# Connexive implications in Substructural Logics

Davide Fazio<EMAIL_ADDRESS>Gavin St. John<EMAIL_ADDRESS>
###### Abstract

This paper is devoted to the investigation of term-definable connexive
implications in substructural logics with exchange and, on the semantical
perspective, in sub-varieties of commutative residuated lattices
(FL${}_{\mbox{\scriptsize e}}$-algebras). In particular, we inquire into
sufficient and necessary conditions under which generalizations of the
connexive implication-like operation defined in [6] for Heyting algebras still
satisfy connexive theses. It will turn out that, in most cases, connexive
principles are equivalent to the equational Glivenko property with respect to
Boolean algebras. Furthermore, we provide some philosophical upshots like
e.g., a discussion on the relevance of the above operation in relationship
with G. Polya’s logic of plausible inference, and some characterization
results on weak and strong connexity.

##  1 Introduction

The basic ideas of connexive logic can be traced back to Aristotle’s _Prior
Analytics_ and to Boethius’ _De hypotheticis syllogismis_ (see [18]).
Connexive logic developed as an underground stream along the whole history of
logic until the sixties of the last century. Since then, it has become a well
established subject of investigation in non-classical logic. Connexive
principles reflect a _connection_ , or _compatibility_ , between the
antecedent and consequent of sound conditionals. Specifically, they establish
that a conditional statement “if $A$, then $B$” is sound provided that the
negation of $B$ is incompatible with $A$. Such a connection can be expressed
in a language containing a unary (negation) connective $\neg$ and a binary
(implication) connective ${\to}$ by means of axioms in Table 1.

Aristotle’s Theses |  |  | $\neg(A{\to}\neg A)$
---
$\neg(\neg A{\to}A)$
Boethius’ Theses |  |  | $(A{\to}B){\to}\neg(A{\to}\neg B)$
---
$(A{\to}\neg B){\to}\neg(A{\to}B)$
Table 1: Basic connexive laws

A connexive logic is nothing but a logic having the above formulas as theorems
with respect to a negation $\neg$ and a non-symmetric implication ${\to}$. The
latter requirement, which we call _the principle of non-symmetry_ , is
essential, since ${\to}$ must be understood as a genuine implication rather
than as an equivalence. Apparently, the above formulas are falsified in
classical logic whenever implications with _false_ antecedents are considered.
Therefore, classical logic is not connexive with material implication and
negation. On the other hand, while material biconditional does satisfy the
theses in Table 1, it is obviously symmetric.

Connexive theses have been motivated by different considerations depending on
the specific meaning given to implication. One motivation comes from relevance
logic and the idea that semantic consequence is a content relationship (see
e.g. [26]). [5] argues that connexive systems can formalize indicative natural
language conditionals. Some authors (e.g. [15]) suggest that a connexive
implication is suitable for modelling counterfactual conditionals (see also
[32]). Moreover, [17] proposes an interpretation of connexive conditionals in
terms of physical or “causal” implications. Also, the results of empirical
research on the interpretation of negated conditionals (see [18, 20, 21, 29])
suggest that speakers having no previous knowledge of formal logic are
inclined to consider connexive conditionals as the sound ones. Recently, A.
Kapsner has argued that the intuitive appeal of connexive principles is
rooted, _at least in the case of indicative conditionals_ , in the _use_ of
implication in concrete argumentation. In particular, it seems to depend on
the assumption that a conditional statement of the form “if $A$, then $B$” is
sound provided that $A$ is “epistemically possible” for the speaker asserting
it. Quoting Kapsner ([13]):

> There is a presupposition for indicatives (which I called IP) that says that
> the antecedent needs to be epistemically possible for the speaker: If I say
> “If today is Monday, Susy will come home tomorrow”, then this is only
> correct if I don’t know for sure that today is not Monday.

The interested reader is referred to [29] for an exhaustive survey on these
topics.

As stated in [30, p. 381], “[…] the central concern of connexive logic
consists of developing connexive systems that are naturally motivated
conceptually or in terms of applications, that admit of a simple and plausible
semantics, and that can be equipped with proof systems possessing nice proof-
theoretical properties […]”. Indeed, the literature on this subject offers a
large amount of connexive logical systems “built from scratch” having
interesting proof-theoretical features and, in some cases, transparent and
elegant semantics. However, less has been said on the possibility of defining
connexive implications within well established sub-logics of classical logic
or some expansion thereof. In [22], C. Pizzi shows that a connexive arrow,
called _consequential implication_ , can be defined by means of the modal
notions of necessity and possibility within ordinary modal logic systems
S1-S5, plus the system T. This approach has the advantage of suggesting a well
studied framework like ordinary modal logic as a conceptual and formal basis
for connexive semantics (cf. [18]). In the same spirit, G. Gherardi and E.
Orlandelli introduce super-strict implications [11]. Very recently, D. Fazio,
A. Ledda, and F. Paoli [6] have investigated _Connexive Heyting Logic_ (CHL)
which is algebraizable in the sense of Blok & Pigozzi (see e.g., [2, 8]) with
respect to a subvariety of H. P. Sankappanavar’s semi-Heyting algebras
satisfying an equational rendering of Aristotle’s and Boethius’ theses, i.e.
_Connexive Heyting algebras_ (CHA). It turns out that CHA’s are term-
equivalent to Heyting algebras, and so CHL is _deductively equivalent_ to
Intuitionistic Logic. More precisely, a CHA $\mathbf{A}$ is a semi-Heyting
algebra $(A,\land,\lor,{\Rightarrow},0,1)$ satisfying, among other identities,
the following:

$\displaystyle 1\approx(x{\Rightarrow}y){\Rightarrow}\neg(x{\Rightarrow}\neg
y)$ $\displaystyle 1\approx(x{\Rightarrow}\neg
y){\Rightarrow}\neg(x{\Rightarrow}y)$ $\displaystyle
1\approx\neg(x{\Rightarrow}\neg x)$ $\displaystyle 1\approx\neg(\neg
x{\Rightarrow}x)$

where $\neg x\coloneq x{\Rightarrow}0$. Remarkably enough, ${\Rightarrow}$ is,
in general, not symmetric. Setting $x{\to}y\coloneq x{\Rightarrow}(x\land y)$,
it follows by general results on semi-Heyting algebras that
$\mathbb{H}(\mathbf{A})=(A,\land,\lor,{\to},0,1)$ is a Heyting algebra.
Conversely, given a Heyting algebra $\mathbf{A}$, upon setting

$x{\Rightarrow}y\coloneq(x{\to}y)\land(y\to\neg\neg x)$ (1.1)

(where $\neg x$ is defined as expected), one has that
$\mathbb{C}(\mathbf{A})=(A,\land,\lor,{\Rightarrow},0,1)$ is a CHA.
Furthermore, $\mathbb{C}$ and $\mathbb{H}$ are mutually inverse mappings.

Due to the above features, CHL, which is the $1$-assertional logic of CHAs,
can be entitled as a full-fledged connexive logic. Moreover, it enjoys
properties which are indeed rare within connexive logics literature. For
example, CHL is _strongly connexive_ in the sense of [14]. Furthermore, it
allows one to investigate connexivity with well-known mathematical tools and
with the conceptual “arsenal” provided by Intuitionistic Logic (think e.g. to
the BHK semantics).

A somewhat suggestive interpretation of the above results might be the
following. For any Heyting algebra $\mathbf{A}$, the operation $\neg\neg\colon
A\to A$ is a nucleus over the $\ell$-monoid $(A,\land,\lor,\cdot,1)$, where
$\cdot\ \coloneq\ \land$. Therefore, $\neg\neg$ can be regarded as a modal
operator (cf. e.g. [33]). If we read $\neg\neg x$ as “$x$ is not absolutely
false”, or “$x$ is plausible”, then a well behaving connexive arrow
${\Rightarrow}$ can be obtained from intuitionistic implication $\rightarrow$
by strengthening the latter in such a way that no true (or at least plausible)
statement can be implied by an _a priori_ false/implausible one. Therefore,
connexive Heyting implication might be seen as encoding, at a semantic level,
a weak version of Kapsner’s presupposition (IP) recalled above.

Given this observation, we frame this technique in a broader class of non-
classical logics, Substructural Logics, which are understood as the external
logics induced by axiomatic extensions of the Full Lambek Calculus
$\mathtt{FL}$, a sequent system introduced to model natural language. In this
manuscript we confine ourselves to the commutative setting, namely to
$\mathtt{FL}_{\\_}\mathtt{e}$, which is obtained from $\mathtt{FL}$ by adding
the exchange structural rule $(\mathtt{e})$. Its extensions include many of
the most well-studied non-classical logics: intuitionistic logic, relevance
logic, linear logic, many-valued logics, with classical logic as a limit case.
They find applications to areas as diverse as linguistics, philosophy, and
theoretical computer science. Moreover, extensions of
$\mathtt{FL}_{\\_}\mathtt{e}$ turn out to be particularly appealing as they
are algebraizable, in the sense of Blok & Pigozzi, w.r.t. (pointed)
commutative residuated lattices.

A thorough treatment of substructural logics and residuated lattices can be
found in [9].

In this paper, we work from the semantical perspective and investigate the
class of pointed commutative residuated lattices (FL${}_{\mbox{\scriptsize
e}}$-algebras) for which an implication-type connective ${\Rightarrow}$
satisfying equational renderings of Boethius’ and Aristotle’s theses can be
term-defined. Specifically, we define connectives similar to ${\Rightarrow}$
in (1.1) by considering the following candidates:

$x{\Rightarrow}_{\wedge}y\coloneq(x{\to}y)\land(y{\to}\neg\neg
x)\quad\text{and}\quad
x{\Rightarrow}_{\circ}y\coloneq(x{\to}y)\cdot(y{\to}\neg\neg x),$

where $\neg x\coloneq x{\to}0$. This choice is motivated not only by the
intrinsic interest for investigating mathematical properties of Heyting
algebras underlying connexive principles, but also from more “philosophical”
considerations. In fact, we will argue that the above interpretation of
${\Rightarrow}_{\wedge}$ can be deepened in the broader framework of
FL${}_{\mbox{\scriptsize e}}$-algebras and put into relationship with G.
Polya’s theory of plausible inferences in mathematics developed in [25].
Indeed, we will see that, under a reasonable notion of “being more
credible/plausible/likely to be true”, ${\Rightarrow}_{\wedge}$ might
formalize the kind of conditionals involving a conjecture and one of its
consequences as the antecedent and consequent, respectively. More precisely,
${\Rightarrow}_{\wedge}$ is the semantical counterpart of the _weakest_ term-
definable (in the external logic of ${\mathsf{FL}_{\mathsf{e}}}$) implication-
like connective $\leadsto$ satisfying axiomatic renderings of _modus ponens_
and Polya’s _fundamental inductive pattern_ : from the truth of “$A$ implies
$B$” and $B$, one deduces that “$A$ is more credible”.

This work aims at showing that, for an FL${}_{\mbox{\scriptsize e}}$-algebra
$\mathbf{A}$, a condition ensuring that ${\Rightarrow}_{\wedge}$ and
${\Rightarrow}_{\circ}$ defined over $\mathbf{A}$ satisfy Aristotle’s and
Boethius’ Theses is that $\mathbf{A}$ has the equational Glivenko property
w.r.t. Boolean algebras (see below and e.g. [9]). Such a requirement becomes
also necessary when one deals with ${\Rightarrow}_{\wedge}$. As a consequence,
we obtain a novel characterization of FL${}_{\mbox{\scriptsize e}}$-algebras
enjoying the equational Glivenko property relative to BA’s by establishing a
new link between connexivity and concepts of well-known mathematical depth.

Moreover, we turn our attention to _integral_ FL${}_{\mbox{\scriptsize
e}}$-algebras. In this case we obtain somewhat surprising and strong results.
Indeed, we consider the binary operations ${\Rightarrow}$ in the (pointwise
ordered) interval
$[{\Rightarrow}_{\circ}^{\delta},{\Rightarrow}_{\wedge}^{\delta}]$ (where
$\delta$ is an increasing map replaying the role of $\neg\neg$) over an
integral FL${}_{\mbox{\scriptsize e}}$-algebra $\mathbf{A}$, and we show that
they satisfy connexive axioms if and only if at least one of them does and
$\delta=\neg\neg$. In this case any of the ${\Rightarrow}$’s is symmetric if
and only if $\mathbf{A}$ is a Boolean algebra. Note that the latter result
still holds, at least for ${\Rightarrow}_{\wedge}$, if integrality is dropped.
Therefore, we conclude that, apart from Boolean algebras themselves,
FL${}_{\mbox{\scriptsize e}}$-algebras with the equational Glivenko property
w.r.t. BA’s (and their $1$-assertional logics, see below) can be regarded as
suitable environments in which connexive implications of a certain type can be
defined.

Finally, we focus on the concepts of strong and _weak_ (see below and e.g.
[32]) connexivity. In fact, on the one hand we will show that, in some cases,
weak connexivity and connexivity can be regarded as one and the same thing. On
the other, it will turn out that, although strong connexivity has often been
associated to contra-classical (cf. [12]). theses (see e.g. the notion os
_superconnexivity_ in [14]), in the framework of extensions of
$\mathbf{FL}_{\\_}{\mathbf{e}}$ for which ${\Rightarrow}_{\wedge}$ and
${\Rightarrow}_{\circ}$ are connexive, it is nevertheless equivalent to a
rather “classical” inference schema: _ex falso quodlibet_. Therefore, also in
view of characterization theorems outlined above, this work yields an overall
picture which is somewhat surprising: although connexive theses (and
weak/strong connexivity) make connexive logics incomparable with classical
logic, in the context we deal with, they mirror some distinguishing traits of
intuitionistic logic.

