2 RNmatrices for the minimal modal logic M

Consider a modal propositional signature $\Sigma $ with unary connectives $\neg $ and ${\ominus }$ (classical negation and modality, respectively) and a binary connective $\to $ (material implication). Let $\mathcal {V}$ be a denumerable set of propositional variables $\mathcal {V}=\{p_0,p_1, \ldots \}$ and let $For(\Sigma )$ be the algebra of formulas over $\Sigma $ freely generated by $\mathcal {V}$ . As usual, conjunction $\land $ , disjunction $\vee $ and bi-implication ${\leftrightarrow }$ are defined from $\neg $ and $\to $ as follows: $A \land B := \neg (A \to \neg B)$ , $A \vee B;=\neg A \to B$ and $A {\leftrightarrow } B:=(A \to B) \land (B \to A)$ .Footnote ⁶

In this section, we consider a set of four-valued non-deterministic matrices (Nmatrices, for short) defined from swap structures (see for instance [Reference Carnielli and Coniglio7, chap. 6] and [Reference Coniglio and Golzio13]) in which each truth-value is an ordered pair (or snapshot) $z=(z_1,z_2)$ in $\mathbf {2}^2$ , for $\mathbf {2}=\{0,1\}$ . Here, $z_1$ and $z_2$ represent, respectively, the truth value of A and of ${\ominus } A$ for a given formula A over $\Sigma $ . From now on, given a snapshot $z=(z_1,z_2)$ , $z_1$ and $z_2$ represent the first and second coordinate of the truth-value. This produces four truth-values $\textsf {t}^-:=(1,0)$ , $\textsf {t}^+:=(1,1)$ , $\textsf {f}^+:=(0,1)$ and $\textsf {f}^-:=(0,0)$ . Let $V_4$ be the set of such truth-values. Accordingly, the set of designated values will be $D_4=\{z \in V_4 \ : \ z_1=1\}=\{\textsf {t}^-,\textsf {t}^+\}$ . On the other hand, the set of non-designated values is given as $ND_4=\{\textsf {f}^-,\textsf {f}^+\}$ .

Because of the intended meaning of the snapshots, negation and implication between snapshots are computed over 2 in the first coordinate, while the second one can take an arbitrary value, which will be denoted by $\ast $ . That is:

$$ \begin{align*}\begin{array}{rll} \tilde{\neg}\, z &\!\!:=&\!\! ({\sim} z_1,\ast);\\ z \,\tilde{\to}\, w &\!\!:=&\!\! (z_1 {\Rightarrow} w_1,\ast). \end{array} \end{align*} $$

Here, $\sim $ and ${\Rightarrow }$ denote the Boolean negation and the implication in 2. Observe that the second coordinate is arbitrary since at this moment ${\ominus }$ remains uninterpreted, i.e., there are no axioms ruling the value of ${\ominus } \neg A$ and the value of ${\ominus }(A \to B)$ .

The interpretation of ${\ominus }$ is a multioperatorFootnote ⁷ which simply ‘reads’ the second coordinate, while the second coordinate (corresponding to ${\ominus }{\ominus } A$ ) will be arbitrary at this point, as well:

$$ \begin{align*}\begin{array}{rll} \tilde{{\ominus}}\, z&\!\!:=&\!\! (z_2,\ast). \end{array} \end{align*} $$

Let $\mathcal {M}= \langle V_4, D_4, \mathcal {O}\rangle $ be the obtained four-valued Nmatrix, where $\mathcal {O}(\#)=\tilde {\#}$ for every connective $\#$ in $\Sigma $ . The truth-tables for $\mathcal {M}$ can be displayed as follows:

The logic M generated by the Nmatrix $\mathcal {M}$ is then defined as follows: $\Gamma \vDash _{\mathbf {M}} A$ iff, for every $v \in \mathcal {F}$ : if $v(B) \in D_4$ for every $B \in \Gamma $ then $v(A) \in D_4$ .

Let $\mathcal {H}$ be the standard Hilbert calculus for CPL, presented in the signature $\Sigma $ (that is, no axioms nor rules for ${\ominus }$ are given, being Modus Ponens, denoted by MP, the only inference rule). It is easy to prove the following result (a sketch of the proof will be given in the next subsection).

In the next section, the valuations considered over the Nmatrix $\mathcal {M}$ will be restricted to certain conditions, in order to satisfy particular modal axioms which will characterize specific modal logics. It should be observed that a similar investigation was already done in [Reference Omori and Skurt28], but with a different approach: instead of restricting the set of valuations, there the multioperators of the Nmatrix $\mathcal {M}$ were restricted.

3 RNmatrices for extensions of M

In what follows, we will introduce and exploit what in [Reference Coniglio and Toledo14] was called restricted Nmatrices (RNmatrices).

As proved in [Reference Coniglio and Toledo14], any RNmatrix as above generates a Tarskian and structural consequence relation defined as expected: $\Gamma \vDash _{\mathcal {RM}} A$ iff, for every $v \in \mathcal {F}'$ : if $v(B) \in D_4$ for every $B \in \Gamma $ then $v(A) \in D_4$ . The aim of the restriction is to satisfy certain modal axiom(s) and, later on, modal rules.

We start by considering the well-known axiom $\textsf {(K)}$ , introduce important notions and prove soundness and completeness of the standard Hilbert-calculus w.r.t. RNmatrices. Then we will discuss other well-known axioms with one modal operator.

Consider

$$ \begin{align*}\textsf{(K)}:{\ominus}(A \to B) \to ({\ominus} A \to {\ominus} B).\end{align*} $$

Then, a valuation $v \in \mathcal {F}$ satisfies (K) iff, by Remark 4, $v_1({\ominus }(A \to B) \to ({\ominus } A \to {\ominus } B))=1$ for every $A,B$ iff $v_1({\ominus }(A \to B)) {\Rightarrow } (v_1({\ominus } A) {\Rightarrow } v_1({\ominus } B))=1$ for every $A,B$ iff $v_1({\ominus }(A \to B)) \leq v_1({\ominus } A) {\Rightarrow } v_1({\ominus } B)$ for every $A,B$ iff $v_2(A \to B) \leq v_2(A) {\Rightarrow } v_2(B)$ for every $A,B$ . Hence, the logic $\mathbf {M}_{\textsf {K}}$ satisfying axiom (K) is characterized by the RNmatrix $\mathcal {RM}_{\textsf {K}}=\langle \mathcal {M}, \mathcal {F}_{\textsf {K}}\rangle $ such that

$$ \begin{align*}\mathcal{F}_{\textsf{K}}=\{v \in \mathcal{F} \ : \ v_2(A \to B) \leq v_2(A) {\Rightarrow} v_2(B) \ \mbox{ for every } A,B \}.\end{align*} $$

Clearly, $\mathcal {RM}_{\textsf {K}}$ is structural. Indeed, let $\rho $ be a substitution and let $v \in \mathcal {F}_{\textsf {K}}$ . Observe that $v \circ \rho =(v_1 \circ \rho ,v_2\circ \rho )$ . Then, for every $A,B$ : $v_2\circ \rho (A \to B)= v_2(\rho (A \to B))= v_2(\rho (A) \to \rho (B)) \leq v_2(\rho (A)) {\Rightarrow } v_2((\rho (B))= v_2\circ \rho (A) {\Rightarrow } v_2\circ \rho (B)$ . Hence ${v \circ \rho \in \mathcal {F}_{\textsf {K}}}$ .

