M -fuzzifying basic inquisitive semantics

Abstract

The basic system of inquisitive semantics (InqB) established by Groenendijk et al. is a general inquisitive semantic theory which doesn’t concern fuzziness. To explain the fuzzy phenomena in natural languages, this paper extends InqB into the framework of M-fuzzifying setting and establishes a basic system of M-fuzzifying inquisitive semantics. To begin with, the notion of M-fuzzifying supporting mapping is defined, where M is a completely distributive lattice with an involution operator and each subset of the universal set of all possible worlds can be regarded as a support of any well-formed formula to some degree. Then the notions of M-fuzzifying entailment order, M-fuzzifying truth mappings, M-fuzzifying informative content mappings and M-fuzzifying inquisitive content mappings are introduced and their properties are discussed. Further, the degrees of assertiveness, informativeness, inquisitiveness and questioning of a well-formed formula are defined, by which the M-fuzzifying assertive projection operator and the M-fuzzifying questioning projection operator are introduced and characterized. Finally, a necessary and sufficient condition is obtained, where a well-formed formula is exactly the disjunction of its unique M-fuzzifying assertive projection and unique M-fuzzifying questioning projection.

1 Introduction and preliminaries

Traditionally, the meaning of a (declarative) sentence is presented by a set of possible worlds, i.e., a proposition, identifying with the informative content of the sentence. In dynamic semantics, the meaning of a (declarative) sentence is identified with its potential to change the common ground, which is also construed as a set of possible worlds, representing the information shared by all conversational participants. That is, the central task of the informative content of a sentence is to enhance the corresponding common ground by eliminating some of its possible worlds [15]. However, sentences in natural languages are used not only to provide information by asserting facts, but also to request information by rising issues. Considering this fact, Groenendijk and Mascarenhas introduced a basic system of inquisitive semantics, where the proposition expressed by a (declarative or interrogative) sentence not only embodies its informative content, but also its inquisitive content [5, 9]. Later, Roelofsen further developed an algebraic foundation for inquisitive semantics [12]. In recent years, many scholars have been devoting on this topic [1–4 , 18].

In inquisitive semantics, a proposition expressed by a sentence (no matter interrogative or declarative) is presented as a nonempty and downward closed set of possibilities, where a possibility is a maximal set of possible worlds. As a result, a proposition always consists of one or more possibilities and participants are requested to respond in a cooperative way to choose among these possibilities. In each response, some possibilities are excluded and the common ground is enhanced [5]. Consequently, an entailment order occurs: if each possibility of a proposition is contained in a possibility of an another proposition, then the former proposition entails the latter. This entailment order gives rise to a complete Heyting algebra, where meet, join and implication stand for conjunction, disjunction and condition of propositions respectively [12].

Inquisitive semantics established by Groenendijk et al. doesn’t take fuzziness into consideration. That is, each possibility of a proposition is either true or false, because the possibilities always clearly divide the set of possible worlds. However, things become markedly different and more complex when it comes to fuzzy terms. Let’s compare the following two examples.

Situation 1: There is a butterfly and Tom is not sure whether it is male or female (since it’s hard to recognize a butterfly’s sex). So Tom may ask:

(1a) Is this butterfly male or not?

A satisfactory answer to (1a) is either (1b) or (1c), which are mutually exclusive.

(1b) It is male.

(1c) It is female.

Situation 2: Tom is holding in hand an apple that is partly red, and he isn’t sure whether it can be regarded as red or not (since it is partly red). So he asks:

(2a) Is this apple red or not?

However, in this situation, it is possible that neither (2b) nor (2c) is the true answer.

(2b) It is red.

(2c) It is not red.

Sentences (1a) and (2a) are both inquisitive. However, the propositions expressed by sentence (1a) are crisp, since the possibilities expressed by (1b) and (1c) are either absolutely true or absolutely false, while the propositions expressed by sentence (2a) are fuzzy, because the possibilities expressed by (2b) and (2c) are neither completely true nor completely false.

The differences between the two examples come from the fact that the application boundary of fuzzy terms like the predicate ‘red’ is not clear-cut. This can be seen from the fact that each of the sentences (2d)-(2g) below may be a true answer to (2a), depending on the degree of the red color. (Note that we say ‘may be’ because whether a partly red apple is a red one depends on local customs. People in some regions may think a partly red apple is a red one, while people in other regions may not think so. This is not our concern.)

(2d) It is almost red.

(2e) It is not very red.

(2f) It is half red.

(2g) It is a little red.

From Example 2, we see that whether an object does or does not have an attribute can often be a matter of degree. Zadeh had defined fuzzy sets and pointed out that it is possible to develop a mathematical model of semantics to express the fuzziness of natural languages [20 –23]. His approach has received much attention in the literature [10 , 13]. In this paper, we extend inquisitive semantics into M-fuzzifying setting. We start with a definition of M-fuzzifying support mappings which assigns a degree to each subset of the universal set of all possible worlds for a given well-formed formula. Based on this definition, we further define M-fuzzifying truth (resp. informative content, inquisitive content) mapping. Then we introduce degrees of assertiveness, informativeness, inquisitiveness and questioning of a well-formed formula and discuss their relations. After this, we define M-fuzzifying assertive (resp. questioning) projection mappings and show that a well-formed formula has a unique M-fuzzifying assertive projection and a unique M-fuzzifying questioning projection. Finally, we give a necessary and sufficient condition, under which a proposition is exactly the conjunction of its M-fuzzifying assertive projection and its M-fuzzifying questioning projection. Besides, if M is a Boolean algebra, then this condition can be removed.

The following are some basic notions and denotations in inquisitive semantics (other notions not being mentioned can be seen in [5 , 18]).

The framework of the basic inquisitive semantics (briefly InqB) is a propositional languages structure $(L_{P}, W_{P}, ⊨)$ , where $L_{P}$ is a propositional languages over a nonempty set of propositional variables $P$ ( $P$ is either finite or countably infinite), $W_{P}$ is the universal set of suitable possible worlds with its power set $2^{W_{P}}$ , and ⊨ is both the support symbol and the entailment order. Elements of $L_{P}$ are called well-formed formulas (briefly wffs), which are logical representations of propositions, which in turn can be expressed by sentences. Elements of $2^{W_{P}}$ are called states or supports, corresponding to pieces of information expressed by sentences [12, 18].

Atomic propositional variables in $L_{P}$ are denoted by lower-cast bold letters such as p, q, and wffs in $L_{P}$ are denoted by Capital letters such as $A, B, C$ (atomic propositions are simple wffs; the negation of $A$ is also a wff, denoted by $\neg A$ ). In particular, the contradiction and the tautology in $L_{P}$ are respectively denoted by ⊥ and ⊤. The primitive logical constants of the languages are ⊥, ∨ , ∧ , →, where → is the implication operator [6 , 18]. These symbols are written in bold typefaces to distinguish the operators in the lattice M.

The definition of supports in InqB given by Groenendijk [5] and Wiśniewski [18] is a subset $σ \subseteq W_{P}$ satisfying the following conditions: for all $p, A, B \in L_{P}$ and $σ \subseteq W_{P}$ ,

(S1) σ ⊨ p iff p is true in each w ∈ σ;

(S2) σ⊨ ⊥ iff σ =∅.

(S3) $σ ⊨ (A \land B)$ iff $σ ⊨ A$ and $σ ⊨ B$ ;

(S4) $σ ⊨ (A \lor B)$ iff $σ ⊨ A$ or $σ ⊨ B$ ;

(S5) $σ ⊨ (A \to B)$ iff for each τ ⊆ σ, if $τ ⊨ A$ , then $τ ⊨ B$ .

The negation $\neg A$ of an wff $A \in W_{P}$ is defined by: $\neg A = A \to ⊥$ . Thus, for each $σ \subseteq W_{P}$ , $σ ⊨ \neg A$ iff $τ ⊭ A$ for each ∅ ≠ τ ⊆ σ. Besides, since a proposition in InqB is defined as a nonempty, downward closed set of states, it is easy to check that the following items are valid for any $A \in L_{P}$ [12]:

(1) $\emptyset ⊨ A$ ;

(2) If $σ ⊨ A$ then $τ ⊨ A$ for all τ ⊆ σ.