The results discussed above suggest that a systematic (programmatic)
investigation of connexive implication connectives which are _term-definable_
within well known systems of non-classical logic might have interesting
consequences. On the one hand, it would provide (eventually strongly)
connexive logics with transparent semantics and implications with intuitive
meanings. Therefore, such investigations would perhaps shed some light on the
“semantic source” of connexive principles. On the other, this line of research
would establish relationships between connexive systems and logics which, like
substructural logics, have relevant applications in logico-philosophical
investigations as well as in mathematics, computer science, and Artificial
Intelligence. This work aims at being a small step in that direction.

Let us summarize the discourse of the paper. In Section 2 we dispatch the
basic notions needed for the development of our arguments, and we provide some
preliminary results. Section 3 is devoted to alternative characterizations of
those FL${}_{\mbox{\scriptsize e}}$-algebras in which ${\Rightarrow}_{\circ}$,
${\Rightarrow}_{\wedge}$ and (in the integral case) any binary operation
${\Rightarrow}$ “in the between”, satisfy connexive theses. We prove that, for
${\Rightarrow}_{\wedge}$ (and under a slightly stronger assumption
${\Rightarrow}_{\circ}$), the _largest_ class of algebras of this sort
coincides with the variety of FL${}_{\mbox{\scriptsize e}}$-algebras with the
equational Glivenko property w.r.t. Boolean algebras. In Section 4, building
on results obtained in previous sections, we provide some philosophical
upshots like e.g. a discussion on the relevance of ${\Rightarrow}_{\wedge}$, a
characterization of substructural logics for which ${\Rightarrow}_{\wedge}$
(${\Rightarrow}_{\circ}$) is strongly connexive, and some remarks on the
relationship between weak connexivity and connexivity. We conclude in Section
5.

##  2 Basic notions and preliminary results

In this section we provide the necessary notions for the contents of this
manuscript. We will assume the reader has a working understanding in basic
concepts of universal algebra, referring them to [3] for further edification.

### 2.1 Residuated structures

Here we recall the definitions and properties of residuated lattices and FL-
algebras, but only define their commutative versions as their full generality
is not needed for the purpose of this work. For a thorough treatment of these
structures and their connection to substructural logics, we refer the reader
to [9].

A commutative residuated lattice (CRL) is an algebra $\mathbf{R}=\langle
R,\wedge,\vee,\cdot,\to,1\rangle$ such that $\langle R,\wedge,\vee\rangle$ is
a lattice, $\langle R,\cdot,1\rangle$ is a commutative monoid, and
$\mathbf{R}$ satisfies the law of residuation, i.e., for all $x,y,z\in R$,

$x\cdot y\leq z\iff x\leq y\to z,$ (residuation)

where $\leq$ is the induced lattice order; i.e., $x\leq y$ iff $x\wedge y=x$.
The law of residuation can be written equationally, and therefore the class of
commutative residuated lattices form a variety denoted by $\mathsf{CRL}$. We
will often abbreviate $\cdot$ by concatenation, i.e. $xy\coloneq x\cdot y$,
with the convention that concatenation binds tighter than the remaining
connectives in the signature. We also use a bi-implication abbreviation
$x\leftrightarrow y\coloneq(x\to y)\wedge(y\to x)$. A residuated lattice is
called integral if the monoid unit is the greatest element, i.e., it satisfies
the identity ($\mathsf{i}$): $x\wedge 1\approx x$. We note that, if there is a
greatest element $\top$ (not necessarily $1$), then $x\leq y$ implies
$x{\to}y\approx\top$. If there is a least element $\bot$, then there must be a
greatest element, namely $\bot{\to}\bot$, and $\bot$ is _absorbing_ , i.e.,
$\bot\cdot x\approx\bot$.

Below we recall some basic properties of CRLs, which we will mostly utilize
without reference throughout this article.

######  Proposition 2.1.

The following identities hold in $\mathsf{CRL}$:

  1. 1.

$x(y\vee z)\approx xy\vee xz$

  2. 2.

$x\to(y\wedge z)\approx(x\to y)\wedge(x\to z)$

  3. 3.

$(x\vee y)\to z\approx(x\to z)\wedge(y\to z)$

  4. 4.

$1\to x\approx x$

  5. 5.

$x\to(y\to z)\approx yx\to z$

Consequently, $\cdot$ is order-preserving in both coordinates, while ${\to}$
is order-preserving (reversing) in its right (left) coordinate.

An FL${}_{\mbox{\scriptsize e}}$-algebra is simply a $0$-pointed CRL, i.e., a
CRL whose signature is expanded by a constant $0$, and the variety of
FL${}_{\mbox{\scriptsize e}}$-algebras is denoted by
${\mathsf{FL}_{\mathsf{e}}}$. An FL${}_{\mbox{\scriptsize e}}$-algebra is _$0$
-bounded_ if $0$ is the least element, i.e., it satisfies ($\mathsf{o}$):
$0\wedge x\approx 0$. By ($\mathsf{w}$) we denote weakening, defined via
$(\mathsf{w})=(\mathsf{i})+(\mathsf{o})$. We denote the variety of
FL${}_{\mbox{\scriptsize e}}$-algebras satisfying integrality (weakening) by
$\mathsf{FL}_{\mathsf{ei}}$ ($\mathsf{FL}_{\mathsf{ew}}$).

Given that $0$ is a constant in the signature of FL${}_{\mbox{\scriptsize
e}}$-algebras, we define the unary operation $\neg x\coloneq x\to 0$, and use
the convention that $\neg$ binds tighter than any connective. An
FL${}_{\mbox{\scriptsize e}}$-algebra is said to be involutive if it satisfies
$x\approx\neg\neg x$. An FL${}_{\mbox{\scriptsize e}}$-algebra is called
pseudo-complemented if it satisfies the identity

$x\wedge\neg x\leq 0.$ (pc)

We say an FL${}_{\mbox{\scriptsize e}}$-algebra is strongly pseudo-
complemented if the following identity holds:

$\neg x\wedge\neg(x{\to}y)\leq 0$ (spc)

Clearly $\mathrm{(spc)}$ entails (pc) as an instance, however the converse
need not hold as is witnessed by Figure 1 below.

$\bullet$$1$$\bullet$$0=0^{2}$$\bullet$$\bot$

$\begin{array}[]{c | c c c}{\to}&1&0&\bot\\\ \hline\cr 1&1&0&\bot\\\ 0&1&1&\bot\\\ \bot&1&1&1\end{array}$

Figure 1: The Hasse diagram of a pseudo-complemented integral
FL${}_{\mbox{\scriptsize e}}$-algebra which does not satisfy $\mathrm{(spc)}$
since $\neg 0\wedge\neg(0{\to}\bot)=\neg 0\wedge\neg\bot=1\wedge 1=1\neq 0$.

######  Remark 2.2.

The converse does hold for $0$-bounded FL${}_{\mbox{\scriptsize e}}$-algebras,
as being $0$-bounded entails $\neg x\leq x\to y$, and hence $\neg
x\wedge\neg(x{\to}y)\leq\neg x\wedge\neg\neg x\leq 0$.

It is easy to see that, for FL${}_{\mbox{\scriptsize e}}$-algebras, the
operation $\neg$ is order-reversing and satisfies $\neg x\approx\neg\neg\neg
x$. Additionally, the map $x\mapsto\neg\neg x$ is a nucleus; i.e., a map
$\gamma\colon G\to G$ on a partially-ordered groupoid that is a closure
operator [namely, a map $\gamma$ that is increasing: $x\leq\gamma(x)$;
monotone: $x\leq y$ implies $\gamma(x)\leq\gamma(y)$; and idempotent:
$\gamma\circ\gamma=\gamma$] which further satisfies the identity
$\gamma(x)\cdot\gamma(y)\leq\gamma(xy)$ [or equivalently,
$\gamma(\gamma(x)\cdot\gamma(y))\approx\gamma(xy)$]. It is well known that,
for any FL${}_{\mbox{\scriptsize e}}$-algebra $\mathbf{A}$, if $\gamma\colon
A\to A$ is a nucleus then
$\mathbf{A}_{\\_}\gamma\coloneq\langle\gamma[A],\wedge,\vee_{\\_}\gamma,\cdot_{\\_}\gamma,\to,\gamma(0),\gamma(1)\rangle$
is also an FL${}_{\mbox{\scriptsize e}}$-algebra, where $x\cdot_{\\_}\gamma
y\coloneq\gamma(x\cdot y)$ and $x\vee_{\\_}\gamma y\coloneq\gamma(x\vee y)$
[E.g., see Chapter 3.4.11 in [9]].

For simplicity, we use the notation ${\diamond}x\coloneq\neg\neg x$
henceforth, and summarize the basic properties of ${\diamond}$, all of which
hold for any nucleus, in the proposition below, which we may often utilize
without reference.

######  Proposition 2.3.

The following (quasi-) identities hold in ${\mathsf{FL}_{\mathsf{e}}}$:

  1. 1.

$x\leq{\diamond}x$

  2. 2.

$x\leq y\implies{\diamond}x\leq{\diamond}y$

  3. 3.

${\diamond}{{\diamond}x}\approx{\diamond}{x}$

  4. 4.

${\diamond}x\cdot{\diamond}y\leq{\diamond}(xy)$

  5. 5.

${\diamond}x\wedge{\diamond}y\approx{\diamond}({\diamond}x\wedge{\diamond}y)$

  6. 6.

${\diamond}x\to{\diamond}y\approx
x\to{\diamond}y\approx{\diamond}(x\to{\diamond}y)$

Consequently, for any $\mathbf{A}\in{\mathsf{FL}_{\mathsf{e}}}$,
${\diamond}{\mathbf{A}}\coloneq\mathbf{A}_{\\_}{\diamond}$ is an involutive
FL${}_{\mbox{\scriptsize e}}$-algebra.

The following example will be a useful counter-model throughout this article.

######  Example 2.4.

Let $\mathbb{Z}$ be the set of integers. It is well known that $\mathbb{Z}$
forms a (totally-ordered) commutative residuated lattice where
$x\wedge_{\\_}\mathbb{Z}y\coloneq\min\\{x,y\\}$;
$x\vee_{\\_}\mathbb{Z}y\coloneq\max\\{x,y\\}$,
$x\cdot_{\\_}\mathbb{Z}y\coloneq x+y$; $x{\to}_{\\_}\mathbb{Z}y:=y-x$; and
$1_{\\_}\mathbb{Z}\coloneq 0$. For fixed $n\in\mathbb{Z}$, by $\mathbb{Z}(n)$
we denote the FL${}_{\mbox{\scriptsize e}}$-algebra as an expansion of
$\mathbb{Z}$ by taking $0_{\\_}{\mathbb{Z}(n)}\coloneq n$. In $\mathbb{Z}(n)$,
it is clear that for all $x\in\mathbb{Z}$, $\neg x=n-x$ and $\neg\neg x=x$, so
$\mathbb{Z}(n)$ is involutive, i.e., ${\diamond}\mathbb{Z}(n)=\mathbb{Z}(n)$.

### 2.2 Algebraization for substructural logics

A _logic_ $\mathbf{L}$ over an algebraic language $\mathcal{L}$ is a
_structural consequence relation_ (i.e., closed under substitutions and is
reflexive, transitive, and monotone)
${\vdash}\subseteq\wp(\mbox{{Fm}}_{\\_}\mathcal{L})\times\mbox{{Fm}}_{\\_}\mathcal{L}$.
The interested reader is referred to [8] for details.

The name “FL${}_{\mbox{\scriptsize e}}$” comes from the fact that
FL${}_{\mbox{\scriptsize e}}$-algebras are the (unique) equivalent algebraic
semantics of the _external_ logic $\mathbf{FL}_{\\_}\mathbf{e}$ induced by the
Full Lambek Calculus with exchange $\mathtt{FL}_{\\_}{\mathtt{e}}$. Precisely,
we have that, for any $\Phi\cup\psi\subseteq\mbox{{Fm}}_{\\_}\mathcal{L}$ of
formulas over the language $\mathcal{L}=\\{\wedge,\vee,\cdot,{\to},0,1\\}$,

$\Phi\vdash_{\\_}{\mathbf{FL}_{\\_}\mathbf{e}}\psi\quad\text{iff}\quad\\{\operatorname{\blacktriangleright}\varphi:\varphi\in\Phi\\}\vdash_{\\_}{\mathtt{FL}_{\\_}{\mathtt{e}}}\operatorname{\blacktriangleright}\psi.$

${\mathsf{FL}_{\mathsf{e}}}$ is the equivalent algebraic semantics of
$\mathbf{FL}_{\\_}{\mathbf{e}}$ means that there exists a pair of “mutually
inverse” mappings (called _transformers_)
$\tau:\mbox{{Fm}}_{\\_}{\mathcal{L}}\to\wp(\mbox{{Fm}}_{\\_}{\mathcal{L}}^{2})$
from formulas to sets of equations, and
$\rho:\mbox{{Fm}}_{\\_}{\mathcal{L}}^{2}\to\wp(\mbox{{Fm}}_{\\_}{\mathcal{L}})$
from equations to sets of formulas such that, for any
$\Phi\cup\psi\subseteq\mbox{{Fm}}_{\\_}\mathcal{L}$:

  * •

$\Phi\vdash_{\\_}{\mathbf{FL}_{\\_}{\mathbf{e}}}\psi$ if and only if
$\tau(\Phi)\models_{\\_}{{\mathsf{FL}_{\mathsf{e}}}}\tau(\psi)$;

  * •

$x\approx y\leftmodels\models\tau\rho(x\approx y)$,

or, equivalently, for any set of equations
$\\{\epsilon_{\\_}{i}\approx\delta_{\\_}{i}:i\in
I\\}\cup\\{\epsilon\approx\delta\\}\subseteq Fm^{2}_{\\_}{\mathcal{L}}$:

  * •

$\\{\epsilon_{\\_}{i}\approx\delta_{\\_}{i}:i\in
I\\}\models_{\\_}{{\mathsf{FL}_{\mathsf{e}}}}\epsilon\approx\delta$ iff
$\\{\rho(\epsilon_{\\_}{i}\approx\delta_{\\_}{i}):i\in
I\\}\vdash_{\\_}{\mathbf{FL}_{\\_}{\mathbf{e}}}\rho(\epsilon\approx\delta)$;

  * •

$x\dashv\vdash_{\\_}{\mathbf{FL}_{\\_}{\mathbf{e}}}\rho(\tau(x))$.