The consequence relation $\vDash _2^{\textsf {K}}$ w.r.t. bivaluations for $\mathcal {H}_{\textsf {K}}$ is defined as follows: $\Gamma \vDash _2^{\textsf {K}} A$ iff, for every b: if $b(B)=1$ for every $B \in \Gamma $ then $b(A)=1$ .

We will now consider the following well-known axioms with one modal operator and present to corresponding restrictions on the RNmatrices in order to show their versatility. Note that, this list can be easily extended:

• • $\textsf {(Kw)}:A \to ({\ominus }(A \to B) \to ({\ominus } A \to {\ominus } B)).$

• • $\textsf {(T)}: {\ominus } A \to A. $

• • $\textsf {(Tc)}: A \to {\ominus } A.$

• • $\textsf {(4)}: {\ominus } A \to {\ominus }{\ominus } A.$

• • $\textsf {(4c)}: {\ominus }{\ominus } A \to {\ominus } A.$

• • $\textsf {(GL)}:{\ominus }({\ominus } A \to A) \to {\ominus } A.$

By analysis similar to the one given for $\textsf {(K)}$ , it is immediate to see that the conditions imposed by these axioms on valuations and bivaluations are, respectively, the following, where $\sqcap $ is the boolean meet:

A general method for obtaining conditions for valuations and bivaluations from a given axiom will be given after Example 14.

Now, let $\mathsf {Ax} \in \{\mathsf {Kw, T, Tc, 4, 4c, GL}\}$ , then it is easy to see that all RNmatrices $\mathcal {RM}_{\textsf {Ax}}$ are structural. Furthermore, the Hilbert calculi $\mathcal {H}_{\textsf {Ax}}$ for $\mathbf {M}_{\textsf {Ax}}$ are obtained by adding the axiom $\mathsf {(Ax)}$ to $\mathcal {H}$ . The consequence relation w.r.t. bivaluations will be denoted by $\vDash _2^{\textsf {Ax}}$ .

An A-saturated set $\Delta $ in $\mathcal {H}_{\textsf {Ax}}$ satisfies conditions (1) and (2) of Proposition 1 plus the following, respectively:

• • if $A \in \Delta $ , ${\ominus }(A \to B) \in \Delta $ and ${\ominus } A \in \Delta $ then ${\ominus } B \in \Delta $

• • if ${\ominus } A \in \Delta $ , then $A \in \Delta $

• • if $A \in \Delta $ , then ${\ominus } A \in \Delta $

• • if ${\ominus } A \in \Delta $ , then ${\ominus }{\ominus } A \in \Delta $

• • if ${\ominus }{\ominus } A \in \Delta $ , then ${\ominus } A \in \Delta $

• • if ${\ominus }({\ominus } A \to A) \in \Delta $ then ${\ominus } A \in \Delta .$

From this, it is easy to adapt the corresponding proofs for $\mathcal {H}_{\textsf {K}}$ in order to obtain the following results.

We end this section by showing that it is possible to extend Theorem 5 to any axiomatic extension of the basic system $\mathcal {H}$ (which corresponds to the Nmatrix M). That is, any modal axiom (or, in general any finite set $\textsf {Ax}$ of modal axioms) can be added to $\mathcal {H}$ , in such a manner that it is possible to obtain conditions on the valuations (and on the bivaluations) faithfully characterizing the resulting Hilbert calculus $\mathcal {H}_{\textsf {Ax}}$ by the corresponding bivaluation semantics and Nmatrix $\mathcal {RM}_{\textsf {Ax}}$ . In order to do this, observe first that, in the given modal signature, any formula A satisfies one and only one of the following conditions:

• - A is atomic; or

• - $A= \neg B$ for a unique formula B; or

• - $A= {\ominus } B$ for a unique formula B; or

• - $A= A_1 \to (A_2 \to (\ldots \to (A_n \to B) \ldots ))$ for a unique $n \geq 1$ and unique formulas $A_i$ (for $1 \leq i \leq n$ ) and B such that either B is atomic, or $B= \neg C$ or $B= {\ominus } C$ for a unique formula C.

Then, starting from the equation $v_1(A)=1$ , it is possible to reduce, in a finite number of steps, any axiom A to a finite set of conditions over valuations in $\mathcal {F}$ .

Clearly, and by using a similar technique, it is possible to obtain in a finite number of steps a finite set of conditions on bivaluations from any axiom or finite set of axioms. Specifically, given a bivaluation b: $b(A)$ and $b(A)=j$ reduce, respectively, to $b(A)$ and $b(A)=j$ , if $A \in \mathcal {V}$ or $A={\ominus } B$ for some B, for $j=0,1$ ; $b(\neg A)$ and $b(\neg A)=j$ reduce, respectively, to ${\sim }b(A)$ and $b(A)={\sim }j$ , for $j=0,1$ ; $b(A_1 \to (A_2 \to (\ldots \to (A_n \to B) \ldots )))$ and $b(A_1 \to (A_2 \to (\ldots \to (A_n \to B) \ldots )))=1$ reduce, respectively, to $(b(A_1) \sqcap \ldots \sqcap b(A_n)) {\Rightarrow } b(B)$ and $b(A_1) \sqcap \ldots \sqcap b(A_n) \leq b(B)$ ; and $b(A_1 \to (A_2 \to (\ldots \to (A_n \to B) \ldots )))=0$ reduces to the conditions $b(A_k)=1$ (for $1 \leq k \leq n$ ) and $b(B)=0$ . Of course these reductions are applied iteratively, arriving in a finite number of steps to a set of conditions over b starting from $b(A)=1$ for a given axiom A.

In general, let $\textsf {Ax}$ be a finite set of modal axioms. By adapting the results above, it follows that the logic generated from the bivaluation semantics obtained from $\textsf {Ax}$ by means of the reductions above coincides with the one generated from the restricted Nmatrix obtained from $\textsf {Ax}$ by the corresponding reductions.

In order to prove completeness of $\mathcal {H}_{\textsf {Ax}}$ w.r.t. the bivaluation semantics generated from Ax, the following observation is crucial: if $\Delta $ is a C-saturated set for $\mathcal {H}_{\textsf {Ax}}$ then it is a closed theory that contains any instance of any axiom in Ax. This, combined with conditions (1) and (2) of Proposition 1, produces general conditions for $\Delta $ (by considering a generic instance of each axiom). Such conditions correspond to the conditions for bivaluations obtained from Ax; that is, Corollary 1 can be adapted to $\mathcal {H}_{\textsf {Ax}}$ . This shows that the proposed method works for any set $\textsf {Ax}$ of modal axioms.

