-dependency.

Focusing for simplicity on the case of a finite set of premises, we can phrase the relation $\mathbb {D}_T(x_1,\dots ,x_n;y)$ equivalently in terms of the existence of a functional dependency f that yields the value of y from the values of $x_1,\dots ,x_n$:

$$\begin{aligned} \mathbb {D}_T(x_1,\dots ,x_n;y)\iff & {} \exists f:D^n\rightarrow D\text { such that }\forall g\in T:\\{} & {} g(y)=f(g(x_1),\dots ,g(x_n)). \end{aligned}$$

Thus, for instance, in the team $X_1$ of Fig. 7.1:

• $\mathbb {D}_T(y;z)$ holds;

• $\mathbb {D}_T(z;y)$ does not hold;

• $\mathbb {D}_T(x,z; y)$ holds.

The relation of v-dependency was well-studied long before the rise of dependence logic, especially in the context of database theory [see 11, for an overview]. The most celebrated result about this relation is that the following three principles, known as Armstrong’s axioms, completely characterize the logic of v-dependency in a natural sense [12].²

• 1. $\mathbb {D}_T(X;x)$ for any $x\in X$;

• 2. $\mathbb {D}_T(X;y)$ implies $\mathbb {D}_T(X';y)$ for all $X'\supseteq X$;

• 3. $\mathbb {D}_T(Y;z)$ and $\mathbb {D}_T(X;y)$ for all $y\in Y$ implies $\mathbb {D}_T(X;z)$.

These axioms are formally analogous to Tarski’s axioms for a consequence relation, with the difference that a consequence relation is defined on formulas rather than variables. We will come back to this point in Sect. 7.3.4.

7.2 Dependence Logic

7.2.1 Historical Notes

The line of work leading to dependence logic originates with Henkin’s observation that certain patterns of quantification over individuals are not expressible in first-order logic. For instance, it is impossible to write a first-order formula expressing that for every x and $x'$, there exist a y determined only by x and a $y'$ determined only by $x'$, such that a certain formula $\phi (x,x',y,y')$ holds. To provide the tools to express such patterns, Henkin [13] introduced so-called branching quantifiers, and Hintikka and Sandu [14] later developed this work in the framework of Independence Friendly (IF) logic, which allows for quantified variables to be explicitly marked as independent of other variables. IF logic was claimed by Hintikka not to allow for a compositional semantics based on a recursive definition of truth/satisfaction. However, Hodges [2, 3] showed that such a semantics could in fact be given in the framework of team semantics, where formulas are evaluated relative to a relational structure and a team—a set of variable assignments (Hodges used the term trump instead of team, but the latter term has since become standard).

Building on the ideas of team semantics, Väänänen [1] proposed a new approach to the issue. He noticed that the team semantics context allows us to interpret a new kind of atomic formula $\,=\!\!\!(x_1,\dots ,x_n;y)$ expressing the fact that the value of y is determined by the values of $x_1,\dots ,x_n$. In this way, dependency and quantification may be disentangled. In the Dependence Logic system that he proposed, the syntax of quantification is standard, and the expression of dependencies between quantified variables is delegated to the new dependence atoms. Thus, e.g., the pattern of quantification mentioned above may be expressed as follows:

$$\forall x\forall x'\exists y\exists y'({\;=\!\!(x;y)\,}\wedge {\,=\!\!(x';y')\,}\wedge \,\phi (x,x'\!,y,y')).$$

Due to the similarity between individual variables in predicate logic and propositional variables in propositional logic, dependence atoms have later been considered also in the setting of propositional and modal logic (see, a.o., Väänänen [6], Yang [15], Yang and Väänänen [16]). In this setting, a dependence atom has the form $\,=\!\!(p_1,\dots ,p_n;q)$, and it is interpreted, relative to a set s of possible worlds, as expressing that the truth-value that a world $w\in s$ assigns to q is determined by the truth-values it assigns to $p_1,\dots ,p_n$.

At the same time, it was soon noticed that the basic idea of dependence logic could be used to extend classical predicate logic with other kinds of atoms expressing interesting relations between variables that only become “visible” at the level of teams, such as independence (Grädel and Väänänen [9]) and inclusion (Galliani [7]). In this way, the study of dependence logic evolved into a more general study of team-based logics which extend predicate logic with formulas expressing global properties of teams.

We cannot do justice here to the large amount of recent literature on these topics; for an overview, a good starting point is the Stanford Encyclopedia entry on Dependence Logic [17].

7.2.2 The Standard System D

In this section, we introduce the standard version of dependence logic, a logical system D introduced by Väänänen [1] which conservatively extends classical first-order logic with formulas expressing dependencies between variables.

The language $\mathcal {L}^{\textsf {D}}$ of first-order dependence logic is obtained by introducing, besides the usual atomic formulas of predicate logic, new atomic formulas called dependence atoms, having the form $=\!\!(x_1,\dots ,x_n,y)$, where $x_1,\dots ,x_n,y\in \textsf {Var}$. $\mathcal {L}^{\textsf {D}}$ does not have a primitive negation operator, but instead includes negative versions of the standard atoms of predicate logic, denoted $\lnot R(t_1,\dots ,t_n)$ and $t\ne t'$. Complex formulas can be formed by means of conjunction $\wedge$, a “tensor disjunction” $\otimes$, and two quantifiers $\forall ^d$  and $\exists ^d$.³ Thus, the language $\mathcal {L}^{\textsf {D}}$ is given by the following definition, where $\overline{t}=t_1,\dots ,t_n$ is a tuple of terms matching the arity of R, and $\overline{x}=x_1,\dots ,x_n$ is a tuple of variables:

$$\phi \;:=\; R\overline{t}\;|\;\lnot R\overline{t}\;|\,t=t'\,|\,t\ne t'\,|=\!\!(\overline{x};y)\;|\,\phi \wedge \phi \,|\,\phi \otimes \phi \,|\,\forall ^d x\phi \,|\,\exists ^d x\phi .$$

Intuitively, formulas without dependence atoms correspond to formulas of classical first-order logic in negation normal form (i.e., where negation only occurs in front of atomic sentences). A dependence atom of the form $=\!\!(x_1,\dots ,x_n;y)$ stands for the claim that the values of the variables $x_1,\dots ,x_n$ determine the value of the variable y.

Semantically, the language is interpreted relative to a standard relational structure $M=\langle D,I\rangle$ and a set T of assignments $g:\textsf {Var}\rightarrow D$, i.e., a team over D. In order to state the semantics of D, we first need to introduce some operations on teams.

Definition 7.2.1

(Operations on teams) Let T be a team over a domain D and let $x\in \textsf {Var}$, $d\in D$, and $f:T\rightarrow \wp ^+(D)$, where $\wp ^+(D)=(\wp (D)-\{\emptyset \})$. We define:

• $T[x\mapsto d]=\{g[x\mapsto d]\,|\,g\in T\}$;

• $T[x\mapsto f]=\{g[x\mapsto d]\,|\,g\in T\text { and }d\in f(g)\}$;

• $T[x\mapsto D]=\{g[x\mapsto d]\,|\,g\in T,d\in D\}$.

In words, $T[x\mapsto d]$ is the team that results from setting the value of x to d uniformly throughout the team; $T[x\mapsto f]$ is the team obtained by replacing each $g\in T$ by an x-variant $g[x\mapsto d]$ for each of the values $d\in f(g)$; finally $T[x\mapsto D]$ is the team obtained by taking, for each $g\in T$, all of its x-variants $g[x\rightarrow d]$ for $d\in D$.

The semantics of D can then be stated as follows, where the denotation $[t]_g^M$ of a term is defined as usual.

Definition 7.2.2

(Semantics of D)

• $M\models _T R(t_1,\dots ,t_n)\iff \text {for all }g\in T,\; \langle [t_1]_g^M,\dots ,[t_n]_g^M\rangle \in I(R)$

• $M\models _T {\lnot R(t_1,\dots ,t_n)}\iff \text {for all }g\in T,\; \langle [t_1]_g^M,\dots ,[t_n]_g^M\rangle \not \in I(R)$

• $M\models _T (t=t')\iff \text {for all }g\in T,\;[t]_g^M=[t']_g^M$

• $M\models _T (t\ne t')\iff \text {for all }g\in T,\;[t]_g^M\ne [t']_g^M$

• $M\models _T \;=\!\!\!(x_1,\dots ,x_n;y)\iff \,\mathbb {D}_T(x_1,\dots ,x_n;y)$

• $M\models _T \phi \wedge \psi \iff M\models _T \phi \text { and }M\models _T \psi$

• $M\models _T \phi \otimes \psi \iff T=T'\cup T''\text { for some }T',T''\text { s.t.\ }M\models _{T'} \phi \text { and }M\models _{T''} \psi$

• $M\models _T \forall ^d x\phi \iff M\models _{T[x\mapsto D]}\phi$

• $M\models _T \exists ^d x\phi \iff M\models _{T[x\mapsto f]}\phi$ for some $f:T\rightarrow \wp ^+(D)$

In words, a literal (positive or negative atom) is satisfied with respect to a team T in case it is true under each assignment $g\in T$. A dependence atom $=\!\!\!(x_1,\dots ,x_n;y)$ is satisfied with respect to T if the variables $x_1,\dots ,x_n$ determine y relative to T, in the sense of Definition 7.1.2. A conjunction is satisfied iff both conjuncts are satisfied. A tensor disjunction is satisfied if the team T can be split into two (not necessarily disjoint) sub-teams with each sub-team supporting one of the disjuncts. The clauses for the quantifiers are perhaps best understood by introducing the notion of x-variant of a team. Intuitively, an x-variant of a team T is a team $T'$ that differs from T only in the column corresponding to x.

Definition 7.2.3

(x-variants) Two assignments $g,g'$ are x-variants, notation $g\sim _x g'$, if they coincide on every variable except possibly x (i.e., if $g\!\upharpoonright \!_{\textsf {Var}-\{x\}}=g'\!\upharpoonright \!_{\textsf {Var}-\{x\}}$). Given a team T and a set X of variables, the restriction of T to X is obtained by restricting each element of the team:

$$T\!\upharpoonright \!_{X}\;:=\;\{g\!\upharpoonright \!_{X}\mid g\in T\}.$$

We then say that two teams $T,T'$ are x-variants if their restrictions to variables different from x is the same:

$$ T\sim _x T'\;\iff \;T\!\upharpoonright \!_{\textsf {Var}-\{x\}}=T'\!\upharpoonright \!_{\textsf {Var}-\{x\}}.$$

Equivalently, $T\sim _x T'$ if every $g\in T$ is an x-variant of some $g'\in T'$ and every $g'\in T'$ is an x-variant of some $g\in T$.

The clauses for quantifiers can then be shown to be equivalent (in the context of the present system) to the following ones:

• $M\models _T \forall ^d x\phi \iff \text {for every }T'\sim _x T,\; M\models _{T'} \phi$;

• $M\models _T \exists ^d x\phi \iff \text {for some }T'\sim _x T,\; M\models _{T'} \phi$.

Thus, a team T satisfies $\forall ^d x\phi$ (respectively, $\exists ^d x\phi$) if every (respectively, some) way of re-assigning the interpretation of the variable x leads to a team that satisfies $\phi$.⁴

The satisfaction relation has the same features which are familiar from the support relation, although the relevant information ordering now concerns a team T, rather than a set s of possible worlds: satisfaction is preserved as information grows (persistency property) and in the limit case of inconsistent information, every formula is trivially satisfied (empty team property).

Proposition 7.2.4

For any relational structure M and formula $\phi \in \mathcal {L}^{\textsf {D}}$:

• Persistency property: $M\models _T\phi$ and $Y\subseteq T$ implies $M\models _Y\phi$.

• Empty team property: $M\models _\emptyset \phi$.

