2 Preliminaries

In this work, a logic is identified with a consequence relation [Reference Font24], as opposed to only a set of formulas. While the second approach is more common in the literature of modal logics [Reference Chagrov and Zakharyaschev16], we opt for the former definition because the differences between local and global modal logics are lost if only the tautologies of the logic are considered. Observe that the lack of the Deduction Theorem in the global logic makes the implication and the logical consequence not interchangeable.

Given a set of variables $\mathcal {V}$ and an algebraic language $\mathtt {L}$ , the set $Fm^{\mathtt {L}}(\mathcal {V})$ is the set of formulas built from $\mathcal {V}$ using the symbols from $\mathtt {L}$ . Unless stated otherwise, $\mathcal {V}$ is a fixed denumerable set, and it will be omitted in the notation of the set of formulas, and if the language is clear from the context we will omit it as well. A rule in $Fm$ is a pair . We say a rule is finitary whenever $\varGamma $ is a finite set. A logic $\mathcal {L}$ over $Fm$ is a consequence relation on $Fm$ , that is, a set of rules such that

1. (1) $\mathcal {L}$ is reflexive, i.e., for every $\varGamma \subseteq Fm$ and every $\gamma \in \varGamma $ , ,

2. (2) $\mathcal {L}$ satisfies cut, i.e., if for all $\phi \in \Phi $ , and then ,

3. (3) $\mathcal {L}$ is substitution invariant, i.e., for each substitution $\sigma $ , if then .

Whenever we will write $\varGamma \vdash _{\mathcal {L}} \varphi $ . Given a set of rules R, we will write $R^l$ to denote the minimal logic containing the rules in R. We say that a set of rules R axiomatizes a logic $\mathcal {L}$ whenever $R^l = \mathcal {L}$ . Observe that in this sense, every logic is axiomatized at least by itself.

With computational questions in mind, we will be focused on logics determined by finitary rules. A logic $\mathcal {L}$ is finitary whenever $\varGamma \vdash _{\mathcal {L}} \varphi $ if and only if $\varGamma _0 \vdash _{\mathcal {L}} \varphi $ for some finite $\varGamma _0 \subseteq _\omega \varGamma $ .Footnote ⁸ For convenience, we will denote by $\mathcal {L}^{fin}$ the set of finitary consequences of $\mathcal {L}$ , namely .

As it is most usual, in this work we do not consider infinite inputs for the computability questions. We say that a logic $\mathcal {L}$ is decidable, recursive, or recursively enumerable, respectively, if this is the case for the set $\mathcal {L}^{fin}$ .

When R is a set of finitary rules, the logic $R^l$ can be equivalently characterized through the usual notion of finite proofFootnote ⁹ in R. Given a finite set of formulas $\varGamma \cup \{\varphi \}$ , a proof or derivation of $\varphi $ from $\varGamma $ in R is a finite list of formulas such that $\psi _n = \varphi $ and for each $\psi _i$ in the list, either $\psi _i \in \varGamma $ or there is a rule $\varSigma \vdash \phi $ in R and a substitution $\sigma $ such that $\sigma (\phi ) = \psi _i$ and $\sigma [\varSigma ]$ (possibly empty) is a subset of (or empty if $i = 1$ ). It is well known that .

We will say that a logic is axiomatizable whenever it can be axiomatized using a recursive set of finitary rules. It is clear that an axiomatizable logic is finitary and R.E. On the other hand, a finitary R.E. logic with a definable idempotent n-ary operation for every n is always axiomatizable. This fact is a natural generalization of Craig’s Theorem, and it can be checked in a similar way. For the interested reader we provide the details in Section 7.

In the rest of the paper we will work with logics having such an idempotent operation (which will be simply $\wedge $ ). Thereby we will resort without further notice to the following observation.

Modal many-valued logics arise from Kripke structures evaluated over certain algebras, putting together relational and algebraic semantics in a way adapted to model different reasoning notions. In the next section, the general algebraic setting of these semantics will be the one of $FL_{ew}$ -algebras, the corresponding algebraic semantics of the Full Lambek Calculus with exchange and weakening. This will offer a very general approach to the topic while relying on well-known algebraic structures. We will later focus on modal expansions of MV and product algebras, particular classes of $FL_{ew}$ -algebras.

We will usually write $ab$ instead of $a \cdot b$ , and abbreviate $\overbrace {x \cdot x \cdots x}^{n}$ by $x^n$ for $n \geqslant 1$ . Moreover, as usual, we will define $\neg a$ to stand for $a \rightarrow \overline {0}$ . A chain is a linearly ordered $\mathbf {FL_{ew}}$ -algebra.

In the setting of the previous definition, we will denote by $\mathbf {Fm}'$ the algebra of formulas built over a countable set of variables $\mathcal {V}$ using the language corresponding to the above class of algebras (i.e.,

). We will refer to the bottom and top elements of the algebra, $\overline {0}$ and $\overline {1}$ , simply by $0$ and $1$ . Moreover, we will again write $\varphi \psi $ instead of $\varphi \cdot \psi $ and $\varphi ^n$ for the product of $\varphi $ with itself n times (for $n \geqslant 1$ ), and we let, as usual

For a set of formulas $\varGamma \cup \{\varphi \}$ and a class of $\mathbf {FL_{ew}}$ -algebras $\mathbb {A}$ , we write $\varGamma \models _{\mathbb {A}} \varphi $ if and only if, for each $\mathbf {A} \in \mathbb {A}$ and each $h \in Hom(\mathbf {Fm}',\mathbf {A})$ , if $h(\gamma ) = 1$ for each $\gamma \in \varGamma $ , then $h(\varphi ) = 1$ too. We will write $\models _{\mathbf {A}}$ instead of $\models _{\{\mathbf {A}\}}$ . As expected, $\models _{\mathbb {A}}$ is a logic, and by convenience we will write $\models _{\mathbb {A}}$ instead of $\vdash _{\models _{\mathbb {A}}}$ .

$\mathbb {FL}_{ew}$ , the class of $\mathbf {FL_{ew}}$ -algebras, is a variety studied in depth, see for instance [Reference Galatos, Jipsen, Kowalski and Ono25, Reference Ono34].

Let us introduce some examples of well-known subvarieties of $\mathbb {FL}_{ew}$ . Heyting Algebras, the algebraic counterpart of Intuitionistic logic, are $\mathbf {FL_{ew}}$ -algebras where $\wedge = \cdot $ . The variety of Gödel algebras, $\mathbb {G}$ , (corresponding to intermediate Gödel–Dummett logic $\mathcal {G}$ ) is that of semilinear Heyting algebras, i.e., those satisfying $(a \rightarrow b) \vee (b \rightarrow a)=1$ for all $a,b$ in the algebra. BL algebras, the algebraic counterpart of Hájek Basic Logic $\mathcal {BL}$ , are semilinear $\mathbf {FL_{ew}}$ algebras where $a \cdot (a \rightarrow b) = a \wedge b$ for every $a, b$ in the algebra. The variety of MV algebras $\mathbb {MV}$ , algebraic counterpart of Łukasiewicz logic , is formed by the involutive BL algebras (i.e., satisfying ), and that of Product algebras $\mathbb {P}$ (corresponding to Product Logic $\varPi $ ), is formed by those BL algebras satisfying and .

Particular algebras in the previous classes are the so-called standard ones, whose universe is the standard unit real interval $[0,1]$ and the order (affecting the lattice $\wedge , \vee $ operations) is the standard one. Let us introduce explicitly the operations, which we will denote, for convenience, with the subscripts G, and $\varPi $ (for Gödel, Łukasiewicz, and Product logics, respectively).

• • $[0,1]_G$ , the standard Gödel algebra, defines

  

• • , the standard MV algebra, defines

  

• • $\bf {MV_n}$ , the n-valued MV algebra is the subalgebra of with universe .

• • $[0,1]_\varPi $ , the standard Product ( $\varPi $ ) algebra, defines

  
  with $\times $ being the usual product between real numbers.

