TY - JOUR
AU - Figallo,, Martín
AB - Abstract Involutive Stone algebras (or S-algebras) were introduced by R. Cignoli and M. Sagastume in connection to the theory of |$n$|-valued Łukasiewicz–Moisil algebras. In this work we focus on the logic that preserves degrees of truth associated to S-algebras named Six. This follows a very general pattern that can be considered for any class of truth structure endowed with an ordering relation, and which intends to exploit many-valuedness focusing on the notion of inference that results from preserving lower bounds of truth values, and hence not only preserving the value |$1$|⁠. Among other things, we prove that Six is a many-valued logic (with six truth values) that can be determined by a finite number of matrices (four matrices). Besides, we show that Six is a paraconsistent logic. Moreover, we prove that it is a genuine Logic of Formal Inconsistency with a consistency operator that can be defined in terms of the original set of connectives. Finally, we study the proof theory of Six providing a Gentzen calculus for it, which is sound and complete with respect to the logic. 1 Introduction Let |$A$| be a De Morgan algebra. Denote by |$K(A)$| the set of all elements |$a\in A$| such that the De Morgan negation |$\neg a$| of |$a$| coincides with the complement of |$a$|⁠. For every |$a\in A$|⁠, let |$K_a=\{k\in K(A): a\leq k\}$| and, if |$K_a$| has a least element, denote it by |$\nabla a$| the least element of |$K_a$|⁠. The class of all De Morgan algebras |$A$| such that for every |$a\in A$|⁠, |$\nabla a$| exists, and the map \begin{equation*}a \mapsto \nabla a\end{equation*} is a lattice homomorphism called the class of involutive Stone algebras (S-algebras), denoted by |$\textbf{S}$|⁠. These algebras were introduced by Cignoli and de Gallego [5] in connection to the theory of |$n$|-valued Łukasiewicz–Moisil algebras. Most logics studied in the literature in association with a given class of algebras are defined by taking 1 as the only truth value to be preserved by inference. These truth-preserving logics do not take full advantage of being many-valued, as they focus on the truth value 1 (the truth) and not on other intermediate truth values. A different approach to preserving only truth is going beyond this by considering the notion of inference that results from preserving lower bounds of truth values, and hence not only preserving the value 1. In this setting the language remains the same as in truth-preserving logics, and what changes is the inference relation. This kind of inference corresponds to the so-called logics preserving degrees of truth discussed at length and follows a very general pattern which could be considered for any class of truth structures endowed with an ordering relation. The study of logics that preserves degrees of truth goes back to Wójcicki in his 1988 book [17], in the context of Łukasiewicz logic and then expanded in [1, 12–14], among others. Regarding these logics and their relation to paraconsistency, it is worth citing [4, 7, 11]. In this work, we focus on the logic |$\mathbb{L}_{\boldsymbol{S}}^{\leq }$| that preserves degrees of truth associated to S-algebras. Among other things, we prove that |$\mathbb{L}_{\textbf{S}}^{\leq }$| is a many-valued logic (with six truth values) that can be determined by a finite number of matrices (four matrices). Besides, we show that |$\mathbb{L}_{\boldsymbol{S}}^{\leq }$| is a paraconsistent logic. Moreover, we prove that it is a genuine Logic of Formal Inconsistency (LFI; [3]) with a consistency operator that can be defined in terms of the original set of connectives. Finally, we study the proof theory of |$\mathbb{L}_{\boldsymbol{S}}^{\leq }$| providing a Gentzen calculus for it, which is sound and complete with respect to the intended semantics. 2 Preliminaries In this section, we shall recall some well-known notions and results in order to make this work self-contained. Recall that a De Morgan algebra is a structure |$\langle A, \wedge , \vee , \neg , 0, 1 \rangle $| of type |$(2,2,1,0,0)$| such that the reduct |$\langle A, \wedge , \vee , 0, 1 \rangle $| is a bounded distributive lattice and |$\neg $| satisfies the identities (DM1) |$\neg \neg x \approx x$|⁠, (DM2) |$\neg (x \wedge y) \approx \neg x \vee \neg y$|⁠. Note that it follows that |$\neg (x \vee y) = \neg x \wedge \neg y$|⁠, |$\neg 0 = 1$| and |$\neg 1 = 0$|⁠. We denote by |$\bf M$| the variety of all De Morgan algebras. Let |$A$| be a De Morgan algebra. A set |$F\subseteq A$| is a lattice filter of |$A$| iff (a) |$1\in F$|⁠, (b) |$x\in F$| and |$x \leq y$| imply |$y\in F$| and (c) |$x, y \in F$| implies |$x\wedge y \in F$|⁠. By |${\cal F}i(A)$| we denote the family of all lattice filters of |$A$|⁠. Let |$A\in \textbf{M}$|⁠, |$a \in A$| is said to be complemented if there exists |$b\in A$|⁠, |$a \vee b = 1$| and |$a \wedge b = 0$|⁠, |$b$| is the complement of |$a$| and it is denoted |$a^{\prime }$|⁠. If the complement of an element |$a$| exists this is unique. We denote by |$B(A)$| the center of|$A$|⁠, that is, the set of all complemented elements of |$A$|⁠. It can be seen that |$B(A)$| is an M-subalgebra of |$A$|⁠. Let |$K(A)$| be the subset of |$A$| formed by all elements |$a \in B(A)$| such that the De Morgan negation |$\neg a$| coincides with the complement |$a^{\prime }$|⁠. That is, \begin{equation*}K(A)=\{a\in B(A): \neg a = a^{\prime}\}.\end{equation*} It is easy to check that |$K(A)$| is an M-subalgebra of |$B(A)$|⁠. For every |$a\in A$|⁠, let |$K_a =\{k \in K(A): a\leq k \}$|⁠. If |$K_a$| has a least element, we denote it by |$\nabla a$| as in [5]. If for every |$a\in A$|⁠, |$K_a$| has a least element, it is said that |$A$| is a |$\nabla $|-De Morgan algebra. The class of S-algebras, denoted by S, is the subclass of |$\nabla $|-De Morgan algebras with the property that the map |$a \mapsto \nabla a$| from |$A$| into |$K(A)$| is a lattice homomorphism. Theorem 2.1 (Cignoli and de Gallego [6]). |$\bf S$| is an equational class. Indeed, a De Morgan algebra |$A \in \textbf{S}$| if and only if there is an operator |$\nabla : A\to A$| satisfying the following equations: (IS1) |$\nabla 0 \approx 0$|⁠, (IS2) |$a \wedge \nabla a \approx a$|⁠, (IS3) |$\nabla (a \wedge b) \approx \nabla a \wedge \nabla b$|⁠, (IS4) |$\neg \nabla a \wedge \nabla a \approx 0$|⁠. If |$A\in \textbf{S}$|⁠, then |$a^\star = \neg \nabla a$| is the pseudocomplement of |$A$| (see [16]) and |$a^{\star \star }=\neg \nabla \neg \nabla a =\nabla a$|⁠. Since |$a^{\star } \vee a^{\star \star } = 1$|⁠, |$A$| is a Stone lattice. The dual pseudocomplement |$a^{+}$| of |$a$| also exists, in fact, |$a^{+}= \nabla \neg a$| and therefore |$A$| is also a dual Stone lattice. From this, we have that S is a subclass of the class of double Stone algebras (see [6]). It is well known that in a Stone lattice |$A$|⁠, |$a\in B(A)$| iff |$a=a^{\star \star }$|⁠. Consequently, if |$A$| is a De Morgan algebra that is also a Stone lattice and, if we define |$\nabla a=a^{\star \star }$|⁠, we have that the algebra |$\langle A,\vee ,\wedge ,\neg ,\nabla ,0,1\rangle \in \textbf{S}$| iff |$a^{\star }=\neg (a^{\star \star })$| for each |$a\in A$| iff |$B(A)=K(A)$| (cf. [6]). Motivated by this, the members of S were called S-algebras. Some examples of S-algebras are the following. Examples 2.2 (i) Every Boolean algebra |$\langle A, \wedge , \vee , \neg , 0,1\rangle $| admits a structure of S-algebra. Indeed, taking |$\nabla x =x$| we have that |$\langle A, \wedge , \vee , \neg , \nabla , 0,1\rangle $| is an S-algebra. (ii) Let |$\langle A, \wedge , \vee , \sim , \sigma ^n_1,\ldots ,\sigma ^n_{n-1},0,1\rangle $| be an |$n$|-valued Łukasiewicz–Moisil algebra. Then the reduct |$\langle A, \wedge , \vee , \sim , \sigma ^n_{n-1},0,1 \rangle $| is an S-algebra [5]. In particular, when |$n=3$| we have that |$\langle A, \wedge , \vee , \sim , \sigma ^3_2,0,1 \rangle \in \textbf{S}$| and that the operator |$\sigma ^3_2$| satisfies the relation |$a \wedge \sigma ^3_2 \neg a \leq b \vee \neg \sigma ^3_2 b$| for each |$a,b \in A$|⁠. On the other hand, if |$A\in \textbf{S}$| and the relation |$a\wedge \nabla \neg a \leq b\vee \neg \nabla b$| holds in |$A$|⁠, then |$\langle A,\vee , \wedge , \neg ,\varDelta ,\nabla ,0,1\rangle $| is a three-valued Łukasiewicz algebra, where |$\varDelta =\neg \nabla \neg $|⁠. Therefore, (Cignoli et al.) the class of three-valued Łukasiewicz algebras coincides with the class of S-algebras that satisfies the equation |$a\wedge \nabla \neg a \leq b\vee \neg \nabla b$|⁠. (iii) For each |$n\geq 2$|⁠, |$\mathbf{{^{\tiny{\not}}\kern-1.5pt{\textrm{L}}} }_{n}$| will denote the |$n$|-element chain |$0 < \frac{1}{n-1} < \ldots < \frac{n-2}{n-1} < 1$| with the well-known lattice structure and |$\neg (\frac{i}{n-1}) = \frac{n-i-1}{n-1}$|⁠, where |$i=0,1,\ldots ,n-1$|⁠. The algebras |$\mathbf{{^{\tiny{\not}}\kern-1.5pt{\textrm{L}}} }_{n}$| are examples of S-algebras, where |$\nabla x= 1$|⁠, for all |$x\neq 0$| and |$\nabla 0=0$|⁠. (iv) The following is another important example of an S-algebra, as we shall see. Let |$\mathbb{S}_6=\{0, \frac{1}{3}, N, B, \frac{2}{3}, 1\}$| with the lattice structure given by the negation given by |$\neg 0 =1$|⁠, |$\neg 1 =0$|⁠, |$\neg N =N$|⁠, |$\neg B =B$|⁠, |$\neg \frac{1}{3} =\frac{2}{3}$| and |$\neg \frac{2}{3}=\frac{1}{3}$|⁠. Besides, |$\nabla x= 1$|⁠, for all |$x\not = 0$| and |$\nabla 0=0$|⁠. Remark 2.3 The operator |$\nabla $| enjoys well-known modal properties such as the necessitation rule. In [6], the authors extended the well-known Priestley’s duality for De Morgan algebras (by Cornish and Fowler) to S-algebras. Using this duality, they determined the subdirectly irreducible and simple S-algebras. In this work we focus on the logic that preserves degrees of truth associated to S-algebras. This follows a very general pattern that can be considered for any class of truth structure endowed with an ordering relation, and which intends to exploit many-valuedness focusing on the notion of inference that results from preserving lower bounds of truth values, and hence not only preserving the value |$1$| (see [12–14]). More precisely, if |$K$| is a class of ordered algebras and |$\mathfrak{Fm}$| is the absolutely free algebra of the same type of |$K$|⁠, the notion of inference |$\models ^{\leq }_{K}$| that results from preserving lower bounds of truth values gives rise to the following definition: for all |$\varGamma \cup \{\alpha \}\subseteq Fm$||$\varGamma \models ^{\leq }_{K} \alpha $| iff |$\,\forall A\in \textbf{K}$|⁠, |$\forall v\in Hom_{\textbf{K}}(\mathfrak{Fm},A)$|⁠, |$\forall a\in A$|⁠, if |$v(\alpha _i)\geq a$|⁠, for all |$i\leq n$|⁠, then |$v(\alpha )\geq a$|⁠. 3 The semantic models: S-algebras In this section, we shall exhibit the structure of our semantic models as well as their most important properties. Proposition 3.1 If |$A$| is an S-algebra then (i) |$\nabla A= K(A)$|⁠, (ii) |$x\in \nabla A$| iff |$\,x = \nabla x$|⁠, (iii) |$\nabla (x \vee y)= \nabla x \vee \nabla y$|⁠. Proof. It is a consequence of Theorem 2.1. Then, |$\nabla $| satisfies the following properties. Lemma 3.2 In every S-algebra |$A$| the following identities hold: \begin{align*} (IS5)\phantom{1} &\quad \nabla 1 \approx 1, & (IS6)\phantom{1} & \quad\neg x \vee \nabla x \approx 1, & (IS7) &\quad \nabla \nabla x \approx \nabla x, \\ (IS8)\phantom{1} &\quad \nabla \neg \nabla x \approx \neg \nabla x, & (IS9)\phantom{1} & \quad\nabla( x \vee \neg x) \approx 1,& &\\ (IS10) & \quad x \wedge \neg \nabla x \approx 0,& (IS11) & \quad\nabla(x \vee \nabla y) \approx \nabla x \vee \nabla y.& & \end{align*} Proof. It is a consequence of Theorem 2.1. As usual, we can define the operator |$\varDelta $| as follows: \begin{equation*} \varDelta a =_{def.} \neg \nabla \neg a. \end{equation*} From this definition and Proposition 3.1 (i) we have that |$\varDelta A= \nabla A= K(A)$|⁠. Besides, Lemma 3.3 In every S-algebra |$A$| the following identities hold: \begin{align*} (IS12) & \quad\Delta x \wedge x \approx \Delta x, & (IS13) & \quad\Delta \nabla x \approx \nabla x,\\ (IS14) & \quad\nabla \Delta x \approx \Delta x, & (IS15) &\quad \Delta(x \wedge \neg x)\approx 0, \\ (IS16) & \quad\Delta(x \vee y) \approx \Delta x \vee \Delta y,& (IS17) & \quad\Delta(x \wedge y) \approx \Delta x \wedge \Delta y. \end{align*} Proof. It is a consequence of Theorem 2.1. Taking advantage of the topological duality developed in [6], the authors proved the following. Lemma 3.4 (Cignoli and de Gallego [6]) The subdirectly irreducible algebras of S are |$\mathbf{{^{\tiny{\not}}\kern-1.5pt{\textrm{L}}} }_n$|⁠, for |$2\leq n\leq 5$|⁠, and |$\mathbb{S}_6$|⁠. The simple algebras of S are |$\mathbf{{^{\tiny{\not}}\kern-1.5pt{\textrm{L}}} }_2$| and |$\mathbf{{^{\tiny{\not}}\kern-1.5pt{\textrm{L}}} }_3$|⁠. From the fact that the algebras |$\mathbf{{^{\tiny{\not}}\kern-1.5pt{\textrm{L}}} }_n$|⁠, for |$2\leq n\leq 5,$| are S-subalgebras of |$\mathbb{S}_6$| we have the following. Corollary 3.5 |$\mathbb{S}_6$| generates the variety S. This last result indicates that a given equation holds in every S-algebra iff it holds in |$\mathbb{S}_6$|⁠. This will be very useful in what follows. Remark 3.6 Belnap’s four-valued logic |$FOUR$| ([3, 4]) is a logical system that is well known for many applications it has found in several fields. This logic intends to model the following situation: faced with a situation where one has several conflicting pieces of information on the truth of a sentence, or where one has no information about it, the classical truth values (true and false) must be treated as being mutually independent, thus giving birth to four non-classical epistemic values: 1 (true and not false), 0 (false and not true), N (neither true nor false) and B (both true and false). This gives rise to the well-known four-element De Morgan algebra |$\mathfrak{B}_4$|⁠, where |$\neg 0=1$|⁠, |$\neg 1 =0$|⁠, |$\neg N=N$| and |$\neg B=B$|⁠. On the other hand, many-valued logics (Łukasiewicz |$n$|-valued logics, for instance) intend to express the existence of various degrees of truth. Since |$\mathbf{{^{\tiny{\not}}\kern-1.5pt{\textrm{L}}} }_4$| and |$\mathfrak{B}_4$| are both M-subalgebras of |$\mathbb{S}_6$|⁠, we think that any logic based on this particular algebra would combine these two ideas. In the next section we shall study one of them. The lattice |$\mathfrak{B}_4$| gives rise to another well-known logic, namely, the tetravalent modal logic|${\mathcal{TML}}$| which was studied in depth in [15] and [8]. 4 The logic that preserves degrees of truth with respect to S We now focus on the logic that preserves degrees of truth |$\mathbb{L}_{\boldsymbol{S}}^{\leq }$| associated to S-algebras. Let |$\mathfrak{Fm}=\langle Fm, \wedge , \vee , \neg , \nabla , \bot , \top \rangle $| be the absolutely free algebra of type |$(2,2,1,1,0,0)$| generated by some fixed denumerable set of propositional variables |$Var$|⁠. As usual, the letters |$p,q, \dots $| denote propositional variables, the letters |$\alpha ,\beta , \dots $| denote formulas and |$\varGamma ,\varSigma , \dots $| denote sets of formulas. If |$A$| and |$B$| are to algebras of type |$(2,2,1,1,0,0)$| (not necessarily S-algebras), we denote by |$Hom(A,B)$| the set of all homomorphisms from |$A$| into |$B$|⁠. Definition 4.1 The logic that preserves degrees of truth over the variety |$\textbf{S}$| is |$\mathbb{L}^{\leq }_{\boldsymbol{S}}=\langle Fm, \models ^{\leq }_{\boldsymbol{S}}\rangle ,$| where for every set of formulas |$\varGamma \cup \{\alpha \}\subseteq Fm$| (i) if |$\varGamma =\{\alpha _1,\dots , \alpha_{n}\}\neq \emptyset $|⁠, $$\begin{array}{ccc} \qquad\alpha_{1},\dots, \alpha_{n} \models^{\leq}_\boldsymbol{S} \alpha & \textrm{iff} & \forall A\in\textbf{S}, \forall v\in Hom(\mathfrak{Fm},A), \forall a\in A\\ & & \textrm{if}\ v(\alpha_i)\geq a, \textrm{for all} \ i\leq n,\\ & & \textrm{then} \ v(\alpha)\geq a \end{array}$$ (ii) |$\emptyset \models ^{\leq }_{\boldsymbol{S}} \alpha $| iff |$\,\forall A\in \textbf{S}$|⁠, |$\forall v\in Hom(\mathfrak{Fm},A)$|⁠, |$v(\alpha )=1$|⁠; (iii) if |$\varGamma \subseteq Fm$| is non-finite, $$\begin{array}{ccc} \qquad\varGamma \models^{\leq}_\boldsymbol{S} \alpha & \textrm{iff} & \textrm{there are}\ \alpha_1,\dots, \alpha_n \in \varGamma \ \textrm{such that} \\ & & \alpha_1,\dots, \alpha_n \models^{\leq}_\textbf{S} \alpha. \end{array}$$ From the above definition we have that |$\mathbb{L}^{\leq }_{\textbf{S}}$| is a sentential logic, that is, |$\models ^{\leq }_{\textbf{S}}$| is a finitary consequence relation over |$Fm$|⁠. Proposition 4.2 Let |$\{\alpha _1,\dots , \alpha _n, \alpha \}\subseteq Fm$|⁠, |$n\geq 1$|⁠. Then, the following conditions are equivalent: (i) |$\alpha _1,\dots , \alpha _n \models ^{\leq }_{\boldsymbol{S}} \alpha $|⁠, (ii) |$\alpha _1 \wedge \dots \wedge \alpha _n \models ^{\leq }_{\boldsymbol{S}} \alpha $|⁠, (iii) |$\textbf{S} \models \alpha _1 \wedge \dots \wedge \alpha _n \preccurlyeq \alpha $|⁠, that is, the inequality |$\alpha _1 \wedge \dots \wedge \alpha _n \preccurlyeq \alpha $| holds in the variety S. Proof. (i)|$\Rightarrow $|(ii) Let |$\{\alpha _1,\dots , \alpha _n, \alpha \}\subseteq Fm$|⁠, |$n\geq 1$|⁠. Suppose that |$\alpha _1,\dots , \alpha _n \models ^{\leq }_{\textbf{S}} \alpha $|⁠. Then |$\forall A\in \textbf{S}$|⁠, |$\forall v\in Hom(\mathfrak{Fm},A)$|⁠, |$\forall a\in A$|⁠, if |$v(\alpha _i)\geq