Abstract
The aim of this work is to investigate the problem of Logical Omniscience in epistemic logic by means of truthmaker semantics. We will present a semantic framework based on \(\varvec{W}\)-models extended with a partial function, which selects the body of knowledge of the agents, namely the set of verifiers of the agent’s total knowledge. The semantic clause for knowledge follows the intuition that an agent knows some information \(\varvec{\phi }\), when the propositional content that \(\varvec{\phi }\) is contained in her total knowledge. We will argue that this idea mirrors the philosophical conception of immanent closure by Yablo (2014), giving to our proposal a strong philosophical motivation. We will discuss the philosophical implications of the semantics and we will introduce its axiomatization.
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Acknowledgements
I am grateful to Stephan Krämer and Daniele Porello for several valuable comments and discussions regarding the content of this article. I would also like to thank two anonymous referees for carefully reading and commenting on this work.
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Appendices
Appendix A
In this Appendix we prove the completeness theorem of the logic EL with respect to the class of complete epistemic models (Theorem 19).
Theorem 21
(Soundness of EL) EL is sound with respect to the class of epistemic models: for all \(\phi \) and \(\psi \in \mathcal {L}_e\), if \(\phi \vdash _{EL} \psi \), then \(\phi \models \psi \).
Proof
K0. Classical logic is sound and complete with respect to loose consequence. K1. Let \(\phi , \psi \in \mathcal {L}\) and assume that \(\phi \vdash _{AC} \psi \). By soundness of AC, \(\phi >_{AC} \psi \). Consider an arbitrary \( w\in W\) such that \(\mathcal {M},w\models K\phi \). Hence, there is a \(s\in S^\Diamond \), such that \(s\sqsubseteq w\) and \(s\Vdash K\phi \), i.e. \(|\phi |^+\preceq f(s)\). Since \(\phi >_{AC} \psi \), by definition \(|\psi |^+\preceq |\phi |^+\). By transitivity of the relation of containment we can conclude \(|\psi |^+\preceq f(s)\), which proves that \(\mathcal {M},w\models K\psi \). K2 follows from Proposition 12. \(\square \)
In this section, we will develop the proof of the completeness theorem of EL, namely that every consistent formula in \(\mathcal {L}_e\) is satisfiable.
Definition 29
For all \(\alpha ,\beta \in \mathcal {L}_e\), we say that \(\alpha \) is inconsistent, when it classically derives a contradiction, i.e. \(\alpha \vdash _{CL} \beta \wedge \lnot \beta \).
We sometimes abbreviate a classical contradiction with the symbol \(\bot \). Since for all \(\alpha \in \mathcal {L}_e\), \(\beta \wedge \lnot \beta \vdash _{CL}\alpha \), then, \(\alpha \) is inconsistent if and only if \(\alpha \) is provably equivalent in CL to \(\beta \wedge \lnot \beta \). Otherwise it is consistent.
Definition 30
A formula \(\alpha \in \mathcal {L}_e\) is satisfiable when there is an epistemic model \(\mathcal {M}\) and a world \(w\in \mathcal {M}\) which loosely verifies it, i.e. \(\mathcal {M},w\models \alpha \).
We can summarize the strategy of the proof as follows. We will first show that (i) every consistent K-formula – call it \(\delta \) – is satisfiable, and from this it will straightforwardly follow that (ii) every consistent formula \(\alpha \) in \(\mathcal {L}_e\) is satisfiable. To prove (i), it suffices to show that \(\delta \) has a model, which amounts to showing that there is an epistemic model with a possible state that exactly verifies \(\delta \). Indeed, in an epistemic model, every possible state is part of a possible world. Hence, if a possible state verifies \(\delta \), then there is a possible world that loosely verifies it. In particular, our goal is to build a syntactic model for \(\delta \), which we prove to be indeed an epistemic model (in Section A.2). In order to accomplish this result, we first bring \(\delta \) into a specific disjunctive normal form, which we call Maximal K-form.
1.1 A.1 Normal Forms
As mentioned our completeness proof draws on the idea of disjunctive normal forms. We will proceed in two steps. We first shall identify a suitable class of disjunctive normal forms in \(\mathcal {L}\), namely suitable for the non-epistemic formulas. Secondly, we shall identify a particular class of disjunctive normal forms for any epistemic formulas. We call the former the class of closed disjunctive forms, the latter the class of maximal K-forms.
Definition 31
(Descriptions) A description is a conjunction \(\lambda _1 \wedge \lambda _2 \wedge ... \wedge \lambda _m\) of literals. As a limiting case a literal is also considered a description.
Definition 32
(Sub-description) Let \(\phi = \lambda _1 \wedge \lambda _2 \wedge ... \wedge \lambda _m\) and \(\psi \) be descriptions, then \(\psi \) is a sub-description of \(\phi \), written \(\psi \Subset \phi \), if \(\psi \) is a conjunction of some of \(\lambda _1, \lambda _2, ..., \lambda _m\).
Note that the order and the repetition of the literals is not relevant for a conjunction of literals to be a sub-description of another one. Note also that, given the descriptions \(\phi \) and \(\psi \), \(\phi \Subset \psi \) is equivalent to say that \(Lit(\phi ) \subseteq Lit(\psi )\).