It can be seen that, in our framework, a suitable pair of transformers is
given by $\tau(\varphi):=\\{1\leq\varphi\\}$, while
$\rho(\epsilon\approx\delta):=\\{(x\to y)\land(y\to x)\\}$. Cf. [10] for
details. For this reason, $\mathbf{FL}_{\\_}{\mathbf{e}}$ is often referred to
as the $1$-_assertional_ logic of ${\mathsf{FL}_{\mathsf{e}}}$. Now, the
lattice of axiomatic extensions of $\mathbf{FL}_{\\_}\mathbf{e}$, in this
paper called _substructural logics_ 111Typically, “substructural logic” refers
to any axiomatic extension of the logic $\mathbf{FL}$, i.e. the external
consequence relation induced by the Full Lambek Calculus $\mathtt{FL}$ without
the axiom of exchange (commutativity in the corresponding algebras), where the
same algebraization results hold. See [9]., is dually isomorphic to the
lattice of subvarieties of ${\mathsf{FL}_{\mathsf{e}}}$. Indeed, for every
class $\mathsf{K}$ of FL${}_{\mbox{\scriptsize e}}$-algebras and for every set
$\Phi\subseteq\mbox{{Fm}}_{\\_}\mathcal{L}$ of formulas over $\mathcal{L}$,
let
$\mathbf{L}(\mathsf{K}):=\\{\phi\in\mbox{{Fm}}_{\\_}\mathcal{L}:\mathsf{K}\models
1\leq\phi\\}$ and
$\mathsf{V}(\Phi):={\mathsf{FL}_{\mathsf{e}}}\cap\mathrm{Mod}(\\{1\leq\phi:\phi\in\Phi\\})$,
we state the well-known algebraization theorem (e.g., Theorem 2.29 in [9]):

######  Theorem 2.5 (The Algebraization Theorem).

The following hold:

  1. 1.

For every substructural logic $\mathbf{L}$, $\mathbf{L}$ is algebraizable with
$\mathsf{V}(\mathbf{L})$ as its equivalent algebraic semantics. In particular,
for any $\Phi\cup\\{\gamma\\}\subseteq\mbox{{Fm}}_{\\_}\mathcal{L}$:

$\Phi\vdash_{\\_}\mathbf{L}\gamma\quad\text{iff}\quad\\{1\leq\varphi:\varphi\in\Phi\\}\models_{\\_}{\mathsf{V}(\mathbf{L})}1\leq\gamma;$

  2. 2.

For any subvariety $\mathsf{V}\subseteq{\mathsf{FL}_{\mathsf{e}}}$,
$\mathsf{V}$ is the equivalent algebraic semantics of the logic
$\mathbf{L}(\mathsf{V})$. Specifically, for any $E\cup\\{s\approx
t\\}\subseteq\wp({\mbox{{Fm}}_{\\_}{\mathcal{L}}}^{2})$, one has:

$E\models_{\\_}\mathsf{V}s\approx t\quad\text{iff}\quad\\{u\to v,v\to
u:(u\approx v)\in
E\\}\vdash_{\\_}{\mathbf{L}(\mathsf{V})}(s{\to}t)\wedge(t{\to}s).$

### 2.3 Boolean algebras and their Glivenko varieties

It is well known that Boolean algebras are term-equivalent to
FL${}_{\mbox{\scriptsize e}}$-algebras satisfying $x\cdot y\approx x\wedge y$
and $x{\to}y\approx\neg x\vee y$ (e.g., Lem. 8.32 in [9]). By $\mathsf{BA}$ we
will denote the variety of FL${}_{\mbox{\scriptsize e}}$-algebras which are
term-equivalent to Boolean algebras. It is well known that Boolean algebras
are the equivalent algebraic semantics of classical logic
$\mathbf{CPL}$.333I.e., $\mathsf{BA}=\mathsf{V}(\mathbf{CPL})$ and
$\mathbf{CPL}=\mathbf{L}(\mathsf{BA})$. The following proposition is likely
folklore, but we provide its proof as it will be useful later.

######  Proposition 2.6.

Let $\mathbf{A}$ be an FL${}_{\mbox{\scriptsize e}}$-algebra. Then
$\mathbf{A}$ is (term-equivalent to) a Boolean algebra iff $\mathbf{A}$ is
integral, involutive, and pseudo-complemented.

###### Proof.

The forward direction is obvious. For the reverse direction, let $x,y\in L$.
First, by using the properties of integrality, commutativity, and pseudo-
complemented, we compute $\neg(xy)\cdot(x\wedge y)\leq\neg(xy)x\wedge
y=(x{\to}\neg y)x\wedge y\leq\neg y\wedge y\leq 0,$ hence $x\wedge
y\leq\neg\neg(xy)=xy$ by residuation and involutivity. Since $xy\leq x\wedge
y$ by integrality, we obtain $x\cdot y=x\wedge y$. Hence, using this fact and
involutivity, we obtain

$x{\to}y=x{\to}\neg\neg y=\neg(x\cdot\neg y)=\neg(x\wedge\neg y)=\neg(\neg\neg
x\wedge\neg y)=\neg\neg(\neg x\vee\neg\neg y)=\neg x\vee y.$

∎

It is well known that classical propositional logic $\mathbf{CPL}$ is
translatable into intuitionistic propositional logic $\mathbf{IPL}$; namely,
$\vdash_{\\_}\mathbf{CPL}\varphi$ iff
$\vdash_{\\_}\mathbf{IPL}\neg\neg\varphi$, for any formula $\varphi$. This
concept, known as the Glivenko property, has been studied for substructural
logics in general (see [10, 9]), as the property is reflected via their
algebraic semantics via the algebraization theorem. Indeed, for substructural
logics $\mathbf{L}$ and $\mathbf{K}$, we say the _Glivenko property holds for
$\mathbf{K}$ relative to $\mathbf{L}$_ if for any formula $\varphi$,
$\vdash_{\\_}\mathbf{L}\varphi$ iff $\vdash_{\\_}{\mathbf{K}}\neg\neg\varphi$.
Indeed, by Theorem 2.5, this equivalently stated: for any
FL${}_{\mbox{\scriptsize e}}$-term $t$, $\mathsf{V}(\mathbf{L})\models 1\leq
t$ iff $\mathsf{V}(\mathbf{K})\models 1\leq\neg\neg t$.

######  Proposition 2.7 (See [9, 10]).

For subvarieties $\mathsf{V},\mathsf{W}\subseteq{\mathsf{FL}_{\mathsf{e}}}$
and FL${}_{\mbox{\scriptsize e}}$-terms $s,t$, the following are equivalent:

  1. 1.

$\mathsf{V}\models 1\leq t$ iff $\mathsf{W}\models 1\leq\neg\neg t$.

  2. 2.

$\mathsf{V}\models s\leq t$ iff $\mathsf{W}\models\neg t\leq\neg s$.

  3. 3.

$\mathsf{V}\models s\approx t$ iff $\mathsf{W}\models\neg s\approx\neg t$.

Thus, for varieties $\mathsf{V}$ and $\mathsf{W}$ of FL${}_{\mbox{\scriptsize
e}}$-algebras, we say the equational Glivenko property holds for $\mathsf{W}$
relative to $\mathsf{V}$ iff

$\mathsf{V}\models s\approx t\iff\mathsf{W}\models\neg s\approx\neg t$ (2.1)

for any equation $s\approx t$ in the language of ${\mathsf{FL}_{\mathsf{e}}}$.
In particular, it follows that the equational Glivenko property holds for the
variety of Heyting algebras $\mathsf{HA}$ relative to $\mathsf{BA}$, as it is
well known $\mathsf{HA}=\mathsf{V}(\mathbf{IPL})$. Note that the algebras in
$\mathsf{HA}$ are exactly those FL${}_{\mbox{\scriptsize ew}}$-algebras in
which $\cdot$ coincides with $\wedge$.

By $\mathbf{G}_{\\_}{\mathsf{FL}_{\mathsf{e}}}(\mathsf{V})$ we denote the
largest subvariety $\mathsf{W}$ for which the equational Glivenko property
holds for $\mathsf{W}$ relative to $\mathsf{V}$. By
$\mathbf{G}_{\\_}\mathsf{U}(\mathsf{V})$ we denote
$\mathbf{G}_{\\_}{\mathsf{FL}_{\mathsf{e}}}(\mathsf{V})\cap\mathsf{U}$, where
$\mathsf{U}$ is a subvariety of ${\mathsf{FL}_{\mathsf{e}}}$, which we call
_the Glivenko variety of $\mathsf{U}$ relative to $\mathsf{V}$_. Dually, for a
substructural logic $\mathbf{L}$,
$\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{e}}(\mathbf{L})$ denotes the
_least_ (w.r.t. set inclusion) substructural logic $\mathbf{K}$ for which the
Glivenko property holds relative to $\mathbf{L}$. If $\mathbf{K}$ is a
substructural logic, by $\mathbf{G}_{\\_}\mathbf{K}(\mathbf{L})$ we denote
$\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{e}}(\mathbf{L})\vee\mathbf{K}$,
i.e., the _smallest_ extension of $\mathbf{K}$ having the Glivenko property
w.r.t. $\mathbf{L}$, which we call _the Glivenko logic of $\mathbf{K}$
relative to $\mathbf{L}$_.

######  Theorem 2.8 (See [9, 10]).

  1. 1.

$\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$ is axiomatized relative
to ${\mathsf{FL}_{\mathsf{e}}}$ by the following identities:

    1. (a)

$\neg(x\cdot y)\approx\neg(x\wedge y)$

    2. (b)

$1\leq{\diamond}({\diamond}x{\to}x)$ [or, alternatively,
$\neg(x{\to}y)\approx\neg(\neg x\vee y)$]

  2. 2.

$\mathbf{G}_{\mathsf{FL}_{\mathsf{ei}}}(\mathsf{BA})$ is axiomatized relative
to $\mathsf{FL}_{\mathsf{ei}}$ by the following identities:

    1. (a)

$x\wedge\neg x\leq 0$ [or, alternatively, $\neg(x\wedge y)\approx\neg(x\cdot
y)$]

    2. (b)

$1\leq{\diamond}({\diamond}x{\to}x)$

######  Remark 2.9.

Of course,
$\mathsf{BA}\subsetneq\mathbf{G}_{\mathsf{FL}_{\mathsf{ew}}}(\mathsf{BA})\subseteq\mathbf{G}_{\mathsf{FL}_{\mathsf{ei}}}(\mathsf{BA})\subseteq\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$
as any Heyting algebra is in
$\mathbf{G}_{\mathsf{FL}_{\mathsf{ew}}}(\mathsf{BA})$ but there are Heyting
algebras which are not Boolean, see e.g. [9].

As they will be useful in what follows, we provide further characterizations
for the Glivenko varieties of ${\mathsf{FL}_{\mathsf{e}}}$
($\mathsf{FL}_{\mathsf{ei}}$, and $\mathsf{FL}_{\mathsf{ew}}$) relative to
Boolean algebras.

######  Lemma 2.10.

For an FL${}_{\mbox{\scriptsize e}}$-algebra $\mathbf{A}$, the following are
equivalent:

  1. 1.

$\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$.

  2. 2.

$\mathbf{A}$ is pseudo-complemented, satisfies
$1\leq{\diamond}({\diamond}x{\to}x)$, and has ${\diamond}1$ as its greatest
element (i.e., ${\diamond}\mathbf{A}$ is integral).

  3. 3.

${\diamond}\mathbf{A}$ is Boolean and $\mathbf{A}$ satisfies
$1\leq{\diamond}({\diamond}x{\to}x)$.

