Abstract
This paper introduces the inquisitive extension of R, denoted as InqR, which is a relevant logic of questions based on the logic R as the background logic of declaratives. A semantics for InqR is developed, and it is shown that this semantics is, in a precisely defined sense, dual to Routley-Meyer semantics for R. Moreover, InqR is axiomatized and completeness of the axiomatic system is established. The philosophical interpretation of the duality between Routley-Meyer semantics and the semantics for InqR is also discussed.
Similar content being viewed by others
References
Anderson, A.R. (1959). Completeness theorems for the systems E of entailment and EQ of entailment with quantification. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 6, 201–216.
Anderson, A.R., & Belnap, N.D. Jr. (1961). Enthymemes. The Journal of Philosophy, 58, 713–723.
Anderson, A.R., & Belnap, N.D. Jr. (1961). The pure calculus of entailment. The Journal of Symbolic Logic, 27, 19–52.
Anderson, A.R., & Belnap, N.D. Jr. (1961). Tautological entailments. Philosophical studies, 13, 9–24.
Anderson, A.R., & Belnap, N.D. Jr. (1965). Entailment with negation. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 11, 277–289.
Anderson, A.R., & Belnap, N.D. Jr. (1975). Entailment: the logic of relevance and necessity Vol. 1. Princeton: Princeton University Press.
Anderson, A.R., Belnap, N.D. Jr., Dunn, J.M. (1992). Entailment: the logic of relevance and necessity Vol. 2. Princeton: Princeton University Press.
Barwise, J. (1993). Constraints, channels and the flow of information. In Aczel, P., Israel, D., Peters, S., Katagiri, Y. (Eds.) Situation theory and its applications, (Vol. 3 pp. 3–28): CSLI Publications.
Beall, J., Brady, R., Dunn, J.M., Hazen, A.P., Mares, E., Meyer, R.K., Priest, G., Restall, G., Ripley, D., Slaney, J., Sylvan, R. (2012). On the ternary relation and conditionality. Journal of Philosophical Logic, 41, 595–612.
Belnap, N.D. Jr. (1967). Intensional models for first degree formulas. The Journal of Symbolic Logic, 32, 1–22.
Bílkova, M., Majer, O., Peliš, M., Restall, G. (2010). Relevant agents. In Beklemishev, L., Goranko, V., Shehtman, V (Eds.) Advances in modal logic, (Vol. 8 pp. 22–38): College Publications.
Ciardelli, I., Groenendijk, J., Roelofsen, F. (2013). Inquisitive semantics: a new notion of meaning. Language and Linguistics Compass, 7, 459–476.
Ciardelli, I., & Roelofsen, F. (2011). Inquisitive logic. Journal of Philosophical Logic, 40, 55–94.
Ciardelli, I. (2016). Questions in logic. PhD Thesis. University of Amsterdam.
Ciardelli, I. (2016). Dependency as question entailment. In Abramsky, S., Kontinen, J., Väänänen, J., Vollmer, H. (Eds.) Dependence logic: theory and applications (pp. 129–181): Birkhäuser.
Ciardelli, I. (2018). Questions as information types. Synthese, 195, 321–365.
Ciardelli, I., Groenendijk, J., Roelofsen, F. (2019). Inquisitive semantics. Oxford University Press.
Ciardelli, I., Iemhoff, R., Yang, F. (forthcoming). Questions and dependency in intuitionistic logic. Notre Dame Journal of Formal Logic.
Church, A. (1951). The weak theory of implication. In Menne, A., Wilhelmy, A., Angsil, H. (Eds.) Kontroliertes Denken: Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften (pp. 22–37): Kommissions-Verlag Karl Alber.
Copeland, B.J. (1979). On when a semantics is not a semantics: some reasons for disliking the Routley-Meyer semantics for relevance logic. Journal of Philosophical Logic, 8, 399–413.
Došen, K. (1992). The first axiomatization of relevant logic. Journal of Philosophical Logic, 21, 339–356.
Dunn, J.M., & Restall, G. (2002). Relevance logic. In Handbook of philosophical logic, (Vol. 6 pp. 1–128): Kluwer.
Dunn, J.M. (1966). The algebra of intensional logics. PhD thesis: University of Pittsburgh.
Dunn, J.M. (1993). Star and perp: two treatments of negation. Philosophical Perspectives, 7, 331–357.
Dunn, J.M. (2001). Ternary relational semantics and beyond: programs as data and programs as instructions. Logical Studies, 7, 1–20.
Dunn, J.M. (2015). The relevance of relevance to relevance logic. In Banerjee, M., & Krishna, S.N. (Eds.) Logic and its applications, proceedings of the 6th Indian conference (ICLA) (pp. 11–29): Springer.
