# Mortensen Logics

Luis Estrada-González Institute for Philosophical Research
National Autonomous University of Mexico
Mexico City, MexicoThis paper was written during the COVID-19 crisis. The
authors want to thank the reviewers for their precious comments, as well as
the support from the PAPIIT project IG400422 and from the Notre Dame
International-Mexico Faculty Grant Program project “The scope and limits of
non-detachable conditionals”. The first author also wants to acknowledge the
support from the DGAPA-UNAM through a PASPA sabbatical grant and from the
Coimbra Group and the KU Leuven through a scholarship from the Programme for
Young Professors and Researchers from Latin American Universities.Faculty of
Physico-Mathematical Sciences
Meritorious Autonomous University of Puebla
Puebla, Mexico<EMAIL_ADDRESS>Universidad Panamericana
Mexico City Fernando Cano-Jorge Universidad Panamericana
Mexico City<EMAIL_ADDRESS>
###### Abstract

In [24], Mortensen introduced a connexive logic commonly known as ‘M3V’. M3V
is obtained by adding a special conditional to LP. Among its most notable
features, besides its being connexive, M3V is negation-inconsistent and it
validates the negation of every conditional. But Mortensen has also studied
and applied extensively other non-connexive logics, for example, _closed set
logic_ , CSL, and a variant of Sette’s logic, identified and called ‘P2’ by
Marcos in [18].

In this paper, we analyze and compare systematically the connexive variants of
CSL and P2, obtained by adding the M3V conditional to them. Our main
observations are two. First, that the inconsistency of M3V is exacerbated in
the connexive variant of closed set logic, while it is attenuated in the
connexive variant of the Sette-like P2. Second, that the M3V conditional is,
unlike other conditionals, _connexively stable_ , meaning that it remains
connexive when combined with the main paraconsistent negations.

##  1 Introduction

In a paraconsistent context where formulas have three admissible assignments,
and assuming the standard properties with respect to the “classical”
assignments, that is

$A$ | $NA$
---|---
$\\{1\\}$ | $\\{0\\}$
$\\{1,0\\}$ |
$\\{0\\}$ | $\\{1\\}$

there are only two possibilities for paraconsistent negation $N$, namely the
de Morgan negation found in González-Asenjo/Priest’s LP and the negation of
Sette’s P1, respectively:

$A$ | $\sim\\!A$
---|---
$\\{1\\}$ | $\\{0\\}$
$\\{1,0\\}$ | $\\{1,0\\}$
$\\{0\\}$ | $\\{1\\}$
$A$ | $\neg A$
---|---
$\\{1\\}$ | $\\{0\\}$
$\\{1,0\\}$ | $\\{1\\}$
$\\{0\\}$ | $\\{1\\}$

Among his many contributions in logic and philosophy, Chris Mortensen
introduced a connexive logic commonly known as ‘M3V’. M3V is obtained by
adding a special conditional to González-Asenjo/Priest’s LP. Such conditional
is structurally the same as the one used by Anderson and Belnap in [2] 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 $X>(Y>Z)$,
with $>$ an entailment connective, $X$ a contingent truth and $(Y>Z)$ a
logical truth.111A logic containing M3V was developed around the same time by
Peña to cope with comparatives, gradables and vagueness. See [35] for a
summary of his results and [34] for a more friendly exposition of them. Among
its most notable features, besides its being connexive, M3V is negation-
inconsistent and it validates the negation of every conditional.

But Mortensen has also studied and applied extensively other non-connexive
logics. On the one hand there is _closed set logic_ , CSL, a paraconsistent
logic motivated by dualizing open set logic, i.e. intuitionistic logic. CSL
has notoriously been found defective in lacking a conditional connective
because in it there is no connective $\copyright$ such that $A\copyright B$ is
untrue if $A$ is true and $B$ untrue, as one would expect from a conditional.
The two most obvious candidates, $\neg A\vee B$ and $\neg(A\wedge\neg B)$ are
true when $A$ is true and $B$ is untrue, delivering thus countermodels to
Detachment.222Mortensen has always argued that this is not a serious defect,
especially when it comes to doing mathematics with CSL. We will not address
this issue here. The fact is that there is no such connective in the logic;
how bad is that is a different discussion. On the other hand, in [25] he
proposed another logic, which later Marcos [18] modified to obtain a variant
of Sette’s logic, identified and called P2 by Marcos.

