2 Preliminaries

We use this section to fix some notations to be used throughout the article.

We will use IH as the abbreviation for Induction Hypothesis in the inductive proofs, and we will write $\alpha :=\beta $ to mean that we define $\alpha $ as $\beta $ . We will use the usual notations for sets and functions. The set of all subsets of X will be denoted by $\mathcal {P}(X)$ . The natural numbers are understood as the finite von Neumann ordinals, and $\omega $ as the smallest infinite ordinal. Given an $n \in \omega $ and a tuple $\alpha = (x_1,\ldots ,x_n)$ of any sort of objects, we will refer to $\alpha $ by $\bar {x}_n$ and will denote by $init(\alpha )$ and $end(\alpha )$ the initial and final elements of $\alpha $ , that is to say, $x_1$ and $x_n$ , respectively. More generally, given any $i < \omega $ such that $1 \leq i \leq n$ , we set that $\pi ^i(\alpha ):= x_i$ , i.e., that $\pi ^i(\alpha )$ denotes the i-th projection of $\alpha $ . Given another tuple $\beta = \bar {y}_m$ , we will denote by $(\alpha )^\frown (\beta )$ the concatenation of the two tuples, i.e., the tuple $(x_1,\ldots ,x_n,y_1,\ldots ,y_m)$ . The empty tuple will be denoted by $\Lambda $ .

We will extensively use ordered couples of sets, which we will call bi-sets. Relations are understood as sets of ordered tuples. Given binary relations $R \subseteq X \times Y$ and $S\subseteq Y\times Z$ , we denote their composition by $R\circ S:= \{(a,c)\mid \text { for some }b \in Y,\,(a, b)\in R,\,(b,c)\in S\}$ .

We view functions as relations with special properties; we write $f:X \to Y$ to denote a function $f \subseteq X\times Y$ such that its left projection is all of X. If $f: X \to Y$ and $Z\subseteq X,$ then we will denote the image of Z under f by $f(Z)$ . In view of our previous convention for relations, for any given two functions $f:X\to Y$ and $g:Y\to Z$ , we will denote the function ${x\mapsto g(f(x))}$ by $f\circ g$ , even though, in the existing literature, this function is often denoted by $g\circ f$ instead.

Given a set X, we will denote by $id[X]$ the identity function on X, i.e., the function $f:X\to X$ such that $f(x) = x$ for every $x \in X$ . In relation to compositions, functions of the form $id[X]$ have a special importance as a limiting case. More precisely, given a set X and a family F of functions from X to X, we will assume that the composition of the empty tuple of functions from F is just $id[X]$ .

Furthermore, if $f:X\to Y$ is any function, $x \in X$ and $y \in Y$ , we will denote by $f[x/y]$ the unique function $g:X\to Y$ such that, for a given $z \in X$ we have

$$ \begin{align*}g(z):=\begin{cases} y,\text{ if }z = x;\\ f(z),\text{ otherwise. } \end{cases} \end{align*} $$

3 Logics

We are going to give a notion of logic that is wide enough to cover every formal system to be considered below, without aspiring to give any sort of ultimate generalization of this notion.

Logics are based on languages, and in this article we confine ourselves to considering languages of two types: the propositional language with an added conditional or modal operator(s) and the first-order relational languages with equality. Speaking generally, a language $\mathcal {L}$ is simply a certain set. The elements of languages are called their formulas. The languages are often generated from certain sets of atoms by repeated application of connectives and quantifiers. Every language considered in this article will include at least the binary connectives $\to $ , $\wedge $ , and $\vee $ . One consequence of this is that we can assume the standard reading of equivalence $\phi \leftrightarrow \psi $ as the abbreviation for $(\phi \to \psi )\wedge (\psi \to \phi )$ for every language considered below.

In this article, we will treat the sets of formulas generated by different sets of atoms over the same set of logical symbols as different languages rather than different versions of the same language.

Although one and the same logic can be formulated over different versions of the same language (or, in our terminology, over different languages), in this article we will abstract away from such subtleties, and will simply treat a logic as a consequence relation, or, in other words, as a set of consecutions over a particular language $\mathcal {L}$ . More precisely, a logic is a set $\mathsf {L} \subseteq \mathcal {P}(\mathcal {L})\times \mathcal {P}(\mathcal {L})$ , for some language $\mathcal {L}$ , where $(\Gamma , \Delta )\in \mathsf {L}$ iff $\Delta\ \mathsf {L}$ -follows from $\Gamma $ (we will also denote this by $\Gamma \models _{\mathsf {L}}\Delta $ ). We will say that $(\Gamma , \Delta )$ is $\mathsf {L}$ -satisfiable iff ${(\Gamma , \Delta )\notin \mathsf {L}}$ . Given a $\phi \in \mathcal {L}$ , $\phi $ is $\mathsf {L}$ -valid or a theorem of $\mathsf {L}$ (we will also write $\phi \in \mathsf {L}$ ) iff $(\emptyset , \{\phi \}) \in \mathsf {L}$ .

Every logic considered in this article will be introduced either by its intended semantics, or by a complete Hilbert-style axiomatization; for most logics in this article, we will mention both.

The semantics of various logics considered in this article is going to be laid out according to the following general scheme. Recall that, given a classically flavored logic $\mathsf {L}$ , we typically define $\mathsf {L}$ by setting that $(\Gamma , \Delta )\in \mathsf {L}$ iff the truth of every $\phi \in \Gamma $ implies the truth of some $\psi \in \Delta $ . In the more general setting of our article, if $\mathcal {L}$ is a language, then a semantics for $\mathcal {L}$ is a pair $\sigma = (EP_\sigma , \models _\sigma )$ , where $EP_\sigma $ is a (definable) class called the class of $\sigma $ -evaluation points. The other element of the semantics is a (class-)relation $\models _\sigma \subseteq EP_\sigma \times \mathcal {L}$ called the ( $\sigma $ -)satisfaction relation.

In case $(pt, \phi ) \in \models _\sigma $ , we write $pt\models _\sigma \phi $ , and say that $pt\ \sigma $ -satisfies $\phi $ . More generally, given any $\Gamma , \Delta \subseteq \mathcal {L}$ , and a $pt \in EP_\sigma $ , we say that $pt \sigma $ -satisfies $(\Gamma , \Delta )$ and write $pt\models _\sigma (\Gamma , \Delta )$ iff:

$$ \begin{align*}(\forall \phi \in \Gamma)(pt\models_\sigma\phi)\text{ and }(\forall \psi \in \Delta)(pt\not\models_\sigma\psi). \end{align*} $$

More conventionally, we say that $pt$ satisfies $\Gamma $ , and write $pt\models _\sigma \Gamma $ , iff $pt\models _\sigma (\Gamma , \emptyset )$ . This latter format is in fact sufficient to set up a semantics as long as our logics can express Boolean negation. However, this is not the case for many logics to be considered in this article. For such logics the “double-entry” format for satisfaction relation proves to be more convenient and flexible.

Next, given any $\Gamma , \Delta \cup \{\phi \} \subseteq \mathcal {L}$ , we say that $(\Gamma , \Delta )$ (resp., $\Gamma $ , $\phi $ ) is $\sigma $ -satisfiable iff, for some $pt \in EP_\sigma $ , $pt$ satisfies $(\Gamma , \Delta )$ (resp., $\Gamma $ , $\phi $ ). Finally, $\Delta \sigma $ -follows from $\Gamma $ (written $\Gamma \models _\sigma \Delta $ ) iff $(\Gamma , \Delta )$ is $\sigma $ -unsatisfiable, in other words, if every $\sigma $ -evaluation point satisfying every formula from $\Gamma $ , also satisfies at least one formula from $\Delta $ .

We now say that, for a given language $\mathcal {L}$ , a semantics $\sigma $ over $\mathcal {L}$ induces the logic $\mathsf {L}$ over $\mathcal {L}$ and write $\mathsf {L} = \mathbb {L}(\sigma )$ iff for all $\Gamma , \Delta \subseteq \mathcal {L}$ , we have $(\Gamma , \Delta ) \in \mathsf {L}$ iff $(\Gamma , \Delta )$ is $\sigma $ -unsatisfiable; in other words, $\mathsf {L} = \mathbb {L}(\sigma )$ means that, for all $\Gamma , \Delta \subseteq \mathcal {L}$ , $\Delta\ \mathsf {L}$ -follows from $\Gamma $ iff $\Delta\ \sigma $ -follows from $\Gamma $ . In case $\sigma $ is also used to introduce $\mathsf {L}$ by definition, we write $\mathsf {L} := \mathbb {L}(\sigma )$ .

As for the Hilbert-style systems, all of them will be given by a finite number of axiomatic schemas $\alpha _1,\ldots ,\alpha _n$ augmented with a finite number of inference rules $\rho _1,\ldots ,\rho _m$ , such a system will be denoted by $\Sigma (\bar {\alpha }_n; \bar {\rho }_m)$ . Every Hilbert-style system considered below happens to extend a certain minimal system which we will denote by $\mathfrak {IL}^+$ .Footnote ² We have $\mathfrak {IL}^+:= \Sigma (\alpha _1-\alpha _8;\text{(MP)})$ , where

$$ \begin{align*} &\phi \to(\psi\to\phi)\,(\alpha_1),\quad(\phi\to(\psi\to\chi))\to((\phi\to\psi)\to(\phi\to\chi))\,(\alpha_2),\\&\quad(\phi\wedge\psi)\to\phi\,(\alpha_3),\quad(\phi\wedge\psi)\to\psi\,(\alpha_4),\quad \phi\to(\psi\to (\phi\wedge\psi))\,(\alpha_5),\\&\quad\phi\to(\phi\vee \psi)\,(\alpha_6),\quad\psi\to(\phi\vee \psi)\,(\alpha_7),\\ &\quad (\phi\to\chi)\to((\psi \to \chi) \to ((\phi\vee\psi)\to \chi))\,(\alpha_8) \end{align*} $$

and

(MP) $$ \begin{align} \text{From }\phi, \phi \to \psi\text{ infer }\psi. \end{align} $$

It is therefore important for our purposes to be able to refer to Hilbert-style systems as extensions of other systems. If $\mathfrak {Ax} = \Sigma (\bar {\alpha }_n; \bar {\rho }_m)$ , and $\beta _1,\ldots ,\beta _k$ are some new axiomatic schemes and $\sigma _1,\ldots ,\sigma _r$ are some new rules, then we will write $\mathfrak {Ax}+(\bar {\beta }_k;\bar {\sigma }_r)$ to denote the system $\Sigma ((\bar {\alpha }_n)^\frown (\bar {\beta }_k);(\bar {\rho }_m)^\frown (\bar {\sigma }_r))$ .

Axiomatic systems can be viewed as operators generating logics when applied to languages. More precisely, if $\mathfrak {Ax} = \Sigma (\bar {\alpha }_n; \bar {\rho }_m)$ and $\mathcal {L}$ is a language, then $\mathsf {L} = \mathfrak {Ax}(\mathcal {L})$ can be described as follows. We say that a $\phi \in \mathcal {L}$ is provable in $\mathfrak {Ax}(\mathcal {L})$ iff there exists a finite sequence $\bar {\psi }_k\in \mathcal {L}^k$ such that every formula in this sequence is either a substitution instance of one of $\bar {\alpha }_n$ or results from an application of one of $\bar {\rho }_m$ to some earlier formulas in the sequence and $\psi _k = \phi $ ; we will say that $(\Gamma , \Delta ) \in \mathfrak {Ax}(\mathcal {L})$ iff $(\Gamma , \Delta ) \in \mathcal {P}(\mathcal {L})\times \mathcal {P}(\mathcal {L})$ and there exists a sequence $\bar {\chi }_r\in \mathcal {L}^r$ such that every formula in it is either in $\Gamma $ , or is provable in $\mathfrak {Ax}(\mathcal {L})$ or results from an application of (MP) to a pair of earlier formulas in the sequence, and, for some $\theta _1,\ldots ,\theta _s\in \Delta $ we have $\chi _r = \theta _1\vee \cdots \vee \theta _s$ . This definition makes sense in the context of our article, since every language that we are going to consider contains $\vee $ , and every axiomatic system that we are going to consider contains (MP). We will also express the fact that $(\Gamma , \Delta ) \in \mathfrak {Ax}(\mathcal {L})$ by writing $\Gamma \vdash _{\mathfrak {Ax}(\mathcal {L})}\Delta $ . If also $\mathsf {L} = \mathfrak {Ax}(\mathcal {L})$ , then we can write $\Gamma \vdash _{\mathsf {L}}\Delta $ instead of $\Gamma \vdash _{\mathfrak {Ax}(\mathcal {L})}\Delta $ . In the latter case we will also have, for every $\phi \in \mathcal {L}$ , that $\phi \in \mathsf {L}$ iff $\vdash _{\mathfrak {Ax}(\mathcal {L})}\phi $ iff $\phi $ is provable in $\mathfrak {Ax}(\mathcal {L})$ .

4 The first-order languages and their logics

We start by defining a handful of first-order languages (relational with equality; the latter is denoted by $\equiv $ ) according to the scheme laid out in the previous section. First, we let $\Pi $ denote the set $\{p^1_n\mid n\in \omega \}\cup \{S^1, O^1, E^2, R^3\}$ . In case $\Omega \subseteq \Pi $ , we set $\Omega ^\pm := \{(P_+)^n, (P_-)^n\mid P^n\in \Omega \}$ . For instance, if $\Omega = \{S^1, R^3, p^1_0\}$ , then $\Omega ^\pm = \{S^1_+, R^3_+, (p_0)^1_+, S^1_-, R^3_-, (p_0)^1_- \}$ .

Next, we define $Sign:= \Pi \cup \Pi ^\pm \cup \{\epsilon ^2\}$ . The elements of $Sign$ will serve as predicateFootnote ³ letters; their superscripts denote their arities. In order to define the first-order atoms, we also need to supply the set $Ind:= \{v_n\mid n \in \omega \}$ of individual variables.

If now $\Omega \subseteq Sign$ , then the set $At(\Omega ):= \{x \equiv y,\, Q\bar {x}_n\mid Q^n\in \Omega ,\,x,y, \bar {x}_n \in Ind\}$ is called the set of $\Omega $ -atoms. The set $Lit(\Omega ):= At(\Omega )\cup \{\sim \phi \mid \phi \in At(\Omega )\}$ is called the set of $\Omega $ -literals. Our definition of atoms entails, among other things, that the arity superscripts are not a part of predicate letters; strictly speaking, they are assigned by $\Omega $ to the predicate letters, so that, viewed more formally, $\Omega $ must be understood as a function from the set of predicate letters into $\omega $ . This is why the arity superscripts do not appear in actual formulas. To take our previous example, $\Omega = \{S^1, R^3, p^1_0\}$ is, strictly speaking, a function and could have been alternatively written in the form $\Omega = \{(S, 1), (R,3), (p_0, 1)\}$ ; examples of $\Omega $ -atoms in this case are given by $S(v_0)$ and $R(v_0,v_0, v_3)$ rather than, say, $S^1(v_0)$ .

The first-order language $\mathcal {FO}(\Omega )$ is then generated on the basis of $At(\Omega )$ by the following BNF (where $x \in Ind$ ):

$$ \begin{align*}\phi::= At(\Omega)\mid \phi\wedge\phi\mid\phi\vee\phi\mid\phi\to\phi\mid\sim\phi\mid\forall x\phi\mid\exists x\phi. \end{align*} $$

The positive first-order language $\mathcal {FO}^+(\Omega )$ is the ( $\sim $ )-free subset of $\mathcal {FO}(\Omega )$ .

