# From forcing to satisfaction in Kripke models of intuitionistic predicate logic

From forcing to satisfaction in Kripke models of intuitionistic predicate logic Abstract In this paper, we answer a natural question concerning the relation between forcing and satisfaction in Kripke models of intuitionistic predicate logic. We define a class of formulas denoted by $$\mathsf{R}^{*}$$, with the property that forcing of any $$\mathsf{R}^{*}$$-formula in a node of a Kripke model of intuitionistic predicate logic implies its satisfaction in the classical structure attached to that node. We also prove that any formula with this property is an $$\mathsf{R}^{*}$$-formula. 1 Introduction The relationship between classical and intuitionistic satisfiability of formulas in Kripke models of intuitionistic predicate logic and also intuitionistic arithmetic gives rise to challenging questions. We review some of these questions below. A node of a Kripke model of Heyting arithmetic HA is called PA-normal, if the classical world attached to it, satisfies Peano arithmetic PA. Now the natural question one can ask is this: is every Kripke model of HA, PA-normal? In [3], it is shown that every finite Kripke model of HA is PA-normal. Also it is proved that models of HA over the frame $$(\omega , \leq )$$ contains infinitely many Peano nodes. In [9], it is shown that such Kripke models are in fact PA-normal. The problem in general is still open. Is every PA-normal Kripke model a model of HA? The answer is ‘No’. In [1], a PA-normal Kripke model over the frame $$(\omega , \leq )$$ is constructed which does not force HA. In [7], a two-node Kripke model with this property is constructed. Moreover, in [1], for each set T of sentences, an intuitionistic theory $$\mathcal{H}(T)$$ is introduced. This intuitionistic theory is exactly the class of all formulas which are forced in each T-normal Kripke model. In [1], it is shown that $$\mathcal{H}(\mathsf{PA})\neq \mathsf{HA}$$. More generally, similar questions can be asked about Kripke models of intuitionistic predicate logic. In [6], the class of all formulas with the property that their forcing and satisfaction are equivalent in any node (world) is introduced. Moreover, in [5], a class of formulas is defined such that, whenever a formula of that class is forced at a node of a Kripke model, it must be satisfied in the classical structure associated with that node. One may say that this class is sound with respect to the mentioned property. In this paper, we improve this result. We define a class of $$\mathsf{R}^{*}$$-formulas and prove that it is exactly the class of all formulas which have this property, i.e. the class is sound and complete with respect to this property. 2 Preliminaries We consider a first-order language $$\mathcal{L}$$ to be the set of all formulas that can be built from a symbol set (relation, function and constant symbols and variables) using ⊤, ⊥, ∧, ∨, $$\rightarrow$$, ∃ and ∀. Symbols ⊤ and ⊥ are both atoms and nullary connectives. $$\neg \varphi$$ is short for $$\varphi \rightarrow \bot$$ and $$\varphi \leftrightarrow \psi$$ is short for $$(\varphi \rightarrow \psi )\wedge (\psi \rightarrow \varphi )$$. A list of terms $$t_{1}, \ldots , t_{n}$$ is abbreviated as $$\overline{t}$$. If C is an arbitrary set of constant symbols, then $$\mathcal{L}(C)$$ is the language $$\mathcal{L}$$ extended by all constant symbols in C. $$\mathsf{At}\subseteq \mathcal{L}$$ is the set of atomic formulas in $$\mathcal{L}$$. Analogously, $$\mathsf{At}(C)\subseteq \mathcal{L}(C)$$ is the set of atomic formulas in $$\mathcal{L}(C)$$. Let $$\Gamma$$ be a set of formulas in $$\mathcal{L}$$. By $$\Gamma (C)$$, we mean all $$\mathcal{L}(C)$$-sentences of the form $$\varphi (\overline{c})$$, where $$\varphi (\overline{x})$$ is a formula in $$\Gamma$$ and $$\overline{c}\in C$$. We use the symbols $$\vdash _{i}$$ and $$\vdash _{c}$$ to denote derivability in the intuitionistic and classical predicate calculus, respectively. If $$\Gamma \subseteq \mathcal{L}(C)$$ is a set of sentences, then we define $$\mathsf{Th}_{i}[C](\Gamma )=\{\varphi \in \mathcal{L}(C): \Gamma \vdash _{i}\varphi \}$$ to be the deductive closure of $$\Gamma$$ over $$\mathcal{L}(C)$$. By a theory, we mean a set of sentences closed under deduction. The classical theory $$\mathsf{Th}_{c}[C](\Gamma )$$ is defined in a similar way as $$\{\varphi \in \mathcal{L}(C): \Gamma \vdash _{c}\varphi \}$$. An intuitionistic theory $$\Gamma$$ over $$\mathcal{L}(C)$$ is called prime if for all sentences $$\varphi ,\psi \in \mathcal{L}(C)$$, we have $$\Gamma \vdash _{i}\varphi \vee \psi$$ if and only if $$\Gamma \vdash _{i}\varphi$$ or $$\Gamma \vdash _{i}\psi$$. An intuitionistic consistent theory $$\Gamma$$ over $$\mathcal{L}(C)$$ is called C- Henkin if for all sentences of the form $$\exists x \,\varphi (x)\in \mathcal{L}(C)$$, we have $$\Gamma \vdash _{i}\exists x \,\varphi (x)$$ if and only if there is c ∈ C such that $$\Gamma \vdash _{i}\varphi (c)$$. A theory is called C- Henkin prime if it is both C-Henkin and prime. These notions can be defined for classical theories in a similar way. A Kripke model $$\mathcal{A}$$ in the language $$\mathcal{L}$$, is a pair $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$ such that (F, ≤) is a partially ordered set (the frame of $$\mathcal{A}$$) and to each element (node) $$\alpha$$ of F is attached a classical structure $$\mathcal{A}_{\alpha }$$ for $$\mathcal{L}$$ in which the interpretation of equality is an equivalence relation which may properly extend the real equality. For each two nodes $$\alpha ,\beta$$, if $$\beta$$ is accessible from $$\alpha$$ (i.e. $$\alpha \leq \beta$$), then the world at $$\alpha$$ must be a weak substructure of the one at $$\beta$$. By this we mean that $$\mathcal{A}_{\beta }$$ preserves truth in $$\mathcal{A}_{\alpha }$$ of atomic sentences in $$\mathcal{L}(A_{\alpha })$$. The forcing relation is defined as usual. A Kripke model is a $$\Gamma$$-Kripke model if it forces the axioms of $$\Gamma$$ at all nodes. The calligraphic letters $$\mathcal{A}$$, $$\mathcal{B}$$, $$\mathcal{C}, \ldots$$ represent either classical models or Kripke models. If $$\mathcal{A}$$ is a classical model, then the domain of $$\mathcal{A}$$ is denoted by the corresponding Latin letter A, and $$\mathcal{L}(A)$$ is the language $$\mathcal{L}$$ extended by a new constant symbol for every element in A. The symbol ⊧ denotes classical satisfaction in a model and is defined for sentences (closed formulas) only. Definition 2.1 Let $$\varphi (\overline{x})$$ be a formula in the language $$\mathcal{L}$$. We say that $$\varphi (\overline{x})$$ is f-stable (forcing stable) if and only if for any Kripke model $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$, any $$\alpha \in F$$ and any $$\overline{a}\in A_{\alpha }$$, we have $$\alpha\Vdash_{{\mathcal{A}}}\varphi(\overline{a})\Longrightarrow \mathcal{A}_{\alpha}\models\varphi(\overline{a}).$$ We say that $$\varphi (\overline{x})$$ is s-stable (satisfaction stable) if and only if for any Kripke model $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$, any $$\alpha \in F$$ and any $$\overline{a}\in A_{\alpha }$$, we have $$\mathcal{A}_{\alpha}\models\varphi(\overline{a}) \Longrightarrow \alpha\Vdash_{{\mathcal{A}}}\varphi(\overline{a}).$$ We say that $$\varphi (\overline{x})$$ is stable if and only if for any Kripke model $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$, any $$\alpha \in F$$ and any $$\overline{a}\in A_{\alpha }$$, we have $$\alpha\Vdash_{{\mathcal{A}}}\varphi(\overline{a})\Longleftrightarrow \mathcal{A}_{\alpha}\models\varphi(\overline{a}).$$ In [6], Markovic gives a characterisation of stable and also s-stable formulas. Below we bring these results. Moreover, in [5], he introduces a class of f-stable formulas. One may say that this class is sound with respect to the f-stable property. In this paper, we improve this result. We define a class of $$\mathsf{R}^{*}$$-formulas and prove that it is exactly the class of f-stable formulas, i.e. it is sound and complete with respect to this property. Definition 2.2 The set of existential positive formulas $$\mathsf{E^{+}}$$ is the set of formulas containing At and closed under ∨, ∧ and ∃. Fact 2.3 A formula $$\varphi (\overline{x})$$ of $$\mathcal{L}$$ is stable if and only if $$\varphi (\overline{x})$$ is intuitionistically equivalent to an $$\mathsf{E^{+}}$$-formula. Proof. See [6, Theorem 2]. Definition 2.4 We define $$\mathsf{P}^{*}=\lbrace \varphi : \textrm{for some} \psi \in \mathsf{E^{+}},\ \vdash _{c}\psi \leftrightarrow \varphi \textrm{and} \vdash _{i}\psi \rightarrow \varphi \rbrace$$. Fact 2.5 We have $$\varphi (\overline{x})$$ is s-stable if and only if $$\varphi (\overline{x})\in \mathsf{P}^{*}$$. Proof. See [6, Theorem 1]. 3 Our new formula classes In this section, we define a class of formulas denoted by $$\mathsf{R}^{*}$$. In the next section, we prove that a formula is f-stable if and only if it is an $$\mathsf{R}^{*}$$-formula. Frist, we define two classes of formulas that are used in the definition of the target class $$\mathsf{R}^{*}$$. Definition 3.1 We define the classes P and Q of formulas inductively as follows: \begin{array}{lll} \mathsf{At}\subseteq\mathsf{P},&& \lbrace\top, \perp\rbrace\subseteq\mathsf{Q},\\ \varphi,\psi\in\mathsf{P}\ \Rightarrow\ \varphi\vee\psi,\varphi\wedge\psi\in\mathsf{P}, && \varphi,\psi\in \mathsf{Q}\ \Rightarrow\ \varphi\vee\psi,\varphi\wedge\psi\in\mathsf{Q}\\ \varphi\in \mathsf{Q},\psi\in\mathsf{P}\ \Rightarrow\ \varphi\rightarrow\psi\in\mathsf{P}, && \varphi\in \mathsf{P},\psi\in \mathsf{Q}\ \Rightarrow\ \varphi\rightarrow\psi\in\mathsf{Q},\\ \varphi\in\mathsf{P}\ \Rightarrow\ \exists x \,\varphi\in\mathsf{P}, && \varphi\in\mathsf{Q}\ \Rightarrow \quad \forall x \,\varphi\in\mathsf{Q}. \end{array} Note that, the class of Q-formulas does not contain atomic formulas (except ⊤ and ⊥). Proposition 3.2 Let $$\varphi$$ is any formula of $$\mathcal{L}$$. Then the following hold: If $$\varphi \in \mathsf{P}$$, then $$\varphi$$ is s-stable. If $$\varphi \in \mathsf{Q}$$, then $$\neg \varphi$$ is s-stable. Proof. Let $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$ be a Kripke model. We proceed by induction on the complexity of formulas in P ∪ Q, for all $$\alpha$$ simultaneously. Let $$\alpha \in F$$, and let $$\varphi \in \mathsf{P}(A_{\alpha })$$ be a sentence. Suppose $$\varphi \in \mathsf{At}(A_{\alpha })$$, and $$\mathcal{A}_{\alpha }\models \varphi$$. By the definition of forcing, we have $$\alpha \Vdash _{\mathcal{A}}\varphi$$. The induction steps for $$\varphi :=\psi \wedge \theta$$, $$\varphi :=\psi \vee \theta$$ and $$\varphi :=\exists x \,\psi (x)$$ are obvious. Suppose $$\varphi :=\psi \rightarrow \theta$$, where $$\psi \in \mathsf{Q}(A_{\alpha })$$ and $$\theta \in \mathsf{P}(A_{\alpha })$$. Assume that $$\mathcal{A}_{\alpha }\models \psi \rightarrow \theta$$. Let $$\beta \geq \alpha$$ be in F. Suppose that $$\beta \Vdash _{\mathcal{A}}\psi$$. If $$\mathcal{A}_{\alpha }\models \neg \psi$$, since $$\psi \in \mathsf{Q}(A_{\alpha })$$, by the induction hypothesis we