In this paper, we analyze and compare systematically the connexive variants of
CSL and P2, obtained by adding the M3V conditional to them. Our main
observations are two. First, that the inconsistency of M3V is exacerbated in
the connexive variant of closed set logic, while it is attenuated in the
connexive variant of the Sette-like P2. Second, that the M3V conditional is,
unlike other conditionals, _connexively stable_ , meaning that it validates
the core connexive schemas when combined with the main paraconsistent
negations.

The plan of the paper is as follows. In Section 2 we present some preliminary,
general notions that will be useful for the remainder of the paper. In Section
3 we present M3V and mention some of its properties; some of them are already
well-known, but others are noticed here for the first time. In Section 4 we
introduce cCSL3, closed set logic restricted to three admissible
interpretations, like M3V, enriched with the E-conditional. We give some of
its most notable features, including likenesses and differences with M3V.
There we show that, unlike other conditionals, the E-conditional is
connexively stable with respect to both $\sim$ and $\neg$. Finally, in Section
5 we present cP2. It shares the
$\\{\sim,\rightarrow_{\\_}{\tiny{\textbf{E}}}\\}$-fragment with M3V, but still
they differ in ways that are significant for connexive logicians.

##  2 Preliminary notions

Let $A$ and $B$ arbitrary formulas of a given formal language, and $\Gamma$ a
set of formulas of that language. In this paper, logical consequence is
understood as truth-preservation from premises to conclusions in all
interpretations, that is:

  * $\Gamma\models_{\\_}{\tiny{\textbf{L}}}A$ if and only if, for all $\sigma$, if $1\in\sigma(B)$ for all $B\in\Gamma$ then $1\in\sigma(A)$

Now, let $N$ and $>$ be a negation and a conditional, respectively.
_Unrestricted Detachment_ is logically valid in L iff

$A,A>B\models_{\\_}{\tiny{\textbf{L}}}B$

A logic L is _connexive_ iff the following hold:

$\models_{\\_}{\tiny{\textbf{L}}}N\\!(A>N\\!A)$ Aristotle’s Thesis

$\models_{\\_}{\tiny{\textbf{L}}}N\\!(N\\!A>A)$ Variant of Aristotle’s Thesis

$\models_{\\_}{\tiny{\textbf{L}}}(A>B)>N\\!(A>N\\!B)$ Boethius’ Thesis

$\models_{\\_}{\tiny{\textbf{L}}}(A>N\\!B)>N\\!(A>B)$ Variant of Boethius’
Thesis

and

$\not\models_{\\_}{\tiny{\textbf{L}}}(A>B)>(B>A)$ Non-symmetry of implication

A logic L is _hyper-connexive_ iff it is connexive and at least one of the
following holds:

$\models_{\\_}{\tiny{\textbf{L}}}N\\!(A>N\\!B)>(A>B)$ Converse of Boethius’
Thesis

$\models_{\\_}{\tiny{\textbf{L}}}N\\!(A>B)>(A>N\\!B)$ Converse of Variant of
Boethius’ Thesis

A logic L is _nexive_ iff the following hold:

$\models_{\\_}{\tiny{\textbf{L}}}N\\!(A>N\\!A)$ Aristotle’s Thesis

$\models_{\\_}{\tiny{\textbf{L}}}N\\!(N\\!A>A)$ Variant of Aristotle’s Thesis

$\models_{\\_}{\tiny{\textbf{L}}}(N\\!A>B)>N\\!(A>B)$ Francez’s Thesis

$\models_{\\_}{\tiny{\textbf{L}}}(A>B)>N\\!(N\\!A>B)$ Variant of Francez’s
Thesis

and

$\not\models_{\\_}{\tiny{\textbf{L}}}(A>B)>(B>A)$ Non-symmetry of implication

A logic L is _hyper-nexive_ iff it is nexive and at least one of the following
holds:

$\models_{\\_}{\tiny{\textbf{L}}}N\\!(A>B)>(N\\!A>B)$ Converse of Francez’s
Thesis

$\models_{\\_}{\tiny{\textbf{L}}}N\\!(N\\!A>B)>(A>B)$ Converse of Boethius’
Thesis

A logic L is _contradictory_ or _negation-inconsistent_ iff there is an $A$
such that $\models_{\\_}{\tiny{\textbf{L}}}A$ and
$\models_{\\_}{\tiny{\textbf{L}}}N\\!A$.