4 Extensions of M by global inference rules

In this section, the logic M and its axiomatic extensions will be enriched with various global inference rules for ${\ominus }$ , and so we will obtain a vast class of modal logics. In all previous works, e.g., [Reference Coniglio, Del Cerro and Peron9], [Reference Kearns20] or [Reference Omori and Skurt25], and follow-up articles, the only global inference rule was the rule of necessitation. In this section, however, we will show how to generalize John Kearns’ original level-valuations technique, in order to capture any global inference rule. We will present furthermore, a generalized soundness and completeness proof and thus showing that any of the Hilbert systems presented before, are sound and complete w.r.t. the corresponding semantics. On top of that, the results below present a general recipe for constructing semantics for various normal and non-normal modal logics.

Throughout this section, L will denote any of the modal logics considered in §3. That is, L is characterized by a Hilbert calculus $\mathcal {H}_{\textsf {Ax}}$ obtained by adding the (set of) axiom(s) (Ax) to $\mathcal {H}$ . The RNmatrix characterizing L will be denoted by $\mathcal {RM}_{\textsf {Ax}}=\langle \mathcal {M}, \mathcal {F}_{\textsf {Ax}}\rangle $ .

Definition 17. Let L be an extension of M characterized by a Hilbert calculus $\mathcal {H}_{\textsf {Ax}}$ . We say that $\mathcal {H}_{\textsf {Ax}}^R$ is the Hilbert calculus obtained from $\mathcal {H}_{\textsf {Ax}}$ by adding the (global) inference rule

$$ \begin{align*}\frac{E_1 \ \ E_2 \ \ \ldots \ \ E_s }{E} \quad(\mathbf{R})\end{align*} $$

The logic characterized by $\mathcal {H}_{\textsf {Ax}}^R$ will be denoted by $\mathbf {L}^R$ .

The next definition will show the definitions of the notion of derivation in Hilbert calculi that one obtains by adding a global rule. For convenience, some technical notions will be recalled here.

It is worth noting that $\emptyset \vdash _{\mathcal {H}_{\textsf {Ax}}^{R}} A$ iff $\vdash _{\mathcal {H}_{\textsf {Ax}}^{R}} A$ .

Following [Reference Coniglio, Del Cerro and Peron9, Reference Kearns20, Reference Omori and Skurt28] we define the set of valuations over the RNmatrix $\mathcal {RM}_{\textsf {Ax}}=\langle \mathcal {M}, \mathcal {F}_{\textsf {Ax}}\rangle $ as follows.

From Theorems 6 and 7, we obtain that the modal logic $\mathbf {L}^R$ , which is axiomatized by $\mathcal {H}_{\textsf {Ax}}^{R}$ , is semantically characterized by the RNmatrix $\mathcal {RM}_{\textsf {Ax}}^R=\langle \mathcal {M}, \mathcal {F}^R_{\textsf {Ax}}\rangle $ .

In the rest of this section, some well-known modal (and so, global) inference rules, see, for example, [Reference Chellas6], will be considered in the present framework.

Example 23 (Necessitation rule)

Let $\mathcal {H}_{\textsf {Ax}}^N$ the Hilbert calculus obtained from $\mathcal {H}_{\textsf {Ax}}$ by adding the necessitation inference rule:

$$ \begin{align*}\frac{A}{{\ominus} A} \quad(\mathbf{N}). \end{align*} $$

By Definition 21, the set $\mathcal {F}^N_{\textsf {Ax}}$ of level valuations for $\mathbf {L}^N$ can be defined as follows:

• • $\mathcal {F}^0_{\textsf {Ax}} = \mathcal {F}_{\textsf {Ax}}$

• • $\mathcal {F}^{m+1}_{\textsf {Ax}} = \big \{v \in \mathcal {F}^{m}_{\textsf {Ax}} \ : \ \forall B\in For(\Sigma ), \text { if }\forall w\in \mathcal {F}^{m}_{\textsf {Ax}}(w_1(B) = 1) \\[1mm] \text {then } v_2(B)=1 \big \}$

• • $\mathcal {F}^N_{\textsf {Ax}} = \overset {\infty }{\underset {m=0}{\bigcap }} \mathcal {F}^m_{\textsf {Ax.}}$

Observe that $\mathcal {F}^N_{\textsf {Ax}}$ coincides with the original definition of level valuations introduced by Kearns (see [Reference Kearns20], [Reference Coniglio, Del Cerro and Peron9], [Reference Omori and Skurt25]).

Example 24 (Distributive rule)

Let $\mathcal {H}_{\textsf {Ax}}^D$ the Hilbert calculus obtained from $\mathcal {H}_{\textsf {Ax}}$ by adding the following distributive inference rule:

$$ \begin{align*} \frac{{\ominus} A \ \ \ {\ominus} B}{{\ominus} (A\land B)} \quad(\mathbf{D}). \end{align*} $$

By Definition 21, the set $\mathcal {F}^D_{\textsf {Ax}}$ of level valuations for $\mathbf {L}^D$ can be defined as follows:

• • $\mathcal {F}^0_{\textsf {Ax}} = \mathcal {F}_{\textsf {Ax}}$

• • $\mathcal {F}^{m+1}_{\textsf {Ax}} = \big \{v \in \mathcal {F}^{m}_{\textsf {Ax}} \ : \ \forall \, A,B\in For(\Sigma ) \\[1mm] \text {if } \forall w\in \mathcal {F}^{m}_{\textsf {Ax}}(w_2(A) = w_2(B)= 1), \text { then } v_2(A \land B)=1 \big \}$

• • $\mathcal {F}^D_{\textsf {Ax}} = \overset {\infty }{\underset {m=0}{\bigcap }} \mathcal {F}^m_{\textsf {Ax.}}$

Example 25 (Congruentiality rule)

Let $\mathcal {H}_{\textsf {Ax}}^C$ the Hilbert calculus obtained from $\mathcal {H}_{\textsf {Ax}}$ by adding the congruentiality inference rule:

$$ \begin{align*} \frac{A{\leftrightarrow} B}{ {\ominus} A{\leftrightarrow}{\ominus} B} \quad (\mathbf{C}). \end{align*} $$

By Definition 21, the set $\mathcal {F}^C_{\textsf {Ax}}$ of level valuations for $\mathbf {L}^C$ can be defined as follows:

• • $\mathcal {F}^0_{\textsf {Ax}} = \mathcal {F}_{\textsf {Ax}}$

• • $\mathcal {F}^{m+1}_{\textsf {Ax}} = \big \{v \in \mathcal {F}^{m}_{\textsf {Ax}} \ : \ \forall \, A,B\in For(\Sigma ) \\[1mm] \text {if } \forall w\in \mathcal {F}^{m}_{\textsf {Ax}}(w_1(A) = w_1(B)), \text { then } v_2(A) = v_2(B) \big \}$

• • $\mathcal {F}^C_{\textsf {Ax}} = \overset {\infty }{\underset {m=0}{\bigcap }} \mathcal {F}^m_{\textsf {Ax.}}$