###### Proof.

(1) implies (2) is obvious. Since ${\diamond}A$ is involutive by fiat, (2)
implies (3) follows Proposition 2.6. For (3) implies (1), suppose
${\diamond}\mathbf{A}$ is Boolean and $\mathbf{A}$ satisfies
$(\star):1\leq{\diamond}({\diamond}x{\to}x)$. So ${\diamond}1$ is the greatest
element of $\mathbf{A}$ and the following identities hold in $\mathbf{A}$:

${\diamond}(x\cdot y)\approx{\diamond}x\wedge{\diamond}y\quad\mathsf{and}\quad
x{\to}{\diamond}y\approx{\diamond}(\neg x\vee y).$

By Theorem 2.8, we need only verify that $\mathbf{A}$ satisfies $\neg(x\cdot
y)\approx\neg(x\wedge y)$, or equivalently, ${\diamond}(x\cdot
y)\approx{\diamond}(x\wedge y)$. Towards this, we will first show $\mathbf{A}$
satisfies $(\star\star):{\diamond}(x{\to}y)\approx x{\to}{\diamond}y$. Note
that we need only verify $x{\to}{\diamond}y\leq{\diamond}(x{\to}y)$. Let
$x,y\in A$. Since $(x{\to}{\diamond}y)\cdot x\leq{\diamond}y$, we have

${\diamond}y{\to}y\leq[(x{\to}{\diamond}y)\cdot
x]{\to}y=(x{\to}{\diamond}y){\to}(x{\to}y),$

from which it follows that:

$\begin{array}[]{r c l l}{\diamond}1&=&{\diamond}[{\diamond}y{\to}y]&\mbox{By
$(\star)$}\\\ &\leq&{\diamond}[(x{\to}{\diamond}y){\to}(x{\to}y)]\\\
&\leq&(x{\to}{\diamond}y){\to}{\diamond}(x{\to}y).\\\ \end{array}$

Hence $x{\to}{\diamond}y\leq{\diamond}(x{\to}y)$ by residuation.

Now, since ${\diamond}(x\wedge
y)\leq{\diamond}({\diamond}x\wedge{\diamond}y)={\diamond}(xy)$, we need only
verify ${\diamond}(xy)\leq{\diamond}(x\wedge y)$, or equivalently,
$1\leq{\diamond}(xy)\to{\diamond}(x\wedge y)$. First, we observe

$\begin{array}[]{r c l l}{\diamond}1{\to}1&\leq&(x\vee y){\to}1&\mbox{Since
$x,y\leq{\diamond}1$}\\\ &=&[x{\to}1]\wedge[y{\to}1]\\\
&\leq&[x{\to}(y{\to}y)]\wedge[y{\to}(x{\to}x)]\\\
&=&[xy{\to}y]\wedge[xy{\to}x]\\\ &=&xy{\to}(x\wedge y).\end{array}$

Thus we obtain

$\begin{array}[b]{r c l l}1&\leq&{\diamond}[{\diamond}1{\to}1]&\mbox{By
$(\star)$}\\\ &\leq&{\diamond}[xy{\to}(x\wedge y)]&\mbox{By the above}\\\
&=&xy{\to}{\diamond}(x\wedge y)&\mbox{By $(\star\star)$}\\\
&=&{\diamond}(xy){\to}{\diamond}(x\wedge y).\end{array}\qed$

######  Lemma 2.11.

For $\mathbf{A}$ an integral FL${}_{\mbox{\scriptsize e}}$-algebra, the
following are equivalent:

  1. 1.

$\mathbf{A}\in\mathbf{G}_{\mathsf{FL}_{\mathsf{ei}}}(\mathsf{BA})$.

  2. 2.

$\mathbf{A}$ is strongly pseudo-complemented.

  3. 3.

$\mathbf{A}\models\neg(x{\to}y)\approx\neg(\neg x\vee y)$.

###### Proof.

(1) implies (2) is clear since in
$\mathbf{G}_{\mathsf{FL}_{\mathsf{ei}}}(\mathsf{BA})$ we have

$\neg x\wedge\neg(x{\to}y)\approx\neg x\wedge\neg(\neg x\vee y)\leq\neg
x\wedge\neg\neg x\leq 0.$

For (2) implies (3), suppose $\mathbf{A}$ is strongly pseudo-complementedand
let $x,y\in A$. Note that $\neg(\neg x\vee y)={\diamond}x\wedge\neg y$. On the
one hand we have

$\begin{array}[]{rcll}(x{\to}y)\cdot({\diamond}x\wedge\neg
y)&\leq&[(x{\to}y)\cdot{\diamond}x]\wedge\neg y\\\
&\leq&[({\diamond}x{\to}{\diamond}y)\cdot{\diamond}x]\wedge\neg y&\mbox{Since
$x{\to}y\leq x{\to}{\diamond}y={\diamond}x{\to}{\diamond}y$}\\\
&\leq&{\diamond}y\wedge\neg y\\\ &\leq&0&\mbox{By (\ref{spc})}.\end{array}$

Hence ${\diamond}x\wedge\neg y\leq\neg(x{\to}y)$ by residuation. On the other
hand, we have

$\begin{array}[]{r c l l}(\neg x\vee y)\cdot\neg(x{\to}y)&=&[\neg
x\cdot\neg(x{\to}y)]\vee[y\cdot\neg(x{\to}y)]\\\ &\leq&[\neg
x\wedge\neg(x{\to}y)]\vee[y\cdot\neg y]&\mbox{By integrality}\\\ &\leq&[\neg
x\wedge\neg(x{\to}y)]\vee 0\\\ &=&0&\mbox{By (\ref{spc})}.\end{array}$

Hence $\neg(x{\to}y)\leq\neg(\neg x\vee y)$, and we are done.

Lastly, for (3) implies (1), by Theorem 2.8(2), it is enough to show that
$\neg(x\wedge y)\approx\neg(x\cdot y)$ holds in $\mathbf{A}$. Note that the
$\leq$-direction holds by integrality. Let $x,y\in A$. Note that $x\wedge
y\leq{\diamond}x\wedge{\diamond}y=\neg(\neg x\vee\neg y)$, and $\neg(\neg
x\vee\neg y)=\neg(x{\to}\neg y)$ by assumption. Hence

$(x\wedge y)\cdot\neg(xy)\leq\neg(x{\to}\neg y)\cdot\neg(xy)=\neg(x{\to}\neg
y)\cdot(x{\to}\neg y)\leq 0.$

Hence $\neg(x\cdot y)\leq\neg(x\wedge y)$ by residuation and we are done. ∎

######  Corollary 2.12.

$\mathbf{G}_{\mathsf{FL}_{\mathsf{ei}}}(\mathsf{BA})=\mathsf{FL}_{\mathsf{ei}}+\mbox{\em(\ref{spc})}$
and
$\mathbf{G}_{\mathsf{FL}_{\mathsf{ew}}}(\mathsf{BA})=\mathsf{FL}_{\mathsf{ew}}+\mbox{\em(\ref{pc})}$.

###### Proof.

The first claim immediately follows from Lemma 2.11, while the second claim
follows from the additional observation made in Remark 2.2, namely that
${\mathsf{FL}_{\mathsf{e}}}+(\mathsf{o})\models\mbox{\eqref{spc}
$\mathsf{iff}$ \eqref{pc}}$. ∎

##  3 Connexive FL${}_{\mbox{\scriptsize e}}$-algebras

In this section, we lay the groundwork for studying connexivity in
substructural logics through the algebraic lens provided by their algebraic
semantics, i.e. FL${}_{\mbox{\scriptsize e}}$-algebras. We will first begin by
considering the case of when the inherent implication ${\to}$ is connexive,
and after a short discussion, devote our time to studying the operations
${\Rightarrow}_{\wedge}$ and ${\Rightarrow}_{\circ}$ (noted in the
introduction), and their generalizations. We will show that varieties of
FL${}_{\mbox{\scriptsize e}}$-algebras in which these operations satisfy the
(corresponding algebraic notions of) connexive laws are fundamentally linked
to Glivenko varieties relative to Boolean algebras – and therefore their
corresponding substructural logics having the Glivenko property w.r.t.
Classical Logic. In doing so, we generalize a number of results from [6], for
which much of this investigation was inspired. Furthermore, we provide an
entire interval of logics that are _bonafide_ examples of connexive logics,
which moreover are equivalently axiomatized by just any one of the connexive
theses of Boethius (or even Aristotle in the case of weakening).

### 3.1 Connexive principles in FL${}_{\mbox{\scriptsize e}}$-algebras

In light of the Algebraization Theorem for substructural logics [cf., Thm.
2.5], the algebraic analogue for the connexive laws become evident. Let
${\Rightarrow}$ and ${\sim}$ be some binary and unary connectives,
respectively, definable in the language $\\{\wedge,\vee,\cdot,{\to},0,1\\}$.
Then a substructural logic $\mathbf{L}$ has a connexive law (i.e., one of
Aristotle’s or Boethius’ theses from Table 1) as a theorem, in terms of
${\Rightarrow}$ and ${\sim}$, if and only if the corresponding identity in
Table 2 is modeled by subvariety $\mathsf{V}(\mathbf{L})$ of
FL${}_{\mbox{\scriptsize e}}$-algebras.

| Equational
---
Aristotle’s Theses
$\displaystyle 1\leq{\sim}(x{\Rightarrow}{\sim}x)$ (AT) $\displaystyle
1\leq{\sim}({\sim}x{\Rightarrow}x)$ (AT’)
| Equational
---
Boethius’ Theses
$\displaystyle
1\leq(x{\Rightarrow}y){\Rightarrow}{\sim}(x{\Rightarrow}{\sim}y)$ (BT)
$\displaystyle
1\leq(x{\Rightarrow}{\sim}y){\Rightarrow}{\sim}(x{\Rightarrow}y)$ (BT’)
Table 2: Order-algebraic connexive laws

Moreover, the connective ${\Rightarrow}$ satisfies the principle of non-
symmetry in $\mathbf{L}$ iff $\mathsf{V}(\mathbf{L})$ has a member in which
${\Rightarrow}$ is not a symmetric relation, i.e., there exists
$\mathbf{A}\in\mathsf{V}(\mathbf{L})$ such that

$(\exists x,y\in A)[x{\Rightarrow}y\nleq y{\Rightarrow}x\quad\mathsf{or}\quad
y{\Rightarrow}x\nleq x{\Rightarrow}y]$ (NS)

Dually, for a variety $\mathsf{V}\subseteq{\mathsf{FL}_{\mathsf{e}}}$,
$\mathsf{V}$ satisfies an identity from Table 2 if and only if the
corresponding connexive law is a theorem of $\mathbf{L}(\mathsf{V})$.
Similarly, if $\mathsf{V}$ has a member in which ${\Rightarrow}$ is not
symmetric, then the logic $\mathbf{L}(\mathsf{V})$ satisfies the principle of
non-symmetry for ${\Rightarrow}$.

This leads us to the following definition, whose broader generality will be
useful for what follows. Let $\mathbf{A}=\langle A;\leq,{\sim},1\rangle$ be a
pointed ordered-algebra with a unary operation, i.e., $\langle A,\leq\rangle$
is a preorder and $\langle A;{\sim},1\rangle$ is an algebra of type $(1,0)$.
For a function ${\Rightarrow}\colon A^{2}\to A$, we say
$(\mathbf{A},{\Rightarrow})$ is proto-connexive if the identities in Table 2
hold in $\mathbf{A}$, and we say $\mathbf{A}$ is connexive if, furthermore,
the condition (NS) holds in $\mathbf{A}$, i.e., that ${\Rightarrow}$ is not
symmetric.

It is worth observing that our notion of proto-connexivity is very general. In
fact, it does not mention that ${\Rightarrow}$ and $\sim$ should be understood
as a binary and a unary operation satisfying some _minimal requirements_ to be
entitled as (the semantical counterpart of) an implication-like and a
negation-like connective, respectively. Indeed, our notion is aimed at
capturing the behavior of those algebras which may serve _in principle_ as the
(possibly equivalent) algebraic semantics of a $1$-assertional logic which,
satisfies under some respect, connexive theses.

We are now ready to state the primary definitions for what follows. First, for
an FL${}_{\mbox{\scriptsize e}}$-algebra $\mathbf{A}$, by
$\mathbf{A}^{\prime}$ we will denote the pointed ordered-algebra $\langle
A;\leq,\neg,1\rangle$, where $\leq$ is lattice order inherent to $\mathbf{A}$
and $\neg$ is defined as usual via $\neg x:=x\to 0$. In this way, the we
specialize the above concepts to the framework of substructural logics and
FL${}_{\mbox{\scriptsize e}}$-algebras:

######  Definition 3.1.

Let $\mathbf{A}\in{\mathsf{FL}_{\mathsf{e}}}$ and ${\Rightarrow}\colon
A^{2}\to A$ a binary function. We say $(\mathbf{A},{\Rightarrow})$ is _proto-
connexive_ if $(\mathbf{A}^{\prime},{\Rightarrow})$ is proto-connexive, and
similarly, $(\mathbf{A},{\Rightarrow})$ is _connexive_ if
$(\mathbf{A}^{\prime},{\Rightarrow})$ is connexive. If ${\Rightarrow}$ is
term-definable in the language of FL${}_{\mbox{\scriptsize e}}$-algebras, then
for a variety $\mathsf{V}\subseteq{\mathsf{FL}_{\mathsf{e}}}$ we say
$(\mathsf{V},{\Rightarrow})$ is _proto-connexive_ if for every member of
$\mathbf{A}\in\mathsf{V}$, $(\mathbf{A},{\Rightarrow})$ is proto-connexive,
and we say $(\mathsf{V},{\Rightarrow})$ is _connexive_ if furthermore it
contains member $\mathbf{A}$ such that $(\mathbf{A},{\Rightarrow})$ is
connexive [i.e., $\mathsf{V}\nmodels x{\Rightarrow}y\approx y{\Rightarrow}x$].
If $\mathbf{L}$ is a substructural logic, we say $(\mathbf{L},{\Rightarrow})$
is (_proto-_) _connexive_ if $(\mathsf{V}(\mathbf{L}),{\Rightarrow})$ is.