##  3 Mortensen’s three-valued connexive logic

The logic M3V was introduced, although not with that name, in [24] (the name
was given in [21], presumably to mean “Mortensen’s 3-valued connexive logic”).
The following truth tables, with $V_{\\_}{\tiny{\textbf{M3V}}}=\\{2,1,0\\}$
and $D^{+}=\\{2,1\\}$, characterize M3V:

$A$ | $B$ | $\sim\\!A$ | $A\wedge B$ | $A\vee B$ | $A\rightarrow_{\\_}{\tiny{\textbf{E}}}B$
---|---|---|---|---|---
$2$ | $2$ | $0$ | $2$ | $2$ | $1$
$2$ | $1$ | $0$ | $1$ | $2$ | $0$
$2$ | $0$ | $0$ | $0$ | $2$ | $0$
$1$ | $2$ | $1$ | $1$ | $2$ | $1$
$1$ | $1$ | $1$ | $1$ | $1$ | $1$
$1$ | $0$ | $1$ | $0$ | $1$ | $0$
$0$ | $2$ | $2$ | $0$ | $2$ | $1$
$0$ | $1$ | $2$ | $0$ | $1$ | $1$
$0$ | $0$ | $2$ | $0$ | $0$ | $1$

A biconditional can be defined as usual, that is, as
$(A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)\wedge(B\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$.

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 [2] to show the consistency of the logic E, hence the subscript. In
particular, they showed how to block the paradox of necessity, i.e. to avoid
validating formulas of the form $X>(Y>Z)$, where $X$ is a contingent truth and
$(Y>Z)$ is a logical truth.333A logic containing M3V was developed around the
same time by Peña to cope with comparatives, gradables and vagueness. See [35]
for a summary of his results and [34] for a more friendly exposition of them.

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 $\\{\ \\}$ aside as in the two-valued relational
semantics for LP:

$A$ | $B$ | $\sim\\!A$ | $A\wedge B$ | $A\vee B$ | $A\rightarrow_{\\_}{\tiny{\textbf{E}}}B$
---|---|---|---|---|---
$\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1,0\\}$
$\\{1\\}$ | $\\{1,0\\}$ | $\\{0\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{0\\}$
$\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$
$\\{1,0\\}$ | $\\{1\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{1,0\\}$
$\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$
$\\{1,0\\}$ | $\\{0\\}$ | $\\{1,0\\}$ | $\\{0\\}$ | $\\{1,0\\}$ | $\\{0\\}$
$\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1,0\\}$
$\\{0\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$
$\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{1,0\\}$

Applying the mechanical procedure described in [32] 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
$\rightarrow_{\\_}{\tiny{\textbf{E}}}$ are as follows:

  * •

$1\in\sigma(A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)$ if a and only if $1\notin
A$, or $0\notin B$, or both $0\in A$ and $1\in B$

  * •

$0\in\sigma(A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)$ if and only if
$1\in\sigma(A)$ or $0\in\sigma(A)$ and either $1\in\sigma(B)$ or
$0\in\sigma(B)$

We are now in a position to point out some of M3V’s main features.

  * •

Unlike LP, M3V validates unrestricted Detachment.

  * •

It is connexive.

  * •

It is contradictory. As witnesses, consider
$(A\wedge\sim\\!A)\rightarrow_{\\_}{\tiny{\textbf{E}}}A$ and
$\sim\\!((A\wedge\sim\\!A)\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$.

  * •

All conditionals are false in M3V. The falsity condition for the conditional
is but a sophisticated way of expressing
$0\in\sigma(A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)$, which implies that
$\models_{\\_}{\tiny{\textbf{M3V}}}\sim\\!(A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)$,
for any $A$ and $B$.

  * •

Though all conditionals are false in M3V, some of them are true as well.
Simply consider a conditional where both antecedent and consequent are just
true. The conditional is false, yet true as well.

  * •

$\models_{\\_}{\tiny{\textbf{M3V}}}\sim\\!(A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)$
implies
$\models_{\\_}{\tiny{\textbf{M3V}}}\sim\\!(A\rightarrow_{\\_}{\tiny{\textbf{E}}}\sim\\!B)$,
by a simple substitution in the consequent. Due to the validity of the latter,
we say that M3V is _ultra-Abelardian_.444Claudio 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 [19].) 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.

  * •

Almost obvious given the validity of
$\sim\\!(A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)$, but even more overlooked,
is the fact that M3V validates some schemas from Abelian logic, namely the
_Centering_ principles555Nonetheless, it does not validate the _Meyer-Slaney
relativity axiom (schema)_ , characteristic of purely implicative Abelian
logics:
$\not\models_{\\_}{\textbf{\tiny{M3V}}}((A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)\rightarrow_{\\_}{\tiny{\textbf{E}}}B)\rightarrow_{\\_}{\tiny{\textbf{E}}}A$
(For a countermodel, let $\sigma(A)=\\{0\\}$ and $\sigma(B)=\\{1\\}$.) The
validity of $\sim\\!(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$ demands moreover
a comparison with Meyer and Martin’s SI$\sim$I —see [23]—, where such schema
is valid too. In that logic, $(C\rightarrow D)\rightarrow((A\rightarrow
C)\rightarrow(A\rightarrow D))$ and $(A\rightarrow C)\rightarrow((C\rightarrow
D)\rightarrow(A\rightarrow D))$, both object-language expressions of
transitivity, are valid, but their negations are not. Nevertheless, since all
conditionals are false in M3V, the negation of these forms of transitivity is
valid as well.:

$\models_{\\_}{\tiny{\textbf{M3V}}}\sim\\!(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$

$\models_{\\_}{\tiny{\textbf{M3V}}}\sim\\!(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)\leftrightarrow_{\\_}{\tiny{\textbf{E}}}(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$

(This provides other witnesses of negation-inconsistency, namely
$A\rightarrow_{\\_}{\tiny{\textbf{E}}}A$ and
$\sim\\!(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$.

  * •

M3V is not hyper-connexive. Suppose it were, and that the Converse of Boethius
hold. By ultra-Abelardianism and Detachment, $A\rightarrow B$ would be valid,
but it is not. (A similar argument can be run using the Converse of the
Variant of Boethius and the falsity of all conditionals.)

  * •

Francez’s logics (see [11]; see also [12] and [13]) have been the only
recognized nexive logics so far. But M3V is nexive too, as a consequence of
all negated conditionals being true. It is not hyper-nexive, though. (The
proof is similar to the proof that it is not hyper-connexive.)

From the above, perhaps the most surprising feature is the fact that all
conditionals are false in M3V. Indeed, one could argue that M3V is an
interesting logic in so far as having arbitrary false conditionals, among many
otherwise familiar properties, is an interesting feature for a logic to have.
Nonetheless, this may require some philosophical elucidation.

The first thing to be said is that Cantwell’s logic for conditional negation
CN and M3V are inter-definable. In particular, the E-conditional is the
contraposable conditional defined with the conditional in CN; see [33]. Thus,
one could attempt to build upon the intuitive features of CN to obtain some
extra-logical support for M3V. True, the intuitiveness of the basic notions do
not transfer immediately to the derived notions, but it could be a start.

We do not follow that route, though. In our view, it is not unreasonable to
have a logic in which all conditionals are false. On the one hand, tradition
has it that certain syllogisms that are deemed valid often lack some tacit
premise. For example, from “Every human is mortal” infer “I am mortal”, where
premise “I am a human” is tacit, i.e. it is a suppressed or unstated truth or
piece of information not mentioned explicitly yet being part of the argument
so that the conclusion indeed follows. This kind of argument is called
_enthymeme_ by Aristotle (_Rethoric_ , 1357a16-21) and the implication
relation between its premises and its conclusions is called _enthymematic
implication_ by Sylvan [38, p. 142]. Following this line of thought, M3V might
be considered as a logic of enthymematic implication, i.e. as a logic about
conditional arguments that strictly speaking are invalid, since they always
lack some antecedent, premise or background information in order to hold (i.e.
in order to entail the conclusion or consequent), but which may also be
accepted as valid _sotto voce_ , _prima facie_ or _ceteris paribus_.

On the other hand, connexive logic has been intimately attached to
counterfactual notions from its very (contemporary) beginnings. (See [3].).
This is relevant because, for example, Alan Hájek has long argued, in still
unpublished but much read work, for the idea that most counterfactuals are
false. (See [15].) According to him, the indeterminism and indeterminacy
associated with most counterfactuals entail their falsehood. Yet,
counterfactual reasoning seems to play an important role in science, and
ordinary speakers judge many counterfactuals that they utter to be true. Thus,
M3V could be regarded as both a (zero-order) formalization of a radical
version of Hájek’s ideas on the falsity of counterfactual conditionals, while
also capturing the idea that some of them need to be true.666There are of
course many ways to address Hájek’s challenge, and many of them that do not
require a contradictory logic. Here we simply suggest that M3V can be taken as
a formalization of a certain form of that debate. For another proposal in the
connexive vicinity to address Hájek’s challenge, see [16].