Example 26 (Monotonicity rule)

Let $\mathcal {H}_{\textsf {Ax}}^M$ the Hilbert calculus obtained from $\mathcal {H}_{\textsf {Ax}}$ by adding the monotonicity inference rule:

$$ \begin{align*}\frac{A\to B}{ {\ominus} A\to {\ominus} B} \quad(\mathbf{ M}).\end{align*} $$

By Definition 21, the set $\mathcal {F}^M_{\textsf {Ax}}$ of level valuations for $\mathbf {L}^M$ can be defined as follows:

• • $\mathcal {F}^0_{\textsf {Ax}} = \mathcal {F}_{\textsf {Ax}}$

• • $\mathcal {F}^{m+1}_{\textsf {Ax}} = \big \{v \in \mathcal {F}^{m}_{\textsf {Ax}} \ : \ \forall \, A,B\in For(\Sigma ) \\[1mm] \text {if } \forall w\in \mathcal {F}^{m}_{\textsf {Ax}}(w_1(A) \leq w_1(B)), \text { then } v_2(A) \leq v_2(B) \big \}$

• • $\mathcal {F}^M_{\textsf {Ax}} = \overset {\infty }{\underset {m=0}{\bigcap }} \mathcal {F}^m_{\textsf {Ax.}}$

5 RNmatrices for the minimal bimodal logic Mb

Modal logics usually contain more than one modal operator in their language, e.g., alethic modal operators $\Box $ and $\Diamond $ , deontic modal modalities O and P or the multitude of temporal modal operators. While usually those operators are interdefinable, typically $\Diamond A := \neg \Box \neg A$ , there are systems where this does not need to be the case, for example, in intuitionistic modal logics. In this section, we introduce bimodal systems, where the different modal operators can be independent of each other or, depending on the modal axioms, are interdefinable.

We will consider a bimodal version of the minimal modal logic M, namely, the minimal bimodal logic Mb.Footnote ¹⁰ This logic is defined over a signature $\Sigma _b$ obtained from $\Sigma $ by replacing ${\ominus }$ with two modal operators, which will be denoted by ${\ominus }$ and ${\otimes }$ .Footnote ¹¹ As expected, the snapshots are now triples $z=(z_1,z_2,z_3)$ over 2 in which each coordinate represents a possible truth-value for the formulas A, ${\ominus } A$ and ${\otimes } A$ , respectively. Hence, eight truth-values are obtained in this way, namely, $\textsf {t}^{--}:=(1,0,0)$ , $\textsf {t}^{-+}:=(1,0,1)$ , $\textsf {t}^{+-}:=(1,1,0)$ , $\textsf {t}^{++}:=(1,1,1)$ , $\textsf {f}^{++}:=(0,1,1)$ , $\textsf {f}^{+-}:=(0,1,0)$ , $\textsf {f}^{-+}:=(0,0,1)$ and $\textsf {f}^{--}:=(0,0,0)$ . Let $V_8$ be the set of such truth-values, and let $D_8=\{z \in V_8 \ : \ z_1=1\}=\{\textsf {t}^{-+},\textsf {t}^{++},\textsf {t}^{--}, \textsf {t}^{+-}\}$ be the set of designated values. Hence, $ND_8=\{\textsf {f}^{-+},\textsf {f}^{++},\textsf {f}^{--}, \textsf {f}^{+-}\}$ .

The definition of the multioperators over $V_8$ interpreting the connectives of $\Sigma _b$ is a natural generalization of the four-valued case:

$$ \begin{align*} \begin{array}{rll} \tilde{\neg}\, z&\!\!:=&\!\! ({\sim} z_1,\ast,\ast);\\ \tilde{{\ominus}}\, z&\!\!:=&\!\! (z_2,\ast,\ast);\\ \tilde{{\otimes}}\, z&\!\!:=&\!\! (z_3,\ast,\ast);\\ z \,\tilde{\to}\, w &\!\!:=&\!\! (z_1 {\Rightarrow} w_1,\ast,\ast). \end{array} \end{align*} $$

Let $\mathcal {M}_b= \langle V_8, D_8, \mathcal {O}_b\rangle $ be the obtained eight-valued Nmatrix, where $\mathcal {O}_b(\#)=\tilde {\#}$ for every connective $\#$ in $\Sigma _b$ . Thus, the truth-tables for $\mathcal {M}_b$ are the following:

Let $\mathcal {F}_b$ be the set of all the valuations over the Nmatrix $\mathcal {M}_b$ . Thus, $v \in \mathcal {F}_b$ iff $v:For(\Sigma _b) \to V_8$ is a function satisfying the following properties:

• • $v(\# A) \in \tilde {\#}\, v(A)$ for $\# \in \{\neg , {\ominus },{\otimes }\}$ ;

• • $v(A \to B) \in v(A) \, \tilde {\to }\, v(B)$ .

Now, any valuation $v \in \mathcal {F}_b$ will be written as $v=(v_1,v_2,v_3)$ such that $v_1,v_2,v_3: For(\Sigma _b) \to V_8$ and so $v(A)=(v_1(A),v_2(A),v_3(A))$ for every formula A. This means that, for every formulas A and B:

• • $v(A) \in D_8$ iff $v_1(A)=1$ ;

• • $v_1(\neg A)={\sim } v_1(A)$ ;

• • $v_1({\ominus } A)= v_2(A)$ ;

• • $v_1({\otimes } A)= v_3(A)$ ;

• • $v_1(A \to B)= v_1(A) {\Rightarrow } v_1(B)$ .

The logic Mb generated by the Nmatrix $\mathcal {M}_b$ is defined as follows: $\Gamma \vDash _{\mathbf {Mb}} A$ iff, for every $v \in \mathcal {F}_b$ , $v(A) \in D_8$ whenever $v(B) \in D_8$ for every $B \in \Gamma $ .

The proof of the following result can be adapted from the respective one for $\mathcal {M}$ . So, let $\mathcal {H}_b$ be the standard Hilbert calculus for CPL, presented in the signature $\Sigma _b$ (that is, no axioms nor rules for ${\ominus }$ nor for ${\otimes }$ are given, being MP the only inference rule).

Axiomatic extensions of $\mathcal {H}_b$ (over the signature $\Sigma _b$ ) can be considered, obtaining in each case an eight-valued and structural characteristic RNmatrix. The definition of the restricted set of valuations, in each extension, is exactly as in the case of $\mathcal {M}$ , taking obviously into account that they are subsets of $\mathcal {F}_b$ instead of subsets of $\mathcal {F}$ and that the signature is $\Sigma _b$ instead of $\Sigma $ , where ${\ominus }$ is replaced by ${\ominus }$ or ${\otimes }$ in each axiom of the previous examples (of course axioms containing both ${\ominus }$ and ${\otimes }$ can also be considered).

Observe that the technique for reducing terms and equations for valuations and bivaluations presented at the end of §3, see Definitions 10–13, can be easily adapted to the bimodal setting.