Edgington, D. (1986). Do conditionals have truth-conditions? Critica, 18, 3–30.
Fine, K. (1974). Models for entailment. Journal of Philosophical Logic, 3, 347–372.
Harrah, D. (2002). The logic of questions. In Gabbay, D.M., & Guenthner, F. (Eds.) Handbook of philosophical logic, (Vol. 8 pp. 1–60): Springer.
Lewis, C.I. (1912). Implication and the algebra of logic. Mind, 21, 522–531.
Mares, E.D. (2004). Relevant logic: a philosophical interpretation. Cambridge University Press.
Mares, E. (2014). Relevance logic. In Zalta, E.N. (Ed.) The Stanford encyclopedia of philosophy, Spring 2014, https://plato.stanford.edu/archives/spr2014/entries/logic-relevance/.
Orlov, I.F. (1928). The calculus of compatibility of propositions (in Russian). Matematicheskii Sbornik, 35, 263–286.
Peliš, M. (2016). Inferences with ignorance: logics of questions. Inferential Erotetic Logic & Erotetic Epistemic Logic. Karolinum.
Punčochář, V. (2016). A generalization of inquisitive semantics. Journal of Philosophical Logic, 45, 399–428.
Punčochář, V. (2017). Algebras of information states. Journal of Logic and Computation, 27, 1643–1675.
Punčochář, V. (2018). Substructural inquisitive logics. Review of Symbolic Logic, 12, 296–330.
Ramsey, F.P., & Ramsey, F.P. (1929). General propositions and causality. In Mellor, D.H. (Ed.) Philosophical papers (p. 1990). Cambridge: Cambridge University Press.
Restall, G. (1996). Information flow and relevant logics. In Seligman, J., & Westerståhl, D. (Eds.) Logic, language and computation (pp. 463–478): CSLI Publications.
Restall, G. (2000). An introduction to substructural logics. Routledge.
Routley, R., & Meyer, R.K. (1973). Semantics of entailment. In Leblanc, H. (Ed.) Truth, syntax and modality, proceedings of the temple university conference on alternative semantics (pp. 194–243). North Holland.
Shaw-Kwei, M. (1950). The deduction theorems and two new logical systems. Methodos, 2, 56–75.
Urquhart, A. (1972). The completeness of weak implication. Theoria, 37, 274–282.
Urquhart, A. (1972). Semantics for relevant logics. Journal of Symbolic Logic, 37, 159–169.
Urquhart, A. (1972). The semantics of entailment. PhD thesis: University of Pittsburgh.
van Benthem, J., & Minica, S. (2012). Toward a Dynamic Logic of Questions. Journal of Philosophical Logic, 41, 633–669.
Wiśniewski, A. (1995). The posing of questions: logical foundations of erotetic inferences. Kluwer.
Wiśniewski, A. (2013). Questions, inferences, and scenarios. College Publications.
Acknowledgments
The work on this paper was supported by grant no. 17-15645S of the Czech Science Foundation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
Proof of Proposition 5
(1) Assume that a is a completely prime state of an re-model. Due to I9 there is a set of basic states X such that \(a = \bigsqcup X\). Since a is completely prime, a ∈ X. Thus, a is basic.
(2) Now assume that a is basic, that is, there is b such that aCb and for every c, if bCc then \(a \sqsubseteq c\). Assume that \(a = \bigsqcup X\) for some set of states X. Since (due to I6) bCa, there is (due to I8) c ∈ X such that bCc. It follows that \(a \sqsubseteq c\), but since it also hods that \(c \sqsubseteq a\), we obtain a = c.
□
Proof of Proposition 6
First, assume that I7 holds. Then aCb iff aC(b ⋅ 1) iff (a ⋅ b)C1. So, I7∗ also holds. Second, assume that I7∗ holds. Then aC(b ⋅ c) iff (a ⋅ (b ⋅ c))C1 iff ((a ⋅ b) ⋅ c)C1 iff (a ⋅ b)Cc. So, in that case, I7 holds. □
Proof of Proposition 7
(1) Assume that \(\langle Info, \sqsubseteq , C, \cdot , 0, 1, V \rangle \) is an re-model and let \(X \subseteq Info\). Then it holds \(\bigsqcup X \in K^{C}\) iff \((\bigsqcup X) C1\) iff there is a ∈ X such that aC1 iff there is a ∈ X such that a ∈ KC. So, the condition I10 is satisfied for KC and thus \(\langle Info, \sqsubseteq , K^{C}, \cdot , 0, 1, V \rangle \) is an re∗-model. Moreover, \(aC^{K^{C}}b\) iff a ⋅ b ∈ KC iff (a ⋅ b)C1 iff aCb.
