TY - JOUR
AU - Osorio Galindo,, Mauricio
AB - Abstract In memoriam José Arrazola Ramírez (1962–2018) The logic |$\textbf{G}^{\prime}_3$| was introduced by Osorio et al. in 2008; it is a three-valued logic, closely related to the paraconsistent logic |$\textbf{CG}^{\prime}_3$| introduced by Osorio et al. in 2014. The logic |$\textbf{CG}^{\prime}_3$| is defined in terms of a multi-valued semantics and has the property that each theorem in |$\textbf{G}^{\prime}_3$| is a theorem in |$\textbf{CG}^{\prime}_3$|⁠. Kripke-type semantics has been given to |$\textbf{CG}^{\prime}_3$| in two different ways by Borja et al. in 2016. In this work, we continue the study of |$\textbf{CG}^{\prime}_3$|⁠, obtaining a Hilbert-type axiomatic system and proving a soundness and completeness theorem for this logic. 1 Introduction Broadly speaking, we can say that paraconsistent logics are systems that do not satisfy ex contradictione quodlibet i.e. there exist some formulas |$\varphi $| and |$\psi $|⁠, such that |$\varphi ,\neg \varphi \not \vdash \psi $|⁠. A logic is paraconsistent if and only if its logical consequence relation is not explosive; this consequence relation could be considered either semantic or proof theoretic [18]. Although there is no general consensus about the definition of paraconsistent logic in [4, 17–19], we can find exhaustive discussions about paraconsistency and some examples of paraconsistent systems. The |$\textbf{CG}^{\prime}_3$| logic is a three-valued paraconsistent logic of recent creation, introduced in [14] by means of multi-valued semantics. The definitions of |$\textbf{CG}^{\prime}_3$| and |$\textbf{G}^{\prime}_{3}$| by means of three-valued matrices are very close; actually, the only difference is that the sets of designated values are different. In [10], the authors define a Kripke-type semantics for |$\textbf{CG}^{\prime}_{3}$| in two different ways. The first one is based on the Kripke-type semantics of |$\textbf{G}^{\prime}_{3}$| following the modelling relation: |$(M,w)\models _{\textbf{CG}^{\prime}_3}\varphi $| iff there exist |$wRw^{\prime}$| such that |$(M,w^{\prime})\models _{\textbf{G}^{\prime}_3}\varphi $|⁠. The second one is achieved by changing the notion of validity; authors propose an alternative way of defining the modelling relation by considering that Kripke models in |$\textbf{CG}^{\prime}_3$| are a subset of those models of |$\textbf{G}^{\prime}_3$| validating existentially the formulas. This paper continues and extends the work in [6, 10, 14]. In [6], the authors present a study about relations between |$\textbf{CG}^{\prime}_3$| and some other logics; nevertheless, they did not present a Hilbert-type axiomatization for |$\textbf{CG}^{\prime}_3$|⁠. Generally speaking, counting with an axiomatization for a logic is desirable, since axiomatizations make the process of comparison with other logics easier, besides allowing a better comprehension on the nature of its connectives. Our main contributions are briefly summarized as follows: |$\bullet $| A list of properties that |$\textbf{CG}^{\prime}_3$| satisfies, from the multi-valued semantics approach, is presented; see Section 3.1. |$\bullet $| A formal axiomatic theory |$\mathbb{L}$|⁠, for |$\textbf{CG}^{\prime}_3$|⁠, is proposed in 3.2. This theory has three primitive connectives, ten axioms and modus ponens as the only inference rule. |$\bullet $| We show a list of properties that are satisfied in |$\mathbb{L}$| among these properties one can find: monotonicity, deduction theorem, rules-AND and cut; see Theorem 1. In addition, properties such as transitivity, introduction of conjunction and change of hypothesis are fulfilled in |$\mathbb{L}$|⁠; see Lemma 1. Also a series of valid properties in |$\mathbb{L}$| are verified; see Lemmas 2 and 4, Proposition 2 and Lemmas 10 and 11. |$\bullet $| It is shown that the formal axiomatic theory |$\mathbb{L}$| is sound and complete with respect to |$\textbf{CG}^{\prime}_3$|⁠; see Theorems 2 and 3. To prove completeness theorem for |$\textbf{CG}^{\prime}_3$|⁠, we proceed analogously to the method presented in [12] for classical logic. |$\bullet $| It is exhibited that the substitution theorem is not satisfied in a general way, by means of an example. However, a restricted substitution theorem is verified by defining a congruence relation between formulas; see Section 3.4. |$\bullet $| In Section 4, it is shown that |$\textbf{CG}^{\prime}_3$| and |$\mathbb{Z}$| are incomparable; see Corollary 1. In this section, we also show that the logic |$\textbf{CG}^{\prime}_3$| and Ł3 have the same expressive power; this means that the connectives of |$\textbf{CG}^{\prime}_3$| can be defined from those of Ł3 and vice versa. The structure of our paper is as follows. In Section 2, we present the necessary background to read the paper. In Section 3, we present the definition of |$\textbf{CG}^{\prime}_3$| from the point of view of multi-valued semantics, then we propose a formal axiomatic theory |$\mathbb{L}$| for |$\textbf{CG}^{\prime}_3$| and examine some interesting properties of |$\mathbb{L}$|⁠. We prove that |$\mathbb{L}$| is sound and complete with respect to |$\textbf{CG}^{\prime}_3$|⁠. We close the section examining the substitution property in |$\textbf{CG}^{\prime}_3$|⁠. In Section 4, we compare |$\textbf{CG}^{\prime}_3$| with some important multi-valued logics. Finally, in Section 5, we present our conclusions and ideas for future work. Proofs are presented in the Appendix. 2 Background We first introduce the syntax of the logical formulas considered in this article, in addition to some basic definitions and concepts. We assume that the reader is familiar with basic concepts of logic such as those of chapter one in [12]. 2.1 Logical system We consider a formal (propositional) language |$\mathcal{L}$| built from a denumerable set of elements called atoms, |$atom(\mathcal{L})=\{p,q,r,$|…|$\};$| the set of connectives |$\mathcal{C}$| formed by the binary connectives |$\wedge ,\vee ,\rightarrow $| and the unary connective |$\neg $|⁠. Formulas are constructed as usual and will be denoted by lowercase Greek letters. The set of all formulas of the language |$\mathcal{L}$| is denoted as |$Form(\mathcal{L})$|⁠. Theories are sets of formulas and will be denoted by uppercase Greek letters. In this paper, a logic is simply a set of formulas closed under modus ponens. All the logics involved in this article are propositional and all of them have the same language, |$\mathcal{L}$|⁠. The elements of a logic |$X$| are called theorems and the notation |$\vdash _X \varphi $| 1 is used to state that the formula |$\varphi $| is a theorem of |$X$|⁠, i.e. |$\varphi \in X$|⁠. We say that a logic |$X$| is weaker than or equal to a logic |$Y$|⁠, if |$X\subseteq Y$|⁠. Equivalently, |$Y$| extends |$X$|⁠. Similarly, we say that |$X$| is stronger than or equal to |$Y,$| if |$Y\subseteq X$|⁠. In logic, a Hilbert-type axiomatic system2 is a type of formal deduction system attributed to Gottlob Frege and David Hilbert. Hilbert-type axiomatic systems are characterized by having a large number of logical axiom schemas and a small set of inference rules. 