For instance, consider the following axioms:Footnote ¹²

1. (Di) ${\otimes } A \leftrightarrow \neg {\ominus } \neg A$

2. (Bo) ${\ominus } A \leftrightarrow \neg {\otimes } \neg A $

3. (D) ${\ominus } A \to {\otimes } A$

4. (B) $ A \to {\ominus }{\otimes } A$

5. (5) ${\otimes } A \to {\ominus } {\otimes } A$

6. (C) ${\otimes }{\ominus } A \to {\ominus }{\otimes } A$ .

The conditions imposed by these axioms on valuations and bivaluations are, respectively, the following:

Let us analyze the case of (Di) in detail: if $B={\otimes } A \leftrightarrow \neg {\ominus } \neg A=\neg (({\otimes } A \to \neg {\ominus } \neg A) \to \neg (\neg {\ominus } \neg A \to {\otimes } A))$ then $v_1(B)=1$ reduces to $v_1(({\otimes } A \to \neg {\ominus } \neg A) \to \neg (\neg {\ominus } \neg A \to {\otimes } A))=0$ , which reduces to $v_1({\otimes } A \to \neg {\ominus } \neg A)=1$ , $v_1(\neg (\neg {\ominus } \neg A \to {\otimes } A))=0$ , which reduces to $v_1({\otimes } A) \leq v_1(\neg {\ominus } \neg A)$ , $v_1(\neg {\ominus } \neg A \to {\otimes } A)=1$ . The latter reduces to $v_3(A) \leq {\sim }v_1({\ominus } \neg A)$ , $v_1(\neg {\ominus } \neg A) \leq v_1({\otimes } A)$ , which reduces to $v_3(A) \leq {\sim }v_2(\neg A)$ , ${\sim }v_1({\ominus } \neg A) \leq v_3(A)$ , which reduces to $v_3(A) \leq {\sim }v_2(\neg A)$ , ${\sim }v_2(\neg A) \leq v_3(A)$ . The latter is equivalent to the condition $v_2(\neg A) = {\sim }v_3(A)$ .

Concerning bivaluations, $b(B)=1$ reduces to $b(({\otimes } A \to \neg {\ominus } \neg A) \to \neg (\neg {\ominus } \neg A \to {\otimes } A))=0$ , which reduces to $b({\otimes } A \to \neg {\ominus } \neg A)=1$ , $b(\neg (\neg {\ominus } \neg A \to {\otimes } A))=0$ , which reduces to $b({\otimes } A) \leq b(\neg {\ominus } \neg A)$ , $b(\neg {\ominus } \neg A \to {\otimes } A)=1$ . The latter reduces to $b({\otimes } A) \leq {\sim }b({\ominus } \neg A)$ , $b(\neg {\ominus } \neg A) \leq b({\otimes } A)$ , which reduces to $b({\otimes } A) \leq {\sim }b({\ominus } \neg A)$ , ${\sim }b({\ominus } \neg A) \leq b({\otimes } A)$ . The latter is equivalent to the condition $b({\ominus } \neg A) = {\sim }b({\otimes } A)$ .

It is worth noting that, if axioms (Di) and (Bo) are taken together then $v(\neg A)=({\sim }v_1(A), {\sim }v_3(A),{\sim }v_2(A))$ for every A. That is, the restricted valuations are deterministic for negated formulas.

Let Ax be a finite set of modal axioms in the signature $\Sigma _b$ . Let $\mathbf {Lb}$ be the bimodal logic given by means of the Hilbert calculus $\mathcal {H}_{b\textsf {Ax}}$ , obtained by $\mathcal {H}_b$ by adding all the axioms in Ax. By adapting the proofs for the monomodal case, it is easy to prove that this logic is semantically characterized by the RNmatrix $\mathcal {RM}_{b\textsf {Ax}}=\langle \mathcal {M}, \mathcal {F}_{b\textsf {Ax}}\rangle $ . The adaptations to be done with respect to the monomodal case are simple.

As before it is possible, as well, to add global inference rules to any bimodal logic $\mathbf {Lb}$ as defined above. It is worth noting that the technique for defining level valuation semantics from a given global inference rule R presented in §4 does not depend on the modal signature of the underlying logic. This being so, the soundness and completeness results obtained in §4 can be easily adapted to the bimodal (or, in general, the multimodal) case. The only difference will appear in concrete examples of modal inference rules, in which the reduction of the conditions for the level valuations must be done for the bimodal case. By adapting the proofs of Theorems 6 and 7 to the bimodal setting, we obtain the following result

Example 27 (Necessitation rules)

Consider the following global inference rules:

$$ \begin{align*}\mathbf{N}_{\ominus}{\mbox{-rule}} \qquad \frac{A}{{\ominus} A} \qquad\qquad\qquad \mathbf{ N}_{\otimes}{\mbox{-rule}} \qquad \frac{A}{{\otimes} A.}\end{align*} $$

Let $\mathcal {H}^{N_{\ominus }}_{b\textsf {Ax}},\mathcal {H}^{N_{\otimes }}_{b\textsf {Ax}},\mathcal {H}^{N_{{\ominus }{\otimes }}}_{b\textsf {Ax}}$ be the extensions of the Hilbert calculus $\mathcal {H}_{b\textsf {Ax}}$ by adding the $\mathbf {N}_{\ominus }$ -rule, the $\mathbf {N}_{\otimes }$ -rule, or both. The notion of derivation is then analogous to the one given in Definition 19. The addition of one or both of those rules will be denoted by $\mathbf {Lb}^{N_{\ominus }}, \mathbf { Lb}^{N_{\otimes }}$ or $\mathbf {Lb}^{N_{{\ominus }{\otimes }}}$ , respectively. The semantics for those system will be given, as in the previous cases, by considering level valuations over the corresponding RNmatrices. The set of level valuations will be given, respectively, by $\mathcal {F}^{N_{\ominus }}_{b\textsf {Ax}} = \overset {\infty }{\underset {m=0}{\bigcap }} \mathcal {F}^{{\ominus },m}_{b\textsf {Ax}}$ , $\mathcal {F}^{N_{\otimes }}_{b\textsf {Ax}} = \overset {\infty }{\underset {m=0}{\bigcap }} \mathcal {F}^{{\otimes },m}_{b\textsf {Ax}}$ and $\mathcal {F}^{N_{{\ominus }{\otimes }}}_{b\textsf {Ax}} = \overset {\infty }{\underset {m=0}{\bigcap }} \mathcal {F}^{{\ominus }{\otimes },m}_{b\textsf {Ax}}$ . The sets of level valuations are such that $\mathcal {F}^{{\ominus },0}_{b\textsf {Ax}}= \mathcal {F}^{{\otimes },0}_{b\textsf {Ax}}=\mathcal {F}_{b\textsf {Ax}}$ , and

