Blocking the Routes to Triviality with Depth Relevance

Abstract

In Rogerson and Restall’s (J Philos Log 36, 2006, p. 435), the “class of implication formulas known to trivialize (NC)” (NC abbreviates “naïve comprehension”) is recorded. The aim of this paper is to show how to invalidate any member in this class by using “weak relevant model structures”. Weak relevant model structures verify deep relevant logics only.

Appendices

Appendix 1: Some Relevant and Deep Relevant Logics

The following logics are formulated in the propositional language described in Definition 1.1. Firstly, we shall define Routley and Meyer’s basic logic B (cf. Routley et al. 1982, Chapter 4). The logic B can be axiomatized with the following axioms and rules

Axioms:

$$\begin{aligned} \text {a1. }&A\rightarrow A \\ \text {a2. }&(A\wedge B)\rightarrow A\text { / }(A\wedge B)\rightarrow B \\ \text {a3. }&[(A\rightarrow B)\wedge (A\rightarrow C)]\rightarrow [A\rightarrow (B\wedge C)] \\ \text {a4. }&A\rightarrow (A\vee B)\text { / }B\rightarrow (A\vee B) \\ \text {a5. }&[(A\rightarrow C)\wedge (B\rightarrow C)]\rightarrow [(A\vee B)\rightarrow C]\\ \text {a6. }&[A\wedge (B\vee C)]\rightarrow [(A\wedge B)\vee (A\wedge C)] \\ \text {a7. }&A\rightarrow \lnot \lnot A \\ \text {a8. }&\lnot \lnot A\rightarrow A \end{aligned}$$

Rules

$$ \begin{aligned} \text {Adjunction (Adj) }&A~ \& ~B\Rightarrow A\wedge B \\ \text {Modus Ponens (MP) }&A~ \& ~A\rightarrow B\Rightarrow B \\ \text {Suffixing (Suf) }&A\rightarrow B\Rightarrow (B\rightarrow C)\rightarrow (A\rightarrow C) \\ \text {Prefixing (Pref) }&B\rightarrow C\Rightarrow (A\rightarrow B)\rightarrow (A\rightarrow C) \\ \text {Contraposition (Con) }&A\rightarrow B\Rightarrow \lnot B\rightarrow \lnot A \end{aligned}$$

Then, we shall consider the extensions of B defined by adding to it some of the following axioms and rules (notice that a36 and a37 are not classical tautologies).

$$\begin{aligned} \text {a9. }&(B\rightarrow C)\rightarrow [(A\rightarrow B)\rightarrow (A\rightarrow C)] \\ \text {a10. }&(A\rightarrow B)\rightarrow [(B\rightarrow C)\rightarrow (A\rightarrow C)] \\ \text {a11. }&[(A\rightarrow A)\rightarrow B]\rightarrow B \\ \text {a12. }&A\rightarrow [(A\rightarrow B)\rightarrow B] \\ \text {a13. }&[A\rightarrow (A\rightarrow B)]\rightarrow (A\rightarrow B) \\ \text {a14. }&A\rightarrow (A\rightarrow A) \\ \text {a15. }&(A\rightarrow B)\rightarrow [A\rightarrow (A\rightarrow B)] \\ \text {a16. }&(B\rightarrow A)\rightarrow (A\rightarrow A) \\ \text {a17. }&(A\rightarrow B)\rightarrow (A\rightarrow A) \\ \text {a18. }&[(A\rightarrow B)\rightarrow A]\rightarrow A \\ \text {a19. }&[(A\rightarrow B)\wedge (B\rightarrow C)]\rightarrow (A\rightarrow C) \\ \text {a20. }&[(A\rightarrow B)\wedge A]\rightarrow B \\ \text {a21. }&[A\rightarrow (B\rightarrow C)]\rightarrow [(A\wedge B)\rightarrow C] \\ \text {a22. }&(A\rightarrow B)\rightarrow [A\rightarrow (A\wedge B)] \\ \text {a23. }&(A\rightarrow B)\rightarrow [(A\wedge C)\rightarrow (B\wedge C)]\\ \text {a24. }&(A\rightarrow B)\rightarrow [(A\vee B)\rightarrow B] \\ \text {a25. }&(A\rightarrow B)\rightarrow [(A\vee C)\rightarrow (B\vee C)] \\ \text {a26. }&(A\rightarrow B)\vee (B\rightarrow A) \\ \text {a27. }&A\vee (A\rightarrow B) \\ \text {a28. }&A\vee \lnot A \\ \text {a29. }&\lnot (A\wedge \lnot A) \\ \text {a30. }&(A\rightarrow B)\rightarrow (\lnot B\rightarrow \lnot A) \\ \text {a31. }&(A\rightarrow \lnot A)\rightarrow \lnot A \\ \text {a32. }&(A\wedge \lnot B)\rightarrow \lnot (A\rightarrow B) \\ \text {a33. }&[(A\rightarrow B)\wedge \lnot B)]\rightarrow \lnot A \\ \text {a34. }&[(A\rightarrow B)\wedge (A\rightarrow \lnot B)]\rightarrow \lnot A \\ \text {a35. }&\lnot B\vee (A\rightarrow B) \\ \text {a36. }&B\vee \lnot (A\rightarrow B) \\ \text {a37. }&\lnot A\vee \lnot (A\rightarrow B) \end{aligned}$$