2.2 Multi-valued semantics The most adequate manner to define the multi-valued semantics of a logic is by means of a matrix. In this section, we introduce the definition of deterministic matrix, also known as logical matrix or simply as matrix. Definition 1 Given a logic |$L$| in the language |$\mathcal{L}$|⁠, the matrix of |$L$| is a structure |$M:=\langle D,D^*,F\rangle $|⁠, where |$\bullet $| |$D$| is a non-empty set of truth values (domain). |$\bullet $| |$D^*$| is a subset of |$D$| (set of designated values). |$F:=\{f_c| c\in \mathcal{C}\}$| is a set of truth functions, with one function for each logical connective |$c$| of |$\mathcal{L}$|⁠. Definition 2 Given a logic |$L$| in the language |$\mathcal{L}$|⁠, an interpretation |$t$|⁠, is a function |$ t:atom(\mathcal{L})\rightarrow D$| that maps atoms to elements in the domain. Any interpretation |$t$| can be extended to a function on all formulas in |$Form(\mathcal{L})$| as usual, i.e. applying recursively the truth functions of logical connectives in |$F$|⁠. If |$t$| is a valuation in the logic |$L$|⁠, then we will say that |$t$| is an |$L$|-valuation. Interpretations allow us to define the notion of validity in this type of semantics as follows: Definition 3 Given a formula |$\varphi $| and an interpretation |$t$| in a logic |$L$|⁠, we say that the formula |$\varphi $| is valid under |$t$| in |$L$|⁠, if |$t(\varphi )\in D^*$|⁠, and we denote it as |$t\vDash _L \varphi $|⁠. Let us note that validity depends on the interpretation, but if we want to talk about ‘logical truths’ in the system, then the validity should be absolute, as stated in next definition: Definition 4 Given a formula |$\varphi $| in the language of a logic |$L$|⁠, we say |$\varphi $| is a tautology in |$L$|⁠, if for every possible interpretation, the formula |$\varphi $| is valid, and we denote it as |$\vDash _L \varphi $|⁠. If |$\varphi $| is a tautology in the logic |$L$|⁠, then we say that |$\varphi $| is an |$L$|-tautology. When a logic is defined via a multi-valued semantics, it is usual to define the set of theorems of |$L$| as the set of tautologies obtained from the multi-valued semantics, i.e. |$\varphi \in $| |$L$| if and only if |$\vDash _{L} \varphi $|⁠. In [11], we can find an exhaustive discussion about many-valued logics and some examples. 3 A Hilbert-type axiomatic system for the |$\textbf{CG}^{\prime}_{3}$| logic In this section, we provide a Hilbert-type axiomatic system for the logic |$\textbf{CG}^{\prime}_3$|⁠, introduced in 2014 by Osorio et al. First, in Section 3.1, we give the definition of |$\textbf{CG}^{\prime}_3$| by means of multi-valued semantics as it is given in [14]; we also recall some properties of this logic presented in [6], then in Section 3.2 we define a formal axiomatic theory for |$\textbf{CG}^{\prime}_{3}$| and prove some basic properties of this theory. In Section 3.3, we introduce some needed results to prove that the formal axiomatic theory proposed is sound and complete with respect to |$\textbf{CG}^{\prime}_{3}$| logic. Finally in Section 3.4 we show that the substitution property does not hold in general, but a restricted version of it is presented. 3.1 The |$\textbf{CG}^{\prime}_{3}$| logic The paraconsistent logic |$\textbf{CG}^{\prime}_{3}$| is introduced in [14], and it is given by the matrix |$M=\langle D,D^*,F\rangle $|⁠, where |$D=\{0,1,2\}$| is the domain, we use these truth values instead of |$\{0,i,1\}$|⁠, as other authors use, to continue with the terminology used in [6, 8, 10, 13], |$D^*=\{1,2\}$| is the set of designated values and |$F$| is the set of truth functions for the connectives |$\{\wedge $|⁠, |$\vee $|⁠, |$\rightarrow , \neg \}$| and consists of the functions shown in Table 1. Table 1 Truth functions for the connectives |$\wedge $|⁠, |$\vee $|⁠, |$\rightarrow $| and |$\neg $| in |$\textbf{CG}^{\prime}_{3}$|⁠. |$f_\wedge $| . 0 . 1 . 2 . . |$f_\vee $| . 0 . 1 . 2 . . |$f_\rightarrow $| . 0 . 1 . 2 . . |$f_\neg $| . . 0 0 0 0 0 0 1 2 0 2 2 2 0 2 1 0 1 1 1 1 1 2 1 0 2 2 1 2 2 0 1 2 2 2 2 2 2 0 1 2 2 0 |$f_\wedge $| . 0 . 1 . 2 . . |$f_\vee $| . 0 . 1 . 2 . . |$f_\rightarrow $| . 0 . 1 . 2 . . |$f_\neg $| . . 0 0 0 0 0 0 1 2 0 2 2 2 0 2 1 0 1 1 1 1 1 2 1 0 2 2 1 2 2 0 1 2 2 2 2 2 2 0 1 2 2 0 Open in new tab Table 1 Truth functions for the connectives |$\wedge $|⁠, |$\vee $|⁠, |$\rightarrow $| and |$\neg $| in |$\textbf{CG}^{\prime}_{3}$|⁠. |$f_\wedge $| . 0 . 1 . 2 . . |$f_\vee $| . 0 . 1 . 2 . . |$f_\rightarrow $| . 0 . 1 . 2 . . |$f_\neg $| . . 0 0 0 0 0 0 1 2 0 2 2 2 0 2 1 0 1 1 1 1 1 2 1 0 2 2 1 2 2 0 1 2 2 2 2 2 2 0 1 2 2 0 |$f_\wedge $| . 0 . 1 . 2 . . |$f_\vee $| . 0 . 1 . 2 . . |$f_\rightarrow $| . 0 . 1 . 2 . . |$f_\neg $| . . 0 0 0 0 0 0 1 2 0 2 2 2 0 2 1 0 1 1 1 1 1 2 1 0 2 2 1 2 2 0 1 2 2 2 2 2 2 0 1 2 2 0 Open in new tab It is worth mentioning that the matrix of |$\textbf{CG}^{\prime}_{3}$| is obtained by adding to the matrix of |$\textbf{G}^{\prime}_{3}$| the value |$1$| as a designated value, also note that each |$\textbf{G}^{\prime}_3$|-valuation is a |$\textbf{CG}^{\prime}_3$|-valuation. The logic |$\mathbb{Z}$|⁠, introduced by Béziau [3], is a paraconsistent logic with an intuitive semantics and a strong negation (see [6, 7]). From the point of view of multi-valued semantics, |$\textbf{CG}^{\prime}_3$| fulfills some properties that may share with other logics, such as |$\mathbb{Z}$| and |$\textbf{G}^{\prime}_3$|⁠. In Table 2, we show these comparisons among |$\textbf{G}^{\prime}_3$|⁠, |$\mathbb{Z}$| and |$\textbf{CG}^{\prime}_3$|⁠. Table 2 Logics |$\textbf{G}^{\prime}_3$|⁠, |$\mathbb{Z}$| and |$\textbf{CG}^{\prime}_3$| are pairwise incomparable. |$((\varphi \rightarrow \psi )\rightarrow \varphi )\rightarrow \varphi $| |$\neg \neg $|-Necessitation |$(\neg \varphi \wedge \neg \psi )\rightarrow \neg (\varphi \vee \psi )$| |$\textbf{G}^{\prime}_3$| No Yes Yes |$\textbf{CG}^{\prime}3$| Yes No Yes |$\mathbb{Z}$| Yes Yes No |$((\varphi \rightarrow \psi )\rightarrow \varphi )\rightarrow \varphi $| |$\neg \neg $|-Necessitation |$(\neg \varphi \wedge \neg \psi )\rightarrow \neg (\varphi \vee \psi )$| |$\textbf{G}^{\prime}_3$| No Yes Yes |$\textbf{CG}^{\prime}3$| Yes No Yes |$\mathbb{Z}$| Yes Yes No Open in new tab Table 2 Logics |$\textbf{G}^{\prime}_3$|⁠, |$\mathbb{Z}$| and |$\textbf{CG}^{\prime}_3$| are pairwise incomparable. |$((\varphi \rightarrow \psi )\rightarrow \varphi )\rightarrow \varphi $| |$\neg \neg $|-Necessitation |$(\neg \varphi \wedge \neg \psi )\rightarrow \neg (\varphi \vee \psi )$| |$\textbf{G}^{\prime}_3$| No Yes Yes |$\textbf{CG}^{\prime}3$| Yes No Yes |$\mathbb{Z}$| Yes Yes No |$((\varphi \rightarrow \psi )\rightarrow \varphi )\rightarrow \varphi $| |$\neg \neg $|-Necessitation |$(\neg \varphi \wedge \neg \psi )\rightarrow \neg (\varphi \vee \psi )$| |$\textbf{G}^{\prime}_3$| No Yes Yes |$\textbf{CG}^{\prime}3$| Yes