Given an $\Omega \subseteq Sign$ and a formula $\phi \in \mathcal {FO}(\Omega )$ , we denote by $Sub(\phi )$ the set of subformulas of $\phi $ assuming its standard definition by induction on the construction of $\phi $ . Furthermore, we can inductively define for $\phi $ its set of free variables in a standard way (see, e.g., [Reference van Dalen4, p. 64]). This set, denoted by $FV(\phi )$ , is always finite. Given an $n \in \omega $ , and a $\bar {x}_n \in Ind^n$ , we will denote by $\mathcal {FO}(\Omega )^{\bar {x}_n}$ the set $\{\phi \in \mathcal {FO}(\Omega )\mid FV(\phi ) \subseteq \{\bar {x}_n\}\}$ . If $\phi \in \mathcal {FO}(\Omega )^\emptyset $ , then $\phi $ is called an $\Omega $ -sentence. Finally, given some $x,y \in Ind$ , we assume a standard definition for the property of y being substitutable for x in $\phi $ ; in case this property holds, we define the result $\phi [x/y]$ of this substitution simply as the result of replacing all free occurrences of x in $\phi $ with the occurrences of y.

We are going to define three first-order logics, the classical logic $\mathsf {QCL}$ , the positive intuitionistic logic $\mathsf {QIL}^+$ , and, finally, the logic $\mathsf {QN4}$ , which represents the paraconsistent variant of Nelson’s first-order logic of strong negation. Each of these logics will be defined over a different set of the first-order language variants. Among the three logics, the classical logic has the simplest semantics. We describe it as follows.

We will also need the following notion relating classical models.

The class $EP_c(\Omega )$ of classical $\Omega $ -evaluation points is then the class

$$ \begin{align*}\{(\mathcal{M}, f)\mid \mathcal{M}\in \mathbb{C}(\Omega),\,f:Ind\to U^{\mathcal{M}}\}. \end{align*} $$

We now define $\mathsf {QCL}(\Omega ):=\mathbb {L}(EP_c(\Omega ), \models _c)$ , where $\models _c$ is the classical first-order satisfaction relation. We will often write $\mathcal {M}\models _{c}\phi [f]$ instead of $(\mathcal {M}, f)\models _{c}\phi $ (also for the non-classical first-order logics). The relation itself is defined by the following induction on the construction of $\phi $ :

$$ \begin{align*} \mathcal{M}\models_{c}P\bar{x}_n[f] &\text{ iff } f(\bar{x}_n)\in P^{\mathcal{M}} &&P^n\in \Omega\\\mathcal{M}\models_{c}x\equiv y[f] &\text{ iff } f(x) = f(y)\\\mathcal{M}\models_{c}(\psi\wedge\chi)[f] &\text{ iff } \mathcal{M}\models_{c}\psi[f]\text{ and }\mathcal{M}\models_{c}\chi[f]\\\mathcal{M}\models_{c}(\psi\vee\chi)[f] &\text{ iff } \mathcal{M}\models_{c}\psi[f]\text{ or }\mathcal{M}\models_{c}\chi[f]\\\mathcal{M}\models_{c}(\psi\to\chi)[f] &\text{ iff } \mathcal{M}\not\models_{c}\psi[f]\text{ or }\mathcal{M}\models_{c}\chi[f]\\\mathcal{M}\models_{c}\sim\psi[f] &\text{ iff } \mathcal{M}\not\models_{c}\psi[f]\\\mathcal{M}\models_{c}\exists x\psi[f] &\text{ iff } (\exists a\in U^{\mathcal{M}})(\mathcal{M}\models_{c}\psi[f[x/a]])\\\mathcal{M}\models_{c}\forall x\psi[f] &\text{ iff } (\forall a\in U^{\mathcal{M}})(\mathcal{M}\models_{c}\psi[f[x/a]]). \end{align*} $$

Our next logic is the positive intuitionistic logic $\mathsf {QIL}^+$ . Its semantics (defined here for every $\Omega \subseteq Sign$ over the language $\mathcal {FO}^+(\Omega )$ ) is somewhat more involved, and exists in several variants. By far the most popular one is the so-called Kripke semantics (see, e.g., [Reference Gabbay, Shehtman and Skvortsov6, Chapter 3]). However, the proof of our main result proceeds more conveniently on the basis of a somewhat involved semantics of Kripke sheaves which we define next.

For any $\Omega \subseteq Sign$ , the class $EP_{i}(\Omega )$ of intuitionistic evaluation points is the class

$$ \begin{align*}\{(\mathcal{S}, \mathbf{w}, f)\mid \mathcal{S}\in \mathbb{I}(\Omega),\,\mathbf{w} \in W,\,f:Ind \to U^{\mathtt{M}_{\mathbf{w}}}\}. \end{align*} $$

The satisfaction relation $\models _{i}$ is then defined by the following induction:

$$ \begin{align*} \mathcal{S},\mathbf{w}\models_{i}P\bar{x}_n[f] &\text{ iff } \mathtt{M}_{\mathbf{w}}\models_{c}P(\bar{x}_n)[f] \text{ iff } f(\bar{x}_n)\in P^{\mathtt{M}_{\mathbf{w}}} && P^n\in \Omega\\\mathcal{S},\mathbf{w}\models_ix \equiv y &\text{ iff } f(x) = f(y)\\\mathcal{S},\mathbf{w}\models_i(\psi\wedge\chi)[f] &\text{ iff } \mathcal{S},\mathbf{w}\models_i\psi[f]\text{ and }\mathcal{S},\mathbf{w}\models_i\chi[f]\\\mathcal{S},\mathbf{w}\models_i(\psi\vee\chi)[f] &\text{ iff } \mathcal{S},\mathbf{w}\models_i\psi[f]\text{ or }\mathcal{S},\mathbf{w}\models_i\chi[f]\\\mathcal{S},\mathbf{w}\models_i(\psi\to\chi)[f] &\text{ iff } (\forall \mathbf{v}\geq \mathbf{w})(\mathcal{S},\mathbf{v}\not\models_i\psi[f\circ\mathtt{H}_{\mathbf{w}\mathbf{v}}]\text{ or }\mathcal{S}, \mathbf{v}\models_i\chi[f\circ\mathtt{H}_{\mathbf{w}\mathbf{v}}])\\\mathcal{S},\mathbf{w}\models_i\exists x\psi[f] &\text{ iff } (\exists a\in U^{\mathtt{M}_{\mathbf{w}}})(\mathcal{S},\mathbf{w}\models_i\psi[f[x/a]])\\\mathcal{S},\mathbf{w}\models_i\forall x\psi[f] &\text{ iff } (\forall \mathbf{v}\geq \mathbf{w})(\forall a\in U^{\mathtt{M}_{\mathbf{v}}})(\mathcal{S},\mathbf{v}\models_i\psi[(f\circ\mathtt{H}_{\mathbf{w}\mathbf{v}})[x/a]]). \end{align*} $$

We now define $\mathsf {QIL}^+(\Omega ) := \mathbb {L}(EP_{i}(\Omega ), \models _i)$ for any $\Omega \subseteq Sign$ . See [Reference Gabbay, Shehtman and Skvortsov6, Section 3.6 ff] for a proof that we indeed get a correct semantics for the positive intuitionistic logic in this way, the fact is also mentioned in [Reference van Dalen4, Corollary 5.3.16].

Sheaf semantics is a generalization of Kripke semantics in that the latter can be obtained from sheaf semantics as long as we assume in Definition 3 that $\mathtt {H}_{\mathbf {w}\mathbf {v}} = id[U^{\mathtt {M}_{\mathbf {w}}}]$ for all $\mathbf {w},\mathbf {v} \in W$ such that $\mathbf {w}\leq \mathbf {v}$ . Therefore, most of the properties and constructions available in the usual Kripke semantics have obvious counterparts in the sheaf semantics. The following lemma mentions some of these properties.

Alternatively, one can define $\mathsf {QIL}^+$ by its complete Hilbert-style axiomatization. More precisely, consider the following set of axiomatic schemes:

(α9) $$ \begin{align} \forall x\phi&\to \phi[x/y] \end{align} $$

(α10) $$ \begin{align} \phi[x/y]&\to \exists x\phi \end{align} $$

(α11) $$ \begin{align} &x \equiv x \end{align} $$

(α12) $$ \begin{align} y\equiv z &\to (\phi[x/y]\to\phi[x/z]) \end{align} $$

plus the following rules of inference:

(R∀) $$ \begin{align} \text{From }\psi\to\phi[x/y]&\text{ infer }\psi \to \forall x\phi \end{align} $$

(R∃) $$ \begin{align} \text{From }\phi[x/y]\to\psi&\text{ infer }\exists x\phi \to \psi, \end{align} $$

where $x, z \in Ind$ and $y \in Ind\setminus FV(\psi )$ are such that $z, y$ are substitutable for x in $\phi $ . We then let $\mathfrak {QIL}^+:= \mathfrak {IL}^++((\alpha_9),\ldots ,(\alpha_12);(R\forall),(R\exists))$ . It is well-known that for every $\Omega \subseteq Sign$ , we have $\mathsf {QIL}^+(\Omega ) = \mathfrak {QIL}^+(\mathcal {FO}(\Omega ))$ .

We now turn to the paraconsistent variant of Nelson’s logic of strong negation, which we denote by $\mathsf {QN4}$ and which we only define over $\mathcal {FO}(\Omega )$ for $\Omega \subseteq \Pi $ . Our main goal in this section is to set up a sheaf semantics also for $\mathsf {QN4}$ . Since no variants of sheaf semantics were yet (to the best of our knowledge) proposed for this logic in the existing literature, we cannot use our proposed semantics to define $\mathsf {QN4}$ . Instead, we extend $\mathfrak {QIL}^+$ with the following axiomatic schemes:

(An1) $$ \begin{align} \sim\sim\phi &\leftrightarrow \phi \end{align} $$

(An2) $$ \begin{align} \sim(\phi\wedge \psi) &\leftrightarrow (\sim\phi\vee \sim\psi) \end{align} $$

(An3) $$ \begin{align} \sim(\phi\vee \psi) &\leftrightarrow (\sim\phi\wedge \sim\psi) \end{align} $$

(An4) $$ \begin{align} \sim(\phi\to \psi) &\leftrightarrow (\phi\wedge \sim\psi) \end{align} $$

(An5) $$ \begin{align} \sim\exists x\theta &\leftrightarrow \forall x\sim\theta \end{align} $$

(An6) $$ \begin{align} \sim\forall x\theta &\leftrightarrow \exists x\sim\theta. \end{align} $$

In doing so, we obtain the system $\mathfrak {QN4}:= \mathfrak {QIL}^++\textrm{(An1)},\ldots ,\textrm{(An6)};)$ which represents the most standard way to axiomatize $\mathsf {QN4}$ known in the existing literature on the subject (see, e.g., [Reference Odintsov, Wansing, Hendricks and Malinowski9, p. 313]). We can therefore set $\mathsf {QN4}(\Omega ):= \mathfrak {QN4}(\mathcal {FO}(\Omega ))$ for every $\Omega \subseteq \Pi $ .

This definition makes it obvious that $\mathsf {QN4}$ is a sublogic of $\mathsf {QCL}$ and extends $\mathsf {QIL}^+$ . We retain these observations for future reference.

Lemma 2. Let $\Omega \subseteq \Pi $ . Every substitution instance of a $\mathsf {QIL}^+(\Omega )$ -theorem is a theorem of $\mathsf {QN4}(\Omega )$ and every inference rule that is deducible in $\mathsf {QIL}^+(\Omega )$ is also deducible in $\mathsf {QN4}(\Omega )$ . In particular, the following derived rules hold:

(DT) $$ \begin{align} \Gamma \cup \{\phi\}\vdash_{\mathsf{QN4}} \psi&\text{ iff }\Gamma\vdash_{\mathsf{QN4}} \phi\to\psi \qquad\qquad\qquad\qquad\qquad\end{align} $$

(B∀) $$ \begin{align} \Gamma\vdash_{\mathsf{QN4}} \phi\to \psi&\text{ implies }\Gamma\vdash_{\mathsf{QN4}} \phi\to\forall x\psi&&x\notin FV(\Gamma \cup \{\phi\}) \end{align} $$

(B∃) $$ \begin{align} \Gamma\vdash_{\mathsf{QN4}} \phi\to \psi&\text{ implies }\Gamma\vdash_{\mathsf{QN4}} \exists x\phi\to\psi&&x\notin FV(\Gamma \cup \{\psi\}) \end{align} $$

(Gen) $$ \begin{align} \Gamma\vdash_{\mathsf{QN4}}\phi&\text{ implies }\Gamma\vdash_{\mathsf{QN4}}\forall x\phi &&\ \ x \notin FV(\Gamma). \end{align} $$

We observe, further, that, for every $\Omega \subseteq \Pi $ , $\mathsf {QN4}(\Omega )$ is embeddable into $\mathsf {QIL}^+(\Omega ^\pm \cup \{\epsilon ^2\})$ and that the corresponding embedding $Tr:\mathcal {FO}(\Omega )\to \mathcal {FO}^+(\Omega ^\pm \cup \{\epsilon ^2\})$ can be defined by the following induction on the construction of $\phi \in \mathcal {FO}(\Pi )$ :

$$ \begin{align*} Tr(P\bar{x}_n)&:= P_+\bar{x}_n;&&Tr(\sim P\bar{x}_n):= P_-\bar{x}_n;\\Tr(x\equiv y)&:= x\equiv y;&&Tr(\sim(x \equiv y)):= \epsilon xy;\\Tr(\sim\sim\phi)&:= Tr(\phi);\\Tr(\phi\star\psi)&:= Tr(\phi)\star Tr(\psi);&&Tr(\sim(\phi\star\psi)):= Tr(\sim\phi)\ast Tr(\sim\psi);\\Tr(\phi\to\psi)&:= Tr(\phi)\to Tr(\psi);&& Tr(\sim(\phi\to\psi)):= Tr(\phi)\wedge Tr(\sim\psi);\\Tr(Qx\phi)&:= QxTr(\phi);&& Tr(\sim Qx\phi):= Q'xTr(\sim\phi) \end{align*} $$

for all $n \in \omega $ , all $P^n \in \Pi $ , all $\bar {x}_n, x, y \in Ind$ , and all $\star , \ast , Q,$ and $Q'$ such that both $\{\star , \ast \} = \{\wedge , \vee \}$ and $\{Q,Q'\} = \{\forall , \exists \}$ . More precisely, the following proposition holds.

Proposition 1 is a well-known result about $\mathsf {QN4}$ (see, e.g., [Reference Odintsov, Wansing, Hendricks and Malinowski9, Proposition 7] for a sketch of a proof). To keep this article reasonably self-contained, we also give its proof in Appendix A.1.

We proceed to define a sheaf semantics for $\mathsf {QN4}$ that we will presently show to be adequate for this logic. We start by defining the structures that we will call Nelsonian sheaves.

