Basic Quasi-Boolean Expansions of Relevance Logics

Abstract

The basic quasi-Boolean negation (QB-negation) expansions of relevance logics included in Anderson and Belnap’s relevance logic R are defined. We consider two types of QB-negation: H-negation and D-negation. The former one is of paraintuitionistic or superintuitionistic character, the latter one, of dual intuitionistic nature in some sense. Logics endowed with H-negation are paracomplete; logics with D-negation are paraconsistent. All logics defined in the paper are given a Routley-Meyer ternary relational semantics.

Appendices

Appendix: A

The following sets of truth-tables t1-t13 are used to prove some claims made throughout the paper (designated values are starred). Let L be a logic defined upon the language ${\mathscr{L}}$ (cf. Definitions 2.1 and 2.2), Γ a set of wffs and A a wff of ${\mathscr{L}}$. On the other hand, let t be a set of truth-tables and v an assignment to the propositional variables of ${\mathscr{L}}$ built upon t. v verifies A if it assigns a designated value to A; and v verifies the rule Γ ⇒ A if it assigns a designated value to A, provided it assigns a designated value to each B ∈Γ. Then, t verifes L if every assignment v verifies all axioms and rules of L. The sets t1-t13 have been found by using MaGIC (cf. [26]; each set of tables is the simpler one justifying the respective claim). (In case a tester is needed, the reader can use that in [15].) In what follows, p,q and r are distinct propositional variables.

This set verifies all axioms and rules of RM3₊ (i.e., B₊ plus b2, b4, b6, b7 and b8. Cf. Definitions 2.4 and 2.7) plus A7-A10 and Con$\sim$, but falsifies A11: $v[(p\rightarrow q)\rightarrow (\sim q\rightarrow \sim p)]=0$ for any assignment v such that v(p) = v(q) = 2.

t2 verifies RM3₊ plus A7, A8 and Con$\sim$ but falsifies A10: $v(\sim \sim p\rightarrow p)=0$ for any assignment v such that v(p) = 0.

t3 verifies classical propositional logic C and A12, A13, A15, A16, A18, A19-A21, A23 and A24 but falsifies A14 and A22: $v[p\rightarrow \lbrack q\rightarrow \lnot (r\wedge \lnot r)]]=0$ for any assignment v such that v(p) = v(q) = v(r) = 1; and $v[p\rightarrow \lbrack \overset {\bullet }{\lnot } (r\wedge \overset {\bullet }{\lnot }r)\rightarrow q]]=0$ for any assignment such that v(p) = 1 and v(q) = v(r) = 0.

t4 verifies the logic G (i.e., B plus the PEM axiom $A\vee \sim A$) plus A12, A13, A15, A17-A20, A22, A24, Con$\lnot$ and Con$\overset {\bullet }{ \lnot }$ but falsifies A14 and A21: $v[p\rightarrow \lbrack q\rightarrow \lnot (r\wedge \lnot r)]]=v[p\rightarrow \lbrack q\rightarrow (r\vee \overset { \bullet }{\lnot }r)]]=0$ for any assignment v such that v(p) = v(q) = v(r) = 3.

The tables for $\wedge ,\vee ,\sim ,\lnot$ and $\overset {\bullet }{\lnot }$ are the same as in t4.

t5 verifies the logic G plus A12-A14, A17-A21, A24, Con$\lnot$ and Con$\overset {\bullet }{\lnot }$ but falsifies A15 and A22: $v[p\rightarrow \lbrack (q\wedge \lnot q)\rightarrow r]]=v[p\rightarrow \lbrack \overset { \bullet }{\lnot }(r\vee \overset {\bullet }{\lnot }r)\rightarrow r]]=0$ for any assignment v such that v(p) = v(q) = v(r) = 1.

The tables for ∧,∨ and $\sim$ are the same as in t4.

t6 verifies the logic ticket entailment T (cf. Section 2) plus A12, A13, A15-A18, but falsifies A14: $v[p\rightarrow \lbrack q\rightarrow \lnot (r\wedge \lnot r)]]=0$ for any assignment v such that v(p) = v(q) = v(r) = 1.