No Yes |$\mathbb{Z}$| Yes Yes No Open in new tab 3.2 Hilbert-type axiomatic system for |$\textbf{CG}^{\prime}_{3}$| Let us consider |$\mathbb{L}$|⁠, a formal axiomatic theory for |$\textbf{CG}^{\prime}_{3}$| formed by the primitive logical connectives: |$\neg $|⁠, |$\rightarrow $| and |$\wedge $|⁠. Some logical connectives defined in terms of the primitive ones are $$\begin{equation*} \begin{array}{rcl} \sim\varphi &:=& \varphi\rightarrow(\neg\varphi\wedge\neg\neg\varphi)\\ \nabla\varphi &:=& \sim\sim\varphi \wedge \neg\varphi\\ \varphi\vee\psi &:=& \Big((\varphi\rightarrow\psi)\rightarrow\psi\Big)\wedge \Big((\psi\rightarrow\varphi)\rightarrow\varphi\Big)\\ \varphi\leftrightarrow\psi &:=& (\varphi\rightarrow\psi)\wedge(\psi\rightarrow\varphi). \end{array} \end{equation*}$$ The well-formed formulas are constructed as usual. The axiom schemas are |$\textbf{Pos1:} \quad \varphi \rightarrow (\psi \rightarrow \varphi )$| |$\textbf{Pos2:} \quad \Big (\varphi \rightarrow (\psi \rightarrow \sigma )\Big ) \rightarrow \Big ((\varphi \rightarrow \psi ) \rightarrow (\varphi \rightarrow \sigma )\Big )$| |$\textbf{Pos3:} \quad (\varphi \wedge \psi ) \rightarrow \varphi $| |$\textbf{Pos4:} \quad (\varphi \wedge \psi ) \rightarrow \psi $| |$\textbf{Pos5:} \quad \varphi \rightarrow \Big (\psi \rightarrow (\varphi \wedge \psi )\Big )$| |$\textbf{Peirce}^{\prime}\textbf{s law:} \quad ( (\varphi \rightarrow \psi )\rightarrow \varphi )\rightarrow \varphi $| |$\textbf{Cw1:} \quad \varphi \lor \neg \varphi $| |$\textbf{E2:} \quad \neg \neg (\varphi \rightarrow \psi )\leftrightarrow \Big ((\varphi \rightarrow \psi )\wedge (\neg \neg \varphi \rightarrow \neg \neg \psi )\Big )$| |$\textbf{E3:} \quad \neg \neg (\varphi \wedge \psi )\leftrightarrow (\neg \neg \varphi \wedge \neg \neg \psi )$| |$\textbf{WE:} \quad \neg \neg \varphi \rightarrow (\neg \varphi \rightarrow \psi )$| and modus ponens (MP) as the only inference rule. We use |$\varLambda \vdash \lambda $| to mean that there exists a deduction of |$\lambda $| in |$\mathbb{L}$| having as hypothesis the formulas in the set |$\varLambda $|⁠. As we can note, the list of axioms given above has only the first five axioms of the positive fragment of intuitionistic logic, besides |$\textbf{Peirce}^{\prime}\textbf{s law}$|⁠, |$\textbf{Cw1}$|⁠, E2, E3 and the axiom WE. The next theorem shows some basic but useful properties of the formal theory |$\mathbb{L}$|⁠. The demonstration is direct from the definition of |$\varLambda \vdash \lambda $|⁠. Theorem 1 Let |$\varGamma $| and |$\varDelta $| be sets of formulas, and let |$\varphi $| and |$\psi $| be arbitrary formulas, then the following properties hold in |$\mathbb{L}$|⁠. $$\begin{equation*} \begin{array}{ll} Monotonicity & \textrm{If}\ \varGamma\vdash\varphi, \textrm{then}\ \varGamma\cup\varDelta\vdash\varphi \\ Deduction\ theorem & \varGamma, \varphi\vdash\psi\ \textrm{if and only if}\ \varGamma\vdash\varphi\rightarrow\psi\\ Rules-AND & \varGamma\vdash\varphi\wedge\psi\ \textrm{if and only if}\ \varGamma\vdash \varphi\ \textrm{and}\ \varGamma\vdash\psi \\ Cut & \textrm{If}\ \varGamma\vdash\varphi\ \textrm{and}\ \varDelta,\varphi\vdash\psi, \textrm{then}\ \varGamma\cup\varDelta\vdash\psi\\ \end{array} \end{equation*}$$ Lemma 1 For any formulas |$\varphi , \psi ,\sigma $| and |$\xi $|⁠, the following are theorems in |$\mathbb{L}$|⁠. $$\begin{equation*}\begin{array}{ll} (a) & {\vdash \varphi\rightarrow\varphi} \\ (b) & {\varphi\rightarrow\psi}, {\psi\rightarrow\sigma\vdash \varphi\rightarrow\sigma} \\ (c) & {\varphi \rightarrow\psi}, {\sigma \rightarrow\xi } {\vdash (\varphi \wedge\sigma) \rightarrow (\psi \wedge\xi)} \\ (d) & {\vdash (\varphi \rightarrow(\psi\rightarrow\gamma)) \rightarrow ((\varphi \wedge \psi )\rightarrow\gamma)} \\ (e) & {\varphi \rightarrow(\psi \rightarrow\gamma)} {\vdash\psi \rightarrow(\varphi \rightarrow\gamma)} \\ (f) & {\vdash (\varphi \wedge\psi)\leftrightarrow(\psi\wedge\varphi)} \\ (g) & {\vdash\varphi\rightarrow\Big((\varphi\rightarrow\psi)\rightarrow\psi\Big)} \\ (h) & {\varphi\rightarrow\sigma,(\varphi\rightarrow\psi)\rightarrow\sigma,\sigma\rightarrow\psi\vdash\sigma} \\ \end{array}\end{equation*}$$ Proof. Each item can be demonstrated using |$\textbf{Pos1}$|-|$\textbf{Pos5}$|⁠, MP and deduction theorem. 3.3 Soundness and completeness of |$\textbf{CG}^{\prime}_3$| It is worth mentioning that in the list of axiom schemas of |$\mathbb{L}$| not all of the eight axioms of positive fragment of intuitionistic logic are included; however, they can be derived from this list as well as some other well-known results; some of these are shown in the following lemma; in Appendix 1 is the proof of the |$\textbf{Cw2}$| and |$\textbf{CG}^{\prime}3$|⁠; the proofs of the remaining items are straightforward. Lemma 2 The following results hold in |$\mathbb{L}$|⁠. $$\begin{equation*} \begin{array}{lll} \textbf{Pos6}: & & \varphi \rightarrow (\varphi \vee \psi)\\ \textbf{Pos7}: & & \psi \rightarrow (\varphi \vee \psi)\\ \textbf{Pos8}: & & (\varphi \rightarrow \sigma) \rightarrow \Big((\psi \rightarrow \sigma) \rightarrow (\varphi \vee \psi \rightarrow \sigma)\Big)\\ \textbf{Cw2:} & & \neg\neg\varphi \rightarrow \varphi \\ \textbf{CG}^{\prime}\textbf{3:} & & \nabla\varphi \rightarrow\varphi \\ \textbf{E1}: & & (\neg\varphi\rightarrow\neg\psi)\leftrightarrow(\neg\neg\psi\rightarrow\neg\neg\varphi) \\ \textbf{ON}: & & \neg\varphi\leftrightarrow\neg\neg\neg\varphi \\ \textbf{SPC}:& & \varGamma,\varphi\vdash\psi \textrm{and} \varGamma,\neg\varphi\vdash\psi \textrm{if and only if} \varGamma\vdash\psi\\ \end{array}\end{equation*}$$ It is easy to prove that |$\textbf{CG}^{\prime}_3$| is sound with respect to |$\mathbb{L}$|⁠. It suffices to verify that each axiom is a tautology in |$\textbf{CG}^{\prime}_3$| and MP preserves tautologies. Theorem 2 (Soundness). Let |$\varphi $| be a formula. If |$\varphi $| is a theorem in |$\mathbb{L}$|⁠, then |$\varphi $| is a tautology in |$\textbf{CG}^{\prime}_{3}$|⁠, i.e. if |$\vdash _{\mathbb{L}}\varphi $|⁠, then |$\vDash _{\textbf{CG}^{\prime}_{3}}\varphi $|⁠. Now we prove that |$\textbf{CG}^{\prime}_3$| is complete with respect to |$\mathbb{L}$|⁠. To prove that each tautology in |$\textbf{CG}^{\prime}_3$| is a theorem in |$\mathbb{L}$|⁠, we are going to follow the strategy of the proof of completeness used for propositional classical logic given in [12] originally due to Kalmár. It is worth mentioning that this method has been used in other many-valued logics; see [1, 9]. Let us start defining a transformation of formulas in |$\textbf{CG}^{\prime}_3$| in the next definition. Definition 5 Given a valuation |$v$| in |$\textbf{CG}^{\prime}_3$| and an atomic formula |$p$|⁠, we define the atomic formula |$p_{v}$| called the image of |$p$|⁠, as follows: $$\begin{equation*} p_{v} =\left\{\begin{array}{ll} \neg\neg p & \mbox{if} \; v(p)=2\\ \nabla p & \mbox{if} \; v(p)=1\\ \sim p & \mbox{if} \; v(p)=0; \end{array}\right. \end{equation*}$$ this definition can be extended to formulas