It is known that the standard Gödel, MV, and Product algebras generate their corresponding varieties. They do so also as quasi-varieties, which implies the completeness of the logics (understood as consequence relations) with respect to the logical matrices over the respective standard algebra. In the case of Gödel, it is also the case that the variety is generated as a generalized quasi-variety, while this fails for MV and Product algebras. The first claim amounts to saying that for each set of formulas $\varGamma \cup \{ \varphi \}$ , it holds that $\varGamma \vdash _{\mathcal {G}} \varphi $ if and only if $\varGamma \models _{[0,1]_G} \varphi $ (and if and only if $\varGamma \models _{\mathbb {G}} \varphi $ ). For a finite set of formulas $\varGamma \cup \{\varphi \}$ it holds that if and only if (if and only if $\varGamma \models _{\mathbb {MV}} \varphi $ ); and $\varGamma \vdash _{\varPi } \varphi $ if and only if $\varGamma \models _{[0,1]_{\varPi }} \varphi $ (if and only if $\varGamma \models _{\mathbb {P}} \varphi $ ). The last conditions might fail for infinite $\varGamma $ .

Let us introduce some other families of $\mathbf {FL_{ew}}$ -algebras that will be of use later on.

Observe that if $\mathbf {A}$ is n-contractive, the element $a^n$ is idempotent (namely ${a^n\cdot a^n = a^n}$ ) for every $a \in A$ . Simple examples of these algebras include Heyting algebras ( $1$ -contractive), or $MV_n$ algebras ( $(n-1)$ -contractive). On the other hand, the standard MV-algebra and product algebra are not n-contractive for any n. Regarding weakly saturation, observe that if the element $\inf \{b^n\colon n \in \mathbb {N}\}$ exists in a weakly saturated algebra, then it is an idempotent element. Examples of weakly saturated algebras are the standard MV-algebra, the standard product algebra, as well as the algebras belonging to the generalised quasi-varieties generated by them.

The algebra of modal formulas $\mathbf {Fm}$ is built in the same way as $\mathbf {Fm'}$ , expanding the language of $\mathbf {FL_{ew}}$ -algebras with two unary operators $\square $ and $\Diamond $ . While it is clear how to lift an evaluation from the set of propositional variables $\mathcal {V}$ into an $FL_{ew}$ -algebra to $\mathbf {Fm'}$ , the semantic definition of the modal operators depends on the relational structure in the following way.

Definition 2.3. Let $\mathbf {A}$ be an $\mathbf {FL_{ew}}$ -algebra. An $\mathbf {A}$ -Kripke model is a structure

such that

• • is a Kripke frame. That is to say, W is a non-empty set of so-called worlds and $R\subseteq W \times W$ is a binary relation over W, called an accessibility relation. We will often write $Rvw$ instead of ;

• • e is a map from $W \times \mathcal {V}$ to A.

The evaluation e of an $\mathbf {A}$ -Kripke model

is uniquely extended to a map from $W \times Fm$ to A by letting

A model is safe whenever the values of $e(v, \square \varphi )$ and $e(v, \Diamond \varphi )$ are defined for every formula $\varphi $ at each world $v \in W$ . The class of $\mathbf {FL_{ew}}$ -Kripke models is the (set) union of all safe $\mathbf {A}$ -Kripke models, for $\mathbf {A} \in \mathbf {FL_{ew}}$ .

We call a model $\mathfrak {M}$ directed whenever there is some world $u \in W$ in it such that, for each $v \in W$ , there is some path from u to v Footnote ¹⁰ in $\mathfrak {M}$ .

Regarding notation, given a class of models $\mathbb {C}$ , we denote by $\omega \mathbb {C}$ the class of finite models in $\mathbb {C}$ (namely, the models Footnote ¹¹ ). On the other hand, for a class of algebras $\mathbb {A}$ (or a single algebra $\mathbf {A}$ ) we write $K{\mathbb {A}}$ ( $K{\mathbf {A}}$ ) to denote the class of safe Kripke models over the algebras in the class (or over the single algebra specified). Finally, in order to lighten the reading, we will let and $K{\varPi }$ to denote respectively and $K{[0,1]_\varPi }$ .

Towards the definition of modal logics over $\mathbf {FL_{ew}}$ -algebras relying on the notion of $\mathbf {FL_{ew}}$ -Kripke models, it is natural to use the notion of truth world-wise being $\{1\}$ (in order to obtain, world-wise, the propositional $\mathbb {FL}_{ew}$ logic). With this in mind, for each $\mathbf {A}$ -Kripke model $\mathfrak {M}$ and $v \in W$ we say that $\mathfrak {M}$ satisfies a formula $\varphi $ in v, and write $\mathfrak {M},v \models \varphi $ whenever $e(v, \varphi ) = 1$ . Similarly, we simply say that $\mathfrak {M}$ satisfies a formula $\varphi $ , and write $\mathfrak {M} \models \varphi $ whenever for all $v \in W \mathfrak {M},v \models \varphi $ . The same definitions apply to sets of formulas.

As in the classical case, the previous definition of satisfiability gives rise to two different logics: the local logic and the global one. In this work we will focus on the study of the global logic, but in Section 5 we will point out some results involving the local modal logic as well.

For arbitrary $\varGamma \cup \{ \varphi \} \subseteq Fm$ , we let $ \varGamma \vdash _{\mathbb {C}} \varphi \text { whenever there is } \varGamma _0 \subseteq _\omega \varGamma \text { such that} \varGamma _0 \vdash _{\mathbb {C}} \varphi $ , and the analogous for the local logic.

For a single Kripke model $\mathfrak {M}$ , we write $\varGamma \vdash _{\mathfrak {M}} \varphi $ instead of $\varGamma \vdash _{\{\mathfrak {M}\}} \varphi $ . In a similar way, for a model $\mathfrak {M}$ and a world $u \in W$ we write to denote that $\mathfrak {M} \models \varGamma $ and $\mathfrak {M},u \not \models \varphi $ (namely, $\varphi $ does not follow globally from $\varGamma $ in $\mathfrak {M}$ , and world u witnesses this fact). In a more general setting, fixing a Kripke frame $\mathfrak {F}$ and an algebra $\mathbf {A}$ , we write $\varGamma \vdash _{\mathfrak {F}_{\mathbf {A}}} \varphi $ whenever $\varGamma \vdash _{\mathfrak {M}} \varphi $ for every safe $\mathbf {A}$ -Kripke model $\mathfrak {M}$ with underlying Kripke frame $\mathfrak {F}$ . Analogously, for a class of frames $\mathbb {F}$ and a class of algebras $\mathbb {C}$ , we write $\varGamma \vdash _{\mathbb {F}_{\mathbb {C}}} \varphi $ whenever $\varGamma \vdash _{\mathfrak {F}_{\mathbf {A}}} \varphi $ for each $\mathfrak {F} \in \mathbb {F}$ and each $\mathbf {A} \in \mathbb {C}$ .

Tautologies (formulas following from $\emptyset $ ) of $ \vdash ^l_{\mathbb {C}}$ and $\vdash _{\mathbb {C}}$ coincide, and $\vdash ^l_{\mathbb {C}}$ is strictly weaker than $\vdash _{\mathbb {C}}$ , a trivial separating case being the usual necessitation rule $\varphi \vdash \square \varphi $ (valid in the global case and not in the local one). Observe that $ \vdash ^l_{\mathbb {C}}$ and $\vdash _{\mathbb {C}}$ are, by their definition, determined by the safe directed models generated from the models in $\mathbb {C}$ .

Also, the unraveling and filtrationFootnote ¹² techniques can be applied to an arbitrary directed model, obtaining a directed tree that, from the logical point of view, behaves in its root as the original model (i.e., satisfies exactly the same global and local derivations). Even if the resulting tree might be infinite, all worlds in the tree are, by construction, at a finite distance from the root. Thus, $\vdash _{K\mathbb {C}} {{\kern-2pt}={\kern-2pt}} \vdash_{K\mathbb {C}^T}$ , for $K\mathbb {C}^T$ being the class of safe directed trees generated by models in $\ K\mathbb {C}$ .

Some useful notions concerning Kripke models are the following ones.

Definition 2.5. Given a Kripke model $\mathfrak {M}$ and $w \in W$ , we let the height of w, $\mathbf {\mathtt {h}}(w)$ be the element in $\mathbb {N}\cup \{\infty \}$ Footnote ¹³ given by

Observe that if there exists some cycle in the model, all worlds involved in that cycle (and every predecessor of each world in that cycle in the model) have infinite height.