For $σ \subseteq W_{P}$ and $A \in L_{P}$ , the notion $σ ⊨ A$ means that σ supports $A$ , indicating that σ is a piece of information contained in the sentence represented by $A$ . Further, for all $A, B \in L_{P}$ , $A$ entails $B$ , denoted by $A ⊨ B$ , is defined by: for all $σ \subseteq W_{P}$ , if $σ ⊨ A$ , then $σ ⊨ B$ . In particular, $A$ is equivalent to $B$ , denoted by $A = B$ , if $A$ and $B$ are mutually entailed [12].

Next, we recall some notions and results in lattices.

M is a completely distributive lattice with an involution operator ^′. The minimal element and the maximal element in M is respectively denoted by ⊥_M and ⊤_M (or simply, ⊥ and ⊤). For φ ⊆ M, ⋁_a∈φa and ⋀_a∈φa are denoted by ⋁φ and ⋀φ respectively. Further, M is called a Boolean algebra, if a∧ a′ = ⊥ and a∨ a′ = ⊤ for all a, b ∈ M. If M = {⊥ , ⊤}, then we simply write 2 for M. Clearly, 2 is a Boolean algebra.

An element a ∈ M is called a prime element if for all b, c ∈ M, b ∧ c ≤ a implies b ≤ a or c ≤ a. The set of all prime elements in M ∖ {⊤} is denoted by P (M). An element a ∈ M is called a co-prime element if its complement a′ is a prime element. The set of all co-prime elements in M ∖ {⊥} is denoted by J (M). For any a ∈ M, there are φ ⊆ P (M) and ψ ∈ J (M) such that a = ⋀ φ = ⋁ ψ [17].

A binary relation ≺ on M is defined by: for all a, b ∈ M, a ≺ b iff for each φ ⊆ M, b ≤ ⋁ φ always implies the existence of d ∈ φ such that a ≤ d. The mapping β : M → 2^M, defined as: β (a) = {b : b ≺ a} for all a ∈ M, satisfies β (⋁ _i∈Ωa_i) = ⋃ _i∈Ωβ (a_i) for all {a_i} _i∈Ω ⊆ M [14 , 19]. It is true that β (⊥) = ∅ and a = ⋁ β (a) = ⋁ β^* (a) for each a ∈ M, where β^* (a) = β (a) ∩ J (M) [14, 19]. The implication operator on M is a binary operator →_M : M × M → M (simply denoted by →), defined by: a → _Mb = ⋁ {c ∈ M : a ∧ c ≤ b} for all a, b ∈ M [16]. The following are some equivalent conditions of the partial order ‘≤’ on M which can be seen in [14, 19]:

(1) b ≤ d;

(2) For each a ∈ M, a ≤ b implies a ≤ d;

(3) For each a ∈ J (M), a ≤ b implies a ≤ d;

(4) For each a ∈ β (⊤), a ≤ b implies a ≤ d;

(5) For each a ∈ β^* (⊤), a ≤ b implies a ≤ d.

An M-fuzzy set U on X is a mapping U : X → M. The set of all M-fuzzy sets on X is denoted by M^X. The support set of an M-fuzzy U ∈ M^X is defined by Supp (U) = {x ∈ X : U (x) ≠ ⊥}.

2 M-fuzzifying supports and M-fuzzifying entailments

Since the support mapping is a basic notion in InqB, we first extend this notion to the M-fuzzifying support mapping. Before this, let’s illustrate that the lattice 2 can be applied to InqB.

As we know, InqB doesn’t take fuzziness into consideration. That is, given a wff $A \in L_{P}$ and $w \in W_{P}$ , the truth of $A$ in w is either 1 or 0. Thus the truth value of $A$ in w is contained in 2.

Further, in InqB $(L_{P}, W_{P}, ⊨)$ , we have either $σ ⊨ A$ or $σ ⊭ A$ for all $σ \in W_{P}$ and $A \in L_{P}$ . That is, σ either supports or does not support $A$ . Thus it is natural to define a mapping from $L_{P} \times 2^{W_{P}}$ to 2 by: $S (A, σ) = ⊤$ iff $σ ⊨ A$ , and $S (A, σ) = ⊥$ iff $σ ⊭ A$ . Hence we can interpret this mapping by: $σ ⊨ A$ iff the degree that σ supports $A$ is ⊤; $σ ⊭ A$ iff the degree that σ supports $A$ is ⊥. In this sense, ‘support’ in InqB also has degrees that contained in 2.

Now, we replace 2 by M. Based on the support mapping in InqB, we define the M-fuzzifying supporting mapping as follows.

Definition 2.1. A mapping $S : L_{P} \times 2^{W_{P}} \to M$ is called the M-fuzzifying support mapping if for all $p, A, B \in L_{P}$ and $σ \subseteq W_{P}$ ,

(MS1) S (p, σ) = ⋀ _w∈σS (p, {w});

(MS2) S (⊥ , σ) = ⊥ for each σ≠ ∅;

(MS3) $S (A \land B, σ) = S (A, σ) \land S (B, σ)$ ;

(MS4) $S (A \lor B, σ) = S (A, σ) \lor S (B, σ)$ ;

(MS5) $S (A \to B, σ) = ⋀_{τ \subseteq σ} [S (A, τ)^{'} \lor S (B, τ)]$ .

Remark 2.2. Let $A, B \in L_{P}$ and $σ \subseteq W_{P}$ .

(1) In InqB, since $\emptyset ⊨ A$ , and $σ ⊨ A$ implies $τ ⊨ A$ for each τ ⊆ σ, it is natural to think the following items are necessary:

(i) $S (A, \emptyset) = ⊤$ ;

(ii) $S (A, σ) \leq ⋀_{τ \subseteq σ} S (A, τ)$ for all $σ \in 2^{W_{P}}$ .

However, the inverse of (ii) may fail. It can be shown by the example in (3) of Remark 4.16.

(2) For any $A \in L_{P}$ any any $σ \in W_{P}$ , the value $S (A, σ)$ can be regarded as the degree that σ supports $A$ . Further, the mapping $S (A) : 2^{W_{P}} \to M$ , defined by $S (A) (σ) = S (A, σ)$ for any $σ \subseteq W_{P}$ , can be regarded as the M-fuzzifying of A. Clearly, $S (A) \in M^{2^{W_{P}}}$ .

(3) For all $A, B \in L_{P}$ , we call that $A$ M-entails $B$ , denoted by $A ⊨_{M} B$ , if $S (A) \leq S (B)$ . Further, we call $A$ M-equivalent to $B$ , denoted by $A =_{M} B$ , if $A$ and $B$ are mutually M-entailed.

The following are some properties of the M-fuzzifying support mapping.

Theorem 2.3. Let $A, B, C, p \in L_{P}$ , $σ, η, θ \subseteq W_{P}$ and $m \in ℕ$ .

(1) $S (\neg A, σ) = ⋀_{τ \in 2^{σ} ∖ {\emptyset}} S (A, τ)^{'}$ .

(2) $S (\neg A, σ) = ⋀_{w \in σ} S (\neg A, {w})$ .

(3) $A \to (B \land C) =_{M} (A \to B) \land (A \to C)$ .

(4) $(A \lor B) \to C =_{M} (A \to C) \land (B \to C)$ .

(5) $(A \to B) \lor (A \to C) ⊨_{M} A \to (B \lor C)$ .

(6) $A ⊨_{M} \neg \neg A$ and ¬¬ p = _Mp.

(7) $\neg^{2 m + 1} A =_{M} \neg A$ and $\neg^{2 m} A =_{M} \neg \neg A$ .

(8) $S (\neg A, η) \land S (\neg A, θ) = S (\neg A, η \cup θ)$ .

(9) $\neg (A \lor B) =_{M} \neg A \land \neg B$ .

(10) $\neg A \lor \neg B ⊨_{M} \neg (A \land B)$ .

(11) $\neg A \to (B \lor C) =_{M} (\neg A \to B) \lor (\neg A \to C)$ .

