A firstorder inquisitive semantics
A firstorder inquisitive semantics
Abstract and Keywords
Chapter 4 defines an inquisitive semantics for the language of firstorder logic, which includes the logical connectives (conjunction, disjunction, negation, and implication) as well as universal and existential quantifiers.The semantics of these connectives and quantifiers is given in terms of the algebraic operations on propositions identified in the previous chapter. The main features of the system are highlighted, and illustrated with a range of examples. In Chapters 5–10, a number of linguistic applications of this logical framework are discussed.
Keywords: firstorder logic, connectives, quantifiers, inquisitive semantics, algebraic operations
In this chapter we define an inquisitive semantics for the language of firstorder logic, making use of the operations on propositions identified in the previous chapter. We will highlight some of the main features of the system, and illustrate it with a range of examples. In the following chapters we will use this logical framework for the semantic analysis of a number of linguistic constructions.
4.1 Logical language and models
We will consider a standard firstorder language $\mathcal{L}$, based on a signature that consists of a set of function symbols ${\mathcal{F}}_{\mathcal{L}}$ and a set of relation symbols ${\mathcal{R}}_{\mathcal{L}}$, each with an associated arity n ≥ 0. As usual, 0place function symbols will be referred to as individual constants. We assume that the language has ¬, ∨, ∧, →, ∃, and ∀ as its basic logical constants.
We will interpret $\mathcal{L}$ with respect to firstorder information models. Such models consist of a set of possible worlds W, each associated with a standard firstorder model. A standard firstorder model, in turn, consists of a domain of individuals D and an interpretation function I which maps any function symbol in ${\mathcal{F}}_{\mathcal{L}}$ to a function over D and every relation symbol in ${\mathcal{R}}_{\mathcal{L}}$ to a relation over D.
In order to avoid certain issues arising from quantification across different possible worlds, we will restrict our attention to rigid firstorder information models, in which the domain of individuals as well as the interpretation of function symbols is fixed across worlds. The only thing that may differ from world to world is the interpretation of relation symbols.
Definition 4.1 (Rigid firstorder information models)
A rigid firstorder information model for $\mathcal{L}$ is a triple 〈W, D, I〉, where: (p.60)
• W is a set, whose elements are referred to as possible worlds;
• D is a nonempty set, whose elements are referred to as individuals;
• I is a map that associates every w ∈ W with a firstorder structure I_{w} s.t.:
– for every w ∈ W, the domain of I_{w} is D;
– for every nary function symbol $f\in {\mathcal{F}}_{\mathcal{L}}$, ${I}_{w}(\phantom{\rule{0.3em}{0ex}}f):{D}^{n}\to D$; with the condition that for every w, v ∈ W, I_{w}( f) = I_{v}( f);
– for every nary relation symbol $R\in {\mathcal{R}}_{\mathcal{L}}$, ${I}_{w}(R)\subseteq {D}^{n}$.
Unless specified otherwise, we will assume a fixed model throughout our discussion and we will often omit explicit reference to it. So, while in the previous chapters, where we were not yet considering a concrete logical language, we simply assumed a set of possible worlds W as our logical space, we now consider a triple 〈W, D, I〉, where W is still a set of possible worlds, and the other elements specify the interpretation of the function symbols and relation symbols in our language with regard to these possible worlds.
In order not to have assignments in the way, we will assume that for any d ∈ D, our language $\mathcal{L}$ contains an individual constant d′ such that I_{w}(d′) = d for all w ∈ W: if this is not the case, we simply expand the language by adding new constants, and we expand the map I accordingly. In this way we can work only with sentences, i.e., formulas without free variables, and we can do without assignments altogether. This move is of course not essential, but it simplifies notation and terminology considerably.
Finally, it will be convenient to have a notation for the set of worlds in our model in which a given sentence φ is classically true, in the standard sense.