Definition 33
(Disjunctive normal form) A disjunctive normal form is a disjunction \(\phi _1 \vee \phi _2 \vee ...\vee \phi _m\) of descriptions \(\phi _1, \phi _2, . . . , \phi _m\), with \(m \ge 0\).
Definition 34
(Closed disjunctive form) A disjunctive form \(\phi \) is closed iff for any set \(\Psi \) of disjuncts in \(\phi \), \(\phi \) includes a disjunct \(\psi \) with \(Lit(\psi ) = \bigcup \{Lit(\phi _i) \mid \phi _i\in \Psi \}\).
Lemma 22
-
1.
Any formula \(\phi \) in \(\mathcal {L}\) is provably equivalent in AC to a disjunctive normal form.
-
1.
Any disjunctive normal form in \(\mathcal {L}\) is provable equivalent in AC to a closed disjunctive form.
Proof
(1) is analogous to a standard result in classical logic. (2) is an adaptation of lemma 14 in [41, p. 23]. \(\square \)
Now that we have a suitable normal form for the non-epistemic formulas, which are the arguments for the modality K, we shall now introduce the normal form we need for epistemic formulas.
Definition 35
(Maximal K-form) A maximal K-form \(K^M(\phi )\) abbreviates the conjunction
(with \(0\le m\)), where \(\phi = \phi _1 \vee ...\vee \phi _{n}\) is in closed disjunctive form, and each negated atom \(\lnot K(\psi _k)\) (with \(k\le m\)), is such that \(\psi _k=\psi ^{k}_{1} \vee ... \vee \psi ^k_{m_k}\) is in closed disjunctive form and:
-
1.
either there is \(\psi ^k_i\) (\(i\le m_k\)) such that for all \(\phi _j\) (\(j\le n \))
;
-
2.
or there is \(\phi _j\) (\(j\le n\)) such that for all \(\psi ^k_{i}\) (\(i\le m_k\)),
;
This normal form aims at spelling out the relationship between K-atoms and negated K-atoms, with respect to their arguments. K-literals are, in fact, limit cases of Maximal K-forms: \(K\phi \) satisfies vacuously the disjunct (2) of the definition, because \(m=0\), namely there are no negated K-atoms; similarly, \(\lnot K\psi \) satisfies vacuously the disjunct (1), because \(n=0\).
The following lemma generalizes the reasoning just applied in the previous example.
Lemma 23
Each consistent K-formula \(\delta \) is provably equivalent in EL to a disjunction, where each disjunct is equivalent to a maximal K-form.
Proof
Let \(\delta \) be an arbitrary consistent K-formula. We firstly put \(\delta \) in disjunctive normal form by means of the classical rules of EL, i.e. we obtain a disjunction of conjunctions of K-literals, then we agglomerate all the K-atoms:
Since \(\delta \) is consistent, then there is at least a consistent disjunct \(\delta _i\) in \(\delta \), which will be of the form \(\delta _i= K(\phi ^i_1\wedge ...\wedge \phi ^i_n) \wedge ... \wedge \lnot K(\psi ^i_{1}) \wedge ... \wedge \lnot K (\psi ^i_{m})\).
Let \(\chi ^i\) be a closed disjunctive form (Def. 34) such that \(\chi ^i = \chi ^i_1\vee ...\vee \chi ^i_k \dashv \vdash _{AC} \phi ^i_1 \wedge ... \wedge \phi ^i_n\) (we obtain it by E9 and Lemma 22), and each disjunct in \(\chi ^i\) is a description (def. 3.2). Then \(K(\chi ^i) \dashv \vdash _{EL} K(\phi ^i_1\wedge ... \wedge \phi ^i_n)\) by K6 and thus \(\delta _i\) is provably equivalent to a conjunction of \(K(\chi ^i)\) and the negated K-atoms, that is \(\delta _i\dashv \vdash _{EL} K(\chi ^i_1 \vee .. \vee \chi ^i_k) \wedge ... \wedge \lnot K(\psi ^i_{1}) \wedge ... \wedge \lnot K (\psi ^i_{m})\).
If \(\delta _i\) is equivalent to a formula with only one negated atom \(\lnot K (\psi ^i)\), we will iterate the reasoning that follows for each negated atom. Therefore, we can suppose w.l.o.g. that \(\delta _i\) is equivalent to a formula with only one negated atom \(\lnot K (\psi ^i)\) and \(\psi ^i= \psi ^i_1 \vee ... \vee \psi ^i_j\) is a closed disjunctive form.
It suffices to prove that \(K(\chi ^i_1 \vee .. \vee \chi ^i_k)\wedge \lnot K (\psi ^i)\) is a normal K-form. It amounts to prove that either (1) or (2) in Definition 35 is the case. We assume by contradiction that both (1) and (2) are false, thus:
-
1
for all \(\psi ^i_h\) ( with \(h \le j\)), there is a \(\chi ^i_l\) ( with \(l \le k\)), such that \(\psi ^i_h\Subset \chi ^i_l\).
-
2
for all \(\chi ^i_l\) ( with \(l \le k\)), there is a \(\psi ^i_h\) ( with \(h \le j\)), such that \(\psi ^i_h\Subset \chi ^i_l\).