have $$\alpha \Vdash _{\mathcal{A}}\neg \psi$$. This is a contradiction. So $$\mathcal{A}_{\alpha }\models \psi$$. Thus, $$\mathcal{A}_{\alpha }\models \theta$$. Since $$\theta \in \mathsf{P}(A_{\alpha })$$, by the induction hypothesis again, $$\alpha \Vdash _{\mathcal{A}}\theta$$. By monotonicity of forcing, $$\beta \Vdash _{\mathcal{A}}\theta$$. Thus, $$\alpha \Vdash _{\mathcal{A}}\psi \rightarrow \theta$$. Let $$\alpha \in F$$, and let $$\varphi \in \mathsf{Q}(A_{\alpha })$$ be a sentence. The case of $$\varphi :=\top$$, $$\varphi :=\perp$$ and the induction steps for $$\varphi :=\psi \wedge \theta$$ and $$\varphi :=\psi \vee \theta$$ are obvious. Suppose $$\varphi :=\psi \rightarrow \theta$$, where $$\psi \in \mathsf{P}(A_{\alpha })$$ and $$\theta \in \mathsf{Q}(A_{\alpha })$$. Assume $$\mathcal{A}_{\alpha }\models \neg (\psi \rightarrow \theta )$$. Thus, $$\mathcal{A}_{\alpha }\models \psi$$ and $$\mathcal{A}_{\alpha }\models \neg \theta$$. Since $$\psi \in \mathsf{P}(A_{\alpha })$$, by the induction hypothesis, $$\alpha \Vdash _{\mathcal{A}}\psi$$. Since $$\theta \in \mathsf{Q}(A_{\alpha })$$, by the induction hypothesis again, $$\alpha \Vdash _{\mathcal{A}}\neg \theta$$. So $$\alpha \Vdash _{\mathcal{A}}\neg (\psi \rightarrow \theta )$$. Now suppose $$\varphi :=\forall x \,\psi (x)$$, where $$\psi \in \mathsf{Q}(A_{\alpha })$$. Assume that $$\mathcal{A}_{\alpha }\models \neg \forall x \,\psi (x)$$. So there is an $$a\in A_{\alpha }$$ such that $$\mathcal{A}_{\alpha }\models \neg \psi (a)$$. Since $$\psi \in \mathsf{Q}(A_{\alpha })$$, by the induction hypothesis, we get $$\alpha \Vdash _{\mathcal{A}}\neg \psi (a)$$. Thus, $$\alpha \Vdash _{\mathcal{A}}\neg \forall x \,\psi (x)$$. In this paper, we define a class of f-stable formulas denoted by $$\mathsf{R}^{*}$$. We prove soundness and completeness theorems for this class with respect to the mentioned property. Definition 3.3 The class of formulas $$\mathsf{R}\subseteq \mathcal{L}$$ is defined by use of P-formulas as follows: At ⊆ R, $$\varphi ,\psi \in \mathsf{R}\ \Rightarrow \ \varphi \vee \psi ,\varphi \wedge \psi \in \mathsf{R}$$, $$\varphi \in \mathsf{P},\psi \in \mathsf{R}\ \Rightarrow\! \ \varphi \rightarrow \psi \in \mathsf{R}$$, $$\varphi \in \mathsf{R}\ \!\Rightarrow \ \exists x \,\varphi , \forall x \,\varphi \in \mathsf{R}$$. Definition 3.4 We define $$\mathsf{R}^{*}=\lbrace \varphi : \textrm{for some}\ \psi \in \mathsf{R}, \vdash _{c}\psi \leftrightarrow \varphi \ \textrm{and} \vdash _{i}\varphi \rightarrow \psi \rbrace$$. Definition 3.5 A formula $$\varphi$$ of $$\mathcal{L}$$ is called semi-positive if, whenever $$\psi \rightarrow \theta$$ is a subformula of $$\varphi$$, $$\psi$$ is atomic. It is easy to see that each semi-positive formula is f-stable. Fact 3.6 Let $$\Gamma \subseteq \Delta$$ be intuitionistic theories over $$\mathcal{L}$$. Then $$\Delta$$ is axiomatisable by semi-positive sentences over $$\Gamma$$ if and only if $$\Delta$$ is preserved under $$\Gamma$$-Kripke submodels. Proof. See Definition 3.5 and [8, Section 3]. In [8], it is shown that the class of those formulas of intuitionistic predicate logic which are preserved under taking submodels of Kripke models is precisely the class of semi-positive formulas. A Kripke model $$\mathcal{A}$$ is a submodel of a Kripke model $$\mathcal{B}$$ if the frame of $$\mathcal{A}$$ is a substructure of the frame of $$\mathcal{B}$$ in the classical sense. As mentioned above, each semi-positive formula is f-stable. It is easy to see that the class of $$\mathsf{R}^{*}$$-formulas is a proper extension of the class of semi-positive formulas. Our example below, shows that $$\mathsf{ R}^{*}$$ strictly extends the class of formulas equivalent to a semi-positive one in intuitionistic predicate logic. Example 3.7 Let $$\mathcal{L}_{1}$$ be a first-order language that includes p and q as two nullary relation symbols. Let $$\mathcal{A}$$ and $$\mathcal{B}$$ be two Kripke models of the pictures in the language $$\mathcal{L}_{1}$$ such that p and q are forced only in the node $$\alpha _{1}$$ in both $$\mathcal{A}$$ and $$\mathcal{B}$$, i.e. the other nodes do not force these atoms. Then, $$\mathcal{A}$$ is a submodel of $$\mathcal{B}$$. The formula $$\neg \neg p\rightarrow q$$ is an $$\mathsf{R}^{*}$$-formula, that is forced in the Kripke model $$\mathcal{B}$$, but not in its submodel $$\mathcal{A}$$. 4 Soundness and completeness theorems for $$\mathsf{R}^{*}$$-formulas In this section, we prove the soundness and completeness theorems for $$\mathsf{R}^{*}$$-formulas. In the other words, we show that the class of $$\mathsf{R}^{*}$$-formulas is exactly the class of f-stable formulas. Lemma 4.1 Let $$\varphi$$ is any formula of $$\mathcal{L}$$. If $$\varphi \in \mathsf{R}$$, then $$\varphi$$ is f-stable. Proof. Let $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$ be a Kripke model. We proceed by induction on the complexity of formulas in R, for all $$\alpha$$. Let $$\alpha \in F$$, and let $$\varphi \in \mathsf{R}(A_{\alpha })$$ be a sentence. Suppose $$\varphi \in \mathsf{At}(A_{\alpha })$$, and $$\alpha \Vdash _{\mathcal{A}}\varphi$$. By definition of forcing, we have $$\mathcal{A}_{\alpha }\models \varphi$$. The induction steps for $$\varphi :=\psi \wedge \theta$$, $$\varphi :=\psi \vee \theta$$ and $$\varphi :=\exists x \,\psi (x)$$ are obvious. Suppose $$\varphi :=\psi \rightarrow \theta$$, where $$\psi \in \mathsf{P}(A_{\alpha })$$ and $$\theta \in \mathsf{R}(A_{\alpha })$$. Assume that $$\alpha \Vdash _{\mathcal{A}}\psi \rightarrow \theta$$. Suppose that $$\mathcal{A}_{\alpha }\models \psi$$. Since $$\psi \in \mathsf{P}(A_{\alpha })$$, by Proposition 3.2, we have $$\alpha \Vdash _{\mathcal{A}}\psi$$. So $$\alpha \Vdash _{\mathcal{A}}\theta$$. By the induction hypothesis, $$\mathcal{A}_{\alpha }\models \theta$$. Thus, $$\mathcal{A}_{\alpha }\models \psi \rightarrow \theta$$. Now suppose that $$\varphi :=\forall x \,\psi (x)$$, where $$\psi \in \mathsf{R}(A_{\alpha })$$. Assume that $$\alpha \Vdash _{\mathcal{A}}\forall x \,\psi (x)$$. Let $$a\in A_{\alpha }$$. So $$\alpha \Vdash _{\mathcal{A}}\psi (a)$$. By the induction hypothesis, we get $$\mathcal{A}_{\alpha }\models \psi (a)$$. Thus, $$\mathcal{A}_{\alpha }\models \forall x \,\psi (x)$$. Theorem 4.2 (Soundness) Let $$\varphi$$ is any formula of $$\mathcal{L}$$. If $$\varphi \in \mathsf{R}^{*}$$, then $$\varphi$$ is f-stable. Proof. Let $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$ be a Kripke model. Let $$\alpha \in F$$, and let $$\varphi \in \mathsf{R}^{*}(A_{\alpha })$$ be a sentence. Suppose $$\alpha \Vdash _{\mathcal{A}}\varphi$$. By Definition 3.4, there is $$\psi \in \mathsf{R}$$ such that $$\vdash _{i}\varphi \rightarrow \psi$$ and $$\vdash _{c}\psi \leftrightarrow \varphi$$. Thus, $$\alpha \Vdash _{\mathcal{A}}\psi$$. By Lemma 4.1, we have $$\mathcal{A}_{\alpha }\models \psi$$. Thus, $$\mathcal{A}_{\alpha }\models \varphi$$. Definition 4.3 Let C be a set of constants. Let $$\Gamma$$ be a classical consistent theory, and let $$\Delta$$ be an intuitionistic consistent theory over $$\mathcal{L}(C)$$. The triple $$\langle \Gamma ,C,\Delta \rangle$$ is called acceptable if $$\Gamma \cap \mathsf{P}(C)\subseteq \Delta$$ and $$\Delta \cap \mathsf{R}(C)\subseteq \Gamma$$. Lemma 4.4 Let C be a set of constants. Let $$\Gamma$$ be a classical consistent theory, and let $$\Delta$$ be an intuitionistic consistent theory over $$\mathcal{L}(C)$$. We have, if $$\Delta \cap \mathsf{R}(C)\subseteq \Gamma$$, then the triple $$\langle\Gamma,C,\mathsf{Th}_{i}[C](\Delta\cup(\Gamma\cap\mathsf{P}(C)))\rangle$$ is acceptable. Proof. Let $$\Delta^{\prime}=\mathsf{Th}_{i}[C](\Delta\cup(\Gamma\cap\mathsf{P}(C))).$$ Obviously, $$\Gamma \cap \mathsf{P}(C)\subseteq \Delta ^{\prime }$$. We must show that $$\Delta ^{\prime }\cap \mathsf{R}(C)\subseteq \Gamma$$. Let $$\varphi \in \Delta ^{\prime }\cap \mathsf{R}(C)$$. Then $$\Delta\cup(\Gamma\cap\mathsf{P}(C))\vdash_{i}\varphi.$$ It follows that $$\Delta \cup \{\varrho \}\vdash _{i}\varphi$$, where $$\varrho$$ is a formula in $$\Gamma \cap \mathsf{P}(C)$$. So $$\Delta \vdash _{i}\varrho \rightarrow \varphi$$. Since $$\varrho \in \mathsf{P}(C)$$ and $$\varphi \in \mathsf{R}(C)$$, we have $$\varrho \rightarrow \varphi \in \mathsf{R}\,(C)$$. Thus, $$\varrho \rightarrow \varphi \in \Delta \cap \mathsf{R}(C)\subseteq \Gamma$$. So $$\Gamma \vdash _{c}\varrho \rightarrow \varphi$$. Also, $$\Gamma \vdash _{c}\varrho$$. So $$\Gamma \vdash _{c}\varphi$$. Since ⊥ ∈ R and $$\Gamma$$ is consistent, by $$\Delta ^{\prime }\cap \mathsf{R}(C)\subseteq \Gamma$$, $$\Delta ^{\prime }$$ is also consistent. Proposition 4.5 Let C be a set of constants, and let $$\Gamma$$ and $$\Delta$$ be theories such that $$\langle \Gamma ,C,\Delta \rangle$$ is acceptable. Let D be a set of constants not in $$\mathcal{L}(C)$$ with $$\vert D\vert \geq \vert \mathcal{L}(C)\vert$$. Let $$\varphi \in \mathcal{L}(C)$$ be such that $$\Gamma \nvdash _{c}\varphi$$. Then there is an acceptable triple $$\langle \Gamma ^{\prime },C^{\prime },\Delta ^{\prime }\rangle$$ such that $$\Gamma ^{\prime }$$ is a $$C^{\prime }$$-Henkin prime classical theory, $$\Delta ^{\prime }$$ is a $$C^{\prime }$$-Henkin prime intuitionistic theory, $$\Gamma \subseteq \Gamma ^{\prime },\ \Delta \subseteq \Delta ^{\prime }, C\subseteq C^{\prime }\subseteq C\cup D$$ and $$\Gamma ^{\prime }\nvdash _{c}\varphi$$. Proof. We construct a increasing chain of acceptable triples $$\langle \Gamma _{n},C_{n}, \Delta _{n} \rangle$$ with $$\Gamma _{n}\nvdash _{c}\varphi$$ such that for all $$n\in{\mathbb{N}}$$: $$\Gamma _{3n+1}$$ and $$\Delta _{3n+1}$$ are prime; If $$\Gamma _{3n+1}\vdash _{c}\exists x \,\psi (x)$$, then $$\Gamma _{3n+2}\vdash _{c}\psi (e)$$ for some $$e\in C_{3n+2}$$; If $$\Delta _{3n+2}\vdash _{i}\exists x \,\psi (x)$$, then $$\Delta _{3n+3}\vdash _{i}\psi (e)$$ for some $$e\in C_{3n+3}$$. Set $$\Gamma _{0}=\Gamma$$, $$C_{0}=C$$ and $$\Delta _{0}=\Delta$$. We proceed by induction on $$n\in{\mathbb{N}}$$. Step $$3n+1$$: Suppose that $$\langle \Gamma _{3n},C_{3n},\Delta _{3n}\rangle$$ is acceptable, and $$\Gamma _{3n}\nvdash _{c}\varphi$$. Let S be the set of all acceptable triples $$\langle \Gamma ^{*},C_{3n},\Delta ^{*}\rangle$$ such that $$\Gamma _{3n}\subseteq \Gamma ^{*}$$, $$\Delta _{3n}\subseteq \Delta ^{*}$$ and $$\Gamma ^{*}\nvdash _{c}\varphi$$. We define a partial order on S by set inclusion: $$\langle \Gamma ,C_{3n},\Delta \rangle \leq \langle \Gamma ^{^{\prime }},C_{3n},\Delta ^{^{\prime }}\rangle$$ if and only if $$\Gamma \subseteq \Gamma ^{^{\prime }}$$ and $$\Delta \subseteq \Delta ^{^{\prime }}$$. It is clear from the definition of acceptable triples and compactness that S is closed under unions of chains. Thus, by Zorn’s Lemma, there is a maximal element $$\langle \Gamma _{3n+1},C_{3n},\Delta _{3n+1}\rangle \in \mathbf{S}$$. Set $$C_{3n+1}=C_{3n}$$. Let $$\Gamma _{3n+1}\vdash _{c} \psi \vee \theta$$. Assume $$\Gamma _{3n+1}\cup \lbrace \psi \rbrace \vdash _{c}\varphi$$ and $$\Gamma _{3n+1}\cup \lbrace \theta \rbrace \vdash _{c}\varphi$$. Then $$\Gamma_{3n+1}\vdash_{c}(\psi\rightarrow\varphi)\wedge(\theta\rightarrow\varphi).