Weak Negation in Inquisitive Semantics

Abstract

This paper introduces and explores a conservative extension of inquisitive logic. In particular, weak negation is added to the standard propositional language of inquisitive semantics, and it is shown that, although we lose some general semantic properties of the original framework, such an enrichment enables us to model some previously inexpressible speech acts such as weak denial and ‘might’-assertions. As a result, a new modal logic emerges. For this logic, a Fitch-style system of natural deduction is formulated. The main result of this paper is a theorem establishing the completeness of the system with respect to inquisitive semantics with weak negation. At the conclusion of the paper, the possibility of extending the framework to the level of first order logic is briefly discussed.

Appendices

Appendix 1

We will prove all the statements of Lemma 1. (i) If $\varphi \vdash \lnot p$ then $\lozenge \varphi \vdash \mathord {\sim }p$.¹⁶

(ii) $\mathord {\sim }\blacklozenge \varphi \vdash \lnot \lozenge \varphi$.

(iii) $\square \varphi \vdash \lozenge \varphi$.

(iv) $\vdash \lozenge \varphi \vee \lnot \lozenge \varphi$.

(v) $\lozenge (\varphi \vee \psi ) \vdash \lozenge \varphi \vee \lozenge \psi$.

(vi) $\square (\varphi \vee \psi ), \lnot \lozenge \psi \vdash \square \varphi$.

(vii) $\varphi _{1} \oplus \cdots \oplus \varphi _{n}, \varphi _{1} \rightarrow p, \ldots , \varphi _{n} \rightarrow p \vdash p$. Notice that here is the only place in the completeness proof where the rule (R1) is used.

Appendix 2

We will provide a proof that Kreisel–Putnam schema (KP) restricted to the language $L$ is derivable in our system. This result can be easily obtained by semantic considerations using Theorem 1. Here, we will give a purely syntactic proof of this fact.

In the following text, $\varphi , \psi , \chi$ (possibly indexed) will range over all formulas from $L$ and $\alpha , \beta , \gamma$ (possibly indexed) only over disjunction-free formulas from $L$.

Lemma 6

For any $\alpha , \varphi _{1},\ldots ,\varphi _{n}, \varphi , \psi$:

• (i) $\square \alpha \vdash \alpha$,

• (ii) $\square \lnot \varphi \vdash \lnot \varphi$,

• (iii) $\mathord {\sim }(\varphi _{1} \vee \cdots \vee \varphi _{i} \vee \cdots \vee \varphi _{n}) \vdash \mathord {\sim }\varphi _{i}$,

• (iv) $\mathord {\sim }(\lnot \varphi \rightarrow \alpha ) \vdash \lozenge (\lnot \varphi \wedge \lnot \alpha )$,

• (v) $\varphi , \lozenge \lnot \varphi \vdash \bot$,

• (vi) $\square (\varphi \wedge \psi ) \vdash \square \varphi$,

• (vii) $\lozenge (\varphi \wedge \psi ) \vdash \lozenge \varphi$,

• (viii) if $\varphi \vdash \psi$ then $\blacklozenge \varphi \vdash \blacklozenge \psi$.

Proof

(i) Induction on the complexity of $\alpha$. For atomic formulas, the claim holds due to the rule (R1). For $\bot$ there is the following derivation:

Suppose that the claim holds for some $\beta$ and $\gamma$. We will prove that then it holds also for $\beta \wedge \gamma$ and $\beta \rightarrow \gamma$.

(ii) The following derivation can be constructed:

(iii) The following derivation can be constructed:

(iv) The following derivation can be constructed:

(v) The following derivation can be constructed:

(vi) The following derivation can be constructed:

(vii) The following derivation can be constructed:

(viii) Suppose that $\varphi \vdash \psi$.

The following lemma shows that a restricted version of (KP) schema is derivable in the natural deduction system. Let us remind that $\alpha _{1},\ldots ,\alpha _{n}$ range over disjunction-free formulas from $L$.

Lemma 7

$\vdash (\lnot \varphi \rightarrow (\alpha _{1} \vee \cdots \vee \alpha _{n})) \rightarrow ((\lnot \varphi \rightarrow \alpha _{1}) \vee \cdots \vee (\lnot \varphi \rightarrow \alpha _{n}))$.

Proof

The following derivation can be constructed:

The following lemma shows that every formula from $L$ is provably equivalent with some disjunction of disjunction-free formulas from $L$. An analogous result (concerning, however, different calculi) was exploited also in Punčochář (2009, (2012, (2014).

Lemma 8

For any $\varphi$ there are some $\alpha _{1}, \ldots , \alpha _{n}$ such that $\varphi \dashv$ $\vdash \alpha _{1} \vee \cdots \vee \alpha _{n}$.

Proof

This can be proved by induction on the complexity of the formula $\varphi$. We will show only the inductive step for implication which is most difficult and in which Lemma 7 is used. Suppose that

• (a) $\psi \dashv$ $\vdash \beta _{1} \vee \cdots \vee \beta _{k}$ ($k \ge 1$),

• (b) $\chi \dashv$ $\vdash \gamma _{1} \vee \cdots \vee \gamma _{l}$ ($l \ge 1$).

Then the following formulas are provably equivalent:

• 1. $\psi \rightarrow \chi$,

• 2. $(\beta _{1} \vee \cdots \vee \beta _{k}) \rightarrow (\gamma _{1} \vee \cdots \vee \gamma _{l})$,

• 3. $\bigwedge ^{k}_{i=1}(\beta _{i} \rightarrow (\gamma _{1} \vee \cdots \vee \gamma _{l}))$,

• 4. $\bigwedge ^{k}_{i=1}(\lnot \lnot \beta _{i} \rightarrow (\gamma _{1} \vee \cdots \vee \gamma _{l}))$,

• 5. $\bigwedge ^{k}_{i=1}((\lnot \lnot \beta _{i} \rightarrow \gamma _{1}) \vee \cdots \vee (\lnot \lnot \beta _{i} \rightarrow \gamma _{l}))$,

• 6. $\bigvee ^{l}_{i_{1}=1} \ldots \bigvee ^{l}_{i_{k}=1}((\lnot \lnot \beta _{1} \rightarrow \gamma _{i_{1}}) \wedge \ldots \wedge (\lnot \lnot \beta _{k} \rightarrow \gamma _{i_{k}}))$.

The equivalence of 1. and 2. follows from the induction hypotheses (a) and (b). 3. can be easily derived from 2. with the help of the rules $(\wedge I)$, $(\vee I)$, $(\rightarrow$ $E)$, and $(\rightarrow$ $I)^{*}$. 2. can be derived from 3. using the rules $(\wedge E)$, $(\vee E)$, $(\rightarrow$ $E)$, and $(\rightarrow$ $I)^{*}$. The equivalence between 3. and 4. is a consequence of Lemma 6 (i). The derivation of 5. from 4. is an application of Lemma 7 and in the derivation from 5. to 4. it suffices to use the rules $(\vee E)$, $(\vee I)$, $(\rightarrow$ $E)$, and $(\rightarrow$ $I)^{*}$. The equivalence of 5. and 6. can be established with the help of the derivable distributive laws.

Theorem 2 is a consequence of Lemmas 7 and 8. Let us stress once again that in this theorem $\varphi ,\,\psi$ and $\chi$ range over formulas from $L$.

Theorem 2

$\vdash (\lnot \varphi \rightarrow (\psi \vee \chi )) \rightarrow ((\lnot \varphi \rightarrow \psi ) \vee (\lnot \varphi \rightarrow \chi ))$.