An Easy Road to Multi-contra-classicality

Abstract

A contra-classical logic is a logic that, over the same language as that of classical logic, validates arguments that are not classically valid. In this paper I investigate whether there is a single, non-trivial logic that exhibits many features of already known contra-classical logics. I show that Mortensen’s three-valued connexive logic M3V is one such logic and, furthermore, that following the example in building M3V, that is, putting a suitable conditional on top of the $\{\sim , \wedge , \vee \}$-fragment of LP, one can get a logic exhibiting even more contra-classical features.

1 Introduction

Until very recently, most of the more well-known non-classical logics—constructive logics, relevance logics, paraconsistent logics, and so on—were subclassical. Contra-classical logics are logics with valid arguments that are invalid in classical logic over the same underlying language.¹ Examples of such contra-classical logics are Aristotelian syllogistic—as it validates Subalternation, Darapti or Camestros—or connexive logics, which validate Aristotle’s, that is, $\sim \! (A\rightarrow \sim \! A)$ and $\sim \! (\sim \! A\rightarrow A)$, and Boethius’ Theses, that is, $(A \rightarrow B) \rightarrow \sim \! (A \rightarrow \sim \! B)$ and $(A \rightarrow \sim \! B) \rightarrow \sim \! (A \rightarrow B)$.

No contra-classical logics are subclassical, even if they can be built upon some subclassical logics. Wansing (2005) obtained the connexive logic C by modifying the falsity condition for the conditional in Nelson’s logic N4, with the feature that the converses of Boethius’ Theses also hold. Then Omori (2016) used the same idea, changing the falsity condition for the conditional on top of LP to get another connexive logic, dLP. After that, he has shown (cf. Omori Unpublished typescript) that a number of well-known and new paraconsistent and relevant logics can be obtained also by changing appropriately the falsity condition for some connectives while leaving the FDE-like truth and falsity conditions for the remaining ones, in most cases even negation, fixed.² More recently, Wansing and Unterhuber (2019) modified the falsity condition of Chellas’ basic conditional logic CK and they obtained a (weakly) connexive logic. Even more recently, Omori and Wansing (2019) have put forward a systematization of connexive logics based on certain controlled modifications in the conditional’s truth and falsity conditions, showing that, in general, changing the truth condition has led only to weak connexivity (Boethius’ Theses hold only in rule form, if at all), whereas changing the falsity condition has led to hyper-connexivity (i.e. not only do Boethius’ Theses hold, but also their converses).

Let me call that general approach to non-classical logic ‘the Bochum Plan’, in analogy with the Australian, American and Scottish plans for relevance. But the analogy can only go so far, because the scope of the Bochum Plan is far broader than paraconsistency and relevance, as the connexive logics obtained show. In fact, Omori and Wansing (2018) have generalized that idea of modifying a falsity condition to get other contra-classical logics, not only connexive logics in the ballpark of Wansing’s C and Omori’s dLP. Roughly, the recipe they followed for crafting a contra-classical logic from a given logic L is as follows:

• Provide a (model-theoretic) semantics for L, in the FDE-like style;

• take at least one connective, leave its truth condition fixed and change its falsity condition by making it almost the falsity condition of some other connective in L;

• leave the truth and falsity conditions for the remaining connectives fixed.

What counts as “almost the falsity condition of another connective” is decidedly vague. However, I hope it is clear in the examples below, where the falsity conditions are the usual ones for other connectives. Other, more familiar non-classical logics, can be obtained by making less radical changes.³

One striking and to some extent unexpected outcome obtained by modifying the falsity condition of the conditional á la Wansing is that the logics obtained are not only connexive but also contradictory (or negation-inconsistent) because they validate formulas of the forms X and $\sim \! X$. Thus, the modifications in the falsity condition for the conditional might induce the presence of different brands of contra-classicality into a single logic. This is important because the different sorts of contra-classicality are, in general, logically independent. For example, connexive logics in the early contemporary era—for example, those in Angell (1962), McCall (1966), Pizzi (1977, 1991) and Priest (1999)—were not contradictory.⁴ Or consider Abelian logics, logics that validate

$$\begin{aligned} \begin{array}{ll} (A\rightarrow A)&{}\quad \text {Identity}\\ (A\rightarrow A)\leftrightarrow \sim \! (A\rightarrow A)&{}\quad \text {Centering}\\ ((A\rightarrow B)\rightarrow B)\rightarrow A&{}\quad \text {Meyer--Slaney Relativity} \end{array} \end{aligned}$$

and which are contradictory but not connexive.⁵ And even if Abelian logics also combined two sorts of contra-classicality—contradictoriness and relativity—, Meyer and Martin’s logic SI$\sim$I in Meyer and Martin (2019) also validates both of Identity and Centering without validating Meyer–Slaney Relativity.