Definition 4.2 (Truthset)
For any $\phi \in \mathcal{L}$, the set of worlds where φ is classically true is called the truthset of φ and denoted as φ. In particular, for an atomic formula $R({t}_{1},\dots ,{t}_{n})$:^{1}
$$\begin{array}{rll}\mathcal{R}({t}_{1},\dots ,{t}_{n})=\{w\in W\mid \u3008{I}_{w}({t}_{1}),\dots ,{I}_{w}({t}_{n})\u3009\in {I}_{w}(R)\}& & \end{array}$$
(p.61) 4.2 Semantics
We are now ready to recursively associate each sentence φ of our firstorder language with an inquisitive proposition [φ]. We will take atomic sentences to behave classically: $R({t}_{1},\dots ,{t}_{n})$ provides the information that the relation R holds for the individuals ${t}_{1},\dots ,{t}_{n}$, and does not raise any issue. Thus, $R({t}_{1},\dots ,{t}_{n})$ will have as its informative content the set $R({t}_{1},\dots ,{t}_{n})$, and it will not be inquisitive. By Fact 2.19, this implies that $[R({t}_{1},\dots ,{t}_{n})]$ must amount to $\wp \phantom{\rule{0.3em}{0ex}}(R({t}_{1},\dots ,{t}_{n}))$. As for the connectives and quantifiers, we will take them to express the basic algebraic operations that we identified in Section 3.1.
Definition 4.3 (Firstorder inquisitive semantics)
1. [R(t_{1}, …, t_{n})]:=℘(R(t_{1}, …, t_{n}))
2. [¬φ]:=[φ]^{*}
3. $[\phi \wedge \psi ]:=[\phi ]\cap [\psi ]$
4. $[\phi \vee \psi ]:=[\phi ]\cup [\psi ]$
5. $[\phi \to \psi ]:=[\phi ]\Rightarrow [\psi ]$
6. $[\forall x.\phi (x)]:={\bigcap}_{d\in D}\phantom{\rule{0.3em}{0ex}}[\phi ({d}^{\prime})]$
7. $[\exists x.\phi (x)]:={\bigcup}_{d\in D}\phantom{\rule{0.3em}{0ex}}[\phi ({d}^{\prime})]$
We refer to this firstorder system as InqB, where B stands for basic. We refer to [φ] as the proposition expressed by φ. The clauses of InqB constitute a proper inquisitive semantics in the sense that they indeed associate every sentence $\phi \in \mathcal{L}$ with a proposition in the sense of inquisitive semantics, i.e., a nonempty downward closed set of information states.
Fact 4.4 (Suitability of the semantics)
For any $\phi \in \mathcal{L}$, $[\phi ]\in \mathcal{P}$.
All the notions that were introduced in Chapter 2 with reference to propositions can now be formulated with reference to the sentences in our logical language. For instance, we define the informative content of a sentence φ, info(φ), as the informative content of the proposition it expresses, info([φ]). Similarly, the set of alternatives induced by φ, alt(φ), is the set of alternatives in alt([φ]); and the issue raised by φ is the issue embodied by [φ], which is resolved by an information state s just in case s ∈ [φ]. (p.62)
Definition 4.5 (Informative content, alternatives, and issues)
For any $\phi \in \mathcal{L}$:
• $\text{info}(\phi ):=\bigcup [\phi ]$
• alt(φ):=alt([φ])
• The issue raised by φ is one that is resolved by a state s just in case s ∈ [φ].
Moreover, we say that one sentence φ entails another sentence ψ, φ⊧ψ, just in case the proposition expressed by φ entails the proposition expressed by ψ, and we say that φ and ψ are equivalent, φ ≡ ψ, just in case they express exactly the same proposition.^{2}
Definition 4.6 (Entailment and equivalence)
For any $\phi ,\psi \in \mathcal{L}$:
• φ⊧ψ just in case $[\phi ]\subseteq [\psi ]$
• φ ≡ ψ just in case [φ] = [ψ]
Finally, we say that φ is true in a world w in case the proposition it expresses is true in w, i.e., w ∈info(φ); and we say that φ is supported by an information state s, s⊧φ, in case the proposition it expresses is supported by s, i.e., s ∈ [φ].
Definition 4.7 (Truth and support)
For any $\phi \in \mathcal{L}$:
• φ is true in w if and only if w ∈ info(φ)
• φ is supported by s, notation s⊧φ, if and only if s ∈ [φ]
Notice that, just like the proposition expressed by φ in classical logic is the set of worlds where φ is true, the proposition expressed by φ in InqB is the set of states where φ is supported. As a consequence of this fact, InqB is completely characterized by the support conditions of the sentences in the language. These support conditions are as follows.