Finally, Meyer and Martin wanted to provide a logic for Aristotle’s
syllogistic, which was irreflexive. In their logic SI$\sim$I777They do not
call it in that way, though. However, we simply indicate what further axiom
schemas are added to the basis S, with ‘I’ standing for $A\rightarrow A$, and
‘$\sim$I’ for $\sim\\!(A\rightarrow A)$., $A\rightarrow A$ was treated as a
borderline case, both a fallacy with no valid instances (due to the
irreflexivity of entailment) and a validity (because of the truth-preservation
account of entailment), hence the validity of both $A\rightarrow A$ and
$\sim\\!(A\rightarrow A)$. One could explore the idea that implication or
entailment are relations so demanding that no sentences can be ever in that
relation, hence the validity of $\sim\\!(A\rightarrow B)$. However, as in the
case of SI$\sim$I, one could argue that, for theoretical simplicity, in this
case, the functional approach, the truth of some instances of $A\rightarrow B$
are required as well.

We know that all what we have said is far from convincing. However, making a
full case for the conceptual usefulness of M3V is beyond our aims. We merely
expressed some ideas to take this logic as more than a mathematical curiosity.

##  4 Connexive closed set logic

The logic that we call ‘closed set logic’ was introduced algebraically in [26]
and subsequently studied in [27], [28] and [29].888Although the ideas
underlying it are older, going back at least to [22]. The first systematic
treatment of that logic on its own was the proof-theoretical analysis in [14].
We focus here in the restriction to three interpretations, CSL3, defined by
the following tables:

$A$ | $B$ | $\neg A$ | $A\wedge B$ | $A\vee B$
---|---|---|---|---
$\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{1\\}$ | $\\{1,0\\}$ | $\\{0\\}$ | $\\{1,0\\}$ | $\\{1\\}$
$\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$
$\\{1,0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1,0\\}$ | $\\{1\\}$
$\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$
$\\{1,0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1,0\\}$
$\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$
$\\{0\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1,0\\}$
$\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$

It is common wisdom that there is no conditional in CSL3. Consider a
connective defined as follows:

$A\rightarrow B\coloneqq\neg A\vee B\ (=\neg(A\wedge\neg B))$

This connective does not validate Detachment.999Although, in all fairness, it
validates a restricted version, due to Beall [4], [5] in the context of LP,
namely,
$A,~{}A~{}\rightarrow~{}B~{}\models_{\\_}{\tiny{\textbf{CSL3}}}~{}B\vee(A\wedge\neg
A)$. There are several ways to expand CSL3 with a conditional connective that
validates Detachment. In fact, 24 non-connexive conditionals could do the job;
see [8, p. 72].

But consider the expansion cCSL3, which adds the E-conditional, one of the 24
mentioned above, to CSL3. Let us point out some of cCSL3’s main features,
starting with those involving just its
$\\{\neg,\rightarrow_{\\_}{\tiny{\textbf{E}}}\\}$-fragment.

  * •

cCSL3 validates unrestricted Detachment.

  * •

All conditionals are false in cCSL3 and so
$\models_{\\_}{\tiny{\textbf{cCSL3}}}\neg(A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)$,
for any $A$ and $B$.

  * •

It is connexive.

  * •

It follows that if cCSL3 is connexive and all conditionals are false in it,
cCSL3 is contradictory, just as M3V. As witnesses, consider
$A\rightarrow_{\\_}{\tiny{\textbf{E}}}A$ and
$\neg(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$.

  * •

cCSL3 is ultra-Abelardian.

  * •

cCSL3 does not validate exactly the same centering principles as M3V. One has

$\models_{\\_}{\tiny{\textbf{cCSL3}}}\neg(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$

$\models_{\\_}{\tiny{\textbf{cCSL3}}}(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)\rightarrow_{\\_}{\tiny{\textbf{E}}}\neg(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$

but also

$\not\models_{\\_}{\tiny{\textbf{cCSL3}}}\neg(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)\rightarrow_{\\_}{\tiny{\textbf{E}}}(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$

  * •

The above implies that cCSL3 also lacks the Deduction Property. In fact, every
logical truth in cCSL3 entails any other logical truth, in particular,
$\neg(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)\models_{\\_}{\tiny{\textbf{cCSL3}}}A\rightarrow_{\\_}{\tiny{\textbf{E}}}A$,
yet
$\not\models_{\\_}{\tiny{\textbf{cCSL3}}}\neg(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)\rightarrow_{\\_}{\tiny{\textbf{E}}}(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$.