Whenever no danger of confusion will be impending, we will say that a binary
operation ${\Rightarrow}$ over $\mathbf{X}$ [taken to be an algebra, variety,
or logic], is (proto-) connexive if $(\mathbf{X},{\Rightarrow})$ is.

The next proposition introduces some identities that will be useful for the
development of our arguments.

######  Proposition 3.2.

Let $\mathbf{A}$ be an FL${}_{\mbox{\scriptsize e}}$-algebra and let
${\Rightarrow}\colon A^{2}\to A$. If $(\mathbf{A},{\Rightarrow})$ satisfies
the identities

$\displaystyle 1$ $\displaystyle\leq{x{\Rightarrow}\neg\neg x}$ (P1)
$\displaystyle\neg\neg(x{\Rightarrow}y)$
$\displaystyle\approx\neg(x{\Rightarrow}\neg y)$ (P2)

then $(\mathbf{A},{\Rightarrow})$ is proto-connexive.

###### Proof.

Note that (P2) is equivalent to
$\neg(x{\Rightarrow}y)\approx\neg\neg(x{\Rightarrow}\neg y)$ in the context of
FL${}_{\mbox{\scriptsize e}}$-algberas by Proposition 2.3. Using this
observation, by (P1) and Proposition 2.3, we obtain (AT) via $1\leq
x{\Rightarrow}\neg\neg x\leq\neg\neg(x{\Rightarrow}\neg\neg
x)\approx\neg(x{\Rightarrow}\neg x).$ Similarly, we obtain (AT’) via
$1\leq\neg x{\Rightarrow}\neg\neg(\neg x)\leq\neg\neg(\neg x{\Rightarrow}\neg
x)\approx\neg(\neg x{\Rightarrow}x).$ (BT) is derived by a single application
of (P1) followed by (P2):

$1\leq(x{\Rightarrow}y){\Rightarrow}\neg\neg(x{\Rightarrow}y)\approx(x{\Rightarrow}y){\Rightarrow}\neg(x{\Rightarrow}\neg
y),$

and (BT’) is calculated similarly using the equivalent reformation of (P2). ∎

### 3.2 Connexive implications in ${\mathsf{FL}_{\mathsf{e}}}$-algebras

The most natural question to address first, in the context of substructural
logics, is to ask what happens when we assume that the residual operation
${\to}$ satisfies the connexive laws. The ramifications of such an assumption
are almost immediate.

######  Proposition 3.3.

The following are equivalent for $\mathbf{A}=\langle
A,\wedge,\vee,\cdot,{\to},0,1\rangle\in{\mathsf{FL}_{\mathsf{e}}}$:

  1. 1.

$(\mathbf{A},{\to})$ is proto-connexive.

  2. 2.

$(\mathbf{A},{\to})\models\mbox{\eqref{AT}}$.

  3. 3.

$0$ is the greatest element in $\mathbf{A}$.

###### Proof.

(1) implies (2) holds by definition. Suppose (2), then $\mathbf{A}\models
x{\to}\neg x\leq 0$ by residuation. Now, it is easily verified that
${\mathsf{FL}_{\mathsf{e}}}\models x\leq(y\land 1){\to}x$ using Prop. 2.1,
from which it follows $0\leq(x\land 1){\to}0$ and $(x\land 1){\to}0\leq(x\land
1){\to}((x\land 1){\to}0)\leq 0$. Hence $\mathbf{A}\models\neg(x\land
1)\approx 0$. So for any $x\in A$, we find $x\leq\neg\neg x\leq\neg(\neg
x\wedge 1)\leq 0$ via Prop. 2.3, establishing (3). Lastly, suppose (3) holds,
and note that this entails $\mathbf{A}\models\neg x\approx 0$, as $0\leq x\to
0$ by residuation of $x\cdot 0\leq 0$, from which it immediately follows
$(\mathbf{A},{\to})\models\mbox{\eqref{P2}}$. Since $1\leq x{\to}\neg\neg x$
is satisfied in any FL${}_{\mbox{\scriptsize e}}$-algebra, the desired result
follows from Proposition 3.2. ∎

It is worth noting that an FL${}_{\mbox{\scriptsize e}}$-algebra $\mathbf{A}$
having $0$ as its largest element is equivalently stated via
$\mathbf{A}\models\neg x\approx\neg y$, as the forward direction is shown in
the proof above and the reverse direction follows the valuation $x\mapsto\neg
z$ and $y\mapsto 1$. Therefore, the variety of FL${}_{\mbox{\scriptsize
e}}$-algebras satisfying $x\leq 0$ is equal to the variety
$\mathbf{G}_{\\_}{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{V}_{\\_}\emptyset)$, the
Glivenko variety of ${\mathsf{FL}_{\mathsf{e}}}$ relative to the trivial
variety $\mathsf{V}_{\\_}\emptyset$. In fact:

######  Proposition 3.4.

The variety
$(\mathbf{G}_{\\_}{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{V}_{\\_}\emptyset),{\to})$
is connexive.

###### Proof.

From Proposition 3.3 we have that
$(\mathbf{G}_{\\_}{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{V}_{\\_}\emptyset),{\to})$
is proto-connexive. Thus it suffices to show that it has a member in which
${\to}$ is not symmetric. Indeed, suppose
$\mathbf{A}\in\mathbf{G}_{\\_}{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{V}_{\\_}\emptyset)$
satisfies $x\to y\approx y\to x$. Then, in particular, $1{\to}0\leq 0{\to}1$,
and so by residuation $0=1\cdot 0=1\cdot(1{\to}0)\leq 1$, i.e., $0=1$ and
$\mathbf{A}$ is a trivial algebra. Now, since expanding any non-trivial
integral commutative residuated lattice by a new constant $0$ such that $0:=1$
yields a non-trivial algebra in
$\mathbf{G}_{\\_}{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{V}_{\\_}{\emptyset})$,
it follows that
$\mathbf{G}_{\\_}{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{V}_{\\_}{\emptyset})\neq\mathsf{V}_{\\_}{\emptyset}$
and therefore
$(\mathbf{G}_{\\_}{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{V}_{\\_}\emptyset),{\to})\models\mbox{\eqref{NS}}$.
∎

However, the logic
$\mathbf{L}=\mathbf{L}(\mathbf{G}_{\\_}{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{V}_{\\_}\emptyset))$
is far from behaving as a desirable connexive logic. In fact, although
$\mathbf{L}$ has, for example, Aristotle’s theses among its theorems, one has
also that $\vdash_{\\_}\mathbf{L}\varphi\to\neg\varphi$, for any formula
$\varphi$, since the identity $1\leq x{\to}\neg x$ holds in
$\mathbf{G}_{\\_}{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{V}_{\\_}{\emptyset})$.
The latter fact implies that $\varphi\vdash_{\\_}{\mathbf{L}}\neg\varphi$, for
any formula $\varphi$. Therefore, $\neg$ does not satisfy the minimal
requirement for a connective to be entitled as a negation (see e.g. [1, 16,
31]).

In the light of the above discussion, it seems reasonable to wonder if there
exists other _term-definable_ binary operations on FL${}_{\mbox{\scriptsize
e}}$-algebras (or expansion thereof) behaving like a connexive implication
with much more appealing features. To this aim, for the remainder of this
paper, we will study the generalized definition of the connexive implication
defined in [6] as follows. Given a map $\delta\colon A\to A$ on $\mathbf{A}$,
we define the two following operations:

$\displaystyle x{\Rightarrow}_{\wedge}^{\delta}y$
$\displaystyle\coloneq(x{\to}y)\wedge(y{\to}\delta{x})$ $\displaystyle
x{\Rightarrow}_{\circ}^{\delta}y$
$\displaystyle\coloneq(x{\to}y)\cdot(y{\to}\delta{x})$

We will pay special attention to the case where $\delta={\diamond}$, as the
operations ${\Rightarrow}_{\circ}^{{\diamond}}$ and
${\Rightarrow}_{{\wedge}}^{{\diamond}}$ are term-definable in the language of
${\mathsf{FL}_{\mathsf{e}}}$. Moreover, we will omit the superscript when no
confusion can arise; i.e.,
${\Rightarrow}_{\wedge}\coloneq{\Rightarrow}_{{\wedge}}^{{\diamond}}$ and
${\Rightarrow}_{\circ}\coloneq{\Rightarrow}_{\circ}^{{\diamond}}$.

It is worth noticing that, for
${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$, the
identity $1\leq x{\Rightarrow}x$ holds in all FL${}_{\mbox{\scriptsize
e}}$-algebras as $1\leq x{\to}x\leq x{\to}{\diamond}x$. Now, while
${\Rightarrow}$ is neither order-preserving in its right argument nor order-
reversing in its left argument, generally speaking, it does satisfy a weakened
version of these properties:

######  Proposition 3.5.

The following quasi-identities hold in ${\mathsf{FL}_{\mathsf{e}}}$ for
${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$:

$\displaystyle x\leq y~{}\mathsf{and}~{}{\diamond}x\approx{\diamond}y$
$\displaystyle\quad\mathsf{implies}\quad z{\Rightarrow}x\leq z{\Rightarrow}y$
$\displaystyle x\leq y~{}\mathsf{and}~{}{\diamond}x\approx{\diamond}y$
$\displaystyle\quad\mathsf{implies}\quad y{\Rightarrow}z\leq x{\Rightarrow}z$

###### Proof.

Let $\mathbf{A}\in{\mathsf{FL}_{\mathsf{e}}}$ and let $x,y,z\in A$. Suppose
$x\leq y$ and ${\diamond}x={\diamond}y$. The former implies $y{\to}z\leq
x{\to}z$ and $z{\to}x\leq z{\to}y$, by Prop. 2.1(2,3), while the latter
implies $x{\to}{\diamond}z=y{\to}{\diamond}z$, by Prop. 2.3(6). Since
$*\in\\{\cdot,\wedge\\}$ is order-preserving in both arguments, we obtain:

$\begin{array}[b]{l}z{\Rightarrow}x=(z{\to}x)*(x{\to}{\diamond}z)=(z{\to}x)*(y{\to}{\diamond}z)\leq(z{\to}y)*({\diamond}y{\to}{\diamond}z)=z{\Rightarrow}y;\\\
y{\Rightarrow}z=(y{\to}z)*(z{\to}{\diamond}y)=(y{\to}z)*(z{\to}{\diamond}x)\leq(x{\to}z)*(z{\to}{\diamond}x)=x{\Rightarrow}z.\end{array}\qed$

In terms of their relationship to (equational) connexive principles, we now
prove the following observation which will be important throughout this paper.

######  Lemma 3.6.

Let $\mathbf{A}$ be an FL${}_{\mbox{\scriptsize e}}$-algebra and
${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$. Then
$(\mathbf{A},{\Rightarrow})$ satisfies (P1). Consequently, if
$(\mathbf{A},{\Rightarrow})\models\mbox{\em(\ref{P2})}$ then
$(\mathbf{A},{\Rightarrow})$ is proto-connexive.

###### Proof.

The first claim follows from Proposition 3.5, setting $y,z:=x$ and in the
first quasi-identity, as $x\leq{\diamond}x$,
${\diamond}x={\diamond}{\diamond}x$, and $1\leq x{\Rightarrow}x$ always hold.
Hence $(\mathbf{A},{\Rightarrow})\models(\mbox{{\ref{P1}}})$. The second claim
follows by Proposition 3.2. ∎

In fact, even more can be said for the connective ${\Rightarrow}_{\wedge}$, as
we see below.

######  Lemma 3.7.

For any FL${}_{\mbox{\scriptsize e}}$-algebra $\mathbf{A}$, the following
hold:

  1. 1.

For all $x,y\in A$, $1\leq x{\Rightarrow}_{\wedge}y$ iff $x\leq y$ and
${\diamond}x={\diamond}y$.

  2. 2.

$\mathbf{A}\models{\diamond}(x{\Rightarrow}_{\wedge}{\diamond}y)\approx
x{\Rightarrow}_{\wedge}{\diamond}y\approx{\diamond}x{\Rightarrow}_{\wedge}{\diamond}y$.

###### Proof.

For the first claim, towards the forward direction, $1\leq x{\to}y$ and $1\leq
y{\to}{\diamond}x$ immediately follows by assumption. Hence $x\leq
y\leq{\diamond}x$, and furthermore this give
${\diamond}x\leq{\diamond}y\leq{\diamond}{\diamond}x={\diamond}x$. The reverse
direction follows from Prop. 3.5, setting $z:=x$ in the first quasi-identity,
since $1\leq x{\Rightarrow}x$ always holds. The second claim immediately
follows by Prop. 2.3(5,6). ∎

Furthermore, we obtain a (stronger) converse of Lemma 3.6 for
${\Rightarrow}_{\wedge}$.

######  Lemma 3.8.

Let $\mathbf{A}$ be an FL${}_{\mbox{\scriptsize e}}$-algebra. Then the
following are equivalent:

  1. 1.

$(\mathbf{A},{\Rightarrow}_{\wedge})$ is proto-connexive.

  2. 2.

$(\mathbf{A},{\Rightarrow}_{\wedge})\models\mbox{\em(\ref{BT})}$.

  3. 3.

$(\mathbf{A},{\Rightarrow}_{\wedge})\models\mbox{\em(\ref{BT'})}$.

  4. 4.

$(\mathbf{A},{\Rightarrow}_{\wedge})\models\mbox{\em(\ref{P2})}$.

  5. 5.