Let $C_1$,..., $C_n$ be the sets of defining properties of each certain n kinds of contra-classicality. For the most part, I will assume that, for any $C_j$ and $C_k$ such that $j \ne k$, $C_j\nsubseteq C_k$. This is primarily because of the following two reasons. First, there remains a (limit) case of contra-classicality in the form of triviality and, should we assign an index $C_i$ to triviality, we would immediately have that for some $C_j$, $C_j\subseteq C_i$. Also, when studying the better-known kinds of contra-classical logics (viz. connexive, Abelian, negation-inconsistent), an underlying implicit assumption is that they must be non-trivial to be of philosophical interest. Thus, in keeping with this implicit assumption, the expectation is that there be multi-contra-classical logics that incorporate as many non-standard validities as possible while staying non-trivial. A logic L, which for simplicity will be understood as a collection of valid arguments, is then multi-contra-classical if it satisfies three or more of the $C_n$’s, none of which corresponds to triviality in this paper.

The question I am interested in here is, then, how far one could go into multi-contra-classicality, beyond connexivity and contradictoriness, by changing the conditional alone.⁶ I start by studying a logic exhibiting such multi-contra-classicality already offered in the connexive literature: Mortensen’s three-valued connexive logic, from Mortensen (1984). It is not only connexive and contradictory, its two more well-known properties, but also has important overlaps with Abelian logic and it validates $\sim \! (A\rightarrow B)$. Then, I will show that a logic exhibiting even more contra-classical features than Mortensen’s logic can be obtained by a simple application of the Bochum Plan: take LP and keep all of it fixed except the falsity condition for the conditional; change it to the falsity condition of conjunction and voilà!, a logic exhibiting more than the usual number of contra-classical features arises. Because one is able to obtain Mortensen’s conditional by an application of the Bochum Plan, it belongs to a family of modified conditionals put on top of the implicationless fragment of LP. A couple of members of this family are going to be discussed in some detail too.

The plan for the rest of the paper is as follows. In Sect. 2, I present Mortensen’s three-valued connexive logic, M3V, at tutorial speed. In Sect. 3, I compare the M3V conditional with other two conditionals typically added on top of the $\{\sim , \wedge , \vee \}$-fragment of LP (or of a logic including such fragment). In passing, I make some original comments on what can one learn from those conditionals about hyper-connexivity and Kapsner strength, the latter a property suggested as a stronger demand logics should satisfy to better capture the core intuitions behind the notion of connexivity. Then, in Sect. 4 I present one further variation for the conditional, not yet studied in the literature, which yields even more contra-classical features than M3V, and discuss its philosophical significance beyond the already interesting feature of exhibiting multi-contra-classicality.

Two final disclaimers are in order. Most of the logics, and especially the conditionals, studied here, have already well-known applications to ordinary language issues, and I mention some of them here and there. Nonetheless, my study is almost purely formal (which does not make it less philosophical!); applications of the notions introduced here are left for another occasion. Second, the paper abounds with names of connectives, principles and logics. Keeping track of them might be a bit tiring, but this is unavoidable to some extent: part of the import of the present investigation is to make explicit connections between several topics hitherto developed independently.

2 Mortensen’s Three-Valued Connexive Logic

The logic M3V was introduced, although not with that name, in Mortensen (1984) (the name was given in McCall (2012), presumably to mean “Mortensen’s 3-valued connexive logic”). The following truth tables, with $V_{\tiny \mathbf{M3V }}=\{2, 1, 0\}$ and $D^{+}=\{2, 1\}$, characterize M3V:

A	B	$\sim A$	$A\wedge B$	$A\vee B$	$A\rightarrow B$
2	2	0	2	2	1
2	1	0	1	2	0
2	0	0	0	2	0
1	2	1	1	2	1
1	1	1	1	1	1
1	0	1	0	1	0
0	2	2	0	2	1
0	1	2	0	1	1
0	0	2	0	0	1

A biconditional can be defined as usual, that is, as $(A\rightarrow B)\wedge (B\rightarrow A)$. It must be noted that Mortensen’s satisfiability conditions for the conditional are structurally the same as the ones used by Anderson and Belnap in Anderson and Belnap (1975) to show the consistency of the logic E and, in particular, to show how to block the paradox of necessity, i.e. to avoid validating formulas of the form $X\rightarrow (Y\rightarrow Z)$, where X is a contingent truth and $(Y\rightarrow Z)$ is a logical truth.⁷

The three-valued nature of Mortensen’s logic, along with the number of elements in $D^{+}$ and the evaluation conditions for negation motivate the representation of Mortensen’s 2, 1, 0 as three subsets of the set of classical values $\{1, 0\}$, namely $\{1\}$, $\{1, 0\}$ and $\{0\}$, respectively, leaving the remaining subset $\varnothing$ aside as in the two-valued relational semantics for LP:

A	B	$\sim \! A$	$A\wedge B$	$A\vee B$	$A\rightarrow B$
$\{1\}$	$\{1\}$	$\{0\}$	$\{1\}$	$\{1\}$	$\{1, 0\}$
$\{1\}$	$\{1, 0\}$	$\{0\}$	$\{1, 0\}$	$\{1\}$	$\{0\}$
$\{1\}$	$\{0\}$	$\{0\}$	$\{0\}$	$\{1\}$	$\{0\}$
$\{1, 0\}$	$\{1\}$	$\{1, 0\}$	$\{1, 0\}$	$\{1\}$	$\{1, 0\}$
$\{1, 0\}$	$\{1, 0\}$	$\{1, 0\}$	$\{1, 0\}$	$\{1, 0\}$	$\{1, 0\}$
$\{1, 0\}$	$\{0\}$	$\{1, 0\}$	$\{0\}$	$\{1, 0\}$	$\{0\}$
$\{0\}$	$\{1\}$	$\{1\}$	$\{0\}$	$\{1\}$	$\{1, 0\}$
$\{0\}$	$\{1, 0\}$	$\{1\}$	$\{0\}$	$\{1, 0\}$	$\{1, 0\}$
$\{0\}$	$\{0\}$	$\{1\}$	$\{0\}$	$\{0\}$	$\{1, 0\}$