In analogy to what we did in inquisitive semantics, we can recover a notion of truth relative to single assignment g by defining it in terms of satisfaction by the corresponding singleton team⁵:

$$M\models _g\phi \overset{def}{\iff }\ M\models _{\{g\}}\phi .$$

If we spell out the semantic clauses with respect to singletons, we find the following truth conditions.

Remark 7.2.5

(Truth conditions for D)

• $M\models _g R(t_1,\dots ,t_n)\iff \langle [t_1]_g^M,\dots ,[t_n]_g^M\rangle \in I(R)$

• $M\models _g (t=t')\iff [t]_g^M=[t']_g^M$

• $M\models _g \lnot {R(t_1,\dots ,t_n)}\iff \langle [t_1]_g^M,\dots ,[t_n]_g^M\rangle \not \in I(R)$

• $M\models _g (t\ne t')\iff [t]_g^M\ne [t']_g^M$

• $M\models _g \;=\!\!\!(x_1,\dots ,x_n;y)$ always

• $M\models _g \phi \wedge \psi \iff M\models _g \phi \text { and }M\models _g \psi$

• $M\models _g \phi \otimes \psi \iff M\models _g \phi \text { or }M\models _{g} \psi$

• $M\models _g \forall ^d x\phi \iff M\models _{\{g[x\mapsto d]\,|\,d\in D\}}\phi$

• $M\models _g \exists ^d x\phi \iff M\models _{g[x\mapsto d]}\phi$ for some $d\in D$

Here, it is important to notice that the truth conditions for a universal formula $\forall ^d\phi$ relative to g depend on the satisfaction conditions of $\phi$ at a non-singleton team obtained by taking all x-variants of g. This means that, unlike in the other systems we encountered so far (but similarly to systems of inquisitive modal logic, see e.g. Ciardelli and Roelofsen [18]), truth does not admit a direct recursive characterization; rather, computing the truth conditions of some formula $\phi$ in general requires computing the satisfaction conditions of some sub-formula $\psi$ with respect to non-singleton teams.

A formula is said to be flat if satisfaction at a team T reduces to truth under each assignment $g\in T$. Clearly, the notion of flatness is the counterpart of the notion of truth-conditionality that we encountered in inquisitive logic.

Definition 7.2.6

(Flatness) We call a formula $\phi \in \mathcal {L}^{\textsf {D}}$ flat if for any model M and team T:

$$M\models _T\phi \iff M\models _g\phi \text { for all }g\in T.$$

It is easy to check by induction that all formulas of D without dependence atoms are flat. This means that their semantics is fully captured by their truth conditions relative to single assignments.

Moreover, these truth conditions are simply the ones familiar from the Tarskian semantics of first-order logic, when $\otimes$ is identified with disjunction and $\forall ^d$ and $\exists ^d$  with the quantifiers of first-order logic. This can be proved by induction. The key case is that of a universal formula $\forall ^d x\phi$ (the remaining cases are obvious from Remark 7.2.5). Since we are assuming $\forall ^d x\phi$ does not contain dependence atoms, neither does $\phi$, and so $\phi$ is flat. Using Remark 7.2.5 and the flatness of $\phi$ we have:

$$\begin{aligned} M\models _g\forall ^d x\phi\iff & {} M\models _{\{g[x\mapsto d]\mid d\in D\}}\phi \\\iff & {} \forall g'\in \{g[x\mapsto d]\mid d\in D\}: \;M\models _{g'}\phi \\\iff & {} \forall d\in D: \;M\models _{g[x\mapsto d]}\phi . \end{aligned}$$

This shows that the semantics of formulas not containing dependence atoms is a global counterpart of standard Tarskian semantics (identifying disjunction with $\otimes$ and the first-order quantifiers with $\exists ^d$  and $\forall ^d$). Moreover, these formulas are representative of all formulas of first-order predicate logic, as every formula $\phi$ of first-order logic can be identified with its negation normal form $\phi ^*$, obtained recursively by pushing negations in front of atoms. This means, by a reasoning analogous to the one we used for inquisitive semantics (cf. Sect. 2.2), that in restriction to the “classical fragment” of the language, consisting of formulas without dependence atoms, the above semantics can be seen as a non-standard semantics for classical first-order logic.

Thus, dependence logic and inquisitive first-order logic both extend classical first-order logic by using a similar strategy. We saw that inquisitive first-order logic is a conservative extension of classical first-order logic with questions, obtained by first giving a state-based semantics for classical first-order logic and then exploiting this semantics to interpret new question-forming operators. Similarly, dependence logic can be seen as a conservative extension of classical first-order logic, obtained by first giving a team-based semantics for classical first-order logic and then exploiting this semantics to interpret a new kind of atoms capturing dependencies. Moreover, in both cases the new semantics is obtained in a similar way, by moving from single “points of evaluation” to sets of such points, ensuring that for classical formulas the semantics is distributive, in the sense that satisfaction at a set of points boils down to satisfaction at each element of the set.

One difference is that, whereas in the case of standard inquisitive logic the generalization targets the model of interpretation (thus moving from a single relational structure to a set of possible worlds, each associated with such a structure), in the case of dependence logic it targets the assignment function (thus moving from a single assignment to a set of assignments). This is not an irreconcilable difference, however: as we will see in the next sections, it is possible to implement the key ideas of inquisitive logic in the setting of team semantics, and it is also possible to give a semantic framework that is a common generalization of both information state semantics and team semantics.

Before turning to that, let us look at some of the key features of D. As we expect, the support-conditions for a formula $\phi$ relative to a team T are only sensitive to the values that the assignments in T assign to the variables that occur free in $\phi$.

Proposition 7.2.7

Let $\phi \in \mathcal {L}^{\textsf {D}}$. If T and $T'$ are teams such that $T\!\upharpoonright \!_{\text {FV}(\phi )}=T'\!\upharpoonright \!_{\text {FV}(\phi )}$, then $M\models _T\phi \iff M\models _{T'}\phi$.

In particular, consider the case in which $\phi$ is a sentence, i.e., FV$(\phi )=\emptyset$. Then for every non-empty team T we have $T\!\upharpoonright \!_{\text {FV}(\phi )}=\{\emptyset \}$ (since $\emptyset$ is the only function from the empty set of variables to D). Thus, for any non-empty teams $T,T'$ we have $T \!\upharpoonright \!_{\text {FV}(\phi )}=T' \!\upharpoonright \!_{\text {FV}(\phi )}$, and therefore $M\models _T\phi \iff M\models _{T'}\phi$. Thus, if $\phi \in \mathcal {L}^{\textsf {D}}$ is a sentence, we can simply write $M\models \phi$ as a shorthand for $M\models _T\phi$, where T is an arbitrary non-empty team.

In this way, sentences of dependence logic, just as sentences in classical first-order or second-order logic, define classes of relational structures. This naturally raises the question of how the expressive power of these different systems compares.

This question was answered by Väänänen [1], who showed that, as far as sentences are concerned, D has the same expressive power as $\Sigma _1^1$, the existential fragment of second-order logic, consisting of second-order formulas of the form

$$\exists T_1\dots \exists T_n\phi $$

where $T_1,\dots ,T_n$ are second-order variables, and $\phi$ contains no second-order quantifiers. To state Väänänen’s result, let us introduce the following terminology: if $\phi$ is a sentence in $\mathcal {L}^{\textsf {D}}$ and $\psi$ a sentence in $\Sigma _1^1$ (over the same signature), we will say that $\phi$ and $\psi$ are equivalent, and write $\phi \equiv \psi$, in case for any model M we have $M\models \phi \iff M\models \psi$. Then, we have the following theorem.

Theorem 7.2.8

(Väänänen [1]) There exist computable maps $(\cdot )^{eso}:\mathcal {L}^{\textsf {D}}\rightarrow \Sigma _1^1$ and $(\cdot )^d:\Sigma _1^1\rightarrow \mathcal {L}^{\textsf {D}}$ such that:

• for any sentence $\phi \in \mathcal {L}^{\textsf {D}}$, $\,\phi \equiv \phi ^{eso}$;

• for any sentence $\phi \in \Sigma _1^1$, $\;\phi \equiv \phi ^{d}$.

This theorem implies that first-order dependence logic is not recursively axiomatizable. If it were, then the set of its valid sentences would be recursively enumerable. But by the previous theorem, this would imply that the set of valid $\Sigma _1^1$ sentences is recursively enumerable, which is not the case. In fact, Väänänen [1] shows that the set of (Gödel numbers of) theorems of D is not only not recursively enumerable, but not even arithmetical, i.e., the property of being a code of a valid D-sentence is not expressible in the language of Peano arithmetic.

For a simple example of a dependence logic sentence that is not equivalent to any sentence in classical first-order logic, consider the following:

$$\exists ^d x\forall ^d y\exists ^d z(=\!\!(z,y)\wedge z\ne x).$$

Spelling out the semantic clauses, one can verify that this sentence is satisfied in a model $M=\langle D,I\rangle$ iff there exists a function $f:D\rightarrow D$ which is injective but not surjective. Such a function exists iff D is infinite. Thus, the above formula is satisfied by exactly those models whose domain is infinite—thus expressing a property which is not expressible in standard first-order logic.

This example also shows that D is not entailment compact: if $\xi _n$ is a first-order formula that says that there are at least n individuals in D, then the above sentence is entailed by the set $\{\xi _n\mid n\in \mathbb {N}\}$ (since if all $\xi _n$ are true, D must be infinite), but not by any finite subset of this set (since the truth of finitely many $\xi _n$ is compatible with the finiteness of D).⁶

7.3 Q-Dependency

In the previous sections we discussed the relation of v-dependency in a team and we saw how first-order logic can be extended with formulas that express v-dependencies. We will now see that adopting the ideas of inquisitive logic in the team semantics setting yields another perspective on the notion of dependency. Under this perspective, dependency is viewed, not as a relation between variables, but as a relation between questions (in the way familiar from the previous chapters). We will refer to this notion of dependency as q-dependency. In this section, we will consider the relation between the two perspectives on dependency, and we will discuss a number of attractions of the question-based perspective.

7.3.1 Inquisitive Logic in Team Semantics

The basic ideas of inquisitive logic, as laid out in Chap. 2, apply straightforwardly to the setting of team semantics. For instance, it is natural to consider a statement$\alpha$ as supported in the context of a team T if it is true under all assignments in T. Thus, we expect the following analogue of the Truth-Support Bridge to hold for a team T and a statement $\alpha$:

$$M\models _T\alpha \iff (M\models _g\alpha \text { for all }g\in T).$$

For an illustration, consider the team:

This team supports the statement $\alpha _1$ below, but not $\alpha _2$ or $\alpha _3$.

However, $\alpha _2$ is supported by the sub-team $\{g_3,g_4\}$ and all subsets of this team, while $\alpha _3$ is supported by $\{g_1,g_2,g_3\}$ and its subsets.

We may refer to the maximal sub-teams of a team T supporting a sentence $\phi$ as the alternatives for $\phi$ in T, denoted $\textsc {Alt}_T(\phi )$. Then the alternatives for our three statements in our team $T_2$ are the blocks depicted in Fig. 7.2.

Fig. 7.2

The alternatives for three statements within the team $T_2$

It is equally natural to interpret questions involving variables in the context of a team. As an example, let us first consider two particular sorts of questions.