Fact 4.8 (Support conditions)
1. $s\vDash R({t}_{1},\dots ,{t}_{n})$ iff $s\subseteq R({t}_{1},\dots ,{t}_{n})$
2. s⊧¬φ iff $\forall t\subseteq s:$ if t≠∅ then t⊭φ
3. s⊧φ ∧ ψ iff s⊧φ and s⊧ψ
5. $s\vDash \phi \to \psi $ iff $\forall t\subseteq s:$ if t⊧φ then t⊧ψ
6. s⊧∀xφ(x) iff s⊧φ(d′) for all d ∈ D
7. s⊧∃xφ(x) iff s⊧φ(d′) for some d ∈ D
In much work on inquisitive semantics (Ciardelli, 2009; Groenendijk and Roelofsen, 2009; Ciardelli and Roelofsen, 2011; Ciardelli et al., 2015; Ciardelli, 2016d, 2018), InqB is in fact characterized directly in terms of support conditions. The proposition expressed by a sentence is then defined in terms of these support conditions, i.e., as the set of all states that support the sentence. An advantage of this approach is that it parallels the usual presentation of classical logic, with truth conditions as the basic notion. Another advantage, at least for certain purposes, is that it allows for a very efficient presentation of the system, bypassing many of the more abstract notions that we introduced here before even starting to consider a concrete logical language. In this book, the supportbased perspective will play a key role in Chapter 8, allowing for a perspicuous presentation of the inquisitive account of propositional attitudes.
There are two main reasons why we have chosen a less direct route here, following Ciardelli et al. (2013a) and Roelofsen (2013a). First, the current presentation of the new inquisitive notion of propositions (Chapter 2) brings out very explicitly how the standard informationcentred notion of semantic content is enriched, why the new notion is shaped exactly the way it is, and that it naturally allows for various further extensions and refinements (see the references in section 2.6 as well as in the Further Reading section.) Second, the algebraic perspective adopted here (Chapter 3) makes it possible to motivate the treatment of the connectives and quantifiers in InqB in a solid way, relying only on the structure of our new space of propositions. Moreover, it shows that InqB is, in a very precise sense, the exact counterpart of classical logic in the inquisitive setting. Thus, unlike a supportbased exposition, this mode of presentation flows directly from the abstract motivations and conceptual underpinnings of the system to its concrete implementation.
4.3 Semantic categories and projection operators
We say that a sentence is informative, inquisitive, a hybrid, or a tautology just in case the proposition that it expresses is. This amounts to the following. (p.64)
Definition 4.9 (Semantic categories)
We say that a sentence $\phi \in \mathcal{L}$ is:
• informative iff $\text{info}(\phi )\ne \mathcal{W}$.
• inquisitive iff info(φ)∉[φ].
• a hybrid iff it is both informative and inquisitive;
• a tautology iff it is neither informative nor inquisitive.
Fact 4.10 (Direct characterization of trivial sentences)
• φ is noninquisitive ⇔ [φ] = ℘(info(φ)) ⇔ [φ] has a greatest element.
• φ is noninformative $\phantom{\rule{2.77695pt}{0ex}}\iff \phantom{\rule{2.77695pt}{0ex}}\text{info}(\phi )=\mathcal{W}$.
• φ is a tautology $\phantom{\rule{2.77695pt}{0ex}}\iff \phantom{\rule{2.77695pt}{0ex}}[\phi ]=\wp (\mathcal{W})$.
In Section 3.2 we characterized two projection operators on propositions, which trivialize either the informative or the inquisitive content of any given proposition. Now that we are considering a concrete logical language, we will introduce two oneplace connectives that express these projection operators. We will denote these connectives as ! and ?, just like the operators they express.
Definition 4.11 (Projection operators)
For any $\phi \in \mathcal{L}$:
• [!φ]:=![φ]
• [?φ]:=?[φ]
Recall from Fact 3.15 that the projection operators on propositions can be characterized algebraically:
• !P = P^{**}
• $?P=P\cup {P}^{*}$
Since negation expresses absolute pseudocomplementation and disjunction expresses the join operation, this means that the connectives ! and ? can be characterized in terms of negation and disjunction.
Fact 4.12 (Projection operators in terms of negation and disjunction)
For any $\phi \in \mathcal{L}$:
• !φ ≡¬¬φ
• ?φ ≡ φ ∨¬φ
(p.65) This means that ! and ? do not have to be added to our logical language as primitive connectives; !φ and ?φ can simply be regarded as abbreviations of ¬¬φ and φ ∨¬φ, respectively.