Proof. (1) By (MS2), (MS5) and (1) of Remark 2, $\begin{matrix} S (\neg A, σ) & = & S (A \to ⊥, σ) \\ = & S (⊥, \emptyset) \land ⋀_{τ \in 2^{σ} ∖ {\emptyset}} [S (A, τ)^{'} \lor S (⊥, τ)] \\ = & ⋀_{τ \in 2^{σ} ∖ {\emptyset}} S (A, τ)^{'} . \end{matrix}$

(2) If σ =∅, then $\begin{matrix} S (\neg A, σ) = ⊤ = ⋀_{w \in σ} S (\neg A, {w}) . \end{matrix}$

If σ≠ ∅, then $\begin{matrix} S (\neg A, σ) & = & ⋀_{τ \in 2^{σ} ∖ {\emptyset}} S (A, τ)^{'} \\ \leq & ⋀_{w \in σ} S (A, {w})^{'} = ⋀_{w \in σ} S (\neg A, {w}) . \end{matrix}$

Conversely, since $S (A, τ) \leq S (A, {w})$ for any τ≠ ∅ and any w ∈ τ by (1) of Remark 2, we have

$\begin{matrix} S (\neg A, σ) & = & ⋀_{τ \in 2^{σ} ∖ {\emptyset}} S (A, τ)^{'} \\ \geq & ⋀_{w \in τ} S (\neg A, {w}) = ⋀_{w \in σ} S (\neg A, {w}) . \end{matrix}$

Therefore $S (\neg A) (σ) = ⋀_{w \in σ} S (\neg A, {w})$ .

(3) By (MS3) and (MS5), we have $\begin{matrix} S (A \to (B \land C), σ) \\ = & ⋀_{τ \subseteq σ} [S (A, τ)^{'} \lor (S (B, τ) \land S (C, τ))] \\ = & ⋀_{τ \subseteq σ} [S (A, τ)^{'} \lor S (B, τ)] \land [S (A, τ)^{'} \lor S (C, τ))] \\ = & S (A \to B, σ) \land S (A \to C, σ) . \end{matrix}$

(4) By (MS4) and (MS5), we have $\begin{matrix} S ((A \lor B) \to C, σ) \\ = & ⋀_{τ \subseteq σ} [(S (A, τ)^{'} \land S (B, τ)^{'}) \lor S (C, τ)] \\ = & ⋀_{τ \subseteq σ} [S (A, τ)^{'} \lor S (C, τ)] \land [S (B, τ)^{'} \lor S (C, τ))] \\ = & S (A \to C, σ) \land S (B \to C, σ) . \end{matrix}$

(5) By (MS5), we have $\begin{matrix} S ((A \to B) \lor (A \to C), σ) \\ = & S (A \to B, σ) \lor S (A \to C, σ) \\ \leq & ⋀_{τ \subseteq σ} [S (A, τ)^{'} \lor S (B, τ)] \lor [S (A, τ)^{'} \lor S (C, τ)] \\ = & ⋀_{τ \subseteq σ} [S (A, τ)^{'} \lor S (B \lor C, τ)] \\ = & S (A \to (B \lor C), σ) . \end{matrix}$

(6) Let $σ \subseteq W_{P}$ . If σ =∅, then $S (\neg \neg A, \emptyset) = ⊤ = S (A, \emptyset)$ . If σ≠ ∅, then $\begin{matrix} S (\neg \neg A, σ) & = & ⋀_{τ \in 2^{σ} ∖ {\emptyset}} S (\neg A, τ)^{'} \\ = & ⋀_{τ \in 2^{σ} ∖ {\emptyset}} (⋀_{w \in τ} S (A, {w})^{'})^{'} \\ = & ⋀_{τ \in 2^{σ} ∖ {\emptyset}} ⋁_{w \in τ} S (A, {w}) \\ \geq & ⋀_{τ \in 2^{σ} ∖ {\emptyset}} S (A, τ) \geq S (A, σ) . \end{matrix}$

Thus $S (A) \leq S (\neg \neg A)$ . Hence S (p) ≤ S (¬¬ p). Conversely, by (MS1), $\begin{matrix} S (\neg \neg p) (σ) & = & ⋀_{τ \in 2^{σ} ∖ {\emptyset}} ⋁_{η \in 2^{τ} ∖ {\emptyset}} S (p, η) \\ \leq & ⋀_{w \in σ} S (p, {w}) = S (p, σ) . \end{matrix}$

Hence S (¬¬ p) = S (p).

(7) We prove that $S (\neg \neg \neg A) = S (\neg A)$ and $S (\neg \neg \neg \neg A) = S (\neg \neg A)$ . By (2), we have $\begin{matrix} S (\neg \neg \neg A) (σ) & = & ⋀_{τ \in 2^{σ} ∖ {\emptyset}} ⋁_{η \in 2^{τ} ∖ {\emptyset}} S (\neg A) (η) \\ = & ⋀_{τ \in 2^{σ} ∖ {\emptyset}} ⋁_{w \in τ} S (\neg A) ({w}) \\ = & ⋀_{w \in σ} S (\neg A) ({w}) = S (\neg A) (σ) . \end{matrix}$

Thus $S (\neg \neg \neg A) = S (\neg A)$ and $\begin{matrix} S (\neg \neg \neg \neg A) = S (\neg \neg \neg (\neg A)) = S (\neg \neg A) \end{matrix}$

Hence $\begin{matrix} \neg^{2 m + 1} A =_{M} \neg^{2 m - 1} A =_{M} \cdot \cdot \cdot =_{M} \neg A \end{matrix}$ and $\begin{matrix} \neg^{2 m} A =_{M} \neg^{2 m - 2} A =_{M} \cdot \cdot \cdot =_{M} \neg \neg A . \end{matrix}$

(8) It directly follows from (2).

(9) By (1) and (2), we have $\begin{matrix} S (\neg (A \lor B), σ) & = & ⋀_{w \in σ} S (A \lor B, {w})^{'} \\ = & ⋀_{w \in σ} [S (A, {w})^{'} \land S (B, {w})^{'}] \\ = & S (\neg A, σ) \land S (\neg B, σ) \\ = & S (\neg A \land \neg B, σ) . \end{matrix}$

(10) By (1) and (2), we have $\begin{matrix} S (\neg (A \land B), σ) & = & ⋀_{w \in σ} S (A \land B, {w})^{'} \\ = & ⋀_{w \in σ} [S (A, {w})^{'} \lor S (B, {w})^{'}] \\ \geq & S (\neg A, σ) \lor S (\neg B, σ) \\ = & S (\neg A \lor \neg B, σ) . \end{matrix}$

(11) By (5), we have $\begin{matrix} S ((\neg A \to B) \lor (\neg A \to C)) \leq S (\neg A \to (B \lor C)) . \end{matrix}$

Conversely, suppose that $\begin{matrix} S (\neg A \to (B \lor C)) ≰ S ((\neg A \to B) \lor (\neg A \to C)) . \end{matrix}$

Then there is $σ \in 2^{W_{P}}$ such that $\begin{matrix} S (\neg A \to (B \lor C), σ) \\ ≰ & S (\neg A \to B, σ) \lor S (\neg A \to C, σ) . \end{matrix}$

Thus there is b ∈ J (M) such that $b \leq S (\neg A \to (B \lor C), σ)$ , but $\begin{matrix} b ≰ S (\neg A \to B, σ) \lor S (\neg A \to C, σ) . \end{matrix}$

Since $\begin{matrix} b ≰ S (\neg A \to B, σ) = ⋀_{τ \subseteq σ} [S (\neg A, τ)^{'} \lor S (B, τ)], \end{matrix}$ there is τ₀ ∈ 2^σ ∖ {∅} such that $b ≰ S (\neg A, τ_{0})^{'} \lor S (B, τ_{0})$ . Similarly, since $\begin{matrix} b ≰ S (\neg A \to C, σ) = ⋀_{η \subseteq σ} [S (\neg A, η)^{'} \lor S (C, η)], \end{matrix}$ there is η₀ ∈ 2^σ ∖ {∅} such that $b ≰ S (\neg A, η_{0})^{'} \lor S (C, η_{0})$ . Thus $b ≰ S (B, τ_{0}) \lor S (C, η_{0})$ .

Further, by (1) of Remark 2, we have $\begin{matrix} S (B \lor C, τ_{0} \cup η_{0}) & = & S (B, τ_{0} \cup η_{0}) \lor S (C, τ_{0} \cup η_{0}) \\ \leq & S (B, τ_{0}) \lor S (C, η_{0}) . \end{matrix}$

Hence $b ≰ S (B \lor C, τ_{0} \cup η_{0})$ . Now, by (7), we have $\begin{matrix} b & ≰ & S (\neg A, τ_{0})^{'} \lor S (\neg A, η_{0})^{'} \\ = & [S (\neg A, τ_{0}) \land S (\neg A, η_{0})]^{'} \\ = & S (\neg A, τ_{0} \cup η_{0})^{'} . \end{matrix}$

However, by τ₀ ∪ η₀ = σ, we have $\begin{matrix} b & \leq & S (\neg A \to (B \lor C), σ) \\ = & S (\neg A, τ_{0} \cup η_{0})^{'} \lor S (B \lor C, τ_{0} \cup η_{0}) \end{matrix}$ which is a contradiction. Therefore $S (\neg A \to$ $(B \lor C)) \leq S ((\neg A \to B) \lor (\neg A \to C))$ . □

Remark 2.4. (3) and (4) in Theorem 2 are the M-fuzzifications of Distributive law 1 and 2, respectively [7]. In addition, (12) of Theorem 2 is the M-fuzzification of Kreisel-Putnam equivalence [12].