Example 7.3.1

(Identification questions, $\lambda x$) With any variable x we can associate an identification question $\lambda x$, standing for the question what the value of x is. A team T settles $\lambda x$ if it settles what the value of x is, i.e., if every assignment $g\in T$ assigns to x the same value:

$$M\models _T\lambda x\iff \forall g,g'\in T: g(x)=g'(x).$$

Example 7.3.2

(Polar questions, $?\alpha$) With any statement $\alpha$ we can associate a corresponding polar question $?\alpha$, standing for the question whether $\alpha$ is true or false. A team T settles the question $?\alpha$ if it determines what the value of $\alpha$ is, i.e., if every assignment $g\in T$ assigns to $\alpha$ the same truth value:

$$M\models _T{?\alpha }\iff \forall g,g'\in T: (M\models _g\alpha \iff M\models _{g'}\alpha ).$$

Figure 7.3 illustrates these examples by showing the alternatives for two identification questions and a polar question in the context of the team $T_2$.

Fig. 7.3

The alternatives for three questions within the team $T_2$

All notions, facts, and considerations that we discussed in Chap. 2 carry over straightforwardly to the team semantics setting. We will only restate explicitly those facts and notions that play a special role in our discussion below.

7.3.2 Q-Dependency

The inquisitive approach we just described yields a natural analysis of dependency as a relation holding between questions in the context of a team. Within a team T, a question $\mu$ is fully determined by a set $\Lambda$ of questions if $\mu$ is settled in any sub-team of T which settles all questions in $\Lambda$. This gives us the central notion of q-dependency.

Definition 7.3.3

(Q-dependency) A set $\Lambda$ of questions determines a question $\mu$ in the context of a team T, denoted $\Lambda \models _T\mu$, if for every $T'\subseteq T$, $M\models _{T'}\lambda$ for all $\lambda \in \Lambda$ implies $M\models _{T'}\mu$. We write $\lambda _1,\dots ,\lambda _n\models _{T}\mu$ as a shorthand for $\{\lambda _1,\dots ,\lambda _n\}\models _{T}\mu$.

It is easy to see that v-dependency is a special case of q-dependency involving identification questions. This is made precise by the following fact, which the reader is invited to check.

Proposition 7.3.4

(v-dependencies are q-dependencies) Let $X\cup \{y\}$ be a set of variables and T a team. Let $\lambda X$ stand for the set of identification questions $\{\lambda x\mid x\in X\}$. We have:

$$\mathbb {D}_T(X;y)\;\iff \; \lambda X\models _T\lambda y.$$

Thus, v-dependencies correspond to a special class of q-dependencies—those which involve only identification questions. However, the notion of q-dependency is much more general: besides v-dependencies, there are many other patterns that can be recognized and captured naturally as cases of q-dependency, as we discuss in the next section.

7.3.3 Generality

Consider the following team, in which the values of x and y always differ by one:

$T_3$	x	y
$g_1$	1	0
$g_2$	1	2
$g_3$	2	1
$g_4$	2	3
$g_5$	3	2
$g_6$	3	4
$g_7$	4	3
$g_8$	4	5

In this team, no non-trivial v-dependencies hold: the value of x neither determines nor is determined by the value of y. Yet, in this table we can still recognize many interesting patterns that one would naturally regard as dependencies. For instance, here are some facts:

• the value of x determines the parity of y (i.e., whether y is even or odd);

• the parity of x determines the parity of y;

• the value of x and whether $x<y$ determines the value of y.

These facts can be captured straightforwardly as q-dependencies involving not only identification questions, but also polar questions, as follows:

• $\lambda x\;\models _{T_3}\;?\text {Even}(y)$;

• $?\text {Even}(x)\;\models _{T_3}\;?\text {Even}(y)$;

• $\lambda x,\,?(x<y)\;\models _{T_3}\;\lambda y$.

One can also give other examples where the sort of questions involved are of different kinds, for instance mention-some or mention-all wh-questions. For instance, consider the following team:

$T_4$	x	y	z
$g_1$	1	2	2
$g_2$	1	4	6
$g_3$	2	3	3
$g_4$	2	9	6
$g_5$	3	6	5
$g_6$	3	12	10
$g_7$	3	18	15
$g_8$	3	24	15

Although the value of x determines neither the value of y nor the value of z, in this table we have the following dependence patterns:

• the value of x determines the set PF(y) of prime factors of y (if $x=1$ then $PF(y)=\{2\}$; if $x=2$ then $PF(y)=\{3\}$; if $x=3$ then $PF(y)=\{2,3\}$);

• the value of x yields some prime factor of z (if $x=1$ then $2\in PF(z)$; if $x=2$ then $3\in PF(z)$; if $x=3$ then $5\in PF(z)$).

These facts can be captured naturally as cases of q-dependency involving the following wh-questions:

• The question $\mu _\forall (y):=\text {`what are the prime factors of}$y$'$, supported by a team just in case the team determines the exact set of prime factors of y:

  $$M\models _T\mu _\forall (y)\;\iff \;\forall g,g'\in T: PF(g(y))=PF(g'(y)).$$

• The question $\mu _\exists (z):=\text {`what is one prime factor of}$z$'$, supported by a team T just in case T implies of some number n that it is a prime factor of z:

  $$M\models _T\mu _\exists (z)\iff \exists n\in \mathbb {N}\forall g\in T: n\in PF(g(z)).$$

The two dependence patterns noticed above amount to the q-dependencies:

• $\lambda x\;\models _{T_4}\;\mu _\forall (y)$;

• $\lambda x\;\models _{T_4}\;\mu _\exists (z)$.

We will see in the next section how the relevant questions can be expressed in a formal language by adapting the tools of standard inquisitive predicate logic. For now, the important point is that the notion of q-dependency allows us to view v-dependencies as a special case of a much broader spectrum of logical facts that share common features and that are naturally analyzed in a uniform way. This includes all claims of the form ‘such-and-such information about x yields such-and-such information about y’, where the relevant information need not be the complete information giving the exact value of the variable, but could instead be partial information concerning, e.g., parity, set of prime factors, etc.⁷

In fact, coming back to the idea of information types discussed in detail in Chap. 2, I would like to suggest that we have a dependency whenever information of certain types is guaranteed to yield information of another type. Call these the input types and the output type of the dependency.

As we discussed in Chap. 2, questions can be seen as names for information types. Thus, e.g., $\lambda x$ stands for information of type ‘value of x’, while $?\text {Even}(x)$ stands for information of type ‘parity of x’, and $\mu _\forall (x)$ defined above for information of type ‘prime factors of x’. Now to represent an arbitrary dependency as an instance of q-dependency, we just need to find questions $\mu _1,\dots ,\mu _n$ that correspond to the input types of the dependency and a question $\nu$ that stands for the output type. Then the dependency amounts precisely to the fact that $\mu _1,\dots ,\mu _n\models _T\nu$. In this sense, to the extent that we accept the above idea of what a dependency is, the question approach is bound to be a fully general one.⁸

7.3.4 Taking Dependencies to the Core of Logic

Perhaps the greatest merit of the question-based perspective on dependency is that it brings out the deep connections existing between dependency and logical notions like entailment, conjunction, implication, and proof. In this section, we briefly review these manifold connections, with a focus on how they play out in the team semantics setting.

Q-dependency and Tarskian consequence. First, q-dependency in a team is a Tarskian consequence relation. This means that it is a relation between formulas which satisfies the following three properties:

• Reflexivity: $\Lambda \models _T\lambda$ for all $\lambda \in \Lambda$;

• Weakening: $\Lambda \models _T\mu$ implies $\Lambda '\models _T\mu$ for $\Lambda '\supseteq \Lambda$;

• Transitivity: $\Lambda \models _T\mu$ and $\Lambda '\models _T\lambda$ for all $\lambda \in \Lambda$ implies $\Lambda '\models _T\mu$.

Thus, q-dependency in a given team is a consequence relation among questions. Since v-dependency can be seen as a special case of q-dependency via the equivalence

$$\mathbb {D}_T(X;y)\;\iff \; \lambda X\models _T\lambda y, $$

this means in particular that Armstrong’s axioms discussed in Sect. 7.1 can be seen as a special case of the axioms for consequence.

Q-dependency and logical operators. The fact that the relation of q-dependency connects formulas, rather than variables, has important repercussions as well. Formulas, unlike variables, can be combined by means of logical operations, which gives us important tools to manipulate dependency claims. For an illustration, reproducing standard inquisitive semantics in the team setting we can introduce two connectives $\wedge$ and $\rightarrow$ which work as follows:

• $M\models _T\phi \wedge \psi \iff M\models _T\phi \text { and }M\models _T\psi$;

• $M\models _T\phi \rightarrow \psi \iff \forall T'\subseteq T: M\models _{T'}\phi \text { implies }M\models _{T'}\psi$.

These connectives interact with the q-dependency relation in the way conjunction and implication standardly interact with a consequence relation. We have:

• $\Lambda ,\mu _1,\mu _2\models _T\nu \iff \Lambda ,\mu _1\wedge \mu _2\models _T\nu$;

• $\Lambda ,\mu \models _T\nu \iff \Lambda \models _T\mu \rightarrow \nu$.

Thus, we can always trade multiple determining questions for a single conjunctive one, and we can always drop one of the determining questions and correspondingly weaken our conclusion to a conditional question. This is just an illustration of the fact that q-dependency is a consequence relation that is naturally related to a set of well-behaved logical operations on questions.

Interestingly, moreover, the relevant operations are generalizations to questions of the standard operations of classical logic: when applied to statements$\alpha$ and $\beta$, the connectives $\wedge$ and $\rightarrow$ we just defined yield formulas $\alpha \wedge \beta$ and $\alpha \rightarrow \beta$ which behave as conjunction and material conditional in classical logic. Thus, by working with questions we can handle the premises and the conclusion of a dependence relation by means of logical operators that obey familiar properties and which, moreover, generalize the familiar operators of classical logic. This is a significant finding.

Notice also that, as we discussed in Sect. 2.5, implication yields a fully general way to express q-dependencies in the object language. Indeed, one can check that we have:

$$\begin{aligned} \lambda _1,\dots ,\lambda _n\models _T\mu\iff & {} M\models _T\lambda _1\wedge \dots \wedge \lambda _n\rightarrow \mu \\\iff & {} M\models _T\lambda _1\rightarrow (\dots \rightarrow (\lambda _n\rightarrow \mu )). \end{aligned}$$

So, the fact that $\lambda _1,\dots ,\lambda _n$ determine $\mu$ is expressed in the object language by the formula $\lambda _1\wedge \dots \wedge \lambda _n\rightarrow \mu$, or equivalently by $\lambda _1\rightarrow (\dots \rightarrow (\lambda _n\rightarrow \mu ))$. This brings out a deep connection existing between q-dependency and the implication connective of inquisitive logic.

Conditional q-dependencies for free. Consider the following team:

$T_5$	x	y
$g_1$	1	0
$g_2$	1	2
$g_3$	2	3
$g_4$	2	3
$g_5$	3	2
$g_6$	3	4
$g_7$	4	5
$g_8$	4	5

In this team, the value of x does not generally determine the value of y, but it does so in restriction to those assignments in which the value of x is even. This is an example of a conditional dependency.

In the question-based perspective on dependency, conditional dependencies are captured straightforwardly by allowing statements, in addition to questions, as premises of a q-dependence relation. Since statements and questions can both be interpreted in terms of the same notion of support, the definition of the relation $\models _T$ does not need to be generalized to accommodate statements, but can be applied directly. Following a reasoning analogous to the one in Sect. 2.3.3, we can then verify that the following holds.

Proposition 7.3.5

Let $\Gamma$ be a set of statements and $\Lambda \cup \{\mu \}$ a set of questions. Let T be a team and let $|\Gamma |_M=\{g\mid M\models _g\gamma \text { for all }\gamma \in \Gamma \}$. We have:

$$\Gamma ,\Lambda \models _T\mu \iff \Lambda \models _{T\cap |\Gamma |_M}\mu .$$

This means that the relation $\Gamma ,\Lambda \models _T\mu$ captures a conditional q-dependency: the questions in $\Lambda$ determine the question $\mu$, not (necessarily) relative to the entire team T, but relative to those assignments in T that satisfy $\Gamma$. We can read the relation $\Gamma ,\Lambda \models _T\mu$ as ‘$\Lambda$ determines $\mu$ given $\Gamma$’.