Finally, we have that a sentence φ is always equivalent to the conjunction of its two ‘pure components’ !φ and ?φ (the analogue of Fact 3.14).
Fact 4.13 (Division)
For any φ:
• φ ≡ !φ ∧ ?φ
4.4 Examples
Now let us consider some concrete sentences in InqB and the propositions that they express. We will assume that our language contains just one unary predicate symbol, R, and two individual constants, a and b. Accordingly, we will assume that the domain of discourse consists of just two objects, denoted by a and b, respectively. Our logical space consists of four worlds, one in which both Ra and Rb are true, one in which Ra is true but Rb is false, one in which Rb is true but Ra is false, and one in which neither Ra nor Rb is true. These worlds will be labeled 11, 10, 01, and 00, respectively. As usual, in order to keep the pictures orderly we display only the maximal elements of a proposition. For concreteness, we will read Ra as ‘Ann is in Rome’, and Rb as ‘Bob is in Rome’.
Atomic sentences. Let us first consider the proposition expressed by the atomic sentences Ra and Rb. According to the clause for atomic sentences, [Ra] consists of all states s such that every world in s makes Ra true, i.e., the state {11, 10} and all substates thereof. Thus, as depicted in Figure 4.1(a), [Ra] has a greatest element, {11, 10}. Fact 4.10 therefore ensures that Ra is noninquisitive. It provides the information that Ann is in Rome, and it does not request any further information. So it behaves just as in classical logic. Analogously, Rb provides the information that Bob is in Rome, without requesting any further information. The proposition expressed by Rb is depicted in Figure 4.1(b).
Disjunction. Next, consider the disjunction Ra ∨ Rb. According to the clause for disjunction, [Ra ∨ Rb] consists of those states that are in [Ra] or in [Rb]. These are {11, 10}, {11, 01}, and all substates thereof, as depicted in Figure 4.1(c). (p.66)
Since $\text{info}(Ra\vee Rb)=\bigcup [Ra\vee Rb]=\{11,10,01\}\ne \mathcal{W}$, the disjunction Ra ∨ Rb is informative. It provides the information that at least one of Ann and Bob is in Rome. However, unlike in the case of atomic sentences, in this case there is no unique greatest element in [Ra ∨ Rb] that includes all the others. Instead, there are two maximal elements, Ra = {11, 10} and Rb = {11, 01}, which together contain all the others. Thus, besides being informative, Ra ∨ Rb is also inquisitive. In order to settle the issue that it raises, one has to establish either that Ann is in Rome, or that Bob is in Rome.
A note of caution is perhaps in order here: it is important to keep in mind that InqB does not directly embody an analysis of sentences in natural language: it only provides the tools to formulate such analyses. In particular, a disjunctive sentence in InqB like Ra ∨ Rb does not necessarily correspond to a disjunctive declarative sentence in English like (1) below, or to a disjunctive interrogative sentence like (2) for that matter.
(1)
(2)
In Chapter 6 we will present a concrete analysis of such sentences using InqB. On that analysis, (1) corresponds to !(Ra ∨ Rb) and (2) corresponds either to ?(Ra ∨ Rb) or to Ra ∨ Rb, depending on intonation.
Negation. Let us now turn to negation. According to the clause for negation, [¬Ra] consists of all states s such that s does not have any world in common with any state in [Ra]. Thus, [¬Ra] consists of all states that do not contain the worlds 11 and 10, which are ¬Ra = {01, 00} and all substates thereof, as depicted in Figure 4.1(d). Since this set of states has a greatest element, Fact 4.10 ensures that ¬Ra is noninquisitive. It provides the information that Ann is not in Rome, and does not request any further information.
Now let us consider the negation of an inquisitive disjunction, ¬(Ra ∨ Rb). According to the clause for negation, [¬(Ra ∨ Rb)] consists (p.67) of all states which do not have a world in common with any state in [Ra ∨ Rb]. Thus, [¬(Ra ∨ Rb)] consists of all states that do not contain the worlds 11, 10, and 01, which are {00} and ∅, as depicted in Figure 4.1(e). Again, there is a unique maximal element, namely ¬(Ra ∨ Rb) = {00}. Thus, ¬(Ra ∨ Rb) provides the information that neither Ann nor Bob is in Rome, just like in classical logic, and does not request any further information.