(2) Assume that \(\langle Info, \sqsubseteq , K, \cdot , 0, 1, V \rangle \) is an re∗-model. Then I6 and I7 are satisfied for CK due to commutativity and associativity of fusion. Moreover, I8 is satisfied for CK due to I10 for K. Therefore, \(\langle Info, \sqsubseteq , C^{K}, \cdot , 0, 1, V \rangle \) is an re-model. It also holds that \(a \in K^{C^{K}}\) iff aCK1 iff a ⋅ 1 ∈ K iff a ∈ K.
□
Proof of Proposition 8
For any re-model \(\mathcal {N}\) the following equivalences hold: ψ is a consequence of φ1,…,φn in \(\mathcal {N}\) iff for any a from \(\mathcal {N}\), if \(a \vDash \varphi _{1} \wedge {\ldots } \wedge \varphi _{n}\) then \(a \cdot 1 \vDash \psi \) (since a ⋅ 1 = a) iff \(1 \vDash (\varphi _{1} \wedge {\ldots } \wedge \varphi _{n}) \rightarrow \psi \) in \(\mathcal {N}\) iff \((\varphi _{1} \wedge {\ldots } \wedge \varphi _{n}) \rightarrow \psi \) is valid in \(\mathcal {N}\).□
Proof of Proposition 9
Straightforward induction on the complexity of φ. For the illustration, we prove just the inductive step for negation in (a). Note that since \(\bigsqcup \emptyset = 0\), it follows from I8 that there is no a ∈ Info such that aC0. Thus \(0 \vDash \neg \varphi \), for any φ. (It turns out that in the step for negation we do not need to use any induction hypothesis.) □
Proof of Proposition 10
We will proceed by induction on α. The statement obviously holds for atomic formulas. The induction hypothesis says that the statement holds for some given \({\mathcal{L}}\)-formulas α and β.
- Negation: :
-
It holds that \(\bigsqcup X \nvDash \neg \alpha \) iff (semantic clause for negation) there is b s.t. \(b C (\bigsqcup X)\) and \(b \vDash \alpha \) iff (I8) there is b and there is a ∈ X s.t. bCa and \(b \vDash \alpha \) iff (semantic clause for negation) there is a ∈ X s.t. \(a \nvDash \neg \alpha \).
- Implication: :
-
\(\bigsqcup X \nvDash \alpha \rightarrow \beta \) iff (semantic clause for implication) there is b s.t. \(b \vDash \alpha \) and \(b \cdot \bigsqcup X \nvDash \beta \) iff (I3) there is b s.t. \(b \vDash \alpha \) and \(\bigsqcup \{b \cdot a \) | \( a \in X \} \nvDash \beta \) iff (induction hypothesis) there is a ∈ X and there is b s.t. \(b \vDash \alpha \) and \(b \cdot a \nvDash \beta \) iff (semantic clause for implication) there is a ∈ X s.t. \(a \nvDash \alpha \rightarrow \beta \).
- Conjunction: :
-
\(\bigsqcup X \vDash \alpha \wedge \beta \) iff (semantic clause for conjunction) \(\bigsqcup X \vDash \alpha \) and \(\bigsqcup X \vDash \beta \) iff (induction hypothesis) for all a ∈ X, \(a \vDash \alpha \) and \(a \vDash \beta \) iff (semantic clause for conjunction) for all a ∈ X, \(a \vDash \alpha \wedge \beta \).
□
Proof of Proposition 12
Assume that \({\mathcal{M}} =\langle Sit, R, l, \phantom { }^{*}, v \rangle \) is a normal ro-model that determines the structure \({\mathcal{M}}^{e} = \langle Info, \sqsubseteq , C, \cdot , 0, 1, V \rangle \). We have to verify that the conditions I1-I9 are satisfied. But first observe that due to the persistency requirement for v, V (p) ∈ Info. Moreover, 0, 1 ∈ Info and Info is closed under fusion: the empty set is obviously ⊲-closed, the case of 1 and fusion is due to the monotonicity of the relation R in the third component which is guaranteed by normality.