Definition 2.6. Let $\varphi $ be a formula of $Fm$ . We let the subformulas of $\varphi $ be the set inductively defined by

We let the propositional subformulas of $\varphi $ be the set inductively defined by

For $\varGamma $ a set of formulas we let

Let us finish the preliminaries by stating a well-known undecidable problem, that will be used in the next sections to show undecidability of some of the modal logics introduced above. Recall that given two numbers $\mathtt { x}, \mathtt { y}$ in base $s \in \mathbb {N}$ , their concatenation $\mathtt { x_\smile y}$ is given by $\mathtt { x}s^{\parallel \mathtt { y} \parallel } + \mathtt { y}$ , where $\parallel \mathtt {y}\parallel $ is the number of digits of $\mathtt { y}$ in base s.

Definition 2.7 (Post Correspondence Problem (PCP)).

An instance P of the PCP consists of a list of pairs of numbers in some base $s \geqslant 2$ . A solution for P is a sequence of indexes with such that

$$ \begin{align*}\mathtt{x_{i_1}}{_\smile} \dots _\smile \mathtt{x_{i_k}}= \mathtt{y_{i_1}}{_\smile} \dots{_\smile} \mathtt{y_{i_k}}. \end{align*} $$

The decision problem for PCP is, given a PCP instance, to decide whether such a solution exists or not. This question is undecidable [Reference Post35].

3 Undecidability of global modal logics

In this section, unless stated otherwise, we let $\mathbb {A}$ be a class of weakly saturated $\mathbf {FL_{ew}}$ chains such that for every $n \in \mathbb {N}$ there is some $\mathbf {A}_n \in \mathbb {A}$ such that $\mathbf {A}_n$ is non n-contractive. That is to say, there is some $a \in A_n$ such that $a^{n+1} < a^n.$

Examples of such classes of algebras are and $\{[0,1]_{\varPi }\}$ . Natural examples of classes of algebras not satisfying the above conditions are $\{[0,1]_G\}$ and the variety generated by it, and the varieties of MV and product algebras (since these are not classes of weakly saturated chains).

Let the class of frames $\mathbb {L}$ be the isomorphic copies of the frames in the set

Namely, $\mathbb {L}$ is given by the frames whose structure is isomorphic to the one depicted in Figure 1, for any $k \in \mathbb {N}$ .

Figure 1 Structure of the frames in $\mathbb {L}$ .

The previous theorem follows as a direct consequence of the next result.

Trivially, $(4) \Rightarrow (3)$ and $(3) \Rightarrow (2)$ for any $\varGamma _P \cup \{\varphi _P\} \subseteq _{\omega } Fm$ . In what remains of this section we will first show that $(1) \Rightarrow (4)$ , and afterwards, that both $(2) \Rightarrow (4)$ and $(4) \Rightarrow (1)$ . To this aim, let us begin by explicitly defining a suitable set of formulas $\varGamma _P \cup \{\varphi _P\}$ .

For list of pairs of numbers in base s, we let $\varGamma _P$ be the set with formulas in variables $\mathcal {V} = \{x, y, z\}$ :

1. (1) for each $p \in \mathcal {V}$ .

2. (2) .

3. (3) .

Finally, let

Roughly speaking, variables x and y will store information on the concatenation of the corresponding elements of the PCP, while z will have a technical role.

Given a solution of P, it is not hard to construct a finite model globally satisfying $\varGamma _P$ and not $\varphi _P$ .

Proof. (Of Theorem 3.2, $(1) \Rightarrow (4)$ ).

Let be a solution for P, so $\mathtt { x_{i_1}} {_\smile } \dots {_\smile } \mathtt { x_{i_k}} = \mathtt { y_{i_1}} {_\smile } \dots {_\smile } \mathtt { y_{i_k}} = r$ for some $r \in \mathbb {N}$ . Pick some non r-contractive algebra $\mathbf {A} \in \mathbb {A}$ and $a \in A$ such that $a^{r+1} < a^r$ , and define a finite $\mathbf {A}$ -Kripke model $\mathfrak {M}$ as follows:

• • .

• • .

• • For each let

  1. – $\ e(v_j, z) = a$ ,

  2. – $\ e(v_j, x) = a^{{\mathtt {x_{i_1}}}{_\smile } \dots {_\smile } {\mathtt {x_{i_j}} }}$ ,

  3. – $\ e(v_j, y) = a^{\mathtt { y_{i_1}} {_\smile } \dots {_\smile } \mathtt { y_{i_j}}}$ .

The formula $\neg \square 0$ is evaluated to $0$ in $v_1$ , and to $1$ in all other worlds of the model. Thus, since z is evaluated to the same value in all worlds of the model, and each world has exactly one successor except for $v_1$ (which has none), clearly the family of formulas in $(1)$ and in $(2)$ from $\varGamma _P$ are satisfied in all worlds of the model.

To check that formula $(3)$ from $\varGamma _P$ is satisfied in all worlds of the model we reason by induction on the height of the world (Definition 2.5). For $v_1$ (with height equal to $0$ ), given that it does not have any successors, it is clear that

For all other $v_r$ with $r>1$ , recall that its only successor is $v_{r-1}$ . Applying the definition of concatenation, and the fact that for every $\mathbf {A} \in \mathbb {A}$ and each $a \in A$ and $n,m\in \mathbb {N}$ , trivially $a^n a^m = a^{n+m}$ and $(a^n)^m = a^{nm}$ , we can prove that

With the above, we have proven that $\mathfrak {M} \models \varGamma _P$ .

On the other hand, since $i_1, \dots , i_k$ was a solution for P, $e(v_k, x) = e(v_k, y)$ . Moreover, $e(v_k, z) = a < 1$ , and $e(v_k, xz) = a^{r+1} < a^r = e(v_k, x)$ , so $e(v_k, xz \rightarrow x) < 1$ . This implies that , proving that $\varGamma _P \not \vdash _{\omega K{\mathbb {A}}} \varphi _P$ .⊣

In order to prove the other implications of Theorem 3.2, let us first show some technical characteristics of the models satisfying $\varGamma _P$ and not $\varphi _P$ .

A first easy observation is that in every model satisfying $\varGamma _P$ , the variable z takes the same value in all connected worlds of the model. Relying on the completeness with respect to trees, we can prove that, in these models, z is evaluated to the same value in the whole model.

Proof. Let $\alpha _z = e(u, z)$ . It is easy to prove the lemma by induction on the distance of v from u, which is always finite because $K\mathbf {A}^T$ is a class of directed trees.

If $v = u$ then the claim follows trivially. Otherwise, assume that there are with $w_0 = u, w_{k+1} = v$ and such that $R w_i w_{i+1}$ for all . Since $e(w_k, (1)) = e(w_k, (2)) = 1 $ and $R w_kw_{k+1}$ , then we know

$$ \begin{align*} e(w_k, \square z) = e(w_k, \Diamond z)\qquad \text{ and }\qquad e(w_k, z) = e(w_k, \square z). \end{align*} $$

From the first equality we get that $e(v_1, z) = e(v_2,z)$ for all $v_1,v_2\in W$ such that $R w_k v_1$ and $R w_k v_2$ . In particular, this yields that $e(w_k, \square z) = e(w_{k+1}, z)$ . Together with the second equality, it follows that $e(w_k, z) = e(w_{k+1}, z) = e(v,z)$ . Applying the Induction Hypothesis, we conclude $e(u, z) = e(w_k, z) = e(v, z)$ .⊣

The fact that algebras in $\mathbb {A}$ are linearly ordered and weakly saturated allows us to also prove that such models can be assumed to have finite height.

Proof. From Lemma 3.3 we know that in each world v of $\mathfrak {M}$ it holds that $e(u, z) = e(v, z) = \alpha _z$ . Moreover, from $(3)$ in $\varGamma _P$ it follows that

If u was of infinite height, by weak saturation of $\mathbf {A}$ , it would follow that $e(u,x) e(u,z) = e(u,x)$ . However, since $e(u, \varphi _P) <1$ , necessarily $e(u, xz) < e(u, x)$ , and thus u must be of finite height.⊣

As a corollary, we get that the values of x and y at each world are powers of $\alpha _z$ .

