A Basic Dual Intuitionistic Logic and Some of its Extensions Included in G3$_{\text {DH}}$

Abstract

The logic DHb is the result of extending Sylvan and Plumwood’s minimal De Morgan logic B$_{\text {M}}$ with a dual intuitionistic negation of the type Sylvan defined for the extension CC$\omega$ of da Costa’s paraconsistent logic C$\omega$. We provide Routley–Meyer ternary relational semantics with a set of designated points for DHb and a wealth of its extensions included in G3$_{\text {DH}}$, the expansion of G3$_{+}$ with a dual intuitionistic negation of the kind considered by Sylvan (G3$_{+}$ is the positive fragment of Gödelian 3-valued logic G3). All logics in the paper are paraconsistent.

Appendix

The logic C$\omega$ is axiomatized as follows (cf. da Costa (1974); Sylvan (1990), Definition 2.4 and Lemma 2.13):

Axioms: a7, a27, a30, A2, A4 and A5, PEM and DNE.

Rule of inference: MP

Then, we define the following extensions of C$\omega$ (cf. Osorio et al. (2015)): CC$\omega$ : C$\omega$ plus Con; daC$^{\prime }$: C$\omega$ plus a43; daC: CC$\omega$ plus a44; PH$_{1}$: daC plus a42.

Then, the logics C$\omega _{2}$, CC$\omega _{2}$, daC$_{2}^{\prime }$ and daC $_{2}$ are the result of adding A7, $(\lnot A\wedge \lnot B)\rightarrow \lnot (A\vee B)$, to C$\omega$, CC$\omega$, daC$^{\prime }$ and daC, respectively. The relations these logics maintain to each other are summarized in the following diagram (for any logics L, L$^{\prime }$, L $\rightarrow$ L$^{\prime }$ means that L$^{^{\prime }}$ is an extension of L, but not conversely).

The logic H$_{\text {+}}$ is axiomatized with a7, a27, a30, A2, A4, A5 and MP. The logics B$_{\text {M}}$ and G3$_{\text {DH}}$ are defined in section 2 and 6, respectively.

Proposition A.1

(A7 is not CC$\omega$ valid) The De Morgan law A7, $(\lnot A\wedge \lnot B)\rightarrow \lnot (A\vee B)$, is not CC$\omega$-valid.

Proof

We use the semantics defined by Sylvan in his paper (Sylvan 1990). Let $p_{i},p_{m}$ be distinct propositional variables and M a CC$\omega$-model where $a,b,c\in K,Sab$ and Sac and v an assignment such that $v(p_{i},a)=v(p_{m},a)=1$; $v(p_{i},b)=0,v(p_{m},b)=1$; $v(p_{i},c)=1,v(p_{m},c)=0$. Then, $v(\lnot p_{i},a)=1,v(\lnot p_{m},a)=1$, whence $v(\lnot p_{i}\wedge \lnot p_{m},a)=1$. But $v(\lnot (p_{i}\vee p_{m}),a)=0$ since $v(p_{i}\vee p_{m},a)=1,v(p_{i}\vee p_{m},b)=1$ and $v(p_{i}\vee p_{m},c)=1$. Consequently, $v[(\lnot p_{i}\wedge \lnot p_{m})\rightarrow \lnot (p_{i}\vee \lnot p_{m}),a]=0$. $\square$

Finally, the matrix MS5$_{\text {DH}}$ determining the logic S5$_{\text {DH}}$ is defined. The logic S5$_{\text {DH}}$ is a 3-valued extension of positive modal logic S5$_{+}$ (cf. Hacking (1963)). S5$_{\text {DH}}$ is not included in G3$_{ \text {DH}}$ but it could be interpreted, as well as its subsystems, with a reduced RM-semantics similarly as G3$_{\text {DH}}$ and the subsystems of this logics considered in the present paper have been interpreted.

Definition A.2

(The matrix MS5$_{\text {DH}}$) The matrix MS5$_{\text {DH}}$ is the structure $({\mathcal {V}},D,{\mathcal {F}})$ where ${\mathcal {V}},D,{\mathcal {F}}$ are defined similarly as in MG3, except that now $D=\{1,2\}$ and $f_{\rightarrow }$ is defined according to the following truth table:

$$\begin{aligned} \begin{array}{l|lll} \rightarrow &{} 0 &{} 1 &{} 2 \\ \hline 0 &{} 2 &{} 2 &{} 2 \\ 1 &{} 0 &{} 2 &{} 2 \\ 2 &{} 0 &{} 0 &{} 2 \end{array} \end{aligned}$$

Concerning the logic S5$_{\text {DH}}$, that is, the logic determined by MS5$_{\text {DH}}$, we have not axiomatized it and we ignore if it has been axiomatized somewhere in the literature. However, we remark that two logics related to it, i.e., the logic determined by MS5$_{\text {DH}}$ when 2 is the only designated value, and the logic determined by the matrix resulting of replacing the truth-table $\overset{\bullet }{\lnot }$ for negation with truth-table $\sim$, have been axiomatized in Yang (2012) and Robles and Méndez (2019), respectively.

We remark that the following items in Lemma 2.13 are verified by MS5$_{ \text {DH}}$: a1–a13, a20–a26, a32, a33, a35–a39, a41–a43. But, of course, there are many other theses and rules verified by MS5$_{\text {DH}}$, which are not provable in G3$_{\text {DH}}$, for instance, disjunctive Peirce’s law, i.e., $A\vee (A\rightarrow B)$. It is worth-remarking that S5$_{\text {DH}}$ and all the logics included in it can be proved paraconsistent similarly as DH-logics are shown in Proposition 6.5.