The tables for $\wedge ,\vee ,\sim ,\lnot$ and $\overset {\bullet }{\lnot }$ are as in t4.

t7 verifies the logic E (cf. Section 2) plus T5, T8 (cf. Proposition 3.4), A12, A13, A16-A20, A23, A24, the ECQ axiom, $(A\wedge \lnot A)\rightarrow B$, and the CPEM axiom, $B\rightarrow (A\vee \overset {\bullet }{\lnot }A)$, but falsifies A14, A15, A21 and A22: $v[p\rightarrow \lbrack q\rightarrow \lnot (r\wedge \lnot r)]]=v[p\rightarrow \lbrack (r\wedge \lnot r)\rightarrow q]]=v[p\rightarrow \lbrack q\rightarrow (r\vee \overset {\bullet }{\lnot }r)]]=v[p\rightarrow \lbrack \overset { \bullet }{\lnot }(r\vee \overset {\bullet }{\lnot }r)\rightarrow q]]=0$ for any assignment v such that v(p) = v(q) = v(r) = 1.

t8 verifies RB, that is, the logic R plus $A\leftrightarrow --A$, $B\rightarrow (A\vee -A)$, $(A\wedge -A)\rightarrow B$ and Con− (i.e., $A\rightarrow B\Rightarrow -B\rightarrow -A$) but falsifies $(A\rightarrow B)\rightarrow (-B\rightarrow -A)$: $v[(p\rightarrow q)\rightarrow (-q\rightarrow -p)]=0$ for any assignment v such that v(p) = 5 and v(q) = 1.

t9 verifies RM3 (cf. Section 2) plus A12-A24 but falsifies $q\rightarrow (p\vee \lnot p)$ (for any assignment v such that v(p) = 1 and v(q) = 2) and $(p\wedge \overset {\bullet }{\lnot }p)\rightarrow q$ (for any assignment v such that v(p) = 1 and v(q) = 0).

Also notice that $v(\sim p\rightarrow \lnot p)=v(\overset {\bullet }{\lnot } p\rightarrow \sim p)=0$ for any assignment v such that v(p) = 1.

Tables for $\sim ,\wedge ,\vee$ are as in t1.

t10 verifies R plus A12-A16 and A18, but falsifies A17: $v(p\vee \lnot p)=0$ for any assignment v such that v(p) = 1.

t11 verifies R plus A12-A15, A17-A22 and A24, but falsifies A16 and A23: $v[(p\rightarrow q)\rightarrow (\lnot q\rightarrow \lnot p)]=0$ for any assignment v such that v(p) = 2 and v(q) = 4; $v[(p\rightarrow q)\rightarrow (\overset {\bullet }{\lnot }q\rightarrow \overset {\bullet }{ \lnot }p)]=0$ for any assignment v such that v(p) = 1 and v(q) = 3.

The tables for $\wedge ,\vee ,\sim ,\lnot$ and $\overset {\bullet }{\lnot }$ are as in t9.

t12 verifies the 3-valued expansion of the negationless fragment of Lewis’ modal logic S5 (cf. [16, 24] and references in the last item) plus A12-A24 but falsifies $(p\wedge \sim p)\rightarrow q$ and $(p\wedge \overset {\bullet }{\lnot }p)\rightarrow q$ (for any assignment v such that v(p) = 1 and v(q) = 0) and $q\rightarrow (p\vee \sim p)$ and $q\rightarrow (p\vee \lnot p)$ (for any assignment v such that v(q) = 2 and v(p) = 1).

The tables for $\wedge ,\vee ,\sim ,\lnot$ and $\overset {\bullet }{\lnot }$ are as in t4.

t13 verifies the logic RW (cf. Section 2) plus A12-A24 but falsifies $(\sim p\rightarrow p)\rightarrow p$, $(\lnot p\rightarrow p)\rightarrow p$ and $(\overset {\bullet }{\lnot }p\rightarrow p)\rightarrow p$ (for any assignment v such that v(p) = 2), and $(p\rightarrow \sim p)\rightarrow \sim p$, $(p\rightarrow \lnot p)\rightarrow \lnot p$ and $(p\rightarrow \overset { \bullet }{\lnot }p)\rightarrow \overset {\bullet }{\lnot }p$ (for any assignment v such that v(p) = 1).

Appendix: B

Proposition B.1 (Ant, DNE are deriv. in FDE₊ plus ECQ & CPEM)

The axiom DNE, $--A\rightarrow A$, and the rule Ant, $(A\wedge B)\rightarrow -C\Rightarrow (A\wedge C)\rightarrow -B$ are derivable in FDE₊ plus the axioms ECQ, $(A\wedge -A)\rightarrow B$, and CPEM, $B\rightarrow (A\vee -A)$.