Since the family $\mathtt {H}$ of canonical homomorphisms is shared by $\mathcal {S}^+$ and $\mathcal {S}^-$ , it follows that $U^{\mathtt {M}^+_{\mathbf {w}}} = U^{\mathtt {M}^-_{\mathbf {w}}}$ for every $\mathbf {w} \in W$ ; we can therefore set $U_{\mathbf {w}}:= U^{\mathtt {M}^+_{\mathbf {w}}} = U^{\mathtt {M}^-_{\mathbf {w}}}$ for every $\mathbf {w} \in W$ . As for the Nelsonian sheaf evaluation points, they are defined similarly to the intuitionistic ones: $EP_{n}(\Omega ):= \{(\mathcal {S}, \mathbf {w}, f)\mid \mathcal {M}\in \mathbb {N}4(\Omega ),\,(\mathcal {S}^+, \mathbf {w}, f)\in EP_{i}(\Omega )\}$ .

One peculiarity of $\mathsf {QN4}$ consists in the fact that its semantics is usually constructed on the basis of two satisfaction relations, $\models ^+_{n}$ and $\models ^-_{n}$ , instead of just one.Footnote ⁴ The informal interpretation of the two relations is that whenever $\mathcal {S}, \mathbf {w}\models ^+_{n}\phi [f]$ holds, $\phi $ is verified at $(\mathcal {S}, \mathbf {w}, f)$ , and when $\mathcal {M}, \mathbf {w}\models ^-_{n}\phi [f]$ holds, $\phi $ is falsified at the same triple. These two relations are defined by simultaneous induction on the construction of $\phi \in \mathcal {FO}(\Omega )$ :

$$ \begin{align*} \mathcal{S},\mathbf{w}\models^\ast_nP\bar{x}_n[f] &\text{ iff } \mathcal{S}^\ast,\mathbf{w}\models_iP\bar{x}_n[f]\text{ iff } \mathtt{M}^\ast_{\mathbf{w}}\models_c P\bar{x}_n[f]\text{ iff } f(\bar{x}_n)\in P^{\mathtt{M}^\ast_{\mathbf{w}}}\\& \qquad\qquad\qquad\qquad\qquad\qquad\qquad P^n\in \Omega,\,\ast\in \{+,-\}\\\mathcal{S},\mathbf{w}\models^+_nx \equiv y[f] &\text{ iff } \mathcal{S}^+,\mathbf{w}\models_ix \equiv y[f]\text{ iff } \mathtt{M}^+_{\mathbf{w}}\models_{c}x \equiv y[f]\text{ iff } f(x) = f(y)\\\mathcal{S},\mathbf{w}\models^-_nx \equiv y[f] &\text{ iff } \mathcal{S}^-,\mathbf{w}\models_i\epsilon xy[f]\text{ iff } \mathtt{M}^-_{\mathbf{w}}\models_{c}\epsilon xy[f]\\\mathcal{S},\mathbf{w}\models^+_n(\psi\wedge\chi)[f] &\text{ iff } \mathcal{S},\mathbf{w}\models^+_n\psi[f]\text{ and }\mathcal{S},\mathbf{w}\models^+_n\chi[f]\\\mathcal{S},\mathbf{w}\models^-_n(\psi\wedge\chi)[f] &\text{ iff } \mathcal{S},\mathbf{w}\models^-_n\psi[f]\text{ or }\mathcal{S},\mathbf{w}\models^-_n\chi[f]\\\mathcal{S},\mathbf{w}\models^+_n(\psi\vee\chi)[f] &\text{ iff } \mathcal{S},\mathbf{w}\models^+_n\psi[f]\text{ or }\mathcal{S},\mathbf{w}\models^+_n\chi[f]\\\mathcal{S},\mathbf{w}\models^-_n(\psi\vee\chi)[f] &\text{ iff } \mathcal{S},\mathbf{w}\models^-_n\psi[f]\text{ and }\mathcal{S},\mathbf{w}\models^-_n\chi[f]\\\mathcal{S},\mathbf{w}\models^+_n(\psi\to\chi)[f] &\text{ iff } (\forall \mathbf{v}\geq \mathbf{w})(\mathcal{S},\mathbf{v}\not\models^+_n\psi[f\circ\mathtt{H}_{\mathbf{w}\mathbf{v}}]\text{ or }\mathcal{S}, \mathbf{v}\models^+_n\chi[f\circ\mathtt{H}_{\mathbf{w}\mathbf{v}}])\\\mathcal{S},\mathbf{w}\models^-_n(\psi\to\chi)[f] &\text{ iff } \mathcal{S},\mathbf{w}\models^+_n\psi[f]\text{ and }\mathcal{S},\mathbf{w}\models^-_n\chi[f]\\\mathcal{S},\mathbf{w}\models^+_n\sim\psi[f] &\text{ iff } \mathcal{S},\mathbf{w}\models^-_n\psi[f]\\\mathcal{S},\mathbf{w}\models^-_n\sim\psi[f] &\text{ iff } \mathcal{S},\mathbf{w}\models^+_n\psi[f]\\\mathcal{S},\mathbf{w}\models^+_n\exists x\psi[f] &\text{ iff } (\exists a\in U_{\mathbf{w}})(\mathcal{S},\mathbf{w}\models^+_n\psi[f[x/a]])\\\mathcal{S},\mathbf{w}\models^-_n\exists x\psi[f] &\text{ iff } (\forall \mathbf{v}\geq \mathbf{w})(\forall a\in U_{\mathbf{v}})(\mathcal{S},\mathbf{v}\models^-_n\psi[(f\circ\mathtt{H}_{\mathbf{w}\mathbf{v}})[x/a]])\\\mathcal{S},\mathbf{w}\models^+_n\forall x\psi[f] &\text{ iff } (\forall \mathbf{v}\geq \mathbf{w})(\forall a\in U_{\mathbf{v}})(\mathcal{S},\mathbf{v}\models^+_n\psi[(f\circ\mathtt{H}_{\mathbf{w}\mathbf{v}})[x/a]])\\\mathcal{S},\mathbf{w}\models^-_n\forall x\psi[f] &\text{ iff } (\exists a\in U_{\mathbf{w}})(\mathcal{S},\mathbf{w}\models^-_n\psi[f[x/a]]). \end{align*} $$

However, the negative satisfaction relation $\models ^-_{n}$ is often viewed within this pair as a subsidiary one, which, among other things, is due to the fact that the satisfaction clauses for negation allow to completely reflect the structure of $\models ^-_{n}$ within $\models ^+_{n}$ . Therefore, one can still capture $\mathsf {QN4}$ according to our usual pattern. In other words, we are going to prove the following.

The proof of this proposition makes a substantial use of the embedding $Tr$ defined above.Footnote ⁵ We sketch it in Appendix A.2. Another consequence of the tight relation between $\mathsf {QN4}$ and $\mathsf {QIL}^+$ is that Lemma 1 carries over to Nelsonian sheaves.

Proof of the lemma is relegated to Appendix A.3.

In the remaining part of the section we develop $\mathsf {QN4}$ to an extent that is sufficient for the subsequent sections. As usual, it follows from our semantic definitions that the truth value of a formula $\phi \in \mathcal {FO}$ only depends on the values assigned by f to the values of the variables in $FV(\phi )$ . We will therefore write, for any $\ast \in \{+, -\}$ , $\mathcal {S}, \mathbf {w}\models ^\ast _{n}\phi [x_1/a_1,\ldots ,x_n/a_n]$ iff $\mathcal {S} \in \mathbb {N}4$ , $\mathbf {w} \in W$ , $\phi \in \mathcal {FO}^{\bar {x}_n}$ , and $\mathcal {S}, \mathbf {w}\models ^\ast _{n}\phi [f]$ for every (equivalently, any) f such that both $(\mathcal {S}, \mathbf {w}, f) \in EP_{n}$ and $f(x_i) = a_i$ for every $1 \leq i \leq n$ . In particular, we will write $\mathcal {S}, \mathbf {w}\models ^\ast _{n}\phi $ iff $\phi \in \mathcal {FO}^\emptyset $ and we have $\mathcal {S}, \mathbf {w}\models ^\ast _{n}\phi [f]$ for every (equivalently, any) function f such that $(\mathcal {S}, \mathbf {w}, f) \in EP_{n}$ ; we will write $\mathcal {S}\models ^\ast _{n}\phi $ iff $\mathcal {S}, \mathbf {w}\models ^\ast _{n}\phi $ for every $\mathbf {w}\in W$ . The conventions of this paragraph also extend to sets and bi-sets of formulas in an obvious way; furthermore, in what follows, we will also apply these notational conventions to both intuitionistic and classical semantics of the first-order languages.

Next, we introduce the following abbreviations for all $\phi , \psi \in \mathcal {FO}$ :

• • $\phi \Rightarrow \psi $ (strong implication) for $(\phi \to \psi )\wedge (\sim \psi \to \sim \phi )$ .

• • $\phi \Leftrightarrow \psi $ (strong equivalence) for $(\phi \Rightarrow \psi )\wedge (\psi \Rightarrow \phi )$ , or, (equivalently, in view of Lemma 2), for $(\phi \leftrightarrow \psi )\wedge (\sim \phi \leftrightarrow \sim \psi )$ .

• • $\phi \,\&\,\psi $ (ampersand) for $\sim (\phi \to \sim \psi )$ .

The following lemma lists some properties of these derived connectives.

Lemma 5. Let $\phi , \psi , \chi , \theta \in \mathcal {FO}$ be chosen arbitrarily. Then all of the following theorems hold in $\mathsf {QN4}$ :

(T1) $$ \begin{align} \vdash_{\mathsf{QN4}}(\phi \Rightarrow \psi)&\to(\phi\to\psi) \end{align} $$

(T2) $$ \begin{align} \vdash_{\mathsf{QN4}}(\phi \Leftrightarrow \psi)&\to(\phi\leftrightarrow\psi)\end{align} $$

(T3) $$ \begin{align} \vdash_{\mathsf{QN4}}(\phi \Rightarrow \psi)&\Leftrightarrow(\sim\psi\Rightarrow\sim\phi)\end{align} $$

(T4) $$ \begin{align} \vdash_{\mathsf{QN4}}(\phi \Leftrightarrow \psi)&\Leftrightarrow(\sim\phi \Leftrightarrow\sim\psi) \end{align} $$

(T5) $$ \begin{align} \vdash_{\mathsf{QN4}}\phi &\Leftrightarrow \phi \end{align} $$

(T6) $$ \begin{align}\vdash_{\mathsf{QN4}}(\phi \Leftrightarrow \psi)&\Leftrightarrow(\psi \Leftrightarrow\phi)\end{align} $$

(T7) $$ \begin{align} \vdash_{\mathsf{QN4}}(\phi \Leftrightarrow \psi)&\Leftrightarrow((\psi \Leftrightarrow\chi)\Leftrightarrow(\phi \Leftrightarrow\chi)) \end{align} $$

(T8) $$ \begin{align} \vdash_{\mathsf{QN4}}\sim\sim\phi &\Leftrightarrow \phi \end{align} $$

(T9) $$ \begin{align} \vdash_{\mathsf{QN4}}\sim(\phi\wedge \psi) &\Leftrightarrow (\sim\phi\vee \sim\psi)\end{align} $$

(T10) $$ \begin{align} \vdash_{\mathsf{QN4}}\sim(\phi\vee \psi) &\Leftrightarrow (\sim\phi\wedge \sim\psi)\end{align} $$

(T11) $$ \begin{align} \vdash_{\mathsf{QN4}}\sim\exists x\theta &\Leftrightarrow \forall x\sim\theta \end{align} $$

(T12) $$ \begin{align} \vdash_{\mathsf{QN4}}\sim\forall x\theta &\Leftrightarrow \exists x\sim\theta \end{align} $$

(T13) $$ \begin{align} \kern-10pt \vdash_{\mathsf{QN4}}((\phi \Leftrightarrow \psi) \wedge (\chi \Leftrightarrow \theta))&\to ((\phi\ast\chi)\Leftrightarrow(\psi\ast\theta)) \quad\ast\in \{\wedge,\vee,\to\}\end{align} $$

(T14) $$ \begin{align} \vdash_{\mathsf{QN4}}(\phi \Leftrightarrow \psi)&\to(Qx\phi\Leftrightarrow Qx\psi) \qquad Q\in \{\forall, \exists\} \end{align} $$

(T15) $$ \begin{align} \vdash_{\mathsf{QN4}}(\phi\,\&\,\psi)&\leftrightarrow (\phi\wedge\psi)\end{align} $$

(T16) $$ \begin{align} \vdash_{\mathsf{QN4}}\sim(\phi\,\&\,\psi)&\leftrightarrow (\phi\to\sim\psi). \end{align} $$

Observe that (T9)–(T12) strengthen (in view of (T2)) the axioms (An1)–(An2) and (An5)–(An6), respectively. That this strengthening is non-trivial, follows from the fact that the converses of both (T1) and (T2) fail in general; moreover, one cannot replace $\leftrightarrow $ with $\Leftrightarrow $ in (An4), (T15), or (T16).

Finally, note that the following schemes are not, in general, valid in $\mathsf {QN4}$ :

$$ \begin{align*} (\phi \to \psi)\to(\sim\psi\to\sim\phi),\, (\phi \leftrightarrow \psi)\to(\sim\phi \leftrightarrow\sim\psi). \end{align*} $$

We sketch the proof in Appendix A.4.

Thus, even though our defined connectives wouldn’t make much sense in $\mathsf {QCL}$ as they are classically equivalent to $\to $ , $\leftrightarrow $ , and $\wedge $ , respectively, we see that the situation is different in the context of $\mathsf {QN4}$ . Indeed, whereas $\to $ and $\leftrightarrow $ in $\mathsf {QN4}$ cannot be contraposed, $\Rightarrow $ and $\Leftrightarrow $ define stronger (and contraposable) analogues to $\to $ and $\leftrightarrow $ , respectively. Finally, the ampersand is especially convenient for handling the falsity conditions of restricted existential quantification in the context of $\mathsf {QN4}$ . We provide a motivation for this use of ampersand in Appendix A.5.

Note, furthermore, that the failure of theorems like $(\phi \leftrightarrow \psi )\to (\sim \phi \leftrightarrow \sim \psi )$ implies that provably equivalent formulas in general fail to be substitutable for one another in $\mathsf {QN4}$ . However, this failure does not extend to the strong equivalence; to the contrary, one can even prove that a substitution of strongly equivalent formulas results in a formula that is also strongly equivalent to the original one.

As is well-known, a substitution of one formula for another in a first-order context can lead to rather complicated definitions. For the purposes of our article it is sufficient to confine ourselves to the following one simple case.

If $\phi , \psi \in \mathcal {FO}$ then we denote by $\phi /\psi $ the result of replacing every occurrence of $v_0\equiv v_0$ in $\phi $ by $\psi $ . The replacement operationFootnote ⁶ then commutes with all the connectives and quantifiers, and, for $\phi \in At$ , we stipulate that:

$$ \begin{align*}\phi/\psi:=\begin{cases} \psi,\text{ if }\phi = (v_0\equiv v_0)\\ \phi,\text{ otherwise.} \end{cases} \end{align*} $$

Then the following holds (see Appendix A.6 for a proof).

Lemma 6. For all $\phi , \psi , \chi \in \mathcal {FO}$ , we have

(T17) $$ \begin{align} \vdash_{\mathsf{QN4}}(\phi\Leftrightarrow\psi)\to((\chi/\phi)\Leftrightarrow(\chi/\psi)). \end{align} $$

An immediate consequence of Lemma 6 is that disjunction can also be understood as defined connective.