a$|⁠, for all |$i\leq n$|⁠, then |$v(\alpha )\geq a$|⁠. Since |$A$| has a lattice structure, we have that |$v(\alpha _i)\geq a$|⁠, for all |$i\leq n$| iff |$\bigwedge _{i=1}^{n}v(\alpha _i)\geq a$| iff |$v\left (\bigwedge _{i=1}^{n}(\alpha _i)\right )\geq a$|⁠. Then if |$v\left (\bigwedge _{i=1}^{n}(\alpha _i)\right )\geq a$| we have |$v(\alpha )\geq a$|⁠; that is, |$\alpha _1 \wedge \dots \wedge \alpha _n \preccurlyeq \alpha $| holds in S. Analogously, it is proved (ii)|$\Rightarrow $|(i). Finally, (iii) is just a different way of writing (ii), assuming (i). Remark 4.3 The equivalence (i)|$\Leftrightarrow $|(ii) says that |$\mathbb{L}^{\leq }_{\boldsymbol{S}}$| is a conjunctive logic. On the other hand, (ii)|$\Leftrightarrow $|(iii) expresses that the consequence relation |$\models ^{\leq }_{\boldsymbol{S}}$| internalizes the order in S. As a consequence of Proposition 4.2 we have the following. Corollary 4.4 Let |$\alpha ,\beta \in Fm$|⁠. Then (i) |$\alpha \models ^{\leq }_{\boldsymbol{S}} \beta $| iff |$\textbf{S} \models \alpha \preccurlyeq \beta $|⁠, (ii) |$\alpha \models ^{\leq }_{\boldsymbol{S}} \beta $| iff |$\textbf{S} \models \alpha \approx \beta $|⁠. Proof. (i) Immediate taking |$n=1$|⁠. (ii) Immediate from (i). From Corollary 3.5 we have the following. Proposition 4.5 For every |$\varGamma \cup \{\alpha \} \subseteq Fm$|⁠, |$\varGamma \models ^{\leq }_{\textbf{S}} \alpha $| if and only if for every |$h \in {Hom}(\mathfrak{Fm}, \mathbb{S}_{6})$|⁠, |$\bigwedge \{ h(\gamma ) : \ \gamma \in \varGamma \} \leq h(\alpha )$|⁠. In particular, |$\emptyset \models ^{\leq }_{\boldsymbol{S}} \alpha $| if and only if |$h(\alpha )=1$| for all |$h \in {Hom}(\mathfrak{Fm}, \mathbb{S}_{6})$|⁠. 5 |${\mathbb{L}}^{\leq }_{\boldsymbol{S}}$| as a matrix logic In this section we shall see that |$\mathbb{L}^{\leq }_{\boldsymbol{S}}$| can be characterized in terms of matrices and generalized matrices (g-matrices). Recall that a logic matrix (or simply a matrix) is a pair |$\langle A, F\rangle $| where |$A$| is an algebra and |$F$| is a non-empty subset of |$A$|⁠. A generalized matrix is a pair |$\langle A,{\mathcal{C}}\rangle $| where |$A$| is an algebra and |${\mathcal{C}}$| is a family of subsets of |$A$| which constitutes a closed set system of subsets of |$A$|⁠. That is, |${\mathcal{C}}$| verifies the following: (C1) |$A\in{\mathcal{C}}$|⁠, (C2) If |$\{X_i\}_{i\in I}\subseteq{\mathcal{C}}$| then |$\bigcup _{i\in I} X_i \in{\mathcal{C}}$|⁠. Let |$M=\langle A, F\rangle $| be a matrix. The logic |$\mathbb{L}_{M}=\langle \mathfrak{Fm}, \models _{M}\rangle $| determined by |$M$| is defined as follows: let |$\varGamma \cup \{\alpha \}\subseteq Fm$| $$\begin{array}{ccc} \varGamma \models_{M} \alpha & \textrm{iff} & \forall v\in Hom(\mathfrak{Fm},A)\\ & & \textrm{if}\ v(\gamma) \in F, \textrm{for all}\ \gamma\in \varGamma, \textrm{then}\ v(\alpha)\in F.\end{array}$$ If |${\mathcal{F}}=\{M_i\}_{i\in I}$| is a family of matrices, then the logic |$\mathbb{L}_{\mathcal{F}}=\langle \mathfrak{Fm}, \models _{\mathcal{F}}\rangle $| determined by the family |${\mathcal{F}}$| is the defined by |$\models _{\mathcal{F}} = \bigcap \limits _{i\in I} \models _{M_i}$|⁠. On the other hand, the logic determined by a g-matrix |$G=\langle A,{\mathcal{C}}\rangle $| is |$\mathbb{L}_{G}=\langle \mathfrak{Fm}, \models _{G}\rangle $|⁠, where let |$\varGamma \cup \{\alpha \}\subseteq Fm$| $$\begin{array}{ccc} \varGamma \models_{G} \alpha & \textrm{iff} & \forall v\in Hom(\mathfrak{Fm},A), \forall F\in{\mathcal{C}}\\ & & \textrm{if} \ v(\gamma) \in F, \textrm{for all}\ \gamma\in \varGamma, \textrm{then} \ v(\alpha)\in F.\end{array}$$ The logic determined by a family of g-matrices is defined in a similar way to the logic determined by a family of matrices. Now, consider the following families of matrices and g-matrices and their associated logics. Let \begin{equation*} {\mathcal{F}}=\{\langle A, F\rangle: A\in\textbf{S}\ \textrm{and}\ F\, \textrm{is a lattice filter of}\ A \}\end{equation*} and \begin{equation*}{\mathcal{F}}_g=\{\langle A,{\mathcal{F}}i(A)\rangle: A\in\textbf{S}\},\end{equation*} where |${\mathcal{F}}i(A)$| is the family of all lattice filters of |$A$|⁠. Then, Theorem 5.1 The logic |$\mathbb{L}^{\leq }_{\boldsymbol{S}}$| coincides with the logic |$\mathbb{L}_{\mathcal{F}}$| and also with the logic |$\mathbb{L}_{{\mathcal{F}}_g}$|⁠. Proof. The fact that |$\mathbb{L}_{\mathcal{F}}$| and |$\mathbb{L}_{{\mathcal{F}}_g}$| coincide is a direct consequence of the definition of logic determined by a family of matrices and g-matrices, respectively. Besides, since |$\textbf{S}$| is a variety and the notion of filter is elementary definable, the class |${\mathcal{F}}_g$| is closed under ultraproducts. Then |$\mathbb{L}_{{\mathcal{F}}_g}$| is finitary. Then to prove that |$\mathbb{L}_{{\mathcal{F}}_g}$| and |$\mathbb{L}^{\leq }_{\boldsymbol{S}}$| coincide we just must prove it using a finite number of premises. Let |$\alpha _1,\dots ,\alpha _n,\alpha \in Fm$| such that |$\alpha _1,\dots , \alpha _n \models ^{\leq }_{\boldsymbol{S}} \alpha $|⁠. Then by Lemma 4.1, for all |$v\in Hom(Fm,A)$|⁠, (1) |$v(\alpha _1) \wedge \dots \wedge (\alpha _n) \leq v(\alpha )$|⁠. Let |$F\in Fi(A)$| and suppose that |$v(\alpha _1), \dots ,v(\alpha _n) \in F$|⁠, then (by the definition of filter) we have |$v(\alpha _1)\wedge \dots \wedge v(\alpha _n) \in F$|⁠. By (1) and |$F$| filter, |$v(\alpha )\in F$|⁠. Then |$\alpha _1,\dots , \alpha _n \models _{{\mathcal{F}}_g} \alpha $|⁠. Conversely, suppose that (2) |$\alpha _1,\dots , \alpha _n \models _{{\mathcal{F}}_g} \alpha $| and let |$v\in Hom(Fm,A)$|⁠. Consider the filter generated by |$v(\alpha _1), \dots , v(\alpha _n)$|⁠, |$F=Fi(v(\alpha _1), \dots , v(\alpha _n))$|⁠. Clearly |$v(\alpha _i)\in F$| for all |$i\leq n$|⁠, and then by (2), |$v(\alpha )\in F$|⁠. Then, by properties of generated filters, |$v(\alpha _1)\wedge \dots \wedge v(\alpha _n)\leq v(\alpha )$|⁠. Therefore, by Lemma 4.1, |$\alpha _1,\dots , \alpha _n \models ^{\leq }_{\boldsymbol{S}} \alpha $|⁠. On the other hand, the filters of |$\mathbb{S}_6$| are precisely the principal filters generated by each element of |$\mathbb{S}_6$|⁠. Then |$\mathcal{F}i(\mathbb{S}_6)=\{[0), [\frac{1}{3}), [N), [B), [\frac{2}{3}), [1)\}$|⁠. Now, consider the following family of matrices \begin{equation*}{\mathcal{F}}^{6}=\Biggl\{\!\langle \mathbb{S}_6, [0)\rangle, \bigg\langle \mathbb{S}_6, \left[\frac{1}{3}\right)\!\bigg\rangle, \langle \mathbb{S}_6, [N)\rangle, \langle \mathbb{S}_6, [B)\rangle, \bigg\langle \mathbb{S}_6, \left[\frac{2}{3}\right)\!\bigg\rangle, \langle \mathbb{S}_6, [1)\rangle\! \Biggr\}\end{equation*} and the generalized matrix \begin{equation*}{\mathcal{F}}^{6}_g=\langle\mathbb{S}_6,{\mathcal{F}}i(\mathbb{S}_6)\rangle.\end{equation*} Theorem 5.2 The logics |$\mathbb{L}_{{\mathcal{F}}^{6}}$|⁠, |$\mathbb{L}_{{\mathcal{F}}^{6}_g}$| and |$\mathbb{L}^{\leq }_{\boldsymbol{S}}$| coincide. Proof. Since the logic determined by a finite set of matrices is finitary, again we shall prove that they coincide using a finite number of premises. Besides, it is clear that |$\mathbb{L}_{{\mathcal{F}}^{6}}$| and |$\mathbb{L}_{{\mathcal{F}}^{6}_g}$| coincide. On the other hand, let |$\alpha _1,\dots ,\alpha _n,\alpha \in Fm$| such that |$\alpha _1,\dots , \alpha _n \models ^{\leq }_{\boldsymbol{S}} \alpha $|⁠. By Proposition 4.5, this is equivalent to |$\mathbb{S}_6 \models \alpha _1 \wedge \dots \wedge \alpha _n \preccurlyeq \alpha $|⁠, that is, for all |$h \in Hom(\mathfrak{Fm}, \mathbb{S}_{6})$|⁠, |$h(\alpha _1)\wedge \dots \wedge h(\alpha _n)\leq h(\alpha )$|⁠. Then, by the same properties of filters used before, we have |$\alpha _1,\dots , \alpha _n \models _{{\mathcal{F}}^{6}_g} \alpha $|⁠. The converse is analogous to the converse showed in the proof of Theorem 5.1. So far, we have proved that |$\mathbb{L}^{\leq }_{\boldsymbol{S}}$| is a matrix logic determined by a family of six finite matrices. Next we shall see that two of them are superfluous. In the first place, observe that the matrix |$\langle \mathbb{S}_6, [0)\rangle = \langle \mathbb{S}_6, \{0,\frac{1}{3},N,B,\frac{2}{3}, 1\}\rangle $| is trivial. Indeed, for all |$\alpha \in Fm$| and all |$h \in Hom(\mathfrak{Fm}, \mathbb{S}_{6})$| we have that |$h(\alpha )\in [0)$|⁠. Then for every |$\varGamma \subseteq Fm$| it holds |$\varGamma \models _{\langle \mathbb{S}_6, [0)\rangle } \alpha $| and so we can discard it. On the other hand, we shall see that |$\langle \mathbb{S}_6, [N)\rangle $| and |$\langle \mathbb{S}_6, [B)\rangle $| are isomorphic matrices. For this, the next technical result will be useful. Lemma 5.3 Let |$h: \mathfrak{Fm} \to \mathbb{S}_{6}$|⁠, |${\mathcal{V}}^{\prime } \subseteq Var$| and |$h^{\prime }:\mathfrak{Fm} \to \mathbb{S}_{6}$| such that, for all |$p\in{\mathcal{V}}^{\prime }$|⁠, \begin{equation*} h^{\prime}(\,p)= \begin{cases} h(\,p) & \textrm{if}\ h(\,p)\in \big\{0,\frac{1}{3},\frac{2}{3},1\big\}\\ N & \textrm{if}\ h(\,p)=B\\ B & \textrm{if}\ h(\,p)=N. \end{cases}\qquad \textrm{Then}\ h^{\prime}(\alpha)= \begin{cases} h(\alpha) & \textrm{if}\ h(\alpha)\in \big\{0,\frac{1}{3},\frac{2}{3},1\big\}\\ N & \textrm{if}\ h(\alpha)=B\\ B & \textrm{if}\ h(\alpha)=N \end{cases} \end{equation*} for all |$\alpha \in \mathfrak{Fm}$| such that |$Var(\alpha )\subseteq{\mathcal{V}}^{\prime }$|⁠. Proof. We use induction on the complexity of the formula |$\alpha $|⁠, |$c(\alpha )$|⁠. First, observe that |$\{0,\frac{1}{3},\frac{2}{3},1\}\simeq \mathbf{{^{\tiny{\not}}\kern-1.5pt{\textrm{L}}} }_4$| is an S-subalgebra of |$\mathbb{S}_6$|⁠, that is, it is closed under all the operations of S-algebras. If |$\alpha $| is |$p$|⁠, where |$p$| is a propositional variable, the lemma holds by hypothesis. Suppose that the lemma holds for every formula with complexity less than |$k$|⁠, |$k\geq 1$|⁠; and let |$\alpha $| a formula with complexity |$k$|⁠. Then, we have the following cases: Case I :|$\alpha $| is |$\neg \beta $|⁠, with |$\beta \in Fm$|⁠. If |$h(\alpha )\in \{0,\frac{1}{3},\frac{2}{3},1\}$|⁠, since |$h(\alpha )=h(\neg \beta )=\neg h(\beta )$|⁠, we have that |$h(\beta )\in \{0,\frac{1}{3},\frac{2}{3},1\}$|⁠. On the other hand, |$h^{\prime }(\beta )\in \{0,\frac{1}{3},\frac{2}{3},1\}$|⁠. By I.H., |$h(\beta )=h^{\prime}(\beta )$|⁠. Since |$h^{\prime }$| is a homomorphism |$h^{\prime }(\alpha )=h^{\prime }(\neg \beta )=\neg h(\beta )=\neg h^{\prime }(\beta )=h^{\prime }(\neg \beta )=h^{\prime }(\alpha )$|⁠. If |$h(\alpha )= N$|⁠, from |$\neg N=N$|⁠, we have that |$h(\beta )=N$|⁠. By I.H., |$h^{\prime}(\beta )=B$| and since |$\neg B=B$|⁠, we have that |$h^{\prime}(\alpha )=B$|⁠. Analogously, we prove the lemma in case that |$h(\alpha )=B$|⁠. Case II :|$\alpha $| is |$\beta _1 \wedge \beta _2$|⁠, where |$\beta _1,\beta _2\in Fm$|⁠. If |$h(\alpha )=h(\beta _1) \wedge h(\beta _2)\in \{0,\frac{1}{3},\frac{2}{3},1\}$| there are many sub-cases. If |$h(\beta _1), h(\beta _2)\in \{0,\frac{1}{3},\frac{2}{3},1\}$|⁠, by I.H., |$h(\beta _1)=h^{\prime }(\beta _1)$| and |$h(\beta _2)=h^{\prime }(\beta _2)$| and then |$h(\alpha )=h(\beta _1) \wedge h(\beta _2)= h^{\prime }(\beta _1) \wedge h^{\prime }(\beta _2)=h^{\prime }(\alpha )$|⁠. If |$h(\beta _1)=N$| and |$h(\beta _2)\in \{0,\frac{1}{3}\}$|⁠, by I.H., |$h^{\prime }(\beta _1)=B$| and |$h^{\prime }(\beta _2)=h(\beta _2)$| and then |$h^{\prime }(\alpha )= h^{\prime }(\beta _1) \wedge h^{\prime }(\beta _2)= B\wedge h(\beta _2)=h(\beta _2)= N \wedge h(\beta _2)=h(\beta _1) \wedge h(\beta _2)=h(\alpha )$|⁠. The other sub-cases are studied similarly. Case III :|$\alpha $| is |$\beta _1 \vee \beta _2$|⁠, where |$\beta _1,\beta _2\in Fm$|⁠. It is analogous to case II. Case IV :|$\alpha $| is |$\nabla \beta $|⁠, with |$\beta \in Fm$|⁠. Then |$h(\alpha )=h(\nabla \beta )=\nabla h(\beta )\in \{0,1\}$|⁠. If, |$h(\alpha )=0$|⁠, then |$h(\beta )=0$|⁠, by the definition of |$\nabla $| in |$\mathbb{S}_{6}$|⁠. By I.H., |$h^{\prime }(\beta )=h(\beta )=0$| and therefore |$h^{\prime }(\alpha )=0=h(\alpha )$|⁠. If |$h(\alpha )=1$|⁠, then |$h(\beta )\in \{\frac{1}{3},N,B,\frac{2}{3}, 1\}$|⁠. In any of these cases we have that |$h^{\prime }(\alpha )=1$|⁠. Now, we can see that the matrices |$\langle \mathbb{S}_6, [N)\rangle $| and |$\langle \mathbb{S}_6, [B)\rangle $| determine the same logic. Lemma 5.4 The logics |$\models _{\langle \mathbb{S}_6, [N)\rangle }$| and |$\models _{\langle \mathbb{S}_6, [B)\rangle }$| coincide. Proof. Since |$\models _{\langle \mathbb{S}_6, [N)\rangle }$| (and |$\models _{\langle \mathbb{S}_6, [B)\rangle }$|⁠) is a finitary consequence relation and conjunctive, we only need to consider inferences of the form |$\alpha \models _{\langle \mathbb{S}_6, [N)\rangle } \beta $| (if |$\beta $| is logically valid we consider |$\alpha $| as |$\neg \bot $|⁠). Suppose that (1)|$\alpha \models _{\langle \mathbb{S}_6, [N)\rangle } \beta $| and let |$h \in Hom(\mathfrak{Fm}, \mathbb{S}_{6})$| such that |$h(\alpha ) \in \{B,\frac{2}{3}, 1\}$|⁠. If |$h(\alpha )=B$|⁠, consider the homomorphism |$h^{\prime }$| as in Lemma 5.3. Then |$h^{\prime }(\alpha )=N \in \{N,\frac{2}{3}, 1\}$| and by (1), |$h^{\prime }(\beta )\in \{N,\frac{2}{3}, 1\}$|⁠. If |$h^{\prime }(\beta )=N$|⁠, then |$h(\beta )=B \in \{B,\frac{2}{3}, 1\}$|⁠. Otherwise, if |$h^{\prime }(\beta )\in \{\frac{2}{3}, 1\}$| then |$h^{\prime }(\beta )=h(\beta ) \in \{\frac{2}{3}, 1\} \subseteq \{B,\frac{2}{3}, 1\}$|⁠. If |$h(\alpha )\in \{\frac{2}{3}, 1\}$|⁠, by (1), |$h(\beta )\in \{N,\frac{2}{3}, 1\}$|⁠. If |$h(\beta )=N$| then |$h^{\prime }(\beta )=B$|⁠. But, in this case, |$h(\alpha )=h^{\prime }(\alpha )\in \{\frac{2}{3}, 1\}$|⁠, that is, |$h^{\prime }\in Hom(\mathfrak{Fm}, \mathbb{S}_{6})$| verifies |$h^{\prime }(\alpha )\in \{N,\frac{2}{3}, 1\}$| but |$h^{\prime }(\beta )\notin \{N,\frac{2}{3}, 1\}$|⁠, which contradicts (1). Then |$h(\beta )\in \{\frac{2}{3}, 1\}\subseteq \{B,\frac{2}{3}, 1\}$|⁠. From analysing all these cases we conclude that |$\alpha \models _{\langle \mathbb{S}_6, [B)\rangle } \beta $|⁠. Analogously, we prove that if |$\alpha \models _{\langle \mathbb{S}_6, [B)\rangle } \beta $| then |$\alpha \models _{\langle \mathbb{S}_6, [N)\rangle } \beta $|⁠. From the results above mentioned, we have the following. Theorem 5.5 The logic that preserves degrees of truth associated to S-algebras, |$\mathbb{L}^{\leq }_{\boldsymbol{S}}$|⁠, is a six-valued logic determined by the matrices |$\langle \mathbb{S}_6, [\frac{1}{3})\rangle $|⁠, |$\langle \mathbb{S}_6, [N)\rangle $|⁠, |$\langle \mathbb{S}_6, [\frac{2}{3})\rangle $| and |$\langle \mathbb{S}_6, [1)\rangle $|⁠. In what follows, we shall denote by Six the logic |$\mathbb{L}^{\leq }_{\textbf{S}}$|⁠. Recall that we say that a set of connectives in a given logic is functionally complete if every possible truth table on the matrix semantic, which is sound and complete with respect to the logic, can be expressed in terms of these connectives. We say that the logic is functionally complete if it has a functionally complete set of connectives. Lemma 5.6 Let |$h: \mathfrak{Fm} \to \mathbb{S}_{6}$|⁠, |$h^{\prime}:\mathfrak{Fm} \to \mathbb{S}_{6}$| and |$p \in Var$| such that |$h(\,p)=B$| and |$h^{\prime}(\,p)= N$|⁠. Then, |$h(\alpha )\not =N$| and |$h^{\prime}(\alpha )\not =B$| for all |$\alpha \in \mathfrak{Fm}$| such that |$Var(\alpha ) \subseteq \{\,p\}$|⁠. Proof. It is a direct consequence of the fact that |$\{0,\frac{1}{3},B, \frac{2}{3},1\}$| and |$\{0,\frac{1}{3},N, \frac{2}{3},1\}$| are S-subalgebras of |$\mathbb{S}_{6}$|⁠. Corollary 5.7 Six is not functionally complete. Proof. By Lemma 5.6, it is not possible to define the function |$f:\mathbb{S}_{6} \to \mathbb{S}_{6}$| such that |$f(x)=N$|⁠, for all |$x$|⁠, in terms of the connectives. 6 Six as a paraconsistent logic In this section, we shall study Six under the perspective of paraconsistency. We shall see that contradictions (with respect to |$\neg $|⁠) do not necessarily trivialize the inferences, and therefore, Six is a paraconsistent logic in the sense of da Costa (cf. [9, 10]). Moreover, we shall see that Six is an LFI; and that it is gently explosive with respect to a proper set of formulas |$\bigcirc (\,p)$| which depends on the propositional variable |$p$| (cf. [2, 3]). Proposition 6.1 Six is non-trivial and non-explosive. Proof. Let |$p$| and |$q$| be two different propositional variables and consider the homomorphism |$h: \mathfrak{Fm} \to \mathbb{S}_{6}$| such that |$h(\,p)=N$| and |$h(q)=B$|⁠. Then |$p, \neg p \not \models _{\textbf{{Six}}} q$|⁠. Considering the same homomorphism as in the proof above we conclude also that |$\not \models _{\bf {Six}} q \vee \neg q$|⁠, and therefore Proposition 6.2 Six is paracomplete. On the other side, in the language of Six, put |$\bigcirc (\,p)=\{ \varDelta p \vee \varDelta \neg p \}$|⁠. Then, Lemma 6.3 Six is finitely gently explosive with respect to |$\bigcirc (\,p)$| and |$\neg $|⁠. Proof. Let |$p$| and |$q$| be two different propositional variables. It is easy to check that \begin{equation*}\bigcirc(\,p), p \not \models_{\bf {Six}} q \ \ \textrm{and} \ \ \bigcirc(\,p), \neg p \not \models_{\bf {Six}} q.