Proof

(Sketch)

• (a) $(A\wedge B)\rightarrow -C\Rightarrow (A\wedge C)\rightarrow -B$:

  Suppose (1) $(A\wedge B)\rightarrow -C$ (Hyp) and (2) $(C\wedge -C)\rightarrow -B$ (ECQ). By 1, 2 and FDE₊, we have (3) $[(A\wedge C)\wedge B]\rightarrow -B$. On the other hand, we obviously have (4) $[(A\wedge C)\wedge -B]\rightarrow -B$. By 3, 4 and FDE₊, we get (5) $[(A\wedge C)\wedge (B\vee -B)]\rightarrow -B$. Now, we use (6) $C\rightarrow (B\vee -B)$ (CPEM), whence we obtain (7) $[(A\wedge C)\wedge (A\wedge C)]\rightarrow \lbrack (A\wedge C)\wedge (B\vee -B)]$. Finally, by 5, 7 and FDE₊, we get (8) $(A\wedge C)\rightarrow -B$, as was to be proved.

• (b) $--A\rightarrow A$:

  We have (1) $--A\rightarrow (A\vee -A)$ (CPEM). By 1 and FDE₊, we have (2) $(--A\wedge --A)\rightarrow \lbrack (--A\wedge A)\vee (--A\wedge -A]$. We use now (3) $(--A\wedge -A)\rightarrow A$ (ECQ). By 3 and FDE₊, we get (4) $[(--A\wedge A)\vee (--A\wedge -A)]\rightarrow A$. Finally, by 2, 4 and FDE₊, we have (5) $--A\rightarrow A$, as was to be proved.

□

Proposition B.2 (Non-independence of A15 and A21)

A15 (resp., A21) is derivable from DW plus A12-A14, A18 and Con$\lnot$ (resp., A19, A20, A22, A24 and Con$\overset {\bullet }{\lnot }$) (cf. Section 2 on the logic DW).

Proof

(a) A15: By A18, we have (1) $\lnot (C\wedge \lnot C)\rightarrow \sim (C\wedge \lnot C)$, whence by A14, we get (2) $A\rightarrow \lbrack \sim B\rightarrow \sim (C\wedge \lnot C)]$. Then, by A11, (3) $A\rightarrow \lbrack \sim \sim (C\wedge \lnot C)\rightarrow \sim \sim B]$ follows. Finally, (4) $A\rightarrow \lbrack (C\wedge \lnot C)\rightarrow B]$ is derivable by A9 and A10. (b) The proof of A21 is similar. □

Proposition B.3 (Non-independence of A14)

A14 is derivable from the logic E (cf. Section 2) plus A12, A13 and A15-A18.

Proof

Firstly, the derivability of $B\rightarrow \lnot (A\wedge \lnot A)$ is proved. We use the rule assertion, $A\Rightarrow (A\rightarrow B)\rightarrow B$, admissible in E (cf. [21]).

We have (1) $B\rightarrow \lbrack \lnot C\rightarrow \lnot (A\wedge \lnot A)]$ and (2) $B\rightarrow \lbrack \lnot \lnot C\rightarrow \lnot (A\wedge \lnot A)]$ by A15 and A16. Then, (3) $B\rightarrow \lbrack (\lnot C\vee \lnot \lnot C)\rightarrow \lnot (A\wedge \lnot A)]$ is provable by the E∨ axiom (A5 of B₊ —cf. Definition 2.4). Next, (4) $[(\lnot C\vee \lnot \lnot C)\rightarrow \lnot (A\wedge \lnot A)]\rightarrow \lnot (A\wedge \lnot A)$ follows by the rule assertion and A17. Finally, we get (5) $B\rightarrow \lnot (A\wedge \lnot A)$ from (3) and (4).