• • $\mathcal {F}^{{\ominus },m+1}_{b\textsf {Ax}} = \big \{v \in \mathcal {F}^{{\ominus },m}_{b\textsf {Ax}} \ : \ \forall B\in For(\Sigma _b), \text { if }\forall w\in \mathcal {F}^{{\ominus },m}_{b\textsf {Ax}}(w_1(B) = 1) \\[1mm] \text {then } v_2(B)=1 \big \}$

• • $\mathcal {F}^{{\otimes },m+1}_{b\textsf {Ax}} = \big \{v \in \mathcal {F}^{{\otimes },m}_{b\textsf {Ax}} \ : \ \forall B\in For(\Sigma _b), \text { if }\forall w\in \mathcal {F}^{{\otimes },m}_{b\textsf {Ax}}(w_1(B) = 1) \\[1mm] \text {then } v_3(B)=1 \big \}$

• • $\mathcal {F}^{{\ominus }{\otimes },m}_{b\textsf {Ax}}=\mathcal {F}^{{\ominus },m}_{b\textsf {Ax}} \cap \mathcal {F}^{{\otimes },m}_{b\textsf {Ax}}$ for every m, then $\mathcal {F}^{N_{{\ominus }{\otimes }}}_{b\textsf {Ax}} =\mathcal {F}^{N_{{\ominus }}}_{b\textsf {Ax}} \cap \mathcal {F}^{N_{{\otimes }}}_{b\textsf {Ax}}$ .

The above corollary does not entail that any possible extension of the logics discussed will be a subsystem of S5; in particular, trivial logics are not excluded as possible extensions.

6 Decidability of non-normal modal logics by means of truth-tables

Despite being defined over finite (four-valued) Nmatrices, the RNmatrices for S4 and S5 introduced by Kearns in [Reference Kearns20] do not give origin to a decision procedure for these logics by means of truth-tables. The reason is that, in order to check the validity of a formula A, the semantical information provided by A and all of its subfomulas in its truth-table generated by the corresponding Nmatrix is not enough to obtain a full criterion to delete the rows which do not correspond to level valuations. That is, there is no mechanical procedure to determine what rows should be deleted. A solution to this problem, for modal logics KT and S4, was obtained by L. Grätz in [Reference Grätz17], where a mechanical procedure was found to delete the unsound rows from the three-valued non-deterministic truth-tables generated by each formula being tested. Using a related by somewhat different approach, O. Lahav and Y. Zohar proposed in [Reference Lahav, Zohar, Blanchette, Kovács and Pattinson22] decision procedures for K and KT based on finite-valued RNmatrices.

It is well-known that not every modal system with global inference rules is decidable. However, even for decidable systems, it is not immediate how to obtain a decision procedure (by means of truth-tables) from the characteristic RNmatrix obtained within our framework. That is, decidability techniques by truth-tables for such logics require a case-by-case analysis, by generalizing the techniques introduced in [Reference Grätz17] and [Reference Lahav, Zohar, Blanchette, Kovács and Pattinson22].

It seems clear that the lack of a decision procedure for a modal logic by means of truth-tables, obtained from its characteristic RNmatrix, is due to the existence of global inference rules, which forces defining level valuations. This infinitary procedure can make it difficult to extract an algorithm from it. This would suggest that any finite axiomatic extension L of M, without global rules, is decidable by means of truth-tables. That is, the characteristic RNmatrix of L would allow to define an algorithm to delete unsound rows from the truth-table of any formula being tested.

In this section, it will be shown how to obtain a truth-table decision procedure, obtained from its RNmatrix, for some of the (non-normal) systems analyzed here, both monomodal and bimodal. The extension of the techniques to be introduced below to the other systems without global rules appears to be immediate.

However, as it will be shown in Example 29, there is a purely axiomatic extension of M (called M $_{E}$ ) that is not decidable by the truth-table procedure obtained from the corresponding RNmatrix. This problem will be overcome by means of a detailed analysis of the conditions for the permitted valuations for M $_{E}$ , which allows to obtain an improved set of conditions to delete unsound rows in the truth-tables (see Theorem 14). This example shows that, even for modal systems without global rules, the definition of a decision procedure by truth-tables must be analyzed case-by-case.

Let us start our analysis. Let $A(p_1,\ldots ,p_n)$ be a formula depending on propositional variables $p_1,\ldots ,p_n$ . From the examples considered in the previous sections, it is immediate to see that in order to check the validity of A in a non-normal system L (Lb) characterized by an RNmatrix $\mathcal {RM}=\langle \mathcal {M}, \mathcal {F}'\rangle $ (in the monomodal case) or $\mathcal {RM}_b=\langle \mathcal {M}_b, \mathcal {F}^{\prime }_b\rangle $ (in the bimodal case),Footnote ¹³ a finite row-branching truth table needs to be constructed. This truth-table is obtained by considering the multioperators of the Nmatrix $\mathcal {M}$ (or $\mathcal {M}_b$ ), starting from the atomic formulas $p_1,\ldots ,p_n$ .

To be more specific, define the complexity $l(B)$ of a formula $B \in For(\Sigma )$ as follows: $l(B)=0$ if B is a variable; $l(\# B)=l(B)+1$ if $\# \in \{\neg ,{\ominus }\}$ ; and $l(B \to C)=l(B)+l(C)+1$ . In $For(\Sigma _b),$ the complexity is defined analogously, but now $l(\# B)=l(B)+1$ if $\# \in \{\neg ,{\ominus },{\otimes }\}$ .

Then, consider the sequence $A_1\ldots A_k=A$ formed by all the subformulas of A, linearly ordered by complexity (formulas with the same complexity are ordered arbitrarily, with the only restriction of putting $p_1,\ldots ,p_n$ in the first n places). From this, $l(A_i) \leq l(A_{i+1})$ for $1 \leq i \leq k$ . The table starts by considering all the possible combination of truth-values for the first n-columns $p_1,\ldots ,p_n$ (that is, $4^n$ rows are generated in the monomodal case, and $8^n$ in the bimodal case). Then, the possible values for the formula $A_{n+1}(p_1,\ldots ,p_n)$ are computed in each row, according to the semantics given in $\mathcal {M}$ (or in $\mathcal {M}_b$ ) to the (unique) conective occurring in $A_{n+1}$ . Consider the four-valued case. By definition of the multioperators of $\mathcal {M}$ , each row splits into two ones (since both sets $D_4$ and $ND_4$ have elements). In the bimodal case, each row splits into four ones. The procedure continues, by considering the possible values for $A_{n+2}$ on the $2\cdot 4^n$ (or $4\cdot 8^n$ ) rows generated in the previous step, and so on. This procedure will eventually terminate by computing all the possible values for A in all the rows, generating in this way a finite truth-table.