Let f be a function such that for each \(h\le j\) of \(\psi ^i\), \(\psi ^i_h\Subset \chi ^i_{f(h)}\). Then by (E14), and from \(1^*\), it follows that for all \(h\le j\):
\(\chi ^i_{f(1)}\vdash _{AC} \psi ^i_1\)
\(\vdots \)
\(\chi ^i_{f(j)}\vdash _{AC} \psi ^i_j\)
Thus, by (E17), \(\chi ^i_{f(1)} \vee ... \vee \chi ^i_{f(j)} \vdash _{AC} \psi ^i_1\vee ... \vee \psi ^i_j\), which is \(\chi ^i_{f(1)} \vee ... \vee \chi ^i_{f(j)} \vdash _{AC} \psi ^i\). Moreover, let g be a function such that for each \(l\le k\) of \(\chi ^i\), \(\psi ^i_{g(l)}\Subset \chi ^i_l\). Then by (E14), and from 2, it follows that for all \(l\le k\):
\(\chi ^i_1\vdash _{AC} \psi ^i_{g(1)}\)
\(\vdots \)
\(\chi ^i_k\vdash _{AC} \psi ^i_{g(k)}\)
Thus, by (E17), \(\chi ^i_1 \vee ... \vee \chi ^i_k \vdash _{AC} \psi ^i_{g(1)}\vee ... \vee \psi ^i_{g(k)}\) which is \(\chi ^i \vdash _{AC} \psi ^i_{g(1)}\vee ... \vee \psi ^i_{g(k)}\). It follows from \(\chi ^i_{f(1)} \vee ... \vee \chi ^i_{f(j)} \vdash _{AC} \psi ^i\) and \(\chi ^i \vdash _{AC} \psi ^i_{g(1)}\vee ... \vee \psi ^i_{g(k)}\) by (E17) and (E4) that \(\chi ^i \vdash _{AC} \psi ^i\) and thus, \(K(\chi ^i) \vdash _{EL} K(\psi ^i)\) by K1. Then \(\delta ^i\vdash _{EL} K(\psi ^i) \wedge \lnot K(\psi ^i)\), against our assumption that \(\delta ^i\) is consistent. Hence, we can conclude that either (1) or (2) is true and so \(K(\chi ^i)\wedge \lnot K(\psi ^i)\) is a maximal K-form. \(\square \)
1.2 A.2 Syntactic Models and Completeness
In the present section, we will define a ‘canonical model’ in which the states are taken to be sets of the literals of the language, and we can read the epistemic function off an arbitrary consistent K-formula \(\delta \). To be precise, the model we are going to built is not canonical in the sense that it satisfies in a world all the consistent formulas of our language. On the contrary, we will show that for each formula there is a model which satisfies it. It is convenient, then, to call it syntactic model, to distinguish it from a model which verifies all the formulas, because, as mentioned, we will built it up from the set of literals of our language.
Recall that \(\delta \) is provably equivalent to a disjunction, each disjunct of which is a maximal K-form \(K^M(\phi ^i)\):
In particular, as it will became apparent soon, the epistemic function is based on the set of literals in such \(\phi \).
Definition 36
(Syntactic Epistemic Model) A syntactic epistemic model for \(\delta \) is \(\mathfrak {M}^\delta = (S,S^\Diamond , \sqsubseteq , f, |.|^+, |.|^-)\) where:
-
\(S = \mathcal {P}(Lit)\)
-
\(S^\Diamond = \{s\in S \mid \{p,\lnot p\}\not \subseteq s, \textit{ for all } p\in Prop\}\)
-
\(\sqsubseteq = \subseteq \)
-
\(\textsf{dom}(f) = S^\Diamond \) and \(f(s) = \{\bigcup Lit(\phi ^i_j)_{\phi ^i_j\in \Psi } \mid \text { for } \Psi \text { a set of disjuncts in } \phi ^i\}\), for each \(s\in S^\Diamond \).
-
\(|p|^+ = \{\{ p\}\}\), \(|p|^- = \{\{ \lnot p\}\}\), for all \(p\in \mathcal {L}\).
Note that the value of f is defined in the same way for all the possible states in the domain. Moreover, it is easy to see that f(s) is the closure under fusion of the sets of literals \(Lit(\phi ^i_j)\) for \(j\le n\), i.e. \(f(s) = \{ Lit(\phi ^i_j) \mid \) for any \(j\le n\}^f\).
Lemma 24
\(\mathfrak {M}^\delta = (S,S^\Diamond , \sqsubseteq , f, |.|^+, |.|^-)\) is a epistemic model.
Proof
\(\mathfrak {M}^\delta = (S,S^\Diamond , \sqsubseteq ,)\) is W-space, see [5, p. 647]. It suffices to show that f is an epistemic function. It is easy to see that f is a partial function, closed under union by definition. Also, since it is defined in the same way for all \(s\in S^\Diamond \), Condition 3.1 holds. Moreover, for the same reason, every possible s is compatible with a state for which f is defined, namely itself, hence also Condition 3.2 holds. \(\square \)
Note that in the syntactic model the valuation functions pick always a singleton for each propositional letter. On the basis of this definition, we shall prove some results regarding the valuation functions, which will be useful later on.
Lemma 25
Let \(mathfrak{M}^\delta \) be a syntactic model as defined in Def. 36 and \(\lambda _1, ..., \lambda _n\) be literals such that \(\{\lambda _1, ...,\lambda _n\}\in S\), then \(|\lambda _1 \wedge ... \wedge \lambda _n|^+ = \{\{\lambda _1,...,\lambda _n \}\}\).