I1. It is evident that \(\sqsubseteq \) is a partial order. Moreover, it is complete since the union and intersection of a set X of ⊲-closed sets are ⊲-closed and thus they are the least upper bound and the greatest lower bound of X, respectively. I2. It is obvious that \(\langle Info, \subseteq \rangle \) is a completely distributive lattice. I3. \(b \cdot \bigcup X = \bigcup \{ b \cdot a\) | a ∈ X}, due to the definition of fusion. I4. Commutativity and associativity of fusion are due to commutativity and associativity of the relation R. Moreover \(a \subseteq a \cdot a\) due to the idempotence of R. I5. Obviously, ∅∪ a = a and ∅⋅ a = ∅. 1 ⋅ a = a is due to the identity condition for R in one direction, and to monotonicity for the other direction. I6. Commutativity of the compatibility relation corresponds to the condition S7 for the Routley star. I7. We will prove this condition in detail. The following conditions are equivalent:
-
1.
(a ⋅ b)Cc,
-
2.
there is z ∈ a ⋅ b such that z∗∈ c,
-
3.
there is z and there are x ∈ a, y ∈ b such that Rxyz and z∗∈ c,
-
4.
there is z and there are x ∈ a, y ∈ b such that Rxz∗y∗ and z∗∈ c,
-
5.
there are y ∈ b, x ∈ a, w ∈ c such that Rxwy∗,
-
6.
there is y ∈ b such that y∗∈ a ⋅ c,
-
7.
bC(a ⋅ c).
I8. It holds that \(bC(\bigcup X)\) iff there is \(z \in \bigcup X\) such that z∗∈ b iff there is a ∈ X and z ∈ a such that z∗∈ b iff there is a ∈ X such that bCa. I9. First, observe that if z∗ ⊲ y then y∗ ⊲ z. For any z ∈ Sit we will use the following notation:
-
az = {y ∈ Sit | z ⊲ y}.
We will show that for every z ∈ Sit, the state az is basic. Since z ∈ az and \(z^{*} \in a_{z^{*}}\), it holds that \(a_{z}Ca_{z^{*}}\). To prove that az is basic, it will suffice to show that if \(a_{z^{*}}Cb\), then \(a_{z} \subseteq b\). Assume \(a_{z^{*}}Cb\). Then for some \(y \in a_{z^{*}}\), y∗∈ b. Since z∗ ⊲ y, y∗ ⊲ z. Assume that x ∈ az, i.e. z ⊲ x. Then y∗ ⊲ x, and, therefore, x ∈ b. So, indeed, \(a_{z} \subseteq b\), and it follows that az is basic. It holds for every a ∈ Info that \(a = \bigcup \{a_{z}\) | z ∈ a}, so every a ∈ Info is the least upper bound of some set of basic states.
We have shown that \({\mathcal{M}}^{e}\) is an re-model. Since 1 is completely prime and thus basic, \({\mathcal{M}}^{e}\) is also basic. □
Proof of Proposition 13
We will prove the claim (b). The claim (a) follows from (b) and persistence. Let \({\mathcal{M}} =\langle Sit, R, l, \phantom { }^{*}, v \rangle \) and \({\mathcal{M}}^{e} = \langle Info, \sqsubseteq , \cdot , 0, 1, C, V \rangle \). Let \(\Vdash \) and \(\vDash \) be related to the models \({\mathcal{M}}\) and \({\mathcal{M}}^{e}\), respectively.
We will proceed by induction on the complexity of α. First consider the cases of atomic formulas: For any atom p, \(a \vDash p\) iff \(a \subseteq V(p)\) iff \(a \subseteq v(p)\) iff for all z ∈ a, \(z \Vdash p\).
The induction hypothesis is that for every b ∈ S it holds that \(b \vDash \alpha \) iff for all z ∈ b, \(z \Vdash \alpha \), and \(b \vDash \beta \) iff for all z ∈ b, \(z \Vdash \beta \). We will use the symbols ∀, ∃, ⇒, and & as the abbreviations for the metalinguistic expressions “for all”, “for some”, “implies”, and “and”, respectively.
- Negation: :
-
The following statements are equivalent:
-
1.
\(a \vDash \neg \alpha \),
-
2.
∀b ∈ Info \((aCb \Rightarrow b \nvDash \alpha )\),
-
3.
∀b ∈ Info (∃z ∈ a (z∗∈ b) ⇒∃y ∈ b \((y \nVdash \alpha ))\),
-
4.
∀z ∈ a ∀b ∈ Info (z∗∈ b ⇒∃y ∈ b \((y \nVdash \alpha ))\),
-
5.
∀z ∈ a \((z^{*} \nVdash \alpha )\),
-
6.
∀z ∈ a \((z \Vdash \neg \alpha )\).
We will prove the implication from 4 to 5. Assume that 5 does not hold. Then there is z ∈ a such that \(z^{*} \Vdash \alpha \). Take b = {y ∈ Sit | z∗ ⊲ y}. It holds that z∗∈ b and for all y ∈ b, \(y \Vdash \alpha \). So 4 does not hold.
-
1.
