Strengthening Brady’s Paraconsistent 4-Valued Logic BN4 with Truth-Functional Modal Operators

Abstract

Łukasiewicz presented two different analyses of modal notions by means of many-valued logics: (1) the linearly ordered systems Ł3,..., ,..., $\hbox {L}_{\omega }$; (2) the 4-valued logic Ł he defined in the last years of his career. Unfortunately, all these systems contain “Łukasiewicz type (modal) paradoxes”. On the other hand, Brady’s 4-valued logic BN4 is the basic 4-valued bilattice logic. The aim of this paper is to show that BN4 can be strengthened with modal operators following Łukasiewicz’s strategy for defining truth-functional modal logics. The systems we define lack “Łukasiewicz type paradoxes”. Following Brady, we endow them with Belnap–Dunn type bivalent semantics.

Appendix

1.1 Brady’s Original Axiomatization of BN4

Brady formulated BN4 with the following axioms, rules and definition.

Axioms:

Rules:

Definition:

$$\begin{aligned} \hbox {Disjunction. }A\vee B=_{\mathrm {df}}\lnot (\lnot A\wedge \lnot B) \end{aligned}$$

Let us label $\mathrm{BN}4_{0}$ Brady’s original formulation. Then, we have:

Proposition 6.1

($\mathrm{BN}4_{0}$ and BN4 are deductively equivalent) The logic $\mathrm{BN}4_{0}$ and BN4 (as defined in Definition 2.10) are deductively equivalent. That is, for $A\in {\mathcal {F}}$, $\vdash _{\mathrm {BN4}}A$ iff $\vdash _{\mathrm {BN4}_{0}}A.$

Proof

(a) If $\vdash _{\mathrm {BN4}}A$, then $\vdash _{\mathrm {BN4}_{0}}A$. It follows immediately from the completeness of $\mathrm{BN}4_{0}$ w.r.t. MBN4-validity (cf. Corollary in p. 28 of Brady (1982); cf. Definition 2.7 above) since all axioms and rules of BN4 (as defined in Definition 2.10) hold in the matrix MBN4. (b) If $\vdash _{\mathrm {BN4}_{0}}A$, then $\vdash _{\mathrm {BN4}}A$: it suffices to prove that a11, a12 and Aff are provable in BN4 (Definition 2.10). a11 $A\rightarrow [(A\rightarrow \lnot A)\rightarrow \lnot A]$ is immediate by A3 ($A\rightarrow [(A\rightarrow B)\rightarrow B]$); a12 $A\vee [\lnot A\rightarrow (A\rightarrow B)]$ is easy by A14 ($A\vee [\lnot (A\rightarrow B)\rightarrow B]$), and T8 ($(\lnot A\rightarrow B)\rightarrow (\lnot B\rightarrow A)$). Finally, Aff is immediate by A2 and the thesis $(B\rightarrow C)\rightarrow [(A\rightarrow B)\rightarrow (A\rightarrow C)]$ which is in its turn immediate by A2 and T5 ($[A\rightarrow (B\rightarrow C)]\rightarrow [B\rightarrow (A\rightarrow C)]$). $\square$

1.2 Łukasiewicz’s Matrix MŁ

Let us define our (version of) Łukasiewicz’s matrix MŁ (cf. Font and Hajek 2002; Tkaczyk 2011 and Méndez et al. 2015).

Definition 6.2

(The matrix MŁ) The proposition language consists of the connectives $\rightarrow ,\lnot ,L$ . The matrix MŁ is the structure $({\mathcal {V}},D,f_{\rightarrow },f_{\lnot },f_{L})$ where ${\mathcal {V}}=\{0,1,2,3\}$ and it is partially ordered as in Belnap–Dunn’s matrix MB4 (Definition 2.5), $D=\{3\}$ and $f_{\rightarrow },f_{\lnot }$ and $f_{L}$ are defined according to the following tables:

The related notions of MŁ-interpretation, etc. are defined according to the general Definition 2.4.

1.3 Smiley’s Matrix MSm4

Smiley’s matrix MSm4 can be defined as follows (cf. Anderson and Belnap 1975, pp. 161–162).

Definition 6.3

(Smiley’s matrix MSm4) The propositional language consists of the connectives $\wedge$, $\vee$, $\lnot$ and $\rightarrow$. Smiley’s matrix MSm4 is the structure $(\mathcal { V},D,$ F) where (1) ${\mathcal {V}}$ and D are defined as in the matrix MŁ and F $=\{f_{\wedge },f_{\vee },f_{\lnot },f_{\rightarrow }\}$ where $f_{\wedge }$, $f_{\vee }$ and $f_{\lnot }$ are defined as in MB4 and $f_{\rightarrow }$ according to the following table:

1.4 Anderson and Belnap’s Matrix $\mathrm{M}_{0}$

Anderson and Belnap’s $\mathrm{M}_{0}$ can be defined as follows (cf. Belnap 1960; Anderson and Belnap 1975, §22.1.3).

Definition 6.4

(Anderson and Belnap’s matrix $\mathrm{M}_{0}$) The propositional language consists of the connectives $\wedge$, $\vee$, $\lnot$ and $\rightarrow$. Anderson and Belnap’s matrix $\mathrm{M}_{0}$ is the structure $( {\mathcal {V}},D,$ F) where (1) ${\mathcal {V}}=\{0,1,2,3,4,5,6,7\}$ , $D=\{4,5,6,7\}$ and $f_{\wedge },f_{\vee },f_{\lnot }$ and $f_{\rightarrow }$ in F are defined according to the following truth tables:

The matrix $\mathrm{M}_{0}$ is axiomatized in Brady (2003). Anderson and Belnap use $-0,-1,-2,-3,+0,+1,+2$ and $+3$ instead of 0, 1, 2, 3, 4, 5, 6 and 7, respectively.

1.5 The Basic Logic $\mathrm{GBL}_{\supset }$ is BN4

As remarked in the introduction to this paper, the basic logic $\mathrm{GBL}_{\supset }$ is the $\{\rightarrow ,\wedge ,\vee ,\lnot \}$ fragment of the bilattice logic $\mathrm{GBL}_{\supset }$ (cf. Arieli and Avron 1996). The “weak implication” $\supset$ is defined as follows (cf. Arieli and Avron 1996, p. 22): $x\supset y=_{\mathrm {df}}\left\{ \begin{array}{l} \text {t if }x\notin D \\ \text {y if }x\in D\end{array}\right\}$. So, the “weak implication” in Belnap–Dunn matrix MB4 (Definition 2.5) is interpreted according to the following table:

which satisfies all classical implicative tautologies. On the other hand, the “strong implication” ($\rightarrow$) is defined as follows: $A\rightarrow B=_{\mathrm {df}}(A\supset B)\wedge (\lnot B\supset \lnot A)$, which gives us the following truth table for $\rightarrow$:

that is, the conditional truth table of Brady’s BN4. Now, given that $\supset$ is definable from the $\{\rightarrow ,\vee \}$ fragment of GBL (Arieli and Avron 1996, Proposition 3.31). the $\{\supset ,\wedge ,\vee ,\lnot \}$ fragment of $\mathrm{GBL}_{\supset }$ is actually Brady’s logic BN4.