following the definition of connectives; see Table 1. For a set |$\varPhi $| of formulas with |$\varPhi _v$| we denote the set |$\{\varphi _v|\varphi \in \varPhi \}$|⁠; particularly, |$Atoms(\varphi )_v$| is the set of images under |$v$| of the atomic formulas in |$\varphi $|⁠, always that |$Atoms(\varphi )$| be the set of atomic formulas in |$\varphi $|⁠. To prove the next lemma, which is essential to demonstrate completeness, we need Lemma 11, whose proof requires some additional results, namely, Proposition 2 and Lemma 10. Lemma 3 Given a formula |$\varphi $| and a valuation |$v$| in |$\textbf{CG}^{\prime}_{3}$|⁠, |$Atoms(\varphi )_v \vdash \varphi _{v}$|⁠. Proof. Let |$\varphi $| be a formula and let |$v$| be a valuation in |$\textbf{CG}^{\prime}_{3}$|⁠. The proof is done by induction on the complexity of |$\varphi $|⁠. Base case: if |$\varphi =p$|⁠, where |$p$| is an atomic formula, then we need to show that |$Atoms(\varphi )_v\vdash \varphi _v$|⁠, but this is evident, since |$Atoms(\varphi )_v=\varphi _v=p_v$|⁠. Let us see now that for any formula |$\varphi $| the claim is true. Suppose that if formula |$\psi $| has less complexity than |$\varphi $|⁠, then the lemma holds. Inductive step: suppose that |$\varphi $| is a non atomic formula. We have three cases: Case |$\neg $|: suppose that |$\varphi =\neg \beta $|⁠. By inductive hypothesis, we know that |$Atoms(\beta )_v\vdash \beta _{v}$|⁠. Then, we have three subcases: If |$v(\beta )=2$|⁠, then |$\beta _v=\neg \neg \beta $|⁠. By inductive hypothesis, we have |$Atoms(\beta )_v \vdash \neg \neg \beta $|⁠. Note that |$v(\varphi )=v(\neg \beta )=0$|⁠, so |$\varphi _v=\sim \varphi $|⁠. However, |$\varphi =\neg \beta $|⁠, hence |$\varphi _v=\sim \neg \beta $|⁠. Note that |$Atoms(\beta )_v =Atoms(\varphi )_v $| since |$\beta $|⁠, |$\varphi $| have the same atomic formulas. We need to prove that |$Atoms(\varphi )_v \vdash \sim \neg \beta $|⁠. By Lemma 11, we know that |$\vdash \neg \neg \beta \rightarrow \sim \neg \beta $| and by inductive hypothesis, |$Atoms(\varphi )_v \vdash \neg \neg \beta $|⁠. Applying modus ponens to previous statements, we conclude |$Atoms(\varphi )_v \vdash \sim \neg \beta $|⁠. If |$v(\beta )=1$|⁠, then |$\beta _v=\nabla \beta $|⁠. By inductive hypothesis, |$Atoms(\beta )_v\vdash \nabla \beta $|⁠. Note that |$v(\varphi )=v(\neg \beta )=2$|⁠, so |$\varphi _v=\neg \neg \varphi $|⁠. However, |$\varphi =\neg \beta $|⁠, hence |$\varphi _v=\neg \neg \neg \beta $|⁠. Note that |$Atoms(\beta )_v=Atoms(\varphi )_v$| since |$\beta $|⁠, |$\varphi $| have the same atomic formulas. We need to prove that |$Atoms(\varphi )_v \vdash \neg \neg \neg \beta $|⁠. By Lemma 11, we know that |$\vdash \nabla \beta \rightarrow \neg \neg \neg \beta $| and by inductive hypothesis, |$Atoms(\varphi )_v\vdash \nabla \beta $|⁠. With the application of modus ponens to previous steps, we conclude |$Atoms(\varphi )_v\vdash \neg \neg \neg \beta $|⁠. If |$v(\beta )=0$|⁠, then |$\beta _v=\sim \beta $|⁠. By inductive hypothesis, |$Atoms(\beta )_v\vdash \sim \beta $|⁠. Note that |$v(\varphi )=v(\neg \beta )=2$|⁠, so |$\varphi _v=\neg \neg \varphi $|⁠. However, |$\varphi =\neg \beta $|⁠, hence |$\varphi _v=\neg \neg \neg \beta $|⁠. Note that |$Atoms(\beta )_v=Atoms(\varphi )_v$| since |$\beta $|⁠, |$\varphi $| have the same atomic formulas. We need to prove that |$Atoms(\varphi )_v \vdash \neg \neg \neg \beta $|⁠. By Lemma 11, we know that |$\vdash \sim \beta \rightarrow \neg \neg \neg \beta $| and by inductive hypothesis, |$Atoms(\varphi )_v \vdash \sim \beta $|⁠. Applying modus ponens to previous statements, we conclude |$Atoms(\varphi )_v\vdash \neg \neg \neg \beta $|⁠.Case |$\rightarrow $|: suppose that |$\varphi =\beta \rightarrow \zeta $|⁠. By inductive hypothesis, we know that |$Atoms(\beta )_v\vdash \beta _{v}$| and |$Atoms(\zeta )_v\vdash \zeta _{v}$|⁠. Then, we have six subcases: If |$v(\beta )=0$|⁠, then |$\beta _v=\sim \beta $|⁠. By inductive hypothesis, |$Atoms(\beta )_v\vdash \sim \beta $|⁠. Note that |$v(\varphi )=v(\beta \rightarrow \zeta )=2$|⁠, so |$\varphi _v=\neg \neg \varphi $|⁠. However, |$\varphi =\beta \rightarrow \zeta $|⁠, hence |$\varphi _v=\neg \neg (\beta \rightarrow \zeta )$|⁠. We need to prove |$Atoms(\varphi )_v\vdash \neg \neg (\beta \rightarrow \zeta )$|⁠. By Lemma 11, we know that |$\vdash \sim \beta \rightarrow \neg \neg (\beta \rightarrow \zeta )$| and by inductive hypothesis, |$Atoms(\beta )_v\vdash \sim \beta $|⁠. With the application of modus ponens to previous statements, we conclude |$Atoms(\beta )_v\vdash \neg \neg (\beta \rightarrow \zeta )$|⁠. Finally, by monotonicity |$Atoms(\varphi )_v\vdash \neg \neg (\beta \rightarrow \zeta )$|⁠. If |$v(\zeta )=2$|⁠, then |$\zeta _v=\neg \neg \zeta $|⁠. By hypothesis, |$Atoms(\zeta )_v\vdash \neg \neg \zeta $|⁠. Note that |$v(\varphi )=v(\beta \rightarrow \zeta )=2$|⁠, so |$\varphi _v=\neg \neg \varphi $|⁠. However, |$\varphi =\beta \rightarrow \zeta $|⁠, then |$\varphi _v=\neg \neg (\beta \rightarrow \zeta )$|⁠. We need to prove |$Atoms(\varphi )_v\vdash \neg \neg (\beta \rightarrow \zeta )$|⁠. By Lemma 11, we know that |$\vdash \neg \neg \zeta \rightarrow \neg \neg (\beta \rightarrow \zeta )$| and by inductive hypothesis, |$Atoms(\zeta )_v\vdash \neg \neg \zeta $|⁠. Applying modus ponens to previous steps, we conclude |$Atoms(\zeta )_v\vdash \neg \neg (\beta \rightarrow \zeta )$|⁠. Finally, by monotonicity |$Atoms(\varphi )_v\vdash \neg \neg (\beta \rightarrow \zeta )$|⁠. If |$v(\beta )=1$| and |$v(\zeta )=0$|⁠, then |$\beta _v=\nabla \beta $| and |$\zeta _v=\sim \zeta $|⁠. By inductive hypothesis, we have that |$Atoms(\beta )_v\vdash \nabla \beta $| and |$Atoms(\zeta )_v\vdash \sim \zeta $|⁠. Note that |$v(\varphi )=v(\beta \rightarrow \zeta )=0$|⁠. So |$\varphi _v=\sim \varphi $|⁠, then |$\varphi _v=\sim (\beta \rightarrow \zeta )$|⁠. We need to prove |$Atoms(\varphi )_v\vdash \sim (\beta \rightarrow \zeta )$|⁠. By Lemma 11, we know that |$\vdash (\nabla \beta \wedge \sim \zeta ) \rightarrow \sim (\beta \rightarrow \zeta )$| and by inductive hypothesis, monotonicity and Rules-AND we have |$Atoms(\varphi )_v\vdash \nabla \beta \wedge \sim \zeta $|⁠. With the application of modus ponens to previous steps, we conclude |$Atoms(\varphi )_v\vdash \sim (\beta \rightarrow \zeta )$|⁠. If |$v(\beta )=1$| and |$v(\zeta )=1$|⁠, then |$\beta _v=\nabla \beta $| and |$\zeta _v=\nabla \zeta $|⁠. By inductive hypothesis, we have that |$Atoms(\beta )_v\vdash \nabla \beta $| and |$Atoms(\zeta )_v\vdash \nabla \zeta $|⁠. Note that |$v(\varphi )=v(\beta \rightarrow \zeta )=2$|⁠, so |$\varphi _v=\neg \neg \varphi $|⁠. However, |$\varphi =\beta \rightarrow \zeta $|⁠, hence |$\varphi _v=\neg \neg (\beta \rightarrow \zeta )$|⁠. We need to prove |$Atoms(\varphi )_v\vdash \neg \neg (\beta \rightarrow \zeta )$|⁠. By Lemma 11, we know that |$\vdash (\nabla \beta \wedge \nabla \zeta )\rightarrow \neg \neg (\beta \rightarrow \zeta )$| and by inductive hypothesis, monotonicity and Rules-AND we