$$ So $$\Gamma _{3n+1}\vdash _{c}(\psi \vee \theta )\rightarrow \varphi$$. Hence, $$\Gamma _{3n+1}\vdash _{c}\varphi$$, contradiction. Thus, without loss of generality, we may suppose that $$\Gamma _{3n+1}\cup \lbrace \psi \rbrace \nvdash _{c}\varphi$$. Let \begin{array}{ccccc} \Gamma^{\prime}=\mathsf{Th}_{c}[C_{3n}](\Gamma_{3n+1}\cup\lbrace\psi\rbrace) & &\textrm{and} & & \Delta^{\prime}=\mathsf{Th}_{i}[C_{3n}](\Delta_{3n+1}\cup(\Gamma^{\prime}\cap\mathsf{P}(C_{3n}))). \end{array} By the acceptability of $$\langle \Gamma _{3n+1},C_{3n},\Delta _{3n+1}\rangle$$, we have $$\Delta_{3n+1}\cap\mathsf{R}(C_{3n})\subseteq\Gamma_{3n+1}\subseteq\Gamma^{\prime}.$$ So by Lemma 4.4, the triple $$\langle \Gamma ^{\prime },C_{3n},\Delta ^{\prime }\rangle$$ is acceptable, hence in S. By maximality, since $$\Gamma _{3n+1}\subseteq \Gamma ^{\prime }$$ and $$\Delta _{3n+1}\subseteq \Delta ^{\prime }$$, we have $$\Gamma _{3n+1}=\Gamma ^{\prime }$$. Thus, $$\Gamma _{3n+1}\vdash _{c}\psi$$. Suppose $$\Delta _{3n+1}\vdash _{i}\chi \vee \xi$$. Assume $$\Gamma_{3n+1}\cup(\mathsf{Th}_{i}[C_{3n}](\Delta_{3n+1}\cup\lbrace\chi\rbrace)\cap\mathsf{R}(C_{3n}))\vdash_{c}\varphi$$ and $$\Gamma_{3n+1}\cup(\mathsf{Th}_{i}[C_{3n}](\Delta_{3n+1}\cup\lbrace\xi\rbrace)\cap\mathsf{R}(C_{3n}))\vdash_{c}\varphi .$$ By compactness, since $$\mathsf{R}(C_{3n})$$ is closed under finite conjunctions, there are \begin{array}{ccccc} \varrho\in\mathsf{Th}_{i}[C_{3n}](\Delta_{3n+1}\cup\lbrace\chi\rbrace)\cap\mathsf{R}(C_{3n}) & & \textrm{and} & & \sigma\in\mathsf{Th}_{i}[C_{3n}](\Delta_{3n+1}\cup\lbrace\xi\rbrace)\cap\mathsf{R}(C_{3n}) \end{array} such that $$\Gamma _{3n+1}\cup \lbrace \varrho \rbrace \vdash _{c}\varphi$$ and $$\Gamma _{3n+1}\cup \lbrace \sigma \rbrace \vdash _{c}\varphi .$$ So $$\Gamma_{3n+1}\vdash_{c}(\varrho\rightarrow\varphi)\wedge(\sigma\rightarrow\varphi).$$ Thus, $$\Gamma _{3n+1}\vdash _{c}(\varrho \vee \sigma )\rightarrow \varphi$$. Also, we have $$\Delta _{3n+1}\cup \lbrace \chi \rbrace \vdash _{i}\varrho$$ and $$\Delta _{3n+1}\cup \lbrace \xi \rbrace \vdash _{i}\sigma$$. Thus, $$\Delta_{3n+1}\vdash_{i}(\chi\rightarrow\varrho)\wedge(\xi\rightarrow\sigma).$$ So $$\Delta _{3n+1}\vdash _{i}(\chi \vee \xi )\rightarrow (\varrho \vee \sigma )$$. Thus, $$\Delta _{3n+1}\vdash _{i}\varrho \vee \sigma$$. Since $$\mathsf{R}(C_{3n})$$ is closed under finite disjunctions, $$\varrho\vee\sigma\in\Delta_{3n+1}\cap\mathsf{R}(C_{3n}).$$ By acceptability of $$\langle \Gamma _{3n+1},C_{3n},\Delta _{3n+1}\rangle$$, we get $$\varrho \vee \sigma \in \Gamma _{3n+1}$$. So $$\Gamma _{3n+1}\vdash _{c}\varphi$$, contradiction. Thus, without loss of generality, we may suppose $$\Gamma _{3n+1}\cup (\mathsf{Th}_{i}[C_{3n}](\Delta _{3n+1}\cup \lbrace \chi \rbrace )\cap \mathsf{R}(C_{3n}))\nvdash _{c}\varphi$$. Let $$\Gamma^{\prime}=\mathsf{Th}_{c}[C_{3n}](\Gamma_{3n+1}\cup(\mathsf{Th}_{i}[C_{3n}](\Delta_{3n+1}\cup\lbrace\chi\rbrace)\cap\mathsf{R}(C_{3n})))$$ and $$\Delta^{\prime}=\mathsf{Th}_{i}[C_{3n}]((\Delta_{3n+1}\cup\lbrace\chi\rbrace)\cup(\Gamma^{\prime}\cap\mathsf{P}(C_{3n}))).$$ By Lemma 4.4, the triple $$\langle \Gamma ^{\prime },C_{3n},\Delta ^{\prime }\rangle$$ is acceptable, hence in S. By maximality, since $$\Gamma _{3n+1}\subseteq \Gamma ^{\prime }$$ and $$\Delta _{3n+1}\subseteq \Delta ^{\prime }$$, we have $$\Delta _{3n+1}=\Delta ^{\prime }$$. Thus, $$\Delta _{3n+1}\vdash _{i}\chi$$. Step $$3n+2$$: Suppose that $$\langle \Gamma _{3n+1},C_{3n+1},\Delta _{3n+1}\rangle$$ is acceptable, and $$\Gamma _{3n+1}\nvdash _{c}\varphi$$. For every sentence $$\exists x \,\psi (x)\in \Gamma _{3n+1}$$, let $$e_{\exists x \,\psi (x)}$$ be a new constant in D. Let $$D^{\prime}=\{e_{\exists x \,\psi(x)}:\exists x \,\psi(x)\in\Gamma_{3n+1}\}.$$ We pick $$D^{\prime }$$ so that $$\vert D\setminus (C_{3n+1} \cup D^{\prime })\vert = \vert D\vert$$. Set $$C_{3n+2}=C_{3n+1}\cup D^{\prime }$$. Set $$\Gamma_{3n+2}=\mathsf{Th}_{c}[C_{3n+2}](\Gamma_{3n+1}\cup\{\psi(e_{\exists x \,\psi(x)}):\exists x \,\psi(x)\in\Gamma_{3n+1}\}).$$ Now we show that $$\Gamma _{3n+2}\nvdash _{c}\varphi$$. Assume $$\Gamma _{3n+2}\vdash _{c}\varphi$$. By compactness, there is a sentence $$\theta (\overline{a}):=\psi _{1}(a_{1})\wedge \cdots \wedge \psi _{m}(a_{m})$$, with $$\overline{a}\in D^{\prime }$$ and $$\exists x \,\psi_{1}(x),\ldots,\exists x \,\psi_{m}(x)\in\Gamma_{3n+1}$$ such that $$\Gamma _{3n+1}\cup \{\theta (\overline{a})\}\vdash _{c}\varphi$$. So $$\Gamma _{3n+1}\vdash _{c}\theta (\overline{a})\rightarrow \varphi$$. Since $$\overline a\not \in C_{3n+1}$$, we have $$\Gamma_{3n+1}\vdash_{c}\forall \overline{x} \,(\theta(\overline{x})\rightarrow\varphi).$$ So $$\Gamma _{3n+1}\vdash _{c}\exists \overline{x}\, \theta (\overline{x})\rightarrow \varphi$$. Since $$\exists x \,\psi_{1}(x),\ldots,\exists x \,\psi_{m}(x)\in\Gamma_{3n+1},$$ we have $$\exists \overline{x} \,\theta (\overline x)\in \Gamma _{3n+1}$$. So $$\Gamma _{3n+1}\vdash _{c}\varphi$$, contradiction. Thus, $$\Gamma _{3n+2}\nvdash _{c}\varphi$$. We claim $$\mathsf{Th}_{i}[C_{3n+2}](\Delta_{3n+1})\cap\mathsf{R}(C_{3n+2})\subseteq\Gamma_{3n+2}.$$ Suppose that $$\varrho (\overline{e})\in \mathsf{Th}_{i}[C_{3n+2}](\Delta _{3n+1})\cap \mathsf{R}(C_{3n+2})$$ such that $$\overline{e}\in D^{\prime }$$. Since $$\overline{e}\not \in C_{3n+1}$$, we have $$\Delta _{3n+1}\vdash _{i}\forall \overline{x} \,\varrho (\overline{x})$$. By the induction hypothesis, $$\forall \overline{x} \,\varrho (\overline{x})\in \Delta _{3n+1}\cap \mathsf{R}(C_{3n+1})\subseteq \Gamma _{3n+1}$$. Thus, $$\Gamma _{3n+1}\vdash _{c}\forall \overline{x} \,\varrho (\overline{x})$$. So $$\Gamma _{3n+2}\vdash _{c}\varrho (\overline{e})$$, which proves the claim. Set $$\Delta_{3n+2}=\mathsf{Th}_{i}[C_{3n+2}](\Delta_{3n+1}\cup(\Gamma_{3n+2}\cap\mathsf{P}(C_{3n+2}))).$$ By Lemma 4.4, $$\langle \Gamma _{3n+2},C_{3n+2},\Delta _{3n+2}\rangle$$ is acceptable. Step $$3n+3$$: Suppose $$\langle \Gamma _{3n+2},C_{3n+2},\Delta _{3n+2}\rangle$$ is acceptable, and $$\Gamma _{3n+2}\nvdash _{c}\varphi$$. For every sentence $$\exists x \,\psi (x)\in \Delta _{3n+2}$$, let $$e_{\exists x \,\psi (x)}$$ be a new constant in D. Let $$D^{\prime}=\{e_{\exists x \,\psi(x)}:\exists x \,\psi(x)\in\Delta_{3n+2}\}.$$ We pick $$D^{\prime }$$ so that $$\vert D\setminus (C_{3n+2} \cup D^{\prime })\vert = \vert D\vert$$. Set $$C_{3n+3}=C_{3n+2}\cup D^{\prime }$$. Set $$\Delta^{\prime}=\mathsf{Th}_{i}[C_{3n+3}](\Delta_{3n+2}\cup\{\psi(e_{\exists x \,\psi(x)}):\exists x \,\psi(x)\in\Delta_{3n+2}\}).$$ We claim $$\mathsf{Th}_{c}[C_{3n+3}](\Gamma_{3n+2}\cup(\Delta^{\prime}\cap\mathsf{R}(C_{3n+3})))\nvdash_{c}\varphi.$$ Assume $$\mathsf{Th}_{c}[C_{3n+3}](\Gamma _{3n+2}\cup (\Delta ^{\prime }\cap \mathsf{R}(C_{3n+3})))\vdash _{c}\varphi$$. Since $$\mathsf{R}(C_{3n+2})$$ is closed under finite conjunction, by compactness there is a sentence $$\theta (\overline{a})\in \Delta ^{\prime }\cap \mathsf{R}(C_{3n+3})$$ with $$\overline{a}\in D^{\prime }$$, such that $$\Gamma _{3n+2}\cup \lbrace \theta (\overline{a})\rbrace \vdash _{c}\varphi$$. By compactness again, there is a sentence $$\varrho (\overline{e}):=\psi _{1}(e_{1})\wedge \cdots \wedge \psi _{m}(e_{m})$$ with $$\overline{e}\in D^{\prime }$$ and $$\exists x \,\psi_{1}(x),\ldots,\exists x \,\psi_{m}(x)\in\Delta_{3n+2}$$ such that $$\Delta _{3n+2}\cup \lbrace \varrho (\overline{e})\rbrace \vdash _{i}\theta (\overline{a})$$. So $$\Delta _{3n+2}\cup \lbrace \varrho (\overline{e})\rbrace \vdash _{i}\exists \overline{y} \,\theta (\overline{y})$$. So $$\Delta _{3n+2}\vdash _{i}\varrho (\overline{e})\rightarrow \exists \overline{y} \,\theta (\overline{y})$$. Since $$\overline{e}\not \in C_{3n+2}$$, we have $$\Delta _{3n+2}\vdash _{i}\forall \overline{x} \,(\varrho (\overline{x})\rightarrow \exists \overline{y} \,\theta (\overline{y}))$$. Thus, $$\Delta _{3n+2}\vdash _{i}\exists \overline{x} \,\varrho (\overline{x})\rightarrow \exists \overline{y} \,\theta (\overline{y})$$. Since $$\exists x \,\psi_{1}(x),\ldots,\exists x \,\psi_{m}(x)\in\Delta_{3n+2},$$ we have $$\exists \overline{x} \,\varrho (\overline{x})\in \Delta _{3n+2}$$. Thus, $$\Delta _{3n+2}\vdash _{i}\exists \overline{y} \,\theta (\overline{y})$$. Since $$\mathsf{R}(C_{3n+2})$$ is closed under existential quantification, we have $$\exists \overline{y} \,\theta (\overline{y})\in \mathsf{R}(C_{3n+2}).$$ By the induction hypothesis, $$\Delta _{3n+2}\cap \mathsf{R}(C_{3n+2})\subseteq \Gamma _{3n+2}$$. So $$\Gamma _{3n+2}\vdash _{c}\exists \overline{y} \,\theta (\overline{y})$$. Since $$\Gamma_{3n+2}\cup\lbrace\theta(\overline{a})\rbrace\vdash_{c}\varphi$$ and $$\overline{a}$$ does not appear in $$\Gamma _{3n+2}$$ and $$\varphi$$, we have $$\Gamma _{3n+2}\cup \lbrace \exists \overline{y} \,\theta (\overline{y})\rbrace \vdash _{c}\varphi$$. So $$\Gamma _{3n+2}\vdash _{c}\varphi$$, contradiction. Set $$\Gamma_{3n+3}=\mathsf{Th}_{c}[C_{3n+3}](\Gamma_{3n+2}\cup(\Delta^{\prime}\cap\mathsf{R}(C_{3n+3}))),$$ and $$\Delta_{3n+3}=\mathsf{Th}_{i}[C_{3n+3}](\Delta^{\prime}\cup(\Gamma_{3n+3}\cap\mathsf{P}(C_{3n+3}))).$$ By Lemma 4.4$$\langle \Gamma _{3n+3},C_{3n+3},\Delta _{3n+3}\rangle$$ is acceptable. This completes the construction and the inductive verification of its correctness. Set $$\Gamma ^{\prime }=\underset{n\in{\mathbb{N}}}\bigcup \Gamma _{n}$$, $$C^{\prime }=\underset{n\in{\mathbb{N}}}\bigcup C_{n}$$ and $$\Delta ^{\prime }=\underset{n\in{\mathbb{N}}}\bigcup \Delta _{n}$$. By compactness, $$\Gamma ^{\prime }$$ is a $$C^{\prime }$$-Henkin prime classical theory, $$\Delta ^{\prime }$$ is a $$C^{\prime }$$-Henkin prime intuitionistic theory, and $$\Gamma ^{\prime }\nvdash _{c}\varphi$$. Clearly, $$\langle \Gamma ^{\prime }, C^{\prime }, \Delta ^{\prime }\rangle$$ is an acceptable triple. A C-Henkin prime intuitionistic theory $$\Pi$$ defines a classical structure with domain C as follows: Definition 4.6 Suppose $$\Pi$$ is a C-Henkin prime intuitionistic theory. Then $$\mathcal{A}_{\Pi }$$ is defined to be the classical structure in the language of $$\mathcal{L}(C)$$ such that the domain of $$\mathcal{A}_{\Pi }$$ is C itself ($$c^{\mathcal{A}_{\Pi }}=c$$) and such that for every atomic C-sentence $$\varphi$$, $$\mathcal{A}_{\Pi }\models \varphi$$ if and only if $$\Pi \vdash _{i}\varphi$$. It is straightforward to check that the definition of $$\mathcal{A}_{\Pi }$$ does indeed specify a unique classical structure. Proposition 4.7 Let C be a set of constants, and $$\Gamma$$ and $$\Delta$$ be theories such that: $$\langle \Gamma ,C,\Delta \rangle$$ is acceptable; $$\Gamma$$ is a C-Henkin prime classical theory; $$\Delta$$ is a C-Henkin prime intuitionistic theory. Then there exists a Kripke model $$\mathcal{A}$$ with the root $$\alpha _{0}=\langle \Gamma , C, \Delta \rangle$$ such that: $$\alpha _{0}\Vdash _{\mathcal{A}}\varphi \Leftrightarrow \Delta \vdash _{i}\varphi$$, for all $$\varphi \in \mathcal{L}(C)$$, $$\mathcal{A}_{\alpha _{0}}\models \varphi \Leftrightarrow \Gamma \vdash _{c}\varphi$$, for all $$\varphi \in \mathcal{L}(C)$$. Proof. Let D be a set of new constants such that $$\vert D\vert \geq \vert \mathcal{L}(C) \vert$$. Let F be the