3 M-fuzzifying truth (resp. informative content, inquisitive content) mappings

In this section, we define notions of M-fuzzifying truth (resp. informative content, inquisitive content) mappings, based on which we further define notions of M-fuzzifying assertiveness (resp. inquisitiveness, questioning, informativeness, hybrid, tautology) of propositions.

As being described in Section 2, the lattice 2 can be applied in defining support mappings in InqB. Now, we illustrate that given a wff $A \in L_{P}$ in InqB, the lattice 2 also can be applied to interpret the truth set, informative content and inquisitive content of $A$ .

Recall that given a wff $A \in L_{P}$ , the truth set, informative content and inquisitive content of $A$ (refer to [12, 18]) are respectively defined by

(1) $| A | = {w \in W_{P} : {w} ⊨ A}$ ;

(2) $Inf (A) = ⋃ {σ \subseteq W_{P} : σ ⊨ A}$ ;

(3) $Inq (A) = {σ \subseteq W_{P} : σ \subseteq Inf (A), σ ⊨ A}$ .

Applying the mapping $S : L_{P} \times 2^{W_{P}} \to 2$ defined in the remark at the beginning of Section 2, we naturally obtain the following conclusions.

(1) $| A |$ is amount to the mapping $| A | : W_{P} \to 2$ , defined by $| A | (w) = ⊤$ whenever $S (A) ({w}) = ⊤$ ;

(2) $Inf (A)$ is amount to the mapping $Inf (A) : 2^{W_{P}} \to 2$ , defined by $Inf (A) (σ) = ⊤$ whenever σ = ⋃ _i∈Iσ_i and $S (A) (σ_{i}) = ⊤$ ;

(3) $Inq (A)$ is amount to the mapping $Inq (A) : 2^{W_{P}} \to 2$ , defined by $Inq (A) (σ) = ⊤$ whenever $Inf (A) (σ) = S (A) (σ) = ⊤$ .

Now, we replace 2 by the lattice M and then introduce the following mappings.

Definition 3.1. Let $A \in L_{P}$ .

(1) The M-fuzzifying truth mapping of $A$ is $| A | : W_{P} \to M$ , defined by $\begin{matrix} \forall w \in W_{P}, | A | (w) = S (A, {w}) . \end{matrix}$

(2) The M-fuzzifying informative content mapping is $I : L_{P} \times 2^{W_{P}} \to M$ , defined by $\begin{matrix} \forall A \in L_{P}, \forall σ \subseteq W_{P}, I (A, σ) = ⋀_{w \in σ} | A | (w) . \end{matrix}$

(3) The M-fuzzifying inquisitive content mapping is $J : L_{P} \times 2^{W_{P}} \to M$ , defined by $\begin{matrix} \forall A \in L_{P}, \forall σ \subseteq W_{P}, J (A, σ) = S (A, σ) \land I (A, σ) . \end{matrix}$

Remark 3.2. Let $A \in L_{P}$ , $w \in W_{P}$ and $σ \subseteq W_{P}$ .

(1) The value $| A | (w)$ can be regarded as the degree that w belongs to the truth set of $A$ .

(2) The value $I (A, {w})$ can be regarded as the degree that w belongs to the informative content of $A$ . Similarly, the value $I (A, σ)$ can be regarded as the degree that σ is contained by the informative content of $A$ . The mapping $I (A) : 2^{W_{P}} \to M$ , defined by $I (A) (σ) = I (A, σ)$ for each $σ \subseteq W_{P}$ , is called the M-fuzzifying informative content of $A$ . Clearly, $I (A) \in M^{2^{W_{P}}}$ .

(3) The value $J (A, σ)$ can be regarded as the degree that σ belongs to the inquisitive content of $A$ . The mapping $J (A) : 2^{W_{P}} \to M$ , defined by $J (A) (σ) = J (A, σ)$ for each $σ \subseteq W_{P}$ , is called the M-fuzzifying informative content of $A$ . Clearly, $J (A) \in M^{2^{W_{P}}}$ .

(4) The system $(L_{P}, W_{P}, ⊨_{M})$ , equipped with Definitions 2.1 and 3.1, is called the basic M-fuzzifying inquisitive semantics, which is denoted by M-InqB.

Now, we discuss the relations among M-fuzzifying truth mapping, M-fuzzifying informative mapping and M-fuzzifying inquisitive mapping.

Theorem 3.3 $I (A, {w}) = | A | (w)$ and $S (A, σ) \leq I (A, σ)$ for all $A \in L_{P}$ , $w \in W_{P}$ and $σ \subseteq W_{P}$ .

Proof. The result $I (A, {w}) = | A | (w)$ is clear. Also, $\begin{matrix} S (A, σ) \leq ⋀_{τ \subseteq σ} S (A, τ) \leq ⋀_{w \in σ} S (A, {w}) = I (A, σ) \end{matrix}$ by (1) of Remark 2.2. □

Corollary 3.4. Let $A \in L_{P}$ and $σ \subseteq W_{P}$ . We have

(1) $J (A, σ) = S (A, σ)$ ;

(2) $I (A, σ) = ⋀_{w \in σ} J (A, {w})$ ;

(3) $I_{a} (A) = ⋃ {σ \subseteq W_{P} : S (A, σ) \geq a}$ for a ∈ J (M), where $I_{a} (A) = {w \in W_{P} : I (A, {w}) \geq a}$ .

Proof. We only prove (3) since (1) and (2) are direct.

If $w \in I_{a} (A)$ , then $S (A, {w}) = I (A, {w}) \geq a$ . Thus $w \in ⋃ {σ \subseteq W_{P} : S (A, σ) \geq a}$ . Conversely, if $w \in ⋃ {σ \subseteq W_{P} : S (A, σ) \geq a}$ , then there is $σ \subseteq W_{P}$ such that $S (A, σ) \geq a$ and w ∈ σ. Thus $\begin{matrix} I (A, {w}) = S (A, {w}) \geq S (A, σ) \geq a . \end{matrix}$

Hence $w \in I_{a} (A)$ . □

In particular, if M = 2, then (1) of Corollary 3 shows that the inquisitive content of $A \in L_{P}$ is exact the set of all $σ \subseteq W_{P}$ supporting $A$ ; (3) of Corollary 3 shows that the informative content of $A \in L_{P}$ is exact the union of all $σ \subseteq W_{P}$ supporting $A$ [12, 18].

The following result directly follows from (1) of Corollary 3.4.

Theorem 3.5. $J (A) \leq J (B)$ implies $I (A) \leq I (B)$ for all $A, B \in L_{P}$ .

Now, for any $A \in L_{P}$ , we define and discuss M-fuzzifying assertiveness (resp. inquisitiveness, questioning, informativeness, hybrid and tautology) of $A$ .

Definition 3.6. Let $A \in L_{P}$ .

(1) The M-fuzzifying assertiveness of $A$ , denoted by $Ass (A)$ , is defined by $\begin{matrix} Ass (A) = I (A) \to_{M} J (A) . \end{matrix}$ That is, $Ass (A) = ⋀_{σ \subseteq W_{P}} I (A, σ) \to_{M} J (A, σ)$ .

(2) The M-fuzzifying inquisitiveness of $A$ , denoted by $Inq (A)$ , is defined by $\begin{matrix} Inq (A) = Ass (A)^{'} . \end{matrix}$

The value $Ass (A)$ (resp. $Inq (A)$ ) can be regard as the degree that $A$ is assertive (resp. inquisitive).

Theorem 3.7. Let $p, A \in L_{P}$ . Then

(1) $Ass (p) = Ass (\neg A) = ⊤$ ;

(2) $Ass (A) \geq S (A, Supp (I (A)))$ .