The formulas of the form $(\forall x)(\chi \to (\phi \Leftrightarrow \psi ))$ will play a prominent role in the subsequent sections. We would like, therefore, also to take a moment to spell out their truth conditions in terms of Nelsonian sheaf semantics.

Corollary 2. Let $x,y \in Ind$ be pairwise distinct, let $\phi \in \mathcal {FO}^{x,y}$ and let $\psi , \chi \in \mathcal {FO}^y$ . For every $\mathcal {S}\in \mathbb {N}4$ , every $\mathbf {w} \in W$ , and every $a \in U_{\mathbf {w}}$ , we have

$$ \begin{align*}\mathcal{S}, \mathbf{w} \models^+_n (\forall y)(\chi(y) \to (\phi(y,x)\Leftrightarrow \psi(y)))[x/a] \end{align*} $$

iff, for every $\mathbf {v} \geq \mathbf {w}$ and every $b \in U_{\mathbf {v}}$ such that $\mathcal {S}, \mathbf {v}\models ^+_n \chi [x/b]$ we have both

$$ \begin{align*}\mathcal{S}, \mathbf{v}\models^+_n \phi[y/b, x/\mathtt{H}_{\mathbf{w}\mathbf{v}}(a)]\text{ iff }\mathcal{S}, \mathbf{v}\models^+_n \psi[y/b] \end{align*} $$

and

$$ \begin{align*}\mathcal{S}, \mathbf{v}\models^-_n \phi[y/b, x/\mathtt{H}_{\mathbf{w}\mathbf{v}}(a)]\text{ iff }\mathcal{S}, \mathbf{v}\models^-_n \psi[y/b]. \end{align*} $$

Moreover, we would like to state an application of Lemma 6 to formulas of this type as a separate corollary.

Corollary 3. For all $\phi , \psi , \chi , \theta \in \mathcal {FO}$ , we have

(T18) $$ \begin{align} \vdash_{\mathsf{QN4}}\forall x(\chi\to (\phi\Leftrightarrow\psi))\to(\forall x(\chi \to (\theta/\phi))\leftrightarrow\forall x(\chi \to (\theta/\psi)). \end{align} $$

Proof We reason as follows:

(1) $$ \begin{align} \vdash_{\mathsf{QN4}}(\chi\to (\phi\Leftrightarrow\psi))&\to (\chi\to((\theta/\phi)\Leftrightarrow(\theta/\psi)))\nonumber\\&\quad \text{by}\ ({T17}), \text{Lm}\ ({2}) \end{align} $$

(2) $$ \begin{align} \vdash_{\mathsf{QN4}}(\chi\to((\theta/\phi)\Leftrightarrow(\theta/\psi)))&\to(\chi\to((\theta/\phi)\leftrightarrow(\theta/\psi)))&&\text{by (T2), Lm ({2})} \end{align} $$

(3) $$ \begin{align} \vdash_{\mathsf{QN4}}(\chi\to((\theta/\phi)\leftrightarrow(\theta/\psi)))&\to((\chi\to(\theta/\phi))\leftrightarrow(\chi\to(\theta/\phi)))&&\text{by Lm ({2})} \end{align} $$

(4) $$ \begin{align} \vdash_{\mathsf{QN4}}(\chi\to (\phi\Leftrightarrow\psi))&\to((\chi\to(\theta/\phi))\leftrightarrow(\chi\to(\theta/\phi)))&&\text{by (1)-(3), Lm(2)}. \end{align} $$

Now (T18) follows from (3) by Lemma 2.

5 Modal logics and their standard translation embeddings

Although modal logics are not the main subject of this article, they are so tightly connected to conditional logics based on the so-called Chellas semantics that a review of their standard translation embedding properties provides the best possible introduction to our main result. We start by setting $Prop:= \Pi \setminus \{S^1, O^1, E^2, R^3\} = \{p^1_n\mid n \in \omega \}$ . The modal language $\mathcal {MD}$ is generated from $Prop$ by the following BNF:

$$ \begin{align*}\phi::= p\mid \phi\wedge\phi\mid\phi\vee\phi\mid\phi\to\phi\mid\sim\phi\mid\Box\phi, \end{align*} $$

where $p^1 \in Prop$ . In case we want to have $\Diamond \phi $ as an elementary modality rather than an abbreviation for $\sim \Box \sim \phi $ , we obtain the language $\mathcal {MD}^\Diamond $ . The Kripke semantics for this language uses the class $pMod$ of pointed Kripke models as evaluation points and the modal satisfaction relation $\models _m$ ; the definitions of both can be easily found in any of the numerous textbooks, e.g., in [Reference Chagrov and Zakharyaschev2, Chapter 3.2]. The minimal normal modal logic $\mathsf {K}$ can then be defined by $\mathsf {K}:= \mathbb {L}(pMod, \models _m)$ .

If we now set $\mu := Prop\cup \{E^2\}$ and assume that, for every $x\in Ind$ we have fixed a $y\in Ind$ which is distinct from x, then the mapping $\sigma \tau _x:\mathcal {MD}\to \mathcal {FO}(\mu )^x$ is definedFootnote ⁷ by induction on the construction of $\phi \in \mathcal {MD}$ and is called the modal standard x-translation:

$$ \begin{align*} \sigma\tau_x(p)&:= px&&p^1 \in Prop\\ \sigma\tau_x(\sim\psi)&:= \sim \sigma\tau_x(\psi)\\ \sigma\tau_x(\psi\ast\chi)&:= \sigma\tau_x(\psi)\ast \sigma\tau_x(\chi)&&\ast\in \{\wedge, \vee, \to\}\\ \sigma\tau_x(\Box\psi) &:= \forall y(Exy\to \sigma\tau_{y}(\psi)). \end{align*} $$

The following lemma is then often presented without any proof in the existing literature.

Lemma 7 comes across as very natural, indeed, almost self-evident: the class of Kripke models for modal logic is just another name for $\mathbb {C}(\mu )$ , $x\in Ind$ chooses a point, turning it into a pointed structure, and the definition of $\sigma \tau _x$ is just a direct formalization of the inductive definition of $\models _m$ in first-order logic.

It is then just a small step to get from Lemma 7 to the following.

In other words, we find that $\sigma \tau _x$ faithfully embeds $\mathsf {K}$ into $\mathsf {QCL}(\mu )$ for every $x\in Ind$ .

The main line of thought presented in the article can be viewed as a natural continuation of this very idea. However, before we get to it, we need to observe that things become more complicated as soon as we replace the classical propositional basis of $\mathsf {K}$ with some non-classical logic. For example, the basic intuitionistic modal logic $\mathsf {IK}$ (see, e.g., [Reference Fischer-Servi and Chiara5]) is supplied with a Kripke semantics of bi-relational models over a classical metalanguage. This immediately results in an analogue of Proposition 3 saying that a straightforward formalization of this semantics amounts to defining an embedding of $\mathsf {IK}$ into a classical first-order correspondence language based on two binary predicates.Footnote ⁸ However, and much more interestingly, $\sigma \tau _x$ turns out to be useful for $\mathsf {IK}$ , too, as it happens to embed it into $\mathsf {QIL}(\mu )$ . This result, which is far from being trivial (one version of its proof can found in [Reference Simpson13, Chapter 5]), suggests that $\mathsf {IK}$ carves out exactly the same fragment of $\mathsf {QIL}(\mu )$ as the one carved out by $\mathsf {K}$ in $\mathsf {QCL}(\mu )$ ; and the latter circumstance provides a strong argument in favor of viewing $\mathsf {IK}$ as the correct intuitionistic analogue of $\mathsf {K}$ .

One may picture this situation as follows: assume that an intuitionist gets interested in modal logic, and asks a classicist colleague to explain them $\mathsf {K}$ . It is natural to expect that this explanation, if it is to be given in precise terms, will end up mentioning $\sigma \tau _x$ at some point. Of course, the intuitionist will read the definition of $\sigma \tau _x$ in terms of $\mathsf {QIL}$ rather than $\mathsf {QCL}$ and thus will end up with $\mathsf {IK}$ . In a sense then, $\mathsf {IK}$ is nothing but $\mathsf {K}$ read intuitionistically.

Moreover, it is easy to see that at most one intuitionistic modal logic can be faithfully embedded by $\sigma \tau _x$ into $\mathsf {QIL}(\mu )$ . In this sense, $\mathsf {IK}$ can be called the correct intuitionistic counterpart to $\mathsf {K}$ . We may even try to elevate this to a general criterion: given a non-classical logic $Q\nu $ such that $Q\nu $ is a proper sublogic of $\mathsf {QCL}$ and its propositional fragment $\nu $ is a proper sublogic of $\mathsf {CL}$ , a modal extension $\nu \mathsf {K}$ of $\nu $ is the $\nu $ -based counterpart of $\mathsf {K}$ iff $\sigma \tau _x$ faithfully embeds $\nu \mathsf {K}$ into $\mathsf {Q}\nu (\mu )$ .

Yet, the very deep and enlightening result about $\mathsf {IK}$ is not without its little annoying wrinkles. Observe that $\mathsf {IK}$ is formulated over $\mathcal {MD}^\Diamond $ rather than $\mathcal {MD}$ as the diamond is no longer definable in terms of box. Therefore, the above definition of $\sigma \tau _x$ is no longer sufficient, as it has to include

(5) $$ \begin{align} \sigma\tau^i_x(\Diamond\psi) &:= \exists y(Exy\wedge \sigma\tau^i_{y}(\psi)) \end{align} $$

as an additional clause. Of course, this clause is implied by the above definition of $\sigma \tau _x$ over $\mathsf {QCL}$ and thus adding it to the definition of $\sigma \tau _x$ results in a classically equivalent reformulation $\sigma \tau ^i_x$ of $\sigma \tau _x$ . However, other classically equivalent formulations of $\sigma \tau _x$ are also possible, and some of them make the analogue of Proposition 3 fail for $\mathsf {IK}$ and $\mathsf {QIL}$ . Think, for example, of the extension $\sigma \tau ^j_x$ of $\sigma \tau _x$ with the following clause:

(6) $$ \begin{align} \sigma\tau^j_x(\Diamond\psi) &:= \sim\forall y(Exy\to \sim\sigma\tau^j_{y}(\psi)). \end{align} $$

The choice of a right classical reformulation of $\sigma \tau _x$ can therefore affect the truth of our analogue of Proposition 3 in the intuitionistic case. Its precise formulation, then, goes as follows.

Of course, the required classical reformulation is pretty clear in the case of $\mathsf {IK}$ ; in fact, it is so clear that one is tempted to dismiss the existential quantifier in the formulation of Proposition 4 as mere pedantry and to speak of just $\sigma \tau _x$ instead. However, we need to be mindful of the fact that our caveat ‘up to a classically equivalent reformulation’ reflects a very real possibility that (to continue with our metaphorical scenario) in explaining the semantics of $\mathsf {K}$ to an interested intuitionist, the classicist modal logician might slip up and explain the semantics of diamond according to (6) rather than (5), which might lead, on the side of the intuitionist colleague, to a logic that is distinct from $\mathsf {IK}$ . In other words, our caveat uncovers the fact that $\mathsf {IK}$ is the result of the intuitionistic reading of $\mathsf {K}$ only up to a certain wording of $\mathsf {K}$ .

Given these considerations, our tentative principle for seeking out non-classical counterparts for $\mathsf {K}$ needs to be corrected as follows: given a non-classical logic $Q\nu $ such that $Q\nu $ is a proper sublogic of $\mathsf {QCL}$ and its propositional fragment $\nu $ is a proper sublogic of $\mathsf {CL}$ , a modal extension $\nu \mathsf {K}$ of $\nu $ is a $\nu $ -based counterpart of $\mathsf {K}$ iff some classically equivalent reformulation of $\sigma \tau _x$ faithfully embeds $\nu \mathsf {K}$ into $\mathsf {Q}\nu (\mu )$ ; the comparative merits of different $\nu $ -based counterparts of $\mathsf {K}$ will then have to be judged on the basis of other properties of $Q\nu $ .

Apart from $\mathsf {IL}$ , $\mathsf {N4}$ provides another possible instantiation of $\nu $ in the above principle; in fact we are not the first to notice this, as the paper [Reference Odintsov, Wansing and Hendricks10] by S. Odintsov and H. Wansing both defines several $\mathsf {N4}$ -based analogues of $\mathsf {K}$ and looks into their standard translation embeddings into $\mathsf {QN4}(\mu )$ . In the context of this article, the most interesting of these results is [Reference Odintsov, Wansing and Hendricks10, Proposition 7], which shows that the $\mathsf {N4}$ -based modal logic $\mathsf {FSK}^d$ is faithfully embedded into $\mathsf {QN4}(\mu )$ by a standard translation $T^{\prime }_x$ which provides yet another classical equivalent to $\sigma \tau _x$ . Although [Reference Odintsov, Wansing and Hendricks10] defines $\mathsf {FSK}^d$ over $\mathcal {MD}^\Diamond $ , the logic also derives the classical definition of $\Diamond $ in terms of $\Box $ which makes it possible to alternatively define $\mathsf {FSK}^d$ and $T^{\prime }_x$ over $\mathcal {MD}$ .Footnote ⁹ The result of [Reference Odintsov, Wansing and Hendricks10] can then be reformulated in the terminology of this article as follows.

Unfortunately, $T^{\prime }_x$ looks a bit clumsy in that it fails to commute (up to a strong equivalence) with the propositional connectives of $\mathsf {N4}$ . For example, given a $p^1 \in Prop$ , we have $T^{\prime }_x(\Box p) = \forall y(Rxy \to py)$ , but also ${T^{\prime }_x(\sim \Box p) = \exists y(Rxy \wedge \sim py)}$ , whereas $\sim T^{\prime }_x(\Box p) = \sim \forall y(Rxy \to py)$ , and the latter formula is clearly equivalent to $\exists y(Rxy\,\&\,\sim py)$ rather than $T^{\prime }_x(\sim \Box p)$ . Whereas (T15) ensures that $\exists y(Rxy\,\&\,\sim py)\leftrightarrow \exists y(Rxy \wedge \sim py)$ , we fail to get the strong equivalence between the two formulas.

Still, Proposition 5 shows that some sort of an $\mathsf {N4}$ -based analogue to Propositions 3 and 4 is possible and that $\mathsf {FSK}^d$ , at least to some extent, can be viewed as an $\mathsf {N4}$ -based counterpart of $\mathsf {K}$ .

6 $\mathsf {CK}$ , the minimal classical logic of conditionals

In this section we will introduce the language of conditional propositional logic (conditional language for short), which we will denote by $\mathcal {CN}$ . The language is generated from $Prop$ by the following BNF:

where $p^1 \in Prop$ . The new connective will be referred to as would-conditional or simply conditional; the elements of $\mathcal {CN}$ will be called conditional formulas. We will apply to the conditional formulas all of the abbreviations introduced earlier for the connectives in $\mathcal {FO}$ , plus the following new one:

• • (might-conditional) for .

One relatively popular and well-researched semantics for $\mathcal {CN}$ is the so-called Chellas semantics. In this article, we will use Chellas semantics to define two logics over $\mathcal {CN}$ , denoted by $\mathsf {CK}$ and $\mathsf {N4CK}$ , respectively. We begin by defining our models.