have that |$Atoms(\varphi )_v\vdash \nabla \beta \wedge \nabla \zeta $|⁠. Applying modus ponens to previous statements, we conclude |$Atoms(\varphi )_v \vdash \neg \neg (\beta \rightarrow \zeta )$|⁠. If |$v(\beta )=2$| and |$v(\zeta )=1$|⁠, then |$\beta _v=\neg \neg \beta $| and |$\zeta _v=\nabla \zeta $|⁠. By inductive hypothesis, we have that |$Atoms(\beta )_v\vdash \neg \neg \beta $| and |$Atoms(\zeta )_v\vdash \nabla \zeta $|⁠. Note that |$v(\varphi )=v(\beta \rightarrow \zeta )=1$|⁠, so |$\varphi _v=\nabla \varphi $|⁠. However, |$\varphi =\beta \rightarrow \zeta $|⁠, then |$\varphi _v=\nabla (\beta \rightarrow \zeta )$|⁠. We need to prove |$Atoms(\varphi )_v\vdash \nabla (\beta \rightarrow \zeta )$|⁠. By Lemma 11, we know that |$\vdash (\neg \neg \beta \wedge \nabla \zeta )\rightarrow \nabla (\beta \rightarrow \zeta )$| and by inductive hypothesis, monotonicity and Rules-AND, |$Atoms(\varphi )_v\vdash \neg \neg \beta \wedge \nabla \zeta $|⁠. With the application of modus ponens to previous statements, we conclude |$Atoms(\varphi )_v\vdash \nabla (\beta \rightarrow \zeta )$|⁠. If |$v(\beta )=2$| and |$v(\zeta )=0$|⁠, then |$\beta _v=\neg \neg \beta $| and |$\zeta _v=\sim \zeta $|⁠. By inductive hypothesis, we have that |$Atoms(\beta )_v\vdash \neg \neg \beta $| and |$Atoms(\zeta )_v\vdash \sim \zeta $|⁠. Note that |$v(\varphi )=v(\beta \rightarrow \zeta )=0$|⁠, so |$\varphi _v=\sim \varphi $|⁠. However, |$\varphi =\beta \rightarrow \zeta $|⁠, hence |$\varphi _v=\sim (\beta \rightarrow \zeta )$|⁠. We need to prove |$Atoms(\varphi )_v \vdash \sim (\beta \rightarrow \zeta )$|⁠. By Lemma 11, we know that |$\vdash (\neg \neg \beta \wedge \sim \zeta )\rightarrow \sim (\beta \rightarrow \zeta )$| and by inductive hypothesis, monotonicity and Rules-AND we have that |$Atoms(\varphi )_v \vdash \neg \neg \beta \wedge \sim \zeta $|⁠. Applying modus ponens to previous statements, we conclude |$ Atoms(\varphi )_v \vdash \sim (\beta \rightarrow \zeta )$|⁠.Case |$\wedge $|: suppose that |$\varphi =\beta \wedge \zeta $|⁠. By inductive hypothesis, we know that |$Atoms(\beta )_v\vdash \beta _{v}$| and |$ Atoms(\zeta )_v \vdash \zeta _{v}$|⁠. Then, we have six subcases: If |$v(\beta )=0$|⁠, then |$\beta _v=\sim \beta $|⁠. By inductive hypothesis, |$ Atoms(\beta )_v \vdash \sim \beta $|⁠. Note that |$v(\varphi )=v(\beta \wedge \zeta )=0$|⁠, so |$\varphi _v=\sim \varphi $|⁠. However, |$\varphi =\beta \wedge \zeta $|⁠, hence |$\varphi _v=\sim (\beta \wedge \zeta )$|⁠. We need to prove |$ Atoms(\varphi )_v \vdash \sim (\beta \wedge \zeta )$|⁠. By Lemma 11, we know that |$\vdash \sim \beta \rightarrow \sim (\beta \wedge \zeta )$| and by hypothesis and monotonicity, |$Atoms(\varphi )_v \vdash \sim \beta $|⁠, with the application of modus ponens to previous statements, we conclude |$ Atoms(\varphi )_v \vdash \sim (\beta \wedge \zeta )$|⁠. If |$v(\zeta )=0$|⁠, then the proof is analogous to the previous case. If |$v(\beta )=1$|⁠, |$v(\zeta )=1$|⁠, then |$\beta _v=\nabla \beta $| and |$\zeta _v=\nabla \zeta $|⁠. By inductive hypothesis, |$Atoms(\beta )_v\vdash \nabla \beta $| and |$Atoms(\zeta )_v \vdash \nabla \zeta $|⁠. Let us note that |$v(\varphi )=v(\beta \wedge \zeta )=1$|⁠. So |$\varphi _v=\nabla \varphi $|⁠. However, |$\varphi =\beta \wedge \zeta $|⁠, hence |$\varphi _v=\nabla (\beta \wedge \zeta )$|⁠. We need to prove |$Atoms(\varphi )_v\vdash \nabla (\beta \wedge \zeta )$|⁠. By Lemma 11, we know that |$\vdash (\nabla \beta \wedge \nabla \zeta ) \rightarrow \nabla (\beta \wedge \zeta )$| and by inductive hypothesis, monotonicity and Rules-AND, |$Atoms(\varphi )_v \vdash \nabla \beta \wedge \nabla \zeta $|⁠. Applying modus ponens to previous steps, we conclude |$ Atoms(\varphi )_v \vdash \nabla (\beta \wedge \zeta )$|⁠. If |$v(\beta )=1$| and |$v(\zeta )=2$|⁠, then |$\beta _v=\nabla \beta $| and |$\zeta _v=\neg \neg \zeta $|⁠. By inductive hypothesis, we have that |$ Atoms(\beta )_v \vdash \nabla \beta $| and |$ Atoms(\zeta )_v \vdash \neg \neg \zeta $|⁠. Note that |$v(\varphi )=v(\beta \wedge \zeta )=1$|⁠, so |$\varphi _v=\nabla \varphi $|⁠. However, |$\varphi =\beta \wedge \zeta $|⁠, hence |$\varphi _v=\nabla (\beta \wedge \zeta )$|⁠. We need to prove |$ Atoms(\varphi )_v \vdash \nabla (\beta \wedge \zeta )$|⁠. By Lemma 11, we know that |$\vdash (\nabla \beta \wedge \neg \neg \zeta ) \rightarrow \nabla (\beta \wedge \zeta )$| and by inductive hypothesis, monotonicity and Rules-AND, |$ Atoms(\varphi )_v \vdash \nabla \beta \wedge \neg \neg \zeta $|⁠. With the application of modus ponens to previous statements, we conclude |$ Atoms(\varphi )_v \vdash \nabla (\beta \wedge \zeta )$|⁠. If |$v(\beta )=2$| and |$v(\zeta )=1$|⁠, then the proof is analogous to the previous case. If |$v(\beta )=2$| and |$v(\zeta )=2$|⁠, then |$\beta _v=\neg \neg \beta $| and |$\zeta _v=\neg \neg \zeta $|⁠. By inductive hypothesis, |$ Atoms(\beta )_v \vdash \neg \neg \beta $| and |$ Atoms(\zeta )_v \vdash \neg \neg \zeta $|⁠. Note that |$v(\varphi )=v(\beta \wedge \zeta )=2$|⁠, so |$\varphi _v=\neg \neg \varphi $|⁠. However, |$\varphi =\beta \wedge \zeta $|⁠, hence |$\varphi _v=\neg \neg (\beta \wedge \zeta )$|⁠. We need to prove |$ Atoms(\varphi )_v \vdash \neg \neg (\beta \wedge \zeta )$|⁠. By |$\textbf{E3}$|⁠, we know that |$\vdash (\neg \neg \beta \wedge \neg \neg \zeta )\rightarrow \neg \neg (\beta \wedge \zeta )$| and by inductive hypothesis, monotonicity and Rules-AND, |$ Atoms(\varphi )_v \vdash \neg \neg \beta \wedge \neg \neg \zeta $|⁠. With the application of modus ponens to previous statements, we conclude |$ Atoms(\varphi )_v \vdash \neg \neg (\beta \wedge \zeta )$|⁠. The next lemma summarizes some important results involving the connectives |$\neg $| and |$\sim $|⁠. Lemma 4 Let |$\varphi $| be a formula. The following are theorems in |$\mathbb{L}$|⁠: (a) |$\vdash \sim \varphi \rightarrow (\neg \neg \varphi \rightarrow \neg \neg \neg \neg \varphi )$| (b) |$\vdash \neg \neg \sim \varphi \rightarrow \neg \neg (\varphi \rightarrow \neg \neg \varphi )$| (c) |$\vdash \neg \neg \varphi \rightarrow (\neg \neg \varphi \rightarrow \neg \neg \neg \neg \varphi )$| (d) |$\vdash \neg \neg \varphi \rightarrow \neg \neg (\varphi \rightarrow \neg \neg \varphi )$| (e) |$\vdash \neg (\varphi \rightarrow \neg \neg \varphi )\rightarrow \neg \sim \varphi $| (f) |$\vdash \neg (\varphi \rightarrow \neg \neg \varphi )\rightarrow \sim \sim \varphi $| (g) |$\vdash \neg (\varphi \rightarrow \neg \neg \varphi )\rightarrow \neg \varphi $| (h) |$\vdash (\neg \varphi \wedge (\varphi \rightarrow \neg \neg \varphi ))\rightarrow \sim \varphi $| We only need one more lemma to give the proof of completeness. This lemma allows to eliminate hypothesis once it is show they are independent of the derivation. It is sufficient to demonstrate that |$\neg \neg \varphi \lor \nabla \varphi \lor \sim \varphi \vdash \psi $|⁠, but to facilitate the reading of the proof, it will be stated as follows: Lemma 5 Let |$\varphi $| and |$\psi $| be formulas and |$\varGamma $| a set of formulas. If |$\varGamma ,\neg \neg \varphi \vdash \psi $|⁠; |$\varGamma ,\nabla \varphi \vdash \psi $| and |$\varGamma ,\sim \varphi \vdash \psi $|⁠, then |$\varGamma \vdash \psi $|⁠. Proof. Applying deduction theorem to |$\vdash (\varphi \rightarrow \neg \neg \varphi )\rightarrow (\varphi \rightarrow \neg \neg \varphi )$|⁠, we have that |$\varphi \rightarrow \neg \neg \varphi , \varphi \vdash \neg \neg \varphi $|⁠. By means of cut from this last formula and the hypothesis |$\varGamma ,\neg \neg \varphi \vdash \psi $|⁠, we obtain |$\varGamma ,(\varphi \rightarrow \neg \neg \varphi ),\varphi \vdash \psi $|⁠. On the other hand, from item (h) of Lemma 4, we have |$\vdash (\neg \varphi \wedge (\varphi \rightarrow \neg \neg \varphi ))\rightarrow \sim \varphi $|⁠. Now, employing Lemma 1, we derive |$\vdash (\varphi \rightarrow \neg \neg \varphi )\rightarrow (\neg \varphi \rightarrow \sim \varphi )$|⁠, and by deduction theorem we have |$(\varphi \rightarrow \neg \neg \varphi ),\neg \varphi \vdash \sim \varphi $|⁠. Because of this formula and hypothesis |$\varGamma ,\sim \varphi \vdash \psi $| using cut, we conclude that |$\varGamma ,(\varphi \rightarrow \neg \neg \varphi ),\neg \varphi \vdash \psi $|⁠. At this time, we have proved |$\varGamma ,(\varphi \rightarrow \neg \neg \varphi ),\varphi \vdash \psi $| and |$\varGamma ,(\varphi \rightarrow \neg \neg \varphi ),\neg \varphi \vdash \psi $|⁠, then applying |$\textbf{SPC}$|⁠, we derive |$\varGamma ,(\varphi \rightarrow \neg \neg \varphi )\vdash \psi $|⁠. On the other hand, using rules-AND to items (f) and (g) of Lemma 4, we obtain |$\neg (\varphi \rightarrow \neg \neg \varphi )\vdash (\sim \sim \varphi \wedge \neg \varphi )$|⁠. Equivalently, because of abbreviation of |$\nabla $|⁠, |$\neg (\varphi \rightarrow \neg \neg \varphi )\vdash \nabla \varphi $|⁠, then applying cut to this formula and hypothesis |$\varGamma ,\nabla \varphi \vdash \psi $|⁠, we conclude that |$\varGamma ,\neg (\varphi \rightarrow \neg \neg \varphi )\vdash \psi $|⁠. Therefore, applying |$\textbf{SPC}$| to |$\varGamma ,(\varphi \rightarrow \neg \neg \varphi )\vdash \psi $| and |$\varGamma ,\neg (\varphi \rightarrow \neg \neg \varphi )\vdash \psi $| we conclude that |$\varGamma \vdash \psi $|⁠. Finally, we present one of the main results of this paper. The proof is a consequence of Lemma 3, Lemma 5 and Theorem 1. Theorem 3 (Completeness). Let |$\varphi $| be a formula. If |$\varphi $| is a tautology in |$\textbf{CG}^{\prime}_{3}$|⁠, then |$\varphi $| is a theorem in |$\mathbb{L}$|⁠, i.e. if |$\vDash _{\textbf{CG}^{\prime}_{3}}\varphi $|⁠, then |$\vdash _{\mathbb{L}}\varphi $|⁠. Proof. Suppose |$\varphi $| is a tautology whose set of atomic formulas is |$\varPhi $|⁠. From Lemma 3, we have |$\varPhi _v \vdash \varphi _v$| for every valuation |$v$|⁠. Then we have two cases. $$\begin{equation*} \begin{array}{ll} Case 1: & \textrm{if}\ v(\varphi)=2, \textrm{then}\ \varPhi_v\vdash\neg\neg\varphi.\\ Case 2: & \textrm{if}\ v(\varphi)=1, \textrm{then}\ \varPhi_v\vdash\nabla\varphi.\\ \end{array} \end{equation*}$$ In the case |$\varPhi _v\vdash \neg \neg \varphi $|⁠, by |$\textbf{Cw2}$| and modus ponens we have |$\varPhi _v\vdash \varphi $|⁠. Let |$p$| be any atomic formula in |$\varPhi $| and let |$\varGamma :=\varPhi \setminus \{p\}$|⁠. Then we have that |$\varGamma _v,p_v \vdash \varphi $| for any valuation |$v$|⁠. Therefore, it must hold that |$\varGamma _v,\neg \neg p\vdash \varphi $|⁠; |$\varGamma _v,\nabla p\vdash \varphi $| and |$\varGamma _v,\sim p \vdash \varphi $|⁠. By Lemma 5, we have |$\varGamma _v\vdash \varphi $|⁠. After |$|\varPhi |$| steps, finally we obtain |$\vdash \varphi $|⁠. In the case |$\varPhi _v\vdash \nabla \varphi $|⁠, by |$\textbf{CG}^{\prime}3$| and modus ponens, we have that |$\varPhi _v\vdash \varphi $|⁠. Let |$p$| be any atomic formula in |$\varPhi $| and let |$\varGamma :=\varPhi \setminus \{p\}$|⁠. Then we have that |$\varGamma _v,p_v\vdash \varphi $| for any |$v$| valuation. Therefore, it must holds that |$\varGamma _v, \neg \neg p\vdash \varphi $|⁠; |$\varGamma _v,\nabla p\vdash \varphi $| and |$\varGamma _v,\sim p \vdash \varphi $|⁠. By Lemma 5, we have |$\varGamma _v\vdash \varphi $|⁠. After |$|\varPhi |$| steps, finally we obtain |$\vdash \varphi $|⁠. 3.4 Some remarks about connective |$\rightarrow $| in |$\textbf{CG}^{\prime}_{3}$| For a pair of formulas |$\sigma $| and |$\varphi $| and an atom |$p$|⁠, we write |$\sigma [\varphi /p]$| to represent the formula obtained by substituting in |$\sigma $| the formula |$\varphi $| for the atomic formula |$p$|⁠. In [2, Theorem 3.5], the authors prove that any propositional paraconsistent logic with a connective |$\rightarrow $| satisfying |$\mathcal{T},\varphi \vdash \psi $| iff |$\mathcal{T}\vdash \varphi \rightarrow \psi $| has not the replacement property, i.e. there exist some formulas |$\varphi $|⁠, |$\psi $| and |$\sigma $| such that |$\varphi \vdash \psi $| and |$\psi \vdash \varphi $| but |$\sigma [\varphi / p] \not \vdash \sigma [\psi / p]$| or |$\sigma [\psi / p] \not \vdash \sigma [\varphi / p]$|⁠. This is the case of |$\textbf{CG}^{\prime}_3$|⁠; it suffices with consider that the formula |$(p\rightarrow p)\leftrightarrow (p\vee \sim p)$| is a theorem, but |$\neg (p\rightarrow p)\leftrightarrow \neg (p\vee \sim p)$| is not. They also show in [2, Proposition 3.3] that if |$\varphi $| and |$\psi $| are formulas such that for any |$v$| valuation |$v(\varphi )=v(\psi )$|⁠, then |$\sigma [\varphi / p] \vdash \sigma [\psi / p]$| and |$\sigma [\psi / p] \vdash \sigma [\varphi / p]$| for any formula |$\sigma $|⁠. Now we define the connective |$\Leftrightarrow $| as follows. We write |$\varphi \Leftrightarrow \psi $| to denote the formula: |$(\varphi \leftrightarrow \psi )\wedge (\neg \varphi \leftrightarrow \neg \psi )$|⁠. The reader can verify that |$\varphi \Leftrightarrow \psi $| is a tautology if and only if for every valuation |$v$|⁠, |$v(\varphi )=v(\psi )$|⁠. Therefore, if |$\varphi \Leftrightarrow \psi $| is a tautology, then |$\sigma [\varphi / p] \vdash \sigma [\psi / p]$| and |$\sigma [\psi / p] \vdash \sigma [\varphi / p]$| for any formula |$\sigma $|⁠. Definition 6 Let |$\varphi $| be a formula in a set |$\varGamma $| and assume that we are given a deduction |$\psi _{1},\ldots ,\psi _{n}$| from |$\varGamma $|⁠, together with a justification for each step in the deduction. We say that |$\psi _{i}$| depends upon |$\varphi $| in this proof if and only if |$\psi _{i}$| is |$\varphi $| and the justification for |$\psi _{i}$| is that it belongs to |$\varGamma $|⁠, or |$\psi _{i}$| is justified as a direct consequence by MP of some preceding formulas of the sequence, where at least one of these preceding formulas depend upon |$\varphi $|⁠. Proposition 1 If |$\psi $| does not depend upon |$\varphi $| in |$\varGamma ,\varphi \vdash _{\textbf{CG}^{\prime}_3} \psi $|⁠, then |$\varGamma \vdash _{\textbf{CG}^{\prime}_3} \psi $|⁠. Proof. The demonstration is similar to the process proposed in [12, Proposition 2.4]. Lemma 6 (Substitution). Let |$\varphi _{1}$| and |$\varphi _{2}$| be two formulas in |$\textbf{CG}^{\prime}_3$|⁠, such that |$\vdash _{\textbf{CG}^{\prime}_3} \varphi _{1}\leftrightarrow \varphi _{2}$| and such that there exists a derivation which does not depend upon |$\nabla \varphi \rightarrow \varphi $|⁠. Let |$\theta $| be a formula and |$p$| an atom. Then |$\vdash _{\textbf{CG}^{\prime}_3}\theta [\varphi _{1}/p]\leftrightarrow \theta [\varphi _{2}/p]$|⁠. Proof. The proof is similar to the one proposed in [16, Lemma 21]. 