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.

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Notes

  1. 1.

    At this moment, it would be reasonable to admit also the empty state, but that would lead to some technical complications in the following section and so we avoid these complications by excluding the empty set from the class of states.

  2. 2.

    ‘Classically true’ means true according to classical logic.

  3. 3.

    The fact that the basic framework of inquisitive semantics is not completely suitable for modeling phenomena related to conditionals and modals seems to motivate an extension of inquisitive semantics called suppositional inquisitive semantics (Groenendijk and Roelofsen 2015). The aim of this paper is also to propose an extension of inquisitive semantics as a reaction to this problem. However, it is evident that our extension differs fundamentally from suppositional inquisitive semantics.

  4. 4.

    As one of the anonymous reviewers rightly pointed out, Grice’s example could also be interpreted in the sense of (b). So the example is probably not the best illustration of (c). However, it is evident that the reading (c) is also possible here, and we will concentrate on this reading simply because it seems to be more straightforward to extend inquisitive semantics in such a way that the negation of conditionals semantically corresponds to (c). It would be more complicated to adjust the framework so that we would be able to formalize the sentence in the sense of (b) without reducing it to (a).

    Grice also admitted (a) as another way of denying conditionals (see Grice 1991, p. 80). Paradoxically, his goal was to defend the material conditional analysis. However, Grice himself admitted that his defense did not lead to a satisfactory explanation of the denial of conditionals (see Grice 1991, p. 83).

  5. 5.

    In the original inquisitive semantics without weak negation, it is natural to admit also the empty state. If the first semantic condition (\(s \nvDash \bot \)) is replaced with the condition \(s \vDash \bot \) iff \(s=\emptyset \), we naturally get that every formula is supported by the empty state. As a result, the presence of the empty state does not have any direct impact on the logic determined by the semantics. The situation changes when weak negation is added to the framework. If the empty state was admitted, it would not naturally support every formula and that would complicate the picture in an undesirable way. Of course, it would be possible to introduce a slightly more complex semantic condition for weak negation: \(s \vDash \mathord {\sim }\varphi \) iff \(s\) is empty or \(s \nvDash \varphi \). However, since the empty state would bring no benefit to us, there seems to be no reason for such a complication and we instead decided to exclude the empty state from the class of states.

  6. 6.

    It is possible to formulate this claim in a stronger way: If \(\left\| \varphi \right\| \) is closed under binary unions, then \(\square \varphi \equiv \blacksquare \varphi \). It holds that if \(\left\| \varphi \right\| \) is closed under binary unions, it is also closed under arbitrary unions. However, we omit the proof of this fact.

  7. 7.

    \(P\) is inquisitive iff \(\bigcup P \notin P\).

  8. 8.

    A formalization of the idea that ‘might’-sentences draw attention to certain possibilities is proposed in Ciardelli et al. (2011).

  9. 9.

    In this and the next section, \(\varphi \), \(\psi \), and \(\chi \) (possibly indexed) will range over the formulas of the language \(L^{\mathord {\sim }}\), unless stated otherwise.

  10. 10.

    In the rest of this paper, ‘hyp’ used in the derivations will stand for ‘hypothetical assumption’.

  11. 11.

    The order of its disjuncts is not important.

  12. 12.

    For different reasons and purposes, an equivalent and almost identical system was independently introduced and investigated in Punčochář (2009, (2012).

  13. 13.

    First order inquisitive semantics was discussed, e.g., in Ciardelli (2010). Sano (2011) is a preliminary attempt to axiomatize this logic. The paper contains an axiomatization of first order inquisitive semantics restricted to states containing no more than two possible worlds.

  14. 14.

    The notation is a standard one. \(t_{1},\ldots ,t_{n}\) are terms and \(ve(t_{n})\) is the value of the term in the structure \(v\) relative to the evaluation \(e\). \(e_{x:a}\) is the variant of \(e\) sending \(x\) to \(a\).

  15. 15.

    We use the letter nabla since these operators can be understood as quantifier versions of the nabla operator and its dual used in coalgebraic logic.

  16. 16.

    The proof of the statement (i) is a ‘meta’-proof. It shows that \(\mathord {\sim }p\) is derivable from \(\lozenge \varphi \) on the assumption that \(\lnot p\) is derivable from \(\varphi \). Thus the meta-proof shows that in every concrete case, that is in every case in which a concrete formula \(\varphi \) and an atom \(p\) are given such that there is a derivation of \(\lnot p\) from \(\varphi \), there is a concrete derivation of \(\mathord {\sim }p\) from \(\lozenge \varphi \). The derivation can be obtained by substituting the derivation of \(\lnot p\) from \(\varphi \) for the step from 5 to 6 in the meta-proof. It is supposed that \(\lnot p\) can be derived from \(\varphi \) alone, without any other assumptions. That guarantees that this derivation does not require any sources from the outer proof even if it is integrated as a subproof of a conditional proof as in the case of the meta-proof of the statement (i). It follows that by such step the use of the rule \((\rightarrow I)^{*}\) cannot be made illegitimate.

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Acknowledgments

The work on this paper was supported by grant no. 13-21076S of the Czech Science Foundation.

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Correspondence to Vít Punčochář.

Appendices

Appendix 1

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

figureg

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

figureh

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

figurei

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

figurej

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

figurek

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

figurel

(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.

figurem

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 \(\alpha , \varphi _{1},\ldots ,\varphi _{n}, \varphi , \psi \):

  1. (i)

    \(\square \alpha \vdash \alpha \),

  2. (ii)

    \(\square \lnot \varphi \vdash \lnot \varphi \),

  3. (iii)

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

  4. (iv)

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

  5. (v)

    \(\varphi , \lozenge \lnot \varphi \vdash \bot \),

  6. (vi)

    \(\square (\varphi \wedge \psi ) \vdash \square \varphi \),

  7. (vii)

    \(\lozenge (\varphi \wedge \psi ) \vdash \lozenge \varphi \),

  8. (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:

figuren

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

figureo
figurep

(ii) The following derivation can be constructed:

figureq

(iii) The following derivation can be constructed:

figurer

(iv) The following derivation can be constructed:

figures

(v) The following derivation can be constructed:

figuret

(vi) The following derivation can be constructed:

figureu

(vii) The following derivation can be constructed:

figurev

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

figurew

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:

figurex

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

  1. (a)

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

  2. (b)

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

Then the following formulas are provably equivalent:

  1. 1.

    \(\psi \rightarrow \chi \),

  2. 2.

    \((\beta _{1} \vee \cdots \vee \beta _{k}) \rightarrow (\gamma _{1} \vee \cdots \vee \gamma _{l})\),

  3. 3.

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

  4. 4.

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

  5. 5.

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

  6. 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

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

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Punčochář, V. Weak Negation in Inquisitive Semantics. J of Log Lang and Inf 24, 323–355 (2015). https://doi.org/10.1007/s10849-015-9219-2

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Keywords

  • Inquisitive semantics
  • Negation
  • Possible worlds
  • Fitch-style natural deduction
  • Denial