Abstract
In the logical context, ignorance is traditionally defined recurring to epistemic logic. In particular, ignorance is essentially interpreted as “lack of knowledge”. This received view has - as we point out - some problems, in particular we will highlight how it does not allow to express a type of content-theoretic ignorance, i.e. an ignorance of φ that stems from an unfamiliarity with its meaning. Contrarily to this trend, in this paper, we introduce and investigate a modal logic having a primitive epistemic operator I, modeling ignorance. Our modal logic is essentially constructed on the modal logics based on weak Kleene three-valued logic introduced by Segerberg (Theoria, 33(1):53–71, 1997). Such non-classical propositional basis allows to define a Kripke-style semantics with the following, very intuitive, interpretation: a formula φ is ignored by an agent if φ is neither true nor false in every world accessible to the agent. As a consequence of this choice, we obtain a type of content-theoretic notion of ignorance, which is essentially different from the traditional approach. We dub it severe ignorance. We axiomatize, prove completeness and decidability for the logic of reflexive (three-valued) Kripke frames, which we find the most suitable candidate for our novel proposal and, finally, compare our approach with the most traditional one.
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We refer the interested reader directly to the introductory textbook [21].
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Acknowledgements
The authors wish to thank Francesco Paoli, Massimiliano Carrara, Pietro Salis, Matteo Plebani, Mario Alai, Alessandro Aldini, Mirko Tagliaferri for their observations and comments on preliminary presentations of this work. We particularly thank Jie Fan for having noticed a mistake which led to an improvement of the paper and Valentin Goranko for his helpful comments and suggestions on possible developments of the ideas proposed in the article.
We finally thank the meticulous work of an anonymous reviewer for the many fruitful comments on previous versions of the paper that brought to a considerable improvements of our ideas and their technical realization.
Funding
Open access funding provided by Università degli Studi di Cagliari within the CRUI-CARE Agreement. The work of Stefano Bonzio was partially supported by the PRIN 2017 project “From models to decisions”, funded by the Italian Ministry of Education, University and Research (MIUR) through the grant n. 201743F9YE . Stefano Bonzio also acknowledges the support of the Marie Curie fellowship within the program “Beatriu de Pinos”, co-funded by Generalitat de Catalunya and the European Union’s Horizon 2020 research and innovation programme under the MSCA grant agreement No. 801370 and GNSAGA (Gruppo Nazionale per le Strutture Algebriche e Geometriche). Vincenzo Fano and Pierluigi Graziani’s works were supported by the Italian Ministry of Education, University and Research through the PRIN 2017 project “The Manifest Image and the Scientific Image” prot. 2017ZNWW7F_004.
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Appendix
Appendix
In this Appendix we provide the details of the proof of strong completeness for Bochvar (external) logic Be (Theorem 7). Moreover, we also show that Be is algebraizable. We start with a preliminary lemma.
Lemma 1
The following elementary facts holds in Be:
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(1)
\(\vdash _{\mathsf {B}_{e}}J_{_0}(\varphi \lor \psi )\leftrightarrow J_{_0}\varphi \wedge _{_0}\psi \);
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(2)
\(\vdash _{\mathsf {B}_{e}}J_{_1}(\varphi \lor \psi )\leftrightarrow J_{_1}\varphi \vee J_{_1}\psi \);
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(3)
\(\vdash _{\mathsf {B}_{e}}J_{_{2}}(\varphi \lor \psi )\leftrightarrow (J_{_{2}}\varphi \wedge J_{_{2}}\psi )\vee (J_{_{2}}\varphi \wedge J_{_{2}}\neg \psi )\vee (J_{_{2}}\neg \varphi \wedge J_{_{2}}\psi ) \);
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(4)
if α is an external formula, then \(\vdash _{\mathsf {B}_{e}} \alpha \leftrightarrow J_{_2}\alpha \);
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(5)
\(\vdash _{\mathsf {B}_{e}}\neg J_{_2}\psi \leftrightarrow (J_{_0}\psi \vee J_{_1}\psi )\).
Proof
Immediate by using Theorem 6. □
Lemma 2
Let \({\Gamma }\cup \{\varphi \}\vdash _{\mathsf {B}_{e}}\psi \). Then \({\Gamma }\vdash _{\mathsf {B}_{e}}J_{_2}\varphi \to J_{_2}\psi \).
Proof
By induction on the length of the derivation of ψ (from Γ). □
The class of Bochvar algebras is introduced by Finn and Grigolia [18, pp. 233-234] as algebraic semantics for Be.