Proof
Let X be a set of literals, then:
\(\begin{array}{lcl} X \in |\lambda _1 \wedge ... \wedge \lambda _m|^+ &{} \text {iff} &{} X = \bigcup _{i\le m} X_i \textit{ and } X_i\in |\lambda _i|^+ \\ &{} \text {iff} &{} X = \bigcup _{i\le m} X_i \textit{ and } X_i = \{\lambda _i\} \\ &{} \text {iff} &{} X = \{\lambda _1\} \cup ... \cup \{\lambda _m\} \\ &{} \text {iff} &{} X= \{\lambda _1, ... , \lambda _m\} \\ \end{array}\)
It follows that \(|\lambda _1 \wedge ... \wedge \lambda _m|^+ = \{\{\lambda _1, ... , \lambda _m\}\}\). \(\square \)
Lemma 26
(Lemma 26 [41]) In \(\mathfrak {M}^\delta \), for any closed disjunctive form \(\phi = \phi _i \vee ... \vee \phi _n\), \(|\phi |^+ = |\phi |_f^+\). To unpack the definition, since in \(\mathfrak {M}^\delta \), \(\bigsqcup X = \bigcup X\), we have that
Recall that we proved (Lemma 23) that a conjunction \(K\phi \wedge \lnot K \psi _1 \wedge \dots \wedge \lnot K\psi _m\) is inconsistent in our system if there is a \(\psi _i\) (\(i\le m\)) such that every disjunct in \(\phi \) has a sub-description that is a disjunct in \(\psi _i\), and every disjunct in \(\psi _i\) is a sub-description of a disjunct in \(\phi \). The next step is to show that the previous conjunction is satisfied by the syntactic model. The idea of the proof is that, in building the syntactic model, we picked a possible state s and let f(s) be the set of verifiers of \(\phi \), and since the conjunction is consistent, we can see that no \(\psi _i\) has a set of verifiers analytically entailed by \(|\phi |^+\). So the state s verifies \(K\phi \) and falsifies each \(K\psi _i\).
In the proof of following theorem, we will spell out all the details of the strategy just sketched.
Lemma 27
Every consistent epistemic formulas \(\delta \) is satisfiable with respect to the class of epistemic models.
Proof
Suppose the K-formula \(\delta \) is consistent. By Lemma 23, \(\delta \) is provably equivalent to a normal K-form \(\delta _1 \vee \delta _2 \vee ...\vee \delta _m, \) for \(m \ge 1\), where each \(\delta _i\) is of the form \(K^M(\phi ^i)\). The syntactic model for \(\delta \) is \(\mathfrak {M}^\delta = (S,S^\Diamond , \sqsubseteq , f, |.|^+, |.|^-)\), defined as in Definition 36. It suffices to show that \(\delta _i\) (for some \(i\le m)\) is true in the model and consequently so is \(\delta \). Let \(\delta _i= K (\phi ^i_1 \vee ... \vee \phi ^i_n) \wedge \lnot K(\psi ^i_{1}) \wedge ... \wedge \lnot K(\psi ^i_{m})\).
In what follows, I will drop the superscript i for readability purposes, and I will consider as before only one negated K-atom, without loss of generality.
Consider an arbitrary state \(s\in S^\Diamond \). By definition of the syntactic model, that there is a possible world w, such that \(s\subseteq w\).
We first prove that \(\mathfrak {M}^\delta ,w \models K(\phi _1 \vee ... \vee \phi _n)\), namely that \(s\Vdash K(\phi _1 \vee ... \vee \phi _n)\), i.e. \(|\phi _1\vee ... \vee \phi _n|^+\preceq f(s)\). Given Lemma 26, and the definition of f(s) in \(\mathfrak {M}^\delta \), we know that \(|\phi _1\vee ... \vee \phi _n|^+ = |\phi _1\vee ... \vee \phi _n|_f^+ = f(s)\), proving immediately our claim.
Secondly, we shall prove that \(\mathfrak {M}^\delta ,w \models \lnot K(\psi )\). Recall the definition of maximal K-form, \(\lnot K(\psi )\) is such that \(\psi =\psi _{1} \vee ... \vee \psi _{m}\) is in closed disjunctive form and:
-
1.
either there is \(\psi _i\) (\(i\le m\)) such that for all \(\phi _j\), (\(j\le n \))
;
-
2.
or there is \(\phi _j\) (\(j\le n \)) such that for all \(\psi _i\) (\(i\le k\)),
;
Suppose (1) is the case. Since both \(\psi _i\) and \(\phi _j\) are descriptions, and , then \(Lit(\psi _i)\not \subseteq Lit(\phi _j)\), for all \(\phi _j\). Since \(\phi \) is a closed normal form, it includes a disjunct \(\phi _l\) such that \(Lit (\phi _l)= \bigcup \{Lit(\phi _j) \mid \phi _j\in \Psi \}\) where \(\Psi \) is a set of disjuncts in \(\phi \). Hence, also \(Lit(\psi _i)\not \subseteq \bigcup \{Lit(\phi _i) \mid \phi _i\in \Psi \}\), for any choice of \(\Psi \). Thus, take an arbitrary \(L\in f(s)\), it is of the form \(\bigcup \{Lit(\phi _i) \mid \phi _i\in \Psi \}\). It follows that, \(Lit(\psi _i)\not \subseteq L\). Hence, since L was arbitrary and \(Lit(\psi _i) \in |\psi |^+\) (Lemma 26), we can conclude that for each \(L\in f(s)\), there is \(L'\in |\psi |^+\) , such that \(L'\not \subseteq L\), i.e. \(|\psi |^+\not \preceq f(s)\).