Notice also that the approach vindicates the connection between conditional dependencies and conditionals. Let us focus for simplicity on the case in which a question $\mu$ determines a question $\nu$ given a statement $\alpha$, i.e., the case in which $\alpha ,\mu \models _T\nu$. This can be expressed in the object language by the formula:

$$\alpha \rightarrow (\mu \rightarrow \nu ).$$

Recall that $\mu \rightarrow \nu$ expresses the fact that $\mu$ determines $\nu$ in the evaluation state. Thus, a conditional dependency is expressed by a conditional having the condition $\alpha$ as antecedent and the dependence formula $\mu \rightarrow \nu$ as consequent.

Summing up, there is no need to further generalize the q-dependency relation $\models _T$ in order to capture conditional dependencies: it suffices to allow statements to be plugged in as determinants alongside questions. In addition, conditional dependencies can be expressed smoothly in the language as conditionals having the relevant conditions as antecedents.

Q-dependency and logical entailment. As usual, the inquisitive approach comes with a general notion of entailment, defined in terms of preservation of support, where the premises and the conclusion can be statements or questions. In the team semantics setting, this is given by the following definition.

Definition 7.3.6

(Entailment)

$\Phi \models \psi \iff \text {for every model}$ M and team $T:M\models _T\Phi \text { implies }M\models _T\psi$.

This general notion of entailment is a conservative extension of the standard entailment relation for statements. That is, if $\Gamma \cup \{\alpha \}$ is a set of statements, for which support amounts to global truth, we have:

$$\Gamma \models \alpha \iff \text {for every model { M} and assignment { g}}:M\models _g\Gamma \text { implies }M\models _g\alpha .$$

At the same time, in the case in which we have question assumptions and a question conclusion, this general notion of entailment captures logical q-dependencies, i.e., q-dependencies which hold in virtue of the logical form of the sentences involved, regardless of the interpretation of non-logical symbols. That is, suppose $\Gamma$ is a set of statements and $\Lambda \cup \{\mu \}$ a set of questions. We have:

$$\Gamma ,\Lambda \models \mu \iff \Gamma ,\Lambda \models _T\mu \text { for any team }T\text { in any model }M.$$

Thus, $\Gamma ,\Lambda \models \mu$ captures the fact that $\Lambda$ logically determines $\mu$ given $\Gamma$.

For a simple example, in the team version of inquisitive first-order logic given in the next section, the following are simple examples of logical q-dependencies (where P, Q are unary predicates and f a unary function symbol):

$$?Px,\;Px\leftrightarrow \lnot Qy\;\models \;?Qy,$$

$$\lambda x,\;y=f(x)\;\models \;\lambda y.$$

Summing up, we saw that q-dependency comes in two versions: a contextual version, relativized to a team, and a logical version, obtained by quantifying over all teams. Logical q-dependencies are those q-dependencies that hold purely on the basis of the logical form of the sentences involved. The main insight of the question-based perspective is that logical q-dependency is nothing but a facet of the central notion of logic, the notion of entailment, once this notion is generalized to apply not just to statements, but also to questions.

Q-dependency and logical proofs. One important repercussion of the fact that logical q-dependencies are logical entailments is that such dependencies can be formally proved if we have a proof system for (a fragment of) our logic. This brings out the connections between dependency and another central concern of logic, namely, proofs.

For an example, consider the logical q-dependency $?Px,\;Px\leftrightarrow \lnot Qy\;\models \;?Qy$ discussed above. We can prove the validity of this dependency in exactly the same way as we can prove the entailment ${?p},p\leftrightarrow \lnot q\models {?q}$ in InqB, using standard inference rules for disjunction and implication (recall that in inquisitive logic, polar questions$?\alpha$ are realized as inquisitive disjunctions). Omitting proof steps which involve only statements, we have the following proof.

The fact we saw above, that q-dependencies are expressed by implications in the object language, also has repercussions in proofs, since it means that we can make inferences with dependence formulas just as we normally do with implications. In order to show that a dependency $\mu \rightarrow \nu$ holds under certain assumptions, we can simply suppose the question $\mu$ and try on that basis to derive the question $\nu$. This corresponds to the standard implication introduction rule.⁹ Moreover, if we have a dependency $\mu \rightarrow \nu$ and we also have the determining question $\mu$, we can conclude the determined question $\nu$. This is just the standard implication elimination rule.¹⁰

Summing up, then, by manipulating questions in inferences we can prove that certain dependencies are logically valid—i.e., hold merely on the basis of the logical form of the sentences involved. Note that in order for this to be possible, it matters that questions—unlike variables—have syntactic structure to them. Since questions are built up by means of certain logical operators, we can make inferences with them by using the inference rules for these operators, as illustrated by the above example of a proof.

7.3.5 Wrapping Up

We saw that the core ideas of inquisitive semantics, as developed in Chap. 2, apply straightforwardly in the setting of team semantics. This allows us to interpret statements and questions involving variables uniformly in the context of a team. This perspective yields a natural notion of dependency as a relation between questions, which encompasses v-dependency as a special case, but which is much more general, capturing not only dependencies of the form ‘the value of $x_1,\dots ,x_n$ yields the value of y’, but also, for instance, all dependencies of the form ‘such-and-such information about $x_1,\dots ,x_n$ yields such-and-such information about y’. Moreover, since on this view dependency is a relation between questions, and since questions are sentences, we can uncover a number of significant connections between dependency and generalized versions of the classical logical operators, consequence relation, and proof system.

The general view discussed in this section can be implemented in many particular formal systems, differing from each other in their set of primitive logical operators. In the next section, we will make the discussion more concrete by considering one particular implementation of these ideas, which stems from interpreting the language of inquisitive first-order logic in the setting of team semantics.

7.4 The System InqBT

In this section, we will see how the language $\mathcal {L}^{\textsf {Q}=}$ of inquisitive first-order logic can be given a natural interpretation in the setting of team semantics, where formulas are interpreted relative to a single model and a set of assignments. We will refer to this system as InqBT, where the letter T marks the fact that formulas are interpreted with respect to teams. In the dependence logic literature, this system has been considered by Yang [16] under the name of WID (for weak intuitionistic dependence logic), and most of the results mentioned here can already be found in her work. However, we will interpret the system in the light of the conceptual picture of inquisitive semantics, as laid out in Chap. 2 and in the previous section.

7.4.1 Syntax and Semantics

The language of InqBT is just the language $\mathcal {L}^{\textsf {Q}=}$ studied in the previous chapter, given by the following syntax:

where p is an atom in the given signature (of the form $R(t_1,\dots ,t_n)$ or $(t=t')$). We regard the inquisitive connectives and $\mathord {\exists \!\!\!\exists }$ as question-forming operators in the way familiar from the previous chapter. Formulas without these operators are called classical and identified with formulas of classical first-order logic. The set of classical formulas is denoted $\mathcal {L}^{\textsf {Q}=}_c$. The operators $\lnot ,\vee ,\exists ,$ and ? are defined as in the previous chapter.

The semantics of InqBT is given in the setting of teams semantics: formulas are evaluated with respect to a relational structure $M=\langle D,I\rangle$ and a team T. The clauses are identical to those for $\textsf {InqBQ}$, except that now it is the team that plays the role of the information state.

Definition 7.4.1

(Semantics of InqBT)

• $M\models _T R(t_1,\dots ,t_n)\iff \text {for all }g\in T,\; \langle [t_1]_g^M,\dots ,[t_n]_g^M\rangle \in I(R)$

• $M\models _T (t=t')\iff \text {for all }g\in T,\;[t]_g^M=[t']_g^M$

• $M\models _T\bot \iff T=\emptyset$

• $M\models _T \phi \wedge \psi \iff M\models _T \phi \text { and }M\models _T \psi$

• $M\models _T \phi \rightarrow \psi \iff \text {for all }T'\subseteq T: M\models _{T'} \phi \text { implies }M\models _{T'} \psi$

• 

• $M\models _T \forall x\phi \iff M\models _{T[x\mapsto d]}\phi$ for every $d\in D$

• $M\models _T \mathord {\exists \!\!\!\exists }x\phi \iff M\models _{T[x\mapsto d]}\phi$ for some $d\in D$

The clauses for atoms are the same as in D: an atomic sentence is settled with respect to a team T if it is true under any assignment $g\in T$. Notice that unlike in D, we do not need negative atoms in the language, since the same result can be produced compositionally by negating atoms by means of the negation operator, defined as $\lnot \phi :=\phi \rightarrow \bot$. The clauses for the connectives are the familiar inquisitive clauses, except that now, the relevant information ordering concerns the team T. The clauses for the quantifiers are also very similar to those we used in $\textsf {InqBQ}$, except that instead of setting the value of x to d in just one assignment, we have to do this for all assignments $g\in T$. Setting the value of x to d throughout T amounts to stipulating that x denotes d; this allows us to look at what is settled in T about the object d, rather than about the variable x.

7.4.2 Basic Properties

Support in InqBT has the usual features.

Proposition 7.4.2

For any relational structure M and $\phi \in \mathcal {L}^{\textsf {Q}=}$ we have:

• Persistence property: $M\models _T\phi$ and $T'\subseteq T$ implies $M\models _{T'}\phi$.

• Empty team property: $M\models _\emptyset \phi$.

We define the notion of truth by setting $M\models _g\phi \iff M\models _{\{g\}}\phi$. We can then check that all the standard operators have the familiar truth-conditions, while the inquisitive operators   and $\mathord {\exists \!\!\!\exists }$ have the same truth-conditions as the corresponding classical operators $\vee$ and $\exists$.

Proposition 7.4.3

(Truth-conditions for InqBT)

• $M\models _g R(t_1,\dots ,t_n)\iff \langle [t_1]_g^M,\dots ,[t_n]_g^M\rangle \in I(R)$

• $M\models _g (t=t')\iff [t]_g^M=[t']_g^M$

• $M\not \models _g\bot$

• $M\models _g \phi \wedge \psi \iff M\models _g \phi \text { and }M\models _g \psi$

• $M\models _g \phi \rightarrow \psi \iff M\not \models _g \phi \text { or }M\models _g \psi$

• 

• $M\models _g \forall x\phi \iff M\models _{g[x\mapsto d]}\phi$ for all $d\in D$

• $M\models _g \mathord {\exists \!\!\!\exists }x\phi \iff M\models _{g[x\mapsto d]}\phi$ for some $d\in D$

Notice that now, unlike in D, the truth-conditions for a formula $\phi \in \mathcal {L}^{\textsf {Q}}$ depend only on the truth-conditions for the sub-formulas of $\phi$: computing the truth-conditions for $\phi$ never requires moving to non-singleton teams. Also, recall from the previous chapter that $\phi ^{cl}$ is the classical formula obtained from $\phi$ by replacing each occurrence of  and $\mathord {\exists \!\!\!\exists }$ with $\vee$ and $\exists$, respectively. Then, it follows from the previous proposition that $\phi$ and $\phi ^{cl}$ always have the same truth-conditions.

Corollary 7.4.4

For any model M, assignment g, and $\phi \in \mathcal {L}^{\textsf {Q}}$, $\;M\models _g\phi \iff M\models _g\phi ^{cl}$.

Thus, any formula in InqBT has the same truth conditions as some classical formula. This is strikingly different from the situation in D, where some formulas have truth conditions which are not shared by any standard first-order formula; an example is the sentence $\exists ^d x\forall ^d y\exists ^d z(=\!\!(z,y)\wedge z\ne x)$ which is true only if the domain of the model is infinite.