After this, some rows in the obtained finite truth-table need to be be removed, in case they violate the conditions stated by $\mathcal {F}'$ (i.e., they correspond to valuations which do not belong to $\mathcal {F}'$ ). For instance, if axiom (K) is valid in a monomodal L, then any row assigning values $x^+$ , $y^-$ and $z^+$ to formulas A, B and $A \to B$ , respectively (for some $x,y,z \in \{\textsf {t},\textsf {f}\})$ , must be removed: indeed in this case $\bar {v}_2(A \to B)=1 \not \leq 0=1 \Rightarrow 0= \bar {v}_2(A) \Rightarrow \bar {v}_2(B)$ , where $\bar {v}$ is the mapping representing the values assigned to the formulas labeling the columns of the given truth-table. That is, $\bar {v}$ does not correspond to a valuation in $\mathcal {F}'$ , given that it does not satisfy axiom (K), hence it must be removed from the table. It is worth noting that the process of eliminating rows can be done “on the fly”, that is, when the table is being constructed. In this way, the first time that a situation is found allowing to delete a row, the row is deleted precisely at that point. This allows to produce smaller truth-tables, turning the process easier to be realized (see, for instance, the case of the bimodal logic characterized by the RNmatrix $\mathcal {RM}_{b\textsf {C}}$ to be analyzed below).

Let $\mathcal {T}'$ be the obtained (reduced) truth table. If A gets a designated value in every row of $\mathcal {T}'$ then A is declared to be valid; otherwise, it is declared non-valid. By observing the examples of non-normal systems considered in the previous sections, it seems clear that this process can be done algorithmically.

In order to prove that this methodology is sound and complete (that is, in order to verify that this procedure indeed represents the RNmatrix semantics given by $\mathcal {RM}'$ ) the proof proposed in [Reference Coniglio and Toledo14, sec. 4.3] for the RNmatrices for the logics $C_n$ will be adapted. Hence, two steps are required:

1. (i) to prove that any row of $\mathcal {T}'$ represents a valuation in $\mathcal {F}'$ ;

2. (ii) to prove that any valuation in $\mathcal {F}'$ , restricted to the formulas occurring in the columns of $\mathcal {T}'$ , is a row of $\mathcal {T}'$ .

From now on, the set of subformulas of a formula A (in $For(\Sigma )$ or $For(\Sigma _b)$ ) will be denoted by $SF(A)$ . A set of formulas $\Gamma $ is said to be closed under subformulas if it satisfies the following: if $A \in \Gamma $ and $B \in SF(A)$ then $B \in \Gamma $ .

Let us start with axiom (GL) in the monomodal scenario.

Consider now axiom (K). A similar algorithm can be given for constructing a (reduced) truth-table for any formula A obeying the restrictions imposed by $\mathcal {F}_{\textsf {K}}$ , namely: $\bar {v}_2(A \to B) \leq \bar {v}_2(A) \Rightarrow \bar {v}_2(B)$ , where $\bar {v}$ represents the values assigned, in an specific row, to the formulas in the columns of the table being constructed, and $\bar {v}(A)$ is written as $(\bar {v}_1(A),\bar {v}_2(A))$ for every A. Hence, the following can be proven

Let $\mathbf {GL}^-$ be the logic obtained by adding the set of axioms $GLK=\{\textsf {(GL)},\textsf {(K)}\}$ to $\mathcal {M}$ . Then, $\mathbf {GL}^-$ is axiomatized by $\mathcal {H}_{GLK}$ and semantically characterized by the RNmatrix $\mathcal {RM}_{GLK}=\langle \mathcal {M}, \mathcal {F}_{GLK}\rangle $ such that

$$ \begin{align*}\begin{array}{ll} \mathcal{F}_{GLK}&\!\!\!\!=\{v \in \mathcal{F} \ : \ v_2(A \to B) \leq v_2(A) \Rightarrow v_2(B) \ \mbox{and}\\[1mm] & \hspace{3.5mm} v_2({\ominus} A \to A) \leq v_2(A) \mbox{ for every } A,B \}. \end{array} \end{align*} $$

In [Reference Boolos16, theorem 18] it was shown that axiom (4): ${\ominus } A \to {\ominus }{\ominus } A$ is derivable in GL, Gödel–Löb logic of provability, which can be obtained from $\mathbf {GL}^-$ by adding the necessitation rule (the ${\ominus }$ -rule in the signature $\Sigma $ of $\mathbf { GL}^-$ ). By using our framework, we can prove that necessitation rule is essentially required to obtain (4) from GL.

Consider now an example of bimodal logic: the logic $\mathbf {Mb}_{\textsf {C}}$ obtained from $\mathbf {Mb}$ by requiring the validity of axiom (C): ${\otimes }{\ominus } A \to {\ominus }{\otimes } A$ , which is is axiomatized by $\mathcal {H}_{b\textsf {C}}$ and semantically characterized by the RNmatrix $\mathcal {RM}_{b\textsf {C}}=\langle \mathcal {M}_b, \mathcal {F}_{b\textsf {C}}\rangle $ such that

$$ \begin{align*}\begin{array}{ll} \mathcal{F}_{b\textsf{C}}=\{v \in \mathcal{F}_b \ : \ v_3({\ominus} A) \leq v_2({\otimes} A) \ \mbox{ for every } A \}. \end{array} \end{align*} $$

As doing before, an algorithm can be defined for constructing a (reduced) truth-table for any formula A by using the restrictions of $\mathcal {F}_{b\textsf {C}}$ . Observe that it is required to check, for each row, the values assigned to pairs of formulas of the form $({\ominus } A, {\otimes } A)$ in order to get rows which are compatible with the definition of $\mathcal {F}_{b\textsf {C}}$ . In order to do this, it may be convenient to exclude the rows that violate the conditions while the truth table is being built, starting from the first pair of the form $({\ominus } A, {\otimes } A)$ found between the columns of the table generated in each step of the process. That is, rows are excluded “on the fly”.

It seems clear that the decidability method described above can be extended to the other systems considered here. However, it can be seen that the set of conditions for valuations obtained from any set of modal axioms does not necessarily produce a decision procedure as it stands. The reason is that certain additional conditions, obtained from the original ones, can be required. Hence, some valid formula can contain in its truth-table some rows which were not deleted by using exclusively the original conditions, and in which the formula gets a non-designated value. Specifically, consider the following example:Footnote ¹⁴

Example 29. Consider the following axioms:

1. (Ax1) ${\ominus } (A \to A)$

2. (Ax2) ${\ominus } (A \to B) \to {\ominus }({\ominus } A \to B)$ .

Let $\textsf {Ax}= \{\textsf {(Ax1)},\textsf {(Ax2)}\}$ , and let M $_E$ be the axiomatic extension of M by adding Ax. Clearly, Ax imposes the following restrictions on the valuations: (1) $v_1({\ominus } (A \to A))=1$ ; (2) $v_1({\ominus }(A \to B)) \leq v_1({\ominus }({\ominus } A \to B))$ . By Remark 2, this is equivalent to the following conditions: (3) $v_2(A \to A)=1$ ; (4) $v_2(A \to B) \leq v_2({\ominus } A \to B)$ . However, although this set of conditions is enough for defining an RNmatrix such that M $_E$ is sound and complete w.r.t. this RNmatrix, it is not enough for defining sound truth-tables for M $_E$ . Indeed, it is immediate to see (taking $A=B=p$ ) that ${\ominus }({\ominus } p \to p)$ is derivable in M $_E$ , hence it is valid w.r.t. the obtained RNmatrix. However, conditions (1)-(2) (or, equivalently, (3)-(4)) are not enough to eliminate in the truth-table all the rows which do not satisfy these conditions: indeed, half of the rows of the truth-table for ${\ominus }({\ominus } p \to p)$ (the ones producing non-designated values) should be deleted.