- Implication: :
-
The following statements are equivalent:
-
1.
\(a \nvDash \alpha \rightarrow \beta \),
-
2.
∃b ∈ Info \((b \vDash \alpha \ \& \ a \cdot b \nvDash \beta )\),
-
3.
∃b ∈ Info (∀w ∈ b \((w \Vdash \alpha ) \ \& \ \exists x \in a \cdot b\)\((x \nVdash \beta ))\).
-
4.
∃b ∈ Info (∀w ∈ b \((w \Vdash \alpha ) \ \& \ \exists x \in Sit\) ∃z ∈ a ∃y ∈ b \((Rzyx \ \& \ x \nVdash \beta ))\),
-
5.
∃z ∈ a ∃b ∈ Info (∀w ∈ b \((w \Vdash \alpha ) \ \& \ \exists x \in Sit\) ∃y ∈ b \((Rzyx \ \& \ x \nVdash \beta ))\),
-
6.
∃z ∈ a ∃y ∈ Sit ∃x ∈ Sit \((Rzyx \ \& \ y \Vdash \alpha \ \& \ x \nVdash \beta )\),
-
7.
∃z ∈ a \((z \nVdash \alpha \rightarrow \beta )\).
We prove the implication from 6 to 5. Assume 6, i.e. that there are z ∈ a, y ∈ Sit, x ∈ Sit such that Rzyx and \(y \Vdash \alpha \) and \(x \nVdash \beta \). Take b = {w ∈ Sit | y ⊲ w}. Then y ∈ b and for all w ∈ b, \(w \Vdash \alpha \). So, 5 holds.
-
1.
- Conjunction: :
-
It holds that \(a \vDash \alpha \wedge \beta \) iff \(a \vDash \alpha \) and \(a \vDash \beta \) iff for all z ∈ a, \(z \Vdash \alpha \), and for all z ∈ a, \(z \Vdash \beta \) iff for all z ∈ a, \(z \Vdash \alpha \wedge \beta \).
□
Proof of Proposition 14
Assume that a is a basic state of an re-model. Then there is b such that aCb and for any c, if bCc then \(a \sqsubseteq c\). Moreover, due to I9 there is a set of basic states X such that \(b=\bigsqcup X\). Then due to I8 there is a basic state d ∈ X such that aCd. We show that the state d is the state comp(a) that we are looking for.
First, we show that for any c, dCc iff \(a \sqsubseteq c\). If dCc, then bCc, and so \(a \sqsubseteq c\). If \(a \sqsubseteq c\) then dCc (since dCa).
Second, we show that for any c, aCc iff \(d \sqsubseteq c\). If \(d \sqsubseteq c\) then aCc (since aCd). We show that aCc implies \(d \sqsubseteq c\). Since d is basic, there is e such that dCe and for any c, if eCc then \(d \sqsubseteq c\). Since eCd, it holds also eCb, and so \(a \sqsubseteq e\). But then if aCc then eCc and so \(d \sqsubseteq c\).
Third, we have to show that d is unique with this property. Assume that there is \(d^{\prime }\) such that for any c, \(d^{\prime }Cc\) iff \(a \sqsubseteq c\), and aCc iff \(d^{\prime } \sqsubseteq c\). Then \(aCd^{\prime }\) since \(d^{\prime } \sqsubseteq d^{\prime }\), and so \(d \sqsubseteq d^{\prime }\). Moreover, \(d^{\prime } \sqsubseteq d\) since aCd. □
Proof of Proposition 15
Let \(\mathcal {N} = \langle Info, \sqsubseteq , \cdot , C, 0, 1, V \rangle \) be a basic re-model. We have to show that \(\mathcal {N}^{o} = \langle Sit, R, l, \phantom { }^{*}, v \rangle \) is a normal ro-model. Note that in \(\mathcal {N}^{o}\) it holds that x ⊲ y iff R1xy iff \(y \sqsubseteq 1 \cdot x\) iff \(y \sqsubseteq x\). So, ⊲ is antisymmetric. We have to show that \(\mathcal {N}^{o}\) satisfies the conditions S1-S8. The conditions S1, S2, S3, S5 and S8 are straightforward consequences of the conditions I1-I9. We prove S4, S6, S7. First, S7: Due to Proposition 14, \(comp(x) \sqsubseteq comp(x)\) iff comp(comp(x))Ccomp(x) iff comp(x)Ccomp(comp(x)) iff \(x \sqsubseteq comp(comp(x))\). Moreover, comp(x)Cx implies \(comp(comp(x)) \sqsubseteq x\). Second, S6: Assume Rxyz, i.e. \(z \sqsubseteq x \cdot y\). This, due to Proposition 14, implies comp(z)C(x ⋅ y). Then also yC(x ⋅ comp(z)). Thus \(comp(y) \sqsubseteq x \cdot comp(z)\), i.e. Rxz∗y∗. Finally, to prove that also S4 is satisfied we will need a Lemma corresponding to the so-called Squeeze Lemma that is usually used in relevant logic in the context of the canonical model construction (see, e.g., [22]). The meaning of the Lemma is illustrated with the picture in Fig. 4. (We will use the lemma also in the next proof.)