We say that $\phi$ is truth-conditional (or flat, in the dependence logic lingo) if $\phi$ is supported by a team T whenever it is true relative to each $g\in T$. As in $\textsf {InqBQ}$, classical formulas are always truth-conditional. This fits the idea that we regard such formulas as statements and that for statements, support is connected to truth via the Truth-Support Bridge.

Proposition 7.4.5

Every $\alpha \in \mathcal {L}^{\textsf {Q}=}_c$ is truth-conditional in InqBT.

Since classical formulas are truth-conditional and their truth conditions are the standard ones, with respect to such formulas what we have given is simply a team semantics for classical first-order logic.

As in the system D, the support conditions for a formula $\phi$ depend only on the values that the assignments in the team T give for those variables which actually occur free in $\phi$.

Proposition 7.4.6

Let $\phi \in \mathcal {L}^{\textsf {Q}=}$. If $T\!\upharpoonright \!_{\text {FV}(\phi )}=T'\!\upharpoonright \!_{\text {FV}(\phi )}$, then

$$M\models _T\phi \iff M\models _{T'}\phi .$$

In particular, a sentence, which has no free variables, is not sensitive to the team of evaluation at all, as long as this team is non-empty. If $\phi$ is a sentence, we can thus write $M\models \phi$ as a shorthand for $M\models _T\phi$, where T is any non-empty team. As a corollary of this fact, we get that all sentences are trivially truth-conditional. Thus, in InqBT, unike in $\textsf {InqBQ}$, no sentence is a question, even if it contains occurrences of   and $\mathord {\exists \!\!\!\exists }$. Notice that, since a sentence $\phi$ and its classical variant $\phi ^{cl}$ have the same truth-conditions (Corollary 7.4.4), and since both are truth-conditional, we always have $\phi \equiv \phi ^{cl}$.

Proposition 7.4.7

If $\phi$ is a sentence, $\;\phi \equiv \phi ^{cl}$ in InqBT.

Thus, any sentence of InqBT is equivalent to a classical first-order sentence. Again, this is very different from what we find in D, where some sentences are equivalent to properly second-order sentences of standard predicate logic.

On the other hand, in InqBT things become interesting as soon as we consider formulas with free variables. As we show in the next section, by using the inquisitive operators   and $\mathord {\exists \!\!\!\exists }$ we can capture many classes of questions concerning the values of variables.

7.4.3 Questions in InqBT

In this section we show how the different sorts of questions concerning variables that came up in our discussion in Sect. 7.3.3 can be captured by formulas in InqBT. For our illustration we will make use of Fig. 7.4, which depicts four teams over a domain of natural numbers. In our examples, the questions we will consider have x as their only free variable. Proposition 7.4.6 ensures that the value of the assignments in T on the variable x is all that matters to decide on the support of these questions, which is why our tables in the figure consist of only one column, the one corresponding to the variable x.

Example 7.4.8

(Polar questions) Given a classical formula $\alpha$, consider the formula . Using Proposition 7.4.5, we have:

$$\begin{aligned} M\models _T{?\alpha }\iff & {} M\models _T \alpha \text { or }M\models _T\lnot \alpha \\\iff & {} [\,M\models _g \alpha \text { for all }g\in T\,]\text { or }[\,M\models _g\lnot \alpha \text { for all }g\in T\,]\\\iff & {} \forall g,g'\in T: [\,M\models _g\alpha \iff M\models _{g'}\alpha \,]. \end{aligned}$$

Thus, $?\alpha$ captures the polar question whether $\alpha$, which is settled relative to a team T if all the assignments in the team agree on whether $\alpha$ is true or false.

For instance, suppose our domain is the set $\mathbb {N}$ of natural numbers, and Even is a predicate symbol interpreted as the set of even numbers. Then, $?\text {Even}(x)$ expresses the question whether the value of x is even or odd, which is supported by a team just in case the parity of x is constant in the team.

Thus, for instance, the question $?\text {Even}(x)$ is settled in the teams $T_a$ and $T_c$ of Fig. 7.4, where it is settled that x is even, and also in $T_b$, where it is settled that x is odd, but not in $T_d$ where it is unsettled whether x is even or odd (Fig. 7.5).

Fig. 7.4

Four teams $T_a,T_b,T_c,T_d$, each consisting of four assignments into the domain $\mathbb {N}$ of natural numbers. For simplicity, we only display the value that each assignment in the team gives for the variable x

Example 7.4.9

(Identification questions) Let t be a term, and let y be any variable which does not occur in t. Consider the formula $\mathord {\exists \!\!\!\exists }y(t=y)$. We have:

$$\begin{aligned} M\models _T{\mathord {\exists \!\!\!\exists }y(t=y)}\iff & {} \text {there is a }d\in D\text { s.t. } M\models _{T[y\mapsto d]} (t=y)\\\iff & {} \text {there is a }d\in D\text { s.t. for all }g\in T, \;[t]_{g[y\mapsto d]}^M=[y]_{g[y\mapsto d]}^M\\\iff & {} \text {there is a }d\in D\text { s.t. for all }g\in T, \;[t]_g^M=d\\\iff & {} \text {for all } g,g'\in T:\,[t]_g^M=[t]_{g'}^M. \end{aligned}$$

Fig. 7.5

The support conditions for the questions in our examples with respect to the four assignments of Fig. 7.4

Thus, $\mathord {\exists \!\!\!\exists }y(t=y)$ is settled in T in case all $g\in T$ assign the same value to t. If, as in the previous chapter, we define the abbreviation

$$\qquad \qquad \lambda t:=\mathord {\exists \!\!\!\exists }y(t=y)\qquad [y\not \in FV(t)]$$

we thus have that $\lambda t$ captures precisely the identification question “what is the value of t”.

This covers, in particular, identification questions of the form $\lambda x$ about the value of variables, which we considered in Sect. 7.3.3, which are settled in a team T if all the assignments $g\in T$ agree on the value of x. Among the teams depicted in Fig. 7.4, $\lambda x$ is settled only in $T_a$.

Identification questions about complex terms are also interesting. For instance, suppose our domain is $\mathbb {N}$, and suppose $\text {mod}_k$ is a unary function symbol such that $\text {mod}_k(x)$ denotes the remainder of the division of x by k. Then the question

$$\lambda \text {mod}_k(x)$$

is settled in a team T in case all the assignments $g\in T$ agree on the value of $\text {mod}_k(x)$, that is, in case the equivalence class of x modulo k is settled in T. Thus, $\lambda \text {mod}_k(x)$ captures the question “what is the value of x, modulo k?”. Note that in the particular case of $k=2$ this is equivalent to the polar question$?\text {Even}(x)$ discussed in the previous example.

Among the teams in Fig. 7.4, the question $\lambda \text {mod}_3(x)$ is settled in teams $T_a$, where it is settled that $\text {mod}_3(x)=1$, and also in team $T_c$, where it is settled that $\text {mod}_3(x)=0$. The question is not settled in team $T_b$ and $T_d$, since the value of x modulo 3 is not constant in these teams.

Finally, it is worth pointing out that identification questions can be expressed in a different way as well. Consider the formula $\forall y?(t=y)$, where y does not occur in t. We have:

$$\begin{aligned} M\models _T{\forall y?(t=y)}\iff & {} \text {for all }d\in D: M\models _{T[y\mapsto d]} ?(t=y)\\\iff & {} \text {for all }d\in D,\text { for all } g,g'\in T:\\{} & {} M\models _{g[y\mapsto d]}(t=y)\iff M\models _{g'[y\mapsto d]}(t=y)\\\iff & {} \text {for all } g,g'\in T,\text { for all }d\in D: \\{} & {} ([t]_g^M=d)\iff ([t]_{g'}^M=d)\\\iff & {} \text {for all } g,g'\in T:[t]_g^M=[t]_{g'}^M. \end{aligned}$$

This is precisely the semantics of the identification question$\lambda t$ defined above. Thus, the identification question about a term t can be expressed equivalently in InqBT as $\mathord {\exists \!\!\!\exists }y(t=y)$ or as $\forall x?(t=x)$. This double route to identification questions is useful, since the two ways to express such questions use a different set of operators, which gives us two alternative perspectives on such questions.

Example 7.4.10

(Mention-some questions) Consider the formula $\mathord {\exists \!\!\!\exists }yR(\overline{x},y)$, where R is a relation symbol. We have:

$$\begin{aligned} M\models _T{\mathord {\exists \!\!\!\exists }yR(\overline{x},y)}\iff & {} \text {there is a }d\in D\text { s.t. }M\models _{T[y\mapsto d]} R(\overline{x},y)\\\iff & {} \text {there is a }d\in D\text { s.t. for all }g\in T: M\models _{g[y\mapsto d]} R(\overline{x},y)\\\iff & {} \text {there is a }d\in D\text { s.t. for all }g\in T: \langle g(\overline{x}),d\rangle \in I(R). \end{aligned}$$

Thus, $\mathord {\exists \!\!\!\exists }y R(\overline{x},y)$ is settled in the team T if there is an object d such that all assignments in T agree that the value of $\overline{x}$ is R-related to d.

To make this more concrete, suppose again that our domain is the set $\mathbb {N}$ of natural numbers, and suppose our language contains a binary relation symbol Pf such that $\text {Pf}(y,x)$ holds iff y is a prime factor of x. Then the question $\mathord {\exists \!\!\!\exists }y\text {Pf}(y,x)$ is settled relative to a team T in case there is a number $n\in \mathbb {N}$ which is a prime factor of x throughout the team. So, this formula captures precisely the mention-some question “what is some prime factor of x?” discussed in Sect. 7.3.3.

Among the teams of Fig. 7.4, the question $\mathord {\exists \!\!\!\exists }y\text {Pf}(y,x)$ is supported in $T_a$, where it is settled that 2 is a prime factor of x, as well as in $T_d$, where it is settled that 5 is a prime factor, and in $T_c$, where it is settled of both 2 and 3 that they are prime factors of x. The question is not supported in $T_b$, since there is no number which is settled in $T_b$ to be a prime factor of x.

Example 7.4.11

(Mention-all questions) Consider the formula $\forall y?R(\overline{x},y)$, where R is a relation symbol. Using the support conditions for polar questions that we have seen above, we have:

$$\begin{aligned} M\models _T{\forall y?R(\overline{x},y)}\iff & {} \text {for all }d\in D: M\models _{T[y\mapsto d]} ?R(\overline{x},y)\\\iff & {} \text {for all }d\in D,\text { for all } g,g'\in T:\\{} & {} M\models _{g[y\mapsto d]}R(\overline{x},y)\iff M\models _{g'[y\mapsto d]}R(\overline{x},y)\\\iff & {} \text {for all } g,g'\in T,\text { for all }d\in D: \\{} & {} \langle g(\overline{x}),d\rangle \in I(R)\iff \langle g'(\overline{x}),d\rangle \in I(R)\\\iff & {} \text {for all } g,g'\in T:\\{} & {} \{d\mid \langle g(\overline{x}),d\rangle \in R\}=\{d\mid \langle g'(\overline{x}),d\rangle \in R\}. \end{aligned}$$

Thus, $\mathord {\exists \!\!\!\exists }y R(\overline{x},y)$ is settled in the team T in case all assignments in T agree on the set of objects d such that $\overline{x}$ is R-related to d.

To make this more concrete, consider the formula $\forall y?\text {Pf}(y,x)$. This is settled relative to a team T in case all assignments in T agree on the set of prime factors of x. Thus, this formula captures the question “what are the prime factors of x?” which we discussed in Sect. 7.3.3.

Among the teams of Fig. 7.4, the formula $\forall y?\text {Pf}(y,x)$ is supported in $T_a$, where it is settled that the only prime factor of x is 2, and in $T_c$, where it is settled that the prime factors of x are 2 and 3. It is not settled in the remaining teams $T_b$ and $T_d$, since the assignments in these teams to not agree with each other on the set of prime factors of x.