The absence of bound variables in $\mathcal {CN}$ allows for a shorter definition of evaluation points compared to the one we had in the first-order case: we simply set

$$ \begin{align*}EP_{ck}:= \{(\mathcal{M}, \mathbf{w})\mid \mathcal{M} \in \mathbb{CK},\,\mathbf{w} \in W\}. \end{align*} $$

The satisfaction relation (denoted by $\models _{ck}$ ) is then supplied by the following definition:

where $\|\psi \|^{ck}_{\mathcal {M}} := \{\mathbf {w}\in W\mid \mathcal {M}, \mathbf {w}\models _{ck}\psi \}$ . Having thus defined the intended semantics for the classical conditional logic, we set $\mathsf {CK} = \mathbb {L}(EP_{ck}, \models _{ck})$ .

It is easy to see that $\mathsf {CK}$ extends $\mathsf {CL}$ . Furthermore, our semantics represents conditionals as a sort of modal sentences where the box is indexed by the antecedent; in other words, it generates a binary $\phi $ -accessibility relation for every $\phi \in \mathcal {CN}$ and reads a conditional ‘if $\phi $ then $\psi $ ’ as something like ‘it is $\phi $ -necessary that $\psi $ ’. This connection leads to some obvious correspondences between normal modal logics and conditional logics based on Chellas semantics: indeed, one can meaningfully ask what kind of modalities are we dealing with when we view the conditionals of a given logic as nothing but formula-indexed modal boxes, and these considerations lead us to the notion of a modal companion of a given conditional logic. For example, the formula-indexed boxes assumed by the conditionals in $\mathsf {CK}$ can be viewed as $\mathsf {K}$ -boxes, which, for every $\phi \in \mathcal {CN}$ , is attested by the existence of the faithful embedding $\eta _\phi $ of $\mathsf {K}$ into $\mathsf {CK}$ which commutes with the propositional connectives and reads modal boxes as conditionals with $\phi $ as the antecedent.

Thus the idea of standard translation, at least for the conditional logics based on Chellas semantics, readily suggests itself. Of course, our accessibility relation is now ternary, so we have to use $R^3$ instead of $E^2$ in the correspondence language. An additional difficulty is that our binary accessibility relations are generated from $R^3$ by truth-sets of the conditional formulas, therefore, our semantics has to say something not only about particular nodes in a Kripke model, but also about certain subsets of this model. Thus we must allow for two kinds of items in the first-order correspondence language, the nodes of the corresponding Kripke model and the subsets thereof; we need, therefore, to add two additional unary predicates to our correspondence language. In our article we denote them $S^1$ and $O^1$ (for ‘sets’ and ‘objects’). Next, we must be able to express that some sets are in fact truth sets of certain conditional formulas, that is to say that an item of the object kind is their element iff this item verifies a given conditional formula, which further means that we have to allow the elementhood relation into the correspondence language. In what follows, we will denote this relation by $E^2$ . To sum up, our correspondence language turns out to be just $\mathcal {FO}(\Pi ) = \mathcal {FO}$ .

Given this informal interpretation of the predicates in $\Pi $ , the following definition is just a straightforward first-order formalization of the inductive definition of $\models _{ck}$ . For every $x \in Ind$ , let us fix $y,z,w \in Ind$ such that $x,y,z,w$ are pairwise distinct. Moreover, for any given $\phi \in \mathcal {FO}$ let $(\forall x)_O\phi $ abbreviate $\forall x(Ox\to \phi )$ . The classical standard x-translation $st_x:\mathcal {CN} \to \mathcal {FO}^x$ is given by induction on the construction of a $\phi \in \mathcal {CN}$ :

Of course, in order for $st_x$ to work correctly, the interaction between S, $O,$ and E must in fact resemble the interaction between sets of worlds and worlds in a classical conditional model. Although E does not need to be a ‘real’ elementhood verifying anything like ZFC, we can at least expect that E satisfies a variant of the extensionality axiom in that we do not have multiple copies of the same truth-set; moreover, sufficiently many instances of set-theoretic comprehension must hold in order to guarantee that at least one copy of a truth-set for any given $\phi \in \mathcal {CN}$ is contained in the extension of the S predicate. The following first-order theory $Th_{ck}\subseteq \mathcal {FO}^\emptyset $ sums up these requirements:

(Thc1) $$ \begin{align} &\forall x\forall y(Sx\wedge Sy \wedge (\forall z)_O(Ezx\leftrightarrow Ezy) \to x \equiv y) \end{align} $$

(Thc2) $$ \begin{align} &\exists x(Sx\wedge (\forall y)_O(Eyx\leftrightarrow py))\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad(p^1\in Prop) \end{align} $$

(Thc3) $$ \begin{align} &\forall x(Sx\rightarrow\exists y(Sy\wedge (\forall z)_O(Ezy\leftrightarrow \sim Ezx)) \end{align} $$

(Thc4) $$ \begin{align} &\forall x\forall y((Sx\wedge Sy)\rightarrow\exists z(Sz\wedge (\forall w)_O(Ewz\leftrightarrow (Ewx\wedge Ewy)))) \end{align} $$

(Thc5) $$ \begin{align} &\forall x\forall y((Sx{\kern-1pt}\wedge{\kern-1pt} Sy){\kern-1pt}\rightarrow{\kern-1pt}\exists z(Sz{\kern-1pt}\wedge{\kern-1pt} (\forall w)_O(Ewz{\kern-1pt}\leftrightarrow{\kern-1pt} \forall u(Rwxu{\kern-1pt}\to{\kern-1pt} Euy)))) \end{align} $$

It is easy to show that this, relatively lightweight, encoding of the subset structure of a classical conditional model is sufficient to turn $st_x$ into a faithful embedding of $\mathsf {CK}$ into $\mathsf {QCL}$ . Although the proof of the fact that $st_x$ faithfully embeds $\mathsf {CK}$ into $\mathsf {QCL}$ under the assumption of $Th_{ck}$ provides no principal difficulty, we could not find it in the existing literature, so we devote the rest of this section to sketching it. In other words,Footnote ¹⁰ we claim the following.

The proof of this proposition employs the following technical lemmas.

Proof (a sketch)

Part 1 is straightforward (even though somewhat tedious) to check. As for Part 2, we argue by induction on the construction of $\phi \in \mathcal {CN}$ in which we only consider a couple of cases.

Basis. Assume that $\phi = p$ where $p^1 \in Prop$ . Then we have

$$ \begin{align*}\mathcal{M}, \mathbf{w} \models_{ck} \phi \text{ iff } \mathbf{w} \in V(p) \text{ iff } \mathbf{w} \in p^{\mathcal{M}^{cl}} \text{ iff } \mathcal{M}^{cl}\models_{c} px = st_x(\phi)[x/\mathbf{w}]. \end{align*} $$

Induction step. The Boolean cases are straightforward. In case for some $\psi , \chi \in \mathcal {CN}$ , and pairwise distinct $x,y,z, w \in Ind$ , we begin by noting that IH for $\psi $ implies that:

(7) $$ \begin{align} \mathcal{M}^{cl}\models_{c} Sy \wedge (\forall z)_O (Ezy\leftrightarrow st_z(\psi))[y/\|\psi\|^{ck}_{\mathcal{M}}]. \end{align} $$

We now reason as follows:

$$ \begin{align*} \mathcal{M}, \mathbf{w} \models_{ck} \phi &\text{ iff } (\forall \mathbf{v} \in W)(R_{\|\psi\|^{ck}_{\mathcal{M}}}(\mathbf{w},\mathbf{v})\text{ implies }\mathcal{M}, \mathbf{v}\models_{ck} \chi)\\&\text{ iff } (\forall \mathbf{v} \in W)(R^{\mathcal{M}^{cl}}(\mathbf{w},\|\psi\|^{ck}_{\mathcal{M}},\mathbf{v})\text{ implies}\\& \quad \mathcal{M}^{cl}\models_{c} st_{w}(\chi)[w/\mathbf{v}])&&\text{by IH}\\&\text{ iff } (\forall \mathbf{v} \in W\ \cup \mathcal{P}(W))(R^{\mathcal{M}^{cl}}(\mathbf{w},\|\psi\|^{ck}_{\mathcal{M}},\mathbf{v})\\& \quad \text{implies }\mathcal{M}^{cl}\models_{c} st_{w}(\chi)[w/\mathbf{v}])&&\text{by def. of } R^{\mathcal{M}^{cl}}\\&\text{ iff } (\forall \mathbf{v} \in U^{\mathcal{M}^{cl}})(R^{\mathcal{M}^{cl}}(\mathbf{w},\|\psi\|^{ck}_{\mathcal{M}},\mathbf{v})\text{ implies}\\& \quad \mathcal{M}^{cl}\models_{c} st_{w}(\chi)[w/\mathbf{v}])\\&\text{ iff } \mathcal{M}^{cl}\models_{c} \forall w(Rxyw \to st_{w}(\chi))[x/\mathbf{w}, y/\|\psi\|^{ck}_{\mathcal{M}}]. \end{align*} $$

The latter implies, by (7), that

(8) $$ \begin{align} \mathcal{M}^{cl}\models_{c} \exists y(Sy \wedge (\forall z)_O (Ezy\leftrightarrow st_z(\psi))\wedge\forall w(Rxyw\to st_{w}(\chi)))[x/\mathbf{w}] \end{align} $$

in other words, that . Conversely, assuming (8), we can find an $X \in U^{\mathcal {M}^{cl}}$ such that $X \in S^{\mathcal {M}^{cl}} = \mathcal {P}(W)$ and we have

(9) $$ \begin{align} \mathcal{M}^{cl}\models_{c} (\forall z)_O (Ezy\leftrightarrow st_z(\psi))\wedge\forall w(Rxyw\to st_{w}(\chi)))[x/\mathbf{w}, y/X]. \end{align} $$

But then, by (7) and Part 1, it follows that $X = \|\psi \|^{ck}_{\mathcal {M}}$ so that $ \mathcal {M}^{cl}\models _{c} \forall w(Rxyw \to st_{w}(\chi ))[x/\mathbf {w}, y/\|\psi \|^{ck}_{\mathcal {M}}]$ follows.

Yet another lemma provides for the converse direction of Proposition 6.

Proof (a sketch)

It is clear that $\mathcal {M}^{ck} \in \mathbb {CK}$ ; as for the main statement of the lemma, we proceed by induction on the construction of $\phi \in \mathcal {CN}$ .

Basis. Assume that $\phi = p$ where $p^1 \in Prop$ . Then we have

$$ \begin{align*}\mathcal{M}^{ck}, \mathbf{w} \models_{ck} \phi = p \text{ iff } \mathbf{w} \in V(p) \text{ iff } \mathbf{w} \in p^{\mathcal{M}} \text{ iff } \mathcal{M}\models_{c} px = st_x(\phi)[x/\mathbf{w}]. \end{align*} $$

Induction step. The Boolean cases are straightforward by IH. If , assume $x,y,z, w \in Ind$ to be pairwise distinct. By Lemma 8, we know that for some $a \in S^{\mathcal {M}}$ we have

(10) $$ \begin{align} \mathcal{M}\models_{c} Sy \wedge (\forall z)_O (Ezy\leftrightarrow st_z(\psi))[y/a]. \end{align} $$

For the left-to-right half of the lemma, assume that $\mathcal {M}^{ck}, \mathbf {w} \models _{ck} \phi $ . Then $(\forall \mathbf {v} \in W^{ck})(R^{ck}_{\|\psi \|^{ck}_{\mathcal {M}^{ck}}}(\mathbf {w},\mathbf {v})\text { implies }\mathcal {M}^{ck}, \mathbf {v}\models _{ck} \chi )$ . If now $\mathbf {v} \in U^{\mathcal {M}} = W^{ck}$ is such that $R^{\mathcal {M}}(\mathbf {w}, a, \mathbf {v})$ , then, by IH and the choice of a, we must have $R^{ck}_{\|\psi \|^{ck}_{\mathcal {M}^{ck}}}(\mathbf {w},\mathbf {v})$ , whence also $\mathcal {M}^{ck}, \mathbf {v}\models _{ck} \chi $ , and, by IH, $\mathcal {M}\models _{c} st_{w}(\chi )[w/\mathbf {v}]$ . Thus we have shown that $\mathcal {M}\models _{c}\forall w(Rxyw\to st_{w}(\chi )))[x/\mathbf {w}, y/a]$ , which, together with (10), yields that .

For the converse, assume that . Then there must exist a $b \in S^{\mathcal {M}}$ such that both

(11) $$ \begin{align} \mathcal{M}\models_{c} Sy \wedge (\forall z)_O (Ezy\leftrightarrow st_z(\psi))[y/b] \end{align} $$

and

(12) $$ \begin{align} \mathcal{M}\models_{c} \forall w(Rxyw\to st_{w}(\chi))[x/\mathbf{w}, y/b]. \end{align} $$

If now $\mathbf {v} \in W^{ck} = U^{\mathcal {M}}$ is such that $R^{ck}_{\|\psi \|^{ck}_{\mathcal {M}^{ck}}}(\mathbf {w},\mathbf {v})$ , then, by IH, we can choose a $c\in U^{\mathcal {M}}$ such that both $R^{\mathcal {M}}(\mathbf {w},c,\mathbf {v})$ and

(13) $$ \begin{align} \mathcal{M}\models_{c} Sy' \wedge (\forall z)_O (Ezy'\leftrightarrow st_z(\psi))[y'/c]. \end{align} $$

Now, (11) and (13), together with (Thc1), imply that $b = c$ so that $R^{\mathcal {M}}(\mathbf {w},b,\mathbf {v})$ also follows. The latter entails $\mathcal {M}\models _{c} st_{w}(\chi )[w/\mathbf {v}]$ by (12). By IH, we must have $\mathcal {M}^{ck}, \mathbf {v}\models _{ck} \chi $ . Thus we have shown that .

If we attempt to adapt the general criterion for seeking out non-classical counterparts to $\mathsf {K}$ that was formulated in the previous section, to the context of conditional logic, the following principle suggests itself: given a non-classical logic $Q\nu $ such that $Q\nu $ is a proper sublogic of $\mathsf {QCL}$ and its propositional fragment $\nu $ is a proper sublogic of $\mathsf {CL}$ , a conditional extension $\nu \mathsf {CK}$ of $\nu $ is a $\nu $ -based counterpart of $\mathsf {CK}$ iff some classically equivalent reformulation of $\sigma \tau _x$ faithfully embeds $\nu \mathsf {CK}$ into $\mathsf {Q}\nu (\Pi )$ modulo the assumption of some classically equivalent reformulation of $Th_{ck}$ ; the comparative merits of different $\nu $ -based counterparts of $\mathsf {CK}$ will then have to be judged on the basis of other properties of $Q\nu $ .

Approaching with this principle to $\mathsf {QIL}$ , we have defined an intuitionistic counterpart $\mathsf {IntCK}$ of $\mathsf {CK}$ in [Reference Olkhovikov12], and we have shown, in [Reference Olkhovikov12, Theorem 2], that $\mathsf {IntCK}$ is an intuitionistic counterpart to $\mathsf {CK}$ according to the above criterion.Footnote ¹¹ Very conveniently, the reformulation $st^i_x$ of $st_x$ used in [Reference Olkhovikov12] is similar to the reformulation $\sigma \tau ^i_x$ of $\sigma \tau _x$ , described in the previous section in that the change consists in adding the standard translation of as a separate clause. Moreover, [Reference Olkhovikov12, Proposition 4] shows that, if we also add a clause for diamonds to the mapping $\eta _\phi $ mentioned in the beginning of this section, then we get an embedding of $\mathsf {IK}$ into $\mathsf {IntCK}$ . Thus, $\mathsf {K}$ , $\mathsf {CK}$ , $\mathsf {IK}$ , and $\mathsf {IntCK}$ are tied to one another by natural embeddings forming, as it were, a tightly connected system. The main aim of the next section, and of this article in general, is to extend this system with $\mathsf {N4}$ -based modal and conditional logics.