Once $B\rightarrow \lnot (A\wedge \lnot A)$ is proved, $A\rightarrow \lnot \lnot A$ is derivable from this thesis and $(A\wedge \lnot A)\rightarrow \lnot \lnot A$, similarly as $--A\rightarrow A$ is demonstrated with $--A\rightarrow (A\vee -A)$ and $(--A\wedge -A)\rightarrow A$ in Proposition Appendix.1. Finally, A14 follows from $C\rightarrow \lbrack (\lnot \lnot B\rightarrow \lnot (A\wedge \lnot A)]$ (cf. (2) above) and $B\rightarrow \lnot \lnot B$. □

Paraconsistency and paracompleteness

Let L be a logic, the negation operator $\overset {\square }{\lnot }$ being one of its connectives; and let a be an L-theory. a is $\overset {\square }{\lnot }$-inconsistent if $A\wedge \overset {\square }{\lnot }A\in a$, for some wff A; and a is $\overset {\square }{\lnot }$-complete if A ∈ a or $\overset {\square }{\lnot }A\in a$ for every wff A. Then, L is $\overset {\square }{\lnot }$-paraconsistent if there is at least one $\overset { \square }{\lnot }$-inconsistent regular L-theory which is not trivial; and L is $\overset {\square }{\lnot }$-paracomplete if there is at least one (non-trivial) prime and regular L-theory which is not complete (notice that if a is a non-trivial regular and $\overset {\square }{\lnot }$ -inconsistent L-theory, in general, it is not difficult to extend a to a prime theory with the same properties).

We prove Propositions Appendix.4 and Appendix.5.

Proposition B.4 (RD _s is $\protect \overset {\bullet }{\lnot }$ -paraconsistent)

The logic RD_s is R (cf. Section 2) plus A19-A24. Then, any logic L included in RD_s is $\overset {\bullet }{\lnot }$-paraconsistent.

Proof

Let p,q be different propositional variables. Consider the set z = {B∣ ⊩_LA & $\vdash _{\text {L}}[A\wedge (p\wedge \overset { \bullet }{\lnot }p)]\rightarrow B\}$. It is easy to show that z is a regular L-theory and that it is $\overset {\bullet }{\lnot }$-inconsistent. Anyway, z is not trivial. Consider t9 (in Appendix Appendix) and any assignment v defined on the set {0, 1, 2} such that v(p) = 1 and v(q) = 0. Clearly, $v[A\wedge (p\wedge \overset {\bullet }{\lnot }p)]=1$ but $v[[A\wedge (p\wedge \overset {\bullet }{\lnot }p)]\rightarrow q]=0$, whence by the soundness theorem of RD_s (cf. Theorem 5.12) we get $\nvdash _{\text {RD}_{\text {s}}}[A\wedge (p\wedge \overset {\bullet }{\lnot } p)]\rightarrow q$. Consequently, q∉z. Then, we apply Lemma 6.16 and there is a prime, regular and non-trivial L-theory x such that $z\subseteq x$ but q∉x. Therefore, x is $\overset {\bullet }{\lnot }$ -inconsistent, but not trivial. □

Proposition B.5 (RH_s is $\lnot$-paracomplete)

The logic RH_s is R (cf. Section 2) plus A12-A16 and A18. Then, any logic L included in RH_s is $\lnot$-paracomplete.

Proof

Similar to (but simpler than) that of Proposition Appendix.4, now using t10 in Appendix Appendix. □

On strong completeness (see [25] and [8])

In Theorem 6.19, a weak completeness theorem is proved for all the QB-logics defined in the paper. Regarding strong completeness, in the context of RM-semantics, we need prime theories closed under all primitive rules of the logic in question. Unfortunately, in general, it is not possible to build up prime L-theories closed under all primitive rules of inference of a QB-logic L if it lacks the MP axiom, $[A\wedge (A\rightarrow B)]\rightarrow B$, or has other primitive rules of inference in addition to MP and Adj. Nevertheless, the required prime L-theories are definable, provided the disjunctive version or the thesis corresponding to each primitive rule of inference of L is added. For instance, suppose Modus Tollens (MT), $A\rightarrow B\And \lnot B\Rightarrow \lnot A$, is a primitive rule of inference of L. Then, the disjunctive version of MT is $C\vee (A\rightarrow B)\And C\vee \lnot B\Rightarrow C\vee \lnot A$; and, of course, the corresponding thesis to MT is $[(A\rightarrow B)\wedge \lnot B]\rightarrow \lnot A$. Consequently, a strong completeness theorem for a QB-logic L is available if (a) L has no primitive rules of inference other than Adj and MP and the MP axiom is an L-theorem, or (b) L has the disjunctive version or the corresponding thesis to each one of its primitive rules of inference.

In addition, it has to be noted that if the required prime theories are available, then a reduced RM₁-semantics, preferable when possible to the unreduced version, can be defined.