The examples we saw are just a small sample of the class of questions expressible in InqBT. Yet, these examples hopefully suffice to illustrate that the sort of picture discussed abstractly in the previous section can be made concrete in the setting of a simple formal language in which questions about the values of variables can be formalized.

7.4.4 Dependencies in InqBT

Let us now illustrate how q-dependencies can be analyzed and expressed naturally in InqBT, in accordance with the general idea described in Sect. 7.3.2. Recall that, if T is a team based on the model M, we obtain a notion of entailment relative to T by letting:

$$\Phi \models _T\psi \iff \forall T'\subseteq T: (M\models _{T'}\Phi \text { implies }M\models _{T'}\psi ).$$

As we saw above, if $\Lambda \cup \{\mu \}$ is a set of questions, then $\Lambda \models _T\mu$ captures a q-dependency relation: in the context of T, the questions in $\Lambda$ determine the question $\mu$. If moreover $\Gamma$ is a set of statements, then $\Gamma ,\Lambda \models \mu$ captures a conditional q-dependency: $\Lambda$ determines $\mu$ relative to those assignments that make $\Gamma$ true.

Recall moreover that, as usual in inquisitive logic, contextual entailments are expressed in the object language by implications:

$$M\models _T\phi \rightarrow \psi \;\iff \; \phi \models _T\psi .$$

We can now see that the q-dependencies discussed in Sect. 7.3.3 can indeed all be captured as relations between questions expressible in the system InqBT, and can be expressed in the object language by corresponding implications. By way of illustration, here are some examples, where we use the abbreviations introduced in the previous section for questions in InqBT.

• The value of $x_1,\dots ,x_n$ determines the value of y.

  • * Meta-language: $\lambda x_1,\dots ,\lambda x_n\models _T\lambda y$.

    (Note: this corresponds to the v-dependency $\mathbb {D}_T(x_1,\dots ,x_n;y)$.)

  • * Object language: $\lambda x_1\wedge \dots \wedge \lambda x_n\rightarrow \lambda y$.

    (Note: this is equivalent to the dependence atom ${=\!\!(x_1,\dots ,x_n;y)}$.)

• The value of x determines the parity of y.

  • * Meta-language: $\lambda x\models _T{?\text {Even}(y)}$.

  • * Object language: $\lambda x\rightarrow {?\text {Even}(y)}$.

• The parity of x determines the parity of y.

  • * Meta-language: $?\text {Even}(x)\models _T{?\text {Even}(y)}$.

  • * Object language: $?\text {Even}(x)\rightarrow {?\text {Even}(y)}$.

• The value of x and whether $x< y$ determines the value of y.

  • * Meta-language: $\;\lambda x,\, {?(x<y)}\models _T\lambda y$.

  • * Object language: $\lambda x\wedge {?(x<y)}\rightarrow \lambda y$.

• The value of x determines the prime factors of y.

  • * Meta-language: $\lambda x\models _T\forall z?\text {Pf}(z,y)$.

  • * Object language: $\lambda x\rightarrow \forall z?\text {Pf}(z,y)$.

• The value of x determines some prime factor of y.

  • * Meta-language: $\lambda x\models _T\mathord {\exists \!\!\!\exists }z\text {Pf}(z,y)$.

  • * Object language: $\lambda x\rightarrow \mathord {\exists \!\!\!\exists }z\text {Pf}(z,y)$.

Fig. 7.6

Five teams, each consisting of six assignments into the domain $\mathbb {N}$. Only the value of the assignments on the variables x and y is displayed

Fig. 7.7

A table that shows in which of the teams of Fig. 7.6 each of the dependencies we consider holds

Figure 7.6 depicts several different teams, and Fig. 7.7 illustrates the above dependencies by showing in which of these teams each dependency holds.

As these examples illustrate, in InqBT we can capture and express a broad range of dependence facts, of which standard functional dependencies are just a particular case. Notice that among the relations that can be captured as q-dependencies, some are weaker than standard functional dependencies; this holds, for instance, when complete information about x yields only some partial information about y (say, the parity of y, its equivalence class modulo k, its set of prime factors, a single prime factor, etc.). Some other dependencies are stronger than the standard ones. This is the case, e.g., when partial information about x suffices to get complete information about y (when, e.g., we can determine the value of y just based on the parity of x). Many other dependencies are neither weaker nor stronger than the standard ones, but simply incomparable. This holds, e.g., when some kind of partial information about x determines some other kind of partial information about y (say, the parity of x determines the set of prime factors of y).

7.4.5 Higher-Order Dependencies

Another interesting class of dependencies that can be captured naturally from the inquisitive perspective are what we might call higher-order dependencies. To illustrate the idea, consider four variables x, y, z, t and a team T over the set $\mathbb {R}$ of real numbers which contains all assignments of the following form, for $a,b\in \mathbb {R}$:

x	y	z	t
a	b	2a	$-b$

In the context of this team, giving a functional dependency f of y on x implies giving a functional dependency $h_f$ of t on z. For suppose we are given the information that y is functionally determined from x via f, i.e., the information that $g(y)=f(g(x))$. Then it follows that $g(t)=-g(y)=-f(g(x))=-f(g(z)/2)$, and so $g(t)=h_f(g(z))$ for the function $h_f(r)=-f(r/2)$.

This means that in any sub-team $T'\subseteq T$ in which there is a functional dependency of y on x, there is also a functional dependency of t on z. Now, given what we have seen above, in $T'$ there is a functional dependency of y on x just in case $M\models _{T'}\lambda x\rightarrow \lambda y$, and there is a functional dependency of t on z just in case $M\models _{T'}\lambda z\rightarrow \lambda t$. So, the observation amounts to the fact that:

$$\forall T'\subseteq T: M\models _{T'}\lambda x\rightarrow \lambda y\text { implies }M\models _{T'}\lambda z\rightarrow \lambda t.$$

This is nothing but the definition of the q-dependency relation:

$$\lambda x\rightarrow \lambda y\;\models _T\;\lambda z\rightarrow \lambda t.$$

Thus, the higher-order dependency that we pointed out in the context of the above team can be analyzed straightforwardly in InqBT as a case of standard q-dependency where the premise and the conclusion are both dependence formulas. (Notice that a dependence formula like $\lambda x\rightarrow \lambda y$ can be seen as a question asking for a functional dependence of y on x; the question is supported in a team just in case such a functional dependence is established.)

As usual, our higher-order dependence can then be expressed by an implication of the relevant formulas:

$$(\lambda x\rightarrow \lambda y)\rightarrow (\lambda z\rightarrow \lambda t).$$

This illustrates a further advantage of question-based approach over the variable-based one: the former, unlike the latter, can be naturally iterated. We first consider dependencies between certain questions, which amount to contextual entailments. These can be expressed in the object language by corresponding implications. These implications can themselves be seen as questions, which can in turn bear q-dependency relations to each other relative to a team. And these higher-order dependencies can be captured simply by adding another implication among the relevant formulas. And of course, this can be iterated further.

7.4.6 Properties of InqBT and Relations to Other Systems

As the previous sections illustrate, InqBT is a natural choice for a system that captures a broad range of dependencies in the team semantics setting. As we saw, this system can be viewed naturally as an inquisitive logic, which extends classical first-order logic with formulas expressing questions about the values of variables. The basic operators of the system are essentially the same as those of the inquisitive first-order logic $\textsf {InqBQ}$, which, as we saw in the previous chapter, have familiar logical properties. In spite of these attractions, InqBT has received relatively little attention in the literature, and its properties are not well-understood. In this section, we mention some facts and some open problems about this logic and its connections to the systems $\textsf {InqBQ}$ and D.

Fundamental open questions. The basic meta-theoretical questions which are open for $\textsf {InqBQ}$ are also open for InqBT. In particular, it is not known whether InqBT is compact in the sense of entailment, i.e., if for every valid entailment $\Phi \models \psi$ there is a finite set $\Phi _0\subseteq \Phi$ such that $\Phi _0\models \psi$.¹¹ Neither is it known whether the set of InqBT-validities is recursively enumerable, or whether the logic admits a sound and complete axiomatization. Finally, it is not known if there exists an entailment-preserving translation from InqBT to classical first-order logic over a suitable signature, nor whether any invalid entailment can be refuted relative to a countable structure and a countable team. More investigation is needed to settle these important (and interrelated) questions.

On the other hand, most of the positive results that we saw in the previous chapter about fragments of $\textsf {InqBQ}$, if not all, have counterparts for InqBT. In the case of the classical antecedent fragment (cf. Sect. 5.7.2), it can be expected that an entailment-preserving translation to classical first-order logic can be given by a strategy analogous to the one discussed in Sect. 5.7.2; the existence of such a translation then implies that in restriction to the fragment, entailment is compact, and the set of validities (as well as the set of valid entailments with finitely many premises) is recursively enumerable. It seems also likely that the completeness proof for the fragment given in Sect. 5.7.2 can be adapted to InqBT, given a suitably adapted version of the proof system. In the case of the finitely coherent fragment (cf. Sect. 5.7.1), it is easy to show by induction that the counterpart of Proposition 5.7.2 also holds in InqBT: every formula in the fragment is n-coherent for some finite n (where coherence is now understood in terms of the team, as in Kontinen [19]). This fact allows us to use a strategy analogous to the one described in Meißner and Otto [20] to define an entailment-preserving translation from the finitely coherent fragment to classical first-order logic. Again, the existence of such a translation implies that, in restriction to the fragment, entailment is compact and the set of InqBT-validities is recursively enumerable. Notice that the finitely coherent fragment includes formulas corresponding to the dependence atoms of D: let us abbreviate by $\kappa t$ the formula $\forall x?(t=x)$, where x is a variable not occurring in t; we have seen in Example 7.4.9 that $\kappa t$ is equivalent to $\lambda t$, and expresses the identification question about t; then the formula $\kappa x_1\wedge \dots \wedge \kappa x_n\rightarrow \kappa y$ has the same semantics as the dependence atom $=\!\!(x_1,\dots ,x_n;y)$, and it belongs to the finitely coherent fragment since it does not contain the operator $\mathord {\exists \!\!\!\exists }$ .

Relations to the inquisitive first-order logic $\textsf {InqBQ}$. The system InqBT can be seen as a counterpart of $\textsf {InqBQ}$ in a setting in which the relevant information state is given by a set of assignments instead of a set of possible worlds. Since the semantics is structurally the same, most of the facts about $\textsf {InqBQ}$ which we established in the previous chapter carry over straightforwardly to InqBT. We will not restate the relevant facts here. Instead, we will point out some respects in which the two logics differ as a result of their different setups.

First, in $\textsf {InqBQ}$ a significant role was played by rigid terms, whose interpretation is fixed across different possible worlds in a state. The counterpart of rigid terms in InqBT is given by closed terms—terms not involving any variables—whose interpretation is fixed across different assignments in a team. These will be the terms to which a universal can be validly instantiated, and from which an inquisitive existential can be introduced.

Proposition 7.4.12

If $\textsf {t}$ is a closed term then for any formula $\phi \in \mathcal {L}^{\textsf {Q}=}$, the entailments $\phi [\textsf {t}/x]\models \mathord {\exists \!\!\!\exists }x\phi$ and $\forall x\phi \models \phi [\textsf {t}/x]$ are valid in InqBT.