$(\mathbf{A},{\Rightarrow}_{\wedge})$ models the following identity,

$\neg(x{\Rightarrow}y)\approx x{\Rightarrow}\neg y.$ (P3)

###### Proof.

For simplicity, set ${\Rightarrow}\coloneq{\Rightarrow}_{\wedge}$. Clearly (1)
implies both (2) and (3), by definition, (4) and (5) are equivalent by Lem.
3.7(2), and (4) implies (1) by Lemma 3.6. It is sufficient then to show that
(2) and (3) each imply (4). Indeed, if $(\mathbf{A},{\Rightarrow}_{\wedge})$
satisfies (BT) then for any $x,y\in A$ we have
$1\leq(x{\Rightarrow}y){\Rightarrow}\neg(x{\Rightarrow}\neg y)$. Hence by
Lemma 3.7(1), $\neg(x{\Rightarrow}y)\approx{\diamond}(x{\Rightarrow}\neg y)$.
A similar calculation shows (3) implies (4). ∎

######  Remark 3.9.

It is worth noticing that (P3) is a well known and motivated falsification
condition for (connexive) implication (see e.g. [27, 29]). For example, [28]
motivates (P3) by the introduction of negated syntactic types in Categorial
Grammar (see e.g. [4]) and the consequent need of providing falsity conditions
for the functor-type forming directional implications. Wansing’s connexive
logic $\mathsf{C}$ is algebraizable w.r.t. a slight modification of the
equivalent algebraic semantics of paraconsistent Nelson’s logic $\mathsf{N4}$
satisfying, among other axioms, (P3) (see [7]).

######  Example 3.10.

The above lemma does not generally hold in ${\mathsf{FL}_{\mathsf{e}}}$ for
the connective ${\Rightarrow}_{\circ}$. Indeed, consider $\mathbb{Z}(n)$ from
Example 2.4, for some arbitrary $n\in\mathbb{Z}$. It is easy to verify that
$(\mathbb{Z}(n),{\Rightarrow}_{\circ})$ satisfies (BT) since
$x{\Rightarrow}_{\circ}y=0=1_{\\_}{\mathbb{Z}(n)}$ for all $x,y\in\mathbb{Z}$.
However, taking $n\neq 0$, we see $\neg(x{\Rightarrow}_{\circ}\neg y)=n\neq
0=x{\Rightarrow}_{\circ}y={\diamond}(x{\Rightarrow}_{\circ}y).$ Moreover,
fixing $n<0$ and $y=x$ above we have $\neg(x{\Rightarrow}_{\circ}\neg
x)=n\not\geq 0=1_{\\_}{\mathbb{Z}(n)}$, so
$(\mathbb{Z}(n),{\Rightarrow}_{\circ})$ does not satisfy (AT). We do, however,
obtain a restricted converse with ${\Rightarrow}_{\circ}$ relative to
FL${}_{\mbox{\scriptsize e}}$-algebras satisfying $x\leq{\diamond}1$ [see Lem.
3.18].

Lastly, we prove the following proposition, whose generality will be useful in
the next section.

######  Proposition 3.11.

Suppose $\mathbf{A}$ is an FL${}_{\mbox{\scriptsize e}}$-algebra in which $0$
is the greatest element. Then for any map $\delta:A^{2}\to A$, the connective
${\Rightarrow}_{\circ}^{\delta}$ is proto-connexive on $\mathbf{A}$. For
$\delta={\diamond}$, then the connective
${\Rightarrow}_{{\wedge}}^{{\diamond}}$ is proto-connexive in $\mathbf{A}$.

###### Proof.

The first claim follows from the fact that $\mathbf{A}$ satisfies $\neg
x\approx 0$, hence also $x\cdot 0\approx 0$ and $x{\to}\neg y\approx 0$, from
which it follows $x{\Rightarrow}_{\circ}^{\delta}\neg y\approx 0$.
Consequently, Aristotle’s and Boethius’ theses hold for
${\Rightarrow}_{\circ}^{\delta}$. For the second claim,
$x{\Rightarrow}_{{\wedge}}^{{\diamond}}\neg y\approx 0$ holds in $\mathbf{A}$,
so ${\Rightarrow}_{{\wedge}}^{{\diamond}}$ is proto-connexive. ∎

### 3.3 Characterizing connexivity for ${\Rightarrow}_{\wedge}$ and
${\Rightarrow}_{\circ}$

We will now investigate the varieties of FL${}_{\mbox{\scriptsize
e}}$-algebras satisfying the connexive laws for both ${\Rightarrow}_{\wedge}$
and ${\Rightarrow}_{\circ}$, and their relationship to the Glivenko variety
relative to Boolean Algebras, which is, at least implicitly, hinted at in [6].
We provide a characterization for ${\Rightarrow}_{\wedge}$ in the general
case, and a restricted one for ${\Rightarrow}_{\circ}$. To that aim, let us
start with the following technical lemmas.

######  Lemma 3.12.

Let $\mathbf{A}$ be an FL${}_{\mbox{\scriptsize e}}$-algebra such that
${\diamond}\mathbf{A}$ is integral. Then:

  1. 1.

$\mathbf{A}$ satisfies $x{\to}y\approx
x{\Rightarrow}_{\wedge}({\diamond}x\wedge y)$

  2. 2.

$\mathbf{A}$ satisfies $x{\Rightarrow}_{\circ}\neg y\leq
x{\Rightarrow}_{\wedge}\neg y$ and
${\diamond}x{\Rightarrow}_{\circ}x\leq{\diamond}({\diamond}x{\Rightarrow}_{\wedge}x)$.

  3. 3.

Let ${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$. Then
the following hold:

(a) If $(\mathbf{A},{\Rightarrow})\models\mbox{\eqref{AT}}$ then $\mathbf{A}$
is pseudo-complemented.

(b) If $(\mathbf{A},{\Rightarrow})\models\mbox{\em(\ref{P2})}$ then
$\mathbf{A}$ satisfies $1\leq{\diamond}({\diamond}x{\to}x)$.

###### Proof.

Let $*\in\\{\wedge,\cdot\\}$, and note that
$\mathbf{A}\models{\diamond}x\approx{\diamond}x*{\diamond}1$ since
${\diamond}1$ is the greatest element (for the $\wedge$ case), and
${\mathsf{FL}_{\mathsf{e}}}\models{\diamond}1\cdot{\diamond}x\approx{\diamond}x$
(for the $\cdot$ case)444${\diamond}x={\diamond}x\cdot
1\leq{\diamond}x\cdot{\diamond}1\leq{\diamond}({\diamond}x\cdot{\diamond}1)={\diamond}(x\cdot
1)={\diamond}x$.. Furthermore, note that ${\diamond}1$ being the greatest
element implies
${\diamond}({\diamond}x{\Rightarrow}_{\wedge}x)\approx{\diamond}({\diamond}x{\to}x)$.

For (1), note that ${\diamond}\mathbf{A}$ being integral entails
${\diamond}1=x{\to}{\diamond}y$ whenever $x\leq{\diamond}y$. So for $x,y\in
A$, we have

$\begin{array}[]{r c l l}x{\Rightarrow}_{\wedge}({\diamond}x\wedge
y)&=&[x{\to}({\diamond}x\wedge y)]\wedge[({\diamond}x\wedge
y){\to}{\diamond}x]\\\ &=&[x{\to}({\diamond}x\wedge y)]\wedge{\diamond}1\\\
&=&(x{\to}{\diamond}x)\wedge(x{\to}y)\\\ &=&{\diamond}1\wedge(x{\to}y)\\\
&=&x{\to}y.\end{array}$

For (2), first observe that ${\diamond}\mathbf{A}$ being integral entails that
$xy\leq{\diamond}x\wedge{\diamond}y$. Hence, from this observation and Prop.
2.3(6),

$x{\Rightarrow}_{\circ}\neg y=(x{\Rightarrow}\neg y)\cdot(\neg
y{\to}{\diamond}x)=(x{\Rightarrow}\neg y)\wedge(\neg
y{\to}{\diamond}x)=x{\Rightarrow}_{\wedge}\neg y.$

For the other identity, observe that
${\diamond}x{\Rightarrow}_{\circ}x\leq({\diamond}x{\to}x)\cdot{\diamond}1\leq{\diamond}({\diamond}x{\to}x)={\diamond}({\diamond}x{\Rightarrow}_{\wedge}x)$.

For (3), let
${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$. Towards
(a), set $c\coloneq x\wedge\neg x$. By order-preservation and residuation, it
is easy to see that $c\leq\neg c$. Since ${\diamond}\mathbf{A}$ is integral,
we have, on the one hand, ${\diamond}1=c{\to}\neg c$, and on the other hand
${\diamond}c={\diamond}1{\to}{\diamond}c\leq\neg c{\to}{\diamond}c$. Now, if
$(\mathbf{A},{\Rightarrow})$ satisfies (AT), then we have

$\begin{array}[b]{r c ll}0&\geq&c{\Rightarrow}\neg c&\mbox{By \eqref{AT} and
residuation}\\\ &\geq&c{\Rightarrow}_{\circ}\neg c&\mbox{By (2)}\\\
&=&(c{\to}\neg c)\cdot(\neg c{\to}{\diamond}c)\\\ &=&\neg
c{\to}{\diamond}c&\mbox{Since ${\diamond}1=c{\to}\neg c$}\\\
&\geq&{\diamond}c\\\ &\geq&c\coloneq x\wedge\neg x.\end{array}$

For (b), by Proposition 3.2 we that
$(\mathbf{A},{\Rightarrow})\models\mbox{(\ref{AT'})}$. So for $x\in A$, we
have

$\begin{array}[b]{rcll}1&\leq&\neg({\diamond}x{\Rightarrow}\neg x)&\mbox{By
{(\ref{AT'})}}\\\ &=&{\diamond}({\diamond}x{\Rightarrow}x)&\mbox{By
(\ref{P2})}\\\ &\leq&{\diamond}({\diamond}x{\Rightarrow}_{\wedge}x)&\mbox{By
(2)}\\\ &=&{\diamond}({\diamond}x{\to}x).\end{array}\qed$

######  Example 3.13.

The converse of Lemma 3.12(1) holds for
${{\Rightarrow}}={\Rightarrow}_{\wedge}$. Indeed, it is easily checked that
${\mathsf{FL}_{\mathsf{e}}}\models x{\Rightarrow}_{\wedge}({\diamond}x\wedge
y)\leq x{\to}{\diamond}x,$ So, if $\mathbf{A}$ satisfies $x{\to}y\approx
x{\Rightarrow}_{\wedge}({\diamond}x\wedge y)$, by taking $x=1$ we obtain
$y=1{\to}y\leq 1{\to}{\diamond}1={\diamond}1.$

######  Lemma 3.14.

Let $\mathbf{A}$ be an FL${}_{\mbox{\scriptsize e}}$-algebra. If
$(\mathbf{A},{\Rightarrow}_{\wedge})\models\mbox{\em(\ref{P2})}$ then
$\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$.

###### Proof.

Note that $(\mathbf{A},{\Rightarrow}_{\wedge})$ satisfies (AT) by Lemma 3.8.
In light of Lemma 2.10(2) and Lemma 3.12(3), it suffices to verify that
${\diamond}\mathbf{A}$ is integral. Indeed, for $x\in A$,

$\begin{array}[]{r c l
l}x{\to}{\diamond}1&\geq&(x{\to}{\diamond}1)\wedge{\diamond}x\\\
&=&(x{\to}{\diamond}1)\wedge({\diamond}1{\to}{\diamond}x)\\\
&=&x{\Rightarrow}_{\wedge}{\diamond}1\\\
&=&\neg(x{\Rightarrow}_{\wedge}0)&\mbox{By (\ref{P3}) since ${\diamond}1=\neg
0$}\\\ &\geq&\neg(x{\to}0)&\mbox{By def. of ${\Rightarrow}_{\wedge}$}\\\
&\geq&x&\mbox{By \lx@cref{creftype~refnum}{dblr}(1)},\end{array}$

and so $x\cdot x\leq{\diamond}1$ by residuation. It easily follows (e.g. by
the substitution $x\mapsto y\vee 1$) that ${\diamond}1$ is the greatest
element of $\mathbf{A}$, i.e. ${\diamond}\mathbf{A}$ is integral. ∎

######  Remark 3.15.

The lemma above does not generally hold for the connective
${\Rightarrow}_{\circ}$. Indeed, consider $\mathbb{Z}(n)$ from Example 2.4 for
$n=0$. Clearly ${\diamond}\mathbb{Z}(0)$ is not integral, but it is easily
checked that $(\mathbb{Z}(0),{\Rightarrow}_{\circ})$ satisfies (P2).

######  Lemma 3.16.

If $\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$ then
$(\mathbf{A},{\Rightarrow}_{\wedge})$ is proto-connexive.

###### Proof.

Boolean algebras are involutive and have ${\wedge}={\cdot}$, so it follows
that

$\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})\models\neg[x{\Rightarrow}y]\approx\neg[(x{\to}y)\wedge(y{\to}x)].$

Furthermore, it is easily verified that Boolean algebras satisfy the
identities:

$(x{\to}\neg y)\wedge(\neg y{\to}\neg\neg x)\approx(x{\to}\neg y)\wedge(\neg
y{\to}x)\approx\neg[(x{\to}y)\wedge(y{\to}x)],$

and hence it follows that
$\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$ satisfies
$\neg(x{\Rightarrow}\neg y)\approx{\diamond}(x{\Rightarrow}y)$. So
$(\mathbf{A},{\Rightarrow}_{\wedge})$ satisfies (P2), and therefore is proto-
connexive by Lemma 3.8. ∎

We now obtain our promised characterization from Lemmas 3.8, 3.16 and 3.14.