\end{equation*} But, since |$h((\varDelta p \vee \varDelta \neg p) \wedge p \wedge \neg p) = h((\neg \nabla \neg p \vee \neg \nabla \neg \neg p) \wedge p \wedge \neg p) = h((\neg \nabla \neg p \vee \neg \nabla p) \wedge p \wedge \neg p)=h((\neg \nabla \neg p \wedge p \wedge \neg p) \vee (\neg \nabla p\wedge p \wedge \neg p))=0 $| (by (IS10)), for all |$h \in Hom(\mathfrak{Fm}, \mathbb{S}_{6})$|⁠. So, |$\bigcirc (\,p), p, \neg p \models _{\textbf{{Six}}} \bot $|⁠. From all the above we have the following. Theorem 6.4 Six is an LFI with respect to |$\neg $| and with consistency operator |$\circ $| defined by |$\circ \alpha = \varDelta \alpha \vee \varDelta \neg \alpha $|⁠, for all |$\alpha \in Fm$|⁠. Besides, as it happens in the systems |$C_n$| of da Costa (cf. [9, 10]), the operator of consistency propagates through the other connectives. Theorem 6.5 In Six it holds the following: $$\begin{array}{ll} {\rm(i)} \models_{\textbf{{Six}}} \circ \bot & \qquad{\rm(ii)} \circ \alpha \models_{\textbf{{Six}}} \circ \nabla \alpha \\ {\rm(iii)} \circ \alpha \models_{\textbf{{Six}}} \circ \neg \alpha &\qquad{\rm(iv)} \circ \alpha, \circ \beta \models_{\textbf{{Six}}} \circ (\alpha \# \beta) \, for \, \# \in \{\wedge, \vee\}. \end{array}$$ Proof. It is routine. Moreover, |$\models _{\textbf{{Six}}} \circ \neg ^{n}{\circ } \alpha $|⁠, for all |$n\geq 0$|⁠. In particular, |$\models _{\textbf{{Six}}} {\circ }{\circ } \alpha $|⁠. So, Six validates all axioms |$(cc)_{n}$| of the logic mCi (see [3]). As it is usual in the framework of LFIs, it is also possible to define an inconsistency operator |$\bullet $| in Six in the following way: \begin{equation*} \bullet \alpha =_{def} \neg \circ \alpha.\end{equation*} Then |$\bullet \alpha $| is logically equivalent to |$\nabla \alpha \wedge \nabla \neg \alpha $|⁠. Note that |$\bullet $| and |$\circ $| can be expressed equivalently as |$\bullet \alpha = \nabla ( \alpha \wedge \neg \alpha )$| and |$\circ \alpha = \varDelta (\alpha \vee \neg \alpha )$|⁠. The truth table of both connectives is the following: Theorem 6.6 In Six it holds the following: (i) |$ \alpha \wedge \neg \alpha \models _{\textbf{{Six}}} \bullet \alpha $| but |$\bullet \alpha \not \models _{\textbf{{Six}}} \alpha \wedge \neg \alpha $|⁠, (ii) |$\bullet \alpha \models _{\textbf{{Six}}} \bullet \neg \alpha $| and |$ \bullet \neg \alpha \models _{\textbf{{Six}}}\bullet \alpha $|⁠, (iii) |$\bullet (\alpha \# \beta ) \models _{\textbf{{Six}}} \bullet \alpha \vee \bullet \beta $| for |$\# \in \{\wedge , \vee \}$|⁠; the converse does not hold. Proof. It is routine. This last result shows us that the concept of consistency on one side, and contradiction on the other, can be differentiated in Six. This is a very valuable characteristic in the universe of LFIs only satisfied by a few of them such as mbC and |$\mathcal{TML}$| (see [8]), the first being the weakest in the hierarchy presented in [3]. In spite of the fact that Six is not functionally complete (cf. Corollary 5.7), it enjoys a nice expressive power. For instance, in Six it is possible distinguish the ‘classical’ truth values (⁠|$0$| and |$1$|⁠) from the ‘non-classical’ ones (⁠|$\frac{1}{3}$|⁠, |$N$|⁠, |$B$| and |$\frac{2}{3}$|⁠). To end this section, we shall show that, in Six, it is possible to reproduce the classical logic. That is, we shall prove a Derivability Adjustment Theorem with respect to the classical propositional logic CPL. We can think of CPL in the language generated by |$\wedge $|⁠, |$\vee $| and |$\neg $|⁠. Let |$\mathfrak{Fm}_{\textbf{CPL}}$| be the algebra of formulas of CPL in this language. Theorem 6.7 Let |$\varGamma \cup \{\alpha \}$| be a finite set of formulas in CPL. Then |$\varGamma \vdash _{\textbf{CPL}} \alpha $| iff |$\varGamma , \circ p_1, \dots , \circ p_n \models _{\textbf{{Six}}} \alpha $|⁠, where |${\textrm{Var}}\ (\varGamma \cup \{ \alpha \})=\{ p_1, \dots , p_n\}$|⁠. Proof. (⁠|$\Leftarrow $|⁠) Let |$\varGamma = \{\alpha _1, \dots , \alpha _k\} \subseteq \mathfrak{Fm}_{\textbf{CPL}}$| and let |$h \in Hom(\mathfrak{Fm}_{\textbf{CPL}}, \mathbb{B}_{1})$| where |$\mathbb{B}_{1}$| is the two-element Boolean algebra (that is, |$\mathbb{B}_{1}=\{0,1\}$|⁠). Since |$\mathbb{B}_{1}$| can be seen as an S-subalgebra of |$\mathbb{S}_{6}$| (if we put |$\nabla (x)=x$|⁠, for all |$x$|⁠, then |$\mathbb{B}_{1}\simeq \mathbf{{^{\tiny{\not}}\kern-1.5pt{\textrm{L}}} }_{2}$|⁠), we have that |$h \in Hom(\mathfrak{Fm}, \mathbb{S}_{6})$|⁠. Then by hypothesis we have \begin{equation*}h\Bigg(\bigwedge \limits_{i=1}^{k} \alpha_i \wedge \bigwedge \limits_{j=1}^{n} \circ p_j\Bigg) \leq h(\alpha).\end{equation*} Since |$h \in Hom(\mathfrak{Fm}, \mathbb{B}_{1})$|⁠, then |$h(p_j) \in \{0,1\}$| for all |$j$|⁠, |$1 \leq j \leq n$|⁠. In this way, by definition of |$\circ $|⁠, |$h(\circ p_j)= \circ h(p_j) = 1$| for all |$j$|⁠, |$1 \leq j \leq n$|⁠. Then if |$h(\alpha _i)=1$| for all |$i$|⁠, |$1\leq i \leq k$| we have |$h(\bigwedge \limits _{i=1}^{k} \alpha _i \wedge \bigwedge \limits _{j=1}^{n} \circ p_j)=1$| and therefore |$h(\alpha )=1$|⁠. That is, |$\varGamma \vdash _{\textbf{CPL}} \alpha $|⁠. (⁠|$\Rightarrow $|⁠) Suppose that |$\varGamma \vdash _{\boldsymbol{CPL}} \alpha $| and let |$h \in Hom(\mathfrak{Fm}, \mathbb{S}_{6})$|⁠. Let’s see that |$\varGamma , \circ p_1, \dots , \circ p_n \models _{M} \alpha $| for each of the matrices |$M$| which determine Six. Let |$M=\langle \mathbb{S}_{6},[\frac{1}{3})\rangle $| and suppose that |$h(\varGamma \cup \{\circ p_1, \dots , \circ p_n\}) \subseteq [\frac{1}{3})$|⁠, where |${\textrm{Var}}\ (\varGamma \cup \{ \alpha \})=\{ p_1, \dots ,p_n\}$|⁠. Then |$h(\circ p_i)\not =0$| which implies that |$h(p_i)\in \{1,0\}$| for all |$i$|⁠. Then we can think that |$h \in Hom(\mathfrak{Fm}_{\textbf{CPL}}, \mathbb{B}_{1})$|⁠. Since |$\varGamma \vdash _{\textbf{CPL}} \alpha $| and |$h(\varGamma ) \subseteq \{1\}$|⁠, then |$h(\alpha )\!=\!1$| and |$h(\alpha ) \in [\frac{1}{3})$|⁠. Then |$\varGamma , \circ p_1, \dots , \circ p_n\! \models _{\langle \mathbb{S}_{6},[\frac{1}{3})\rangle } \alpha $|⁠. Analogously it is proved that |$\varGamma , \circ p_1, \dots , \circ p_n \models _{M} \alpha $| for the other matrices which define Six. Therefore, |$\varGamma , \circ p_1, \dots , \circ p_n \models _{\textbf{{Six}}} \alpha $|⁠. 7 Proof theory for Six So far, we have studied the logic Six by semantical tools. In this section, we shall present a syntactic version of it by means of a sequent calculus. If |$\varSigma $| is a set of formulas, then |$\nabla\! \varSigma =\{\nabla \alpha : \alpha \in \varSigma \}$|⁠. Definition 7.1 We denote by |$\mathfrak{S}$| the sequent calculus whose axioms and rules are the following: Axioms \begin{equation*} \begin{array}{{rl}} \textrm{(structural axiom) } \displaystyle{\alpha \Rightarrow \alpha} & \quad\qquad{(\bot) \, \bot\Rightarrow} \qquad\qquad\qquad{(\top) \, \Rightarrow \top}\\ \textrm{(First modal axiom) }{\alpha\Rightarrow \nabla\alpha} & \qquad\quad \textrm{(Second modal axiom)}\,\,\,\,\, {\Rightarrow \nabla\alpha \vee \neg \nabla \alpha} \end{array} \end{equation*}Structural rules \begin{equation*} \textrm{(Left weakening) } \displaystyle \frac{\varGamma \Rightarrow \varSigma}{\varGamma, \alpha \Rightarrow \varSigma} \qquad\quad\textrm{(Right weakening) } \displaystyle \frac{\varGamma \Rightarrow \varSigma}{\varGamma \Rightarrow \varSigma, \alpha} \end{equation*} \begin{equation*} \textit{(Cut) } \displaystyle \frac{\varGamma \Rightarrow \varSigma, \alpha \alpha, \varGamma \Rightarrow \varSigma}{\varGamma \Rightarrow \varSigma} \end{equation*}Logic Rules \begin{equation*} \begin{array}{{ll}} {(\wedge \Rightarrow)}\, \displaystyle \frac{\varGamma, \alpha, \beta \Rightarrow \varSigma}{\varGamma, \alpha \wedge \beta \Rightarrow \varSigma} \qquad\qquad{(\Rightarrow \wedge)}\, \displaystyle \frac{\varGamma \Rightarrow \varSigma, \alpha \quad \varGamma \Rightarrow \varSigma, \beta}{\varGamma \Rightarrow \varSigma, \alpha \wedge \beta}\\ \\{(\vee \Rightarrow)}\, \displaystyle \frac{\varGamma, \alpha \Rightarrow \varSigma\quad \varGamma, \beta \Rightarrow \varSigma}{\varGamma, \alpha \vee \beta \Rightarrow \varSigma} \qquad\qquad\quad\!\!\!\! {(\Rightarrow \vee)}\, \displaystyle \frac{\varGamma \Rightarrow \varSigma, \alpha, \beta} {\varGamma \Rightarrow \varSigma, \alpha \vee \beta}\\ \\ \qquad\qquad\quad\qquad\qquad\quad\!\!\!\!