Proof

We show only the first entailment, since the proof of the second is similar. Suppose $M\models _T\phi [\textsf {t}/x]$ for some relational structure M and team T. Now let $d\in D$ be the object such that $d=[\textsf {t}]_M$: crucially, this object is assignment-independent, since $\textsf {t}$ is closed. It is straightforward to show by induction that for every formula $\psi$ we have $M\models _T\psi [\textsf {t}/x]\iff M\models _{T[x\mapsto d]}\psi$. Since $M\models _T\phi [\textsf {t}/x]$, it follows that $M\models _{T[x\mapsto d]}\phi$, which by the semantics implies $M\models _T\mathord {\exists \!\!\!\exists }x\phi$. $\Box$

Note that if t is not closed, the relevant entailments are not in general valid. For instance, let $\phi$ be the formula $(x=y)$. Then $\phi [y/x]$ is the formula $(y=y)$, which is a validity. However, $\mathord {\exists \!\!\!\exists }x\phi$ is the formula $\mathord {\exists \!\!\!\exists }x(x=y)$, which is not a validity, but the identification question that we denoted by $\lambda y$, which is supported by a team if all assignments agree on the value of y. Thus in this case we have $\phi [y/x]\not \models \mathord {\exists \!\!\!\exists }x\phi$.

Similarly, if we take $\phi$ to be the formula ?P(x), then the formula $\forall x\phi$ is a validity by Proposition 7.4.7, but $\phi [y/x]$ is the formula ?P(y), which is not a validity (cf. Example 7.4.8). This shows that $\forall x\phi \not \models \phi [y/x]$.

For analogous reasons, the role of free variables as placeholders for arbitrary individuals is taken over in InqBT by fresh constant symbols.

Proposition 7.4.13

If $\textsf {c}$ is a constant not occurring in the set $\Phi \cup \{\phi ,\psi \}$:

• $\Phi \models \forall x\phi \iff \Phi \models \phi [\textsf {c}/x]$;

• $\Phi ,\mathord {\exists \!\!\!\exists }x\phi \models \psi \iff \Phi ,\phi [\textsf {c}/x]\models \psi$.

This proposition also hold in $\textsf {InqBQ}$, provided the constant $\textsf {c}$ is rigid. However, in $\textsf {InqBQ}$ analogous facts hold if instead of a fresh constant c we use a fresh variable y. This is not the case in InqBT: for instance, as we already mentioned, in InqBT $\forall x?P(x)$ is valid even though ?P(y) is not. What underlies this mismatch between free and bound variables is that, in InqBT, bound variables are always interpreted rigidly in the team by the semantic clause for the quantifiers, while free variables may receive different values at different assignments in the team.

This discussion suggests the following strategy to make inferences with quantifiers in InqBT: we first extend the relevant signature with a countably infinite stock of constant symbols (to make sure that we can never run out of fresh variables in a proof) and then we adopt the inference rules for quantifiers given in Fig. 7.8.

Fig. 7.8

Inference rules for quantifiers in InqBT. In these rules, $\alpha$ ranges over classical formulas, while $\phi ,\psi$ range over arbitrary formulas; t is a closed term, while t is an arbitrary term; in the introduction rule for the universal quantifier, $\textsf {c}$ is a constant symbol which does not occur in any undischarged assumption; in the elimination rule for the inquisitive existential quantifier, $\textsf {c}$ is a constant symbol which does not occur free in $\psi$ or in any undischarged assumption

Another significant difference between InqBT and $\textsf {InqBQ}$ stems from Proposition 7.4.7: in InqBT, every sentence $\phi$ is equivalent to its classical variant $\phi ^{cl}$. Thus, at the level of sentences inquisitive operators collapse onto the corresponding classical operators. This reflects the fact that in InqBT, we do not model uncertainty about the state of affairs (since the relational structure is fixed), but only uncertainty about the values of variables, which is only relevant for the interpretation of open sentences.

As a consequence of this collapse of inquisitive operators in the case of sentences, InqBT does not share some of the meta-theoretic properties of $\textsf {InqBQ}$. In particular, we do not have the disjunction property for . To see this, let c be a constant symbol: the formula ?P(c), which abbreviates, is equivalent to $P(c)\vee \lnot P(c)$ in InqBT by Proposition 7.4.7, and thus logically valid; but obviously, neither P(c) nor $\lnot P(c)$ is logically valid. Similarly, we do not have the existence property for $\mathord {\exists \!\!\!\exists }$ . To see this, consider the formula

$$\mathord {\exists \!\!\!\exists }x((P(c)\wedge x=c)\vee (\lnot P(c)\wedge x=c'))$$

where $c,c'$ are two distinct constant symbols. With a reasoning analogous to the one we gave on page 154 we can show that this formula is logically valid in InqBT, but none of the formulas

$$(P(c)\wedge t=c)\vee (\lnot P(c)\wedge t=c')$$

obtained by instantiating the existential to a term t is logically valid.

In spite of these difference existing between $\textsf {InqBQ}$ and InqBT, it seems natural to conjecture that one can give entailment-preserving translations between the two systems. If so, many of the open questions about the properties of InqBT reduce to the corresponding questions about $\textsf {InqBQ}$, and vice versa. Thus, research on these two systems is tightly connected. We leave it as an open problem to establish (or disprove) this conjecture.

Open Problem 7.4.14

(Existence of a translation of InqBT into $\textsf {InqBQ}$) Given a signature $\Sigma$, is there a signature $\Sigma '$, a decidable set $\Theta \subseteq \mathcal {L}^{\textsf {Q}=}(\Sigma ')$ and a computable map $(\cdot )^*:\mathcal {L}^{\textsf {Q}=}(\Sigma )\rightarrow \mathcal {L}^{\textsf {Q}=}(\Sigma ')$ s.t. for all sets $\Phi \cup \{\psi \}\subseteq \mathcal {L}^{\textsf {Q}=}(\Sigma )$ we have $\Phi \models _{\textsf {InqBT}}\psi \iff \Phi ^*,\Theta \models _{\textsf {InqBQ}}\psi ^*$?

Open Problem 7.4.15

(Existence of a translation of $\textsf {InqBQ}$ into InqBT) Given a signature $\Sigma$, is there a signature $\Sigma '$, a decidable set $\Theta \subseteq \mathcal {L}^{\textsf {Q}=}(\Sigma ')$ and a computable map $(\cdot )^*:\mathcal {L}^{\textsf {Q}=}(\Sigma )\rightarrow \mathcal {L}^{\textsf {Q}=}(\Sigma ')$ s.t. for all sets $\Phi \cup \{\psi \}\subseteq \mathcal {L}^{\textsf {Q}=}(\Sigma )$ we have $\Phi \models _{\textsf {InqBQ}}\psi \iff \Phi ^*,\Theta \models _{\textsf {InqBT}}\psi ^*$?

Relations to standard dependence logic D. The systems InqBT and D are defined in the same semantic setting: in both cases, formulas are interpreted relative to a relational structure M and a team T. We can thus ask straightforwardly how the expressive power of these two systems relates. One thing that we can immediately say is the following.

Proposition 7.4.16

There are formulas in D which are not equivalent to any formula in InqBT.

Proof

Take the dependence logic sentence $\exists ^d x\forall ^d y\exists ^d z(=\!\!(z,y)\wedge z\ne x)$ discussed in Sect. 7.1, which is satisfied relative to a structure M and a non-empty team just in case the domain of M is infinite. Let us denote this sentence by $\phi _{\text {inf}}$. We claim that this formula is not equivalent to any formula in InqBT. Towards a contradiction, suppose $\psi$ is an InqBT-formula equivalent to $\phi _{\text {inf}}$. Then in particular, $\psi$ and $\phi _{\text {inf}}$ must be equivalent in terms of truth conditions (i.e., when interpreted relative to single assignments). But we know from Corollary 7.4.4 that $\psi$ has the same truth conditions as its classical variant $\psi ^{cl}$, which are just the truth conditions of $\psi ^{cl}$ in classical first-order logic. It follows that $\phi _{\text {inf}}$ has the same truth conditions as a classical first-order formula, which is not the case. $\Box$

What about the converse? That is, in terms of expressive power, can InqBT be seen as a fragment of D, or are the two systems simply incomparable? This is, to the best of my knowledge, an interesting open question.

Open Problem 7.4.17

Is every formula in InqBT equivalent to some formula in $\textsf {D}$?

In particular, one could ask whether the examples of InqBT-dependence formulas discussed in Sects. 7.4.4 and 7.4.5 are expressible in D.

7.5 A General Framework for First-Order Questions and Dependencies

As we saw, there is a discrepancy between the semantic framework of standard inquisitive first-order logic, based on sets of possible worlds and a single assignment, and the framework used in work on dependence logic, based on a single relational structure and a set of assignments. We saw that the language of inquisitive first-order logic can be interpreted in both settings, leading to different systems $\textsf {InqBQ}$ and InqBT. In these systems we can capture different classes of questions and dependencies. For instance, in $\textsf {InqBQ}$ we can express the question “what is the extension of P?”, and capture the fact that the extension of P determines the extension of Q. In InqBT, by contrast, we can express the question “what is the value of x?” and capture the fact that the value of x determines the value of y. In this section, we show that it is possible to introduce a more general semantic framework, which allows us to capture questions about the state of affairs as well as questions about the values of variables, and questions involving both of these things. As we will see, the systems $\textsf {InqBQ}$ and InqBT can be seen as special cases of a system $\textsf {InqBQ}^+$ formulated in this general setting, each obtained by restricting to a certain kind of evaluation points.

In order to obtain our general framework, we evaluate formulas with respect to objects that capture partial information about both the state of affairs and the value of variables, as well as their correlations. This can be achieved by taking our points of evaluation to be sets of world-assignment pairs. We will refer to such objects as information states with referents, abbreviated as r-states.¹² $^{,}$¹³

Definition 7.5.1

(Indices and r-states) Let $M=\langle W,D,I\rangle$ be a first-order information id-model.¹⁴

• An index is a pair $i=\langle w_i,g_i\rangle$, where $w_i\in W$ and $g_i:\textsf {Var}\rightarrow D$.

• An information state with referents, or r-state for short, is a set s of indices.

In the system $\textsf {InqBQ}^+$, sentences are interpreted relative to r-states. Notice that an r-state determines both an ordinary information state and a team.

Definition 7.5.2

Let s be an information state with referents. Then:

• the state associated with s is $\pi _1[s]=\{w\,|\,\langle w,g\rangle \in s\text { for some }g\,\}$;

• the team associated with s is $\pi _2[s]=\{\,g\,|\,\langle w,g\rangle \in s\text { for some }w\}$.

However, an r-state is not uniquely determined by the state $\pi _1[s]$ and the team $\pi _2[s]$. This is because, in general, s also encodes information about the correlation between the state of affairs and the value of variables, information that is not reflected by the projections $\pi _1[s]$ and $\pi _2[s]$. For instance, the r-states $s_1=\{\langle w,g\rangle ,\langle w',g'\rangle ,\langle w,g'\rangle ,\langle w',g\rangle \}$ and $s_2=\{\langle w,g\rangle ,\langle w',g'\rangle \}$ have the same projections, but the latter also encodes a certain correlation between the state of affairs and the assignment function, which the former does not encode.

The language of our system $\textsf {InqBQ}^+$ is the same first-order language $\mathcal {L}^{\textsf {Q}=}$ that we have for the systems $\textsf {InqBQ}$ and InqBT. The value of a term t at an index i is simply $[t]^i:=[t]_{g_i}^{w_i}$. Moreover, given an r-state s and an individual $d\in D$, we write $s[x\mapsto d]$ for the r-state obtained by modifying the valuation at each index in s from g to $g[x\mapsto d]$:

$$s[x\mapsto d]=\{\langle w,g[x\mapsto d]\rangle \,|\,\langle w,g\rangle \in s\}.$$

The relation of support between r-states s in a model M and formulas $\phi \in \mathcal {L}^{\textsf {Q}}$ is defined as follows.