□
Lemma 1
Let \(\mathcal {N} = \langle Info, \sqsubseteq , C, \cdot , 0, 1, V \rangle \)be an re-model, and a, b, z ∈ Info. Assume that z is basic and \(z \sqsubseteq a \cdot b\). Then there are basic states t, u ∈ Info such that \(t \sqsubseteq a\), \(u \sqsubseteq b\)and \(z \sqsubseteq t \cdot u\).
Proof
Assume \(z \sqsubseteq a \cdot b\) for some basic z. There must be a set of basic elements X such that \(b = \bigsqcup X\). So, \(z \sqsubseteq a \cdot \bigsqcup X = \bigsqcup \{a \cdot u\) | u ∈ X}. Since z is basic, and thus completely prime, there is u ∈ X such that \(z \sqsubseteq a \cdot u\). By the same reasoning we can find \(t \sqsubseteq a\) such that \(z \sqsubseteq t \cdot u\). □
Now we can show that S4 is satisfied. Assume that R2vwyz, i.e. there is x such that \(x \sqsubseteq v \cdot w\) and \(z \sqsubseteq x \cdot y\). Then also \(z \sqsubseteq (v \cdot w) \cdot y = v \cdot (w \cdot y)\). Now apply Squeeze Lemma to a = v and b = w ⋅ y. So there is basic \(u \sqsubseteq w \cdot y\) such that \(z \sqsubseteq v \cdot u\). It follows that R2wyvz.
Proof of Theorem 2
(a) Let \({\mathcal{M}}= \langle Sit, R, l, \phantom { }^{*}, v \rangle \) be a normal ro-model. \({\mathcal{M}}\) is equivalent to \({\mathcal{M}}^{e}\) as a direct consequence of Proposition 13. We show that the function f that assigns to any given x ∈ Sit the set x⊲ = {y ∈ Sit | x ⊲ y} is an isomorphism of the structures \({\mathcal{M}}\) and \({\mathcal{M}}^{eo}\). It holds that the completely prime elements among the ⊲-closed sets are exactly the sets generated by single points. In other words, a is basic (i.e. completely prime) in \({\mathcal{M}}^{e}\) iff a = x⊲ for some x ∈ Sit. This observation together with the normality of \({\mathcal{M}}\) guarantees that f is a bijection beween situations of \({\mathcal{M}}\) and situations of \({\mathcal{M}}^{eo}\). Moreover, it holds: Rxyz in \({\mathcal{M}}\) iff \(f(z) \sqsubseteq f(x) \cdot f(y)\) in \({\mathcal{M}}^{e}\) iff Rf(x)f(y)f(z) in \({\mathcal{M}}^{eo}\); f(l) is the logical situation in \({\mathcal{M}}^{eo}\); for any x ∈ Sit, comp(f(x)) = f(x∗); and, finally, f(x) is in the valuation of p in \({\mathcal{M}}^{eo}\) iff x is in the valuation of p in \({\mathcal{M}}\). So, \({\mathcal{M}}\) and \({\mathcal{M}}^{eo}\) are isomorphic.