Applying the mechanical procedure described in Omori and Sano (2015) for turning truth tables employing three of the four truth values of FDE into Dunn conditions (i.e., pairs of positive and negative conditions in terms of containing or not containing the classical values 0 or 1), we define a relation $\sigma$, which takes formulas as its domain and the set of truth values $\{1, 0\}$ as its codomain.

Then, the positive condition describes the cases in which $1\in \sigma (X)$, and the negative condition describes the cases in which $0\in \sigma (X)$. From the truth tables above we can infer that the conditions for the implication-free fragment of the language are standard, and that the clauses for $\rightarrow$ are as follows:

• $1\in \sigma (A\rightarrow B)$ if a and only if $1\notin A$, or $0\notin B$, or both $0\in A$ and $1\in B$

• $0\in \sigma (A\rightarrow B)$ if and only if [$1\in \sigma (A)$ or $0\in \sigma (A)$] and [$1\in \sigma (B)$ or $0\in \sigma (B)$]

Finally, logical validity in M3V is understood in the usual, forwards truth-preservation way:

• $\Gamma \models _{\tiny \mathbf{M3V }} A$ if and only if, for all $\sigma$, if $1\in \sigma (B)$ for all $B\in \Gamma$ then $1\in \sigma (A)$

We are now in a position to point out some of M3V’s main features. First, we can easily see that M3V validates unrestricted Detachment, i.e.

$$\begin{aligned} A, A\rightarrow B\models _{\tiny \mathbf{M3V }} B \end{aligned}$$

unlike LP, which validates only restricted Detachment, i.e.

$$\begin{aligned} \hbox {If }\models A\hbox { and }\models A\rightarrow B\hbox { then }\models B \end{aligned}$$

It is well-known that M3V is connexive, i.e.

$$\begin{aligned} \begin{array}{ll} \models _{\tiny \mathbf{M3V }} \sim \! (A\rightarrow \sim \! A) &{}\quad \text {Aristotle's Thesis}\\ \models _{\tiny \mathbf{M3V }} \sim \! (\sim \! A\rightarrow A) &{}\quad \text {Variant of Aristotle's Thesis}\\ \models _{\tiny \mathbf{M3V }} (A \rightarrow B) \rightarrow \sim \! (A \rightarrow \sim \! B) &{}\quad \text {Boethius' Thesis}\\ \models _{\tiny \mathbf{M3V }} (A \rightarrow \sim \! B) \rightarrow \sim \! (A \rightarrow B) &{}\quad \text {Variant of Boethius' Thesis}\\ \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{ll} \nvDash _{\tiny \mathbf{M3V }}(A\rightarrow B)\rightarrow (B\rightarrow A)&\quad \text {Non-symmetry of implication} \end{array} \end{aligned}$$

and that it is contradictory, i.e. there is pair of M3V-theorems such that one is the negation of the other, for example, $(A\wedge \sim \! A)\rightarrow A$ and $\sim \! ((A\wedge \sim \! A)\rightarrow A)$.

One of the most overlooked features of M3V is that, in it, the falsity condition for the conditional is but a sophisticated way of expressing

$$\begin{aligned} 0\in \sigma (A\rightarrow B) \end{aligned}$$

which implies that

$$\begin{aligned} \models _{\tiny \mathbf{M3V }} \sim \! (A \rightarrow B) \end{aligned}$$

for any A and B.

Now, it is easy to see, with a simple substitution in the consequent, that $\sim \! (A\rightarrow \sim \! B)$ is valid in M3V too. Because of this, I say that M3V is ultra-Abelardian. Claudio Pizzi has urged the connexive logic community not to multiply the principles with names of ancient philosophers. However, that plays a role in keeping a healthy logical memory. Peter Abelard held that conditionals express natures and that natures are characterized positively. For example, he believed that it would not be part of a human’s nature to not be a stone, although being an animal would be. (For details see Martin 2004.) Thus, for him, no conditional of the form $A\rightarrow \sim \! B$, where A is necessarily positive—that is, its main connective is not a negation—and $\sim \! B$ is not a subformula of A, is true on pain of contradiction. Omitting the constraints on A and $\sim \! B$ would lead to ultra-Abelardianism.