Corollary 3.5. Let $\mathbf {A} \in \mathbb {A}$ , and $\mathfrak {M} \in K\mathbf {A}^T$ with root $u \in W$ be such that . Then for each $v \in W$ there are $a_v, b_v \in \mathbb {N}$ such that

$$ \begin{align*}e(v, x) = \alpha_z^{a_v} \qquad \text{ and } \qquad e(v,y) = \alpha_z^{b_v}.\end{align*} $$

Moreover, if $\mathbf {\mathtt {h}}(v) < \mathbf {\mathtt {h}}(w)$ then $a_v < a_w$ and $b_v < b_w$ .

Another corollary can be proven after observing the way in which the implication behaves between powers of the same element in $FL_{ew}$ chains.

We can now prove that if $\varGamma _P \not \vdash _{K{\mathbb {A}}} \varphi _P$ then this happens in a model whose frame belongs to the class $\mathbb {L}$ , introduced at the beginning of this section (Figure 1).

Lemma 3.8.

$$ \begin{align*}\varGamma_P \vdash_{K{\mathbb{A}}} \varphi_P \text{ if and only if } \varGamma_P \vdash_{\mathbb{L}_{\mathbb{A}}} \varphi_P.\end{align*} $$

At this point, it is possible to obtain a useful characterization of x and y in terms of $\alpha _z$ at each world of a model with a frame isomorphic to one in $\mathbb {L}$ that satisfies $\varGamma _P$ and not $\varphi _P$ in its root. For convenience, in the next result we will invert the labeling of the worlds in the frames isomorphic to those in $\mathbb {L}$ , namely resorting to frames with structure (trivially isomorphic to ).

Lemma 3.9. Let be an $\mathbb {A}$ -Kripke model such that . Then there exist with for each , such that for each ,

$$ \begin{align*}e(u_j, x) = \alpha_z^{\mathtt{ x_{i_1}}{_\smile} \dots{_\smile} \mathtt{ x_{i_j}}} \qquad \text{ and } \qquad e(u_j, y) = \alpha_z^{\mathtt{ y_{i_1}}{_\smile} \dots{_\smile} \mathtt{ y_{i_j}}}.\end{align*} $$

Moreover, for each ,

$$ \begin{align*}e(u_j,x) = e(u_j,y) \text{ if and only if } \mathtt{ x_{i_1}}{_\smile} \dots{_\smile} \mathtt{ x_{i_j}} = \mathtt{ y_{i_1}}{_\smile} \dots{_\smile} \mathtt{ y_{i_j}} .\end{align*} $$

Proof. We will prove the first claim of the lemma by induction on j. The details are only given for the x case, the other one is proven in the same way.

For $j = 1$ , $u_1$ does not have successors. From formula $(3)$ in $\varGamma _P$ (relying on the fact that the algebras in $\mathbb {A}$ are chains) and Lemma 3.3 it follows that there is for which

$$ \begin{align*}e(u_1, x) = e(u_1, \square x)^{s^{\parallel{\mathtt{ x_{i_1}}}\parallel}} e(u_1, z)^{\mathtt{ x_{i_1}}} = 1^{s^{\parallel\mathtt{ x_{i_1}}\parallel}}\alpha_z^{\mathtt{ x_{i_1}}} = \alpha_z^{\mathtt{ x_{i_1}}}.\end{align*} $$

For $j = r+1$ , observe the only successor of $u_j$ in $\mathfrak {M}$ is $u_r$ . Then, from $(3)$ and Lemma 3.3 it follows that there is for which

$$ \begin{align*}e(u_j, x) = e(u_j, \square x)^{s^{\parallel\mathtt{ x_{i_j}}\parallel}} e(u_j, z)^{\mathtt{ x_{i_1}}}= e(u_r, x)^{s^{\parallel\mathtt{ x_{i_j}}\parallel}} \alpha_z^{\mathtt{ x_{i_j}}}.\end{align*} $$

By the Induction Hypothesis, the above value is equal to $ (\alpha _z^{{\mathtt { x_{i_1}}{_\smile } \dots {_\smile } \mathtt { x_{i_r}}}})^{s^{\parallel \mathtt { x_{i_j}}\parallel }} \alpha _z^{\mathtt { x_{i_j}}}$ , and through simple properties of the monoidal operation, to $ (\alpha _z^{{\mathtt { x_{i_1}}{_\smile } \dots {_\smile } \mathtt { x_{i_r}}}})^{s^{\parallel \mathtt { x_{i_j}}\parallel } + \mathtt { x_{i_j}}}$ as well. This value, by definition of the concatenation of numbers in base s, is exactly $\alpha _z^{{\mathtt { x_{i_1}}{_\smile } \dots {_\smile } \mathtt { x_{i_j}}}}$ , concluding the proof of the first claim.

Concerning the second claim, assume towards a contradiction that there is such that $\mathtt { x_{i_1}}{_\smile } \dots {_\smile } \mathtt { x_{i_j}} \neq \mathtt { y_{i_1}}{_\smile } \dots {_\smile } \mathtt { y_{i_j}}$ and $e(u_j, x) = \alpha _z^{\mathtt { x_{i_1}}{_\smile } \dots {_\smile } \mathtt { x_{i_j}}} = \alpha _z^{\mathtt { y_{i_1}}{_\smile } \dots {_\smile } \mathtt { y_{i_j}}} = e(u_j,y)$ . If $\mathtt { x_{i_1}}{_\smile } \dots {_\smile } \mathtt { x_{i_j}} < \mathtt { y_{i_1}}{_\smile } \dots {_\smile } \mathtt { y_{i_j}}$ , it follows that $\alpha _z^{\mathtt { x_{i_1}}{_\smile } \dots {_\smile } \mathtt { x_{i_j}}} \alpha _z^n = \alpha _z^{\mathtt { x_{i_1}}{_\smile } \dots {_\smile } \mathtt { x_{i_j}}} $ for every $n \geqslant 0$ . Thus, in particular, from Corollary 3.5 $e(u_k, x) = e(u_j, x)$ , and also $e(u_k, x) \alpha _z = e(u_k, x) = e(u_j, x)$ . However, $\mathfrak {M}, u_k \not \models \varphi _P$ implies that $e(u_k, x) \alpha _z < e(u_k,x)$ , reaching a contradiction.

The proof is analogous if $\mathtt { x_{i_1}}{_\smile } \dots {_\smile } \mathtt { x_{i_j}}> \mathtt { y_{i_1}}{_\smile } \dots {_\smile } \mathtt { y_{i_j}}$ .⊣

All the previous technical lemmas lead to a simple proof of Theorem 3.2.

Let us conclude this section by pointing out that the undecidability results in [Reference Vidal42], affecting the local deduction over transitive models evaluated over classes of algebras like $\mathbb {A}$ , are immediately translatable to the global logic. Consider the class of frames $\mathbb {L}^4$ given by isomorphic copies of the frames in the set

where $<$ is the usual strict order relation in $\mathbb {N}$ restricted in each case to the corresponding universe.

Proof. It is straightforward, since for any finite $\varGamma \cup \{\varphi \} \subseteq Fm$ and any class of transitive models $\mathbb {C}$ , it is routine to check that

$$ \begin{align*}\varGamma \vdash_{\mathbb{C}} \varphi \text{ if and only if } \varGamma, \square \varGamma \vdash^l_{\mathbb{C}} \varphi. \\[-36pt] \end{align*} $$

⊣

4 Non axiomatizability of modal Łukasiewicz and product logics

The undecidability of the previous family of modal logics over finite models raises the question of their axiomatizability. In particular, it was an open problem how to axiomatize the finitary standard modal Łukasiewicz logic [Reference Diaconescu, Metcalfe and Schnuriger20, Reference Hansoul and Teheux30] and standard modal Product logic [Reference Vidal, Esteva and Godo46]. None of the minimal modal logics over the previous standard algebras has been axiomatized in the literature, including the logics arising from crisp-accessibility frames (nor those from many-valued frames) with either one or both modal operators. In the previous references, infinitary axiomatic systems complete with respect to related (but different) deductive systems have been proposed. For instance, their corresponding infinitary companions (have been axiomatized in some cases, over extended languages).

We close this open problem for the standard Łukasiewicz and Product logics with a negative answer: these logics are in fact not axiomatizable, since their respective sets of valid consequences are not recursively enumerable. We will devote this section to prove the previous claims. For that, three properties turn out to be crucial: undecidability of the global consequence over finite models of the class, decidability of the propositional logic, and completeness of the global consequence with respect to certain well-behaved models (in these cases, in terms of witnessing conditions). We will prove this negative result for the modal expansion of the standard Łukasiewicz logic. Then, the analogous result will follow for the Product logic, relying on the known isomorphism between the standard MV-algebra and a certain Product algebra.