Proof (1) Let $σ \subseteq W_{P}$ . By Theorem 3.3, we have $\begin{matrix} S (p, σ) = ⋀_{w \in σ} S (p, {w}) = I (p, σ) . \end{matrix}$ Thus $J (p) = S (p) = I (p)$ by (1) of Corollary 3.4. Therefore Ass (p) =⊤.

By (2) of Corollary 3 and (2) of Theorem 2.3, $\begin{matrix} J (\neg A, σ) = S (\neg A, σ) = I (\neg A, σ) . \end{matrix}$ Thus $J (\neg A) = I (\neg A)$ followed by $Ass (\neg A) = ⊤$ .

(2) Let $a \in J (S (A, Supp (I (A))))$ . Then $\begin{matrix} a \leq S (A, Supp (I (A))) \leq I (A, Supp (I (A))) \end{matrix}$ which implies $Supp (I (A)) \subseteq I_{a} (A)$ . Also, $I_{a} (A) \subseteq Supp (I (A))$ is clear. So $Supp (I (A)) = I_{a} (A)$ .

Suppose that $a ≰ Asser (A)$ . Then there is $σ \subseteq W_{P}$ such that $a ≰ I (A, σ) \to_{M} J (A, σ)$ . Thus $I (A, σ) \land a ≰ J (A, σ)$ which implies some $b \in J (I (A, σ) \land a)$ such that $b ≰ J (A, σ)$ . Hence $σ \subseteq I_{b} (A)$ and $I_{a} (A) \subseteq I_{b} (A) \subseteq Supp (I (A))$ . But $σ \subseteq I_{b} (A) = Supp (I (A))$ which shows $J (A, σ) \geq a$ . It is a contradiction. So $Ass (A) \geq a$ . Therefore $Ass (A) \geq S (A, Supp (I (A)))$ . □

Definition 3.8. Let $A \in L_{P}$ .

(1) The M-fuzzifying questioning of $A$ , denoted by Que (A), is defined by $\begin{matrix} Que (A) = I (A, W_{P}) . \end{matrix}$

(2) The M-fuzzifying informativeness of $A$ , denoted by $Inf (A)$ , is defined by $\begin{matrix} Inf (A) = Que (A)^{'} . \end{matrix}$

(3) The M-fuzzifying hybrid of $A$ , denoted by $Hyb (A)$ , is defined by $\begin{matrix} Hyb (A) = Inq (A) \land Inf (A) . \end{matrix}$

(4) The M-fuzzifying tautology of $A$ , denoted by $Tau (A)$ , is defined by $\begin{matrix} Tau (A) = Ass (A) \land Que (A) . \end{matrix}$

The value $Que (A)$ (resp. $Inf (A)$ , $Hyb (A)$ and $Tau (A)$ ) can be regarded as the degree that $A$ is questioning (resp. informative, hybrid, tautological).

If M = 2, then the notions defined in Definitions 3.6 and 3.8 are coincide with these defined in Definition 10 and 11 in [12]. Specifically, we have the following conclusions.

(1) $Ass (A) = ⊤$ iff $I (A) \leq J (A)$ . By (3) of Corollary 3.4, $Inf (A) = I_{⊤} (A) \subseteq J_{⊤} (A)$ which shows $Inf (A) ⊨ A$ . Thus $A$ is an assertion with respect to (1) of Definition 3.6 iff $Ass (A) = ⊤$ iff $A$ is an assertion in Roelofsen’s sense. Likewise, $Inq (A) = ⊤$ iff $Inf (A) ⊭ A$ . That is, $A$ is inquisitive with respect to (2) of Definition 3.6 iff $Inq (A) = ⊤$ iff $A$ is inquisitive in Roelofsen’s sense.

(2) $Que (A) = ⊤$ iff $I (A, W_{P}) = ⊤$ . By (3) of Corollary 3.4, $Inf (A) = I_{⊤} (A) = W_{P}$ . Thus $A$ is a question with respect to (1) of Definition 3.8 iff $Que (A) = ⊤$ iff $A$ is a question in Roelofsen’s sense. Likewise, $Inf (A) = ⊤$ iff $Inf (A) = I_{⊤} (A) \neq W_{P}$ . That is, $A$ is informative with respect to (2) of Definition 3.8 iff $Inf (A) = ⊤$ iff $A$ is inquisitive in Roelofsen’s sense.

(3) By (1) and (2) described as above, we conclude that $A$ is a hybrid (resp. tautology) with respect to (3) (resp. (4)) of Definition 3.8 iff $Hyb (A) = ⊤$ (resp. $Tau (A) = ⊤$ ) iff $A$ is a hybrid (resp. tautology) in Roelofsen’s sense.

Remark 3.9. For an $A \in L_{P}$ , $Ass (A)$ , $Inq (A)$ , $Que (A)$ , $Inf (A)$ , $Hyb (A)$ and $Tau (A)$ considerably depend on M. Let’s calculate degrees for the inquisitive sentence (2a) in the introduction section.

Suppose the two possibilities (2b) and (2c) are represented as p and q respectively. Thus (2a) is represented by p ∨ q. Suppose there are four participants in the conversation. Then $W_{P} = {w_{1}, w_{2}, w_{3}, w_{4}}$ . We can apply this situation to the following two examples.

Example 1. Let M = {⊥ , a, b, ⊤} be a diamond lattice with a′ = b and ⊥′ =⊤. Let $W_{P} = {w_{1}, w_{2}, w_{3}, w_{4}}$ and $p, q \in L_{P}$ . We define the M-fuzzifying support mapping by the following table.

S	{w₁}	{w₂}	{w₃}	{w₄}
p	⊤	a	b	⊥
q	⊥	b	a	⊤

Let $A = p \lor q$ . After calculating, we have $\begin{matrix} Ass (A) = Inf (A) = ⊥; Inq (A) = Que (A) = ⊤ . \end{matrix}$

Hence $Tau (A) = Hyb (A) = ⊥$ .

Example 2. Let M = [0, 1] be the unite lattice with a′ = 1 - a for all a ∈ M. Let

W_{P} = {w_{1}, w_{2}, w_{3}, w_{4}}

and

p, q \in L_{P}

. We define the M-fuzzifying support mapping by the following table.

S	{w₁}	{w₂}	{w₃}	{w₄}
p	5/6	3/4	3/5	1/2
q	1/6	1/4	2/5	1/2

After calculating, we have

\begin{matrix} Ass (A) = Tau (A) = \frac{1}{4}; Inq (A) = \frac{3}{4}; \\ Que (A) = Inf (A) = Hyb (A) = \frac{1}{2} . \end{matrix}

Theorem 3.10. $Tau (A) = J (A, W_{P})$ .

Proof. By Theorem 3.3, (2) of Theorem 3.7 and Corollary 3.4, we have $Tau (A) \geq J (A, W_{P})$ . Conversely, $\begin{matrix} Tau (A) \leq [I (A, W_{P}) \to_{M} J (A, W_{P})] \land I (A, W_{P}) . \end{matrix}$

Let a ∈ β (⊤ _M) with $\begin{matrix} a ≺ [I (A, W_{P}) \to_{M} J (A, W_{P})] \land I (A, W_{P}) . \end{matrix}$ Then $a ≺ [I (A, W_{P}) \to_{M} J (A, W_{P})]$ and $a \leq I (A, W_{P})$ . Thus there is b ∈ M such that a ≤ b and $I (A, W_{P}) \land b \leq J (A, W_{P})$ . Hence $a \leq J (A, W_{P})$ which shows $Tau (A) \leq J (A, W_{P})$ . □

In particular, if M = 2, then Theorem 3.10 shows that $A$ is a tautology iff $W_{P}$ supports $A$ . This result is exact Fact 9 in [12].

4 M-fuzzifying assertive (resp. questioning) projection operators

In InqB, propositions can be divided into two directions: one is inhabited by questions, which are always non-informative; the other is inhabited by assertions, which are always non-inquisitive. The two directions form a two-dimensional space: the intersection of them is inhabited by tautologies, which are neither informative nor inquisitive; the space surrounded by them is inhabited by hybrids, which are both informative and inquisitive. Every proposition has a unique assertive projection and a unique questioning projection. Moreover, each proposition is the disjunction of its assertive projection and questioning projection [9, 12].

Now, we extend these projections into M-InqB.

Definition 4.1. Let $A, B \in L_{P}$ . $B$ is called an M-fuzzifying assertive projection (briefly M-AP) of $A$ if

(MAP1) $Ass (B) = ⊤$ ;

(MAP2) $J (B) = I (B) = I (A)$ .