  * •

The invalidity of
$\neg(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)\rightarrow_{\\_}{\tiny{\textbf{E}}}(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$
generalizes. Since any conditional of the form
$X\rightarrow_{\\_}{\tiny{\textbf{E}}}Y$ is false and any conditional of the
form $\neg(W\rightarrow_{\\_}{\tiny{\textbf{E}}}Z)$ is just true, it follows
that no conditional of the form
$\neg(W\rightarrow_{\\_}{\tiny{\textbf{E}}}Z)\rightarrow_{\\_}{\tiny{\textbf{E}}}(X\rightarrow_{\\_}{\tiny{\textbf{E}}}Y)$
is valid.101010And the validity of
$(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)\rightarrow_{\\_}{\tiny{\textbf{E}}}\neg(A\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$
also generalizes: every conditional of the form
$(X\rightarrow_{\\_}{\tiny{\textbf{E}}}Y)\rightarrow_{\\_}{\tiny{\textbf{E}}}\neg(W\rightarrow_{\\_}{\tiny{\textbf{E}}}Z)$
is valid.

  * •

It is clear now that cCSL3 and M3V validate different arguments. As another
witness, consider $A\models_{\\_}{\textbf{M3V}}\sim\sim\\!A$ but
$A\not\models_{\\_}{\textbf{cCSL3}}\neg\neg A$.

  * •

cCSL3 is not hyper-connexive. The argument is as for M3V.

  * •

cCSL3 is nexive, just as M3V. And like M3V, it is not hyper-nexive. Again, the
proof is an adaptation of the proof that M3V is not hyper-connexive.

A natural question at this point is whether $\neg$ is definable in M3V. It is
not. It could be defined as
$\sim\\!\circ(A\rightarrow_{\\_}{\tiny{\textbf{E}}}\sim\\!\circ\circ\\!A)$,
with $\circ$ a consistency connective:

$A$ | $\circ A$
---|---
$\\{1\\}$ | $\\{1\\}$
$\\{1,0\\}$ | $\\{0\\}$
$\\{0\\}$ | $\\{1\\}$

But such a connective is not definable in M3V: The connective is not definable
in CN as per [31], and a connective is definable in M3V iff it is definable in
CN, as per [33].111111What about defining the LP negation in cCSL3? We do not
know, but our guess is that it cannot be defined.

The list of properties above does not highlight enough some features of cCSL3,
especially around connexive principles:

  * •

$\neg(A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)$ is just true in all
interpretations in cCSL3; $\sim\\!(A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)$ is
true in all interpretations in M3V, but it is also false under some of them.
This has consequences for the connexive principles, as we will see.

  * •

Recall that, in M3V, Aristotle’s Theses are true under all interpretations,
although there are some interpretations under which they are also false. That
is not the case in cCSL3: Aristotle’s Theses are just true.

  * •

Boethius’ Theses are true under all interpretations in M3V, although they are
also false under all interpretations. That is the case as well in cCSL3, with
the difference that in this logic, the negations of Boethius’ Theses are just
true.

  * •

Both $(A\wedge\sim\\!A)\rightarrow_{\\_}{\tiny{\textbf{E}}}A$ and
$\sim\\!((A\wedge\sim\\!A)\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$ are valid in
M3V, they are both true and false in all interpretations. But although both
$(A\wedge\neg A)\rightarrow_{\\_}{\tiny{\textbf{E}}}A$ and $\neg((A\wedge\neg
A)\rightarrow_{\\_}{\tiny{\textbf{E}}}A)$ are valid in cCSL3, the latter is
just true in all interpretations.

  * •

More generally: If both $X$ and $\sim\\!X$ are valid in M3V, then $\neg X$ is
just true in cCSL3, unlike $\sim\\!X$ in M3V, even if $X$ fails to be valid in
cCSL3. (The proof is straightforward. For schemas exemplifying this, recall
the ones for the failure of the Deduction Property.)

Finally, an attractive feature of the E-conditional should be mentioned:
unlike some well-known connexive conditionals in the literature, it is stable
under changes of negation. Let us make that more precise.

Let a _standard negation_ be a unary connective $N$ satisfying that
$\sigma(NA)=\\{1\\}$ if $\sigma(A)=\\{0\\}$, and $\sigma(NA)=\\{0\\}$ if
$\sigma(A)=\\{1\\}$. If a standard negation $N$ is such that, in a logic L,
$A,NA\not\models_{\\_}{\tiny{\textbf{L}}}B$, we will call it a _standard
paraconsistent negation_. Let us define the _type of standard paraconsistent
negations_ (TSPN) as the set of all such connectives definable according to a
set of admissible evaluations. In the present context, TSPN only has two
connectives: $\sim$ and $\neg$.