Suppose (2) is the case. Since both \(\psi _i\) and \(\phi _j\) are descriptions, and , then \(Lit(\psi _i)\not \subseteq Lit(\phi _j)\), for all \(\psi _i\). Take an arbitrary \(L\in |\psi |^+\), which will be of the form \(L = \bigcup _{i\le n} Lit(\psi _i)\). If \(n=1\), there is a \(Lit(\phi _j)\in f(s)\), such that \(Lit(\psi _i)\in |\psi |^+\) and \(Lit(\psi _i)\not \subseteq Lit(\phi _j)\). Hence, since \(\psi _i\) was arbitrary, \(|\psi |^+\not \preceq f(s)\). If \(n>1\), it follows from 2 that \(\bigcup _{i\le n} Lit(\psi _i) \not \subseteq Lit(\phi _j)\), and \(\bigcup _{i\le n} Lit(\psi _i)\in |\psi |^+\), by Lemma 26. Hence for the previous reasoning we can conclude that \(|\psi |^+\not \preceq f(s)\). In both cases, \(|\psi |^+\not \preceq f(s)\), so we can conclude that \(\mathfrak {M}^\delta , w\not \models K(\psi )\). \(\square \)
Recall that we called Lit the set of all the propositional letters p and their negation. Let us now call \(Lit_k\) the set of K-literals, namely all the K-atoms. Given classical logic, we know that each \(\alpha \) in \(\mathcal {L}_e\) is provably equivalent to a formula in disjunctive normal form, i.e. \(\alpha \dashv \vdash _{EL} \alpha _1\vee \dots \vee \alpha _n\), where each disjunct is a conjunction of literals either in Lit or in \(Lit_k\). Moreover, we know from Lemma 23, that a conjunction of K-literal is provably equivalent to a maximal K-form \(K^M(\phi )\). Let us call the conjunction of propositional literals \(\Lambda \). Hence, we say w.l.o.g. that each \(\alpha \in \mathcal {L}_e\) is of the form \(\bigvee (\Lambda \wedge K^M(\phi ))\).
Lemma 28
Every consistent formula \(\alpha \in \mathcal {L}_e\) is satisfiable with respect to the class of epistemic models.
Proof
By the previous reasoning \(\alpha \) is provably equivalent to a disjunctive normal form. Since it is consistent, there must be at least a consistent disjunct \(\alpha _i\), which is of the form \(\Lambda \wedge K^M(\phi )\). Since \(\alpha _i\) is consistent then also \(\Lambda \) and \(K^M(\phi )\) are both consistent. From Lemma 27 we know that, for each consistent \(K^M(\phi )\) we can build a syntactic epistemic model that makes it true, call it \(\mathfrak {M}^\delta \). In particular, we know that for all \(s\in S^\Diamond \), \(s\Vdash K^M(\phi )\). Moreover, there is also a \(s' \in S^\Diamond \), that is the set of propositional literals in \(\alpha _i\), i.e. \(s' = Lit(\Lambda )\). Accordingly, \(s'\Vdash \Lambda \wedge K^M(\phi )\). Since \(\mathfrak {M}^\delta \) is an epistemic model, there is a world w such that \(s'\subseteq w\). Hence \(\mathfrak {M}^\delta , w\models \Lambda \wedge K^M(\phi )\) and thereby \(\mathfrak {M}^\delta , w\models \alpha \). \(\square \)
Theorem 29
(Completeness) The system EL is complete: for any epistemic model \(\mathcal {M}\) and for any \(\phi \in \mathcal {L}_e\), if \(\mathcal {M}\models \phi \), then \(\vdash _{EL} \phi \).
Proof
The proof is a straightforward consequence of the previous lemma. \(\square \)
1.3 A.3 Convexity
As we have mentioned, the adoption of regular propositions does not entail any difference on the resulting logic of analytic containment and analytic equivalence, which is the logic AC. On the other hand, closure under convexity is philosophically interesting, because it makes analytic containment antisymmetric, which is arguably a desideratum for a natural characterization of analytic containment. Given these considerations, we might want to consider the introduction of convexity as a closure principle for the epistemic function as well. In this case, we would need to adapt the completeness proof to this new feature, in particular the syntactic model.
The relevant subclass of disjunctive forms in this case is the one that Fine [10] calls maximal. I will call it the regular disjunctive form, in order to avoid mix-ups with the maximal K-form of Definition 35.
Definition 37
(Regular disjunctive form) A disjunctive form \(\phi \) is regular iff for any disjunct \(\phi _i\) in \(\phi \) and any literal \(\lambda \) occurring as a conjunct in a disjunct of \(\phi \), \(\phi \) contains a disjunct \(\phi _j\) with \(Lit(\phi _j) = Lit(\phi _i)\cup \{ \lambda \}\).
Lemma 30
(Lemma 17 in [10]) Every disjunctive form is provably equivalent, within AC, to a regular disjunctive form.