Definition 3
A Bochvar algebra \(\mathbf {A}=\langle {A,\vee ,\wedge , \neg , J_{_0},J_{_1},J_{_2}, 0,1}\rangle \) is an algebra of type 〈2, 2, 1, 1, 1, 1, 0, 0〉 satisfying the following identities and quasi-identities:
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(1)
x ∨ x ≈ x;
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(2)
x ∨ y ≈ y ∨ x;
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(3)
(x ∨ y) ∨ z ≈ x ∨ (y ∨ z);
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(4)
(x ∧ (y ∨ z) ≈ ((x ∧ y) ∨ (x ∧ z));
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(5)
¬(¬x) ≈ x;
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(6)
¬1 ≈ 0;
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(7)
¬(x ∨ y) ≈¬x ∧¬y;
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(8)
0 ∨ x ≈ x;
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(9)
\(J_{_2}J_{_i} x\approx J_{_i} x\), for every i ∈{0, 1, 2};
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(10)
\(J_{_0}J_{_i} x\approx \neg J_{_i} x\), for every i ∈{0, 1, 2};
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(11)
\(J_{_{1}}J_{_i} x \approx 0\), for every i ∈{0, 1, 2};
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(12)
\(J_{_{i}}(\neg x)\approx J_{_{2}-i}x\), for every i ∈{0, 1, 2};
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(13)
\(J_{_{i}}x \approx \neg (J_{_{j}}x\vee J_{_{k}}x)\), for i≠j≠k≠i;
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(14)
\(J_{_{i}}x\vee \neg J_{_{i}}x \approx 1\), for every i ∈{0, 1, 2};
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(15)
\(((J_{_{i}}x\vee J_{_{k}}x)\wedge J_{_{i}}x)\approx J_{_{i}}x\), for i,k ∈{0, 1, 2};
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(16)
\( x\lor J_{_{i}}x \approx x\), for i ∈{1, 2};
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(17)
\(J_{_{0}}(x\lor y)\approx J_{_{0}}x\wedge J_{_{0}}y\);
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(18)
\(J_{_{2}}(x\lor y)\approx (J_{_{2}}x\wedge J_{_{2}}y)\vee (J_{_{2}}x\wedge J_{_{2}}\neg y)\vee (J_{_{2}}\neg x\wedge J_{_{2}}y) \);
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(19)
\(J_{_0} x \approx J_{_0} y \& J_{_1} x \approx J_{_1} y \& J_{_2} x \approx J_{_2} y \Rightarrow x \approx y\).
We denote by \({\mathscr{B}}\mathcal {A}_{3}\) the class of Bochvar algebras. \({\mathscr{B}}\mathcal {A}_{3}\) forms a quasi-variety which is not a variety [18]. Recall that a class \(\mathcal {K}\) of algebras is an algebraic semantics for a logic L provided that: Γ ⊩Lφ iff \(\{\tau (\gamma ): \gamma \in {\Gamma }\}\models _{Eq(\mathcal {K})} \tau (\varphi )\), where τ = {φi(p) ≈ ψi(p)} is a formula-equation transformer and \(Eq(\mathcal {K})\) denotes the usual equational consequence relation relative to the class \(\mathcal {K}\).
Theorem 4
\({\mathscr{B}}\mathcal {A}_{3}\) is an algebraic semantics for Be. In particular, \({\Gamma }\vdash _{\mathsf {B}_{e}}\varphi \) iff \(\{\gamma \approx 1: \gamma \in {\Gamma }\}\models _{Eq({\mathscr{B}}\mathcal {A}_{3})} \varphi \approx 1\).
Proof
(⇒) By induction on the length of the derivation of φ (from Γ), by checking that axioms (A1)-(A29) are evaluated to 1 in every Bochvar algebra A and that the rule (MP) preserves this property.
(⇐) We reason by contraposition. Suppose \({\Gamma }\nvdash _{\mathsf {B}_{e}}\varphi \) and provide a counterexample to such inference by constructing the Lindenbaum-Tarski algebra. Let Δ be the smallest set of formulas including Γ and closed under \(\vdash _{\mathsf {B}_{e}}\) (from now on we will simply write ⊩ instead of \(\vdash _{\mathsf {B}_{e}}\)). For any pair of formulas, define
We claim that:
-
(1)
\(\sim \) is a congruence on Fm;
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(2)
\([1]_{\sim } = {\Delta }\);
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(3)
The quotient algebra \(\mathbf {Fm}_{/_{\sim }}\) is a Bochvar algebra.