The unsound rows in the truth-table of Example 29 were not deleted since none of the formulas involved in conditions (1)-(2) and (3)-(4) can be instantiated with the subformulas of ${\ominus }({\ominus } p \to p)$ in order to apply such conditions.

Another example of a tautology of M $_E$ which is not recognized by the induced truth-tables method is $A:={\ominus }(p \to q) \to {\ominus }({\ominus }{\ominus } p \to q)$ for $p,q \in \mathcal {V}$ , $p\neq q$ . Indeed, no row can be deleted from its truth-table by means of conditions (3)-(4). Because of this, its truth table contains rows in which A gets a non-designated value. For example, consider a row in which $p,q, p \to q, {\ominus } p, {\ominus }{\ominus } p$ and ${\ominus }(p \to q)$ get the value $\textsf {t}^+$ , but ${\ominus }{\ominus } p \to q$ receives the value $\textsf {t}^-$ . Then, ${\ominus }({\ominus }{\ominus } p \to q)$ gets a non-designated value, just like A.

However, from the basic conditions (3)-(4), it is possible to infer additional ones, that expand the possibility of deleting rows, guaranteeing soundness of M $_E$ w.r.t. its truth-tables. Indeed, by taking $A=B$ in (4), and taking into account (3), it follows that $v_2({\ominus } A \to A)=1$ , for every A. Combining this with (4) once again (by instantiating A with ${\ominus } A$ and B with A) it follows that $v_2({\ominus }{\ominus } A \to A)=1$ and, by the same reasoning, that $v_2({\ominus }{\ominus }{\ominus } A \to A)=1$ , and so on, for every A. Let $+:=\{\textsf {t}^+, \textsf {f}^+\}$ . By defining ${\ominus }^0 A:= A$ and ${\ominus }^{n+1}A:={\ominus }{\ominus }^n A$ , the original conditions can be expanded to the following ones:

1. (C1) $v(A \to A)= \textsf {t}^+$ ;

2. (C2) $v({\ominus }^k A \to A)\in +$ , for $k \geq 1$ ;

3. (C3) if $v({\ominus }^k A \to B) \in +$ for some $0 \leq k < n$ and $A\neq B$ then $v({\ominus }^n A \to B) \in +$ .

Let $\mathcal {F}_{\textbf {M}_E}$ be the set of valuations over the Nmatrix $\mathcal {M}$ satisfying conditions (C1)–(C3). It is clear that a valuation v over the Nmatrix $\mathcal {M}$ satisfies conditions (3)-(4) iff it satisfies conditions (C1)–(C3). Thus, the set $\mathcal {F}_{\textbf {M}_E}$ is the set of valuations over the Nmatrix $\mathcal {M}$ which characterizes the logic M $_E$ . In Proposition 7, it will be shown that any partial valuation defined over a finite set closed under subformulas which satisfies these conditions can be extended to a valuation for M $_E$ (i.e., in $\mathcal {F}_{\textbf {M}_E}$ ). Observe that half of the rows of the truth-table in Example 29, seen as partial valuations, do not satisfy property (C2), hence, they do not correspond to valuations for M $_E$ .

It is immediate to see that, given a formula A, the construction of a truth-table for A in M $_E$ can be done in a finite number of steps by means of a mechanical procedure, such that each row is a function from the set $\Gamma $ of subformulas of A to $V_4$ satisfying conditions (V1)–(V5) of Proposition 7. Indeed, when constructing a truth-table for A in $\mathcal {M}$ , if a column corresponds to a formula $B \to B$ then every row must assign to it the value $\textsf {t}^+$ . In turn, if the column corresponds to a formula ${\ominus }^k B \to B$ for some $k \geq 1$ then every row must assign to it a value in $(a \,\tilde {\to }\, b)\cap +$ (a nonempty set), where a and b are the values already assigned to ${\ominus }^k B$ and B in that row. Finally, if the column corresponds to a formula ${\ominus }^n B \to C$ for some $n \geq 1$ and $B\neq C$ such that there is a previous column corresponding to ${\ominus }^k B \to C$ for some $0 \leq k < n$ then every row must assign to ${\ominus }^n B \to C$ a value in $(a \,\tilde {\to }\, b)\cap +$ (a nonempty set), where a and b are the values already assigned to ${\ominus }^n B$ and C in that row, whenever ${\ominus }^k B \to C$ received a value in $+$ in that row. From this, we arrive to the following result.

The examples discussed in this section show that, given an arbitrary (purely) axiomatic extension of M, the definition of a sound and complete decision procedure by truth-tables obtained from the generated RNmatrix is not always guaranteed. As in the case of global rules and level valuations, the decidability must be analyzed case-by-case. It requires a detailed analysis of the set of conditions on the valuations, trying to obtain an equivalent (but improved) set of conditions which guarantees that any partial valuation satisfying the enlarged set of conditions can be extended to a full valuation over the RNmatrix. Finally, it is worth noting that there are in fact undecidable propositional modal logics, cf. [Reference del Cerro and Penttonen8].Footnote ¹⁵

7 Future work and discussion of the results

In this article, we presented a general semantical framework for modal logics that does not rely on the notion of possible worlds. In particular, we have shown that RNmatrices defined from swap structures in combination with a generalized level-valuations technique, originally developed by Kearns, and presented here for the first time for other modal rules than the rule of necessitation, is expressive enough to incorporate various modal logics, normal or non-normal.

We furthermore, when compared to previous approaches using Nmatrices, cf. [Reference Coniglio, Del Cerro and Peron9, Reference Omori and Skurt25], have shown that RNmatrices induced by swap structures allow for a more flexible approach to semantics for many valued logics in general, and modal logics in particular, by providing a simple reduction procedure of formulas to restrictions on the set of valuation, that allowed for systems with for example the Löb axiom or the McKinsey formula, as axioms.

More concretely, we showed that our semantical approach goes beyond Kripke structures and non-deterministic semantics as considered in previous work in the following ways:

1. 1. We provided restricted non-deterministic semantics for non-Sahlquist formulas as axioms, such as the Gödel–Löb axiom or the McKinsey formula, which do not have a (first-order) Kripke semantics.

2. 2. Simple extensions of M with axioms such as $\Box (A \rightarrow A)$ do neither have a Kripke semantics nor a corresponding Nmatrix, cf. [Reference Grätz17]. We have shown that there is RNmatrix semantics for such extensions.

3. 3. We presented a decision procedure for selected modal systems and discussed some restrictions to it.

4. 4. By focusing on the weakest modal logic M and investigating its extensions, we introduced a new hierarchy of (bi-, multi-) modal systems with or without (non-standard, multiple) global rules.