Rules

$$ \begin{aligned} \text {Assertion (Asser) }&A\Rightarrow (A\rightarrow B)\rightarrow B \\ \text {Specialized reductio (sr) }&A\Rightarrow \lnot (A\rightarrow \lnot A)\\ \text {Counterexample (Cnt) }&A\wedge \lnot B\Rightarrow \lnot (A\rightarrow B) \\ \text {Disjunctive Modus Ponens (MPd) }&C\vee A~ \& ~C\vee (A\rightarrow B)\Rightarrow C\vee B \\ \text {Disjunctive Suffixing (Sufd) }&C\vee (A\rightarrow B)\!\Rightarrow \! C\vee [(B\!\rightarrow \! C)\rightarrow (A\!\rightarrow \! C)]\\ \text {Disjunctive Prefixing (Prefd) }&C\vee (B\rightarrow C)\!\Rightarrow \! C\vee [(A\!\rightarrow \! B)\!\rightarrow \! (A\rightarrow C)] \\ \text {Disjunctive Contraposition (Cond) }&C\vee (A\rightarrow B)\Rightarrow C\vee (\lnot B\rightarrow \lnot A) \\ \text {Disjunctive Assertion (Asserd) }&C\vee A\Rightarrow C\vee [(A\rightarrow B)\rightarrow B] \\ \text {Disjunctive specialized reductio (srd) }&C\vee A\Rightarrow C\vee \lnot (A\rightarrow \lnot A) \\ \text {Disjunctive counterexample (Cntd) }&C\vee (A\wedge \lnot B)\Rightarrow C\vee \lnot (A\rightarrow B) \\ \text {Meta-rule Summation (MRs) }&A\Rightarrow B\Rightarrow C\vee A\Rightarrow C\vee B \end{aligned}$$

Deep relevant extensions of B are defined as follows:

DW: it is the result of substituting the rule Con for the corresponding axiom a30.

DJ: DW plus a19.

DK: DJ plus a28.

DR: DJ plus the rule sr.

Each of the deep relevant logics defined can “deep-relevantly” be extended with the metarule MRs.

Next, standard relevant logics can be defined as follows (the rules Suf and Pref of DW are not independent now).

TW: DW plus a9 and a10.

T: TW plus a13 and a31.

E: T plus Asser.

R: T plus a12 (a31 is not independent).

RM: R plus a14.

TW is Contractionless Ticket Entailment; T is Ticket Entailment; E, Logic of Entailment; R, Logic of Relevant Conditional, and finally, RM is R-Mingle (we remark that RM lacks the vsp: in RM the conditional $\rightarrow$ is not actually a relevant conditional. Cf. Anderson and Belnap (1975) and Routley et al. (1982) about the logics defined above).