following poset: The root of F is the acceptable triple $$\langle \Gamma ,C,\Delta \rangle$$ and the other nodes are $$\langle C^{\prime },\Delta ^{\prime }\rangle$$ such that $$C\subseteq C^{\prime } \subseteq C \cup D$$, $$\vert D\setminus C^{\prime }\vert =\vert D\vert$$ and $$\Delta \subseteq \Delta ^{\prime }$$, where $$\Delta ^{\prime }$$ is a consistent $$C^{\prime }$$-Henkin prime intuitionistic theory. The order on F is defined by $$\langle C^{\prime },\Delta ^{\prime }\rangle \leq \langle C^{\prime \prime },\Delta ^{\prime \prime }\rangle$$ if and only if $$C^{\prime }\subseteq C^{\prime \prime }$$ and $$\Delta ^{\prime }\subseteq \Delta ^{\prime \prime }$$. A Kripke model $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$ is defined by: $$A_{\langle \Gamma ,C,\Delta \rangle }=C$$, $$\langle \Gamma ,C,\Delta \rangle \Vdash _{\mathcal{A}} \varphi \Leftrightarrow \Delta \vdash _{i}\varphi$$, for all $$\varphi \in \mathsf{At}(C)$$, $$A_{\langle C^{\prime },\Delta ^{\prime }\rangle }=C^{\prime }$$, $$\langle C^{\prime },\Delta ^{\prime }\rangle \Vdash _{\mathcal{A}} \varphi \Leftrightarrow \Delta ^{\prime }\vdash _{i}\varphi$$, for all $$\varphi \in \mathsf{At}(C^{\prime })$$. It is straightforward to verify that $$\mathcal{A}$$ is a Kripke model with the root $$\alpha _{0}=\langle \Gamma ,C,\Delta \rangle \in F$$. We claim the following: 1. For all $$\alpha \in F$$, i. If $$\alpha =\alpha _{0}=\langle \Gamma ,C,\Delta \rangle$$, then $$\alpha _{0}\Vdash _{\mathcal{A}}\varphi \Leftrightarrow \Delta \vdash _{i}\varphi$$ for all $$\varphi \in \mathcal{L}(C)$$; ii. If $$\alpha =\langle C^{\prime },\Delta ^{\prime }\rangle$$, then $$\alpha \Vdash _{\mathcal{A}}\varphi \Leftrightarrow \Delta ^{\prime }\vdash _{i}\varphi$$ for all $$\varphi \in \mathcal{L}(C^{\prime })$$; 2. $$\mathcal{A}_{\alpha _{0}}\models \varphi \Leftrightarrow \Gamma \vdash _{c}\varphi$$ for all $$\varphi \in \mathcal{L}(C)$$. For the prove of the first assertion, see [2, Section 5.3]. For the prove of second assertion, note that since $$\langle \Gamma ,C,\Delta \rangle$$ is acceptable, we have $$\Gamma \cap \mathsf{At}(C)=\Delta \cap \mathsf{At}(C)$$. So $$\mathcal{A}_{\langle \Gamma ,C,\Delta \rangle }$$ coincides with the canonical model of $$\Gamma$$. Since $$\alpha _{0}=\langle \Gamma ,C,\Delta \rangle \in F$$, we have the following: $$\alpha _{0}\Vdash _{\mathcal{A}}\varphi \Leftrightarrow \Delta \vdash _{i}\varphi$$, for all $$\varphi \in \mathcal{L}(C)$$, $$\mathcal{A}_{\alpha _{0}}\models \varphi \Leftrightarrow \Gamma \vdash _{c}\varphi$$, for all $$\varphi \in \mathcal{L}(C)$$. This completes the proof. Theorem 4.8 (Completeness) Let $$\varphi$$ be a sentence in $$\mathcal{L}$$ such that $$\varphi \notin \mathsf{R}^{*}$$. Then there exists a rooted Kripke model $$\mathcal{A}$$ with the root $$\alpha _{0}$$ such that $$\alpha _{0}\Vdash _{\mathcal{A}}\varphi$$ and $$\mathcal{A}_{\alpha _{0}}\not \models \varphi$$. Proof. Consider the triple $$\langle\mathsf{Th}_{c}(\lbrace\neg\varphi\rbrace\cup(\mathsf{Th}_{i}\lbrace\varphi\rbrace\cap\mathsf{R})), \emptyset, \mathsf{Th}_{i}\lbrace\varphi\rbrace\rangle.$$ We claim $$\mathsf{Th}_{c}(\lbrace \neg \varphi \rbrace \cup (\mathsf{Th}_{i}\lbrace \varphi \rbrace \cap \mathsf{R}))$$ is consistent. Suppose $$\mathsf{Th}_{c}(\lbrace \neg \varphi \rbrace \cup (\mathsf{Th}_{i}\lbrace \varphi \rbrace \cap \mathsf{R}))\vdash _{c}\perp$$. Thus, there exists $$\psi \in \mathsf{Th}_{i}\lbrace \varphi \rbrace \cap \mathsf{R}$$ such that $$\lbrace \neg \varphi \rbrace \cup \lbrace \psi \rbrace \vdash _{c}\perp$$. Then $$\neg \varphi \vdash _{c}\neg \psi$$. Then $$\vdash _{c}\neg \varphi \rightarrow \neg \psi$$. So we have $$\vdash _{c}\psi \rightarrow \varphi$$. Since $$\psi \in \mathsf{Th}_{i}\lbrace \varphi \rbrace$$, we have $$\vdash _{i}\varphi \rightarrow \psi$$. Thus, $$\vdash _{c}\varphi \rightarrow \psi$$. Thus, we have $$\vdash _{c}\psi \leftrightarrow \varphi$$ and $$\vdash _{i}\varphi \rightarrow \psi$$. So $$\varphi \in \mathsf{R}^{*}$$ and it is a contradiction. So $$\mathsf{Th}_{c}(\lbrace \neg \varphi \rbrace \cup (\mathsf{Th}_{i}\lbrace \varphi \rbrace \cap \mathsf{R}))$$ is consistent. By Lemma 4.4, the triple $$\langle\mathsf{Th}_{c}(\lbrace\neg\varphi\rbrace\cup(\mathsf{Th}_{i}\lbrace\varphi\rbrace\cap\mathsf{R})), \emptyset, \mathsf{Th}_{i}(\lbrace\varphi\rbrace\cup\mathsf{Th}_{c}(\lbrace\neg\varphi\rbrace\cup(\textrm{Th}_{i}\lbrace\varphi\rbrace\cap\mathsf{R}))\cap\mathsf{P})\rangle$$ is acceptable. By Proposition 4.5, there exists the acceptable triple $$\langle \Gamma ,C,\Delta \rangle$$ such that $$\Gamma$$ is a C-Henkin prime classical theory, $$\Delta$$ is a C-Henkin prime intuitionistic theory, $$\mathsf{Th}_{c}(\lbrace \neg \varphi \rbrace \cup (\mathsf{Th}_{i}\lbrace \varphi \rbrace \cap \mathsf{R}))\subseteq \Gamma ,\mathsf{Th}_{i}(\lbrace \varphi \rbrace \cup \mathsf{Th}_{c}(\lbrace \neg \varphi \rbrace \cup (\mathsf{Th}_{i}\lbrace \varphi \rbrace \cap \mathsf{R}))\cap \mathsf{P})\subseteq \Delta$$ and $$\Gamma \nvdash _{c}\varphi$$. By Proposition 4.7, there exists a rooted Kripke model $$\mathcal{A}$$ with the root $$\alpha _{0}=\langle \Gamma ,C,\Delta \rangle$$ such that $$\alpha _{0}\Vdash _{\mathcal{A}}\varphi \Leftrightarrow \Delta \vdash _{i}\varphi$$, for all $$\varphi \in \mathcal{L}(C)$$, $$\mathcal{A}_{\alpha _{0}}\models \varphi \Leftrightarrow \Gamma \vdash _{c}\varphi$$, for all $$\varphi \in \mathcal{L}(C)$$. Thus, $$\alpha _{0}\Vdash _{\mathcal{A}}\varphi$$ and $$\mathcal{A}_{\alpha _{0}}\not \models \varphi$$. A natural question one can ask here is this: is there an appropriate syntactic description of the class $$\mathsf{R}^{*}$$? The answer suggested by the reviewer of the paper is ‘No’, if we ask for a decidable desciption. His (her) proof is as follows. Let $$\psi$$ be any sentence of a language with at least one binary predicate symbol R. Let P be a fresh nullary predicate symbol. We extend our language with P. We claim that $$\psi \vee \neg \neg P$$ is f-stable if and only if $$\models \psi$$. The right-to-left part is trivial. We treat the left-to-right part via contraposition. Suppose $$\not \models \psi$$. Let $$\mathcal{A}$$ be a classical model such that $$\mathcal{A}\not \models \psi$$. We now build a Kripke model with two nodes 0 and 1 with 0 ≤ 1. At 0 we have the model $$\mathcal{A}$$ and we declare P not forced and at 1 we have again $$\mathcal{A}$$ and we declare P forced. Clearly, $$0\Vdash \psi \vee \neg \neg P$$ but $$\mathcal{A}_{0}\not \models \psi \vee \neg \neg P$$. So, $$\psi \vee \neg \neg P$$ is not f-stable. Since in this way we can reduce classical validity for a language with a binary predicate symbol to f-stability, the property of f-stability is not decidable. Hence, the class of f-stable formulas, i.e. $$\mathsf{R}^{*}$$, has no decidable description. 5 Appendix In this part, we review some classes of formulas introduced by Markovic. Although we did not use them in this paper, they motivated some of our definitions. In trying to give a syntactic characterisation of $$\mathsf{P}^{*}$$, the following class $$\mathsf{S}_{\omega }$$ of formulas is defined in [6]. Definition 5.1 Let $$\mathsf{S}_{0}=\mathsf{E^{+}}$$. Define $$\mathsf{S}_{n+1}$$ as an extension of $$\mathsf{S}_{n}$$ such that for each $$\varphi , \psi \in \mathsf{S}_{n}$$ we have $$\neg \varphi \rightarrow \psi \in \mathsf{S}_{n+1}$$, $$\varphi \vee \psi , \varphi \wedge \psi , \exists x \,\varphi \in \mathsf{S}_{n+1}$$. Finally, define $$\mathsf{S}_{\omega }=\underset{n\in{\mathbb{N}}}\bigcup \mathsf{S}_{n}$$. In [6], it is shown that $$\mathsf{S}_{\omega }\subseteq \mathsf{P}^{*}$$. Consider the formula $$\theta :=(\varphi \rightarrow (\neg \psi \vee \neg \chi ))\rightarrow \xi$$, where $$\varphi , \psi , \chi , \xi$$ are in $$\mathsf{E^{+}}$$. It is easy to see that $$\theta$$ is in $$\mathsf{P}^{*}$$ but is not in $$\mathsf{S}_{\omega }$$. Thus, $$\mathsf{S}_{\omega }\subsetneqq \mathsf{P}^{*}$$. Our class of P-formulas (see Definition 3.1) is an extension of $$\mathsf{S}_{\omega }$$ and it is contained in $$\mathsf{P}^{*}$$. Clearly, the formula $$\theta$$ is in P. So, $$\mathsf{S}_{\omega }\subsetneqq \mathsf{P}$$. In [5], a class R of f-stable formulas is defined. Definition 5.2 Let $$\mathbf{R}_{0} = \mathsf{E^{+}}\cup \lbrace \neg \varphi : \varphi \in \mathsf{P}^{*}\rbrace$$. We define $$\mathbf{R}_{n+1}$$ as an extension of $$\mathbf{R}_{n}$$ such that: If $$\varphi \in \mathsf{P}^{*}$$ and $$\psi \in \mathbf{R}_{n}$$ then $$\varphi \rightarrow \psi \in \mathbf{R}_{n+1}$$, If $$\varphi , \psi \in \mathbf{R}_{n}$$ then $$\varphi \vee \psi , \varphi \wedge \psi , \exists x \,\varphi , \forall x \,\varphi \in \mathbf{R}_{n+1}$$. Finally, define $$\mathbf{R}_{\omega }=\underset{n\in{\mathbb{N}}}\bigcup \mathbf{R}_{n}$$. Definition 5.3 We define $$\mathbf{R}=\lbrace \varphi : \textrm{for some} \ \psi \in \mathbf{R}_{\omega }, \vdash _{c}\psi \leftrightarrow \varphi \ \textrm{and}\vdash _{i}\varphi \rightarrow \psi \rbrace$$. Fact 5.4 If $$\varphi (\overline{x})\in \mathbf{R}$$, then $$\varphi (\overline{x})$$ is f-stable. Proof. See [5, Corollary 1]. Acknowledgements We would like to thank the anonymous referee of the paper for his (her) invaluable comments and suggestions. References [1] S. R. Buss . Intuitionistic validity in T-normal Kripke structures . Annals of Pure and Applied Logic , 55 , 159 – 173 , 1993 . Google Scholar Crossref Search ADS [2] D. van Dalen . Logic and Structure , 4th edn. Springer-Verlag , 2004 . [3] D. van Dalen , H. Mudler , E. C. W. Krabbe and A. Visser . Finite Kripke models of HA are locally PA . Notre Dame Journal of Formal Logic, 27 , 528 – 532 , 1986 . Google Scholar Crossref Search ADS [4] B. Ellison , J. Fleischmann , D. McGinn and W. Ruitenburg . Kripke submodels and universal sentences . Mathematical Logic Quarterly, 53 , 311 – 320 , 2007 . Google Scholar Crossref Search ADS [5] Z. Markovic . Intuitionistic and classical satisfiability in Kripke models . Institute of Mathematics, 13 , 1 – 5 , 1998 . [6] Z. Markovic . Some preservation results for classical and intuitionistic satisfiability in Kripke models . Notre Dame Journal of Formal Logic , 24 , 395 – 398 , 1983 . Google Scholar Crossref Search ADS [7] M. Moniri . $$\mathcal H$$-theories, fragments of HA and PA-normality . Mathematical Logic Quarterly , 49 , 250 – 254 , 2003 . Google Scholar Crossref Search ADS [8] A. Visser . Submodels of Kripke models . Archive for Mathematical Logic , 40 , 277 – 295 , 2001 . Google Scholar Crossref Search ADS [9] K. Wehmeier . Classical and intuitionistic models of arithmetic . Notre Dame Journal of Formal Logic, 37 , 452 – 461 , 1996 . Google Scholar Crossref Search ADS © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Logic Journal of the IGPL Oxford University Press

# From forcing to satisfaction in Kripke models of intuitionistic predicate logic

Logic Journal of the IGPL, Volume 26 (5) – Sep 25, 2018
11 pages

/lp/ou_press/from-forcing-to-satisfaction-in-kripke-models-of-intuitionistic-3GzGJmINRy
Publisher
Oxford University Press
ISSN
1367-0751
eISSN
1368-9894
D.O.I.