######  Theorem 3.17.

Let $\mathsf{V}$ be a variety of FL${}_{\mbox{\scriptsize e}}$-algebras. Then
$(\mathsf{V},{\Rightarrow}_{\wedge})$ is proto-connexive [or satisfies any of
the identities (BT), (BT’), or (P2)] if and only if
$\mathsf{V}\subseteq\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$.
Consequently, the largest variety of FL${}_{\mbox{\scriptsize e}}$-algebras
for which ${\Rightarrow}_{\wedge}$ is proto-connexive is exactly
$\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$.

We now remark upon the connexivity of other connectives in
$\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$. We start by proving
the following technical lemma, whose generality will be useful in the sequel.

######  Lemma 3.18.

Let $\mathbf{A}$ be an FL${}_{\mbox{\scriptsize e}}$-algebra such that
${\diamond}\mathbf{A}$ is integral. Let ${\Rightarrow}$ be a binary operation
over $\mathbf{A}$ such that $(\mathbf{A},{\Rightarrow})$ satisfies (i) (P1)
and the identities (ii)
$x{\Rightarrow}_{\circ}y\leq{\diamond}(x{\Rightarrow}y)\leq
x{\Rightarrow}_{\wedge}{\diamond}y$. Then the following are equivalent:

  1. 1.

$(\mathbf{A},{\Rightarrow})$ is proto-connexive.

  2. 2.

$(\mathbf{A},{\Rightarrow})$ models any one of (BT), (BT’), or (P2).

  3. 3.

$(\mathbf{A},{\Rightarrow}_{\wedge})$ is proto-connexive [equivalently,
$\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$].

In particular, the above holds for ${\Rightarrow}:={\Rightarrow}_{\circ}$.

###### Proof.

Let us begin with (1) implies (2). Clearly, (1) implies both (BT) and (BT’) by
definition. We show that either of these are sufficient to obtain (P2). Let
$x,y\in A$ and define $a:=x{\Rightarrow}y$ and $b:=x{\Rightarrow}\neg y$. If
(BT) holds, then $1\leq a{\Rightarrow}\neg b$, and we observe

$\begin{array}[]{r c l l}1&\leq&a{\Rightarrow}\neg b\\\
&\leq&{\diamond}[a{\Rightarrow}\neg b]&\mbox{By (ii)}\\\
&\leq&{\diamond}[a{\Rightarrow}_{\wedge}\neg b]&\mbox{By assumption on
${\Rightarrow}$}\\\ &=&a{\Rightarrow}_{\wedge}\neg b&\mbox{By
\lx@cref{creftype~refnum}{charlanfcnxone}(2)}\\\ \end{array}$ (3.1)

Hence, by Lemma 3.7(1), $\neg\neg a=\neg b$. So
$(\mathbf{A},{\Rightarrow})\models\mbox{\eqref{P2}}$. The very same argument
(3.1) follows for the (BT’)-case by swapping the roles of $a$ and $b$. Hence
we have concluded (1) implies (2).

For (2) implies (3), we have that, by the arguments above,
$(\mathbf{A},{\Rightarrow})$ satisfies (P2) in any case. So (3) follows from
that fact that $(\mathbf{A},{\Rightarrow})$ is proto-connexive by (i) and
Proposition 3.2, i.e., it satisfies (BT), and so by the same argument (3.1)
above (by setting $a:=x{\Rightarrow}_{\wedge}y$ and
$b:=x{\Rightarrow}_{\wedge}\neg y$) we conclude
$(\mathbf{A},{\Rightarrow}_{\wedge})\models\mbox{\eqref{P2}}$, and the result
follows from Lemma 3.8.

Now suppose (3) holds. On the one hand, by Lemma 3.8
$(\mathbf{A},{\Rightarrow}_{\wedge})\models\mbox{(P3)}$, so
${\diamond}(x{\Rightarrow}_{\wedge}y)=x{\Rightarrow}_{\wedge}{\diamond}y$. On
the other hand,
$\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$ by Theorem
3.17, so
${\diamond}(x{\Rightarrow}_{\circ}y)={\diamond}(x{\Rightarrow}_{\wedge}y)$ by
Theorem 2.8(1a). Thus
${\diamond}(x{\Rightarrow}y)={\diamond}(x{\Rightarrow}_{\wedge}y)$ follows the
assumption that
${\diamond}(x{\Rightarrow}_{\circ}y)\leq{\diamond}(x{\Rightarrow}y)\leq
x{\Rightarrow}_{\wedge}{\diamond}y$. Hence
$(\mathbf{A},{\Rightarrow})\models\mbox{\eqref{P2}}$ follows from the fact
$(\mathbf{A},{\Rightarrow}_{\wedge})$ does. Since
$(\mathbf{A},{\Rightarrow})\models\mbox{(\ref{P1})}$ by assumption,
$(\mathbf{A},{\Rightarrow})$ is proto-connexive by Proposition 3.2. This
completes the equivalences.

Lastly, clearly the above holds for ${\Rightarrow}:={\Rightarrow}_{\circ}$
since (i) and (ii) follow from Lemma 3.6 and Lemma 3.12(2). ∎

######  Corollary 3.19.

Let $\mathsf{V}$ be a subvariety of FL${}_{\mbox{\scriptsize e}}$-algebras.
Then $(\mathsf{V},{\Rightarrow}_{\wedge})$ is proto-connexive iff
$(\mathsf{V},{\Rightarrow}_{\circ})$ is proto-connexive and $\mathsf{V}$
satisfies the identity $x\leq{\diamond}1$.

######  Example 3.20.

The converse to the above does not hold if the assumption of
$x\leq{\diamond}1$ is dropped; e.g., $\mathbb{Z}(0)$ has no largest element,
$(\mathbb{Z}(0),{\Rightarrow}_{\circ})$ is proto-connexive, but
$(\mathbb{Z}(0),{\Rightarrow}_{\wedge})\nmodels\mbox{\eqref{BT}}$ (take $x=0$
and $y=1$).

Having established the proto-connexitivity of ${\Rightarrow}_{\wedge}$ and
${\Rightarrow}_{\circ}$ (and other related operations) in the Glivenko variety
relative to Boolean algebras, we now investigate the principle of non-symmetry
(NS).

######  Lemma 3.21.

Let $\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$ and
let ${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$. Then
$(\mathbf{A},{\Rightarrow})$ falsifes (NS), i.e., ${\Rightarrow}$ is symmetric
on $\mathbf{A}$, if and only if $\mathbf{A}$ is a Boolean algebra.

###### Proof.

Of course, if $\mathbf{A}$ is a Boolean algebra, then
${\Rightarrow}_{\circ}={\leftrightarrow}={\Rightarrow}_{\wedge}$ which is
indeed symmetric. For the converse direction, first note that ${\diamond}1$ is
the largest element of $\mathbf{A}$. Let $*\in\\{\wedge,\cdot\\}$ and
${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$

${\diamond}1=(1\to{\diamond}1)*({\diamond}1\to{\diamond}1)=1{\Rightarrow}{\diamond}1={\diamond}1{\Rightarrow}1=({\diamond}1\to
1)*(1\to{\diamond}1)=({\diamond}1\to 1)*{\diamond}1.$

On the one hand, if ${*}={\cdot}$, then $({\diamond}1\to 1)*{\diamond}1\leq
1$, so $1={\diamond}1$. On the other hand, for ${*}={\wedge}$ we have
$({\diamond}1\to 1)*{\diamond}1={\diamond}1\to 1$, so one has
${\diamond}1\cdot{\diamond}1\leq 1$ and we compute $x=x\cdot 1\leq
x\cdot{\diamond}1\leq{\diamond}1\cdot{\diamond}1\leq 1$. So $1={\diamond}1$ in
either case, and consequently $1=x{\to}1$.

Now, $x=(1\to x)*(x\to{\diamond}1)=1{\Rightarrow}x=x{\Rightarrow}1=(x\to
1)*(1\to{\diamond}x)={\diamond}x.$ The involutivity of $\neg$, together with
$\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$, yield
directly that $\mathbf{A}$ is Boolean. ∎

Therefore, by Thm. 3.17, Cor. 3.19, Remark 2.9, and the lemma above:

######  Corollary 3.22.

Let $\mathsf{V}$ be any variety of FL${}_{\mbox{\scriptsize e}}$-algebras in
the interval between $\mathsf{HA}$ and
$\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$. Then the connectives
${\Rightarrow}_{\wedge}$ and ${\Rightarrow}_{\circ}$ are both connexive in
$\mathsf{V}$.

### 3.4 Generalizations in the integral case

We now consider the special case when dealing with _integral_ (and
$0$-_bounded_) FL${}_{\mbox{\scriptsize e}}$-algebras, and show that the
results from Section 3.3 can be strengthened. In fact, Theorem 3.25 shows
that, in the setting of $\mathsf{FL}_{\mathsf{ei}}$
($\mathsf{FL}_{\mathsf{ew}}$), a binary operation in the pointwise ordered
interval $[{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}]$ is proto-connexive
if and only if _any_ operation in the interval is. Moreover, ${\diamond}$ is
the only increasing mapping for which this holds. Furthermore, we demonstrate
that satisfying any one of Aristotle’s theses is sufficient to guarantee
proto-connexity for subvarieties of $\mathsf{FL}_{\mathsf{ew}}$.

Since integral FL${}_{\mbox{\scriptsize e}}$-algebras satisfy the identity
$x\cdot y\leq x\wedge y$, given such an algebra $\mathbf{A}$ and map $\delta$,
it follows $x{\Rightarrow}_{\circ}^{\delta}y\leq
x{\Rightarrow}_{\wedge}^{\delta}y$ for all $x,y\in A$. We define the
(nonempty) interval of functions
$[{\Rightarrow}_{\circ}^{\delta},{\Rightarrow}_{\wedge}^{\delta}]\subseteq
A^{2}\times A$ via:

$f\in[{\Rightarrow}_{\circ}^{\delta},{\Rightarrow}_{\wedge}^{\delta}]\iff(\forall
x,y\in A)[x{\Rightarrow}_{\circ}^{\delta}y\leq f(x,y)\leq
x{\Rightarrow}_{\wedge}^{\delta}y].$

######  Lemma 3.23.

Let $\mathbf{A}$ be an integral FL${}_{\mbox{\scriptsize e}}$-algebra,
$\delta$ be an increasing map on $\mathbf{A}$, and
${\Rightarrow}\in[{\Rightarrow}_{\circ}^{\delta},{\Rightarrow}_{\wedge}^{\delta}]$.

  1. 1.

For all $x\in A$, $1{\Rightarrow}x=x$ and $x{\Rightarrow}1=\delta{x}$.

  2. 2.

If $(\mathbf{A},{\Rightarrow})$ models either (AT) or (AT’), then $\mathbf{A}$
is pseudo-complemented.

  3. 3.

If $(\mathbf{A},{\Rightarrow})\models\mbox{\eqref{BT}}$ then $\neg x\approx
x{\Rightarrow}0$, ${\delta}={{\diamond}}$, and
$\mathbf{A}\in\mathbf{G}_{\mathsf{FL}_{\mathsf{ei}}}(\mathsf{BA})$.

###### Proof.

(1) easily follows from the following calculations, by integrality and virtue
of the fact that
${\Rightarrow}\in[{\Rightarrow}_{\circ}^{\delta},{\Rightarrow}_{\wedge}^{\delta}]$
and $\delta$ is increasing: For $x\in A$,

$\displaystyle 1{\Rightarrow}_{\circ}x=(1{\to}x)\cdot(x{\to}\delta{1})=x\cdot
1=x=x\wedge 1=(1{\to}x)\wedge(x{\to}\delta{1})=1{\Rightarrow}_{\wedge}x;$
$\displaystyle
x{\Rightarrow}_{\circ}1=(x{\to}1)\cdot(1{\to}\delta{x})=1\cdot\delta{x}=\delta{x}=1\cdot\delta{x}=(x{\to}1)\wedge(1{\to}\delta{x})=x{\Rightarrow}_{\wedge}1.$

For (2), following essentially the same argument as Lemma 3.12(3a). Suppose
$(\mathbf{A},{\Rightarrow})$ satisfies (AT), and let $x\in A$. Set $c\coloneq
x\wedge\neg x$ and recall that $c\leq\neg c$. Since $\mathbf{A}$ is integral,
we have, on the one hand, $1=c{\to}\neg c$, and on the other hand
$\delta{c}\leq\neg c{\to}\delta{c}$. Hence

$c\leq\delta{c}\leq\neg c\to\delta{c}=(c{\to}\neg c)\cdot(\neg
c\to\delta{c})=c{\Rightarrow}_{\circ}^{\delta}\neg c\leq c{\Rightarrow}\neg
c\leq 0,$

where the last line follows from (AT) by residuation. So $\mathbf{A}$ is
pseudo-complemented. The same argument follows for (AT’) by considering
$\neg\neg c{\Rightarrow}\neg c$, as $c\leq\neg c$ implies ${\diamond}c\leq\neg
c$, so $c\leq\delta({\diamond}c)\leq({\diamond}c{\to}\neg c)\cdot(\neg
c{\to}\delta({\diamond}c))={\diamond}c{\Rightarrow}_{\circ}\neg
c\leq{\diamond}c{\Rightarrow}\neg c\leq 0$.