Theorem 4.2. For any $A \in L_{P}$ , $A \in L_{P}$ is an M-AP of A iff $J (A) = I (A)$ .

Proof. If B is the M-AP of $A$ , then (MAP2) directly implies that $J (B) = I (A)$ . Conversely, let $J (B) = I (A)$ . We next show that $B$ satisfies (MAP1) and (MAP2).

(MAP2). By (2) of Corollary 3.4, for all $σ \subseteq W_{P}$ , $\begin{matrix} I (B, σ) = ⋀_{w \in σ} J (B, {w}) = I (A, σ) . \end{matrix}$

Thus $J (B) = I (A) = I (B)$ .

(MAP1). $Ass (B) = ⊤$ by (MAP2). □

Theorem 4.3. For each $A \in L_{P}$ , $\neg \neg A$ is the unique M-AP of $A$ with respect to M-equivalence.

Proof. It is clear that $J (\neg \neg A, \emptyset) = ⊤ = I (A, \emptyset)$ . If $σ \in 2^{W_{P}} ∖ {\emptyset}$ , then $J (\neg \neg A, σ) = S (\neg \neg A, σ) = I (A, σ)$ by (2) of Theorem 2.3 and (2) of Corollary 3.4. Thus $J (\neg \neg A) = I (\neg \neg A)$ . Therefore $\neg \neg A$ is an M-AP by Theorem 4.2.

To prove that $\neg \neg A$ is the unique M-AP of $A$ , let $B, E \in L_{P}$ be M-APs of $A$ . Then for all $σ \subseteq W_{P}$ , $S (B, σ) = I (A, σ) = S (E, σ)$ by (MAP2). Thus $B =_{M} E$ . □

By Theorem 4.3, we can define the M-AP operator in M-InqB as follows.

Definition 4.4. The operator $! : L_{P} \to L_{P}$ , assigning each $A \in W_{P}$ to its M-AP, is called the M-AP operator in M-InqB.

Theorem 4.5. For each $A \in L_{P}$ , the following conditions are equivalent.

(1) $A =! A$ .

(2) $Ass (A) = ⊤$ .

(3) $J (A) = I (A)$ .

(4) $J (A, σ \cup τ) = J (A, σ) \land J (A, τ)$ .

Proof (1) ⇒ (2). It directly follows from (MAP1).

(2) ⇒ (3). Let $Ass (A) = ⊤$ and $σ \subseteq W_{P}$ . Then $J (A) \leq I (A)$ by Theorem 3.3. Conversely, we have $a ≺ ⊤ = I (A, σ) \to_{M} J (A, σ)$ for each a ∈ β (⊤) with $a ≺ β (I (A, σ))$ . Thus $a = I (A, σ) \land a \leq J (A, σ)$ . Hence $I (A, σ) \leq J (A, σ)$ . Therefore $I (A) \leq J (A)$ by arbitrariness of $σ \subseteq W_{P}$ .

(3) ⇒ (4). Let $J (A) = I (A)$ and $σ, τ \subseteq W_{P}$ . Then $\begin{matrix} J (A, σ \cup τ) & = & I (A, σ \cup τ) = I (A, σ) \land I (A, τ) \\ = & J (A, σ) \land J (A, τ) . \end{matrix}$

(4) ⇒ (3). For any $σ \subseteq W_{P}$ , we have $\begin{matrix} I (A, σ) = ⋀_{w \in σ} J (A, {w}) = J (A, σ) \end{matrix}$ by (2) of Corollary 3.4. Thus $I (A) = J (A)$ .

(3) ⇒ (1). By Theorem 4.2, $A$ is the M-AP of $A$ . Thus $! A = A$ . □

In order to discuss M-fuzzifying questioning projection of a wff $A \in L_{P}$ , we define and study some subsets of $W_{P}$ as follows.

Definition 4.6. Let $A \in L_{P}$ and a ∈ J (M). Denote

(1) $U_{a} (A) = {w \in W_{P} : S (A \land \neg A, {w}) \geq a, or, S (A \lor \neg A, {w}) ≱ a}$ ;

(2) $Γ (A) = ⋃_{a \in J (M)} U_{a} (A)$ and $W_{P}^{A} = W_{P} ∖ Γ (A)$ ;

(3) $P_{a} (A) = {σ \subseteq W_{P}^{A} : S (A, σ) \geq a}$ ;

(4) $Q_{a} (A) = {σ \subseteq W_{P}^{A} : σ \cap ⋃ P_{a} (A) = \emptyset}$ ;

(5) $D_{a} (A) = P_{a} (A) \cup Q_{a} (A)$ .

Theorem 4.7. Let $A \in L_{P}$ and a ∈ J (M). Then

(1) $U_{a} (\neg A) = U_{a} (A) = U_{a} (A \lor \neg A)$ ;

(2) $W_{P}^{A \lor \neg A} = W_{P}^{\neg A} = W_{P}^{A}$ ;

(3) $P_{a} (\neg A) = Q_{a} (A)$ .

Proof. (1) We have $U_{a} (\neg A) = U_{a} (A)$ by (1) of Theorem 2.3. By (9) of Theorem 2.3, $\begin{matrix} S (\neg (A \lor \neg A), {w}) & = & S (\neg \neg A \land \neg A, {w}) \\ = & S (A \land \neg A, {w}) \end{matrix}$ for each $w \in W_{P}$ . Thus $\begin{matrix} S (A \lor \neg A, {w}) \land S (\neg (A \lor \neg A), {w}) \\ = & S (A \land \neg A, {w}) \end{matrix}$ and $\begin{matrix} S (A \lor \neg A, {w}) \lor S (\neg (A \lor \neg A), {w}) \\ = & S (A \lor \neg A, {w}) . \end{matrix}$

Hence $U_{a} (A) = U_{a} (A \lor \neg A)$ .

(2) The result directly follows from (1).

(3) Suppose that $σ \in P_{a} (\neg A) ∖ Q_{a} (A)$ .

Since $σ \notin Q_{a} (A)$ , there is $w \in σ \cap ⋃ P_{a} (A)$ . Thus $w \notin U_{a} (A)$ and $S (A, {w}) \geq a$ . Also, by $σ \in P_{a} (\neg A)$ , we have $\begin{matrix} S (\neg A, {w}) \geq S (\neg A, σ) \geq a \end{matrix}$

Hence $S (A \land \neg A, {w}) \geq a$ which shows $w \in U_{a} (A)$ . It is a contradiction. So $P_{a} (\neg A) \subseteq Q_{a} (A)$ .

Suppose that $σ \in Q_{a} (A) ∖ P_{a} (\neg A)$ .

Since $σ \notin P_{a} (\neg A)$ , $S (\neg A, σ) = ⋀_{v \in σ} S (A, {v})^{'} ≱ a$ . Thus there is w ∈ σ such that $S (A, {v})^{'} ≱ a$ . Since $σ \in Q_{a} (A)$ , $σ \subseteq W_{P}^{A}$ and $σ \cap ⋃ P_{a} (A) = \emptyset$ . Hence $S (A, {v}) ≱ a$ . So $S (A \lor \neg A, {w}) ≱ a$ which shows $w \in U_{a} (A)$ . It is a contradiction. Therefore $Q_{a} (A) \subseteq P_{a} (\neg A)$ . □

In InqB, given $A \in W_{P}$ , $A \lor \neg A$ is the questioning projection of $A$ , which is always a question. However, $Que (A \lor \neg A)$ may not to be ⊤ in M-InqB. That is, $A \lor \neg A$ may not be an absolute question in M-InqB. So we use: ${Que}_{W_{P}^{A}} (A) = I (A, W_{P}^{A})$ for all $A \in L_{P}$ .

Definition 4.8. Let $A, B \in L_{P}$ . $B$ is called a M-fuzzifying questioning projection (briefly, M-QP) of $A$ if

(MQP1) ${Que}_{W_{P}^{B}} (B) = ⊤_{M}$ ;

(MQP2) $U_{a} (B) = U_{a} (A)$ and $D_{a} (B) = D_{a} (A)$ for all a ∈ J (M);

(MQP3) $S (B, σ) = S (A \lor \neg A, σ)$ for all $σ \in 2^{W_{P}}$ with $σ \cap Γ (A) \neq \emptyset$ .

The following two theorems show that there is an unique M-QP of $A \in L_{P}$ .

Theorem 4.9. For each $A \in L_{P}$ , $A \lor \neg A$ is an M-QP.