Let me say that a logic L is hyper-connexive if for some of the Boethius’ Theses, its converse holds as well. Although the converses were known and held by Boethius himself, they deserve another name precisely because of Wansing’s efforts in stressing of the philosophical import of the falsity condition for the conditional, i.e. the circumstances under which a proposition of the form $\sim \! (A\rightarrow B)$ is true⁸:

$$\begin{aligned} \begin{array}{ll} \sim \! (A \rightarrow B) \rightarrow (A \rightarrow \sim \! B) &{}\quad \text {Wansing's Thesis}\\ \sim \! (A \rightarrow \sim \! B) \rightarrow (A \rightarrow B) &{}\quad \text {Variant of Wansing's Thesis} \end{array} \end{aligned}$$

Given the validity of Detachment and $\sim \! (A\rightarrow B)$ (and Uniform Substitution), neither Wansing’s Thesis nor its variant are valid in M3V, on pain of conditional triviality, i.e. the validity of any conditional. (For a countermodel to both the thesis and its variant, let $\sigma (A) = \sigma (B) = \{1\}$.)

Almost obvious given the validity of $\sim \! (A\rightarrow B)$, but even more overlooked, is the fact that M3V shares valid axiom schemas with many standard Abelian logics, namely the Centering axiom schemas:

$$\begin{aligned}&\models _{\tiny \mathbf{M3V }} \sim \! (A\rightarrow A)\\&\models _{\tiny \mathbf{M3V }} \sim \! (A\rightarrow A)\leftrightarrow (A\rightarrow A) \end{aligned}$$

Nonetheless, it does not validate the Meyer–Slaney relativity axiom (schema), characteristic of purely implicative Abelian logics:

$$\begin{aligned} \nvDash _\mathbf{\tiny {M3V} }((A\rightarrow B)\rightarrow B)\rightarrow A \end{aligned}$$

(For a countermodel, let $\sigma (A) = \{0\}$ and $\sigma (B) = \{1\}$.)

The validity of $\sim \! (A\rightarrow A)$ demands moreover a comparison with Meyer and Martin’s SI$\sim$I.⁹ It is axiomatized as follows (names of the axioms are taken, as usual, from the combinatory logic literature):

Axioms

• B. $(C\rightarrow D)\rightarrow ((A\rightarrow C)\rightarrow (A\rightarrow D))$

• $\hbox {B}^{'}$. $(A\rightarrow C)\rightarrow ((C\rightarrow D)\rightarrow (A\rightarrow D))$

• I. $A\rightarrow A$

• $\sim \! I$. $\sim \! (A\rightarrow A)$

Rules

• (BX) If $\models _{\tiny \mathbf{SI\sim \! I }}C\rightarrow D$ then $\models _{\tiny \mathbf{SI\sim \! I }}(A\rightarrow C)\rightarrow (A\rightarrow D)$

• ($\hbox {B}^{'}\hbox {X}$) If $\models _{\tiny \mathbf{SI\sim \! I }}A\rightarrow C$ then $\models _{\tiny \mathbf{SI\sim \! I }}(C\rightarrow D)\rightarrow (A\rightarrow D)$

• (BXY) If $\models _{\tiny \mathbf{SI\sim \! I }}C\rightarrow D$ then, if $\models _{\tiny \mathbf{SI\sim \! I }}A\rightarrow C$ then $\models _{\tiny \mathbf{SI\sim \! I }}A\rightarrow D$

However, in a sense, M3V destroys the underlying idea behind SI$\sim$I. Meyer and Martin wanted to provide a logic for Aristotle’s syllogistic, which was not reflexive. In their logic SI$\sim$I, $A\rightarrow A$ was treated as a borderline case, both a fallacy and a validity, hence the validity of both $A\rightarrow A$ and $\sim \! (A\rightarrow A)$. But transitivity, in its different manifestations, like B and $\hbox {B}^{'}$, was not such a borderline case. Validating their negations, as M3V does, could be interpreted as formalizing the idea that the validity of transitivity is a borderline case too, something that Meyer and Martin would surely reject.

3 M3V and Some Three-Valued Conditionals

Comparing the M3V conditional with some non-contraposible conditionals will lead us to a better understanding of M3V. The three-valued conditional defined by the following clauses

• $1\in \sigma (A\rightarrow B)$ if and only if $1\notin \sigma (A)$ or $1\in \sigma (B)$

• $0\in \sigma (A\rightarrow B)$ if and only if $1\notin \sigma (A)$ or $0\in \sigma (B)$

is a conditional, first introduced in Olkhovikov (2016) and then introduced independently in Cantwell (2008) and Omori (2016). Here it is its truth table, symbolized with ‘$A\rightarrow _{OCO} B$’, for more visibility:

$A\rightarrow _{OCO} B$	$\{1\}$	$\{1, 0\}$	$\{0\}$
$\{1\}$	$\{1\}$	$\{1, 0\}$	$\{0\}$
$\{1, 0\}$	$\{1\}$	$\{1, 0\}$	$\{0\}$
$\{0\}$	$\{1, 0\}$	$\{1, 0\}$	$\{1, 0\}$

Mortensen’s conditional is just the contraposible conditional built upon the OCO conditional, i.e. $(A\rightarrow _{OCO} B)\wedge (\sim \! B\rightarrow _{OCO} \sim \! A)$, as it can be easily verified.¹⁰ For definiteness, let me call ‘CN’ the logic resulting from adding the OCO conditional to the $\{\sim , \wedge , \vee \}$-fragment of LP.¹¹

Consider now the conditional as defined in RM3:¹²

• $1\in \sigma (A\rightarrow B)$ if and only if $0\in \sigma (A)$ or [$1\in \sigma (B)$ and $0\notin \sigma (B)$