These examples of negative sentences exemplify the general observation that we made above concerning pseudocomplementation (just below Fact 3.4): the absolute pseudocomplement of a proposition always contains a greatest element. This means that a negative sentence ¬φ is never inquisitive; it simply provides the information that φ is false, and does not request any further information.
Projection operators. Next let us consider !(Ra ∨ Rb), which abbreviates ¬¬(Ra ∨ Rb). We have just seen that ¬(Ra ∨ Rb) expresses the proposition depicted in Figure 4.1(e). Applying negation again, we arrive at the proposition depicted in Figure 4.2(a), which has Ra ∨ Rb as its unique alternative. Notice that !(Ra ∨ Rb) is not equivalent with Ra ∨ Rb. The two sentences have the same informative content, but the former is purely informative, while the latter is also inquisitive. This exemplifies the general nature of !: for any sentence φ, !φ is a noninquisitive proposition with the same informative content as φ. If φ itself is already noninquisitive, then !φ and φ are equivalent; if φ is inquisitive, as in the example just considered, the two express different propositions.
Let us now turn to ?. Consider ?Ra, which is an abbreviation of Ra ∨¬Ra. We have already seen what [Ra] and [¬Ra] are. According to the clause for disjunction, [?Ra] = [Ra ∨¬Ra] consists of all states that are either in [Ra] or in [¬Ra]. These states are Ra, ¬Ra, and all substates thereof, as depicted in Figure 4.2(b). Since $\text{info}(?Ra)=\mathcal{W}$, ?Ra is noninformative. On the other hand, since [?Ra] contains two alternatives, it is inquisitive. In order to settle the issue that it raises, one has to establish either that Ann is in Rome, or that Ann is not in Rome. (p.68) That is, one has to establish whether Ann is in Rome. Thus, while ?Ra is shorthand for Ra ∨¬Ra, perhaps the most famous classical tautology, it is not a tautology in InqB: instead, it corresponds to the polar question ‘whether Ra’. Analogously, ?Rb, depicted in Figure 4.2(c), corresponds to the polar question ‘whether Rb’.
If ? applies to the disjunction Ra ∨ Rb, which is already inquisitive, then it yields the proposition depicted in Figure 4.2(d). [Ra ∨ Rb] already contains two alternatives, Ra and Rb; ? adds a third alternative, which is the set of worlds that are neither in Ra nor in Rb. Thus, in order to resolve the issue raised by ?(Ra ∨ Rb), one has to establish either that Ann is in Rome, or that Bob is, or that neither is.
Finally, let us consider a case where ! and ? both apply, one after the other: ?!(Ra ∨ Rb). As we already saw above, [!(Ra ∨ Rb)] contains a single alternative, consisting of all worlds where at least one of a and b is in Rome. As depicted in Figure 4.2(e), ? adds a second alternative, which is the set of worlds where neither Ann nor Bob is in Rome. Notice that the resulting proposition differs from that expressed by ?(Ra ∨ Rb), which contains three alternatives rather than two. In order to settle the issue expressed by ?!(Ra ∨ Rb) it is sufficient to establish that at least one of Ann and Bob is in Rome. In order to settle the issue expressed by ?(Ra ∨ Rb) this is not sufficient; rather, it needs to be established for one of Ann and Bob that he or she is in Rome, or that neither of them is. We will see in Chapter 6 that the ability to capture such subtle differences is crucial in order to account for various kinds of disjunctive questions in natural languages.
Conjunction. Next, let us consider conjunction. First, let us look at the conjunction of our two atomic, noninquisitive sentences, Ra and Rb. According to the clause for conjunction, [Ra ∧ Rb] consists of those states that are both in [Ra] and in [Rb]. These are {11} and ∅. Thus, [Ra ∧ Rb] has a greatest element, namely {11}, and accordingly Ra ∧ Rb provides the information that both Ann and Bob are in Rome, just like in the classical case, and does not request any further information.
(p.69) Now let us look at the conjunction of two inquisitive sentences, ?Ra and ?Rb. As depicted in Figure 4.3(b), the proposition [?Ra ∧ ?Rb] contains four alternatives, Ra ∧ Rb, Ra ∧¬Rb, ¬Ra ∧ Rb, and ¬Ra ∧¬Rb. Since these alternatives together cover the entire logical space ?Ra ∧ ?Rb is noninformative. On the other hand, since there is more than one alternative, ?Ra ∧ ?Rb is inquisitive. In order to settle the issue that it raises, one has to establish one of Ra ∧ Rb, Ra ∧¬Rb, ¬Ra ∧ Rb, ¬Ra ∧¬Rb. Thus, our conjunction is a purely inquisitive sentence which requests enough information to settle both the issue whether Ann is in Rome, contributed by ?Ra, and the issue whether Bob is in Rome, contributed by ?Rb.