7 $\mathsf {N4CK}$ , the basic Nelsonian conditional logic

Another logic based on $\mathcal {CN}$ extends $\mathsf {N4}$ rather than $\mathsf {CL}$ . It was introduced in [Reference Olkhovikov11], and its semantics is based on the following notion.

Conditions (c1) and (c2) can be reformulated as requirements to complete the dotted parts of each of the two diagrams given in Figure 1 once the respective straight-line part is given.

Figure 1 Conditions (c1) and (c2).

The evaluation points for $\mathsf {N4CK}$ are then defined as in $\mathsf {CK}$ , in other words, we set $EP:= \{(\mathcal {M}, \mathbf {w})\mid \mathcal {M}\in \mathbb {NC},\,\mathbf {w} \in W\}$ .

Just as in other Nelsonian logics, we find in $\mathsf {N4CK}$ two satisfaction relations $\models ^+$ and $\models ^-$ , representing verifications and falsifications of the conditional formulas, respectively. Their inductive definition runs as follows:

where we assume, for any given $\phi \in \mathcal {CN}$ , that:

$$ \begin{align*}\|\phi\|_{\mathcal{M}}: = (\|\phi\|^+_{\mathcal{M}}, \|\phi\|^-_{\mathcal{M}}) = (\{\mathbf{w} \in W\mid \mathcal{M}, \mathbf{w}\models^+\phi\}, \{\mathbf{w} \in W\mid \mathcal{M}, \mathbf{w}\models^-\phi\}). \end{align*} $$

To set up the semantics for $\mathsf {N4CK}$ it only remains to define that $\mathsf {N4CK}:= \mathbb {L}(EP, \models ^+)$ .

The following lemma is then straightforward to prove.

Among other things, [Reference Olkhovikov11] considered the question of a complete Hilbert-style axiomatization of $\mathsf {N4CK}$ . It turns out (see [Reference Olkhovikov11, Theorem 1]) that we have $\mathsf {N4CK} = \mathfrak {N4CK}(\mathcal {CN})$ for $\mathfrak {N4CK} = \mathfrak {IL}^+ +((An1)\text {--}(An6),(Ax1)\text {--}(Ax4);(RA\Box),(RC\Box1),(RC\Box2))$ , where we assume that:

(Ax1)

(Ax2)

(Ax3)

(Ax4)

(RA□)

(RC□1)

(RC□2)

In what follows, we will write $\vdash $ to denote $\vdash _{\mathfrak {N4CK}}$ .

The completeness of $\mathfrak {N4CK}$ relative to $\mathsf {N4CK}$ was shown in [Reference Olkhovikov11] by the rather standard method of constructing a universal model; we would like to recall this construction in some detail as we plan to use it below. More precisely, we say that, for any given $\Gamma , \Delta \subseteq \mathcal {CN}$ , the pair $(\Gamma , \Delta )$ is maximal iff $\Gamma \not \vdash \Delta $ and $\Gamma \cup \Delta = \mathcal {CN}$ . Our universal model can now be defined as follows.

It follows from [Reference Olkhovikov11, Proposition 7] that:

The main result of this article is that an analogue of Proposition 6 can be proven for $\mathsf {N4CK}$ and $\mathsf {QN4}$ , respectively, which considerably strengthens the claim that $\mathsf {N4CK}$ is the correct minimal conditional logic on the Nelsonian propositional basis. In order to do that, we first need to define the right classical equivalents for $Th_{ck}$ and $st_x$ , respectively. Since these equivalents are at least syntactically different from $Th_{ck}$ and $st_x$ , we will introduce for them separate notations, namely, $Th$ and $ST_x$ .

As for the first component in this pair, we assume that $Th \subseteq \mathcal {FO}^\emptyset $ contains all and only the following first-order sentences:

(Th1) $$ \begin{align} &\forall x\forall y(Sx\wedge Sy \wedge (\forall z)_O(Ezx\Leftrightarrow Ezy) \to x \equiv y) \end{align} $$

(Th2) $$ \begin{align} &\exists x(Sx\wedge (\forall y)_O(Eyx\Leftrightarrow py))\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad(p^1\in Prop) \end{align} $$

(Th3) $$ \begin{align} &\forall x(Sx\rightarrow\exists y(Sy\wedge (\forall z)_O(Ezy\Leftrightarrow \sim Ezx)) \end{align} $$

(Th4) $$ \begin{align} &\forall x\forall y((Sx\wedge Sy)\rightarrow\exists z(Sz\wedge (\forall w)_O(Ewz\Leftrightarrow (Ewx\ast Ewy))))\quad(\ast\in\{\wedge, \to\}) \end{align} $$

(Th5) $$ \begin{align} &\forall x\forall y((Sx\wedge Sy)\rightarrow\exists z(Sz\wedge (\forall w)_O(Ewz\Leftrightarrow \forall u(Rwxu\to Euy)))). \end{align} $$

It is easy to see that $Th$ is classically equivalent to $Th_{ck}$ . The only difference between the two theories is the replacement of equivalences encoding the set extensions in $Th_{ck}$ with the strong equivalences which allows for a replacement of a definable set with its definition in $\mathsf {QN4}$ . In view of the considerations given in Appendix A.5, one could also argue for replacing conjunctions with ampersands in (Th2)–(Th5); however, this is not necessary, since $Th$ only imposes the truth of its components and never says anything about their falsity conditions. We have seen, however, that the differences between $\wedge $ and $\&$ only become apparent when the treatment of the falsity component of a restricted existential quantification comes into question. Therefore, the distinction between $\wedge $ and $\&$ can no longer be overlooked when it comes to the definition of the standard translation which is supposed to faithfully represent both the truth and the falsity conditions of a translated formula. As a result, we obtain, for every given $x \in Ind$ , the standard translation $ST_x:\mathcal {CN}\to \mathcal {FO}^x$ defined by the following induction on the construction of a $\phi \in \mathcal {CN}$ (assuming the same choice of $x,y,z,w$ as in $st_x$ ):

The following proposition provides the ‘easy’ direction of the faithfulness claim.

Its proof is analogous to the proof of the corresponding half of Proposition 6 given in the previous section. We start by proving the following technical lemmas.

The proof of the lemma is relegated to Appendix B.

In order to avoid the clutter, we introduce the notation $\Xi ((\mathbf {w},c), (X,Y))$ for the main universally quantified bi-conditional in Definition 8.3; this part of the definition can then be re-written as

$$ \begin{align*}&R_{\mathcal{S}}((\mathbf{w},a), (X, Y), (\mathbf{v},b))\\& \quad \text{ iff }\mathbf{w} = \mathbf{v}\text{ and }(\exists c \in S^{\mathtt{M}^+_{\mathbf{w}}})(\Xi((\mathbf{w},c), (X,Y))\text{ and }R^{\mathtt{M}^+_{\mathbf{w}}}(a,c,b)). \end{align*} $$

Proof The transitivity and reflexivity of $\leq _{\mathcal {S}}$ easily follow from the same properties of $\leq $ . As for the monotonicity of $V^+_{\mathcal {S}}$ relative to $\leq _{\mathcal {S}}$ , assume that $p^1 \in Prop$ , and that $(\mathbf {w},a), (\mathbf {v},b) \in W_{\mathcal {S}}$ are such that $(\mathbf {w},a)\leq _{\mathcal {S}}(\mathbf {v},b)$ . Then $\mathbf {w} \leq \mathbf {v}$ and $\mathtt {H}_{\mathbf {w}\mathbf {v}}(a) = b$ . If now $(\mathbf {w},a) \in V^+_{\mathcal {S}}(p)$ , then $a \in p^{\mathtt {M}^+_{\mathbf {w}}}$ , and, since $\mathtt {H}_{\mathbf {w}\mathbf {v}}\in Hom(\mathtt {M}^+_{\mathbf {w}}, \mathtt {M}^+_{\mathbf {v}})$ , also $b = \mathtt {H}_{\mathbf {w}\mathbf {v}}(a) \in p^{\mathtt {M}^+_{\mathbf {v}}}$ , which, in turn, means that $(\mathbf {v},b) \in V^+_{\mathcal {S}}(p)$ . We argue similarly for the monotonicity of $V^-_{\mathcal {S}}$ .

It only remains to check the conditions imposed on R. As for (c1), assume that $(\mathbf {w},a), (\mathbf {v},b), (\mathbf {u},c) \in W_{\mathcal {S}}$ and $X, Y \subseteq W_{\mathcal {S}}$ are such that both $(\mathbf {w},a)\leq _{\mathcal {S}}(\mathbf {v},b)$ and $R_{\mathcal {S}}((\mathbf {w},a), (X, Y), (\mathbf {u},c))$ . Then $\mathbf {w} = \mathbf {u}$ and $\mathbf {w} \leq \mathbf {v}$ , whence $(\mathbf {u},c) = (\mathbf {w},c)\leq _{\mathcal {S}}(\mathbf {v},\mathtt {H}_{\mathbf {w}\mathbf {v}}(c))\in W_{\mathcal {S}}$ . On the other hand, $\mathtt {H}_{\mathbf {w}\mathbf {v}}(a) = b$ , and there exists a $d \in S^{\mathtt {M}^+_{\mathbf {w}}}$ such that both $R^{\mathtt {M}^+_{\mathbf {w}}}(a,d,c)$ and, for all $\mathbf {w}'\geq \mathbf {w}$ and all $g \in O^{\mathtt {M}^+_{\mathbf {w}'}}$ we have

(14) $$ \begin{align}\kern-6pt ((\mathbf{w}',g)\in X\text{ iff }E^{\mathtt{M}^+_{\mathbf{w}'}}(g,\mathtt{H}_{\mathbf{w}\mathbf{w}'}(d))) \text{ and }((\mathbf{w}',g)\in Y\text{ iff }E^{\mathtt{M}^-_{\mathbf{w}'}}(g,\mathtt{H}_{\mathbf{w}\mathbf{w}'}(d))). \end{align} $$

Now, consider $\mathtt {H}_{\mathbf {w}\mathbf {v}}(d)\in S^{\mathtt {M}^+_{\mathbf {v}}}$ . Since $\mathtt {H}_{\mathbf {w}\mathbf {v}}\in Hom(\mathtt {M}^+_{\mathbf {w}}, \mathtt {M}^+_{\mathbf {v}})$ , we must have that $R^{\mathtt {M}^+_{\mathbf {v}}}(\mathtt {H}_{\mathbf {w}\mathbf {v}}(a),\mathtt {H}_{\mathbf {w}\mathbf {v}}(d),\mathtt {H}_{\mathbf {w}\mathbf {v}}(c))$ , in other words, that $R^{\mathtt {M}^+_{\mathbf {v}}}(b,\mathtt {H}_{\mathbf {w}\mathbf {v}}(d),\mathtt {H}_{\mathbf {w}\mathbf {v}}(c))$ . Moreover, if $\mathbf {v}'\geq \mathbf {v}$ and $k\in O^{\mathtt {M}^+_{\mathbf {v}'}}$ , then transitivity implies $\mathbf {v}'\geq \mathbf {w}$ , and Definition 4 implies that $\mathtt {H}_{\mathbf {v}\mathbf {v}'}(\mathtt {H}_{\mathbf {w}\mathbf {v}}(d)) = \mathtt {H}_{\mathbf {w}\mathbf {v}'}(d)$ , whence, by (14), we must have

$$ \begin{align*} ((\mathbf{v}',k)\in X\text{ iff }E^{\mathtt{M}^+_{\mathbf{v}'}}(k,\mathtt{H}_{\mathbf{v}\mathbf{v}'}(\mathtt{H}_{\mathbf{w}\mathbf{v}}(d)))) \text{ and }((\mathbf{v}',k)\in Y\text{ iff }E^{\mathtt{M}^-_{\mathbf{v}'}}(k,\mathtt{H}_{\mathbf{v}\mathbf{v}'}(\mathtt{H}_{\mathbf{w}\mathbf{v}}(d)))). \end{align*} $$

Therefore, we must also have $R_{\mathcal {S}}((\mathbf {v},b), X, Y, (\mathbf {v}, \mathtt {H}_{\mathbf {w}\mathbf {v}}(c)))$ , and (c1) is shown to hold.

The argument for (c2) is similar.

Since Definition 8.3 is relatively involved, we look a bit deeper into its consequences, before proceeding further.

Lemma 14. Let $\mathcal {S}\in \mathbb {N}4$ and let $\phi \in \mathcal {CN}$ . Then the following statements hold:

1. 1. Assume that, for every $\star \in \{+, -\}$ , every $x \in Ind$ , and every $(\mathbf {w},a) \in W_{\mathcal {S}}$ , we have $\mathcal {M}_{\mathcal {S}}, (\mathbf {w},a) \models ^\star \phi $ iff $\mathcal {S}, \mathbf {w}\models ^\star _{n}ST_x(\phi )[x/a]$ . Then, for every $(\mathbf {v}, b)\in W_{\mathcal {S}}$ and all distinct $y,z \in Ind$ we have

   $$ \begin{align*}\mathcal{S}, \mathbf{v}\models^+_n (\forall z)_O(Ezy \Leftrightarrow ST_z(\phi))[y/b] \text{ iff }\Xi((\mathbf{v},b), \|\phi\|_{\mathcal{M}_{\mathcal{S}}}). \end{align*} $$

2. 2. For all $(\mathbf {w},a), (\mathbf {v}, b) \in W_{\mathcal {S}}$ , if $(\mathbf {w},a)\leq _{\mathcal {S}}(\mathbf {v}, b)$ and $\Xi ((\mathbf {w},a), \|\phi \|_{\mathcal {M}_{\mathcal {S}}})$ , then $\Xi ((\mathbf {v},b), \|\phi \|_{\mathcal {M}_{\mathcal {S}}})$ .

Proof (Part 1) Assume the premises. We have $\mathcal {S}, \mathbf {v}\models ^+_n (\forall z)_O(Ezy \Leftrightarrow ST_z(\phi ))[y/b]$ iff, by Corollary 2, we have

$$ \begin{align*} (\forall \mathbf{u} \geq \mathbf{v})(\forall d \in O^{\mathtt{M}^+_{\mathbf{u}}})&((\mathcal{S},\mathbf{u}\models^+_n ST_z(\phi)[z/d]\text{ iff }\mathcal{S},\mathbf{u}\models^+_n Ezy[y/\mathtt{H}_{\mathbf{w}\mathbf{u}}(b), z/d])\nonumber\\ &\text{ and }(\mathcal{S},\mathbf{u}\models^-_n ST_z(\phi)[z/d]\text{ iff }\mathcal{S},\mathbf{u}\models^-_n Ezy[y/\mathtt{H}_{\mathbf{w}\mathbf{u}}(b), z/d])). \end{align*} $$

By our hypothesis about $\phi $ , the latter statement is equivalent to

$$ \begin{align*} (\forall \mathbf{u} \geq \mathbf{v})(\forall d \in O^{\mathtt{M}^+_{\mathbf{u}}})(((\mathbf{u}, z)&\in \|\phi\|^+_{\mathcal{M}_{\mathcal{S}}}\text{ iff } E^{\mathtt{M}^+_{\mathbf{u}}}(d,\mathtt{H}_{\mathbf{w}\mathbf{u}}(b)))\nonumber\\ &\text{ and }((\mathbf{u}, z)\in \|\phi\|^-_{\mathcal{M}_{\mathcal{S}}}\text{ iff }E^{\mathtt{M}^-_{\mathbf{u}}}(d,\mathtt{H}_{\mathbf{w}\mathbf{u}}(b)))) \end{align*} $$

that is to say, to $\Xi ((\mathbf {v},b), \|\phi \|_{\mathcal {M}_{\mathcal {S}}})$ .