• $0\in \sigma (A\rightarrow B)$ if and only if $1\in \sigma (A)$ and $0\in \sigma (B)$

As in the previous case, this conditional can also be turned into a conditional with contra-classical features by changing its falsity condition to the Wansing falsity condition. Then, the pair

• $1\in \sigma (A\rightarrow B)$ if a and only if $1\notin A$, or $0\notin B$, or [$0\in A$ and $1\in B$]

• $0\in \sigma (A\rightarrow B)$ if and only if $0\notin \sigma (A)$ or $0\in \sigma (B)$

defines the conditional introduced by in Belikov and Leginov (2019). The resulting truth table for $A \rightarrow _{BL} B$ is as follows:

$A\rightarrow _{BL} B$	$\{1\}$	$\{1, 0\}$	$\{0\}$
$\{1\}$	$\{1\}$	$\{0\}$	$\{0\}$
$\{1, 0\}$	$\{1\}$	$\{1, 0\}$	$\{0\}$
$\{0\}$	$\{1, 0\}$	$\{1, 0\}$	$\{1, 0\}$

Again, Mortensen’s conditional is just the contraposible conditional built upon the BL conditional, i.e. $(A\rightarrow _{BL} B)\wedge (\sim \! B\rightarrow _{BL}\sim \! A)$, as it can be easily verified. For definiteness, let me call ‘dRM3’ the logic resulting from adding the BL conditional to the implicationless fragment of RM3.

However, note that when either the OCO or the BL conditionals are, so to speak, the official conditional in the logic, the validity of ultra-Abelardianism and the Centering axiom schemes are lost, even if connexivity and contradictoriness are retained and hyper-connexivity is gained. An easy inspection of the evaluation conditions reveals that what tells M3V apart from CN and dRM3 is that, in M3V, $0\in \sigma (A\rightarrow B)$ even when $\sigma (A)=\sigma (B)=\{1\}$, which is what warrants the validity of $\sim \! (A\rightarrow B)$ and provides the counterexamples to hyper-connexivity. On the other hand, what makes them similar is, first, that $\sigma (A\rightarrow B)=\{0\}$ when $1\in \sigma (A)$ and $\sigma (B)=\{0\}$; second, that $0\in \sigma (A\rightarrow B)$ when $\sigma (A)=\{0\}$. This latter fact –together with the LP negation, of course– is what brings connexivity in such a straightforward manner.¹³

Kapsner has insisted that in order to meet the intuitive connexive requirements, it is not sufficient to validate Aristotle’s and Boethius’ Theses, but to guarantee that both $A\rightarrow \sim \! A$ and $\sim \! A\rightarrow A$ are unsatisfiable, and that for all interpretations, either $A\rightarrow B$ or $A\rightarrow \sim \! B$ is unsatisfiable. (See for example Kapsner 2012, 2019, 2020.) That a formula X is unsatisfiable means that there is no interpretation under which it is true. Classically, this is equivalent to say that X is false under all interpretations. These two ways of being unsatisfiable can come apart in non-classical contexts, though, and so there is at least two ways of cashing Kapsner’s requirements out.¹⁴

Following Omori and Wansing (2020), let me say that a logic L is truth-relative Kapsner-strong if and only if, for no valuation $\sigma$,

(tKSa):

$1\in \sigma (A\rightarrow \sim \! A)$ and $1\in \sigma (\sim \! A\rightarrow A)$, and

(tKSb):

$1\in \sigma (A\rightarrow B)$ and $1\in \sigma (A\rightarrow \sim \! B)$.

It was under this understanding of Kapsner-strength that CN, M3V and other logics were evaluated in Estrada-González and Ramírez-Cámara (2016), and the result was that none of the contradictory logics was Kapsner-strong. However, let me say that a logic L is falsity-relative Kapsner-strong if and only if, for any valuation $\sigma$,

(fKSa):

$0\in \sigma (A\rightarrow \sim \! A)$ and $0\in \sigma (\sim \! A\rightarrow A)$, and

(fKSb):

$1\in \sigma (A\rightarrow B)$ iff $0\in \sigma (A\rightarrow \sim \! B)$, and $0\in \sigma (A\rightarrow B)$ iff $1\in \sigma (A\rightarrow \sim \! B)$.

When Kapsner-strength is understood in the latter way, M3V, CN and dRM3 are Kapsner-strong.¹⁵

4 A Multi-contra-classical Variant of LP

We have seen that M3V is a logic that encompasses many contra-classical features in a single framework. It only took adding the three-valued E conditional to the $\{\sim , \wedge , \vee \}$-fragment of LP to get that. We saw other non-trivial contra-classical logics—that is, logics that are both connexive and contradictory—extending the $\{\sim , \wedge , \vee \}$-fragment of LP with a conditional connective; in some cases, this meant changing the falsity condition of other well-known conditionals.