Appendix 2: Variations on Meyer’s Crystal Lattice CL

In this “Appendix”, we display particular wr-matrices upon which wr-model structures can be defined as indicated in Sect. 3. We exemplify each one of the wr-matrices considered in Sects. 2 and 3. We begin by recalling, for definiteness, the notion of a “logical matrix” as well as the standard notions related to it (in case a tester is needed, the reader can use that in González (2012)).

Definition 7.1

(Logical matrices) A logical matrix M is a structure $(K,T,F,\,f_{\rightarrow },f_{\wedge }, f_{\vee },f_{\lnot })$ where (1) $K$ is a set; (2) $T$ and $F$ are non-empty subsets of $K$ such that $T\cup F=K$ and $T\cap F=\emptyset$; (3) $f_{\rightarrow },f_{\wedge }$ and $f_{\vee }$ are binary functions (distinct from each other) on $K$ and $f_{\lnot }$ is a unary function on $K$ .

Remark 7.2

(On the set $F$) The set $F$ has been remarked in Definition 7.1 only because it eases the definition of “weak relevant matrices” and “weak relevant model structures”.

In addition to Definition 7.1 we set (cf. Definition 1.1):

Definition 7.3

(Verification, Falsification) Let M be a logical matrix and $A$ a wff. (1) M verifies $A$ iff for any assignment, $v_{m}$, of elements of $K$ to the propositional variables of $A$, $v_{m}(A)\in T$. M falsifies $A$ iff M does not verify $A$. (2) If $A_{1},\ldots ,A_{n}\Rightarrow B$ is a rule of derivation of a logic S, M verifies $A_{1},\ldots A_{n}\Rightarrow B$ iff for any assignment, $v_{m}$, of elements of $K$ to the variables of $A_{1},\ldots A_{n},B$, if $v_{m}(A_{1})\in T,\ldots ,v_{m}(A_{n})\in T$, then $v_{m}(B)\in T$. M falsifies $A_{1},\ldots A_{n}\Rightarrow B$ iff M does not verify it. (3) Let S be a propositional logic. M verifies S iff M verifies all axioms and rules of derivation of S.

The matrices to follow (except the last one) can be considered as variations on the conditional characteristic of Meyer’s Crystal lattice CL [wr-matrices of a different structure are displayed in Robles and Méndez (2012, 2014b)]. The tables for $\wedge$, $\vee$ and $\lnot$ are as follows (all values but 0 are designated). The structure of all matrices is:

Diagram 1

The tables for the conditional are:

We now introduce Matrix 8. The tables are as follows (all values but 0 are designated):

We record the theses and the rules (in the preceding “Appendix”) verified by each matrix. (It has to be understood that the theses and rules omitted are falsified).

Matrix 1 ($M1$): Meyer’s Crystal lattice CL, $M$ CL . Meyer’s $M$CL is a simplification of Belnap’s matrix $M_{0}$ used in Belnap (1960) for proving for the first time that the logic of Entailment E has the vsp [$M_{o}$ is also used in Anderson and Belnap (1975) and in Routley et al. (1982), and it is axiomatized as well as $M_{\text {CL}}$, in Brady (2003)]. CL verifies relevant logic R (so, it verifies the logics TW, T and E (cf. “Appendix 1”)). $M$CL verifies all rules in “Appendix 1” and a9–a13, a19–a21, a27–a35. $M$CL is a wr(1, 2, 3)-matrix.

Matrix 2 ($M2$): $M_\mathrm{RMO}$. $M_\mathrm{RMO}$ is a simplification of the eight-element tables used in Méndez (1988) (see also Méndez (2010)) to prove that the logic RMO has the vsp. The logic RMO is axiomatized by (1) changing a30 for the corresponding rule, Con, in the formulation of R and (2) adding the axiom “mingle” (a14). $M_\mathrm{RMO}$ verifies all rules in “Appendix 1” and a9–a15, a19–a21, a27–a29, a31–a35. $M_\mathrm{RMO}$ is a wr(1, 2)-matrix (it is not a wr(3)-matrix).