Lemma 31
Every consistent K-formula \(\delta \) is provably equivalent to a normal K-form \(\delta _1 \vee ... \vee \delta _m\), where each \(\delta _i\) is a maximal K-formula, i.e. \(\delta _i\) is of the form \(K^M (\phi )\), and additionally \(\phi \) is a regular normal form.
Proof
By Lemmas 23 and 30. \(\square \)
Definition 38
A syntactic regular epistemic model \(\mathfrak {M}_R^\delta \) for \(\delta \) is a syntactic epistemic model \(\mathfrak {M}^\delta = (S,S^\Diamond , \sqsubseteq , f', |.|^+, |.|^-)\), where \(f'(s) = f(s)^r\), where f(s) is defined as in Definition 36, i.e. \(f'(s)\) is closed under fusion and convexity.
It is easy to check that \(\mathfrak {M}_R^\delta \) is an epistemic model. Moreover, it can be shown that for all regular disjunctive forms \(\phi \), \(|\phi |^+ = |\phi |^+_r\) (see [41, p. 30]). Then, the proof of the completeness of the EL system is an adaptation of the previous one.
Appendix B Factivity
In this Appendix we prove the completeness theorem of the logic EL\(^+\) with respect to the class of factive (Theorem 20) epistemic models.
Theorem 32
(Soundness) For all \(\phi ,\psi \in \mathcal {L}_e\), if \(\phi \vdash _{EL^+} \psi \), then \(\phi \models \psi \).
Proof
K7. The proof is analogous to the right-left direction of Lemma 10. K8. It derives from K7, because, suppose by contradiction that \(\vdash _{EL^+} K(\phi \wedge \lnot \phi )\), then by K7, \(K(\phi \wedge \lnot \phi )\vdash _{EL^+} \phi \wedge \lnot \phi \), which contradicts classical logics. \(\square \)
Now, we just need to adapt the previous results concerning the completeness of EL to the new extended logic. It turns out that a syntactic model with a reflexive epistemic function looks quite different from the previous one and we need to build a slightly more sophisticated structure.
We need to be able to talk of the negations of the literals, both the positive and negative ones. Therefore, with each literal \(\lambda \in Lit\), we associate a unique shadow \(\bar{\lambda }\), which acts as follows: if \(\lambda = p\), then \(\bar{\lambda } = \lnot p\); if \(\lambda = \lnot p\), then \(\bar{\lambda } = p\).
Lemma 33
Let \(K^M(\phi )\) be a consistent maximal K-form in \(\mathcal {L}_e\). Since \(\phi = \phi _1 \vee ... \vee \phi _n\), there is at least a disjunct \(\phi ^*\) in \(\phi \) such that , for any \(p\in \mathcal {L}\).
Proof
Suppose by contradiction that for each disjunct \(\phi _i\) (\(i\le n\)) in \(\phi \), there is some \(p^i\in Lit\), such that \(p^i\wedge \lnot p^i \Subset \phi _i\). Hence \(\phi _1\vee ... \vee \phi _n \vdash _{AC} (p^i\wedge \lnot p^i) \vee ... \vee (p^n\wedge \lnot p^n)\), by E14 and E17. Hence, \(K(\phi )\vdash _{EL^+} K( (p^i\wedge \lnot p^i) \vee ... \vee (p^n\wedge \lnot p^n))\), by K1. Therefore,
which contradicts the fact that \(K^M(\phi )\) is consistent. Hence, there is at least a disjunct \(\phi _i\) in \(\phi \) which does not have any contradicting atomic letters as sub-description. \(\square \)
Since \(\phi ^*\) is a description, then it follows that \(Lit(\lambda \wedge \bar{\lambda }) \not \subseteq Lit(\phi ^*)\), for any \(\lambda \in Lit\) which is equivalent to say that \(\{\lambda ,\bar{\lambda } \}\not \subseteq Lit (\phi ^*)\).
Our goal, now, is to read an epistemic syntactic model out of this consistent description \(\phi ^*\). In particular, the set of possible states of our syntactic model will be based on the set of literals in \(\phi ^*\).
A syntactic state space is a tuple \(\mathfrak {S} = (S, \sqsubseteq )\), where \(S= \mathcal {P}(Lit)\) and \(\sqsubseteq = \subseteq \). Now, consider the state \(s = Lit(\phi ^*)\). We shall construct a maximally consistent state w(s), namely a possible world, which contains s as a subset. In other words, we will extend s to a possible world w(s) and we will adopt it as a basis for the set of possible states \(S^\Diamond \) in our syntactic model.
Definition 39
In a syntactic state space \(\mathfrak {S} = (S, \sqsubseteq )\), let s be a state in S, s is consistent when for any \(\lambda \in Lit\), \(\{\lambda ,\bar{\lambda } \}\not \subseteq s\). It is inconsistent otherwise.
As we have already discussed, Lemma 33 shows that \(s = Lit (\phi ^*)\) is consistent. From this consistent state we will build a maximal state, namely a state that, for all atoms \(p\in Prop\), either contains a part which verifies p or contains a part which verifies \(\lnot p\). To do so, it is necessary to be able to talk of every literal of the language. Since they are countable, we can simply enumerate them, by associating them with an index \(i\in \mathbbm {N}\).