The first one of the above properties was proven in Section 3. Let us show how decidability of the underlying propositional logic $\models _{\mathbb {A}}$ implies that the set is R.E., which will allow us to conclude there is no possible axiomatization for the logics of finite models over those classes of algebras.

We first see that the global consequence relation over a finite frame is decidable as long as the underlying propositional consequence relation is decidable too.

Proof. Let

a Kripke frame, $v \in W$ a world in the frame, $\psi $ a modal formula with variables in a finite set $\mathcal {V}$ , and x not in $\mathcal {V}$ . Consider the extended set of propositional variables

We recursively define the non-modal formula

over $\mathcal {V}^*$ as follows:

Then, for $\varSigma $ a set of formulas we let

, where set of original variables is

. Moreover, consider the formulasFootnote ¹⁴

From those, define the set of formulas

We will now prove that

which implies the lemma.

To prove the right-to-left direction, assume $\varGamma \not \vdash _{\mathfrak {F}_{\mathbb {A}}} \varphi $ . Then there is $\mathbf {A} \in \mathbb {A}$ , and an $\mathbf {A}$ -Kripke model over $\mathfrak {F}$ in which $e(v, \varGamma ) \subseteq \{1\}$ for all v and $e(v_0, \varphi ) < 1$ for some $v_0 \in W$ . Consider then the mapping $h \colon \mathcal {V}^* \rightarrow A$ defined by $h(p^v) = e(v, p)$ , $h(x_{\square \psi }^v) = e(v, \square \psi )$ and $h(x_{\Diamond \psi }^v) = e(v, \Diamond \psi )$ . It is easy to see that the extension of this mapping to a homomorphism into $\mathbf {A}$ satisfies for every $\psi \in SFm(\varGamma , \varphi )$ . Thus, it satisfies the premises in the derivation at the right side, since $e(v, \varGamma ) \subseteq \{1\}$ for all v and by the semantical definition of $\square $ and $\Diamond $ in the model. On the other hand, it does not satisfy the consequence, since $e(v_0, \varphi ) < 1$ .

Conversely, given a propositional homomorphism h over some algebra $\mathbf {A} \in \mathbb {A}$ satisfying

and not satisfying

, we can consider the $\mathbf {A}$ -Kripke model over $\mathfrak {F}$ that lets $e(v, p) = h(p^v)$ . Since $h(\varDelta ^v(\varGamma , \varphi )) = \{1\}$ , then

for every $\psi \in SFm(\varGamma , \varphi )$ , concluding the proof.⊣

Exhibiting a recursive procedure enumerating the elements not belonging to $\vdash _{\omega K{\mathbb {A}}}$ is now easy.

The previous lemma implies that, for every class of algebras $\mathbb {C}$ satisfying the premises of Theorem 3.1 for which $\models _{\mathbb {C}}$ is decidable, the logic $\vdash _{\omega K{\mathbb {C}}}$ is not recursively enumerable. Otherwise, since the previous lemma proves that is recursively enumerable, the logic $\vdash _{\omega K{\mathbb {C}}}$ would be decidable, contradicting Theorem 3.1. Since and $\varPi $ are decidable logics [Reference Hájek26], their standard completeness implies that and $\models _{[0,1]_\varPi }$ are decidable too, leading to the following corollary.

However, since it is not a general fact that the logics $\vdash _{K{\mathbb {C}}}$ are complete with respect to finite models, the lack of axiomatization of the previous logics does still not close the problems mentioned in the beginning of this section.

4.1 Modal Łukasiewicz logic is not axiomatizable

We can show that, even if the global modal Łukasiewicz logic might not enjoy the finite model property, if the global modal (standard) Łukasiewicz logic is R.E. then (the analogous logic over finite models) would be R.E. too. Since we saw before the latter does not hold, we will conclude that is not R.E. and so, not axiomatizable.

Lemma 3 from [Reference Hájek27] allows us to prove completeness of with respect to witnessed models, in a similar way to how it is done for tautologies of fuzzy description logic (FDL) over Łukasiewicz logic in the same publication. We do not introduce details of F.O. (standard) Łukasiewicz logic here, we refer the interested reader to, e.g., [Reference Hájek26]. Just recall that:

• • A standard Łukasiewicz F.O. model is a structure where W is a non-empty set and for each $i \in I$ and $ar(i)$ the arity of $P_i$ , $P_i \colon W^{ar(i)} \rightarrow [0,1]$ .

• • An evaluation in an (F.O.) model is a mapping $v \colon \mathcal {V} \mapsto W$ . Moreover, we write $v[x \mapsto m]$ to denote the evaluation v where the mapping of the variable x is overwritten and x is mapped to m (and simply $[x \mapsto m]$ denotes that the evaluation of x is m and the other variables are irrelevant).

• • The value of a formula $\varphi $ in an (F.O.) model $\mathfrak {M}$ under an evaluation v, denoted by $\|\varphi \|_{\mathfrak {M}, v}$ is inductively defined by

  1. – .

  2. – $\ \|\varphi _1 \star \varphi _2\|_{\mathfrak {M}, v} = \|\varphi _1\|_{\mathfrak {M}, v} \star \|\varphi _2\|_{\mathfrak {M}, v}$ for $\star $ propositional (Ł) operation.

  3. – $\ \|\forall x \varphi (x)\|_{\mathfrak {M}, v} = \bigwedge _{m \in W} \|\varphi \|_{\mathfrak {M}, v[x \mapsto m]}$ .

  4. – $\ \|\exists x \varphi (x)\|_{\mathfrak {M}, v} = \bigvee _{m \in W} \|\varphi \|_{\mathfrak {M}, v[x \mapsto m]}$ .

Observe that the value of a sentence (i.e., closed formula, without free variables) $\varphi $ in a model is constant under any evaluation, so we can simply write $\|\varphi \|_{\mathfrak {M}}$ to denote its value in a model. Moreover, we say that a model $\mathfrak {M}$ is witnessed whenever for every sentence $Qx \varphi (x)$ for $Q \in \{\forall , \exists \}$ there is some $m \in W$ such that

$$ \begin{align*} \|Qx \varphi(x)\|_{\mathfrak{M}} = \|\varphi(x)\|_{\mathfrak{M}, [x \mapsto m]}.\end{align*} $$

The consequence relation over standard Łukasiewicz F.O. models, , is defined for sentences by letting whenever for every standard Łukasiewicz F.O. model $\mathfrak {M}$ , if $\|\varGamma \|_{\mathfrak {M}} \subseteq \{1\}$ then $\|\varphi \|_{\mathfrak {M}} = 1$ .

Lemma 4.5 [Reference Hájek27, Lemma 3].

Let $\mathfrak {M}$ be a standard Łukasiewicz F.O. model. Then there is a (standard Łukasiewicz F.O.) witnessed model $\mathfrak {M}'$ such that $\mathfrak {M}$ is a submodel of $\mathfrak {M}'$ and for every sentence $\alpha $ it holds that

$$ \begin{align*}\|\varphi\|_{\mathfrak{M}} = 1 \text{ if and only if } \|\varphi\|_{\mathfrak{M}'} = 1.\end{align*} $$

From here, we can easily prove completeness of with respect to witnessed Kripke models, i.e., those for which, for every modal formula $\triangledown \varphi $ (with $\triangledown \in \{\square , \Diamond \}$ ) and every world v there is some world w such that $Rvw$ and

$$ \begin{align*}e(v, \triangledown\varphi) = (w, \varphi). \end{align*} $$

Lemma 4.6. If there is a witnessed standard Łukasiewicz Kripke model $\mathfrak {M}$ and $v \in W$ such that .

Proof. We can use the usual translation from modal to F.O. logics in order to move from a Kripke model to a suitable F.O. model. For a modal formula $\varphi $ consider the F.O. language $\{R/2, \{P_p/1\colon p\text { variable in }\varphi \}\}$ . For an arbitrary natural number $i \in \mathbb {N}$ , let us define the translation recursively by letting:

• • ,

• • for a propositional connective $\star $ ,

• • ,

• • .