(Part 2) Assume that $(\mathbf {w},a)\leq _{\mathcal {S}}(\mathbf {v}, b)$ ; then $\mathbf {w} \leq \mathbf {v}$ and $\mathtt {H}_{\mathbf {w}\mathbf {v}}(a) = b$ . If now $\Xi ((\mathbf {w},a), \|\phi \|_{\mathcal {M}_{\mathcal {S}}})$ , this means, by Part 1, that $\mathcal {S}, \mathbf {w}\models ^+_n (\forall z)_O(Ezy \Leftrightarrow ST_z(\phi ))[y/a]$ , whence, by Lemma 4.1, $\mathcal {S}, \mathbf {v}\models ^+_n (\forall z)_O(Ezy \Leftrightarrow ST_z(\phi )) [y/\mathtt {H}_{\mathbf {w}\mathbf {v}}(a)]$ , whence, again by Part 1, $\Xi ((\mathbf {v},b), \|\phi \|_{\mathcal {M}_{\mathcal {S}}})$ .

Proof We argue by induction on the construction of $\phi \in \mathcal {CN}$ .

Basis. If $\phi = p$ for some $p^1 \in Prop$ , then we have, for every $\star \in \{+, -\}$ and $(\mathbf {w},a) \in W_{\mathcal {S}}$ , that $\mathcal {M}_{\mathcal {S}}, (\mathbf {w},a) \models ^\star p$ iff $(\mathbf {w},a)\in V^\star _{\mathcal {S}}(p)$ iff $a \in p^{\mathtt {M}^\star _{\mathbf {w}}}$ iff $\mathcal {S}, \mathbf {w}\models ^\star _n px[x/a]$ .

Induction step. The following cases arise.

Case 1. If $\phi = \psi \wedge \chi $ , then we have, on the one hand,

$$ \begin{align*} \mathcal{M}_{\mathcal{S}}, (\mathbf{w},a) \models^+ \psi \wedge \chi &\text{ iff } \mathcal{M}_{\mathcal{S}}, (\mathbf{w},a) \models^+ \psi\text{ and }\mathcal{M}_{\mathcal{S}}, (\mathbf{w},a) \models^+\chi&&\text{ by IH}\\&\text{ iff } \mathcal{S}, \mathbf{w}\models^+_n ST_x(\psi)[x/a]\text{ and }\\& \quad\ \, \mathcal{S}, \mathbf{w}\models^+_n ST_x(\chi)[x/a]\\&\text{ iff } \mathcal{S}, \mathbf{w}\models^+_n ST_x(\psi)\wedge ST_x(\chi)[x/a]\\&\text{ iff } \mathcal{S}, \mathbf{w}\models^+_n ST_x(\psi\wedge\chi)[x/a]. \end{align*} $$

On the other hand, we have

$$ \begin{align*} \mathcal{M}_{\mathcal{S}}, (\mathbf{w},a) \models^- \psi \wedge \chi &\text{ iff } \mathcal{M}_{\mathcal{S}}, (\mathbf{w},a) \models^- \psi\text{ or }\mathcal{M}_{\mathcal{S}}, (\mathbf{w},a) \models^-\chi\\&\text{ iff } \mathcal{S}, \mathbf{w}\models^-_n ST_x(\psi)[x/a]\text{ or } \\& \quad\ \, \mathcal{S}, \mathbf{w}\models^-_n ST_x(\chi)[x/a]&&\text{ by IH}\\&\text{ iff } \mathcal{S}, \mathbf{w}\models^-_n ST_x(\psi)\wedge ST_x(\chi)[x/a]\\&\text{ iff } \mathcal{S}, \mathbf{w}\models^-_n ST_x(\psi\wedge\chi)[x/a]. \end{align*} $$

Case 2 and Case 3, where we assume that $\phi = \psi \vee \chi $ and $\phi = \sim \psi $ , respectively, are solved similarly to Case 1.

Case 4. If $\phi = \psi \to \chi $ , then assume that $ \mathcal {M}_{\mathcal {S}}, (\mathbf {w},a) \models ^+ \psi \to \chi $ , and let $\mathbf {v}\geq \mathbf {w}$ be such that $\mathcal {S}, \mathbf {v}\models ^+_n ST_x(\psi )[x/\mathtt {H}_{\mathbf {w}\mathbf {v}}(a)]$ . Then, by IH, $\mathcal {M}_{\mathcal {S}}, (\mathbf {v},\mathtt {H}_{\mathbf {w}\mathbf {v}}(a)) \models ^+ \psi $ , and we also have $(\mathbf {w},a)\leq _{\mathcal {S}}(\mathbf {v},\mathtt {H}_{\mathbf {w}\mathbf {v}}(a))$ , whence, by our assumption, $\mathcal {M}_{\mathcal {S}}, (\mathbf {v},\mathtt {H}_{\mathbf {w}\mathbf {v}}(a)) \models ^+ \chi $ . Again by IH, $\mathcal {S}, \mathbf {v}\models ^+_n ST_x(\chi )[x/\mathtt {H}_{\mathbf {w}\mathbf {v}}(a)]$ . Since $\mathbf {v}\geq \mathbf {w}$ was chosen arbitrarily, we conclude that

(15) $$ \begin{align} \mathcal{S}, \mathbf{w}\models^+_n (ST_x(\psi)\to ST_x(\chi)) = ST_x(\psi\to\chi)[x/a]. \end{align} $$

In the other direction, assume that (15) holds. If $(\mathbf {v},b) \in W_{\mathcal {S}}$ is such that $(\mathbf {w},a)\leq _{\mathcal {S}}(\mathbf {v},b)$ , then $\mathbf {w} \leq \mathbf {v}$ and $\mathtt {H}_{\mathbf {w}\mathbf {v}}(a) = b$ . If, moreover, we have $\mathcal {M}_{\mathcal {S}}, (\mathbf {v},b) \models ^+ \psi $ , then, by IH, $\mathcal {S}, \mathbf {v}\models ^+_n ST_x(\psi )[x/\mathtt {H}_{\mathbf {w}\mathbf {v}}(a)]$ , whence, by (15), $\mathcal {S}, \mathbf {v}\models ^+_n ST_x(\chi )[x/\mathtt {H}_{\mathbf {w}\mathbf {v}}(a)]$ . Applying IH again, we get that $\mathcal {M}_{\mathcal {S}}, (\mathbf {v},\mathtt {H}_{\mathbf {w}\mathbf {v}}(a)) \models ^+ \psi $ , in other words, that $\mathcal {M}_{\mathcal {S}}, (\mathbf {v},b) \models ^+ \psi $ . By the choice of $(\mathbf {v},b) \in W_{\mathcal {S}}$ , we obtain that $ \mathcal {M}_{\mathcal {S}}, (\mathbf {w},a) \models ^+ \psi \to \chi $ , as desired.

Finally, observe that we have

$$ \begin{align*} \mathcal{M}_{\mathcal{S}}, (\mathbf{w},a) \models^- \psi \to \chi &\text{ iff } \mathcal{M}_{\mathcal{S}}, (\mathbf{w},a) \models^+ \psi\text{ and }\mathcal{M}_{\mathcal{S}}, (\mathbf{w},a) \models^-\chi\\&\text{ iff } \mathcal{S}, \mathbf{w}\models^+_n ST_x(\psi)[x/a]\text{ and } \\& \quad\ \, \mathcal{S}, \mathbf{w}\models^-_n ST_x(\chi)[x/a]&&\text{ by IH}\\&\text{ iff } \mathcal{S}, \mathbf{w}\models^-_n ST_x(\psi)\to ST_x(\chi)[x/a]\\&\text{ iff } \mathcal{S}, \mathbf{w}\models^-_n ST_x(\psi\to\chi)[x/a]. \end{align*} $$

Case 5. If , then assume that

(16)

By Lemma 12, we can choose a $b \in U_{\mathbf {w}}$ such that:

(17) $$ \begin{align} \mathcal{S}, \mathbf{w}\models^+_n Sy\wedge (\forall z)_O(Ezy \Leftrightarrow ST_z(\psi))[y/b]. \end{align} $$

Moreover, choose any $\mathbf {v} \in W$ and any $c \in U_{\mathbf {v}}$ such that:

(18) $$ \begin{align} &\mathbf{v}\geq\mathbf{w} \end{align} $$

(19) $$ \begin{align} &\mathcal{S},\mathbf{v}\models^+_n R(x,y,w)[x/\mathtt{H}_{\mathbf{w}\mathbf{v}}(a), y/\mathtt{H}_{\mathbf{w}\mathbf{v}}(b), w/c]. \end{align} $$

We now reason as follows:

(20) $$ \begin{align} &(\mathbf{w},b)\leq_{\mathcal{S}}(\mathbf{v},\mathtt{H}_{\mathbf{w}\mathbf{v}}(b))&&\text{ by }({18}) \end{align} $$

(21) $$ \begin{align} &b \in S^{\mathtt{M}^+_{\mathbf{w}}}&&\text{ by }(17) \end{align} $$

(22) $$ \begin{align} &\Xi((\mathbf{w},b), \|\psi\|_{\mathcal{M}_{\mathcal{S}}})&&\text{ by }(17), Lm. 14.1,\text{ IH for }\psi \end{align} $$

(23) $$ \begin{align} &\Xi((\mathbf{v},\mathtt{H}_{\mathbf{w}\mathbf{v}}(b)), \|\psi\|_{\mathcal{M}_{\mathcal{S}}})&&\text{ by }({20}),({22}), Lm. {14}.2 \end{align} $$

(24) $$ \begin{align} &\mathtt{H}_{\mathbf{w}\mathbf{v}}(b)\in S^{\mathtt{M}^+_{\mathbf{v}}}&&\text{ by }(21), \mathtt{H}_{\mathbf{w}\mathbf{v}}\in Hom(\mathtt{M}^+_{\mathbf{w}}, \mathtt{M}^+_{\mathbf{v}}) \end{align} $$

(25) $$ \begin{align} &R_{\mathcal{S}}((\mathbf{v},\mathtt{H}_{\mathbf{w}\mathbf{v}}(a)), \|\psi\|_{\mathcal{M}_{\mathcal{S}}}, (\mathbf{v},c))&&\text{ by }(23), (24), (19) \end{align} $$

(26) $$ \begin{align} &(\mathbf{w},a)\leq_{\mathcal{S}}(\mathbf{v},\mathtt{H}_{\mathbf{w}\mathbf{v}}(a))&&\text{ by }(18) \end{align} $$

(27) $$ \begin{align} &\mathcal{M}_{\mathcal{S}}, (\mathbf{v},c)\models^+ \chi&&\text{ by }(16), (25), (26) \end{align} $$

(28) $$ \begin{align} &\mathcal{S}, \mathbf{v}\models^+_n ST_w(\chi)[w/c]&&\text{ by }(27),\text{ IH for }\chi. \end{align} $$

By the choice of $\mathbf {v}$ and c, we obtain that:

(29) $$ \begin{align} \mathcal{S}, \mathbf{w}\models^+_n \forall w(R(x,y,w)\to ST_w(\chi))[x/a, y/b]. \end{align} $$

Now (29), together with (17), allows us to conclude that

(30)

For the converse, assume (30) and choose any $b \in U_{\mathbf {w}}$ such that both (17) and (29) hold. Consider any $(\mathbf {v},c),(\mathbf {u},d) \in W_{\mathcal {S}}$ such that

(31) $$ \begin{align} &(\mathbf{w},a)\leq_{\mathcal{S}}(\mathbf{v},c) \end{align} $$

By (32) and Definition 8.3, there must be an $e \in U_{\mathbf {v}}$ such that all of the following holds:

(33) $$ \begin{align} &e \in S^{\mathtt{M}^+_{\mathbf{v}}} \end{align} $$

(34) $$ \begin{align} &\Xi((\mathbf{v}, e), \|\psi\|_{\mathcal{M}_{\mathcal{S}}}) \end{align} $$

(35) $$ \begin{align} &R^{\mathtt{M}^+_{\mathbf{v}}}(c, e, d). \end{align} $$

We now reason as follows:

(36) $$ \begin{align} &\kern-8pt\mathcal{S}, \mathbf{v}\models^+_n (\forall z)_O(Ezw \Leftrightarrow ST_z(\psi))[w/e]&&\text{ by}\ ({34}), \text{Lm}\ {14}.1,\ \text{IH} \end{align} $$

(37) $$ \begin{align} &\mathcal{S}, \mathbf{v}\models^+_n (\forall z)_O(Ezy \Leftrightarrow ST_z(\psi))[y/\mathtt{H}_{\mathbf{w}\mathbf{v}}(b)]&&\text{ by Lm 4.1, (17)} \end{align} $$

(38) $$ \begin{align} &\mathcal{S}, \mathbf{v}\models^+_n (\forall z)_O(Ezw \Leftrightarrow Ezy)[y/\mathtt{H}_{\mathbf{w}\mathbf{v}}(b), w/e]&&\text{ by (36), (37), (T18)} \end{align} $$

(39) $$ \begin{align} &e = \mathtt{H}_{\mathbf{w}\mathbf{v}}(b)&&\text{ by (Th1), (38)} \end{align} $$

(40) $$ \begin{align} &c = \mathtt{H}_{\mathbf{w}\mathbf{v}}(a)&&\text{ by (31)} \end{align} $$

(41) $$ \begin{align} &\mathcal{S}, \mathbf{v}\models^+_n R(x,y,w)[x/\mathtt{H}_{\mathbf{w}\mathbf{v}}(a), y/\mathtt{H}_{\mathbf{w}\mathbf{v}}(b), w/d]&&\text{ by (35), (39), (40)} \end{align} $$

(42) $$ \begin{align} &\mathcal{S}, \mathbf{v}\models^+_n ST_w(\chi)[w/d]&&\text{ by (41), (29)} \end{align} $$

(43) $$ \begin{align} &\mathcal{M}_{\mathcal{S}}, (\mathbf{v}, d)\models^+ \chi&&\text{ by IH, (42)} \end{align} $$

(44) $$ \begin{align} &\mathbf{v} = \mathbf{u}&&\text{ by (32), Df 8.3} \end{align} $$

(45) $$ \begin{align} &\mathcal{M}_{\mathcal{S}}, (\mathbf{u}, d)\models^+ \chi&&\text{ by (43), (44)}. \end{align} $$

Since $(\mathbf {v},c),(\mathbf {u},d) \in W_{\mathcal {S}}$ were chosen arbitrarily under the conditions (31) and (32), it follows that , as desired.