Let s be our base case, thus \(s = s_0\) and consider the first \(\lambda _0\) of our enumerated list of literals in Lit. Then, the following state \(s_1\) will be \(s_1 = s_0\cup \{\lambda _0\}\), just in case \(\bar{\lambda }_0 \not \in s_0\), and \(s_1 = s_0\) otherwise. Indeed, if \(\bar{\lambda }_0 \in s_0\), we cannot build a new consistent state \(s_1\) by adding \(\lambda _0\) to \(s_0\), because we would obtain \(\{\lambda _0,\bar{\lambda }_0\} \subseteq s_1\), i.e. \(s_1\) would be inconsistent.
We shall replicate the same operation, until we obtain a \(w(s) = \bigcup \limits _{n\in \mathbbm {N}} s_n \), such that nothing more can be added without thereby resulting in an impossible state. In symbols:
Lemma 34
w(s) is a consistent state.
Proof
w(s) is obviously a state, because it is a set of literals. Moreover it is consistent by construction. \(\square \)
In order to check whether w(s) is a possible world (and not just a consistent state), we need first to define our syntactic W-space. As before, consider \(\delta \), which is provably equivalent to a disjunction, each disjunct of which is a maximal K-form \(K^M(\phi ^i)\):
We fix additionally \(\phi ^*\) to be a consistent disjunct in \(\phi ^i\). Recall that we say that \(X^\downarrow \) is the smallest downwards closed set (w.r.t. parthood) containing X.
Definition 40
A syntactic factive epistemic space for \(\delta \) is \(\mathfrak {S}_F^\delta = (S,S^\Diamond , \sqsubseteq , f)\), where
-
\((S,\sqsubseteq )\) is a syntactic state space;
-
\(S^\Diamond = \{w(s)\}^\downarrow \) ;
-
\(\textsf{dom}(f) =\{s\}\), where \(s= Lit(\phi ^*)\) and \(f(s) = \{ Lit(\phi ^i_j) \mid \) for any \(j\le n\}^f\).
Lemma 35
\(\mathfrak {S}_F^\delta = (S,S^\Diamond , \sqsubseteq , f)\) is a factive epistemic space.
Proof
The first step it to prove that \((S,S^\Diamond , \sqsubseteq )\) is a W-space. \(S^\Diamond \) is a non empty subset of S and it is closed under parthood by construction. Hence, it suffices to show that w(s) is a possible world, namely it contains every state with which it is compatible, then, we shall check that every possible state in \(S^\Diamond \) is part of w(s). Consider an arbitrary \(t'\) such stat \(t'\cup w(s) \in S^\Diamond \). Then, by construction of \(S^\Diamond \), \(t'\subseteq w(s)\). This shows that w(s) contains all its compatible states. Now, suppose t is possible, i.e. \(t\in w(s)^\downarrow \), then again \(t\subseteq w(s)\) by construction, which completes the proof.
Secondly, we shall check that the function f is indeed an epistemic function. Since it is defined only on the state s, it vacuously satisfies the Compatibility Condition 3.1.
Moreover, take an arbitrary \(t\in S^\Diamond \), it suffices to show that t is compatible with s. Since \(t\in S^\Diamond \), then \(t\subseteq w(s)\). Since also \(s\subset w(s)\), then, \(t\cup s \subseteq w(s)\), and so \(t\cup s\) is in \(S^\Diamond \). This shows that every possible state is compatible with a state s such that \(s\in \textsf{dom}(f)\), thus satisfying the definability Condition 3.2. Lastly it is easy to check that \(s\in f(s)\) (reflexivity). \(\square \)
Definition 41
A syntactic factive epistemic model for \(\delta \) is \(\mathfrak {M}_F^\delta = (S,S^\Diamond , \sqsubseteq , |.|^+, |.|^-)\) where:
-
\((S,S^\Diamond , \sqsubseteq , f)\) is a syntactic factive epistemic space.
-
\(|p|^+ = \{\{ p\}\}\), \(|p|^- = \{\{ \lnot p\}\}\), for all \(p\in \mathcal {L}\).
Lemma 36
\(\mathfrak {M}_F^\delta = (S,S^\Diamond , \sqsubseteq , |.|^+, |.|^-)\) is a W-model.
Proof
It suffices to show that the evaluation functions satisfy the conditions of exclusivity and exhaustivity and closure.
-
Exclusivity. Consider arbitrary \(t\in |p|^+\) and \(t'\in |p|^-\), which amounts to saying that \(t =\{p\}\) and \(t' = \{ \lnot p\}\). It suffices to show that \(t\cup t' \not \in S^\Diamond \). If it was \(\{p,\lnot p\} \subseteq w(s)\), which is contrary to the fact that w(s) is possible.
-
Exhaustivity. Since w(s) is the maximal element in \(S^\Diamond \) it is also the only possible world of the model. Hence, it suffices to check that for all \(p\in \mathcal {L}\), either there is a \(t\subseteq w(s)\) such that \(t\in |p|^+\) or there is a \(t'\subseteq w(s)\) such that \(t' \in |p|^-\). Suppose this is not the case, by contradiction. Then, there is a p such that, for all \(t\subseteq w(s)\), \(t\not \in |p|^+\) and \(t\not \in |p|^-\), which means that \(t\ne \{p\}\) and \(t\ne \{\lnot p\}\). However, since p is a literal there must be a number \(i\in \mathbbm {N}\), such that \(p =\lambda _i\) and \(\lnot p =\bar{\lambda _i}\). Hence, there is a \(s_i \subseteq w(s)\), such that either \(\lambda _i \in s_{i+1}\) or \(\bar{\lambda _i }\in s_{i+1}\). In the former case, \(\{\lambda _i\} \subseteq s_{i+1} \subseteq w(s)\), in the latter case \(\{\bar{\lambda _i }\} \subseteq s_{i+1} \subseteq w(s)\). In both cases, we contradict the assumption.