It is a simple exercise to check that

If there is some F.O. model satisfying the premises of the right side of the above consequence and not . From the previous lemma we know there is a witnessed (F.O.) model $\mathfrak {M}$ in which the same conditions hold. Then, there is some m in the universe for which . At this point, it is only necessary to build a witnessed Kripke model from $\mathfrak {M}$ that is a global model for $\varGamma $ but does not satisfy $\varphi $ at some world. In order to do that, let the universe of the Kripke model be the same universe of $\mathfrak {M}$ , and let the accessibility relation be given by the interpretation of the binary predicate R in $\mathfrak {M}$ . Observe that, since $\forall x\forall y (R(x,y) \vee \neg R(x,y))$ is true in the model, necessarily $R(x,y) \in \{0,1\}$ , and thus the resulting model will be crisp. Finally, let $e(v, p) = \|P_p(v)\|_{\mathfrak {M}}$ for each variable p and each world $v \in W$ .

By induction on the complexity of the formula it is routine to check that for every $\psi \in SFm(\varGamma , \varphi )$ and every $v \in W$ , . Moreover, since the F.O. model is witnessed, the Kripke model is witnessed too. is a global model of $\varGamma $ , while , concluding the proof of the lemma.⊣

We can use the non-idempotency of the Łukasiewicz t-norm to recursively reduce the global consequence relation over finite models to the unrestricted global consequence relation.

Lemma 4.7. Let $\varGamma \cup \{\varphi \} \subseteq _{\omega } Fm$ and $p,q \not \in \mathcal {V}(\varGamma , \varphi )$ . Define

• • ,

• • ,

• • .

Then

Proof. $\Rightarrow $ : Assume by Lemma 4.6, there is a witnessed standard Łukasiewicz Kripke model $\mathfrak {M}$ and $v \in W$ such that $\mathfrak {M} \models \varGamma , \varXi (p), \xi (p,q)$ l (i.e., $e(u, \varGamma , \varXi (p) \cup \xi (p,q)) \subseteq \{1\}$ for all u) and $e(v, \varphi \vee \psi (p,q)) < 1$ . We can assume that $\mathfrak {M}$ is the unraveled tree generated from v. We will now prove that we can define a finite model equivalent to this one for what concerns the formulas in $F = \textit {SFm}(\varGamma \cup \varXi (p) \cup \{\xi (p,q)\} \cup \{\varphi \vee \psi (p,q)\})$ .

First, since $e(u, \varXi (p)) \subseteq \{1\}$ for each $u \in W$ , it follows that there is $a \in [0,1]$ such that for all $u \in W$ , $e(u, p) = a$ , as it was proven in Lemma 3.3. Moreover, from $e(v, p \vee \neg p) < 1$ we have that $a \in (0,1)$ .

On the other hand, from $e(u, \xi (p,q)) =1$ for each $u \in W$ we get that $e(u, q) = e(u, \square q) a$ . Thus, for each world $u \in W$ we have that for all . In particular, if there was any $u \in W$ with $\mathbf {\mathtt {h}}(u) = \infty $ , v would also have infinite height, and so for all $n\in \mathbb {N}$ . Since $a \in (0,1)$ , by the definition of the product in the standard MV algebra, the previous family of inequalities would imply that $e(v, q) = 0$ . Then it would holds that $e(v, \neg q) = 1$ , which is not possible since by the assumption $e(v, \varphi \vee \psi (p,q)) < 1$ . Thus, v—and so, all worlds of the model—must have finite height.

We can apply now a filtration-like transformation to $\mathfrak {M}$ with respect to the set of formulas F in order to obtain a finite directed model. To do this, let us denote by $wit(u, \triangledown \chi )$ an arbitrary witnessing world for the modal formula $\triangledown \chi $ at world u (i.e., such that $e(u, \triangledown \chi ) = e(wit(u, \triangledown \chi ), \chi )$ . Then define the universe

with

Observe that if there is some $i \in \mathbb {N}$ for which the worlds in $W_i$ do not have successors, $W_j = \emptyset $ for any $j> i$ .

We proved above that all worlds in $\mathfrak {M}$ have finite height. Further, F is a finite set of formulas. Henceforth, the model $\mathfrak {M}'$ resulting from restring $\mathfrak {M}$ to the universe $W'$ is a finite directed model with root v. Moreover, it is such that $e'(w, \varGamma , \varXi (p), \xi (p,q)) \subseteq \{1\}$ for each world $w \in W'$ , and $e'(v, \varphi \vee \psi (p,q)) < 1$ . In particular, $e(w, \varGamma ) \subseteq \{1\}$ at each world w, and $e(v, \varphi ) < 1$ .

: Assume , so there is a finite model $\mathfrak {M}$ and a world $v \in W$ such that . Let $\mathfrak {M}'$ be the model with universe equal to that of $\mathfrak {M}$ , such that $e'(u,x) = e(u,x)$ for all $x \in \mathcal {V}(\varGamma , \varphi )$ and each $u \in W$ , and where the values of variables $p,q$ are defined as follows. Pick an arbitrary element $a \in \big(\frac {\mathbf {\mathtt {h}}{v}}{\mathbf {\mathtt {h}}{v}+1}, 1\big)$ and let, for each $u \in W$ ,

• • $e(u, p) = a$ ,

• • $e(u, q) = e(u, \square q) a$ (observe this is well defined since all worlds have finite height, so we can define q inductively from the worlds with height $0$ ).

This evaluation satisfies, in all worlds of the model, all formulas from $\varXi (p)$ and $\xi (p,q)$ , and it forces $e(v, p) \not \in \{0,1\}$ and $e(v, q) \not \in \{0,1\}$ . Moreover, it satisfies the formulas from $\varGamma $ , and $e(v, \varphi ) <1$ , since the evaluation of all the variables appearing in $\varGamma $ and $\varphi $ has been preserved. Thus, either.⊣

The fact that the (finitary) Łukasiewicz global modal logic is not axiomatizable follows as a consequence of previous reduction (which is recursive) and the undecidability of .

Theorem 4.8. is not axiomatizable.

Proof. Assume is axiomatizable, and so, recursively enumerable. We can prove that then is recursively enumerable too, contradicting Corollary 4.4. For that, take a recursive enumeration of all pairs such that . For each pair, let $\mathcal {V} = \mathcal {V}ars(\varGamma , \varphi )$ , and check whether there are some $p,q \in \mathcal {V}$ for which $\varGamma = \varGamma _0(\mathcal {V}\setminus \{p,q\}) \cup \varXi (p) \cup \{\xi (p,q)\}$ and $\varphi = \varphi _0(\mathcal {V}\setminus \{p,q\}) \vee \psi (p,q)$ for some $\varGamma _0, \varphi _0$ . This is a decidable procedure because $\varGamma $ is a finite set and the translations $\varGamma _0$ and $\varphi _0$ are recursive. If that is the case, output , and don’t output anything otherwise. Lemma 4.7 implies that this procedure enumerates .

However, Corollary 4.4 states that is not R.E., a contradiction.⊣

4.2 Modal product logic is not axiomatizable either

In [Reference Baaz, Hájek, Krajíček and Švejda3], the authors observe, roughly speaking, that the standard MV algebra is isomorphic to the standard product algebra restricted to $[a,1]$ for arbitrary fixed $0 < a < 1$ . Relying on the isomorphism there provided, they also show that the tautologies of standard Łukasiewicz propositional and F.O. logicsFootnote ¹⁵ can be recursively reduced to those of the respective (standard) Product logic. In [Reference Hájek26, Lemmas 4.1.14 and 6.3.5] these results are formulated regarding the corresponding logical deduction relations.

We can use a similar argument, slightly modifying the reduction so it works in the modal case.Footnote ¹⁶

Given a finite set of variables $\mathcal {V}$ , let x be a propositional variable not in $\mathcal {V}$ . For each formula $\varphi $ of

in variables $\mathcal {V}$ , define its translation $\varphi ^x$ as follows:

Further, let .

In the spirit of Lemmas 2 and 3 from [Reference Baaz, Hájek, Krajíček and Švejda3] it is possible to prove the following result. The proof is very similar to the one in the previous reference, but for the sake of completeness we provide the details in Section 7.