(b) Let \(\mathcal {N}=\langle Info, \sqsubseteq , \cdot , C, 0, 1, V \rangle \) be a basic re-model. First, we prove that the function g that assigns to every a ∈ Info the set of basic states that are below a (w.r.t. \(\sqsubseteq \)) is an isomorphism between \(\mathcal {N}\) and \(\mathcal {N}^{oe}\). Due to I9, for any a ∈ Info, \(a=\bigsqcup g(a)\). Moreover, since any basic element is completely prime, it also holds for any downward closed set of basic states X that \(g(\bigsqcup X)=X\). So, g is a bijection between the states of \(\mathcal {N}\)and states of \(\mathcal {N}^{oe}\). Moreover, it holds: \(a \sqsubseteq b\) iff \(g(a) \subseteq g(b)\); aCb in \(\mathcal {N}\) iff for some x ∈ g(a), xCb in \(\mathcal {N}\) iff for some x ∈ g(a), \(comp(x) \sqsubseteq b\) in \(\mathcal {N}\) iff for some x ∈ g(a), \(comp(x) \subseteq g(b)\) iff g(a)Cg(b) in \(\mathcal {N}^{oe}\); moreover, z ∈ g(a ⋅ b) (for ⋅ in \(\mathcal {N}\)) iff \(z \sqsubseteq a \cdot b\) iff (due to Lemma 1) there are basic t, u such that \(t \sqsubseteq a\), \(u \sqsubseteq b\) and \(z \sqsubseteq t \cdot u\) iff z ∈ g(a) ⋅ g(b) (for ⋅ in \(\mathcal {N}^{oe}\)); g(1) is the logical situation in \(\mathcal {N}^{oe}\); g(0) = ∅, which is the least element in \(\mathcal {N}^{oe}\); and g(V (p)) is the valuation of p in \(\mathcal {N}^{oe}\). So, \(\mathcal {N}\) and \(\mathcal {N}^{oe}\) are isomorphic. Moreover, \(\mathcal {N}\) is equivalent to \(\mathcal {N}^{o}\), since \(\mathcal {N}^{o}\) is equivalent to \(\mathcal {N}^{oe}\), due to (a), and \(\mathcal {N}^{oe}\) is isomorphic to \(\mathcal {N}\). □
Proof of Proposition 16
Soundness of axioms InqA1-InqA10 and rules InqR1, InqR2 can be proved in the same way as in the case of A1-A10 and R1, R2. Soundness of InqA11-InqA14 is straightforward. Let us show that the split schema InqA15 is valid in every re-model. Assume that a is a state of an re-model such that . So, there are states b, c such that \(b \vDash \alpha \) but \(b \cdot a \nvDash \psi \), and \(c \vDash \alpha \) but \(c \cdot a \nvDash \chi \). But then \(b \sqcup c \vDash \alpha \) (by Proposition 10) but (by Proposition 9-b). Since (b ⋅ a) ⊔ (c ⋅ a) = (b ⊔ c) ⋅ a, . □
Proof of Proposition 18
Induction on the complexity of φ. The proof is completely standard and the same strategy is used on many places in the literature on inquisitive semantics.Footnote 1□
Proof of Proposition 19
It follows from a general result in [37] that to prove the disjunction property of inquisitive disjunction for a semantics based on a class of information models (particular examples of which are re-models) it is sufficient to show that the class of models is closed under products. Then it holds that the product of an re-model that is a counterexample to φ and an re-model that is a counterexample to ψ is an re-model that is a counterexample to .
Let \({\mathcal{M}}_{1} = \langle Info_{1}, \sqsubseteq _{1}, \cdot _{1}, 0_{1}, 1_{1}, C_{1}, V_{1} \rangle \), \({\mathcal{M}}_{2} = \langle Info_{2}, \sqsubseteq _{2}, \cdot _{2}, 0_{2}, 1_{2}, C_{2}, V_{2} \rangle \) be any re-models. As expected, the product of \({\mathcal{M}}_{1}\) and \({\mathcal{M}}_{2}\), denoted as \({\mathcal{M}}_{1} \times {\mathcal{M}}_{2}\), is the structure \(\langle Info, \sqsubseteq , \cdot , 0, 1, C, V \rangle \), where Info = Info1 × Info2 (Cartesian product of Info1 and Info2); \(\langle a, b \rangle \sqsubseteq \langle c, d \rangle \) iff \(a \sqsubseteq _{1} c\) and \(b \sqsubseteq _{2} d\); 〈a, b〉⋅〈c, d〉 = 〈a ⋅1c, b ⋅2d〉; 0 = 〈01, 02〉; 1 = 〈11, 12〉; 〈a, b〉C〈c, d〉 iff aC1c or bC2d; V (p) = 〈V1(p),V2(p)〉. We need to prove the following Lemma. □
Lemma 2
The product of two re-models is an re-model.
Proof
We take two arbitrary re-models \({\mathcal{M}}_{1} = \langle Info_{1}, \sqsubseteq _{1}, \cdot _{1}, 0_{1}, 1_{1}, C_{1}, V_{1} \rangle \), \({\mathcal{M}}_{2} = \langle Info_{2}, \sqsubseteq _{2}, \cdot _{2}, 0_{2}, 1_{2}, C_{2}, V_{2} \rangle \), and consider their product \({\mathcal{M}}_{1} \times {\mathcal{M}}_{2} = \langle Info, \sqsubseteq , \cdot , 0, 1, C, V \rangle \). The product of complete and completely distributive lattices is a complete and completely distributive lattice. So, I1 and I2 hold for \({\mathcal{M}}_{1} \times {\mathcal{M}}_{2}\). It can be easily verified that the conditions I3-I7 are also satisfied. We will discuss only the conditions I8 and I9.