-
Closure is trivially true.
\(\square \)
Lemma 37
Every consistent epistemic formulas \(\delta \) is satisfiable with respect to the class of factive epistemic models.
Proof
Consider a factive epistemic syntactic model \(\mathfrak {M}^\delta _F\) as in Definition 41. It suffices to check that s is a possible state, but this follows directly from Lemma 33. Thus, there is a possible world, namely w(s), such that \(s\subseteq w(s)\). The proof, then, is analogous to the one of Theorem 27, hence we can conclude that \(\mathfrak {M}_F^\delta ,w(s)\models K^M (\phi )\). \(\square \)
As in the previous proof, we need now to make sure that every formula \(\alpha \in \mathcal {L}_e\) is satisfiable with respect to the class of epistemic factive models, and not only the K-formulas. Since our syntactic factive model consist of only one possible world, it is less trivial then the previous case to show that there is a possible state for each selection of literals compatible with each K-formula. In this case we need to build a different possible world, and thus a different epistemic state space, for each \(\alpha \) we consider. Indeed, recall that we can say w.l.o.g. that each \(\alpha \) is of the form of \(\bigvee (\Lambda \wedge K^M(\phi )) \), where \(\Lambda \) is a certain description (of propositional literals) and \(K^M(\phi )\) is an arbitrary maximal normal form.
Now, in the construction of the syntactic factive model we need to chose the one which includes \(s' = Lit(\Lambda )\) as a subset. This is a legit choice as long as \(s'\) is compatible with our initial \(s_0\), namely \(\{\lambda , \bar{\lambda }\}\not \subseteq s\cup s'\) for all \(\lambda \in Lit\). Recall that \(s_0 = Lit(\phi ^*)\), where \(\phi ^*\) is one of the disjuncts in \(\phi \) which is consistent. Note that we can assume for simplicity, without loss of generality, that each \(\phi \), contains only one consistent disjunct. In fact, if there were many (and they excluded each other), we would just need to construct different possible words, each starting from the state corresponding to the set of literals of a different consistent \(\phi _i\) in \(\phi \).
Lemma 38
For all consistent disjunctive normal forms \(\alpha \in \mathcal {L}_e\), there is one of its disjuncts \(\alpha _i\) which is consistent and such that, for all \(\lambda \in Lit\), \(\{\lambda , \bar{\lambda }\}\not \subseteq Lit(\Lambda )\cup Lit (\phi ^*)\).
Proof
That \(\alpha _i\) exists follows by classical logic and it implies that \(\Lambda \) is consistent, and so is \(K^M(\phi )\). As we showed in Lemma 33, since \(K^M(\phi )\) is a consistent maximal K-form, then there is at least a disjunct \(\phi ^*\) in \(\phi \) such that \(\{\lambda , \bar{\lambda }\} \not \subseteq Lit(\phi ^*)\), for any \(\lambda \in Lit\). Hence, it suffices to check that it is not the case that there is a \(\lambda \in Lit(\Lambda )\) and \(\bar{\lambda } \in Lit (\phi ^*)\). Suppose by contradiction that this was the case. Then, by E14 and classical logic, \(\phi ^* \wedge \Lambda \vdash _{EL^+} \lambda \wedge \bar{\lambda }\). Moreover, since we assume that \(\phi ^*\) is the only consistent disjunct in \(\phi \), by K7 we get \(K^M(\phi )\vdash _{EL^+} \phi ^*\). Hence,
contradicting our assumption that \(\alpha _i\) was consistent. Then we can conclude that for all \(\lambda \in Lit\), \(\{\lambda , \bar{\lambda }\}\not \subseteq Lit(\Lambda )\cup Lit (\phi ^*)\). \(\square \)
This lemma shows that, for each formula \(\alpha \) there is in fact a selection of literals, i.e. a state of literals \(s'\), which is compatible with \(s_0 = Lit(\phi ^*)\), so that we can build a possible world up from it and including \(s'\) in a consistent way. The resulting possible world will make true \(\Lambda \) and \(K^M(\phi )\), proving immediately the following lemma:
Lemma 39
Every consistent formula \(\alpha \in \mathcal {L}_e\) is satisfiable with respect to the class of factive epistemic models.
Proof
The proof is analogous to the one of Lemma 28, given the adequate choice of syntactic factive model, as discussed above. \(\square \)
Theorem 40
(Completeness) The system EL is complete: for any factive epistemic model \(\mathcal {M}^F\) and for any \(\phi \in \mathcal {L}_e\), if \(\mathcal {M}^F\models \phi \), then \(\vdash _{EL^+} \phi \).
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Saitta, V. A Truthmaker-based Epistemic Logic. J Philos Logic (2024). https://doi.org/10.1007/s10992-024-09758-3
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DOI: https://doi.org/10.1007/s10992-024-09758-3