10.1093/jigpal/jzy007
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### Abstract

Abstract In this paper, we answer a natural question concerning the relation between forcing and satisfaction in Kripke models of intuitionistic predicate logic. We define a class of formulas denoted by $$\mathsf{R}^{*}$$, with the property that forcing of any $$\mathsf{R}^{*}$$-formula in a node of a Kripke model of intuitionistic predicate logic implies its satisfaction in the classical structure attached to that node. We also prove that any formula with this property is an $$\mathsf{R}^{*}$$-formula. 1 Introduction The relationship between classical and intuitionistic satisfiability of formulas in Kripke models of intuitionistic predicate logic and also intuitionistic arithmetic gives rise to challenging questions. We review some of these questions below. A node of a Kripke model of Heyting arithmetic HA is called PA-normal, if the classical world attached to it, satisfies Peano arithmetic PA. Now the natural question one can ask is this: is every Kripke model of HA, PA-normal? In [3], it is shown that every finite Kripke model of HA is PA-normal. Also it is proved that models of HA over the frame $$(\omega , \leq )$$ contains infinitely many Peano nodes. In [9], it is shown that such Kripke models are in fact PA-normal. The problem in general is still open. Is every PA-normal Kripke model a model of HA? The answer is ‘No’. In [1], a PA-normal Kripke model over the frame $$(\omega , \leq )$$ is constructed which does not force HA. In [7], a two-node Kripke model with this property is constructed. Moreover, in [1], for each set T of sentences, an intuitionistic theory $$\mathcal{H}(T)$$ is introduced. This intuitionistic theory is exactly the class of all formulas which are forced in each T-normal Kripke model. In [1], it is shown that $$\mathcal{H}(\mathsf{PA})\neq \mathsf{HA}$$. More generally, similar questions can be asked about Kripke models of intuitionistic predicate logic. In [6], the class of all formulas with the property that their forcing and satisfaction are equivalent in any node (world) is introduced. Moreover, in [5], a class of formulas is defined such that, whenever a formula of that class is forced at a node of a Kripke model, it must be satisfied in the classical structure associated with that node. One may say that this class is sound with respect to the mentioned property. In this paper, we improve this result. We define a class of $$\mathsf{R}^{*}$$-formulas and prove that it is exactly the class of all formulas which have this property, i.e. the class is sound and complete with respect to this property. 2 Preliminaries We consider a first-order language $$\mathcal{L}$$ to be the set of all formulas that can be built from a symbol set (relation, function and constant symbols and variables) using ⊤, ⊥, ∧, ∨, $$\rightarrow$$, ∃ and ∀. Symbols ⊤ and ⊥ are both atoms and nullary connectives. $$\neg \varphi$$ is short for $$\varphi \rightarrow \bot$$ and $$\varphi \leftrightarrow \psi$$ is short for $$(\varphi \rightarrow \psi )\wedge (\psi \rightarrow \varphi )$$. A list of terms $$t_{1}, \ldots , t_{n}$$ is abbreviated as $$\overline{t}$$. If C is an arbitrary set of constant symbols, then $$\mathcal{L}(C)$$ is the language $$\mathcal{L}$$ extended by all constant symbols in C. $$\mathsf{At}\subseteq \mathcal{L}$$ is the set of atomic formulas in $$\mathcal{L}$$. Analogously, $$\mathsf{At}(C)\subseteq \mathcal{L}(C)$$ is the set of atomic formulas in $$\mathcal{L}(C)$$. Let $$\Gamma$$ be a set of formulas in $$\mathcal{L}$$. By $$\Gamma (C)$$, we mean all $$\mathcal{L}(C)$$-sentences of the form $$\varphi (\overline{c})$$, where $$\varphi (\overline{x})$$ is a formula in $$\Gamma$$ and $$\overline{c}\in C$$. We use the symbols $$\vdash _{i}$$ and $$\vdash _{c}$$ to denote derivability in the intuitionistic and classical predicate calculus, respectively. If $$\Gamma \subseteq \mathcal{L}(C)$$ is a set of sentences, then we define $$\mathsf{Th}_{i}[C](\Gamma )=\{\varphi \in \mathcal{L}(C): \Gamma \vdash _{i}\varphi \}$$ to be the deductive closure of $$\Gamma$$ over $$\mathcal{L}(C)$$. By a theory, we mean a set of sentences closed under deduction. The classical theory $$\mathsf{Th}_{c}[C](\Gamma )$$ is defined in a similar way as $$\{\varphi \in \mathcal{L}(C): \Gamma \vdash _{c}\varphi \}$$. An intuitionistic theory $$\Gamma$$ over $$\mathcal{L}(C)$$ is called prime if for all sentences $$\varphi ,\psi \in \mathcal{L}(C)$$, we have $$\Gamma \vdash _{i}\varphi \vee \psi$$ if and only if $$\Gamma \vdash _{i}\varphi$$ or $$\Gamma \vdash _{i}\psi$$. An intuitionistic consistent theory $$\Gamma$$ over $$\mathcal{L}(C)$$ is called C- Henkin if for all sentences of the form $$\exists x \,\varphi (x)\in \mathcal{L}(C)$$, we have $$\Gamma \vdash _{i}\exists x \,\varphi (x)$$ if and only if there is c ∈ C such that $$\Gamma \vdash _{i}\varphi (c)$$. A theory is called C- Henkin prime if it is both C-Henkin and prime. These notions can be defined for classical theories in a similar way. A Kripke model $$\mathcal{A}$$ in the language $$\mathcal{L}$$, is a pair $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$ such that (F, ≤) is a partially ordered set (the frame of $$\mathcal{A}$$) and to each element (node) $$\alpha$$ of F is attached a classical structure $$\mathcal{A}_{\alpha }$$ for $$\mathcal{L}$$ in which the interpretation of equality is an equivalence relation which may properly extend the real equality. For each two nodes $$\alpha ,\beta$$, if $$\beta$$ is accessible from $$\alpha$$ (i.e. $$\alpha \leq \beta$$), then the world at $$\alpha$$ must be a weak substructure of the one at $$\beta$$. By this we mean that $$\mathcal{A}_{\beta }$$ preserves truth in $$\mathcal{A}_{\alpha }$$ of atomic sentences in $$\mathcal{L}(A_{\alpha })$$. The forcing relation is defined as usual. A Kripke model is a $$\Gamma$$-Kripke model if it forces the axioms of $$\Gamma$$ at all nodes. The calligraphic letters $$\mathcal{A}$$, $$\mathcal{B}$$, $$\mathcal{C}, \ldots$$ represent either classical models or Kripke models. If $$\mathcal{A}$$ is a classical model, then the domain of $$\mathcal{A}$$ is denoted by the corresponding Latin letter A, and $$\mathcal{L}(A)$$ is the language $$\mathcal{L}$$ extended by a new constant symbol for every element in A. The symbol ⊧ denotes classical satisfaction in a model and is defined for sentences (closed formulas) only. Definition 2.1 Let $$\varphi (\overline{x})$$ be a formula in the language $$\mathcal{L}$$. We say that $$\varphi (\overline{x})$$ is f-stable (forcing stable) if and only if for any Kripke model $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$, any $$\alpha \in F$$ and any $$\overline{a}\in A_{\alpha }$$, we have $$\alpha\Vdash_{{\mathcal{A}}}\varphi(\overline{a})\Longrightarrow \mathcal{A}_{\alpha}\models\varphi(\overline{a}).$$ We say that $$\varphi (\overline{x})$$ is s-stable (satisfaction stable) if and only if for any Kripke model $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$, any $$\alpha \in F$$ and any $$\overline{a}\in A_{\alpha }$$, we have $$\mathcal{A}_{\alpha}\models\varphi(\overline{a}) \Longrightarrow \alpha\Vdash_{{\mathcal{A}}}\varphi(\overline{a}).$$ We say that $$\varphi (\overline{x})$$ is stable if and only if for any Kripke model $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$, any $$\alpha \in F$$ and any $$\overline{a}\in A_{\alpha }$$, we have $$\alpha\Vdash_{{\mathcal{A}}}\varphi(\overline{a})\Longleftrightarrow \mathcal{A}_{\alpha}\models\varphi(\overline{a}).$$ In [6], Markovic gives a characterisation of stable and also s-stable formulas. Below we bring these results. Moreover, in [5], he introduces a class of f-stable formulas. One may say that this class is sound with respect to the f-stable property. In this paper, we improve this result. We define a class of $$\mathsf{R}^{*}$$-formulas and prove that it is exactly the class of f-stable formulas, i.e. it is sound and complete with respect to this property. Definition 2.2 The set of existential positive formulas $$\mathsf{E^{+}}$$ is the set of formulas containing At and closed under ∨, ∧ and ∃. Fact 2.3 A formula $$\varphi (\overline{x})$$ of $$\mathcal{L}$$ is stable if and only if $$\varphi (\overline{x})$$ is intuitionistically equivalent to an $$\mathsf{E^{+}}$$-formula. Proof. See [6, Theorem 2]. Definition 2.4 We define $$\mathsf{P}^{*}=\lbrace \varphi : \textrm{for some} \psi \in \mathsf{E^{+}},\ \vdash _{c}\psi \leftrightarrow \varphi \textrm{and} \vdash _{i}\psi \rightarrow \varphi \rbrace$$. Fact 2.5 We have $$\varphi (\overline{x})$$ is s-stable if and only if $$\varphi (\overline{x})\in \mathsf{P}^{*}$$. Proof. See [6, Theorem 1]. 3 Our new formula classes In this section, we define a class of formulas denoted by $$\mathsf{R}^{*}$$. In the next section, we prove that a formula is f-stable if and only if it is an $$\mathsf{R}^{*}$$-formula. Frist, we define two classes of formulas that are used in the definition of the target class $$\mathsf{R}^{*}$$. Definition 3.1 We define the classes P and Q of formulas inductively as follows: \begin{array}{lll} \mathsf{At}\subseteq\mathsf{P},&& \lbrace\top, \perp\rbrace\subseteq\mathsf{Q},\\ \varphi,\psi\in\mathsf{P}\ \Rightarrow\ \varphi\vee\psi,\varphi\wedge\psi\in\mathsf{P}, && \varphi,\psi\in \mathsf{Q}\ \Rightarrow\ \varphi\vee\psi,\varphi\wedge\psi\in\mathsf{Q}\\ \varphi\in \mathsf{Q},\psi\in\mathsf{P}\ \Rightarrow\ \varphi\rightarrow\psi\in\mathsf{P}, && \varphi\in \mathsf{P},\psi\in \mathsf{Q}\ \Rightarrow\ \varphi\rightarrow\psi\in\mathsf{Q},\\ \varphi\in\mathsf{P}\ \Rightarrow\ \exists x \,\varphi\in\mathsf{P}, && \varphi\in\mathsf{Q}\ \Rightarrow \quad \forall x \,\varphi\in\mathsf{Q}. \end{array} Note that, the class of Q-formulas does not contain atomic formulas (except ⊤ and ⊥). Proposition 3.2 Let $$\varphi$$ is any formula of $$\mathcal{L}$$. Then the following hold: If $$\varphi \in \mathsf{P}$$, then $$\varphi$$ is s-stable. If $$\varphi \in \mathsf{Q}$$, then $$\neg \varphi$$ is s-stable. Proof. Let $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$ be a Kripke model. We proceed by induction on the complexity of formulas in P ∪ Q, for all $$\alpha$$ simultaneously. Let $$\alpha \in F$$, and let $$\varphi \in \mathsf{P}(A_{\alpha })$$ be a sentence. Suppose $$\varphi \in \mathsf{At}(A_{\alpha })$$, and $$\mathcal{A}_{\alpha }\models \varphi$$. By the definition of forcing, we have $$\alpha \Vdash _{\mathcal{A}}\varphi$$. The induction steps for $$\varphi :=\psi \wedge \theta$$, $$\varphi :=\psi \vee \theta$$ and $$\varphi :=\exists x \,\psi (x)$$ are obvious. Suppose $$\varphi :=\psi \rightarrow \theta$$, where $$\psi \in \mathsf{Q}(A_{\alpha })$$ and $$\theta \in \mathsf{P}(A_{\alpha })$$. Assume that $$\mathcal{A}_{\alpha }\models \psi \rightarrow \theta$$. Let $$\beta \geq \alpha$$ be in F. Suppose that $$\beta \Vdash _{\mathcal{A}}\psi$$. If $$\mathcal{A}_{\alpha }\models \neg \psi$$, since $$\psi \in \mathsf{Q}(A_{\alpha })$$, by the induction hypothesis we have $$\alpha \Vdash _{\mathcal{A}}\neg \psi$$. This is a contradiction. So $$\mathcal{A}_{\alpha }\models \psi$$. Thus, $$\mathcal{A}_{\alpha }\models \theta$$. Since $$\theta \in \mathsf{P}(A_{\alpha })$$, by the induction hypothesis again, $$\alpha \Vdash _{\mathcal{A}}\theta$$. By monotonicity of forcing, $$\beta \Vdash _{\mathcal{A}}\theta$$. Thus, $$\alpha \Vdash _{\mathcal{A}}\psi \rightarrow \theta$$. Let $$\alpha \in F$$, and let $$\varphi \in \mathsf{Q}(A_{\alpha })$$ be a sentence. The case of $$\varphi :=\top$$, $$\varphi :=\perp$$ and the induction steps for $$\varphi :=\psi \wedge \theta$$ and $$\varphi :=\psi \vee \theta$$ are obvious. Suppose $$\varphi :=\psi \rightarrow \theta$$, where $$\psi \in \mathsf{P}(A_{\alpha })$$ and $$\theta \in \mathsf{Q}(A_{\alpha })$$. Assume $$\mathcal{A}_{\alpha }\models \neg (\psi \rightarrow \theta )$$. Thus, $$\mathcal{A}_{\alpha }\models \psi$$ and $$\mathcal{A}_{\alpha }\models \neg \theta$$. Since $$\psi \in \mathsf{P}(A_{\alpha })$$, by the induction hypothesis, $$\alpha \Vdash _{\mathcal{A}}\psi$$. Since $$\theta \in \mathsf{Q}(A_{\alpha })$$, by the induction hypothesis again, $$\alpha \Vdash _{\mathcal{A}}\neg \theta$$. So $$\alpha \Vdash _{\mathcal{A}}\neg (\psi \rightarrow \theta )$$. Now suppose $$\varphi :=\forall x \,\psi (x)$$, where $$\psi \in \mathsf{Q}(A_{\alpha })$$. Assume that $$\mathcal{A}_{\alpha }\models \neg \forall x \,\psi (x)$$. So there is an $$a\in A_{\alpha }$$ such that $$\mathcal{A}_{\alpha }\models \neg \psi (a)$$. Since $$\psi \in \mathsf{Q}(A_{\alpha })$$, by the induction hypothesis, we get $$\alpha \Vdash _{\mathcal{A}}\neg \psi (a)$$. Thus, $$\alpha \Vdash _{\mathcal{A}}\neg \forall x \,\psi (x)$$. In this paper, we define a class of f-stable formulas denoted by $$\mathsf{R}^{*}$$. We prove soundness and completeness theorems for this class with respect to the mentioned property. Definition 3.3 The class of formulas $$\mathsf{R}\subseteq \mathcal{L}$$ is defined by use of P-formulas as follows: At ⊆ R, $$\varphi ,\psi \in \mathsf{R}\ \Rightarrow \ \varphi \vee \psi ,\varphi \wedge \psi \in \mathsf{R}$$, $$\varphi \in \mathsf{P},\psi \in \mathsf{R}\ \Rightarrow\! \ \varphi \rightarrow \psi \in \mathsf{R}$$, $$\varphi \in \mathsf{R}\ \!\Rightarrow \ \exists x \,\varphi , \forall x \,\varphi \in \mathsf{R}$$. Definition 3.4 We define $$\mathsf{R}^{*}=\lbrace \varphi : \textrm{for some}\ \psi \in \mathsf{R}, \vdash _{c}\psi \leftrightarrow \varphi \ \textrm{and} \vdash _{i}\varphi \rightarrow \psi \rbrace$$. Definition 3.5 A formula $$\varphi$$ of $$\mathcal{L}$$ is called semi-positive if, whenever $$\psi \rightarrow \theta$$ is a subformula of $$\varphi$$, $$\psi$$ is atomic. It is easy to see that each semi-positive formula is f-stable. Fact 3.6 Let $$\Gamma \subseteq \Delta$$ be intuitionistic theories over $$\mathcal{L}$$. Then $$\Delta$$ is axiomatisable by semi-positive sentences over $$\Gamma$$ if and only if $$\Delta$$ is preserved under $$\Gamma$$-Kripke submodels. Proof. See Definition 3.5 and [8, Section 3]. In [8], it is shown that the class of those formulas of intuitionistic predicate logic which are preserved under taking submodels of Kripke models is precisely the class of semi-positive formulas. A Kripke model $$\mathcal{A}$$ is a submodel of a Kripke model $$\mathcal{B}$$ if the frame of $$\mathcal{A}$$ is a substructure of the frame of $$\mathcal{B}$$ in the classical sense. As mentioned above, each semi-positive formula is f-stable. It is easy to see that the class of $$\mathsf{R}^{*}$$-formulas is a proper extension of the class of semi-positive formulas. Our example below, shows that $$\mathsf{ R}^{*}$$ strictly extends the class of formulas equivalent to a semi-positive one in intuitionistic predicate logic. Example 3.7 Let $$\mathcal{L}_{1}$$ be a first-order language that includes p and q as two nullary relation symbols. Let $$\mathcal{A}$$ and $$\mathcal{B}$$ be two Kripke models of the pictures in the language $$\mathcal{L}_{1}$$ such that p and q are forced only in the node $$\alpha _{1}$$ in both $$\mathcal{A}$$ and $$\mathcal{B}$$, i.e. the other nodes do not force these atoms. Then, $$\mathcal{A}$$ is a submodel of $$\mathcal{B}$$. The formula $$\neg \neg p\rightarrow q$$ is an $$\mathsf{R}^{*}$$-formula, that is forced in the Kripke model $$\mathcal{B}$$, but not in its submodel $$\mathcal{A}$$. 4 Soundness and completeness theorems for $$\mathsf{R}^{*}$$-formulas In this section, we prove the soundness and completeness theorems for $$\mathsf{R}^{*}$$-formulas. In the other words, we show that the class of $$\mathsf{R}^{*}$$-formulas is exactly the class of f-stable formulas. Lemma 4.1 Let $$\varphi$$ is any formula of $$\mathcal{L}$$. If $$\varphi \in \mathsf{R}$$, then $$\varphi$$ is f-stable. Proof. Let $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$ be a Kripke model. We proceed by induction on the complexity of formulas in R, for all $$\alpha$$. Let $$\alpha \in F$$, and let $$\varphi \in \mathsf{R}(A_{\alpha })$$ be a sentence. Suppose $$\varphi \in \mathsf{At}(A_{\alpha })$$, and $$\alpha \Vdash _{\mathcal{A}}\varphi$$. By definition of forcing, we have $$\mathcal{A}_{\alpha }\models \varphi$$. The induction steps for $$\varphi :=\psi \wedge \theta$$, $$\varphi :=\psi \vee \theta$$ and $$\varphi :=\exists x \,\psi (x)$$ are obvious. Suppose $$\varphi :=\psi \rightarrow \theta$$, where $$\psi \in \mathsf{P}(A_{\alpha })$$ and $$\theta \in \mathsf{R}(A_{\alpha })$$. Assume that $$\alpha \Vdash _{\mathcal{A}}\psi \rightarrow \theta$$. Suppose that $$\mathcal{A}_{\alpha }\models \psi$$. Since $$\psi \in \mathsf{P}(A_{\alpha })$$, by Proposition 3.2, we have $$\alpha \Vdash _{\mathcal{A}}\psi$$. So $$\alpha \Vdash _{\mathcal{A}}\theta$$. By the induction hypothesis, $$\mathcal{A}_{\alpha }\models \theta$$. Thus, $$\mathcal{A}_{\alpha }\models \psi \rightarrow \theta$$. Now suppose that $$\varphi :=\forall x \,\psi (x)$$, where $$\psi \in \mathsf{R}(A_{\alpha })$$. Assume that $$\alpha \Vdash _{\mathcal{A}}\forall x \,\psi (x)$$. Let $$a\in A_{\alpha }$$. So $$\alpha \Vdash _{\mathcal{A}}\psi (a)$$. By the induction hypothesis, we get $$\mathcal{A}_{\alpha }\models \psi (a)$$. Thus, $$\mathcal{A}_{\alpha }\models \forall x \,\psi (x)$$. Theorem 4.2 (Soundness) Let $$\varphi$$ is any formula of $$\mathcal{L}$$. If $$\varphi \in \mathsf{R}^{*}$$, then $$\varphi$$ is f-stable. Proof. Let $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$ be a Kripke model. Let $$\alpha \in F$$, and let $$\varphi \in \mathsf{R}^{*}(A_{\alpha })$$ be a sentence. Suppose $$\alpha \Vdash _{\mathcal{A}}\varphi$$. By Definition 3.4, there is $$\psi \in \mathsf{R}$$ such that $$\vdash _{i}\varphi \rightarrow \psi$$ and $$\vdash _{c}\psi \leftrightarrow \varphi$$. Thus, $$\alpha \Vdash _{\mathcal{A}}\psi$$. By Lemma 4.1, we have $$\mathcal{A}_{\alpha }\models \psi$$. Thus, $$\mathcal{A}_{\alpha }\models \varphi$$. Definition 4.3 Let C be a set of constants. Let $$\Gamma$$ be a classical consistent theory, and let $$\Delta$$ be an intuitionistic consistent theory over $$\mathcal{L}(C)$$. The triple $$\langle \Gamma ,C,\Delta \rangle$$ is called acceptable if $$\Gamma \cap \mathsf{P}(C)\subseteq \Delta$$ and $$\Delta \cap \mathsf{R}(C)\subseteq \Gamma$$. Lemma 4.4 Let C be a set of constants. Let $$\Gamma$$ be a classical consistent theory, and let $$\Delta$$ be an intuitionistic consistent theory over $$\mathcal{L}(C)$$. We have, if $$\Delta \cap \mathsf{R}(C)\subseteq \Gamma$$, then the triple $$\langle\Gamma,C,\mathsf{Th}_{i}[C](\Delta\cup(\Gamma\cap\mathsf{P}(C)))\rangle$$ is acceptable. Proof. Let $$\Delta^{\prime}=\mathsf{Th}_{i}[C](\Delta\cup(\Gamma\cap\mathsf{P}(C))).$$ Obviously, $$\Gamma \cap \mathsf{P}(C)\subseteq \Delta ^{\prime }$$. We must show that $$\Delta ^{\prime }\cap \mathsf{R}(C)\subseteq \Gamma$$. Let $$\varphi \in \Delta ^{\prime }\cap \mathsf{R}(C)$$. Then $$\Delta\cup(\Gamma\cap\mathsf{P}(C))\vdash_{i}\varphi.$$ It follows that $$\Delta \cup \{\varrho \}\vdash _{i}\varphi$$, where $$\varrho$$ is a formula in $$\Gamma \cap \mathsf{P}(C)$$. So $$\Delta \vdash _{i}\varrho \rightarrow \varphi$$. Since $$\varrho \in \mathsf{P}(C)$$ and $$\varphi \in \mathsf{R}(C)$$, we have $$\varrho \rightarrow \varphi \in \mathsf{R}\,(C)$$. Thus, $$\varrho \rightarrow \varphi \in \Delta \cap \mathsf{R}(C)\subseteq \Gamma$$. So $$\Gamma \vdash _{c}\varrho \rightarrow \varphi$$. Also, $$\Gamma \vdash _{c}\varrho$$. So $$\Gamma \vdash _{c}\varphi$$. Since ⊥ ∈ R and $$\Gamma$$ is consistent, by $$\Delta ^{\prime }\cap \mathsf{R}(C)\subseteq \Gamma$$, $$\Delta ^{\prime }$$ is also consistent. Proposition 4.5 Let C be a set of constants, and let $$\Gamma$$ and $$\Delta$$ be theories such that $$\langle \Gamma ,C,\Delta \rangle$$ is acceptable. Let D be a set of constants not in $$\mathcal{L}(C)$$ with $$\vert D\vert \geq \vert \mathcal{L}(C)\vert$$. Let $$\varphi \in \mathcal{L}(C)$$ be such that $$\Gamma \nvdash _{c}\varphi$$. Then there is an acceptable triple $$\langle \Gamma ^{\prime },C^{\prime },\Delta ^{\prime }\rangle$$ such that $$\Gamma ^{\prime }$$ is a $$C^{\prime }$$-Henkin prime classical theory, $$\Delta ^{\prime }$$ is a $$C^{\prime }$$-Henkin prime intuitionistic theory, $$\Gamma \subseteq \Gamma ^{\prime },\ \Delta \subseteq \Delta ^{\prime }, C\subseteq C^{\prime }\subseteq C\cup D$$ and $$\Gamma ^{\prime }\nvdash _{c}\varphi$$. Proof. We construct a increasing chain of acceptable triples $$\langle \Gamma _{n},C_{n}, \Delta _{n} \rangle$$ with $$\Gamma _{n}\nvdash _{c}\varphi$$ such that for all $$n\in{\mathbb{N}}$$: $$\Gamma _{3n+1}$$ and $$\Delta _{3n+1}$$ are prime; If $$\Gamma _{3n+1}\vdash _{c}\exists x \,\psi (x)$$, then $$\Gamma _{3n+2}\vdash _{c}\psi (e)$$ for some $$e\in C_{3n+2}$$; If $$\Delta _{3n+2}\vdash _{i}\exists x \,\psi (x)$$, then $$\Delta _{3n+3}\vdash _{i}\psi (e)$$ for some $$e\in C_{3n+3}$$. Set $$\Gamma _{0}=\Gamma$$, $$C_{0}=C$$ and $$\Delta _{0}=\Delta$$. We proceed by induction on $$n\in{\mathbb{N}}$$. Step $$3n+1$$: Suppose that $$\langle \Gamma _{3n},C_{3n},\Delta _{3n}\rangle$$ is acceptable, and $$\Gamma _{3n}\nvdash _{c}\varphi$$. Let S be the set of all acceptable triples $$\langle \Gamma ^{*},C_{3n},\Delta ^{*}\rangle$$ such that $$\Gamma _{3n}\subseteq \Gamma ^{*}$$, $$\Delta _{3n}\subseteq \Delta ^{*}$$ and $$\Gamma ^{*}\nvdash _{c}\varphi$$. We define a partial order on S by set inclusion: $$\langle \Gamma ,C_{3n},\Delta \rangle \leq \langle \Gamma ^{^{\prime }},C_{3n},\Delta ^{^{\prime }}\rangle$$ if and only if $$\Gamma \subseteq \Gamma ^{^{\prime }}$$ and $$\Delta \subseteq \Delta ^{^{\prime }}$$. It is clear from the definition of acceptable triples and compactness that S is closed under unions of chains. Thus, by Zorn’s Lemma, there is a maximal element $$\langle \Gamma _{3n+1},C_{3n},\Delta _{3n+1}\rangle \in \mathbf{S}$$. Set $$C_{3n+1}=C_{3n}$$. Let $$\Gamma _{3n+1}\vdash _{c} \psi \vee \theta$$. Assume $$\Gamma _{3n+1}\cup \lbrace \psi \rbrace \vdash _{c}\varphi$$ and $$\Gamma _{3n+1}\cup \lbrace \theta \rbrace \vdash _{c}\varphi$$. Then $$\Gamma_{3n+1}\vdash_{c}(\psi\rightarrow\varphi)\wedge(\theta\rightarrow\varphi).