Now, let us say that a conditional $A>B$ is _connexively stable_ with respect
to TSPN iff

$\models_{\\_}{\tiny{\textbf{L}}}N_{\\_}i\\!(A>N_{\\_}i\\!A)$

$\models_{\\_}{\tiny{\textbf{L}}}N_{\\_}i\\!(N_{\\_}i\\!A>A)$

$\models_{\\_}{\tiny{\textbf{L}}}(A>B)>N_{\\_}i\\!(A>N_{\\_}i\\!B)$

$\models_{\\_}{\tiny{\textbf{L}}}(A>N_{\\_}i\\!B)>N_{\\_}i\\!(A>B)$

and

$\not\models_{\\_}{\tiny{\textbf{L}}}(A>B)>(B>A)$

for each $N_{\\_}i$ in TSPN. From the previous discussion,
$\rightarrow_{\\_}{\tiny{\textbf{E}}}$ is connexively stable with respect to
TSPN. However, the main connexive conditionals in the literature are not
connexively stable. The conditionals defined by the following tables validate
the connexive principles only with $\sim$, but not with $\neg$:

$A\rightarrow_{\\_}{W}B$ | $\\{1\\}$ | $\\{1,0\\}$ | $\\{0\\}$
---|---|---|---
$\\{1\\}$ | $\\{1\\}$ | $\\{1,0\\}$ | $\\{0\\}$
$\\{1,0\\}$ | $\\{1\\}$ | $\\{1,0\\}$ | $\\{0\\}$
$\\{0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$

$A\rightarrow_{\\_}{BL}B$ | $\\{1\\}$ | $\\{1,0\\}$ | $\\{0\\}$
---|---|---|---
$\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$
$\\{1,0\\}$ | $\\{1\\}$ | $\\{1,0\\}$ | $\\{0\\}$
$\\{0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$

They are, respectively, Wansing’s conditional from [39] restricted to three
admissible interpretations —found explicitly for three interpretations in
[30], [7], [31]—, and Belikov and Loginov’s conditional from [6]. Although
Aristotle’s Thesis becomes just true under all interpretations with the first
conditional and $\neg$, Boethius’ Thesis fails: it is just false when $A$ is
just true and $B$ is both true and false. The problem with the second
conditional is a sort of dual: Boethius’ Thesis is valid, but Aristotle’s
Thesis fails when $A$ is both true and false.

Note that Francez’s conditional, from [12], restricted to three admissible
interpretations, i.e.

$A\rightarrow_{\\_}{F}B$ | $\\{1\\}$ | $\\{1,0\\}$ | $\\{0\\}$
---|---|---|---
$\\{1\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$ | $\\{0\\}$
$\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$ | $\\{0\\}$
$\\{0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$

is also stable with respect to standard paraconsistent negations. It can be
easily verified that it does not allow for the countermodels present in the
previous conditionals.121212Angell-McCall’s conditional, found in [3] and
[20], is connexive with respect to Boolean negation, but not with respect to
other standard explosive negations. The definition of this notion, and the
verification of the claim about the Angell-McCall’s conditional are left as an
exercise.

##  5 Connexive P2

There is one more logic due partly to Mortensen, but also partly to Marcos. In
[25], Mortensen introduced a logic called ‘C${}_{\\_}{0.2}$’ characterized by
the following tables:

$A$ | $B$ | $\sim\\!A$ | $A\wedge_{\\_}{\tiny{\textbf{P}}}B$ | $A\vee_{\\_}{\tiny{\textbf{P}}}B$ | $A\rightarrow_{\\_}{\tiny{\textbf{P}}}B$
---|---|---|---|---|---
$\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{1\\}$ | $\\{\ \\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$
$\\{\ \\}$ | $\\{1\\}$ | $\\{\ \\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{\ \\}$ | $\\{\ \\}$ | $\\{\ \\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{\ \\}$ | $\\{0\\}$ | $\\{\ \\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$
$\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{0\\}$ | $\\{\ \\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$

(Mortensen originally used three values, 1, 2, and 3, being 1 the only
designated value. We are taking advantage here of the Dunn semantics, as
mentioned in Section 3.)

Marcos [18] suggested to replace the interpretation $\\{\ \\}$ by the
interpretation $\\{1,0\\}$ —or, in his original terms, to make the value 2
designated along with 1—, and put $\sim$ instead of $\neg$, to get the logic
P2, whose tables look like these:

$A$ | $B$ | $\sim\\!A$ | $A\wedge_{\\_}{\tiny{\textbf{P}}}B$ | $A\vee_{\\_}{\tiny{\textbf{P}}}B$ | $A\rightarrow_{\\_}{\tiny{\textbf{P}}}B$
---|---|---|---|---|---
$\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{1\\}$ | $\\{1,0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$
$\\{1,0\\}$ | $\\{1\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{1,0\\}$ | $\\{0\\}$ | $\\{1,0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$
$\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{0\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$

Now, to get a connexive variant of this, cP2, replace the P-conditional with
the E-conditional, i.e. obtain a logic characterized by the following tables:

$A$ | $B$ | $\sim\\!A$ | $A\wedge_{\\_}{\tiny{\textbf{P}}}B$ | $A\vee_{\\_}{\tiny{\textbf{P}}}B$ | $A\rightarrow_{\\_}{\tiny{\textbf{E}}}B$
---|---|---|---|---|---
$\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1,0\\}$
$\\{1\\}$ | $\\{1,0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$
$\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$
$\\{1,0\\}$ | $\\{1\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1,0\\}$
$\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1,0\\}$
$\\{1,0\\}$ | $\\{0\\}$ | $\\{1,0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$
$\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1,0\\}$
$\\{0\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1,0\\}$
$\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{1,0\\}$

M3V and cP2 coincide in the
$\\{\sim,\rightarrow_{\\_}{\tiny{\textbf{E}}}\\}$-fragment, but they differ in
ways that are significant for connexive logicians. Consider the following
(non-core) connexive principles:

$\sim\\!((A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)\wedge_{\\_}{\tiny{\textbf{P}}}(\sim\\!A\rightarrow_{\\_}{\tiny{\textbf{E}}}B))$
Aristotle’s Second Thesis

$\sim\\!((A\rightarrow_{\\_}{\tiny{\textbf{E}}}B)\wedge_{\\_}{\tiny{\textbf{P}}}(A\rightarrow_{\\_}{\tiny{\textbf{E}}}\sim\\!B))$
Abelard’s Principle

These are valid in M3V, as originally reported in [10], but they are not in
cP2. For a countermodel, consider the case when both $A$ and $B$ are both true
and false. (This will do for both principles.) For the record, these are
countermodels for the principles written in the language of cCSL3.

### Short digression.

There is a different, more direct way of presenting cP2, starting directly
with Sette’s P1 without the detour through Mortensen’s C${}_{\\_}{2.0}$.
Consider Sette’s logic P1, characterized by the following truth tables:

$A$ | $B$ | $\neg A$ | $A\wedge_{\\_}{\tiny{\textbf{P}}}B$ | $A\vee_{\\_}{\tiny{\textbf{P}}}B$ | $A\rightarrow_{\\_}{\tiny{\textbf{P}}}B$
---|---|---|---|---|---
$\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{1\\}$ | $\\{1,0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$
$\\{1,0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{1,0\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{1,0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$
$\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{0\\}$ | $\\{1,0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{1\\}$
$\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{0\\}$ | $\\{0\\}$ | $\\{1\\}$

To obtain P2, simply replace $\neg$ by $\sim$, as it has been already noticed
in [17].131313He anticipated thus Marcos’ formulation of P2. However, Karpenko
wrongly claims that Mortensen’s original logic C${}_{\\_}{0.2}$ is
paraconsistent. Karpenko assumed that the value 2, in Mortensen’s
presentation, is designated, which is not. Marcos correctly noticed that P2
requires a certain amount of dualization in Mortensen’s C${}_{\\_}{0.2}$.
Then, to get cP2, replace the P-conditional with the E-conditional.

##  6 Conclusions

In this paper, we took two interests of Mortensen, connexivity and certain
brands of paraconsistency, and combined them into single logics. Although
connexivity is at least a matter of two, negation and the conditional, the
E-conditional of Mortensen’s M3V excels among other conditional in validating
the connexive schemas even when combined with other (paraconsistent)
negations. Also, some features of M3V are exacerbated when a different
negation is used. For example, in M3V all negated conditionals are true, but
also sometimes false, whereas changing the negation leads to the result that
negated conditionals are just true, never false.

There are at least five ways in which this work can be continued. First, when
presented in slightly different ways, a logic might exhibit more interesting
features. As we mentioned, the logics CN and M3V are inter-definable; it would
be worth take a look at the logics defined here with other conditionals
definable on them. Second, one could enrich the languages here with
consistency connectives to make a comparison with the LFIs. Third, one could
try to get both negations in a single language and study the effect of that on
connexive principles. Fourth, at least in the E-conditional , Mortensen
suggested to couple closed set logic with different notions of logical
consequence. This would allow, among other things, to discriminate between
schemas that are true under all interpretations and those that are just true
under all interpretations. This would give rise to “exactly true” or “non-
falsity” versions of the logics above, which have been studied in the vicinity
of FDE. (See for example [36] and [37], but also [9] for a discussion closer
to the present context.) Speaking of that, and finally, one can move the
entire discussion on top of FDE to work with more admissible interpretations.
That would augment the number of logical and conceptual distinctions available
to work with.

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