Logics for Knowability Paradox with a Non-normal Possibility Operator

Abstract

This paper studies a Glivenko–Kuroda translation in logics for knowability paradox in terms of sequent calculi. Hilbert-style axiomatization and Gentzen-style sequent calculus are proposed for an intuitionistic modal logic with and without the knowability axiom. When the system does not contain the knowability axiom, it is shown to be sound and complete for an intuitionistic version of a combination of Kripke and neighborhood semantics. We define a double negation translation from our intuitionistic modal logics to the corresponding classical modal logics to establish syntactic proofs of Glivenko-style embedding results with and without the knowability axiom. In the setting without the knowability axiom, our embedding result enables us to reduce the cut-elimination theorem, Craig interpolation theorem, and decidability from the classical setting to the intuitionistic setting.

This chapter is in its final form and it is not submitted to publication anywhere else.

Appendix: Strong Completeness of \(\mathbf {HK\Diamond c}\)

Definition 14

The classical system of $\mathbf {HK\Diamond c}$ can be obtained by adding to the $\mathbf {HK\Diamond i}$ the law of excluded middle:

(LEM)     $A\vee \lnot A$.

Moreover, $\mathbf {HK\Diamond c}^+$ can be obtained by adding the (KP)$A\supset \Diamond KA$ to $\mathbf {HK\Diamond c}$.

The notion of a derivation in these systems is defined similarly to those of the intuitionistic systems.

Theorem 11

For any formula A, $\vdash _{\mathbf {HK\Diamond c}} A$ iff $\,\vdash _{\mathbf {G1\Diamond c}} \Rightarrow A$. And $\,\vdash _{\mathbf {HK\Diamond c}^+} A$ iff $\,\vdash _{\mathbf {G1K\Diamond c}^+} \Rightarrow A$.

Definition 15

We say that a model $M=(W,\le ,R,\tau ,V)$ is classical if the preorder $\le$ is the identity relation $=$ (as a graph) on W. We use $\mathbb {C}_{\mathrm {all}}$ to denote the class of all classical models.

It is immediate that for a classical model $M=(W,=,R,\tau ,V)$ the conditions of $=;R\subseteq R$, (NE), (DEC) and the persistency condition hold trivially. Moreover, $=\subseteq R$ gives the reflexivity of R. It is obvious that the truth condition for $A\supset B$ becomes

$$ \begin{array}{lll} M, w\models A\supset B&\,\mathrm {iff}\,&M,w\models A ~\text {implies}~ M,w\models B. \end{array} $$

Theorem 12

If $\vdash _{\mathbf {HK \Diamond c}} A$ then $\models _{\mathbb {C}_{\mathrm {all}} } A$, for all formulas A.

Proof

It is sufficient to show that the law of excluded middle is valid in $\mathbb {C}_{\mathrm {all}}$. The proof is straightforward.   $\square$

The notion of a consistent prime theory in $\mathbf {HK\Diamond c}$ can be similarly defined to that of $\mathbf {HK\Diamond i}$.

Lemma 8

Let $\varGamma \cup \{ B\}$ be a set of formulas. If $\varGamma \nvdash _{ \mathbf {HK \Diamond c} } B$ then there exists a consistent prime theory $\varGamma ^+$ such that $\varGamma \subseteq \varGamma ^+$ and $\varGamma ^+ \nvdash _{ \mathbf {HK \Diamond c} } B$ and $\bot \notin \varGamma ^+$.

By the law of the excluded middle, the following holds immediately.

Proposition 10

Given a consistent prime theory $\varGamma$ in $\mathbf {HK \Diamond c}$, either $A\in \varGamma$ or $\lnot A\in ~\varGamma$, for any formula A.

Definition 16

The canonical model $M^\mathbf {HK\Diamond c}= (W,=, R, \tau , V)$ is defined as follows:

• $W:= \{ \varGamma : \varGamma$ is a consistent prime theory in $\mathbf {HK\Diamond c}$}.

• $\varSigma R \varPi$ iff $KA\in \varSigma$ implies $A \in \varPi$ for all formulas A.

• $\tau (\varGamma ) = \{ \overline{|C|}: \Diamond C \notin \varGamma \}$, where $\overline{|C|} = \{ \varDelta \in W: C \notin \varDelta \}$.

• $\varSigma \in V(p)$ iff $p\in \varSigma$.

Proposition 11

The canonical model $M^\mathbf {HK\Diamond c}= (W,=, R, \tau , V)$ is a classical model, i.e., $M^\mathbf {HK\Diamond c}\in \mathbb {C}_{\mathrm {all}}$.

Proof

It is sufficient to show that: $= \subseteq R$ and $\tau (\varGamma ) \ne \emptyset$ for all $\varGamma \in W$. These can be easily checked.

Lemma 9

For an arbitrary formula A and a consistent prime theory $\varGamma$ in $\mathbf {HK\Diamond c}$,

$M^{\mathbf {HK\Diamond c}},\varGamma \models A$ iff $A \in \varGamma$.

Proof

We proceed by induction on the complexity of A. We only show the case of $B\supset C$, since the other cases are shown similarly to the proof of Lemma 2. Suppose that $B\supset C \in \varGamma$. Then we show that $M^{\mathbf {HK\Diamond c}},\varGamma \models B\supset C$. It is sufficient to show that $M,\varGamma \models B$ implies $M,\varGamma \models C$. Suppose that $M,\varGamma \models B$, we have $B\in \varGamma$ by induction hypothesis. From the supposition $B\supset C \in \varGamma$, we have $C\in \varGamma$. Then $M,\varGamma \models C$ by induction hypothesis.

Conversely, suppose that $B\supset C \notin \varGamma$. We show $M^{\mathbf {HK\Diamond c}},\varGamma \nvDash B\supset C$. It is sufficient to show $M^{\mathbf {HK\Diamond c}},\varGamma \models B$ and $M^{\mathbf {HK\Diamond c}},\varGamma \nvDash C$. Since $C\supset (B \supset C)\in \varGamma$, we have $C\notin \varGamma$. Since $\lnot B \supset (B\supset C)$, it cannot be the case that $\lnot B\in \varGamma$. By Proposition 10, we have $B\in \varGamma$. From $B\in \varGamma$ and $C\notin \varGamma$, we have $M^{\mathbf {HK\Diamond c}},\varGamma \models B$ and $M^{\mathbf {HK\Diamond c}},\varGamma \nvDash C$ by induction hypothesis.

Theorem 13

(Strong Completeness of $\mathbf {HK\Diamond c}$) If $\varGamma \models _{\mathbb {C}_{\mathrm {all}}} A$ then $\varGamma \vdash _{\mathbf {HK\Diamond c}} A$, for every set $\varGamma \cup \{A\}$ of formulas.