$$ So $$\Gamma _{3n+1}\vdash _{c}(\psi \vee \theta )\rightarrow \varphi$$. Hence, $$\Gamma _{3n+1}\vdash _{c}\varphi$$, contradiction. Thus, without loss of generality, we may suppose that $$\Gamma _{3n+1}\cup \lbrace \psi \rbrace \nvdash _{c}\varphi$$. Let \begin{array}{ccccc} \Gamma^{\prime}=\mathsf{Th}_{c}[C_{3n}](\Gamma_{3n+1}\cup\lbrace\psi\rbrace) & &\textrm{and} & & \Delta^{\prime}=\mathsf{Th}_{i}[C_{3n}](\Delta_{3n+1}\cup(\Gamma^{\prime}\cap\mathsf{P}(C_{3n}))). \end{array} By the acceptability of $$\langle \Gamma _{3n+1},C_{3n},\Delta _{3n+1}\rangle$$, we have $$\Delta_{3n+1}\cap\mathsf{R}(C_{3n})\subseteq\Gamma_{3n+1}\subseteq\Gamma^{\prime}.$$ So by Lemma 4.4, the triple $$\langle \Gamma ^{\prime },C_{3n},\Delta ^{\prime }\rangle$$ is acceptable, hence in S. By maximality, since $$\Gamma _{3n+1}\subseteq \Gamma ^{\prime }$$ and $$\Delta _{3n+1}\subseteq \Delta ^{\prime }$$, we have $$\Gamma _{3n+1}=\Gamma ^{\prime }$$. Thus, $$\Gamma _{3n+1}\vdash _{c}\psi$$. Suppose $$\Delta _{3n+1}\vdash _{i}\chi \vee \xi$$. Assume $$\Gamma_{3n+1}\cup(\mathsf{Th}_{i}[C_{3n}](\Delta_{3n+1}\cup\lbrace\chi\rbrace)\cap\mathsf{R}(C_{3n}))\vdash_{c}\varphi$$ and $$\Gamma_{3n+1}\cup(\mathsf{Th}_{i}[C_{3n}](\Delta_{3n+1}\cup\lbrace\xi\rbrace)\cap\mathsf{R}(C_{3n}))\vdash_{c}\varphi .$$ By compactness, since $$\mathsf{R}(C_{3n})$$ is closed under finite conjunctions, there are \begin{array}{ccccc} \varrho\in\mathsf{Th}_{i}[C_{3n}](\Delta_{3n+1}\cup\lbrace\chi\rbrace)\cap\mathsf{R}(C_{3n}) & & \textrm{and} & & \sigma\in\mathsf{Th}_{i}[C_{3n}](\Delta_{3n+1}\cup\lbrace\xi\rbrace)\cap\mathsf{R}(C_{3n}) \end{array} such that $$\Gamma _{3n+1}\cup \lbrace \varrho \rbrace \vdash _{c}\varphi$$ and $$\Gamma _{3n+1}\cup \lbrace \sigma \rbrace \vdash _{c}\varphi .$$ So $$\Gamma_{3n+1}\vdash_{c}(\varrho\rightarrow\varphi)\wedge(\sigma\rightarrow\varphi).$$ Thus, $$\Gamma _{3n+1}\vdash _{c}(\varrho \vee \sigma )\rightarrow \varphi$$. Also, we have $$\Delta _{3n+1}\cup \lbrace \chi \rbrace \vdash _{i}\varrho$$ and $$\Delta _{3n+1}\cup \lbrace \xi \rbrace \vdash _{i}\sigma$$. Thus, $$\Delta_{3n+1}\vdash_{i}(\chi\rightarrow\varrho)\wedge(\xi\rightarrow\sigma).$$ So $$\Delta _{3n+1}\vdash _{i}(\chi \vee \xi )\rightarrow (\varrho \vee \sigma )$$. Thus, $$\Delta _{3n+1}\vdash _{i}\varrho \vee \sigma$$. Since $$\mathsf{R}(C_{3n})$$ is closed under finite disjunctions, $$\varrho\vee\sigma\in\Delta_{3n+1}\cap\mathsf{R}(C_{3n}).$$ By acceptability of $$\langle \Gamma _{3n+1},C_{3n},\Delta _{3n+1}\rangle$$, we get $$\varrho \vee \sigma \in \Gamma _{3n+1}$$. So $$\Gamma _{3n+1}\vdash _{c}\varphi$$, contradiction. Thus, without loss of generality, we may suppose $$\Gamma _{3n+1}\cup (\mathsf{Th}_{i}[C_{3n}](\Delta _{3n+1}\cup \lbrace \chi \rbrace )\cap \mathsf{R}(C_{3n}))\nvdash _{c}\varphi$$. Let $$\Gamma^{\prime}=\mathsf{Th}_{c}[C_{3n}](\Gamma_{3n+1}\cup(\mathsf{Th}_{i}[C_{3n}](\Delta_{3n+1}\cup\lbrace\chi\rbrace)\cap\mathsf{R}(C_{3n})))$$ and $$\Delta^{\prime}=\mathsf{Th}_{i}[C_{3n}]((\Delta_{3n+1}\cup\lbrace\chi\rbrace)\cup(\Gamma^{\prime}\cap\mathsf{P}(C_{3n}))).$$ By Lemma 4.4, the triple $$\langle \Gamma ^{\prime },C_{3n},\Delta ^{\prime }\rangle$$ is acceptable, hence in S. By maximality, since $$\Gamma _{3n+1}\subseteq \Gamma ^{\prime }$$ and $$\Delta _{3n+1}\subseteq \Delta ^{\prime }$$, we have $$\Delta _{3n+1}=\Delta ^{\prime }$$. Thus, $$\Delta _{3n+1}\vdash _{i}\chi$$. Step $$3n+2$$: Suppose that $$\langle \Gamma _{3n+1},C_{3n+1},\Delta _{3n+1}\rangle$$ is acceptable, and $$\Gamma _{3n+1}\nvdash _{c}\varphi$$. For every sentence $$\exists x \,\psi (x)\in \Gamma _{3n+1}$$, let $$e_{\exists x \,\psi (x)}$$ be a new constant in D. Let $$D^{\prime}=\{e_{\exists x \,\psi(x)}:\exists x \,\psi(x)\in\Gamma_{3n+1}\}.$$ We pick $$D^{\prime }$$ so that $$\vert D\setminus (C_{3n+1} \cup D^{\prime })\vert = \vert D\vert$$. Set $$C_{3n+2}=C_{3n+1}\cup D^{\prime }$$. Set $$\Gamma_{3n+2}=\mathsf{Th}_{c}[C_{3n+2}](\Gamma_{3n+1}\cup\{\psi(e_{\exists x \,\psi(x)}):\exists x \,\psi(x)\in\Gamma_{3n+1}\}).$$ Now we show that $$\Gamma _{3n+2}\nvdash _{c}\varphi$$. Assume $$\Gamma _{3n+2}\vdash _{c}\varphi$$. By compactness, there is a sentence $$\theta (\overline{a}):=\psi _{1}(a_{1})\wedge \cdots \wedge \psi _{m}(a_{m})$$, with $$\overline{a}\in D^{\prime }$$ and $$\exists x \,\psi_{1}(x),\ldots,\exists x \,\psi_{m}(x)\in\Gamma_{3n+1}$$ such that $$\Gamma _{3n+1}\cup \{\theta (\overline{a})\}\vdash _{c}\varphi$$. So $$\Gamma _{3n+1}\vdash _{c}\theta (\overline{a})\rightarrow \varphi$$. Since $$\overline a\not \in C_{3n+1}$$, we have $$\Gamma_{3n+1}\vdash_{c}\forall \overline{x} \,(\theta(\overline{x})\rightarrow\varphi).$$ So $$\Gamma _{3n+1}\vdash _{c}\exists \overline{x}\, \theta (\overline{x})\rightarrow \varphi$$. Since $$\exists x \,\psi_{1}(x),\ldots,\exists x \,\psi_{m}(x)\in\Gamma_{3n+1},$$ we have $$\exists \overline{x} \,\theta (\overline x)\in \Gamma _{3n+1}$$. So $$\Gamma _{3n+1}\vdash _{c}\varphi$$, contradiction. Thus, $$\Gamma _{3n+2}\nvdash _{c}\varphi$$. We claim $$\mathsf{Th}_{i}[C_{3n+2}](\Delta_{3n+1})\cap\mathsf{R}(C_{3n+2})\subseteq\Gamma_{3n+2}.$$ Suppose that $$\varrho (\overline{e})\in \mathsf{Th}_{i}[C_{3n+2}](\Delta _{3n+1})\cap \mathsf{R}(C_{3n+2})$$ such that $$\overline{e}\in D^{\prime }$$. Since $$\overline{e}\not \in C_{3n+1}$$, we have $$\Delta _{3n+1}\vdash _{i}\forall \overline{x} \,\varrho (\overline{x})$$. By the induction hypothesis, $$\forall \overline{x} \,\varrho (\overline{x})\in \Delta _{3n+1}\cap \mathsf{R}(C_{3n+1})\subseteq \Gamma _{3n+1}$$. Thus, $$\Gamma _{3n+1}\vdash _{c}\forall \overline{x} \,\varrho (\overline{x})$$. So $$\Gamma _{3n+2}\vdash _{c}\varrho (\overline{e})$$, which proves the claim. Set $$\Delta_{3n+2}=\mathsf{Th}_{i}[C_{3n+2}](\Delta_{3n+1}\cup(\Gamma_{3n+2}\cap\mathsf{P}(C_{3n+2}))).$$ By Lemma 4.4, $$\langle \Gamma _{3n+2},C_{3n+2},\Delta _{3n+2}\rangle$$ is acceptable. Step $$3n+3$$: Suppose $$\langle \Gamma _{3n+2},C_{3n+2},\Delta _{3n+2}\rangle$$ is acceptable, and $$\Gamma _{3n+2}\nvdash _{c}\varphi$$. For every sentence $$\exists x \,\psi (x)\in \Delta _{3n+2}$$, let $$e_{\exists x \,\psi (x)}$$ be a new constant in D. Let $$D^{\prime}=\{e_{\exists x \,\psi(x)}:\exists x \,\psi(x)\in\Delta_{3n+2}\}.$$ We pick $$D^{\prime }$$ so that $$\vert D\setminus (C_{3n+2} \cup D^{\prime })\vert = \vert D\vert$$. Set $$C_{3n+3}=C_{3n+2}\cup D^{\prime }$$. Set $$\Delta^{\prime}=\mathsf{Th}_{i}[C_{3n+3}](\Delta_{3n+2}\cup\{\psi(e_{\exists x \,\psi(x)}):\exists x \,\psi(x)\in\Delta_{3n+2}\}).$$ We claim $$\mathsf{Th}_{c}[C_{3n+3}](\Gamma_{3n+2}\cup(\Delta^{\prime}\cap\mathsf{R}(C_{3n+3})))\nvdash_{c}\varphi.$$ Assume $$\mathsf{Th}_{c}[C_{3n+3}](\Gamma _{3n+2}\cup (\Delta ^{\prime }\cap \mathsf{R}(C_{3n+3})))\vdash _{c}\varphi$$. Since $$\mathsf{R}(C_{3n+2})$$ is closed under finite conjunction, by compactness there is a sentence $$\theta (\overline{a})\in \Delta ^{\prime }\cap \mathsf{R}(C_{3n+3})$$ with $$\overline{a}\in D^{\prime }$$, such that $$\Gamma _{3n+2}\cup \lbrace \theta (\overline{a})\rbrace \vdash _{c}\varphi$$. By compactness again, there is a sentence $$\varrho (\overline{e}):=\psi _{1}(e_{1})\wedge \cdots \wedge \psi _{m}(e_{m})$$ with $$\overline{e}\in D^{\prime }$$ and $$\exists x \,\psi_{1}(x),\ldots,\exists x \,\psi_{m}(x)\in\Delta_{3n+2}$$ such that $$\Delta _{3n+2}\cup \lbrace \varrho (\overline{e})\rbrace \vdash _{i}\theta (\overline{a})$$. So $$\Delta _{3n+2}\cup \lbrace \varrho (\overline{e})\rbrace \vdash _{i}\exists \overline{y} \,\theta (\overline{y})$$. So $$\Delta _{3n+2}\vdash _{i}\varrho (\overline{e})\rightarrow \exists \overline{y} \,\theta (\overline{y})$$. Since $$\overline{e}\not \in C_{3n+2}$$, we have $$\Delta _{3n+2}\vdash _{i}\forall \overline{x} \,(\varrho (\overline{x})\rightarrow \exists \overline{y} \,\theta (\overline{y}))$$. Thus, $$\Delta _{3n+2}\vdash _{i}\exists \overline{x} \,\varrho (\overline{x})\rightarrow \exists \overline{y} \,\theta (\overline{y})$$. Since $$\exists x \,\psi_{1}(x),\ldots,\exists x \,\psi_{m}(x)\in\Delta_{3n+2},$$ we have $$\exists \overline{x} \,\varrho (\overline{x})\in \Delta _{3n+2}$$. Thus, $$\Delta _{3n+2}\vdash _{i}\exists \overline{y} \,\theta (\overline{y})$$. Since $$\mathsf{R}(C_{3n+2})$$ is closed under existential quantification, we have $$\exists \overline{y} \,\theta (\overline{y})\in \mathsf{R}(C_{3n+2}).$$ By the induction hypothesis, $$\Delta _{3n+2}\cap \mathsf{R}(C_{3n+2})\subseteq \Gamma _{3n+2}$$. So $$\Gamma _{3n+2}\vdash _{c}\exists \overline{y} \,\theta (\overline{y})$$. Since $$\Gamma_{3n+2}\cup\lbrace\theta(\overline{a})\rbrace\vdash_{c}\varphi$$ and $$\overline{a}$$ does not appear in $$\Gamma _{3n+2}$$ and $$\varphi$$, we have $$\Gamma _{3n+2}\cup \lbrace \exists \overline{y} \,\theta (\overline{y})\rbrace \vdash _{c}\varphi$$. So $$\Gamma _{3n+2}\vdash _{c}\varphi$$, contradiction. Set $$\Gamma_{3n+3}=\mathsf{Th}_{c}[C_{3n+3}](\Gamma_{3n+2}\cup(\Delta^{\prime}\cap\mathsf{R}(C_{3n+3}))),$$ and $$\Delta_{3n+3}=\mathsf{Th}_{i}[C_{3n+3}](\Delta^{\prime}\cup(\Gamma_{3n+3}\cap\mathsf{P}(C_{3n+3}))).$$ By Lemma 4.4$$\langle \Gamma _{3n+3},C_{3n+3},\Delta _{3n+3}\rangle$$ is acceptable. This completes the construction and the inductive verification of its correctness. Set $$\Gamma ^{\prime }=\underset{n\in{\mathbb{N}}}\bigcup \Gamma _{n}$$, $$C^{\prime }=\underset{n\in{\mathbb{N}}}\bigcup C_{n}$$ and $$\Delta ^{\prime }=\underset{n\in{\mathbb{N}}}\bigcup \Delta _{n}$$. By compactness, $$\Gamma ^{\prime }$$ is a $$C^{\prime }$$-Henkin prime classical theory, $$\Delta ^{\prime }$$ is a $$C^{\prime }$$-Henkin prime intuitionistic theory, and $$\Gamma ^{\prime }\nvdash _{c}\varphi$$. Clearly, $$\langle \Gamma ^{\prime }, C^{\prime }, \Delta ^{\prime }\rangle$$ is an acceptable triple. A C-Henkin prime intuitionistic theory $$\Pi$$ defines a classical structure with domain C as follows: Definition 4.6 Suppose $$\Pi$$ is a C-Henkin prime intuitionistic theory. Then $$\mathcal{A}_{\Pi }$$ is defined to be the classical structure in the language of $$\mathcal{L}(C)$$ such that the domain of $$\mathcal{A}_{\Pi }$$ is C itself ($$c^{\mathcal{A}_{\Pi }}=c$$) and such that for every atomic C-sentence $$\varphi$$, $$\mathcal{A}_{\Pi }\models \varphi$$ if and only if $$\Pi \vdash _{i}\varphi$$. It is straightforward to check that the definition of $$\mathcal{A}_{\Pi }$$ does indeed specify a unique classical structure. Proposition 4.7 Let C be a set of constants, and $$\Gamma$$ and $$\Delta$$ be theories such that: $$\langle \Gamma ,C,\Delta \rangle$$ is acceptable; $$\Gamma$$ is a C-Henkin prime classical theory; $$\Delta$$ is a C-Henkin prime intuitionistic theory. Then there exists a Kripke model $$\mathcal{A}$$ with the root $$\alpha _{0}=\langle \Gamma , C, \Delta \rangle$$ such that: $$\alpha _{0}\Vdash _{\mathcal{A}}\varphi \Leftrightarrow \Delta \vdash _{i}\varphi$$, for all $$\varphi \in \mathcal{L}(C)$$, $$\mathcal{A}_{\alpha _{0}}\models \varphi \Leftrightarrow \Gamma \vdash _{c}\varphi$$, for all $$\varphi \in \mathcal{L}(C)$$. Proof. Let