Matrix 3 ($M3$): $M_{\mathrm{Fac}}$. $M_{\mathrm{Fac}}$ abbreviates “Matrix Factor”. It is used for defining a wr-ms verifying some deep relevant extensions of B with the axiom “Factor” (a23) and related theses such as a22, in Robles and Méndez (in preparation). $M_{\mathrm{Fac}}$ verifies all extensions of B with any selection of the following axioms and rules: a13, a17, a19, a20, a21, a22, a23, a27, a28, a29, a33, a35, a37, MPd, Sufd, Prefd, Cond and MRs. $M_{\text {Fac}}$ is a wr(1)-matrix (it is neither a w(2)-matrix, nor a wr(3)-matrix).

Matrix 4 ($M4$): $M_{\text {SUM}}$. $M_{\text {SUM}}$ abbreviates “Matrix Summation”. It is used in Robles and Méndez (in preparation) for defining a wr-ms verifying some deep relevant extensions of B with the axiom “Summation” (a25) and related theses such as a24. $M_{\text {SUM}}$ verifies all extensions of B with any selection of the following axioms and rules: a11, a13, a14, a15, a16, a18, a19, a20, a21, a24, a25, a27, a28, a29, a31, a32, a33, a34, a35, a36 and all rules in “Appendix 1”. $M_{\text {SUM}}$ is a wr(2)-matrix (it is neither a w(1)-matrix, nor a wr(3)-matrix).

Matrix 5 ($M5$): $M_{\text {SUM}^{\prime }}$. $M_{\text {SUM} ^{\prime }}$ is a modification of $M_{\text {SUM}}$. $M5$ verifies all extensions of DW (i.e., B plus a30—cf. “Appendix 1”) with any selection of the following axioms and rules: a11, a13, a18, a19, a20, a21, a27, a28, a29, a30, a31, a32, a33, a34, a35, a36, a37 and all rules in Appendix 1. $M5$ is interesting because it can be used for defining deep relevant logics extending DR (cf. “Appendix 1”) with a36, a37 and other similar theses that are not classical tautologies. $M_{\text {SUM} ^{\prime }}$ is a wr(3)-matrix (it is neither a w(1)-matrix, nor a wr(2)-matrix).

Matrix 6 ($M6$): $M_{\text {SUM}^{\prime \prime }}$. $M_{ \text {SUM}^{\prime \prime }}$ is also a modification of $M_{\text {SUM}}$. $M6$ verifies all extensions of B with any selection of the following axioms and rules: a11, a13, a18, a19, a20, a21, a27, a28, a29, a31, a32, a33, a34, a35, a36 and all rules in “Appendix 1”. $M_{\text {SUM}^{\prime }}$ is a wr(2, 3)-matrix (it is not a w(1)-matrix).

Matrix 7 ($M7$): $M_{\text {Fac}^{\prime }}$. $M_{\text {Fac} ^{\prime }}$ is a modification of $M_{\text {Fac}}$. $M7$ verifies all extensions of B with any selection of the following axioms and rules: a11, a13, a19, a20, a21, a27, a28, a29, a31, a32, 33, a34, a35, a37 and all rules in “Appendix 1”. $M_{\text {Fac}^{\prime }}$ is a wr(1, 3)-matrix (it is not a w(2)-matrix).

Matrix 8 ($M8$). $M8$ is a simple four-element matrix, which is a w(1, 2, 3)-matrix. Now, let B$^{\prime }$ be the result of changing the rule Con (cf. “Appendix 1”) for Con$^{\prime }$ ($A\rightarrow B~ \& ~\lnot B\Rightarrow \lnot A$) in the axiomatization of B. Then, $M8$ verifies all extensions of B$^{\prime }$ with any selection of the following axioms and rules: a11, a14, a15, a19, a20, a26, a27, a28, a29, a31, a32, a35, a36, Asser, sr, Cnt, MPd, Sufd, Prefd, Asserd and the rule Con$^{\prime }$d. Notice that the negation defined in $M8$ is not a De Morgan negation: $\lnot (A\vee B)\rightarrow (\lnot A\wedge \lnot B)$, $(\lnot A\vee \lnot B)\rightarrow \lnot (A\wedge B)$ and the rule Con are falsified ($v(A)=1$ and $v(B)=2$).