Definition 7.5.3

• $M,s\models R(t_1,\dots ,t_n)\iff \langle [t_1]^i,\dots ,[t_n]^i\rangle \in I_{w_i}(R)$ for all $i\in s$

• $M,s\models (t=t')\iff [t]^i=[t']^i$ for all $i\in s$

• $M,s\models \bot \iff s =\emptyset$

• $M,s\models \phi \wedge \psi \iff M,s\models \phi \text { and }M,s\models \psi$

• $M,s\models \phi \rightarrow \psi \iff$for all $t\subseteq s, M,t\models \phi$ implies $M,t\models \psi$

• 

• $M,s\models \forall x\phi \iff M,s[x\mapsto d]\models \phi$ for every $d\in D$

• $M,s\models \mathord {\exists \!\!\!\exists }x\phi \iff M,s[x\mapsto d]\models \phi$ for some $d\in D$

This system is similar to $\textsf {InqBQ}$ in many respects. First of all, the semantics is persistent, and every formula is supported by the empty r-state. We can define truth with respect to an index i as support with respect to $\{i\}$, and we can call a formula truth-conditional if support for it always amounts to truth at each world. Then, we still have that all classical formulas are truth-conditional and that, moreover, the truth-conditions for a classical formula at an index $i=\langle w,g\rangle$ are the ones given by classical first-order logic with respect to the assignment g and the relational structure $M_w=\langle D,I_w\rangle$ associated with w.

Proposition 7.5.4

For every $\alpha \in {\mathcal {L}^{\textsf {Q}}_c}$ and every model M:

• Truth-conditionality: for every r-state s in M,

  $$M,s\models \alpha \iff M,i\models \alpha \text { for all }i\in s.$$

• Standard truth conditions: for every index $i=\langle w,g\rangle$ in M,

  $$M,i\models \alpha \iff {M_w}\models _{g}\alpha \text { in standard Tarskian semantics}.$$

This shows that, as far as classical formulas are concerned, $\textsf {InqBQ}^+$ is yet another informational semantics for classical first-order logic: in restriction to classical formulas, the entailment relation that arises from $\textsf {InqBQ}^+$ is just the one of classical first-order logic.

Moreover, it is worth remarking that the systems $\textsf {InqBQ}$ and InqBT can both be seen as special cases of $\textsf {InqBQ}^+$, obtained by restricting the semantics of $\textsf {InqBQ}^+$ to particular kinds of r-states. The system $\textsf {InqBQ}$ is obtained by restricting the semantics to r-states in which all indices i have the same assignment component $g_i$.

Proposition 7.5.5

Suppose $\pi _2[s]=\{g\}$. Then $M,s\models \phi \iff M,\pi _1[s]\models _{g}\phi$ in $\textsf {InqBQ}$.

Similarly, the system InqBT is obtained by restricting the semantics to r-states in which all indices i have the same world component $w_i$.

Proposition 7.5.6

Suppose $\pi _1[s]=\{w\}$. Then $M,s\models \phi \iff M_w\models _{\pi _2[s]}\phi$ in InqBT.

As we expect, the interpretation of sentences is insensitive to the assignment component of an r-state, which means that, as far as sentences are concerned, $\textsf {InqBQ}^+$ coincides with $\textsf {InqBQ}$, in the following sense.

Proposition 7.5.7

If $\phi$ is a sentence and s is an r-state, $M,s\models \phi \iff M,\pi _1[s]\models \phi$ in $\textsf {InqBQ}$.

Notice all the questions that we discussed in the previous chapter were sentences: we were only interested in variables insofar as these would ultimately get bound. The previous proposition implies that all those questions receive exactly the same interpretation in $\textsf {InqBQ}^+$ as they did in $\textsf {InqBQ}$. This includes, e.g., polar questions of the form $?\exists xP(x)$, mention-some questions of the form $\mathord {\exists \!\!\!\exists }xP(x)$, and mention-all questions of the form $\forall x{?P(x)}$.

In addition to these questions, however, $\textsf {InqBQ}^+$ can also interpret questions which concern the value of free variables. This includes, in particular, identification questions of the form $\lambda x$ for a variable x (recall that we defined $\lambda x$ to be $\mathord {\exists \!\!\!\exists }y(y=x)$ for some variable y distinct from x), which receive the same interpretation as in InqBT:

$$M,s\models \lambda x\iff \text {for all }g,g'\in \pi _2[s]:\;g(x)=g'(x).$$

Thus, in $\textsf {InqBQ}^+$ we can capture questions that concern only the state of affairs, such as $\forall x?P(x)$, and questions that concern only the value of free variables, such as $\lambda x$. In addition, we can also capture questions that concern both aspects at once. For an example, consider a polar question?P(x). We have:

$$M,s\models {?P(x)}\iff [\,g_i(x)\in P_{w_i}\text { for all }i\in s]\text { or }[\,g_i(x)\not \in P_{w_i}\text { for all }i\in s].$$

Thus, whether ?Px is supported depends on what s settles about the value of x and about the extension of P. An r-state s might determine exactly the value of x, but fail to support ?Px because it does not determine whether the extension of P includes the relevant object; conversely, s might determine exactly the extention of P, yet fail to support ?Px because it does not determine the value of x. On the other hand, an r-state s may support ?Px without determining of any particular object whether it has property P.

Within the system $\textsf {InqBQ}^+$, we obtain a uniform analysis of the different sorts of dependencies that we encountered in the previous chapter and in the present one. As in the previous chapter, we can capture, e.g., the fact that the extensions of $P_1,\dots ,P_n$ determine the extension of Q. This is expressed by the formula:

$$\forall x?P_1(x)\wedge \dots \wedge \forall x?P_n(x)\;\rightarrow \;\forall x?Q(x).$$

As in the previous section, we can capture the fact that the value of $x_1,\dots ,x_n$ determines the value of y. This is expressed by the formula:

$$\lambda x_1\wedge \dots \wedge \lambda x_n\;\rightarrow \;{\lambda y}.$$

Moreover, we can express mixed dependencies. For instance, the following formula, which is logically valid in $\textsf {InqBQ}^+$, expresses the fact that the value of x and the extension of P jointly determine whether x has property P:

$$\forall y?P(y)\,\wedge \, \lambda x\;\rightarrow \; {?P(x)}.$$

We will not delve further here into the study of the system $\textsf {InqBQ}^+$. Our aim in this section was merely to illustrate how one can give a semantic framework which simultaneously represents partial information about the state of affairs, the values of variables, and their correlations, and to show how within such a framework one can define a generalized version of inquisitive first-order logic that encompasses, as special cases, both the standard inquisitive first-order logic $\textsf {InqBQ}$ that we studied in detail in the previous chapter, and its team semantics counterpart InqBT that we discussed in the previous section.

7.6 Summary and Final Considerations

In this chapter, we looked at some of the tight relations existing between inquisitive logic and dependence logic, with a special focus on the analysis of the notion of dependency. We started out by introducing the standard notion of functional dependency in a team, understood as a relation between variables, and we saw how this relation plays a role in the semantics of the standard system of dependence logic. We then discussed how the basic ideas and notions of inquisitive logic, as laid out in Chap. 2, can be applied naturally in the context of team semantics. Doing so yields a new perspective on the notion of dependency, which can be viewed as a relation between questions, rather than as a relation between variables. We emphasized several virtues of this perspective: it is much more general than the one based on variables, since even in a context where very few variables are at stake, there is a broad spectrum of questions about these variables that can be considered, and thus a broad spectrum of dependencies that can be analyzed in terms of such questions. Standard functional dependency is thus found to be a special case of a broad class of relations which share a common logical core and can be handled by the same logical tools. Moreover, the question-based perspective reveals that dependency can be seen as a facet of the central logical notion of entailment, once this is extended to questions. Dependencies are thus directly connected to the central concerns of logic.

We illustrated these general points by means of a specific logical system, InqBT, which is an adaptation to the team semantics setting of the first-order inquisitive logic of the previous chapter. In exploring this system, our main aim was to illustrate the inquisitive approach in the team semantics setting, of which the system InqBT is only an instance. When deciding on a logic to analyze and reason about dependencies, the particular choice of logical repertoire will depend on one’s ultimate purposes. If one’s purpose is to capture dependencies between bound variables, of the kind that play a role in Henkin quantifiers, then InqBT is not sufficient, since this system is equivalent to standard first-order logic with respect to sentences. For this purpose, one may want, e.g., to enrich InqBT with the quantifiers of dependence logic. On the other hand, InqBT is a very natural logic to capture dependencies involving free variables, and thus to reason about features of a given team. As we saw, more research is needed to understand the exact meta-theoretic properties of InqBT—to determine, e.g., whether this system is recursively axiomatizable. If this system turns out to be relatively complex, for some purposes one may well want to restrict to a well-behaved fragment, such as the finitely coherent fragment (cf. Sect. 5.7.1).¹⁵ However, all these different choices concerning the logical repertoire of the system are compatible with the general conception and system architecture that stems from the key ideas of inquisitive logic discussed in Chap. 2.

7.7 Exercises

Exercise 7.7.1

(Formalizing dependencies) Consider a team over the domain $\mathbb {R}$ of real numbers which contains all assignments g such that g(y) equals the square of g(x). The following table depicts some rows of this team.

x	y
$\vdots$	$\vdots$
$-2$	4
$-1$	1
$-\frac{1}{2}$	$\frac{1}{4}$
0	0
$\frac{1}{2}$	$\frac{1}{4}$
1	1
2	4
$\vdots$	$\vdots$

Consider a language equipped with a relation symbol <, two constant symbols 1, 0 interpreted in the natural way, and with a function symbol $|\cdot |$ interpreted as mapping a number a to its absolute value.

The following (conditional) dependencies hold in the context of this team.

Write down formulas of InqBT that express these dependencies.

Exercise 7.7.2

(Inquisitive logic in team semantics) Consider a team over the domain $\mathbb {N}$ of natural numbers which contains all assignments g such that $g(z)=g(x)g(y)$. The following table gives the idea.

Consider a language equipped with predicate symbols Even, <, and Pf, interpreted respectively as “is even”, “is less than”, “is a prime factor of”, as well as constant symbols 0, 1 and a binary function symbol + interpreted in the natural way. Determine whether the following implications are supported in the team.

• 1. $\lambda x\wedge \lambda y\rightarrow \lambda z$

• 2. $\lambda x\wedge \lambda z\rightarrow \lambda y$

• 3. $(x>0)\wedge \lambda x\wedge \lambda z\rightarrow \lambda y$

• 4. ${?\text {Even}(x)}\wedge {?\text {Even}(y)}\rightarrow {?\text {Even}(z)}$

• 5. $(x>0)\wedge {?\text {Even}(x)}\wedge {?\text {Even}(z)}\rightarrow {?\text {Even}(y)}$

• 6. $\lambda x\wedge \lambda (y+z)\rightarrow \lambda z$

• 7. $\mathord {\exists \!\!\!\exists }t\text {Pf}(t,x)\rightarrow \mathord {\exists \!\!\!\exists }t\text {Pf}(t,z)$

• 8.

• 9. $\mathord {\exists \!\!\!\exists }t\text {Pf}(t,z)\rightarrow \mathord {\exists \!\!\!\exists }t(\text {Pf}(t,x)\vee \text {Pf}(t,y))$

• 10. $(x>1)\wedge \forall t?\text {Pf}(t,x)\rightarrow \forall t{?\text {Pf}(t,z)}$

• 11. $(x>1)\wedge (y>1)\wedge \forall t?\text {Pf}(t,x)\wedge \forall t?\text {Pf}(t,y)\rightarrow \forall t{?\text {Pf}(t,z)}$

• 12. $(x>1)\wedge (z>1)\wedge \forall t?\text {Pf}(t,x)\wedge \forall t?\text {Pf}(t,z)\rightarrow \forall t{?\text {Pf}(t,y)}$

• 13. $(x>0)\wedge \forall t?\text {Pf}(t,x)\rightarrow {?\text {Even}(x)}$