{(\neg)}\, \displaystyle \frac{\alpha \Rightarrow \beta} {\neg \beta \Rightarrow \neg \alpha} \\ \\{(\neg \neg \Rightarrow)}\, \displaystyle \frac{\varGamma, \alpha \Rightarrow \varSigma }{\varGamma, \neg \neg \alpha \Rightarrow \varSigma} \qquad\qquad\quad{(\Rightarrow \neg \neg)}\, \frac{\varGamma \Rightarrow \alpha, \varSigma }{\varGamma \Rightarrow \neg \neg \alpha, \varSigma}\\ \\{(\nabla)}\, \displaystyle \frac{\varGamma, \alpha \Rightarrow \nabla \varSigma }{\varGamma, \nabla \alpha \Rightarrow \nabla \varSigma} \qquad\qquad\qquad{(\neg \nabla \Rightarrow)}\, \displaystyle \frac{\varGamma, \neg \nabla \alpha \Rightarrow \varSigma }{\varGamma, \nabla \neg \nabla \alpha \Rightarrow \varSigma}. \end{array} \end{equation*} The notion of derivation (o |$\mathfrak{S}$|-proof) in the Gentzen calculus |$\mathfrak{S}$| is the usual. That is, we say that a sequent |$\varGamma \Rightarrow \varSigma $| is derivable in |$\mathfrak{S}$|⁠, denoted |$\mathfrak{S} \vdash \varGamma \Rightarrow \varSigma $|⁠, when it has a derivation whose initial sequents are instances of the axioms and where the rules used are among those in the above list. We have not explicitly included the rules of Exchange and Contraction because we are using sets of formulas in our sequents, and not just multisets or sequences; thus this Gentzen calculus satisfies all structural rules. If |$\alpha $| is a formula we write |$\mathfrak{S}\vdash \alpha $| to indicate that |$\mathfrak{S} \vdash \Rightarrow \alpha $|⁠. Lemma 7.2 Let |$\alpha _1,\dots , \alpha _n, \beta _1, \dots , \beta _m \in Fm$|⁠. The following conditions are equivalent: (i) |$\alpha _1,\dots , \alpha _n \Rightarrow \beta _1, \dots , \beta _m$| is derivable in |$\mathfrak{S}$|⁠, (ii) |$\alpha _1\wedge \dots \wedge \alpha _n \Rightarrow \beta _1, \dots , \beta _m$| is derivable in |$\mathfrak{S}$| (iii) |$\alpha _1,\dots , \alpha _n \Rightarrow \beta _1\vee \dots \vee \beta _m$| is derivable in |$\mathfrak{S}$| (iv) |$\alpha _1\wedge \dots \wedge \alpha _n \Rightarrow \beta _1\vee \dots \vee \beta _m$| is derivable in |$\mathfrak{S}$|⁠. Proof. It is routine taking into account that the rules (⁠|$\wedge \Rightarrow $|⁠), (⁠|$\Rightarrow \wedge $|⁠), (⁠|$\vee \Rightarrow $|⁠), (⁠|$\Rightarrow \vee $|⁠) are the same rules of the classical case. Remark 7.3 Since the logical rules which govern the behaviour of the conjunction |$\wedge $| and the disjunction |$\vee $| are the classical ones, we know that in |$\mathfrak{S}$| all classical properties concerning these two connectives hold. In particular, the following holds: let |$\alpha , \beta , \gamma \in Fm$|⁠. If |$\mathfrak{S} \vdash \alpha \Leftrightarrow \beta ,$| then (i) if |$\mathfrak{S} \vdash \alpha \Leftrightarrow \beta ,$| then |$\mathfrak{S} \vdash \alpha \vee \gamma \Leftrightarrow \beta \vee \gamma $| and |$\mathfrak{S} \vdash \alpha \wedge \gamma \Leftrightarrow \beta \wedge \gamma $|⁠, and (ii) if |$\mathfrak{S} \vdash \alpha \Rightarrow \beta $| and |$\mathfrak{S} \vdash \gamma \Rightarrow \delta ,$| then |$\mathfrak{S} \vdash \alpha \wedge \gamma \Rightarrow \beta \wedge \delta $| and |$\mathfrak{S} \vdash \alpha \vee \gamma \Rightarrow \beta \vee \delta $|⁠, (iii) |$\mathfrak{S} \vdash \alpha \vee (\beta \wedge \gamma ) \Leftrightarrow (\alpha \vee \beta ) \wedge (\alpha \vee \gamma )$|⁠. In what follows, we write |$\mathfrak{S} \vdash \varGamma \Leftrightarrow \varSigma $| to indicate that both sequents |$\varGamma \Rightarrow \varSigma $| and |$\varSigma \Rightarrow \varGamma $| are derivable in |$\mathfrak{S}$|⁠. Lemma 7.4 Let |$\alpha , \beta , \gamma \in Fm$| such that |$\mathfrak{S} \vdash \alpha \Leftrightarrow \beta $|⁠. Then (i) |$\mathfrak{S} \vdash \neg \alpha \Leftrightarrow \neg \beta $|⁠, (ii) |$\mathfrak{S} \vdash \nabla \alpha \Leftrightarrow \nabla \beta $|⁠. Proof. (i) is immediate. (ii) \begin{array} \text{(cut)} \displaystyle \frac{\alpha \Rightarrow \beta \beta \Rightarrow \nabla\beta}{(\nabla) \displaystyle\frac{\alpha \Rightarrow \nabla \beta}{\nabla\alpha \Rightarrow \nabla \beta}} \end{array} Analogously, it is proved that |$\nabla \beta \Rightarrow \nabla \alpha $|⁠. Also, since |$\neg $| is a De Morgan negation the following result is expected. Lemma 7.5 Let |$\alpha ,\beta \in Fm$|⁠. The following sequents are derivable in |$\mathfrak{S}$|⁠: (i) |$\alpha \Rightarrow \neg \neg \alpha $|⁠, |$\neg \neg \alpha \Rightarrow \alpha $|⁠, (ii) |$ \alpha \vee \beta \Rightarrow \neg \neg \alpha \vee \neg \neg \beta $|⁠, (iii) |$\neg \neg \alpha \wedge \neg \neg \beta \Rightarrow \alpha \wedge \beta $|⁠, (iv) |$\neg (\alpha \vee \beta ) \Rightarrow \neg \alpha \wedge \neg \beta $|⁠, (v) |$\neg \alpha \vee \neg \beta \Rightarrow \neg (\alpha \wedge \beta )$|⁠, (vi) |$\neg \alpha \wedge \neg \beta \Rightarrow \neg (\alpha \vee \beta )$|⁠, (vii) |$\neg (\alpha \wedge \beta ) \Rightarrow \neg \alpha \vee \neg \beta $|⁠. Proof. (i) Immediate from (⁠|$\neg \neg \Rightarrow $|⁠) and (⁠|$\Rightarrow \neg \neg $|⁠). (ii) (iii) Analogous to (ii). (iv) (v) (vi) (vii) From (iii), (⁠|$\neg $|⁠), (i) and the cut rule. Finally, the following lemma shows that |$\nabla $| enjoys the properties of the respective unary operator of S-algebras. Lemma 7.6 Let |$\alpha ,\beta \in Fm$|⁠. Then (i) |$\mathfrak{S} \vdash \nabla (\alpha \vee \beta ) \Leftrightarrow \nabla \alpha \vee \nabla \beta $|⁠, (ii) |$\mathfrak{S} \vdash \nabla ( \alpha \wedge \beta ) \Leftrightarrow \nabla \alpha \wedge \nabla \beta $|⁠, (iii) |$\mathfrak{S} \vdash \nabla \nabla \alpha \Leftrightarrow \nabla \alpha $|⁠, (iv) |$\mathfrak{S} \vdash \nabla \neg \nabla \alpha \Leftrightarrow \neg \nabla \alpha $|⁠, |$\mathfrak{S} \Rightarrow \neg \nabla \bot $|⁠. Proof. (i) (ii) (iii) |$\nabla \alpha \Rightarrow \nabla \nabla \alpha $| is an instance of the first modal axiom. On the other hand, (iv) It is a consequence of the first modal axiom and the rule (⁠|$\neg \nabla $|⁠). (v) 8 Inversion principle, soundness and completeness In order to prove soundness and completeness of |$\mathfrak{S}$| with respect to Six, we need to state the notion of validity. Definition 8.1 Let |$\alpha _1, \dots , \alpha _n, \beta _1, \dots , \beta _m \in Fm$|⁠. We shall say that the sequent |$\alpha _1, \dots , \alpha _n \Rightarrow \beta _1, \dots , \beta _m$| is valid iff for all |$h\in Hom(Fm,\mathbb{S}_6)$| it holds \begin{equation*}h\left( \bigwedge \limits_{i=1}^{n} \alpha_i\right) \leq h\left(\bigvee \limits_{j=1}^{m} \beta_j\right)\!. \end{equation*} If |$n = 0$|⁠, we take |$\bigwedge \limits _{i=1}^{n} \alpha _i = \top $| and if |$m=0$|⁠, we take |$ \bigvee \limits _{j=1}^{m} \beta _j = \bot $|⁠. From the above definition we have the following. Proposition 8.2 |$\alpha _1, \dots , \alpha _n \Rightarrow \beta _1, \dots , \beta _m$| is valid iff |$\bigwedge \limits _{i=1}^{n} \alpha _i \models _{\bf {Six}} \bigvee \limits _{j=1}^{m} \beta _j$|⁠. Besides, we shall say that the rule \begin{equation*}\displaystyle\frac{\varGamma_1 \Rightarrow \varSigma_1, \dots, \varGamma_k \Rightarrow \varSigma_k}{\varGamma \Rightarrow \varSigma}\end{equation*} preserves validity if it holds \begin{equation*}\varGamma_i \Rightarrow \varSigma_i \ \textrm{is valid for} \ 1\leq i\leq k \, \, \textrm{implies} \, \, \varGamma \Rightarrow \varSigma \mbox{ is valid}. \end{equation*} Recall that the complexity of the formula |$\alpha $|⁠, |$c(\alpha )$|⁠, is the number of occurrences of connectives in |$\alpha $|⁠. We denote |$\alpha\,{\models}\kern-4.5pt\models \beta $| to indicate that |$\alpha \models _{\textbf{{Six}}} \beta $| and |$\beta \models _{\textbf{{Six}}} \alpha $|⁠. Theorem 8.3 Let |$\alpha \in Fm$|⁠. Then |$\alpha\, {=}\kern-1.5pt{|}{|}\kern-1.5pt{=}\top $| or |$\alpha\,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bot $| or \begin{equation*}\alpha\,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bigwedge \limits_{i=1}^{n} \bigvee \limits_{j=1}^{m_i} \alpha_{ij},\end{equation*} where each |$\alpha _{ij} \in \{\,p_k, \neg p_k, \nabla p_k, \nabla \neg p_k, \neg \nabla p_k, \neg \nabla \neg p_k \}$| for some propositional variable |$p_k$|⁠, |$n\geq 0$|⁠, |$m_i\geq 0$|⁠. Proof. We use induction on the complexity of |$\alpha $|⁠. If |$c(\alpha )=0$|⁠, then |$\alpha $| is |$p_k$| or |$\top $| or |$\bot $|⁠. In any of these cases the theorem holds. (I.H.) Suppose that the theorem holds for any formula with complexity less than |$k$|⁠, |$k\geq 0$|⁠. Let |$\alpha $| such that |$c(\alpha )=k$|⁠. If |$\alpha $| is |$\alpha ^1\wedge \alpha ^2$|⁠, by (I.H.), |$\alpha ^k \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \top $| or |$\alpha ^k \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bot $| or \begin{equation*}\alpha^k \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bigwedge\limits_{i=1}^{n_k} \bigvee \limits_{j=1}^{m_{k_i}} \alpha^k_{ij}\end{equation*} for |$k=1,2$|⁠. If |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bot $| or |$\alpha ^2 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bot $|⁠, then |$\alpha \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bot $|⁠. If |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \top $|⁠, then |$\alpha \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \alpha ^2$| and by (I.H.) the theorem holds. Analogously if |$\alpha ^2 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \top $|⁠. Otherwise, |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bigwedge \limits _{i=1}^{n_1} \bigvee \limits _{j=1}^{m_{1_i}} \alpha ^1_{ij}$| and |$\alpha ^2 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bigwedge \limits _{i=1}^{n_2} \bigvee \limits _{j=1}^{m_{2_i}} \alpha ^2_{ij}$|⁠. Then |$\alpha =\alpha ^1\wedge \alpha ^2 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bigwedge \limits _{i=1}^{n_1} \bigvee \limits _{j=1}^{m_{1_i}} \alpha ^1_{ij} \wedge \bigwedge \limits _{i=1}^{n_2} \bigvee \limits _{j=1}^{m_{2_i}} \alpha ^2_{ij}$| and this last expression is a conjunction of terms which are disjunction of formulas whose forms are in |$\{\,p_k, \neg p_k, \nabla p_k, \nabla \neg p_k, \neg \nabla p_k, \neg \nabla \neg p_k \}$|⁠. If |$\alpha $| is |$\alpha ^1\vee \alpha ^2$| with |$\alpha ^1$| and |$\alpha ^2$| as in the previous case. Then if |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \top $| or |$\alpha ^2 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \top $|⁠, then |$\alpha \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \top $|⁠. If |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bot $|⁠, then |$\alpha \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \alpha ^2$| and by (I.H.) the theorem holds. Analogously if |$\alpha ^2 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bot $|⁠. Otherwise, |$\alpha =\alpha ^1\vee \alpha ^2 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bigwedge \limits _{i=1}^{n_1} \bigvee \limits _{j=1}^{m_{1_i}} \alpha ^1_{ij} \vee \bigwedge \limits _{i=1}^{n_2} \bigvee \limits _{j=1}^{m_{2_i}} \alpha ^2_{ij}$|⁠. By Corollary 4.4 and having in mind that every S-algebra is, in particular, a distributive lattice, using distributivity as many times as necessary we can transform the expression |$\bigwedge \limits _{i=1}^{n_1} \bigvee \limits _{j=1}^{m_{1_i}} \alpha ^1_{ij} \vee \bigwedge \limits _{i=1}^{n_2} \bigvee \limits _{j=1}^{m_{2_i}} \alpha ^2_{ij}$| into one expression of the form |$\bigwedge \bigvee \beta _t$| where |$\beta _t\in \{\,p_k, \neg p_k, \nabla p_k, \nabla \neg p_k, \neg \nabla p_k, \neg \nabla \neg p_k \}$| for some propositional variable |$p_k$|⁠. If |$\alpha $| is |$\neg \alpha ^1$|⁠, by (I.H.), |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \top $| or |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bot $| or |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bigwedge \limits _{i=1}^{n_1} \bigvee \limits _{j=1}^{m_{1_i}} \alpha ^1_{ij}$|⁠. If |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \top $|⁠, then |$\alpha \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bot $|⁠. If |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bot $|⁠, then |$\alpha \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \top $|⁠. Otherwise, by Lemma 7.4 and Remark 7.3, |$\alpha =\neg \alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \neg \bigwedge \limits _{i=1}^{n_1} \bigvee \limits _{j=1}^{m_{1_i}} \alpha ^1_{ij}$|⁠. Since every S-algebra is, in particular, a De Morgan algebra we have |$\alpha \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \neg \bigwedge \limits _{i=1}^{n_1} \bigvee \limits _{j=1}^{m_{1_i}} \alpha ^1_{ij}\,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bigvee \limits _{i=1}^{n_1} \bigwedge \limits _{j=1}^{m_{1_i}} \neg \alpha ^1_{ij}$|⁠. Again, by distributivity and (DM1), |$\neg \neg x \approx x$|⁠, we have that the expression |$\bigvee \limits _{i=1}^{n_1} \bigwedge \limits _{j=1}^{m_{1_i}} \neg \alpha ^1_{ij}$| is equivalent to one of the type |$\bigwedge \bigvee \beta _t$| where each |$\beta _t\in \{\,p_k, \neg p_k, \nabla p_k, \nabla \neg p_k, \neg \nabla p_k, \neg \nabla \neg p_k \}$| for some propositional variable |$p_k$|⁠. Finally, if |$\alpha $| is |$\nabla \alpha ^1$| and |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \top $|⁠, then |$\alpha \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \top $|⁠. If |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bot $|⁠, then |$\alpha \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bot $|⁠. Otherwise, |$\alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bigwedge \limits _{i=1}^{n_1} \bigvee \limits _{j=1}^{m_{1_i}} \alpha ^1_{ij}$| and from Lemma 3.2 (IS7), (IS8) and |$\nabla $| being a lattice homomorphism we have |$\alpha =\nabla \alpha ^1 \,{=}\kern-1.5pt{|}{|}\kern-1.5pt{=} \bigwedge \limits _{i=1}^{n_1} \bigvee \limits _{j=1}^{m_{1_i}} \nabla \alpha ^1_{ij}$|⁠, where |$\nabla \alpha ^1_{ij}$| has the form of some element of |$\{p_k, \neg p_k, \nabla p_k, \nabla \neg p_k, \neg \nabla p_k, \neg \nabla \neg p_k \}$| for some propositional variable |$p_k$|⁠. Remark 8.4 From Theorem 8.3 we have that every formula |$\alpha \in Fm$| is logically equivalent (in Six) to |$\top $| or |$\bot $| or to a formula of the form |$\bigwedge \limits _{i=1}^{n} \bigvee \limits _{j=1}^{m_i} \alpha _{ij}$|⁠. If |$\alpha $| is not equivalent to |$\bot $| nor to |$\top $|⁠, we shall say that |$\bigwedge \limits _{i=1}^{n} \bigvee \limits _{j=1}^{m_i} \alpha _{ij}$| is one conjunctive form of |$\alpha $|⁠. Every formula has infinite conjunctive forms. We shall denote by |$\bar{\alpha }$| any conjunctive form of |$\alpha $|⁠. If |$\alpha $| is equivalent to |$\bot $| or |$\top $|⁠, then |$\bot $| (⁠|$\top $|⁠) is a conjunctive form of |$\alpha $|⁠. Besides, we say that the |$\alpha _{ij}$|s are the blocks of |$\bar{\alpha }$|⁠. If |$\alpha $| is a conjunctive form, we denote by |$\# \alpha $| the number of blocks of |$\alpha $|⁠. That is, if |$\alpha $| is |$\bot $| or |$\top $|⁠, then |$\# \alpha = 0$|⁠. Otherwise, |$\alpha $| is |$\bigwedge \limits _{i=1}^{n} \bigvee \limits _{j=1}^{m_i} \alpha _{ij}$| and then |$\# \alpha = m_1+\dots +m_n$|⁠. Examples 8.5 Consider the formula |$\nabla ((p_1\wedge \neg \nabla p_2) \vee \nabla p_2)$|⁠. By Lemma 7.6 (i), we have \begin{equation*}\nabla((p_1\wedge \neg \nabla p_2) \vee \nabla p_2) \, \Leftrightarrow \, \nabla(p_1\wedge \neg \nabla p_2) \vee \nabla \nabla p_2\end{equation*} and, by Lemma 7.6 (ii) and Remark 7.3 (i), \begin{equation*}\nabla(p_1\wedge \neg \nabla p_2) \vee \nabla \nabla p_2 \, \Leftrightarrow \, (\nabla p_1\wedge \nabla \neg \nabla p_2) \vee \nabla \nabla p_2.\end{equation*} Now, |$\nabla p_1\wedge \nabla \neg \nabla p_2 \, \Leftrightarrow \, \nabla p_1\wedge \neg \nabla p_2$|⁠, by Remark 7.3 (i) and Lemma 7.6 (iv). Then by Remark 7.3 (i) \begin{equation*}(\nabla p_1\wedge \nabla \neg \nabla p_2) \vee \nabla \nabla p_2 \, \Leftrightarrow \, (\nabla p_1 \wedge \neg \nabla p_2) \vee \nabla \nabla p_2.\end{equation*} From Lemma 7.6 (iv) and Remark 7.3 (i), we have \begin{equation*}(\nabla p_1\wedge \neg \nabla p_2) \vee \nabla \nabla p_2 \, \Leftrightarrow \, (\nabla p_1 \wedge \neg \nabla p_2) \vee \nabla p_2.\end{equation*} Finally, from Remark 7.3 (iii), we have \begin{equation*}(\nabla p_1 \wedge \neg \nabla p_2) \vee \nabla p_2 \, \Leftrightarrow \, (\nabla p_1 \vee \nabla p_2) \wedge (\neg \nabla p_2 \vee \nabla p_2). \end{equation*} This last formula is a conjunctive form logically equivalent to the original one, and we have proved that |$\nabla ((p_1\wedge \neg \nabla p_2) \vee \nabla p_2)$| and |$(\nabla p_1 \vee \nabla p_2) \wedge (\neg \nabla p_2 \vee \nabla p_2)$| are inter-derivable. That is, the following sequents are derivable in |$\mathfrak{S}$|⁠: \begin{equation*}\nabla((p_1\wedge \neg \nabla p_2) \vee \nabla p_2) \, \Leftrightarrow \, (\nabla p_1 \vee \nabla p_2) \wedge (\neg \nabla p_2 \vee \nabla p_2).\end{equation*} Besides, |$\# (\nabla p_1 \vee \nabla p_2) \wedge (\neg \nabla p_2 \vee \nabla p_2) = 4.