These two examples of conjunctive sentences exemplify a general fact: if φ and ψ are noninquisitive, then the conjunction φ ∧ ψ is noninquisitive as well, conveying both the information provided by φ and the information provided by ψ; on the other hand, if φ and ψ are noninformative, then the conjunction φ ∧ ψ is noninformative as well, expressing an issue which is settled just in case both the issue expressed by φ and the one expressed by ψ are settled.
Implication. Next, let us consider implication. Again, we will first consider a simple case, $Ra\to Rb$, where both the antecedent and the consequent are atomic, and therefore noninquisitive. According to the clause for implication, $[Ra\to Rb]$ consists of all states s such that every substate $t\subseteq s$ that is in [Ra] is also in [Rb]. These are all and only those states that are contained in $Ra\to Rb=\{11,01,00\}$, as depicted in Figure 4.3(c). So, $[Ra\to Rb]$ has a unique greatest element, $Ra\to Rb$, which means that the implication $Ra\to Rb$ is a noninquisitive sentence which, just like in the classical setting, provides the information that if Ann is in Rome, then so is Bob.
Now let us consider a more complex case, Ra → ?Rb, where the consequent is noninformative but inquisitive. As depicted in Figure 4.3(d), the proposition [Ra → ?Rb] contains two alternatives, $Ra\to Rb=\{11,01,00\}$, and $Ra\to \neg Rb=\{10,01,00\}$. Since these two alternatives together cover the entire logical space, our implication is noninformative. Moreover, since there is more than one alternative, the implication is inquisitive. In order to settle the issue that it raises, one must either establish $Ra\to Rb$, or Ra →¬Rb. In the former case one establishes that if Ann is in Rome, then so is Bob; in the latter case, that if Ann is in Rome, then Bob isn’t. So $Ra\to ?Rb$ requests enough information to establish whether Bob is in Rome under the assumption that Ann is.
(p.70) Again, these two examples of conditional sentences exemplify a general feature of InqB: if ψ is noninquisitive, then φ → ψ is noninquisitive as well; and similarly, if ψ noninformative, then φ → ψ is noninformative as well.
Quantification. Finally, let us consider existential and universal quantification. As usual, existential quantification behaves essentially like disjunction and universal quantification behaves essentially like conjunction. In fact, since our current domain of discourse consists of only two objects, denoted by a and b, respectively, ∃x.Rx expresses exactly the same proposition as Ra ∨ Rb, depicted in Figure 4.1(c), and ∀x.Rx expresses exactly the same proposition as Ra ∧ Rb, depicted in Figure 4.3(a). Finally, consider the proposition expressed by ∀x.?Rx, depicted in Figure 4.3(e). Notice that this proposition induces a partition on the logical space, where each block of the partition consists of worlds that agree on the extension of R. Thus, ∀x.?Rx asks for an exhaustive specification of the individuals that are in Rome.
4.5 Informative content, truth, and support
Recall that info(φ) is defined as $\bigcup [\phi ]$, which is a set of worlds. In classical logic, the informative content of a sentence φ is also embodied by a set of worlds, namely the set of all worlds where φ is true, φ. Thus, it is natural to ask how these two notions of informative content relate to each other. The answer is that the relation is as direct as it could be: the two always coincide.
Fact 4.14 (Informative content and truth)
For any $\phi \in \mathcal{L}$, info(φ) = φ.
This shows that InqB preserves the classical treatment of informative content. The system only differs from classical logic in that, besides informative content, it takes inquisitive content into consideration as well.
Notice that Facts 2.18 and 4.14 together yield the following characterization of noninformative and noninquisitive sentences in terms of classical truth.
Fact 4.15 (Informational and inquisitive triviality in terms of classical truth)
• φ is noninformative $\phantom{\rule{2.77695pt}{0ex}}\iff \phantom{\rule{2.77695pt}{0ex}}\phi =\mathcal{W}$
• φ is noninquisitive ⇔ φ∈ [φ] ⇔ [φ] = ℘(φ)
(p.71) Thus, noninformative sentences in InqB are precisely those sentences that are classically true at any world. On the other hand, a sentence φ is noninquisitive in InqB just in case the proposition it expresses is fully determined by its classical truthset: it provides the information that φ is true, and does not request any further information. Thus, noninquisitive sentences behave exactly as in classical logic.