Lemma 4.9. For every $x \not \in \mathcal {V}ar(\varGamma \cup \{\varphi \})$ , it holds that if and only if $\varGamma ^x, \varTheta ^x \vdash _{K\varPi } \varphi ^x$ .

Since the reduction is recursive, together with Theorem 4.8, the following is immediate.

Corollary 4.10. $\vdash _{K\varPi }$ is not axiomatizable.

We have proven that and $\vdash _{K\varPi }$ are not in $\varSigma _1$ from the Arithmetical Hierarchy. We leave open the question of whether they are $\varPi _2$ -complete, as it is the case for the tautologies of their F.O. versions, or whether they belong to some other level of the hierarchy. The proofs of Ragaz in [Reference Ragaz36, Reference Ragaz37] heavily rely on the expressive power of F.O. logic, and also in proving the result directly for tautologies of the logic. In the present work, the results affect the logic itself, since for instance, the tautologies of are decidable: they coincide with those of and this logic is decidable [Reference Vidal42, Corollary 4.5].

5 The necessitation rule

Recall that in (classical) modal logic, the global deduction is axiomatized as the local one plus the (unrestricted) necessitation rule $N_\square \colon x \vdash \square x$ . It was asked in [Reference Bou, Esteva, Godo and Rodríguez9] whether this was the case in general, or whether at least, this condition held for modal expansions of fuzzy logics. This was the case in the modal logics known up to now (e.g., in modal Gödel logics, and in the infinitary modal Łukasiewicz and Product logics studied in the literature). The question remained open in full generality over modal many-valued logics. We can give a negative answer to the problem, first by a simple counter-example over the modal expansions of Łukasiewicz logic (using the non-axiomatizability of proven in Theorem 4.8) and later proving that this is a more general fact.

First, it is possible to see that the local deduction is decidable using the version of Lemma 4.6 referring to the local logic. A detailed proof of the decidability of can be found in [Reference Vidal42, Corollary 4.5]. Thus, has a recursive axiomatization (for instance, built by enumerating all possible pairs with $\varGamma \cup \{ \varphi \} \subseteq _{\omega } Fm$ , and then returning the pairs for which ). On the other hand, if the global consequence were to coincide with the local one plus the $N_\square $ rule, the logic axiomatized by adding to the previous system the $N_\square $ rule should produce a recursive axiomatization of , contradicting Theorem 4.8.

As we said, it is possible to widen the scope of the previous result, and produce a constructive proof serving all modal logics built over classes of algebras like the ones in Theorem 3.1. This can be done following an approach different from the previous direct one working for the Łukasiewicz case, and instead providing a derivation that is valid in the global modal logics and not in the local ones extended by the necessitation rule.

For simplicity, let us fix a class of algebras $\mathbb {A}$ like the one from Section 3, and let $\vdash $ and $\vdash ^l$ denote $\vdash _{K_{\mathbb {A}}}$ and $\vdash ^l_{K_{\mathbb {A}}}$ respectively. Further, let $\vdash ^l_{N_\square }$ denote the logic $\vdash ^l$ plus the necessitation rule $x \vdash \square x$ . A natural way to understand this extension is by considering the (possibly non recursive) list of finite derivations valid in $\vdash ^l_{K_{\mathbb {A}}}$ and add to this set the rule schemata $N_\square $ . Let us call this set R. The minimal logic containing R, namely $R^l$ , is the logic $\vdash ^l_{N_\square }$ . Since all rules in R have finitely many premises, the resulting logic is finitary.

Considering R as a (possibly non recursive) axiomatization for $\vdash ^l_{N_\square }$ , all derivations valid in $\vdash ^l$ have a proof in the extended system of length $0$ . Thus, the length of the proofs in the extended system only reflects the applications of the necessitation rule. Since by definition $\vdash ^l$ is a finitary logic, only finitely many applications of the rule are used at each specific derivation. This means that a proof of $\varphi $ from a finite set of premises $\varGamma $ in this axiomatic system is given simply as a finite list of pairs such that

• • $\varGamma _0 = \varGamma $ and $\varphi _N = \varphi $ ,

• • For each , $\varGamma _i \vdash ^l \varphi _i$ ,

• • $\varGamma _{i+1} = \varGamma _i \cup \{\square \varphi _i\}$ .

From here, it is quite simple to prove the following characterization of $\vdash ^l_{N_\square }$ .

Lemma 5.1.

$$ \begin{align*}\varGamma \vdash^l_{N_\square} \varphi \text{ if and only if } \{\square^i\varGamma\}_{i \in \mathbb{N}}\vdash^l \varphi.\end{align*} $$

We can then produce a set of formulas that yields a valid derivation in the global logics, but it does not in the corresponding local logics plus necessitation.

Proof. We claim that both

which proves the theorem.

Regarding the first claim, consider an arbitrary Kripke model satisfying globally the set of premises. In particular, from $\neg \square \perp $ we get that each world in the model has a successor, and so, has infinite height in the sense of Definition 2.5. Moreover, the value of y is constant inside each connected part of the model, as in Lemma 3.3. Consider each connected submodel $\mathfrak {M}$ , and let $\alpha $ be the value of y in it. Then, at each point of the model, for all $i\in \mathbb {N}$ . Then, since the algebras in the class are weakly saturated, we get that $e(u,x) \alpha = e(u,x)$ , proving the formula in the right side.

In order to prove the second claim, let us denote by $\varSigma $ the set of premises. From the previous lemma we have that our claim holds if and only if

$$ \begin{align*}\{\square^i \varSigma\}_{i \in \mathbb{N}} \not \vdash^l x \rightarrow xy.\end{align*} $$

Since $\vdash ^l$ is finitary by definition, this holds if and only if for all $N \in \mathbb {N}$ . We can produce a counter-model for each $N \in \mathbb {N}$ .

Indeed, consider a model with universe

, and the accessibility relation given by

. Regarding the evaluation, pick an arbitrary $\mathbf {A} \in \mathbb {A}$ that is not $(N+1)$ -contractive, and pick $a \in \mathbf {A}$ such that $a^{N+2} < a^{N+1}$ . Then let

It is routine to check that this evaluation satisfies at the world $0$ , i.e., $e(0, \square ^i\varSigma ) = 1$ for all . On the other hand, observe that $e(0, x) = a^{N+1}$ . Due to the way we chose a it holds that $e(0, x y) = a^{N+2} < a^{N+1} = e(0,x)$ , thus falsifying the consequence.⊣

6 Acknowledgments

I am thankful to the anonymous referee for many useful and detailed contributions. This project has received funding from the following sources: the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101027914 and the Grant No. CZ.02.2.69/0.0/0.0/17050/0008361 of the Operational programme Research, Development, Education of the Ministry of Education, Youth and Sport of the Czech Republic. I also wish to thank Lluís Godo, Gavin St. John, and Tommaso Moraschini for their useful help.

7 Appendix

7.1 Craig’s Theorem

It is easy to see that a finitary R.E. logic with a definable idempotent n-ary operation for every n is always axiomatizable. This can be checked as it is done in Craig’s Theorem for classical logic.

Indeed, let $\star _n$ be an idempotent n-ary operation (i.e., such that

and

), as for instance, the classical conjunction $\wedge $ in classical logic. For convenience, denote

by $\star _n(\varphi )$ . Consider a recursive enumeration of the finitary derivations of the logic, i.e.,

(which exists because the logic is R.E), and take the axiomatization given by the set of rules

This is a recursive set: for a finitary rule

, it is first possible to decide whether there is some

(where $symb(\chi )$ denotes the number of symbols appearing in $\chi $ ) such that $\chi $ is of the form $\star _j(\varphi _j)$ and $\varSigma = \varGamma _j$ . If this is the case,

belongs to R. Otherwise, it is possible to decide whether $\varSigma $ is a singleton $\{\psi \}$ and, in this case, whether there is some

such that $\psi $ is of the form $\star _j(\varphi _j)$ and $\chi $ is exactly $\varphi _j$ . If the answer to both questions is yes,

belongs to R. Otherwise, the rule does not belong to R.

Further, it is clear that $\mathcal {L}\subseteq R^l$ . On the other hand, since the rules in R are finitary, it is easy to see (by means of the characterization of $R^l$ as the proofs in R) that $R^l \subseteq \mathcal {L}$ too.