Next, assume that

(46)

This means that, for some $(\mathbf {v}, b) \in W_{\mathcal {S}}$ , we have

(47) $$ \begin{align} &R_{\mathcal{S}}((\mathbf{w},a), \|\psi\|_{\mathcal{M}_{\mathcal{S}}}, (\mathbf{v},b)) \end{align} $$

(48) $$ \begin{align} &\mathcal{M}_{\mathcal{S}}, (\mathbf{v},b) \models^- \chi. \end{align} $$

By (47), we can choose a $c \in U_{\mathbf {w}}$ such that all of the following holds:

(49) $$ \begin{align} &c \in S^{\mathtt{M}^+_{\mathbf{w}}} \end{align} $$

(50) $$ \begin{align} &\Xi((\mathbf{w}, c), \|\psi\|_{\mathcal{M}_{\mathcal{S}}}) \end{align} $$

(51) $$ \begin{align} &R^{\mathtt{M}^+_{\mathbf{w}}}(a, c, b). \end{align} $$

Assume, next, that some $\mathbf {u}'\geq \mathbf {u}\geq \mathbf {w}$ and some $d\in U_{\mathbf {u}}$ are such that

(52) $$ \begin{align} \mathcal{S}, \mathbf{u}\models^+_n Sy\wedge (\forall z)_O(Ezy \Leftrightarrow ST_z(\psi))[y/d]. \end{align} $$

We now reason as follows:

(53) $$ \begin{align}\!\!\! &\mathbf{w} = \mathbf{v}&&\text{ by} \ ({47}) \end{align} $$

(54) $$ \begin{align} &\mathcal{S}, \mathbf{w}\models^-_n ST_w(\chi)[w/b]&&\text{ by IH for }\chi, \text{ (48),(53)} \end{align} $$

(55) $$ \begin{align} &\mathcal{S}, \mathbf{u}'\models^-_n ST_w(\chi)[w/\mathtt{H}_{\mathbf{w}\mathbf{u}'}(b)]&&\text{ by Lm 4.1, (54)} \end{align} $$

(56) $$ \begin{align} &\mathcal{S}, \mathbf{u}'\models^+_n Rxyw[x/\mathtt{H}_{\mathbf{w}\mathbf{u}'}(a), y/\mathtt{H}_{\mathbf{w}\mathbf{u}'}(c), w/\mathtt{H}_{\mathbf{w}\mathbf{u}'}(b)]&&\text{ by Lm 4.1, (51)} \end{align} $$

(57) $$ \begin{align} & \mathcal{S}, \mathbf{u}'\models^-_n Rxyw \to ST_w(\chi) [x/\mathtt{H}_{\mathbf{w}\mathbf{u}'}(a), y/\mathtt{H}_{\mathbf{w}\mathbf{u}'}(c), w/\mathtt{H}_{\mathbf{w}\mathbf{u}'}(b)] &&\text{ by ({55}), ({56})} \end{align} $$

(58) $$ \begin{align} \mathcal{S}, \mathbf{u}'\models^-_n \forall w(Rxyw \to ST_w(\chi)) [x/\mathtt{H}_{\mathbf{w}\mathbf{u}'}(a), y/\mathtt{H}_{\mathbf{w}\mathbf{u}'}(c)]&&\text{ by ({57})} \end{align} $$

(59) $$ \begin{align} &\mathcal{S}, \mathbf{w}\models^+_n Sx\wedge (\forall z)_O(Ezx \Leftrightarrow ST_z(\psi))[x/c]&&\text{ by Lm 14.1, (49), (50)} \end{align} $$

(60) $$ \begin{align} &\mathcal{S}, \mathbf{u}'\models^+_n Sx\wedge (\forall z)_O(Ezx \Leftrightarrow ST_z(\psi))[x/\mathtt{H}_{\mathbf{w}\mathbf{u}'}(c)]&&\text{ by Lm 4.1, (59)} \end{align} $$

(61) $$ \begin{align} &\mathcal{S}, \mathbf{u}'\models^+_n Sy\wedge (\forall z)_O(Ezy \Leftrightarrow ST_z(\psi))[y/\mathtt{H}_{\mathbf{u}\mathbf{u}'}(d)]&&\text{ by Lm 4.1, (52)} \end{align} $$

(62) $$ \begin{align} \mathcal{S}, \mathbf{u}'\models^+_n Sx\wedge Sy\wedge (\forall z)_O(Ezx \Leftrightarrow Ezy) [x/\mathtt{H}_{\mathbf{w}\mathbf{u}'}(c), y/\mathtt{H}_{\mathbf{u}\mathbf{u}'}(d)]&&\text{ by ({T18}), ({60}), ({61})} \end{align} $$

(63) $$ \begin{align} &\mathtt{H}_{\mathbf{u}\mathbf{u}'}(d) = \mathtt{H}_{\mathbf{w}\mathbf{u}'}(c)&&\text{ by (Th1), (62)} \end{align} $$

(64) $$ \begin{align} &\mathcal{S}, \mathbf{u}\models^-_n \forall w(Rxyw \to ST_w(\chi)) [x/\mathtt{H}_{\mathbf{w}\mathbf{u}'}(a), y/\mathtt{H}_{\mathbf{u}\mathbf{u}'}(d)]&&\text{ by ({58}), ({63})} \end{align} $$

(65) $$ \begin{align} & \mathcal{S}, \mathbf{u}\models^+_n \sim\forall w(Rxyw \to ST_w(\chi)) [x/\mathtt{H}_{\mathbf{u}\mathbf{u}'}(\mathtt{H}_{\mathbf{w}\mathbf{u}}(a)), y/\mathtt{H}_{\mathbf{u}\mathbf{u}'}(d)]&&\text{ by Df. {4}, ({64})}, \end{align} $$

It follows from the above reasoning, by the choice of $\mathbf {u}'\geq \mathbf {u}$ , that we must have

$$ \begin{align*}\mathcal{S}, \mathbf{u}\models^+_n (Sy\wedge (\forall z)_O(Ezy \Leftrightarrow ST_z(\psi)))\to \sim\forall w(Rxyw \to ST_w(\chi))[x/\mathtt{H}_{\mathbf{w}\mathbf{u}}(a), y/d]; \end{align*} $$

the latter implies, by the definition of $\&$ , that

$$ \begin{align*}\mathcal{S}, \mathbf{u}\models^-_n (Sy\wedge (\forall z)_O(Ezy \Leftrightarrow ST_z(\psi)))\,\&\,\forall w(Rxyw \to ST_w(\chi))[x/\mathtt{H}_{\mathbf{w}\mathbf{u}}(a), y/d], \end{align*} $$

whence, by the choice of $\mathbf {u}$ and d, it follows that

$$ \begin{align*}\mathcal{S}, \mathbf{u}\models^-_n \exists y((Sy\wedge (\forall z)_O(Ezy \Leftrightarrow ST_z(\psi)))\,\&\,\forall w(Rxyw \to ST_w(\chi)))[x/a], \end{align*} $$

or, in other words, that , as desired.

On the other hand, if we have

(66)

this means that, for every $(\mathbf {v}, b) \in W_{\mathcal {S}}$ , we have

(67) $$ \begin{align} &R_{\mathcal{S}}((\mathbf{w},a), \|\psi\|_{\mathcal{M}_{\mathcal{S}}}, (\mathbf{v},b))\text{ implies }\mathcal{M}_{\mathcal{S}}, (\mathbf{v},b) \not\models^- \chi. \end{align} $$

By Lemma 12, we can choose a $c \in U_{\mathbf {w}}$ , such that:

(68) $$ \begin{align} \mathcal{S}, \mathbf{w}\models^+_n Sy\wedge (\forall z)_O(Ezy \Leftrightarrow ST_z(\psi))[y/c]. \end{align} $$

We choose, next, any $d \in U_{\mathbf {w}}$ and reason as follows:

(69) $$ \begin{align} &R^{\mathtt{M}^+_{\mathbf{w}}}(a, c, d)&&\text{ premise} \end{align} $$

(70) $$ \begin{align} &\Xi((\mathbf{w}, c), \|\psi\|_{\mathcal{M}_{\mathcal{S}}})&&\text{ by Lm 14.1, (68)} \end{align} $$

(71) $$ \begin{align} &R_{\mathcal{S}}((\mathbf{w},a), \|\psi\|_{\mathcal{M}_{\mathcal{S}}}, (\mathbf{w},d))&&\text{ by (69), (70)} \end{align} $$

(72) $$ \begin{align} &\mathcal{M}_{\mathcal{S}}, (\mathbf{w},d) \not\models^- \chi&&\text{ by (67), (71)} \end{align} $$

(73) $$ \begin{align} &\mathcal{S}, \mathbf{w}\not\models^-_n ST_w(\chi)[w/d]&&\text{ by IH for }\chi, \text{ (72)} \end{align} $$

(74) $$ \begin{align} &\mathcal{S}, \mathbf{w}\not\models^-_n Rxyw\to ST_w(\chi)[x/a, y/c, w/d]&&\text{ by (69), (73)}. \end{align} $$

This reasoning, by the choice of $d \in U_{\mathbf {w}}$ , shows that we have

$$ \begin{align*}\mathcal{S}, \mathbf{w}\not\models^-_n \forall w(Rxyw\to ST_w(\chi))[x/a, y/c], \end{align*} $$

which, together with (68), implies that

$$ \begin{align*}\mathcal{S}, \mathbf{w}\not\models^+_n (Sy\wedge (\forall z)_O(Ezy \Leftrightarrow ST_z(\psi))) \to \sim\forall w(Rxyw\to ST_w(\chi))[x/a, y/c], \end{align*} $$

whence we get, by definition of $\&$ , that

$$ \begin{align*}\mathcal{S}, \mathbf{w}\not\models^-_n ((Sy\wedge (\forall z)_O(Ezy \Leftrightarrow ST_z(\psi)))\,\&\,\forall w(Rxyw\to ST_w(\chi)))[x/a, y/c]; \end{align*} $$

the latter implies that

$$ \begin{align*}\mathcal{S}, \mathbf{w}\not\models^-_n \exists y((Sy\wedge (\forall z)_O(Ezy \Leftrightarrow ST_z(\psi)))\,\&\,\forall w(Rxyw\to ST_w(\chi)))[x/a], \end{align*} $$

or, in other words, that , as desired.

We are now in a position to prove Proposition 8.

We now wish to prove a converse of Proposition 8. This task also requires a series of preliminary constructions that develop the potential of the universal model $\mathcal {M}_c$ of $\mathsf {N4CK}$ given in Definition 7.

Definition 9. For any $n \in \omega $ , a sequence $\alpha = ((\Gamma _0,\Delta _0),\phi _1,\ldots ,\phi _n, (\Gamma _n, \Delta _n))$ is called a $(\Gamma _0,\Delta _0)$ -standard sequence of length $n + 1$ iff $(\Gamma _0,\Delta _0),\ldots ,(\Gamma _n, \Delta _n)\in W_c$ , $\phi _1,\ldots ,\phi _n \in \mathcal {CN}$ , and we have

$$ \begin{align*}(\forall i < n)(R_c((\Gamma_i,\Delta_i),\|\phi_{i + 1}\|_{\mathcal{M}_c}, (\Gamma_{i+1}, \Delta_{i+1}))). \end{align*} $$

Given a $(\Gamma , \Delta )\in W_c$ , the set of all $(\Gamma ,\Delta )$ -standard sequences will be denoted by $Seq(\Gamma ,\Delta )$ . The set $Seq$ of all standard sequences is then given by ${Seq := \bigcup \{Seq(\Gamma ,\Delta )\mid (\Gamma , \Delta )\in W_c\}}$ .

Finally, given a $\beta = ((\Xi _0,\Theta _0),\psi _1,\ldots ,\psi _m, (\Xi _m, \Theta _m))\in Seq$ , we say that $\beta $ extends $\alpha $ and will write $ \alpha \mathrel {\prec }\beta $ iff (1) $m = n$ , (2) $\phi _1 = \psi _1,\ldots , \phi _n = \psi _n (= \psi _m)$ , and

(3) $(\Gamma _0,\Delta _0)\leq _c (\Xi _0,\Theta _0),\ldots , (\Gamma _n,\Delta _n)\leq _c (\Xi _n,\Theta _n)(= (\Xi _m,\Theta _m))$ .

Given any $(\Gamma , \Delta ), (\Xi , \Theta )\in W_c$ such that $(\Gamma ,\Delta )\leq _c (\Xi ,\Theta )$ , a function $f:Seq(\Gamma ,\Delta )\to Seq(\Xi ,\Theta )$ is called a local $((\Gamma ,\Delta ),(\Xi ,\Theta ))$ -choice function iff $(\forall \alpha \in Seq(\Gamma ,\Delta ))(\alpha \mathrel {\prec }f(\alpha ))$ . The set of all local $((\Gamma ,\Delta ),(\Xi ,\Theta ))$ -choice functions will be denoted by $\mathfrak {F}((\Gamma ,\Delta ),(\Xi ,\Theta ))$ .

However, $((\Gamma ,\Delta ),(\Xi ,\Theta ))$ -choice functions will mostly interest us as restrictions of global choice functions. More precisely, $F: Seq \to Seq$ is a global choice function iff for every $(\Gamma , \Delta ) \in W_c$ there exists a $(\Xi , \Theta )\in W_c$ such that $(\Gamma ,\Delta )\leq _c (\Xi ,\Theta )$ and $F\upharpoonright Seq(\Gamma ,\Delta ) \in \mathfrak {F}((\Gamma ,\Delta ),(\Xi ,\Theta ))$ . The set of all global choice functions will be denoted by $\mathfrak {G}$ . The following lemma sums up some useful properties of global choice functions.

The proof of Lemma 16 repeats the proof of [Reference Olkhovikov12, Lemma 19] almost word-for-word; however, for the sake of completeness, we also include this proof in Appendix C.

The global choice functions are the basis for another type of sequences, that, along with the standard sequences, is necessary for the main model-theoretic construction of the present section. We will call them global sequences. A global sequence is any sequence of the form $(F_1,\ldots , F_n) \in \mathfrak {G}^n$ , where $n \in \omega $ (thus $\Lambda $ is also a global sequence with $n = 0$ ). Given two global sequences $(F_1,\ldots , F_k)$ and $(G_1,\ldots , G_m)$ , we say that $(G_1,\ldots , G_m)$ extends $(F_1,\ldots , F_k)$ and write $(F_1,\ldots , F_k)\mathrel {\sqsubseteq }(G_1,\ldots , G_m)$ iff $k \leq m$ and $F_1 = G_1,\ldots , F_k = G_k$ . Furthermore, we will denote by $Glob$ the set $\bigcup _{n \in \omega }\mathfrak {G}^n$ , that is to say, the set of all global sequences.

The final item in this series of preliminary model-theoretic constructions is a certain equivalence relation on $\mathcal {CN}$ . Namely, given any $\phi ,\psi \in \mathcal {CN}$ , we define that:

$$ \begin{align*}\phi\simeq\psi\text{ iff } (\phi \Leftrightarrow \psi \in \mathsf{N4CK}). \end{align*} $$

For any $\phi \in \mathcal {CN}$ , we will denote its equivalence class relative to $\simeq $ by $[\phi ]_\simeq $ .

We now proceed to define an $\mathcal {S}_c \in \mathbb {N}4$ induced by $\mathcal {M}_c\in \mathbb {NC}$ of Definition 7.

The first thing to show is that we, indeed, have $\mathcal {S}_c\in \mathbb {N}4$ . Again, we begin by establishing another technical fact.