A natural question at this point is whether there is a non-trivial extension of the $\{\sim , \wedge , \vee \}$-fragment of LP which validates even more kinds of contra-classicality. For example, whether it can validate the characteristic axiom schemas of the following kinds of logics:

• Relativity logics: $((A\rightarrow B)\rightarrow B)\rightarrow A\qquad$ Meyer–Slaney Relativity

• Lovelight logics: Schemas of the form $\Lambda (R_{T})$

  where $R_T$ is any theorem of a relativity logic and, assuming that the language includes not only a conditional but also negation, conjunction and disjunction, $\Lambda (-)$ is defined as follows:

  $$\begin{aligned}&\Lambda (p) = p\\&\Lambda (\sim \! A) = \sim \! \Lambda (A)\\&\Lambda (A\copyright B) = (\Lambda (A)\copyright \Lambda (B)),\text { for }\copyright \in \{\wedge , \vee \}\\&\Lambda (A\rightarrow B) = (\Lambda (B)\rightarrow \Lambda (A)) \end{aligned}$$

• Super-contracting logics: $(A\rightarrow (A\rightarrow B))\rightarrow B\qquad$ Super-absorption

Some remarks regarding these kinds of logics are in order. First, the mainstream approach to relativity logics has been adding the Relativity axiom schema to a basic purely implicative logic and then enriching the resulting logic. Nonetheless, the earliest study of a relativity logic in Meredith and Prior (1968) used, together with Substitution and Detachment, the single axiom

$$\begin{aligned} \begin{array}{ll} (A\rightarrow B)\rightarrow (((C\rightarrow B)\rightarrow A)\rightarrow C&\quad \hbox {Meredith's Axiom} \end{array} \end{aligned}$$

Second, one can take Meyer and Slaney’s idea that the Relativity axiom schema is a generalized version of the arrow version of Double Negation Elimination and then obtain relativity logics from relativizing a certain collection N of basic principles concerning negation.¹⁶

Third, whether the Meyer–Slaney relativization will lead to contra-classicality or not will unsurprisingly depend on the principles about negation one starts with. Suppose that one starts with $A\rightarrow A$ and $A\vee \sim \! A$, such that the latter is the only $\hbox {N}_i$. Then one ends up with $A\rightarrow A$ and $A\vee (A\rightarrow B)$, both of which are classically valid.

Finally, if a relativity logic is required to be based on BCI logic, as it is suggested since Meyer and Slaney (2002), then Assertion is provable and hence super-contracting logics would be a special case of lovelight ones.¹⁷

Let me return to the question whether there is a way to expand the $\{\sim , \wedge , \vee \}$-fragment of LP to get more kinds of contra-classicality—for example, those just mentioned—into a single logic. The answer is affirmative. Actually, it only takes modifying the falsity condition of the LP conditional as follows:

$$\begin{aligned} 0\in \sigma (A\rightarrow _{O} B)\hbox { iff }0\in \sigma (A)\hbox { or }0\in \sigma (B) \end{aligned}$$

In terms of the Bochum Plan, this move amounts to substituting the falsity condition of the conditional with the falsity condition of conjunction. Here it is the truth table for more visibility:

$A\rightarrow _{O} B$	$\{1\}$	$\{1, 0\}$	$\{0\}$
$\{1\}$	$\{1\}$	$\{1, 0\}$	$\{0\}$
$\{1, 0\}$	$\{1, 0\}$	$\{1, 0\}$	$\{1, 0\}$
$\{0\}$	$\{1, 0\}$	$\{1, 0\}$	$\{1, 0\}$

Let me call ‘mccLP’ the logic resulting from adding the O conditional to the $\{\sim , \wedge , \vee \}$-fragment of LP.¹⁸ Like its LP-like relatives in the previous sections, mccLP is contradictory—as witnesses, take $(A\wedge \sim \! A)\rightarrow A$ and $\sim \! ((A\wedge \sim \! A)\rightarrow A)$—, and although it is (falsity-relative) Kapsner strong, it is not properly hyper-connexive, because it fails the Non-symmetry of Implication.¹⁹

Validating this many contra-classical theses surely comes with a cost. Indeed, the failure of Non-symmetry of Implication might be seen as one of them. Moreover, the following features of the O conditional might suggest that we are no longer dealing with a connective recognizable as a conditional:²⁰

• $\models _{\tiny \mathbf{mccLP }} (A\rightarrow _{O} B)\rightarrow _{O} (A\wedge B)$

• $A, A\rightarrow _{O} B\nvDash _{\tiny \mathbf{mccLP }} B$, including the strict version, when both premises and conclusions are theorems. For a counterexample, consider the schema

  $$\begin{aligned} ((((A\rightarrow _{O} \sim \! A)\rightarrow _{O} \sim \! A))\rightarrow _{O}(A\rightarrow _{O} \sim \! A)\rightarrow _{O} \sim \! A))\rightarrow _{O}(A\rightarrow _{O} \sim \! A) \end{aligned}$$

  which is a logical truth in mccLP; moreover, its antecedent is but an instance of Identity, which is a logical truth in the logic too, but the consequent is not.