D be a set of new constants such that $$\vert D\vert \geq \vert \mathcal{L}(C) \vert$$. Let F be the following poset: The root of F is the acceptable triple $$\langle \Gamma ,C,\Delta \rangle$$ and the other nodes are $$\langle C^{\prime },\Delta ^{\prime }\rangle$$ such that $$C\subseteq C^{\prime } \subseteq C \cup D$$, $$\vert D\setminus C^{\prime }\vert =\vert D\vert$$ and $$\Delta \subseteq \Delta ^{\prime }$$, where $$\Delta ^{\prime }$$ is a consistent $$C^{\prime }$$-Henkin prime intuitionistic theory. The order on F is defined by $$\langle C^{\prime },\Delta ^{\prime }\rangle \leq \langle C^{\prime \prime },\Delta ^{\prime \prime }\rangle$$ if and only if $$C^{\prime }\subseteq C^{\prime \prime }$$ and $$\Delta ^{\prime }\subseteq \Delta ^{\prime \prime }$$. A Kripke model $$\mathcal{A}=((\mathcal{A}_{\alpha })_{\alpha \in F},\leq )$$ is defined by: $$A_{\langle \Gamma ,C,\Delta \rangle }=C$$, $$\langle \Gamma ,C,\Delta \rangle \Vdash _{\mathcal{A}} \varphi \Leftrightarrow \Delta \vdash _{i}\varphi$$, for all $$\varphi \in \mathsf{At}(C)$$, $$A_{\langle C^{\prime },\Delta ^{\prime }\rangle }=C^{\prime }$$, $$\langle C^{\prime },\Delta ^{\prime }\rangle \Vdash _{\mathcal{A}} \varphi \Leftrightarrow \Delta ^{\prime }\vdash _{i}\varphi$$, for all $$\varphi \in \mathsf{At}(C^{\prime })$$. It is straightforward to verify that $$\mathcal{A}$$ is a Kripke model with the root $$\alpha _{0}=\langle \Gamma ,C,\Delta \rangle \in F$$. We claim the following: 1. For all $$\alpha \in F$$, i. If $$\alpha =\alpha _{0}=\langle \Gamma ,C,\Delta \rangle$$, then $$\alpha _{0}\Vdash _{\mathcal{A}}\varphi \Leftrightarrow \Delta \vdash _{i}\varphi$$ for all $$\varphi \in \mathcal{L}(C)$$; ii. If $$\alpha =\langle C^{\prime },\Delta ^{\prime }\rangle$$, then $$\alpha \Vdash _{\mathcal{A}}\varphi \Leftrightarrow \Delta ^{\prime }\vdash _{i}\varphi$$ for all $$\varphi \in \mathcal{L}(C^{\prime })$$; 2. $$\mathcal{A}_{\alpha _{0}}\models \varphi \Leftrightarrow \Gamma \vdash _{c}\varphi$$ for all $$\varphi \in \mathcal{L}(C)$$. For the prove of the first assertion, see [2, Section 5.3]. For the prove of second assertion, note that since $$\langle \Gamma ,C,\Delta \rangle$$ is acceptable, we have $$\Gamma \cap \mathsf{At}(C)=\Delta \cap \mathsf{At}(C)$$. So $$\mathcal{A}_{\langle \Gamma ,C,\Delta \rangle }$$ coincides with the canonical model of $$\Gamma$$. Since $$\alpha _{0}=\langle \Gamma ,C,\Delta \rangle \in F$$, we have the following: $$\alpha _{0}\Vdash _{\mathcal{A}}\varphi \Leftrightarrow \Delta \vdash _{i}\varphi$$, for all $$\varphi \in \mathcal{L}(C)$$, $$\mathcal{A}_{\alpha _{0}}\models \varphi \Leftrightarrow \Gamma \vdash _{c}\varphi$$, for all $$\varphi \in \mathcal{L}(C)$$. This completes the proof. Theorem 4.8 (Completeness) Let $$\varphi$$ be a sentence in $$\mathcal{L}$$ such that $$\varphi \notin \mathsf{R}^{*}$$. Then there exists a rooted Kripke model $$\mathcal{A}$$ with the root $$\alpha _{0}$$ such that $$\alpha _{0}\Vdash _{\mathcal{A}}\varphi$$ and $$\mathcal{A}_{\alpha _{0}}\not \models \varphi$$. Proof. Consider the triple $$\langle\mathsf{Th}_{c}(\lbrace\neg\varphi\rbrace\cup(\mathsf{Th}_{i}\lbrace\varphi\rbrace\cap\mathsf{R})), \emptyset, \mathsf{Th}_{i}\lbrace\varphi\rbrace\rangle.$$ We claim $$\mathsf{Th}_{c}(\lbrace \neg \varphi \rbrace \cup (\mathsf{Th}_{i}\lbrace \varphi \rbrace \cap \mathsf{R}))$$ is consistent. Suppose $$\mathsf{Th}_{c}(\lbrace \neg \varphi \rbrace \cup (\mathsf{Th}_{i}\lbrace \varphi \rbrace \cap \mathsf{R}))\vdash _{c}\perp$$. Thus, there exists $$\psi \in \mathsf{Th}_{i}\lbrace \varphi \rbrace \cap \mathsf{R}$$ such that $$\lbrace \neg \varphi \rbrace \cup \lbrace \psi \rbrace \vdash _{c}\perp$$. Then $$\neg \varphi \vdash _{c}\neg \psi$$. Then $$\vdash _{c}\neg \varphi \rightarrow \neg \psi$$. So we have $$\vdash _{c}\psi \rightarrow \varphi$$. Since $$\psi \in \mathsf{Th}_{i}\lbrace \varphi \rbrace$$, we have $$\vdash _{i}\varphi \rightarrow \psi$$. Thus, $$\vdash _{c}\varphi \rightarrow \psi$$. Thus, we have $$\vdash _{c}\psi \leftrightarrow \varphi$$ and $$\vdash _{i}\varphi \rightarrow \psi$$. So $$\varphi \in \mathsf{R}^{*}$$ and it is a contradiction. So $$\mathsf{Th}_{c}(\lbrace \neg \varphi \rbrace \cup (\mathsf{Th}_{i}\lbrace \varphi \rbrace \cap \mathsf{R}))$$ is consistent. By Lemma 4.4, the triple $$\langle\mathsf{Th}_{c}(\lbrace\neg\varphi\rbrace\cup(\mathsf{Th}_{i}\lbrace\varphi\rbrace\cap\mathsf{R})), \emptyset, \mathsf{Th}_{i}(\lbrace\varphi\rbrace\cup\mathsf{Th}_{c}(\lbrace\neg\varphi\rbrace\cup(\textrm{Th}_{i}\lbrace\varphi\rbrace\cap\mathsf{R}))\cap\mathsf{P})\rangle$$ is acceptable. By Proposition 4.5, there exists the acceptable triple $$\langle \Gamma ,C,\Delta \rangle$$ such that $$\Gamma$$ is a C-Henkin prime classical theory, $$\Delta$$ is a C-Henkin prime intuitionistic theory, $$\mathsf{Th}_{c}(\lbrace \neg \varphi \rbrace \cup (\mathsf{Th}_{i}\lbrace \varphi \rbrace \cap \mathsf{R}))\subseteq \Gamma ,\mathsf{Th}_{i}(\lbrace \varphi \rbrace \cup \mathsf{Th}_{c}(\lbrace \neg \varphi \rbrace \cup (\mathsf{Th}_{i}\lbrace \varphi \rbrace \cap \mathsf{R}))\cap \mathsf{P})\subseteq \Delta$$ and $$\Gamma \nvdash _{c}\varphi$$. By Proposition 4.7, there exists a rooted Kripke model $$\mathcal{A}$$ with the root $$\alpha _{0}=\langle \Gamma ,C,\Delta \rangle$$ such that $$\alpha _{0}\Vdash _{\mathcal{A}}\varphi \Leftrightarrow \Delta \vdash _{i}\varphi$$, for all $$\varphi \in \mathcal{L}(C)$$, $$\mathcal{A}_{\alpha _{0}}\models \varphi \Leftrightarrow \Gamma \vdash _{c}\varphi$$, for all $$\varphi \in \mathcal{L}(C)$$. Thus, $$\alpha _{0}\Vdash _{\mathcal{A}}\varphi$$ and $$\mathcal{A}_{\alpha _{0}}\not \models \varphi$$. A natural question one can ask here is this: is there an appropriate syntactic description of the class $$\mathsf{R}^{*}$$? The answer suggested by the reviewer of the paper is ‘No’, if we ask for a decidable desciption. His (her) proof is as follows. Let $$\psi$$ be any sentence of a language with at least one binary predicate symbol R. Let P be a fresh nullary predicate symbol. We extend our language with P. We claim that $$\psi \vee \neg \neg P$$ is f-stable if and only if $$\models \psi$$. The right-to-left part is trivial. We treat the left-to-right part via contraposition. Suppose $$\not \models \psi$$. Let $$\mathcal{A}$$ be a classical model such that $$\mathcal{A}\not \models \psi$$. We now build a Kripke model with two nodes 0 and 1 with 0 ≤ 1. At 0 we have the model $$\mathcal{A}$$ and we declare P not forced and at 1 we have again $$\mathcal{A}$$ and we declare P forced. Clearly, $$0\Vdash \psi \vee \neg \neg P$$ but $$\mathcal{A}_{0}\not \models \psi \vee \neg \neg P$$. So, $$\psi \vee \neg \neg P$$ is not f-stable. Since in this way we can reduce classical validity for a language with a binary predicate symbol to f-stability, the property of f-stability is not decidable. Hence, the class of f-stable formulas, i.e. $$\mathsf{R}^{*}$$, has no decidable description. 5 Appendix In this part, we review some classes of formulas introduced by Markovic. Although we did not use them in this paper, they motivated some of our definitions. In trying to give a syntactic characterisation of $$\mathsf{P}^{*}$$, the following class $$\mathsf{S}_{\omega }$$ of formulas is defined in [6]. Definition 5.1 Let $$\mathsf{S}_{0}=\mathsf{E^{+}}$$. Define $$\mathsf{S}_{n+1}$$ as an extension of $$\mathsf{S}_{n}$$ such that for each $$\varphi , \psi \in \mathsf{S}_{n}$$ we have $$\neg \varphi \rightarrow \psi \in \mathsf{S}_{n+1}$$, $$\varphi \vee \psi , \varphi \wedge \psi , \exists x \,\varphi \in \mathsf{S}_{n+1}$$. Finally, define $$\mathsf{S}_{\omega }=\underset{n\in{\mathbb{N}}}\bigcup \mathsf{S}_{n}$$. In [6], it is shown that $$\mathsf{S}_{\omega }\subseteq \mathsf{P}^{*}$$. Consider the formula $$\theta :=(\varphi \rightarrow (\neg \psi \vee \neg \chi ))\rightarrow \xi$$, where $$\varphi , \psi , \chi , \xi$$ are in $$\mathsf{E^{+}}$$. It is easy to see that $$\theta$$ is in $$\mathsf{P}^{*}$$ but is not in $$\mathsf{S}_{\omega }$$. Thus, $$\mathsf{S}_{\omega }\subsetneqq \mathsf{P}^{*}$$. Our class of P-formulas (see Definition 3.1) is an extension of $$\mathsf{S}_{\omega }$$ and it is contained in $$\mathsf{P}^{*}$$. Clearly, the formula $$\theta$$ is in P. So, $$\mathsf{S}_{\omega }\subsetneqq \mathsf{P}$$. In [5], a class R of f-stable formulas is defined. Definition 5.2 Let $$\mathbf{R}_{0} = \mathsf{E^{+}}\cup \lbrace \neg \varphi : \varphi \in \mathsf{P}^{*}\rbrace$$. We define $$\mathbf{R}_{n+1}$$ as an extension of $$\mathbf{R}_{n}$$ such that: If $$\varphi \in \mathsf{P}^{*}$$ and $$\psi \in \mathbf{R}_{n}$$ then $$\varphi \rightarrow \psi \in \mathbf{R}_{n+1}$$, If $$\varphi , \psi \in \mathbf{R}_{n}$$ then $$\varphi \vee \psi , \varphi \wedge \psi , \exists x \,\varphi , \forall x \,\varphi \in \mathbf{R}_{n+1}$$. Finally, define $$\mathbf{R}_{\omega }=\underset{n\in{\mathbb{N}}}\bigcup \mathbf{R}_{n}$$. Definition 5.3 We define $$\mathbf{R}=\lbrace \varphi : \textrm{for some} \ \psi \in \mathbf{R}_{\omega }, \vdash _{c}\psi \leftrightarrow \varphi \ \textrm{and}\vdash _{i}\varphi \rightarrow \psi \rbrace$$. Fact 5.4 If $$\varphi (\overline{x})\in \mathbf{R}$$, then $$\varphi (\overline{x})$$ is f-stable. Proof. See [5, Corollary 1]. Acknowledgements We would like to thank the anonymous referee of the paper for his (her) invaluable comments and suggestions. References [1] S. R. Buss . Intuitionistic validity in T-normal Kripke structures . Annals of Pure and Applied Logic , 55 , 159 – 173 , 1993 . Google Scholar Crossref Search ADS [2] D. van Dalen . Logic and Structure , 4th edn. Springer-Verlag , 2004 . [3] D. van Dalen , H. Mudler , E. C. W. Krabbe and A. Visser . Finite Kripke models of HA are locally PA . Notre Dame Journal of Formal Logic, 27 , 528 – 532 , 1986 . Google Scholar Crossref Search ADS [4] B. Ellison , J. Fleischmann , D. McGinn and W. Ruitenburg . Kripke submodels and universal sentences . Mathematical Logic Quarterly, 53 , 311 – 320 , 2007 . Google Scholar Crossref Search ADS [5] Z. Markovic . Intuitionistic and classical satisfiability in Kripke models . Institute of Mathematics, 13 , 1 – 5 , 1998 . [6] Z. Markovic . Some preservation results for classical and intuitionistic satisfiability in Kripke models . Notre Dame Journal of Formal Logic , 24 , 395 – 398 , 1983 . Google Scholar Crossref Search ADS [7] M. Moniri . $$\mathcal H$$-theories, fragments of HA and PA-normality . Mathematical Logic Quarterly , 49 , 250 – 254 , 2003 . Google Scholar Crossref Search ADS [8] A. Visser . Submodels of Kripke models . Archive for Mathematical Logic , 40 , 277 – 295 , 2001 . Google Scholar Crossref Search ADS [9] K. Wehmeier . Classical and intuitionistic models of arithmetic . Notre Dame Journal of Formal Logic, 37 , 452 – 461 , 1996 . Google Scholar Crossref Search ADS © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

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Logic Journal of the IGPLOxford University Press

Published: Sep 25, 2018

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