$| Lemma 8.6 Let |$\alpha \in Fm$|⁠. Then there exists a conjunctive form |$\bar{\alpha }$| of |$\alpha $| such that |$\alpha \Leftrightarrow \bar{\alpha }$|⁠. Proof. The proof is by induction on the complexity of |$\alpha $| and using Remark 7.3 and Lemmas 7.4, 7.5 and 7.6. We leave the details to the patient reader. Theorem 8.7 (Soundness) Every sequent |$\varGamma \Rightarrow \varSigma $| derivable in |$\mathfrak{S}$| is valid. Proof. It is easy to check that the axioms are valid sequents and that the structural rules preserve validity. On the other hand, taking into account that |$\mathbb{S}_{6}$| is a De Morgan algebra, (DM1), (IS7) and (IS8) we see clearly that (⁠|$\neg $|⁠), (⁠|$\neg \neg \Rightarrow $|⁠), (⁠|$\Rightarrow \neg \neg $|⁠) and |$(\neg \nabla )$| preserve validity. Finally, let’s see that \begin{equation*} (\nabla) \, \displaystyle \frac{\varGamma, \alpha \Rightarrow \nabla \varSigma} {\varGamma, \nabla \alpha \Rightarrow \nabla \varSigma} \end{equation*} preserves validity. Suppose that (1) |$\bigwedge \limits _{i=1}^{n} h(\alpha _i) \wedge h(\alpha ) \leq \bigvee \limits _{j=1}^{m} h(\nabla \beta _j)$|⁠. Then since |$\nabla $| is a lattice homomorphism, |$\bigwedge \limits _{i=1}^{n} h(\alpha _i) \wedge h(\alpha ) \leq \bigvee \limits _{j=1}^{m} \nabla h(\beta _j) = \nabla \left ( \bigvee \limits _{j=1}^{m} h(\beta _j) \right )$|⁠. However, in |$\mathbb{S}_6$| we have that |$\nabla x\in \{0,1\}$| for all |$x$|⁠. Then |$\nabla \left ( \bigvee \limits _{j=1}^{m} h(\beta _j) \right )\in \{0,1\}$|⁠. If |$\nabla \left ( \bigvee \limits _{j=1}^{m} h(\beta _j) \right )=0$|⁠, then, by (1), |$\bigwedge \limits _{i=1}^{n} h(\alpha _i) \wedge h(\alpha )=0$| and then |$\bigwedge \limits _{i=1}^{n} h(\alpha _i)=0$| or |$ h(\alpha )=0$|⁠. In both cases we have |$\bigwedge \limits _{i=1}^{n} h(\alpha _i) \wedge h(\nabla \alpha )=0$| and therefore |$\bigwedge \limits _{i=1}^{n} h(\alpha _i) \wedge h(\nabla \alpha )\leq \bigvee \limits _{j=1}^{m} h(\nabla \beta _j)$|⁠. If |$\nabla \left ( \bigvee \limits _{j=1}^{m} h(\beta _j) \right )=1$|⁠, trivially we have |$\bigwedge \limits _{i=1}^{n} h(\alpha _i) \wedge h(\nabla \alpha )\leq \bigvee \limits _{j=1}^{m} h(\nabla \beta _j)$|⁠. Next we shall see that in |$\mathfrak{S}$| the Inversion principle holds. Lemma 8.8 (Inversion principle) For every rule |$\frac{S_1,\dots ,S_k}{S}$| of |$\mathfrak{S}$|⁠, except the left and right weakenings, it holds: if |$S$| is valid, then |$S_i$| is valid for |$i\in \{1, \dots , k\}$|⁠. Proof. Since rules (⁠|$\wedge \Rightarrow $|⁠), (⁠|$\Rightarrow \wedge $|⁠), (⁠|$\vee \Rightarrow $|⁠) and (⁠|$\Rightarrow \vee $|⁠) are the same as in the classical case, and taking into account that S-algebras are distributive lattices, the principle holds for them as in the classical case. Since every S-algebra is a De Morgan algebra, rules (⁠|$\neg $|⁠), (⁠|$\neg \neg \Rightarrow $|⁠) and (⁠|$\Rightarrow \neg \neg $|⁠) verify the principle. Rule (⁠|$\neg \nabla \Rightarrow $|⁠) verifies the principle by (IS8). Let’s see that (⁠|$\nabla $|⁠) verifies the principle. Suppose that the sequent |$\varGamma , \nabla \alpha \Rightarrow \nabla \varSigma $| is valid. Then, for every |$h\in Hom(Fm,\mathbb{S}_6)$| we have that \begin{equation*}h\left(\bigwedge \limits_{\gamma \in \varGamma}\gamma\right) \wedge \nabla h(\alpha) = h\left( \bigwedge \limits_{\gamma \in \varGamma}\gamma \wedge \nabla \alpha\right) \leq h\left(\bigvee \limits_{\beta \in \varSigma} \nabla \beta\right) = h\left(\nabla \bigvee \limits_{\beta \in \varSigma} \beta\right) = \nabla h\left(\bigvee \limits_{\beta \in \varSigma} \beta\right)\!.\end{equation*} By the definition of |$\nabla $| in |$\mathbb{S}_6$| we have that |$\nabla h(\bigvee \limits _{\beta \in \varSigma } \beta ) \in \{0,1\}$|⁠. If |$\nabla h(\bigvee \limits _{\beta \in \varSigma } \beta )=0$|⁠, then |$h(\bigwedge \limits _{\gamma \in \varGamma }\gamma ) \wedge \nabla h(\alpha )=0$|⁠. In |$\mathbb{S}_6$|⁠, we can assert that |$h(\bigwedge \limits _{\gamma \in \varGamma }\gamma ) =0$| or |$\nabla h(\alpha )=0$|⁠. If |$h(\bigwedge \limits _{\gamma \in \varGamma }\gamma ) =0$| then |$h( \bigwedge \limits _{\gamma \in \varGamma }\gamma \wedge \alpha ) \leq h(\bigvee \limits _{\beta \in \varSigma } \nabla \beta )$|⁠. If |$\nabla h(\alpha )=0$|⁠, then |$h(\alpha )=0$| and therefore |$h(\bigwedge \limits _{\gamma \in \varGamma }\gamma ) \wedge h(\alpha )=0$| and |$h( \bigwedge \limits _{\gamma \in \varGamma }\gamma \wedge \alpha ) \leq h(\bigvee \limits _{\beta \in \varSigma } \nabla \beta )$|⁠. On the other side, if |$\nabla h(\bigvee \limits _{\beta \in \varSigma } \beta )=1$|⁠, clearly |$h( \bigwedge \limits _{\gamma \in \varGamma }\gamma \wedge \alpha ) \leq h(\bigvee \limits _{\beta \in \varSigma } \nabla \beta )$|⁠. Theorem 8.9 (Completeness) Every sequent valid in |$\mathfrak{S}$| is derivable in |$\mathfrak{S}$|⁠. Proof. We show that every valid sequent |$\varGamma \Rightarrow \varSigma $| has a |$\mathfrak{S}$| proof, by induction on the total number of logical connectives |$\vee $| and |$\wedge $| occurring in |$\varGamma \Rightarrow \varSigma $|⁠. In virtue of Lemma 8.6, we assume that every formula in |$\varGamma \cup \varSigma $| is in conjunctive form. For the base case, every formula in |$\varGamma $| and |$\varSigma $| is a block in |$\{\,p, \neg p, \nabla p, \nabla \neg p, \neg \nabla p, \neg \nabla \neg p \}$| for some propositional variable |$p$| or one of the constants |$\bot $| or |$\top $|⁠. Since the sequent is valid, we have the following cases: (1) |$\bot \in \varGamma $| or |$\top \in \varSigma $|⁠, (2.1) |$p \in \varGamma $| and |$p \in \varSigma $|⁠; (2.2) |$p \in \varGamma $| and |$\nabla p \in \varSigma $|⁠, (3.1) |$\neg p \in \varGamma $| and |$\neg p \in \varSigma $|⁠; (3.2) |$\neg p \in \varGamma $| and |$\nabla \neg p \in \varSigma $|⁠, (4.1) |$\nabla p \in \varGamma $| and |$\nabla p \in \varSigma $|⁠; (4.2) |$\nabla \neg p \in \varGamma $| and |$\nabla \neg p \in \varSigma $|⁠, (5.1) |$\neg \nabla \neg p \in \varGamma $| and |$\neg \nabla \neg p \in \varSigma $|⁠; (5.2) |$\neg \nabla \neg p \in \varGamma $| and |$p \in \varSigma $|⁠, (6) |$\nabla p, \neg \nabla p \in \varSigma $|⁠. In case (1), |$\varGamma \Rightarrow \varSigma $| can be derived from (⁠|$\bot $|⁠) or (⁠|$\top $|⁠) and by weakenings. In cases (2.1), (3.1), (4.1), (4.2) and (5.1), |$\varGamma \Rightarrow \varSigma $| can be derived from the structural axiom and by weakenings. In cases (2.2) and (3.2), |$\varGamma \Rightarrow \varSigma $| can be derived from the first modal axiom and by weakenings. In case (5.2), we can derive |$\neg \nabla \neg p \Rightarrow p$| as follows: and from this we derive |$\varGamma \Rightarrow \varSigma $| by weakenings. In case (6), we derive |$\varGamma \Rightarrow \varSigma $| by using the second modal axiom, Lemma 7.2 and weakenings. For the induction step, let |$\alpha $| be any formula which is not a block and not a constant in |$\varGamma $| or |$\varSigma $|⁠. Then by the definition of propositional formula |$\alpha $| must have one of the forms (⁠|$\beta \vee \gamma $|⁠) or (⁠|$\beta \wedge \gamma $|⁠). If |$\alpha $| has the form (⁠|$\beta \vee \gamma $|⁠) and |$\nabla p$| is a block of |$\beta $| and |$\neg \nabla p$| is a block of |$\gamma $|⁠, then |$\varGamma \Rightarrow \varSigma $| by using the second modal axiom, Lemma 7.2 and weakenings. Otherwise |$\varGamma \Rightarrow \varSigma $| can be derived from |$\vee $|-introduction or |$\wedge $|-introduction, respectively, using either the left case or the right case, depending on whether |$\alpha $| is in |$\varGamma $| or |$\varSigma $|⁠, but not using weakenings. In each case, each top sequent of the rule will have at least one fewer connective |$\wedge $| or |$\vee $| than |$\varGamma \Rightarrow \varSigma $|⁠, and the sequent is valid by the inversion principle. Hence, each top sequent has a derivation in |$\mathfrak{S}$|⁠, by the induction hypothesis. 9 Conclusions This paper was dedicated to studying the logic that preserves degrees of truth with respect to S-algebras, named Six. We showed that this is a six-valued logic that can be defined in terms of four matrices. Six turns out to be to be an interesting logic which combines nice characteristics of Belnap’s four-valued logic |$FOUR$| and |$n$|-valued Łukasiewicz logics (⁠|$2\leq n\leq 5$|⁠). As a paraconsistent logic, it is a genuine LFI with a consistency operator that distinguishes between consistency and contradiction. Finally, we provide a syntactic presentation of Six by means of the sequent calculus |$\mathfrak{S}$|⁠. We think that |$\mathfrak{S}$| deserves a deeper study, for instance, it is an open question wether it admits cut elimination. We presume this is not the case in view of the unusual form of the logical rules that regulate the behaviour of the modal operator |$\nabla $|⁠. In this sense, we think that the framework of hypersequents would be a more suitable tool for presenting a Gentzen-style system for Six with the cut elimination property. Acknowledgements We would like to thank the anonymous referees for their helpful comments and suggestions which allowed us to improve the final version of our paper. References [1] F. Bou , F. Esteva , J. M. Font , A. Gil , L. Godo , A. Torrens and V. Verdú . 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TI - On the logic that preserves degrees of truth associated to involutive Stone algebras
JO - Logic Journal of the IGPL
DO - 10.1093/jigpal/jzy071
DA - 2018-11-29
UR - https://www.deepdyve.com/lp/oxford-university-press/on-the-logic-that-preserves-degrees-of-truth-associated-to-involutive-5JpSXaahAF
SP - 1
VL - Advance Article
IS - 
DP - DeepDyve
ER -