The classical behavior of noninquisitive sentences results in a tight connection between their support conditions and their truth conditions. Namely, such a sentence φ is supported by a state s just in case it is true in every world in s. This holds only for noninquisitive sentences; the moment a sentence becomes inquisitive, the connection between support and truth breaks down.
Fact 4.16 (Support and truth)
The following are equivalent for any sentence $\phi \in \mathcal{L}$:
• φ is noninquisitive
• For every information state s: s⊧φ ⇔ φ is true in every world in s
4.6 Syntactic properties of nonhybrid sentences
Below we provide some syntactic conditions which make it easy to recognize sentences that are either noninformative or noninquisitive, just based on their form, without inspecting their meaning.
Let us start with noninquisitive sentences. The following fact provides some syntactic conditions which guarantee that a sentence is noninquisitive. These conditions generalize some of the more specific observations that were already made in discussing the examples above.
Fact 4.17 (Sufficient conditions for noninquisitivity)
1. Atomic sentences are always noninquisitive;
2. ¬φ is always noninquisitive;
3. !φ is always noninquisitive;
4. If φ and ψ are noninquisitive, then so is φ ∧ ψ;
5. If ψ is noninquisitive, then so is φ → ψ for any antecedent φ;
6. If φ(d′) is noninquisitive for all d ∈ D, then so is ∀xφ(x).
Now let us turn to noninformative sentences. Again we provide some syntactic conditions that guarantee that a given sentence is noninformative, generalizing some of the more specific observations made in discussing the examples in Section 4.4. (p.72)
Fact 4.18 (Sufficient conditions for noninformativity)
1. ?φ is always noninformative;
2. If φ and ψ are noninformative, so is φ ∧ ψ;
3. If ψ is noninformative, then so are φ ∨ ψ and φ → ψ, for any φ;
4. If φ(d′) is noninformative for all d ∈ D, then so is ∀xφ(x);
5. If φ(d′) is noninformative for some d ∈ D, then so is ∃xφ(x).
4.7 Sources of inquisitiveness
The partial syntactic characterization of noninquisitive sentences in Fact 4.17 implies that disjunction, the existential quantifier, and the projection operator ? are the only sources of inquisitiveness in our logical language.
Fact 4.19 (Sources of inquisitiveness)
Any sentence that does not contain ∨, ∃, or ? is noninquisitive.
Note that there is a close connection between disjunction, the existential quantifier, and the ? operator in InqB. Namely, they all behave as join operators: [φ ∨ ψ] is the join of [φ] and [ψ], [∃x.φ(x)] is the join of {[φ(d′)]∣d ∈ D}, and [?φ] is the join of [φ] and [φ]^{*}. In terms of semantic operators, then, the join operator is the essential source of inquisitiveness: without applying this operator, it is impossible to produce inquisitive propositions from noninquisitive ones.
This fact may provide the basis for an explanation of the wellknown observation that in many natural languages, question words are homophonous with words for disjunction and/or existentials (see Jayaseelan, 2001; Bhat, 2005; Haida, 2007; Jayaseelan, 2008; Cable, 2010; AnderBois, 2011; Slade, 2011, among others). For instance, Malayalam oo and Japanese ka are used for all three purposes.
Malayalam 
Japanese 
English 


Existential 
aaroo 
dareka 
someone 
Disjunction 
Annaoo Peteroo 
Annaka Peterka 
Anna or Peter 
Question 
Anna wannu(w)oo 
Anna wa kitaka 
Did Anna come? 
Szabolcsi (2015b) proposes an account of this crosslinguistic phenomenon in inquisitive semantics, suggesting that the inquisitive join operation can indeed be seen as the semantic common core of disjunctive, existential, and interrogative constructions in languages like Malayalam and Japanese.