Proof. (MQP1). Since $S (A \lor \neg A, {w}) \geq a$ for each $w \in W_{P}^{A}$ and for all a ∈ J (M), we have $S (A \lor \neg A, {w}) = ⊤$ for each $w \in W_{P}^{A}$ . In addition, it follows from (1) and (2) of Theorem 4.7 that $Γ (A \lor \neg A) = Γ (A)$ and $W_{P}^{A \lor \neg A} = W_{P}^{A}$ . Thus $\begin{matrix} {Que}_{W_{P}^{A \lor \neg A}} (A \lor \neg A) = I (A \lor \neg A, W_{P}^{A}) = ⊤ . \end{matrix}$

(MQP2). $U_{a} (A \lor \neg A) = U_{a} (A)$ follows from (1) of Theorem 4.7. To prove $D_{a} (A \lor \neg A) = D_{a} (A)$ , we firstly check that $Q_{a} (A \lor \neg A) = {\emptyset}$ .

Suppose that $w \in σ \in Q_{a} (A \lor \neg A)$ . Since $\begin{matrix} {Que}_{W_{P}^{A \lor \neg A}} = I (A \lor \neg A, W_{P}^{A}) = ⊤, \end{matrix}$ we have $\begin{matrix} W_{P}^{A} \subseteq I_{a} (A \lor \neg A) = ⋃ P_{a} (A \lor \neg A) . \end{matrix}$

Further, since $σ \in Q_{a} (A \lor \neg A)$ , we have $\begin{matrix} w \in σ = σ \cap W_{P}^{A} \subseteq σ \cap ⋃ P_{a} (A \lor \neg A) = \emptyset . \end{matrix}$

It is a contradiction. Therefore $Q_{a} (A \lor \neg A) = {\emptyset}$ .

Now, we prove $D_{a} (A \lor \neg A) = D_{a} (A)$ .

Note that $D_{a} (A \lor \neg A) = P_{a} (A \lor \neg A)$ . If $σ \in P_{a} (A \lor \neg A)$ , then $S (A \lor \neg A, σ) \geq a$ . Thus $S (A, σ) \geq a$ or $S (\neg A, σ) \geq a$ . Hence, if $S (A, σ) \geq a$ , then $σ \in P_{a} (A) \subseteq D_{a} (A)$ ; if $S (\neg A, σ) \geq a$ , then $\begin{matrix} σ \in P_{a} (\neg A) \subseteq Q_{a} (A) \subseteq D_{a} (A) \end{matrix}$ by (3) of Theorem 4.7. Thus $D_{a} (A \lor \neg A) \subseteq D_{a} (A)$ .

Conversely, let $σ \in D_{a} (A)$ . If $σ \in P_{a} (A)$ , then $\begin{matrix} σ \in P_{a} (A) \subseteq P_{a} (A \lor \neg A) \subseteq D_{a} (A \lor \neg A) . \end{matrix}$

If $σ \in Q_{a} (A)$ , then $\begin{matrix} σ \in Q_{a} (A) & = & P_{a} (\neg A) \subseteq P_{a} (A \lor \neg A) \\ \subseteq & D_{a} (A \lor \neg A) . \end{matrix}$

Hence $D_{a} (A) \subseteq D_{a} (A \lor \neg A)$ . So (MQP2) holds.

(MQP3). The result is direct.

Therefore $A \lor \neg A$ is the M-QP of $A$ . □

Theorem 4.10. For each $A \in L_{P}$ , all M-QPs of $A$ are M-equivalent.

Proof. Assume that $B, E$ are M-QPs of $A$ . To prove that $B =_{M} E$ , we have to prove $S (B, σ) = S (E, σ)$ for all $σ \subseteq W_{P}$ . It is sufficient to that $P_{a} (B) = P_{a} (E)$ for all a ∈ J (M).

Since $I (B, W_{P}^{B}) = {Que}_{W_{P}^{B}} (B) = ⊤$ by (MQP1), we have $W_{P}^{B} \subseteq I_{a} (B) = ⋃ P_{a} (B)$ by (3) of Corollary 3.4. Next, we prove that $Q_{a} (B) = {\emptyset}$ .

Suppose that $w \in η \in Q_{a} (B)$ . Then $η \subseteq W_{P}^{A}$ and $\begin{matrix} \emptyset = η \cap ⋃ P_{a} (B) \subseteq η \cap W_{P}^{B} . \end{matrix}$

However, $w \in η \cap W_{P}^{B} = \emptyset$ which is a contradiction. Hence $Q_{a} (B) = {\emptyset}$ .

Similarly, $Q_{a} (E) = {\emptyset}$ . Thus $P_{a} (B) = D_{a} (B) = D_{a} (E) = P_{a} (E)$ by (MQP2). Hence $B =_{M} E$ . □

By Theorem 4.9 and 4.10, we can define the M-QP operator in M-InqB as follows.

Definition 4.11. The operator $? : L_{P} \to L_{P}$ , assigning each $A \in L_{P}$ to its M-QP, is called the M-QP operator in M-InqB.

Theorem 4.12. For a wff $A \in L_{P}$ , $? A =_{M} A$ iff ${Que}_{W_{P}^{A}} (A) = ⊤$ and $S (A, σ) \geq S (\neg A, σ)$ for all $σ \in 2^{W_{P}}$ with $σ \cap Γ (A) \neq \emptyset$ .

Proof. The sufficiency is clear.

Necessity. ${Que}_{W_{P}^{A}} (A) = {Que}_{W_{P}^{A}} (? A) = ⊤$ by (MQP1). For $σ \in 2^{W_{P}}$ with $σ \cap Γ (A) \neq \emptyset$ , $S (A, σ) = S (A \lor \neg A, σ) \geq S (\neg A, σ)$ by (MQP3). □

Corollary 4.13. If M is a Boolean algebra, then $! A =_{M} ? A =_{M} A$ iff $Tau (A) = ⊤$ .

Theorem 4.14. For $A \in L_{P}$ , we have $A =_{M}! A \land ? A$ iff $S (\neg \neg A \land \neg A) \leq S (A)$ .

Proof. Sufficiency. Let $σ \in 2^{W_{P}} ∖ {\emptyset}$ . By (6) of Theorem 2.3, we have $\begin{matrix} S (! A \land ? A, σ) \\ = & S (\neg \neg A, σ) \land [S (A, σ) \lor S (\neg A, σ)] \\ = & [S (\neg \neg A, σ) \land S (A, σ)] \lor \\ [S (\neg \neg A, σ) \land S (\neg A, σ)] \\ = & S (A, σ) \lor S (\neg \neg A \land \neg A, σ) = S (A, σ) . \end{matrix}$

Therefore $A =_{M}! A \land ? A$ .

Necessity. If $A =_{M}! A \land ? A$ , then $S (\neg \neg A \land \neg A) \leq S (! A \land ? A) = S (A)$ . □

Corollary 4.15. If M is a Boolean algebra, then $A =_{M}! A \land ? A$ for all $A \in L_{P}$ .

Remark 4.16. (1) Corollary 4.15 and Theorem 4.14 are M-fuzzifying extensions of Fact 14 in [12].

(2) It follows from (MS1) and (8) of Theorem 2.3 that p = _M ! p ∧ ? p and $\neg A =_{M}! (\neg A) \land ? (\neg A)$ for $p, A \in L_{P}$ .

(3) Generally, p ∨ q ≠ _M ! (p ∨ q) ∧ ? (p ∨ q) for $p, q \in L_{P}$ . For example, let $W_{P} = {w_{1}, w_{2}}$ , $p, q \in L_{P}$ and M = [0, 1]. We can define the M-fuzzifying support mapping as follows.

S	{w₁}	{w₂}
p	1/4	1/6
¬p	3/4	5/6
q	1/6	1/5
¬q	5/6	4/5

Let $A = p \lor q$ . Then $S (A, W_{P}) = \frac{1}{6}$ , $S (\neg A, W_{P}) = \frac{3}{4}$ and $S (\neg \neg A, W_{P}) = \frac{1}{5}$ . Thus $S (! A, W_{P}) = \frac{1}{5}$ and $S (? A, W_{P}) = \frac{3}{4}$ . Hence $S (! A \land ? A, W_{P}) \neq S (A, W_{P})$ . Therefore $A \neq_{M}! A \land ? A$ .