For any \(Y \subseteq Info_{1}\), \(Z \subseteq Info_{2}\) let us denote the least upper bound of Z in \({\mathcal{M}}_{1}\) as \(\bigsqcup _{1} Y\) and of Z in \({\mathcal{M}}_{2}\) as \(\bigsqcup _{2} Z\). For any \(X \subseteq Info\), let
-
X1 = {a ∈ Info1| for someb ∈ Info2,〈a, b〉∈ X},
-
X2 = {b ∈ Info2| for somea ∈ Info1,〈a, b〉∈ X}.
It holds that \(\bigsqcup X\) in \({\mathcal{M}}_{1} \times {\mathcal{M}}_{2}\) can be expressed as \(\langle \bigsqcup _{1} X_{1}, \bigsqcup _{2} X_{2} \rangle \). Then \( \langle a_{1}, a_{2} \rangle C (\bigsqcup X)\) iff \(\langle a_{1}, a_{2} \rangle C \langle \bigsqcup _{1} X_{1}, \bigsqcup _{2} X_{2} \rangle \) iff \(a_{1} C_{1} (\bigsqcup _{1} X_{1})\) or \(a_{2} C_{2} (\bigsqcup _{2} X_{2})\) iff for some b1 ∈ X1, a1C1b1 or for some b2 ∈ X2, a2C2b2 iff for some b1 ∈ X1, b2 ∈ X2, 〈a1, a2〉C〈b1, b2〉. So I8 is satisfied.
Now we show that also I9 is satisfied. First observe that
-
(a)
if a1 is basic in \({\mathcal{M}}_{1}\), then 〈a1, 02〉 is basic in \({\mathcal{M}}_{1} \times {\mathcal{M}}_{2}\),
-
(b)
if a2 is basic in \({\mathcal{M}}_{2}\), then 〈01, a2〉 is basic in \({\mathcal{M}}_{1} \times {\mathcal{M}}_{2}\).
We will prove only (a). The proof of (b) is similar. Assume that a1 is basic in \({\mathcal{M}}_{1}\). That means that there is b1 ∈ Info1 such that a1C1b1 and for all c1 ∈ Info1, if b1C1c1, then \(a_{1} \sqsubseteq _{1} c_{1}\). Take the state 〈b1, 02〉∈ Info. It holds that 〈a1, 02〉C〈b1, 02〉. Assume that 〈b1, 02〉C〈c1, c2〉. Then b1C1c1 and so \(a_{1} \sqsubseteq _{1} c_{1}\). It follows that \(\langle a_{1}, 0_{2} \rangle \sqsubseteq \langle c_{1}, c_{2} \rangle \). We have proved that 〈a1, 02〉 is basic.
Now take any 〈a1, a2〉∈ Info. We have to show that there is a set of basic states \(X \subseteq Info\) such that \(\langle a_{1}, a_{2} \rangle = \bigsqcup X\). We will use the fact that there is a set of basic states \(Y \subseteq Info_{1}\) such that \(\bigsqcup _{1} Y = a_{1}\) and there is a set of basic states \(Z \subseteq Info_{2}\) such that \(\bigsqcup _{2} Z = a_{2}\). Let us define:
-
X = {〈b1, 02〉; b1 ∈ Y }∪{〈01, b2〉; b2 ∈ Z}.
Now it holds that \(\bigsqcup X = \langle \bigsqcup _{1} X_{1}, \bigsqcup _{2} X_{2} \rangle = \langle \bigsqcup _{1} Y, \bigsqcup _{2} Z \rangle = \langle a_{1}, a_{2} \rangle \). Moreover, X is a set of basic states. □
Proof of Proposition 20
Assume that φ ⊩InqRψ. Due to Proposition 18 there are \({\mathcal{L}}\)-formulas α1,…,αn and β1,…,βm such that and . Moreover, from the proof of Proposition 18, it is evident that α1,…,αn involve only atomic formulas that occur also in φ and β1,…,βm involve only atomic formulas that occur also in ψ. Take an arbitrary αi (1 ≤ i ≤ n). Then there is some j (1 ≤ j ≤m) such that αi ⊩InqRβj. Since InqR is a conservative extension of R, we have αi ⊩Rβj. Since R has variable-sharing property, there is an atomic formula that occurs in both αi and βj. Thus, this atomic formula occurs in both φ and ψ. □
Rights and permissions
About this article
Cite this article
Punčochář, V. A Relevant Logic of Questions. J Philos Logic 49, 905–939 (2020). https://doi.org/10.1007/s10992-019-09541-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-019-09541-9