(p.73) 4.8 Comparison with alternative semantics
There is a close connection between the treatment of disjunction and existentials in InqB, and their treatment in alternative semantics (Kratzer and Shimoyama, 2002; MenéndezBenito, 2005; Simons, 2005; AlonsoOvalle, 2006; Aloni, 2007, among others). In both frameworks, disjunction and existentials introduce sets of alternatives. In the case of alternative semantics, this treatment is motivated by a number of empirical phenomena, including free choice inferences, exclusivity implicatures, and counterfactual conditionals with disjunctive antecedents. The analysis of disjunction and existentials as introducing sets of alternatives has made it possible to develop new accounts of these phenomena which improve considerably on previous accounts. However, while work on alternative semantics has provided ample empirical motivation for its treatment of disjunction and existentials, its explanatory power would increase substantially if the treatment could be motivated by considerations independent of the linguistic phenomena that it has aimed to capture.
Moreover, the empirical phenomena that have motivated work on disjunction and existentials in alternative semantics have been taken to require a radical departure from the classical algebraic treatment of disjunction and existentials. For instance, AlonsoOvalle (2006) writes in the conclusion section of his dissertation:
This dissertation has investigated the interpretation of counterfactuals with disjunctive antecedents, unembedded disjunctions, and disjunctions under the scope of modals. We have seen that capturing the natural interpretation of these constructions proves to be challenging if the standard analysis of disjunction, under which or is the Boolean join, is assumed.
Similarly, Simons (2005) starts her paper as follows:
In this paper, the meanings of sentences containing the word or and a modal verb are used to arrive at a novel account of the meaning of or coordinations. It has long been known that such sentences […] pose a problem for the standard treatment of or as a Boolean connective equivalent to set union.
The approach we have taken here shows that, once we take both informative and inquisitive content into account, general algebraic considerations lead essentially to the treatment of disjunction that was proposed in alternative semantics, thus providing exactly the independent motivation that has so far been missing (for detailed discussion of this point, see Roelofsen, 2015b). Moreover, it shows that the treatment (p.74) of disjunction as generating sets of alternatives can actually be seen as a natural generalization of the classical treatment, rather than a radical departure from it: as soon as we adopt a notion of meaning that encompasses both informative and inquisitive content, treating disjunction as a join operator automatically gives it the potential to generate multiple alternatives. Thus, we can have our cake and eat it: we can treat disjunction as a join operator and as introducing sets of alternatives at the same time. In inquisitive semantics, the two go hand in hand.^{3}
4.9 Exercises
Exercise 4.1 Propositions in InqB
Using diagrams analogous to those in Figure 4.1, depict the propositions expressed by the following formulas:
1. Ra ∧ ?Rb
2. ?Ra ∨ ?Rb
3. ¬(Ra ∧ Rb)
4. !∃xRx →∃xRx
5. !∃xRx →∀x?Rx
6. ?Ra → ?Rb
Exercise 4.2 De Morgan’s laws
Below are two wellknown classical equivalences, known as De Morgan’s laws:
Do these equivalences also hold in inquisitive semantics? If yes, give a proof. If no, provide a counterexample.
(p.75) Exercise 4.3 The law of double negation
Recall that in classical logic, ¬¬φ → φ is a tautology for any given formula φ. Show that, in InqB, ¬¬φ → φ is a tautology if and only if φ is noninquisitive.
Explain why this difference between classical logic and InqB arises, even though ¬ and → express exactly the same algebraic operations in both frameworks (absolute and relative pseudocomplementation, respectively). (p.76)
Notes:
(^{1}) The interpretation I_{w}(t) of a term t is defined inductively as usual: if t is an individual constant c, then I_{w}(t) = I_{w}(c). If $t=f({t}_{1},\dots ,{t}_{n})$, then ${I}_{w}(t)={I}_{w}(f)({I}_{w}({t}_{1}),\dots ,{I}_{w}({t}_{n}))$. Notice that, since we are working only with formulas without free variables, we do not need to consider the case that t is a variable.
(^{2}) Notice that the notions of entailment and equivalence given here are semantic notions which assume a given information model; as such, they incorporate facts that are encoded by the model as analytical, i.e., true in all possible worlds, but which are not purely logical. The purely logical notion of entailment—studied in inquisitive logic—is obtained by universally quantifying over all information models.
(^{3}) It should be noted that, while both in alternative semantics and in inquisitive semantics disjunction generates alternatives in a similar way, there is also a subtle but important difference. Namely, in inquisitive semantics one alternative can never be nested in another, unlike in alternative semantics. This has certain advantages, as we will discuss briefly in Section 9.1 (for more detailed discussion of this difference, see Ciardelli and Roelofsen, 2017a).