• Now, given Dwyer’s Theorem,²¹ the invalidity of strict Detachment implies that some of the rules (BX), ($\hbox {B}^{'}$X) and (BXY) must be invalid in mccLP as well, indicating that the conditional is not fully transitive. In fact, only (BXY) is invalid in mccLP. For a counterexample to (BXY), consider

  $$\begin{aligned} ((A\rightarrow _{O} \sim \! A)\rightarrow _{O} A)\rightarrow _{O}(A\rightarrow _{O} \sim \! A) \end{aligned}$$

  and

  $$\begin{aligned} A\rightarrow _{O}((A\rightarrow _{O} \sim \! A)\rightarrow _{O} A) \end{aligned}$$

  These two are logical truths in mccLP, however, $A\rightarrow _{O}(A\rightarrow _{O} \sim \! A))$ is not. Moreover,

  $$\begin{aligned} A\rightarrow _{O} B, B\rightarrow _{O} C\nvDash _{\tiny \mathbf{mccLP }} A\rightarrow _{O} C \end{aligned}$$

Let me briefly explain why I do not think that these are insurmountable problems. First, equivalence between two formulas X and Y comes in many degrees: one has

• the validity of a biconditional, $X\leftrightarrow Y$, most likely defined as $(X\rightarrow Y)\wedge (Y\rightarrow X)$;

• inter-derivability, i.e. $X\dashv \vdash Y$;

• generalized equivalence, i.e. $1\in \sigma (X)$ iff $1\in \sigma (Y)$, and $0\in \sigma (X)$ iff $0\in \sigma (Y)$, for all $\sigma$;

• strict equivalence, that is, $\sigma (X) = \sigma (Y)$, for all $\sigma$.

Thus, even if one has $\models _{\tiny \mathbf{mccLP }}(A\rightarrow B)\leftrightarrow (A\wedge B)$, ones does not have any of inter-derivability (since $A \rightarrow B \nvDash _{\tiny \mathbf{mccLP }} A \wedge B$), generalized or strict equivalence: simply consider the case when $\sigma (A) = \{1, 0\}$ and $\sigma (B)=\{0\}$. Hence, contrary to the appearances, and in spite of having many conjunctive-like features, the O conditional does not collapse into a conjunction.

One can be alarmed because of the validity of a certain form of Simplification: $(A\rightarrow _{O} B)\rightarrow _{O} A$ and $(A\rightarrow _{O} B)\rightarrow _{O} B$. Nonetheless, one should give reasons for thinking of the arrow as a conjunction in the antecedent but as a conditional when it is the main connective. If one does not apply that double standard here and thinks that $A \rightarrow _{O} B$ is uniformly a conjunction, it validates something not very conjunction-like (let me use ‘$\otimes$’ as a generic conjunction): $(A\otimes B)\otimes A$ and $(A\otimes B)\otimes B$. (Several forms of transitivity, to be discussed below, do not make sense for a conjunction, either.)

So much for the danger of ending with a more conjunction-like connective. Let me give now some reasons to think that we are still in the presence of a conditional. First, the failure of Detachment, even in the strict form, might appear as a serious drawback for some people. Nonetheless, Beall has stressed several times (see for example Beall 2011, 2015) that even if Detachment is invalid for the conditional in LP, it is default valid in the sense that $A, A\rightarrow B \models _{\tiny \mathbf{LP } } B\vee (A\wedge \sim \! A)$ holds, that is, either Detachment holds or the antecedent is a formula evaluated as $\{1, 0\}$, which arguably is not the case in most situations. The second disjunct internalizes in the conclusion the structure of truth values into the object language: Detachment does not hold when the antecedent is both true and false, linguistically expressed as $A\wedge \sim \! A$. Thus, either Detachment holds for A and $A\rightarrow B$, or A is both true and false, i.e. $A\wedge \sim \! A$. Detachment is default valid exactly in the same way in mccLP, that is, $A, A\rightarrow _{O} B\models _{\tiny \mathbf{mccLP }} B\vee (A\wedge \sim \! A)$.

Second, in the very formulation of the final worry already comes the answer. Even if transitivity of the O conditional is lost in some forms, it is kept in others, for example in the validity of (BX) and (B$'$X). Additionally, the arrow forms of (BX), (B$'$X) and (BXY) hold. But perhaps more importantly, Beall’s strategy to recover Detachment as default valid in LP can be borrowed to recover (BXY) and plain Transitivity, since their invalidity is due to the contradictoriness of the middle term. Thus,

$$\begin{aligned} A\rightarrow _{O} B, B\rightarrow _{O} C\models _{\tiny \mathbf{mccLP }} (A\rightarrow _{O} C)\vee (B\wedge \sim \! B) \end{aligned}$$

If $\models _{\tiny \mathbf{mccLP }}C\rightarrow D$ then, if $\models _{\tiny \mathbf{mccLP }}A\rightarrow C$ then $\models _{\tiny \mathbf{mccLP }}(A\rightarrow D)\vee (C\wedge \sim \! C)$

Third: as it has already been advanced since Footnote 16, mccLP is the same logic as the one named DF/TT in Egré et al. (2021), in the sense that they validate exactly the same set of valid arguments. The authors did not notice all its contra-classical potential, though—besides connexivity, they just discussed the validity of $(A\rightarrow _{O} B)\rightarrow _{O} A$. Moreover, it is a bit mistaken identifying the O conditional with the “De Finetti” conditional: the latter is undetermined, has neither truth value, when the antecedent is not true, whereas the former is overdetermined, has both truth values, when the antecedent is not simply true. To cope with logical consequence in a sensible manner, the “De Finetti” conditional needs to be paired with preservation of non-falsity, whereas the O conditional can be paired with the usual preservation of truth. Although those differences do not alter the set of arguments validated, it does affect the presentation and proper understanding of all the features of the logic, not reducible to the arguments validated.