-
(1)
It is easy to check that \(\sim \) is an equivalence relation. To show that it is a congruence, we check the compatibility of \(\sim \) with the operations in the type of \({\mathscr{L}}_{K^{e}}\).
[¬] Suppose \(\varphi \sim \psi \), then φ ≡ ψ ∈Δ, i.e. \(\displaystyle \bigwedge _{i=0}^{2} J_{_{i}}\varphi \leftrightarrow J_{_{i}} \psi \in {\Delta }\), which is equivalent to \(\displaystyle \bigwedge _{i=0}^{2} J_{_{2-i}}\varphi \leftrightarrow J_{_{2-i}} \psi \in {\Delta }\). Hence, in virtue of (A12), \(\displaystyle \bigwedge _{i=0}^{2} J_{_i}(\neg \varphi )\leftrightarrow J_{_i}(\neg \psi )\in {\Delta }\), i.e. ¬φ ≡¬ψ ∈Δ, showing that \(\neg \varphi \sim \neg \psi \).
[\(J_{_2}\)] Suppose \(\varphi \sim \psi \), thus φ ≡ ψ ∈Δ, i.e. \(\displaystyle \bigwedge _{i=0}^{2} J_{_i}\varphi \leftrightarrow J_{_i}\psi \in {\Delta }\). In particular \({\Delta }\vdash J_{_2}\varphi \leftrightarrow J_{_2}\psi \). In virtue of (A9), we have \(\vdash J_{_2}\varphi \leftrightarrow J_{_2}J_{_2}\varphi \) and \(\vdash J_{_2}\psi \leftrightarrow J_{_2}J_{_2}\psi \), from which \({\Delta }\vdash J_{_2}J_{_2}\varphi \leftrightarrow J_{_2}J_{_2}\psi \), i.e. \(J_{_2}J_{_2}\varphi \leftrightarrow J_{_2}J_{_2}\psi \in {\Delta } \) (as Δ is closed under consequences of ⊩). Analog reasoning, using (A10) and (A11), shows that \(J_{_0}J_{_2}\varphi \leftrightarrow J_{_0}J_{_2}\psi \in {\Delta } \) and \(J_{_1}J_{_2}\varphi \leftrightarrow J_{_1}J_{_2}\psi \in {\Delta } \), from which \(J_{_2}\varphi \sim J_{_2}\psi \).
[∨] Suppose \(\varphi _{1}\sim \psi _{1}\) and \(\varphi _{2}\sim \psi _{2}\). Then \({\Delta }\vdash J_{_0}\varphi _{1}\leftrightarrow J_{_0}\psi _{1}\) and \({\Delta }\vdash J_{_0}\varphi _{2}\leftrightarrow J_{_0}\psi _{2}\), hence \({\Delta }\vdash (J_{_0}\varphi _{1}\wedge J_{_0}\varphi _{2})\leftrightarrow (J_{_0}\psi _{1}\wedge J_{_0}\psi _{2})\). Applying Lemma 31-(1), we have \({\Delta }\vdash J_{_0}(\varphi _{1}\lor \varphi _{2})\leftrightarrow J_{_0}(\psi _{1}\lor \psi _{2})\), from which \(J_{_0}(\varphi _{1}\lor \varphi _{2})\leftrightarrow J_{_0}(\psi _{1}\lor \psi _{2})\in {\Delta }\). Analog reasoning (using Lemma 31-(2,3)) allows to conclude \(\varphi _{1}\lor \varphi _{2}\sim \psi _{1}\lor \psi _{2}\).
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(2)
[\(\subseteq \)] Let \(\psi \in [1]_{\sim }\). Then ψ ≡ 1 ∈Δ, i.e. \(\displaystyle \bigwedge _{i=0}^{2} J_{_i} \psi \leftrightarrow J_{_i} 1\in {\Delta }\). In particular, \(J_{_2}\psi \in {\Delta }\) (since \(\vdash J_{_2} 1\)) and \(J_{_1}\psi \leftrightarrow 0\in {\Delta }\) (as \(\vdash J_{_1} 1\leftrightarrow 0\)), from which we deduce that ψ is an external formula, so, by Lemma 31-(4), ψ ∈Δ.