Theorem 4.17. Let $A, B, p \in L_{P}$ . Then

(1) $W_{P}^{? A} = W_{P}^{A} = W_{P}^{! A} = W_{P}^{! ? A} = W_{P}^{?! A}$ ;

(2) ! p = _Mp, ? ¬ p = _M ? p and $! \neg A =_{M} \neg A$ ;

(3) $A ⊨_{M}! A$ and $A ⊨_{M} ? A$ ;

(4) if $A ⊨_{M} B$ , then $! A ⊨_{M}! B$ ;

(5) $!! A ⊨_{M}! A$ and $? ? A ⊨_{M} ? A$ ;

(6) $! ? A ⊨_{M} ?! A ?! ? A ⊨_{M}! ?! A$ ;

(7) $A \to! B =_{M}! (A \to B)$ ;

(8) $A \to B =_{M} (A \to! B) \land (A \to ? B)$ iff $S (\neg \neg B \land \neg B) \leq S (B)$ ;

(9) $! A \to (B \lor C) =_{M} (! A \to B) \lor (! A \to C)$ ;

(10) $(! A \to B) \lor (? A \to B) ⊨_{M} A \to B$ .

Proof. (1) It follows from (1) and (2) of Theorem 4.7.

(2) By (6) and (9) of Theorem 2.3, ! p = _Mp and $! \neg A =_{M} \neg A$ . Thus $\begin{matrix} ? \neg p =_{M} \neg p \lor! p =_{M} \neg p \lor p =_{M} ? p . \end{matrix}$

(3) It follows from (6) of Theorem 2.3 and Definition 2.1.

(4) By (2) of Theorem 2.3 and Theorem 3.5, $\begin{matrix} S (! A, σ) = I (A, σ) \leq I (B, σ) = S (! B, σ) \end{matrix}$ for all $σ \subseteq W_{P}$ . Thus $! A ⊨_{M}! B$ .

(5) $!! A =_{M}! A$ by (7) of Theorem 2.3. Also, by (9) of Theorem 2.3, $\begin{matrix} ? ? A & =_{M} & ? (A \lor \neg A) =_{M} (A \lor \neg A) \lor \neg (A \lor \neg A) \\ =_{M} & (A \lor \neg A) \lor (\neg A \land \neg \neg A) =_{M} ? A . \end{matrix}$

(6) To prove $! ? A =_{M} ?! A$ , let $σ \subseteq W_{P}$ . By (2), (6) and (7) of Theorem 2.3, $\begin{matrix} S (! ? A, σ) & = & ⋀_{w \in σ} S (? A, {w}) = I (? A, σ) \\ = & S (! A \lor \neg A, σ) = S (! A \lor \neg! A, σ) \\ = & S (?! A, σ) . \end{matrix}$

Thus $! ? A =_{M} ?! A$ . Therefore $?! ? A =_{M}! ? ? A =_{M}! ? A$ and $! ?! A =_{M} ?!! A =_{M} ?! A$ .

(7) For each $σ \in 2^{W_{P}} ∖ {\emptyset}$ , we have $\begin{matrix} S (A \to! B, σ) & \leq & ⋀_{w \in σ} [S (A, {w})^{'} \lor S (! B, {w})] \\ = & ⋀_{w \in σ} [S (A, {w})^{'} \lor S (B, {w})] \\ = & ⋀_{w \in σ} S (A \to B, {w}) \\ = & S (! (A \to B), σ) . \end{matrix}$

Conversely, we have $\begin{matrix} S (A \to! B, σ) & = & ⋀_{τ \subseteq σ} ⋀_{w \in τ} [S (A, τ)^{'} \lor S (B, {w})] \\ \geq & ⋀_{w \in σ} [S (A, {w})^{'} \lor S (B, {w})] \\ = & S (! (A \to B), σ) . \end{matrix}$

Therefore $A \to! B =_{M}! (A \to B)$ .

(8) It follows from (3) of Theorem 2.3 and Theorem 4.14.

(9) The result follows from (11) of Theorem 2.3.

(10) Notice that $A ⊨_{M}! A \land ? A$ . For each $σ \subseteq W_{P}$ , $\begin{matrix} S (A \to B, σ) & \geq & S (! A \land ? A \to B) \\ = & ⋀_{τ \subseteq σ} [S (! A \land ? A, τ)^{'} \lor S (B, τ)] \\ = & ⋀_{τ \subseteq σ} {[S (! A, τ)^{'} \lor S (B, τ)] \lor \\ [S (? A, τ)^{'} \lor S (B, τ)]} \\ \geq & S ((! A \to B) \lor (? A \to B), σ) \end{matrix}$

Therefore $(! A \to B) \lor (? A \to B) ⊨_{M} A \to B$ . □

Remark 4.18. (7) and (8) of Theorem 4.17 are M-fuzzifying extensions of Fact 11 and Fact 14 in [7]. Moreover, (9) of Theorem 4.17 is an M-fuzzifying extension of Mascarenhas Equivalence in [9] (Page 22).

5 Conclusions

(i) The framework of InqB established by Groenendijk, Roelofsen and Mascarenhas et al. can be regarded as a general inquisitive semantic theory which doesn’t concern fuzziness. However, since fuzziness is a universal phenomenon in natural languages, it is quite necessary to take it into account in language studies. By assigning degrees to attributes of InqB, we established the framework of M-InqB, which is not only consistent with the approach of InqB in dealing crisp phenomena in natural languages, but also more adaptable to explain fuzzy phenomena.

For example, given a proposition $p \in L_{P}$ , both of the formulas p∧ ¬ p = ⊥ and p∨ ¬ p = ⊤ in InqB are absolutely true. However, neither of them is necessarily true in M-InqB. This is just the way to explain the following fuzzy phenomenon.

Suppose there is a man at the age of 38. Tom thinks the man is very young, but Jack doesn’t think so. Then there are two possible worlds, say w₁ and w₂, representing the actual worlds of Tom and Jack. The four propositions can be expressed as follows:

p: The man is young;

¬p: The man is not young.

p ∧ ¬ p: The man is young and the man is not young.

p ∨ ¬ p: Is the man young or not ?

In InqB, p ∧ ¬ p is a contradiction. We can see this interpretation is unacceptable in InqB since the logical values in InqB are 0 and 1. However, in the actual world, it is indeed reasonable to regard a man of age 38 as both young and not young. So, in M-InqB where M = [0, 1], we can practically define the M-fuzzifying support mapping as follows.

S	{w₁}	{w₂}
p	3/4	2/5
¬p	1/4	3/5

In this case, we have $S (p \land \neg p, W_{P}) = \frac{1}{4}$ and p∧ ¬ p ≠ _M ⊥. So p ∧ ¬ p can be reasonably interpreted by: The man is young but not very young.

Likewise, in InqB, p ∨ ¬ p is a polar question (i.e., the informative content of p ∨ ¬ p is $W_{P}$ ). However, in M-InqB, we have $Inf (p \lor \neg p) = \frac{2}{3}$ since $S (p \lor \neg p, W_{P}) = \frac{2}{5}$ . That is, the degree that $W_{P}$ supports p ∨ ¬ p is $\frac{3}{5}$ (other than 1). Although p ∨ ¬ p in M-InqB is expressed by the same sentence (Is the man young or not ?) in InqB, it has completely different replies. For instance, ‘The man is neither very young nor very old’ is reasonable in M-InqB, while it is unacceptable in InqB.

(ii) The logical structure of M-InqB is far more complex than that of InqB (e.g., Theorem 3.7, 4.12 and 4.14, and Corollary 3.4 and 4.13). But its logical relations among propositions are direct and apparent (e.g., Theorem 2.3, 4.2, 4.3, 4.5, 4.14 and 4.17).

(iii) M-InqB is exactly InqB provided that M is reduced to 2. Hence InqB can be regarded as a special case of M-InqB.

(iv) This paper mainly focuses on establishing the basic framework of M-InqB in a theoretic viewpoint. But there is still a lot of work needed to be done, such as enriching M-InqB by characterizing M-fuzzifying forms of possibilities, quantifiers, homogeneities and compliances among propositions, studying M-fuzzifying inquisitive pragmatics and so on.

Footnotes

Acknowledgments

We are very grateful to the editor for handling our manuscript. And we sincerely thank the anonymous reviewers for their comments and suggestions.

This work is supported by the Provincial Social Science Foundation in Hunan China (13YBA402), the Key Project of Hunan Educational Committee (18A474) and the Provincial Youth Science Foundation in Hunan China (2018JJ3192).

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