4 Comparisons of |$\textbf{CG}^{\prime}_3$| with other logics Let us now compare |$\textbf{CG}^{\prime}_3$| with other logics. At this point of the analysis, we are in position of treat theorems and tautologies in |$\textbf{CG}^{\prime}_3$| equally. As we have mentioned previously, any tautology in |$\textbf{G}^{\prime}_3$| is a tautology in |$\textbf{CG}^{\prime}_3$|⁠; therefore, any theorem in |$\textbf{G}^{\prime}_3$| is a theorem in |$\textbf{CG}^{\prime}_3$|⁠. Beside this, we have that the formula $$\begin{equation*}((\varphi\rightarrow\psi)\rightarrow\varphi)\rightarrow\varphi\end{equation*}$$ is a theorem in |$\textbf{CG}^{\prime}_3$|⁠. However, this formula is not a theorem in |$\textbf{G}^{\prime}_3$|⁠; it confirms that |$\textbf{CG}^{\prime}_3$| is a proper extension of |$\textbf{G}^{\prime}_3$|⁠. On the other hand, it is well known that in some logics, such as the logic |$\mathbb{Z}$|⁠, developed by Béziau in [3], in the logic |$\textbf{daC}$| of da Costa and in the logic |$\textbf{G}^{\prime}_3$|⁠, the |$\neg \neg $|-necessitation rule is verified (for more details see [6]). This rule establishes that if |$\varphi $| is a theorem in the logic |$X$|⁠, then also |$\neg \neg \varphi $| must be a theorem in |$X$|⁠. This rule, however, is not verified in |$\textbf{CG}^{\prime}_3$|⁠. To see this, it suffices to consider the formula |$p\vee \sim p$|⁠, which is a theorem in |$\textbf{CG}^{\prime}_3$|⁠, nevertheless its double negation |$\neg \neg (p\vee \sim p)$| is not. Other important formula is one of the De Morgan’s laws, namely, the formula |$(\neg \varphi \wedge \neg \psi )\rightarrow \neg (\varphi \vee \psi )$|⁠, which is a theorem in |$\textbf{CG}^{\prime}_3$| but it is not a theorem in |$\mathbb{Z}$|⁠. This properties are relevant, since they are used to define a family of logics to reconstruct logic |$\mathbb{Z}$| (see [16]). As a simple corollary of these examples we have that Corollary 1 |$\textbf{CG}^{\prime}_3$| and |$\mathbb{Z}$| are incomparable. In Figure 1, we present some logics related to |$\textbf{CG}^{\prime}_3$|⁠, where the arrows in the figure mean contention. The logic |$\textbf{daC}^{\prime}$| defined in [13] is a weaker version of the logic |$\textbf{daC}$|⁠. The logic |$\textbf{PH}_{1}$| was proposed in [5] as an extension of the logic |$\textbf{daC}$|⁠. The logic |$L_5$| was introduced in [14]; this logic is weaker than the logic |$\mathbb{Z}$|⁠. The logics |$\mathbb{Z}1$| and |$\mathcal{P}$|-|$\mathcal{FOUR^{\prime}}$| are extensions of |$\mathbb{Z}$|⁠; see [13]. The logic |$\mathcal{P}$|-|$\mathcal{F}\mathcal{O}\mathcal{U}\mathcal{R}$| defined in [15] is a 4-valued logic and it is studied with further detail in [7]. The logic |$\textbf{G}^{\prime}_3$| extends |$\textbf{PH}_{1}$|⁠; see [7]. Figure 1. Open in new tabDownload slide Relationships among logics close to |$\textbf{CG}^{\prime}_3$|⁠. Figure 1. Open in new tabDownload slide Relationships among logics close to |$\textbf{CG}^{\prime}_3$|⁠. Let us see now that the three-valued logic Ł3 of Łukasiewicz and |$\textbf{CG}^{\prime}_3$| have the same expressive power. To construct the three-valued logic Ł3 of Łukasiewicz, we consider a propositional language |$\mathcal{L}$|⁠, a denumerable set of atomic formulas, the set of binary connectives {→Ł3, ∧Ł3, ∨Ł3}, the set of unary connectives and the constant logic ⊥Ł3. Lemma 7 In Ł3, if we consider →Ł3 and the constant ⊥Ł3 (which as interpreted as 0) as primitive connectives, then we can obtain the rest of its connectives: |$ $| Proof. The proof follows straightforward from the truth tables of Ł3. Lemma 8 The connectives of |$\textbf{CG}^{\prime}_3$| are definable in the language of the connectives of Ł3. Proof. It is suffices to show that |$\neg _{\textbf{CG}^{\prime}_3}$| and |$\rightarrow _{\textbf{CG}^{\prime}_3}$| are definable en terms of connectives in Ł3, since the rest of connectives have the same truth tables in both logics. Note that |$\bullet $| |$\neg _{\textbf{CG}^{\prime}_3}\varphi =\neg _L\Box _L\varphi $| |$\bullet $| Lemma 9 The negation and implication of Ł3 can be expressed in terms of the connectives of |$\textbf{CG}^{\prime}_3$|⁠. Note that Theorem 4 The connectives of Ł|$3$| can be expressed in terms of the connectives of |${\textbf{CG}^{\prime}_3}$| and vice versa. Proof. Straightforward from Lemmas 7, 8 and 9. 5 Conclusions This paper provides a formal axiomatic theory for |$\textbf{CG}^{\prime}_3$|⁠. The |$\textbf{CG}^{\prime}_3$| logic was defined in [14] by means of multi-valued semantics and later on, in [10] a Kripke-type semantics for it was given. However, |$\textbf{CG}^{\prime}_3$| lacked of a formal axiomatic theory to develop proof theory. In this work, a Hilbert-type axiomatization for |$\textbf{CG}^{\prime}_3$| is given. This axiomatic system satisfies many good-nature properties such as those presented in Lemma 1 and Theorem 1. Among this properties, we can mention deduction theorem, transitivity, rules-AND, cut, etc. In Lemma 2, we prove that this axiomatization contains the positive fragment of intuitionistic logic. Even more, the positive fragment of classical logic is contained in |$\mathbb{L}$|⁠. It is worth mentioning that this axiomatization is an extension of the axiomatization for |$\textbf{C}_{\omega}$|⁠. We proved that this formal axiomatic theory is sound and complete with respect to |$\textbf{CG}^{\prime}_3$|⁠. By means of this axiomatization, we perform some comparisons with other logics such as |$\mathbb{Z}$| and |$\textbf{G}^{\prime}_3$| and we also show that Ł3 and |$\textbf{CG}^{\prime}_3$| have the same expressive power. As a future work, we plan to analyse if |$\textbf{CG}^{\prime}_3$| is algebraizable in the sense of Block and Pigozzi. On the other hand, an anonymous referee suggested apply Suszko’s thesis to |$\textbf{CG}^{\prime}_3$| and then use regular Kalmár’s proof to classical logic. We think that this is an interesting possibility. We plan to compare both approaches in a future work. Footnotes 1 We drop the subscript |$X$| when the given logic is understood from the context. 2 Sometimes called Hilbert calculus, Hilbert style deductive system or a Hilbert–Ackerman system. References [1] C. Anahit and K. Artur. Generalization of Kalmar’s proof of deducibility in two valued propositional logic into many valued logic . Pure and Applied Mathematics Journal , 6 , 71 – 75 , 2017 . Google Scholar Crossref Search ADS WorldCat [2] A. Avron and J.