[\(\supseteq \)] Let ψ ∈Δ. Observe that 1 ∈Δ (since ⊩ 1), hence, by Lemma 32, \({\Delta }\vdash J_{_2}\psi \leftrightarrow J_{_2} 1\), thus \(J_{_2}\psi \leftrightarrow J_{_2} 1\in {\Delta }\). Moreover, \({\Delta }\vdash \neg J_{_2}\psi \leftrightarrow 0\) and, by Lemma 31, \(\vdash \neg J_{_2}\psi \leftrightarrow (J_{_0}\psi \vee J_{_1}\psi )\), so \({\Delta }\vdash (J_{_0}\psi \vee J_{_1}\psi )\leftrightarrow 0\), from which \({\Delta }\vdash J_{_0}\psi \leftrightarrow 0 \) and \( {\Delta }\vdash J_{_1}\psi \leftrightarrow 0\); therefore \({\Delta }\vdash J_{_0}\psi \leftrightarrow J_{_0} 1 \) and \( {\Delta }\vdash J_{_1}\psi \leftrightarrow J_{_1} 1\) (as \(\vdash J_{_0} 1\leftrightarrow 0\) and \(\vdash J_{_1} 1\leftrightarrow 0\)). This shows that ψ ≡ 1 ∈Δ, i.e. \(\psi \in [1]_{\sim }\).
-
(3)
It is routine to check that \(\mathbf {Fm}_{/_{\sim }}\) is indeed a Bochvar algebra.
To provide a counterexample to the inference \({\Gamma }\nvdash _{\mathsf {B}_{e}}\varphi \), consider the Bochvar algebra \(\mathbf {A}= \mathbf {Fm}_{/_{\sim }}\) and the homomorphism h: Fm →A, \(h(\varphi )= [\varphi ]_{\sim }\). Since \({\Gamma }\subseteq {\Delta }\) and \({\Delta } = [1]_{\sim }\), then h(γ) = 1A, for each γ ∈Γ, but h(φ)≠ 1A (since φ∉Δ).
□
Theorem 7 follows from Theorem 34 by observing that \({\mathscr{B}}\mathcal {A}_{3}\) is the quasi-variety generated by WKe ([18, Theorem 3.3]).
It is natural to wonder whether the quasi-variety of Bochvar algebras is simply an algebraic semantics for Be. Actually the relationship between Be and the class \({\mathscr{B}}\mathcal {A}_{3}\) is tighter. Recall that a logic ⊩ is algebraizable with \(\mathcal {K}\) as equivalent algebraic semantics (where \(\mathcal {K} \) is a class of algebras of the same language as the logic ⊩) if there exists a map τ from formulas to sets of equations, and a map ρ from equations to sets of formulas such that the following conditions hold, for any pair of formulas φ,ψ and set of equations E.
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Γ ⊩ φ iff \(\tau [{\Gamma }]\models _{Eq(\mathcal {K})}\tau (\varphi )\);
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\(E\models _{Eq(\mathcal {K})} \varphi \approx \psi \) iff \(\rho (E)\models _{Eq(\mathcal {K})}\rho (\varphi ,\psi )\);
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φ ⊣⊩ ρ(τ(φ));
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\(\varphi \approx \psi =\joinrel \mathrel |\models _{Eq\left (\mathcal {K}\right )} \tau (\rho (\varphi ,\psi ))\).
Examples of algebraizable logics include, among many others, classical, intuitionistic logic, all substructural logics and global modal logics. Not all logics though are algebraizable: examples of non-algebraizable logics can be found in the realm of Kleene logics, such as the Logic of Paradox (see [37]), Paraconsistent weak Kleene (see [6]) and Bochvar internal logic (see [7,8,9]). The above definition of algebraizable logic can be drastically simplified: ⊩ is algebraizable with equivalent algebraic semantics \(\mathcal {K}\) if and only if it satisfies either ALG1 and ALG4 (or, else ALG2 and ALG3).Footnote 1
Theorem 5
The logic Be is algebraizable with \({\mathscr{B}}\mathcal {A}_{3}\) as equivalent algebraic semantics.
Proof
Consider τ = {φ ≈ 1} and ρ = {φ ≡ ψ}. □
The usefulness of the above result will be explored in a fore-coming paper, focused on a deeper understanding of the properties of Bochvar algebras [10].
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Bonzio, S., Fano, V., Graziani, P. et al. A Logical Modeling of Severe Ignorance. J Philos Logic 52, 1053–1080 (2023). https://doi.org/10.1007/s10992-022-09697-x
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DOI: https://doi.org/10.1007/s10992-022-09697-x