7.2 Proof of Lemma 4.9

The proof of Lemma 4.9 detailed below draws inspiration from the results in [Reference Baaz, Hájek, Krajíček and Švejda3], and relies in the same isomorphic mappings introduced there. However, the approach and details are slightly different here, since we formulate alternative intermediate results, and we propose a more explicit proof using basic arithmetics.

Proof of Lemma 4.9.

(For every $x \not \in \mathcal {V}ar(\varGamma \cup \{\varphi \})$ , it holds that if and only if $\varGamma ^x, \varTheta ^x \vdash _{K\varPi } \varphi ^x$ .)

: Assume

. Then there is some standard Łukasiewicz Kripke model $\mathfrak {M}$ such that $\mathfrak {M} \models \varGamma $ but $\mathfrak {M},v \not \models \varphi $ for some v in the model. Chose an arbitrary $a \in (0,1)$ , and let us define a standard product model $\mathfrak {M}'$ by letting the universe and accessibility relations be those of $\mathfrak {M}$ , and further, for each $w \in W$ , letFootnote ¹⁸

Claim 1. For every formula $\psi $ in variables from $\mathcal {V}ar(\varGamma \cup \{\varphi \})$ , and for each $w \in W$ , it holds that

$$ \begin{align*}e'(w, \psi^x) = a^{1-e(w, \psi)}.\end{align*} $$

The conclusion easily follows, as we proceed to explain. Indeed, Claim 1 implies that $e'(w, \gamma ^x) = a^{1-e(w, \gamma )} = a^0 = 1$ for each $\gamma \in \varGamma $ and $w \in W$ . It is also clear that $e'(w, \varTheta ^x) = 1$ , since x is evaluated to the same element $a> 0$ in all worlds of $\mathfrak {M}'$ . On the other hand, $e'(v, \varphi ^x) = a^{1-e(v, \varphi )} <1$ , since $e(v, \varphi ) <1$ . Thereby, $\mathfrak {M}' \models \varGamma ^x, \varTheta ^x$ and $\mathfrak {M}',v \not \models \varphi ^x$ , and so, $\varGamma ^x, \varTheta ^x \not \vdash _{K\varPi } \varphi ^x$ . Then, it only remains to prove the claim.

Proof of Claim 1. It can be proven by induction on the complexity of the formula.

• • For variables it is straightforward from the definition, since $e'(w,q^x) = e'(w,q \vee x) = a \vee a^{1-e(w,q)}$ , and for every $q \in [0,1]$ .

• • For $\psi = \psi _1 \cdot \psi _2$ , we have the following chain of equalities:

  

• • For $\psi = \psi _1 \rightarrow \psi _2$ , we have the following chain of equalities:

  

• • For $\psi = \square \psi _1$ , we know that

  $$ \begin{align*} e'(w, (\square \psi_1)^x) & = e'(w, \square \psi_1^x) = \bigwedge_{Rwu} e'(u, \psi_1^x) \overset{I.H}{=} \bigwedge_{Rwu} a^{1-e(u, \psi_1)}. \end{align*} $$
  On the one hand, since for each u with $Rwu$ , it holds that .

  On the other hand,

  
  This concludes the proof of the claim.⊣

$\Rightarrow $ : It is proven similarly, using the corresponding inverse of the isomorphism from [Reference Baaz, Hájek, Krajíček and Švejda3]. Assume there is a standard product model $\mathfrak {P}$ such that $\mathfrak {P} \models \varGamma ^x, \varTheta ^x$ and $\mathfrak {P},v \not \models \varphi ^x$ for some v in the model. As proven in Lemma 3.3, there is some element a such that $e(w,x) = a$ for all w in the universe of the model. Moreover, since $e(v, \neg \neg x) = 1$ , necessarily $a> 0$ .

Observe first that if $a = 1$ then $e(w,\psi ^x) = 1$ for every $\psi $ with variables in $\mathcal {V}ar(\varGamma \cup \{\varphi \})$ and each w in the universe. This observation is immediate by induction on the complexity of the formula. Since this would contradict the fact that $e(v, \varphi ^x) < 1$ , necessarily $a < 1$ .

Let us define a standard Łukasiewicz model $\mathfrak {P}'$ as the model whose universe and accessibility relation are those of $\mathfrak {P}$ and for each variable q and each world w let .

Claim 2. For every formula $\psi $ in variables from $\mathcal {V}ar(\varGamma \cup \{\varphi \})$ and for each $w \in W$ it holds that

$$ \begin{align*}e'(w, \psi) = 1-log_a e(w, \psi^x).\end{align*} $$

As in the previous case, the conclusion easily follows. Indeed, If Claim 2 holds, $e'(w, \gamma ) = 1-log_a e(w, \gamma ^x) = 1 - log_a 1 = 1$ for each $\gamma \in \varGamma $ and $w \in W$ , and $e'(v, \varphi ) = 1-log_a e(v, \varphi ^x) = 1 - log_a \alpha $ for some . Since the logarithm in base a of elements in that interval is a value in $(0,1]$ , necessarily $e'(v, \varphi ) < 1$ , concluding the proof of the lemma. Then, it only remains to prove the claim.

Proof of Claim 2. We will prove it by induction on the complexity of the formula. Observe a consequence of Claim 2 is that $e(w, \psi ^x) \geqslant a$ for all $\psi $ and w as before.Footnote ¹⁹ We will (inductively) use this property in the modal step, and refer to it by $I.H'$ .

• • For variables it is immediate, since $e'(w, q) = 1-log_a a \vee e(w, q) = 1-log_a e(w, q^x)$ .

• • For $\psi = \psi _1 \cdot \psi _2$ , we have the following chain of equalities: .

  Now, if $0 < e(w, \psi _1^x \cdot \psi _2^x) < a$ , it holds that $ log_a(e(w, \psi _1^x \cdot \psi _2^x))> 1$ , and thus, $max\{0, 1 - log_a(e(w, \psi _1^x \cdot \psi _2^x))\} = 0 = 1 - log_a(e(w, \psi _1^x \cdot \psi _2^x)\vee a)$ . It follows that

  $$ \begin{align*}max\{0, 1 - log_a(e(w, \psi_1^x \cdot \psi_2^x))\} & = 1-log_a(e(w, \psi_1^x \cdot \psi_2^x)\vee a) \\[3pt] & = 1 - log_a e(w, (\psi_1 \cdot \psi_2)^x).\end{align*} $$

• • For $\psi = \psi _1 \rightarrow \psi _2$ , we have the following chain of equalities:

  $$ \begin{align*} & e'(w, \psi_1 \rightarrow \psi_2) = min\{1, 1 e'(w, \psi_1) + e'(w, \psi_2)\}\\[3pt] & \overset{I.H} = min\{1, 1 - (1-log_a e(w, \psi_1^x)) + 1- log_a e(w, \psi_2^x)\}\\[3pt] &= min\{1, 1 - (log_a e(w, \psi_2^x) - log_a e(w, \psi_1^x))\}\\[3pt] &= min\{1, 1 - log_a e(w, \psi_2^x)/e(w, \psi_1^x)\} \\[3pt] &= 1-log_a(min\{1, e(w, \psi_2^x)/e(w, \psi_1^x)\})\\[3pt] &= 1 - log_a e(w, \psi_1^x \rightarrow \psi_2^x)\\[3pt] &= 1 - log_a e(w, (\psi_1 \rightarrow \psi_2)^x). \end{align*} $$

• • For $\psi = \square \psi _1$ , we have that $e'(w, \square \psi _1) = \bigwedge _{Rwv}e'(v,\psi _1) \overset {I.H}{=} \bigwedge _{Rwv} (1-log_a e(v, \psi _1^x)) = 1 - \bigvee _{Rwv}log_a e(v, \psi _1^x)$ . Now, by $I.H'$ , we know that $e(v, \psi _1^x) \geqslant a$ for all v in the model, so in particular $\bigwedge _{Rwv} e(v, \psi _1^x) \in [a,1]$ . Since the function $log_a()$ is continuous and decreasing in $[a, 1]$ , it follows that $1 - \bigvee _{Rwv}log_a e(v, \psi _1^x) = 1 - log_a \bigwedge _{Rwv} e(v, \psi _1^x) = 1 - log_a e(w, \square \psi _1^x) = 1 - log_a e(w, (\square \psi _1)^x)$ .⊣