-Y. Beziau. Self-extensional three-valued paraconsistent logics have no implication . Logic Journal of the IGPL , 25 , 183 – 194 , 2016 . Google Scholar OpenURL Placeholder Text WorldCat [3] J.-Y. Béziau . 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Proof of Lemma 2: Next, we present a proof for each item, let us recall that |$\sim \varphi $| is an abbreviation of |$\varphi \rightarrow (\neg \varphi \wedge \neg \neg \varphi )$|⁠. $$\begin{align*} \begin{array}{llll} & & \textbf{Cw2} \quad \vdash\neg\neg\varphi \rightarrow \varphi & \\ & & & \\ &1.& \varphi\rightarrow(\neg\neg\varphi\rightarrow\varphi) & \textbf{Pos1}\\ &2.& \neg\varphi\rightarrow(\neg\neg\varphi\rightarrow\varphi) & \textbf{WE} \\ &3.& (\varphi\rightarrow(\neg\neg\varphi\rightarrow\varphi))\rightarrow((\neg\varphi\rightarrow(\neg\neg\varphi\rightarrow\varphi))\rightarrow((\varphi\vee\neg\varphi)\rightarrow\\ &&(\neg\neg\varphi\rightarrow\varphi)))& \textbf{Pos8}\\ &4.& (\neg\varphi\rightarrow(\neg\neg\varphi\rightarrow\varphi))\rightarrow((\varphi\vee\neg\varphi)\rightarrow(\neg\neg\varphi\rightarrow\varphi)) & MP(1,3)\\ &5.& (\varphi\vee\neg\varphi)\rightarrow(\neg\neg\varphi\rightarrow\varphi) & MP(2,4)\\ &6.& \varphi\vee\neg\varphi & \textbf{Cw1} \\ &7.& \neg\neg\varphi\rightarrow\varphi & MP(5,6)\\ & & & \end{array} \end{align*}$$ $$\begin{equation*} \begin{array}{llll} & & \textbf{CG}^{\prime}{3} \quad \vdash\nabla\varphi \rightarrow\varphi & \\ & & & \\ &1.& \nabla\varphi &\qquad\qquad\qquad\ \ \ \ Hyp \\ &2.& \sim\sim\varphi\wedge\neg\varphi &\qquad\qquad\qquad\ \ \ \ Abbr of \nabla \\ &3.& (\sim\sim\varphi\wedge\neg\varphi)\rightarrow\sim\sim\varphi &\qquad\qquad\qquad\ \ \ \ \textbf{Pos3} \\ &4.& \sim\sim\varphi &\qquad\qquad\qquad\ \ \ \ MP(1,2) \\ &5.& \sim\varphi\rightarrow(\neg\sim\varphi\wedge\neg\neg\sim\varphi) &\qquad\qquad\qquad\ \ \ \ Abbr of \sim \\ &6.& (\varphi\rightarrow(\neg\varphi\wedge\neg\neg\varphi))\rightarrow(\neg\sim\varphi\wedge\neg\neg\sim\varphi) &\qquad\qquad\qquad\ \ \ \ Abbr of \sim \\ &7.& \neg\sim\varphi\rightarrow(\neg\neg\sim\varphi\rightarrow\varphi) &\qquad\qquad\qquad\ \ \ \ \textbf{WE} \\ &8.& (\neg\sim\varphi\wedge\neg\neg\sim\varphi)\rightarrow\varphi &\qquad\qquad\qquad\ \ \ \ Lemma\ 1 \\ &9.& (\varphi\rightarrow(\neg\varphi\wedge\neg\neg\varphi))\rightarrow\varphi &\qquad\qquad\qquad\ \ \ \ Lemma\ 1 \\ &10.& ((\varphi\rightarrow(\neg\varphi\wedge\neg\neg\varphi))\rightarrow\varphi)\rightarrow\varphi &\qquad\qquad\qquad\ \ \ \ \textbf{Pierce} \\ &11.& \varphi &\qquad\qquad\qquad\ \ \ \ MP(9,10) \\ &12.& \nabla\varphi\vdash\varphi &\qquad\qquad\qquad\ \ \ \ 1-11 \\ &13.& \nabla\varphi\rightarrow\varphi &\qquad\qquad\qquad\ \ \ \ DT to 12 \\ \end{array} \end{equation*}$$ Proposition 2 For any formulas |$\varphi ,\psi $|⁠, the following are theorems in |$\mathbb{L}$|⁠. (a) |$\vdash \sim \varphi \rightarrow (\varphi \rightarrow \psi )$| (b) |$\vdash (\varphi \rightarrow \psi )\rightarrow ((\varphi \rightarrow \sim \psi )\rightarrow (\varphi \rightarrow (\neg \psi \wedge \neg \neg \psi )))$| (c) |$\vdash (\neg \psi \wedge \neg \neg \psi )\rightarrow (\neg \varphi \wedge \neg \neg \varphi )$| (d) |$\vdash (\varphi \rightarrow \psi )\rightarrow ((\varphi \rightarrow \sim \psi )\rightarrow \sim \varphi )$| (e) |$\vdash (\varphi \rightarrow \psi )\rightarrow (\sim \psi \rightarrow \sim \varphi )$| (f) |$\vdash \sim \sim \psi \rightarrow (\sim \varphi \rightarrow ((\psi \rightarrow \varphi )\rightarrow (\sim \psi \wedge \sim \sim \psi )))$| (g) |$\vdash (\sim \psi \wedge \sim \sim \psi )\rightarrow (\neg (\psi \rightarrow \varphi )\wedge \neg \neg (\psi \rightarrow \varphi ))$| (h) |$\vdash (\sim \sim \psi \wedge \sim \varphi )\rightarrow \sim (\psi \rightarrow \varphi )$| (i) |$\vdash \sim \sim \varphi \rightarrow \sim \sim (\psi \rightarrow \varphi )$| (j) |$\vdash \varphi \rightarrow \sim \sim \varphi $| (k) |$\vdash \psi \rightarrow (\sim \varphi \rightarrow \sim (\psi \rightarrow \varphi ))$| (l) |$\vdash \sim \sim \varphi \rightarrow \varphi $| (m) |$\vdash (\sim \sim \varphi \wedge \sim \sim \psi )\rightarrow \sim \sim (\varphi \wedge \psi )$| Lemma 10 For any formula |$\varphi $|⁠, the following formulas are theorems in |$\mathbb{L}$|⁠. (a) |$\vdash \sim \varphi \rightarrow \neg \varphi $| (b) |$\vdash \sim \neg \varphi \rightarrow \sim \sim \varphi $| (c) |$\vdash \neg \neg \varphi \rightarrow \sim \neg \varphi $| (d) |$\vdash \neg \neg \varphi \rightarrow \sim \sim \varphi $| (e) |$\vdash \sim \varphi \rightarrow \neg \neg \sim \varphi $| (f) |$\vdash \neg \sim \varphi \rightarrow \sim \sim \varphi $| (g) |$\vdash \sim \varphi \rightarrow (\neg \neg \varphi \rightarrow \neg \neg \psi )$| (h) |$\vdash \neg \neg (\varphi \wedge \psi )\rightarrow \neg \neg \varphi $| (i) |$\vdash (\neg \varphi \wedge \neg \psi )\rightarrow \neg (\varphi \wedge \psi )$| (j) |$\vdash (\nabla \varphi \wedge \nabla \psi )\rightarrow \sim \sim (\varphi \wedge \psi )$| (k) |$\vdash (\nabla \varphi \wedge \nabla \psi )\rightarrow \neg (\varphi \wedge \psi )$| (l) |$\vdash (\neg \neg \varphi \wedge \nabla \psi )\rightarrow \sim \sim (\varphi \rightarrow \psi )$| (m) |$\vdash (\neg \neg \varphi \wedge \nabla \psi )\rightarrow (\neg \neg (\varphi \rightarrow \psi )\rightarrow \neg \neg \psi )$| (n) |$\vdash (\neg \neg \varphi \wedge \nabla \psi )\rightarrow \neg (\varphi \rightarrow \psi )$| (o) |$\vdash (\nabla \varphi \wedge \neg \neg \psi )\rightarrow \sim \sim (\varphi \wedge \psi )$| (o) |$\vdash (\nabla \varphi \wedge \neg \neg \psi )\rightarrow \neg (\varphi \wedge \psi )$| Lemma 11 For any formulas |$\varphi $|⁠, the following are theorems in |$\mathbb{L}$|⁠. (a) |$\vdash \neg \neg \varphi \rightarrow \sim \neg \varphi $| (b) |$\vdash \nabla \varphi \rightarrow \neg \neg \neg \varphi $| (c) |$\vdash \sim \varphi \rightarrow \neg \neg \neg \varphi $| (d) |$\vdash \sim \varphi \rightarrow \neg \neg (\varphi \rightarrow \psi )$| (e) |$\vdash \neg \neg \psi \rightarrow \neg \neg (\varphi \rightarrow \psi )$| (f) |$\vdash (\nabla \varphi \wedge \sim \psi )\rightarrow \sim (\varphi \rightarrow \psi )$| (g) |$\vdash (\nabla \varphi \wedge \nabla \psi )\rightarrow \neg \neg (\varphi \rightarrow \psi )$| (h) |$\vdash (\neg \neg \varphi \wedge \nabla \psi )\rightarrow \nabla (\varphi \rightarrow \psi )$| (i) |$\vdash (\neg \neg \varphi \wedge \sim \psi )\rightarrow \sim (\varphi \rightarrow \psi )$| (j) |$\vdash \sim \varphi \rightarrow \sim (\varphi \wedge \psi )$| (k) |$\vdash (\nabla \varphi \wedge \nabla \psi )\rightarrow \nabla (\varphi \wedge \psi )$| (l) |$\vdash (\nabla \varphi \wedge \neg \neg \psi )\rightarrow \nabla (\varphi \wedge \psi )$| © The Author(s) 2020. 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TI - An axiomatic approach to CG′3 logic
JF - Logic Journal of the IGPL
DO - 10.1093/jigpal/jzaa014
DA - 2020-12-11
UR - https://www.deepdyve.com/lp/oxford-university-press/an-axiomatic-approach-to-cg-3-logic-QnBbGWUJY0
SP - 1218
EP - 1232
VL - 28
IS - 6
DP - DeepDyve
ER -