This leads me to discuss the issue of the philosophical import of mccLP more generally and beyond the philosophy of logic. As I have already said, that so many contra-classical features can coexist in a single logic is already worth mentioning. Nonetheless, this might not be seen as a good feature, though, for the different contra-classical features might differ in degree of plausibility. For example, (hyper-)connexivity—and, surprisingly, contradictoriness– seems better motivated than, say, lovelight features, or the simplification of conditionals.²²

I agree with the above remark. Nonetheless, when facing weird yet new or understudied logics, my working assumption is that, if the logic is non-trivial and it contains a bunch of other recognizable principles about the usual connectives, it is very unlikely to get certain principles—contra-classical theses in this case—validated just by chance. Then my working methodology in those cases is that a creative, more charitable stare towards the formalism is needed; not everything should be left to the founding parents nor everything should be judged empty or meaningless without having tried really hard to get something from the apparent dryness. Of course, sometimes this would be an exercise of trying to get blood from a stone—and not everyone should be devoted to do that—, but sometimes the stone might in fact be a lithops. I still do not know whether mccLP is a creature more attractive than a lithops, but I am almost sure that it is not a stone.

The above is not flashy rhetoric. DF/TT in Egré et al. (2021) has been extensively studied as a logic suitable for suppositional reasoning, that is, as a logic where conditionals are read as expressing conditional degrees of belief. Thus, mccLP at least opens the door for studying suppositional reasoning from the point of view of inconsistent degrees of belief. But the possible applications are not exhausted there. $A~\rightarrow _{O}~B$ belongs to two ancient and venerable traditions:

• that of non-vacuously true conditionals, conditionals that are not true (only) when their antecedent is not true;²³

• that of comparative conditionals, where ‘$A\rightarrow B$’ is read ‘B is comparable to A (according to some parameter)’.²⁴

That the conditional is related to these two issues opens further avenues of investigation. Moreover, highlighting the multi-contra-classicality of mccLP points towards a common feature of conditional negation, categorial grammar and constructive falsification—which deliver hyper-connexivity—comparative conditionals—the main application of relativity logics—, suppositional reasoning—which delivers simplification of the conditional—and subjunctive conditionals—one of the contemporary homes of connexivity; see Angell (1962, 2016)—. This is not to say that all these phenomena can be handled at once with mccLP, but merely that they might have important common features that deserves further exploration.

To conclude this section, let me mention one more conditional in this family. From certain remarks in Peña (1995) one can consider an extreme case of a comparative conditional that can be read “B is at least as true as A, provided they have different values and A is not false”:

$A\rightarrow _{P} B$	$\{1\}$	$\{1, 0\}$	$\{0\}$
$\{1\}$	$\{1, 0\}$	$\{1, 0\}$	$\{0\}$
$\{1, 0\}$	$\{1, 0\}$	$\{1, 0\}$	$\{1, 0\}$
$\{0\}$	$\{0\}$	$\{1, 0\}$	$\{1, 0\}$

Put on top of the $\{\sim , \wedge , \vee \}$-fragment of LP, $A\rightarrow _{P} B$ delivers almost the same as $A\rightarrow _{O} B$, with a big difference: it is ultra-Abelardian, i.e. $(A~\rightarrow _{P}~ \sim \! ~ B)$ becomes valid, as in M3V.

Note that this invites comparative readings of the previous conditionals as follows:

• $A~\rightarrow _{OCO}~B$: “B is at least as true as A, provided A is not false”;

• $A~\rightarrow _{BL}~B$: “B is at least as true as A, provided both B and A are not false”;

Note that the provisos may play the same role as suppositions in suppositional conditionals, and that the different conditionals express different reactions to the unsatisfaction of the provisos or the suppositions. Again, this resembles the situation of conditionals with just true antecedents. Nonetheless, a more precise and systematic study of both the non-vacuously true conditionals in the context of contra-classical logics, and of the contra-classical logics following Peña’s ideas on comparative conditionals, is left for further work.

5 Conclusion

I started with the question whether there is a way to expand the $\{\sim , \wedge , \vee \}$-fragment of LP to get more kinds of contra-classicality, not only connexivity and contradictoriness, into a single logic. The answer is affirmative. Mortensen’s three valued connexive logic M3V, which can be described as putting the three-valued E conditional on top of the aforementioned fragment, was an early example of that.

After discussing many properties of M3V, especially in comparison with other connexive and contra-classical logics, I introduced a logic exhibiting even more contra-classical features than M3V. It can be obtained by a simple application of the Bochum Plan: take LP, keep all of it fixed except the falsity condition for the conditional and change it to the falsity condition of conjunction, giving raise to a logic dubbed ‘mccLP’.

The present study leaves open at least the following research avenues:

• The treatment of multi-contra-classicality in a more general setting, for example, on top of FDE rather than LP;

• an integrated study of the different topics that usually motivate certain contra-classical logics and whose formal counterparts coexist in mccLP, namely, suppositional reasoning, conditional negation or comparative conditionals;

• a more systematic study of non-vacuously true conditionals in the context of contra-classical logics.