For (3), suppose $(\mathbf{A},{\Rightarrow})\models\eqref{BT}$. Using (BT) and
(1), we have

$1=(1{\Rightarrow}x){\Rightarrow}\neg(1{\Rightarrow}\neg
x)=x{\Rightarrow}\neg\neg x\leq
x{\Rightarrow}_{\wedge}^{\delta}{\diamond}x=(x\to{\diamond}x)\wedge({\diamond}x\to\delta{x})={\diamond}x{\to}\delta{x},$

where the last equality follows from integrality. So
${\diamond}x\leq\delta{x}$ by residuation. Since $\mathbf{A}$ is integral,
$0\cdot\neg x\leq 0$ and thus $0\leq{\diamond}x\leq\delta{x}$, so it follows
that $0{\to}\delta{x}=1$. Hence

$\neg x=(x{\to}0)\cdot(0{\to}\delta{x})=x{\Rightarrow}_{\circ}^{\delta}0\leq
x{\Rightarrow}0\leq
x{\Rightarrow}_{\wedge}^{\delta}0=(x{\to}0)\wedge(0{\to}\delta{x})=\neg x,$

completing the first claim. Towards the second claim, it suffices to show
$\delta{x}\leq{\diamond}x$, or equivalently $1\leq\delta{x}\to{\diamond}x$.
Indeed, we see

$\begin{array}[b]{ r c l
l}1&=&(x{\Rightarrow}1){\Rightarrow}\neg(x{\Rightarrow}\neg 1)&\mbox{By
\eqref{BT}}\\\ &=&\delta{x}{\Rightarrow}\neg(x{\Rightarrow}0)&\mbox{By (1)}\\\
&=&\delta{x}{\Rightarrow}{\diamond}x&\mbox{By (b)}\\\
&\leq&\delta{x}{\Rightarrow}_{\wedge}^{\delta}{\diamond}x&\mbox{Def. of
${\Rightarrow}$}\\\ &=&\delta{x}{\to}{\diamond}x&\mbox{By integrality since
${\diamond}x{\to}\delta{x}=1$}.\end{array}$

Hence $\delta={\diamond}$, and so
${\Rightarrow}\in[{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}]$. Since
$x{\Rightarrow}_{\circ}y\leq x{\Rightarrow}y$, ${\Rightarrow}$ clearly
satisfies $\eqref{P1}$ by Lemma 3.6. Therefore, ${\Rightarrow}$ satisfies the
assumptions of Lemma 3.18, and hence
$\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$. Since
$\mathbf{A}$ is integral by assumption, the claim follows. ∎

The lemma above allows us to conclude, in contrast to the general case in
Section 3.3, the following stronger theorems in the integral setting.

######  Theorem 3.24.

Let $\mathbf{A}$ be an integral FL${}_{\mbox{\scriptsize e}}$-algebra,
$\delta$ an increasing mapping on $\mathbf{A}$, and
${\Rightarrow}\in[{\Rightarrow}_{\circ}^{\delta},{\Rightarrow}_{\wedge}^{\delta}]$.
Then $(\mathbf{A},{\Rightarrow})$ is proto-connexive but it does not satisfy
$(\mathrm{NS})$ if and only if $\delta={\diamond}$ and $\mathbf{A}$ is a
Boolean algebra.

###### Proof.

Suppose that $(\mathbf{A},{\Rightarrow})$ is proto-connexive but it does not
satisfy $(\mathrm{NS})$. So $(\mathbf{A},{\Rightarrow})$ satisfies (BT), and
hence $\delta={\diamond}$ and
$\mathbf{A}\in\mathbf{G}_{\mathsf{FL}_{\mathsf{ei}}}(\mathsf{BA})$ by Lemma
3.23(3), in particular $\delta\mathbf{A}={\diamond}\mathbf{A}\in\mathsf{BA}$.
Since $(\mathrm{NS})$ is not satisfied and hence, for any $x\in A$,
$x{\Rightarrow}1=1{\Rightarrow}x$. Thus $x=\delta{x}={\diamond}x$ by Lemma
3.23(1), and we conlcude ${\diamond}=\mathrm{id}_{\\_}\mathbf{A}$. So
$\mathbf{A}={\diamond}\mathbf{A}$ is a Boolean algebra.

The converse direction easily follows upon noticing that, if $\mathbf{A}$ is a
Boolean algebra and ${\diamond}=\delta$, then
${{\Rightarrow}}={\leftrightarrow}$ follows from involutivity and fact that
$\cdot$ and $\wedge$ coincide in Boolean algebras. Of course, material
equivalence in Boolean algebras is proto-connexive but fails $(\mathrm{NS})$
by straightforward calculations. ∎

######  Theorem 3.25.

For $\mathbf{A}$ an integral FL${}_{\mbox{\scriptsize e}}$-algebra, the
following are equivalent:

  1. 1.

$\mathbf{A}\in\mathbf{G}_{\mathsf{FL}_{\mathsf{ei}}}(\mathsf{BA})$ (i.e.,
$\mathbf{A}$ is strongly pseudo-complemented).

  2. 2.

For all ${\Rightarrow}\in[{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}]$,
$(\mathbf{A},{\Rightarrow})$ is proto-connexive.

  3. 3.

There exists an increasing $\delta$ and
${\Rightarrow}\in[{\Rightarrow}_{\circ}^{\delta},{\Rightarrow}_{\wedge}^{\delta}]$
such that $(\mathbf{A},{\Rightarrow})\models\mbox{\eqref{BT}}$.

###### Proof.

For (1) implies (2), note that
${\Rightarrow}\in[{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}]$ implies
$x{\Rightarrow}_{\circ}y\leq{\diamond}(x{\Rightarrow}y)\leq{\diamond}(x{\Rightarrow}_{\wedge}y)$
and ${\diamond}(x{\Rightarrow}_{\wedge}y)=x{\Rightarrow}_{\wedge}{\diamond}y$
by the fact that ${\Rightarrow}_{\wedge}$ satisfies (P3) by (1). Since
${\diamond}\mathbf{A}$ is integral, by fiat, (2) follows from Lemma 3.18. (2)
implies (3) is obvious, and (3) implies (1) by Lemma 3.23(3). ∎

In light of Corollary 2.12 and Lemma 3.23(2), the theorem above implies:

######  Corollary 3.26.

Let $\mathbf{A}\in\mathsf{FL}_{\mathsf{ew}}$. Then the following are
equivalent.

  1. 1.

$\mathbf{A}\in\mathbf{G}_{\mathsf{FL}_{\mathsf{ew}}}(\mathsf{BA})$ (i.e.,
$\mathbf{A}$ is pseudo-complemented).

  2. 2.

For all ${\Rightarrow}\in[{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}]$,
$(\mathbf{A},{\Rightarrow})$ is proto-connexive.

  3. 3.

There exists an increasing $\delta$ and
${\Rightarrow}\in[{\Rightarrow}_{\circ}^{\delta},{\Rightarrow}_{\wedge}^{\delta}]$
such that $(\mathbf{A},{\Rightarrow})\models\mbox{\eqref{AT}}$.

######  Example 3.27.

Note that the direction (4) to (1) in Corollary 3.26 does not generally hold
if $0$-boundedness is dropped. Indeed, Figure 1 is an integral
FL${}_{\mbox{\scriptsize e}}$-algebra satisfying Aristotle’s thesis but not
Boethius’ thesis (take $x=0$ and $y=\bot$) for ${\Rightarrow}_{\wedge}$.

### 3.5 Returning to the logics: The main results

In what follows, we put in good use Theorem 2.5 to set substructural logics
mimicking a connexive implication expressed in terms of
${\Rightarrow}_{\wedge}$ and ${\Rightarrow}_{\circ}$ “on the map”. First, as a
consequence of Theorem 3.17, Lemma 3.18, and Corollary 3.22, we obtain the
following theorem.

######  Theorem 3.28.

Let $\mathbf{L}$ be any substructural logic in the interval between
$\mathbf{FL}_{\\_}\mathbf{e}$ and $\mathbf{IPL}$. Then the following are
equivalent:

  1. 1.

$(\mathbf{L},{\Rightarrow}_{\wedge})$ is a connexive logic.

  2. 2.

$(\mathbf{L},{\Rightarrow}_{\wedge})$ is a proto-connexive logic.

  3. 3.

Any one of Boethius’ theses for ${\Rightarrow}_{\wedge}$ are theorems of
$\mathbf{L}$.

  4. 4.

$\mathbf{L}$ is an axiomatic exentision of
$\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{e}}(\mathbf{CPL})$.

The same holds for the connective ${\Rightarrow}_{\circ}$ if it is further
assumed that $\varphi{\to}\neg\neg 1$ are theorems of $\mathbf{L}$.

Furthermore, in the presence of weakening, by Corollary 3.26 the above theorem
specializes:

######  Theorem 3.29.

Let $\mathbf{L}$ be any logic in the interval between
$\mathbf{FL}_{\\_}\mathbf{ew}$ and $\mathbf{IPL}$. Then the following are
equivalent for
${\Rightarrow}\in\\{{\Rightarrow}_{\wedge},{\Rightarrow}_{\circ}\\}$:

  1. 1.

$(\mathbf{L},{\Rightarrow})$ is a connexive logic.

  2. 2.

Any one of Aristotle’s theses for ${\Rightarrow}$ are theorems of
$\mathbf{L}$.

  3. 3.

$\mathbf{L}$ is an axiomatic extension of
$\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{ew}}(\mathbf{CPL})$.

Interestingly enough, in the framework of $\mathbf{FL}_{\\_}{\mathbf{ew}}$,
the above result “does justice” to the idea that “Aristotle’s Thesis is the
cornerstone of the logics belonging to the family of so-called connexive
logics” (cf. [23]).

Since the variety $\mathsf{HA}$ of Heyting algebras is pseudo-complemented, we
recover the following result from [6] as a particular instance of Theorem
3.29.

######  Corollary 3.30.

$(\mathsf{HA},{\Rightarrow}_{\wedge})$ is connexive, and therefore
$(\mathbf{IPL},{\Rightarrow}_{\wedge})$ is connexive.

As the connectives $\cdot$ and $\wedge$ coincide in Heyting algebras, so too
do the operations ${\Rightarrow}_{\wedge}$ and ${\Rightarrow}_{\circ}$. We
therefore conclude with the following corollary.

######  Corollary 3.31.

The connectives ${\Rightarrow}_{\wedge}$ and ${\Rightarrow}_{\circ}$ are
connexive for every logic in the interval between
$\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{e}}(\mathbf{CPL})$ and
$\mathbf{IPL}$.

##  4 Some philosophical considerations

As we have seen through the above sections, the implication
${\Rightarrow}_{\wedge}$ plays a prominent role in our investigation. Indeed,
its intrinsic interest (see p. 1) depends not only on that it behaves as a
connexive implication with a desirable formal behavior, but also on its
particularly smooth interpretation. Section 4.1 is devoted to deepening such a
reading, highlighting interesting links between ${\Rightarrow}_{\wedge}$ and
the theory of plausible reasoning. Subsequently, we will put into good use the
machinery developed so far in order to highlight some interesting features of
${\Rightarrow}_{\wedge}$. The last part of this section provides an
investigation of two variants of connexivity, namely _weak_ and _strong_
connexivity. Interestingly enough, we will show that, when dealing with
$\mathsf{FL}_{\mathsf{ew}}$, these concepts can be regarded as one and the
same thing. Furthermore, we will point out that, for any substructural logic,
the strong connexivity of ${\Rightarrow}_{\wedge}$ and ${\Rightarrow}_{\circ}$
is formally embodied by a well known inference schema of intuitionistic logic:
ex falso quodlibet.

### 4.1 A focus on ${\Rightarrow}_{\wedge}$

In the second volume of his famous _Mathematics and plausible reasoning_ [25],
George Polya aims at formulating several _patterns_ of plausible reasoning
explicitly. Among them, he investigates an inference schema “which is of so
general use that we could extract it from almost any example” ([25, vol.2,
p.3]). Let $A$ be some clearly formulated conjecture which is, at present,
neither proved, nor refuted. Also, let $B$ be some consequence of $A$ which we
have neither proved, nor refuted, as well. For example, if we set $A$ to be
Goldbach’s conjecture

_every even natural number greater than 2 is the sum of two prime numbers_ ,

$B$ might be that $198388=p_{\\_}1+p_{\\_}2$ for suitable primes $p_{\\_}1$
and $p_{\\_}2$. Indeed, although we do not know whether $A$ or $B$ is true, it
is unquestionable that _$A$ implies $B$_. We verify $B$. If $B$ turns out to
be false, then performing _modus tollens_ we can conclude that $A$ is false as
well. Otherwise, if we recognize that $B$ is true (and this is the case),
then, although we do not have a proof of $A$, we can nevertheless conclude
that $A$ is _more credible_. In other words, we have applied the following
_fundamental inductive pattern_ (in brief “inductive pattern” [25, vol.2,
p.4]) or _heuristic syllogism_ :

$A$ implies $B$ $B$ $A$ is more credible

As stated in [24, p.34], a reasonable _logic of plausible inference_ should a)
be general enough to include the use of inductive (in a broad sense) reasoning
in mathematics; b) include the heuristic syllogism among its inference rules;
and c) be fully qualitative, in the sense that “[…] it is not possible to give
a numerical value to the degree of credence attached to any statement
considered”. If one takes into account the possibility of developing this
proposal in a _monotonic_ setting, then it is reasonable to interpret “$x$ is
more credible” (or “$x$ is likely to be true”) as a modal operator. Therefore,
it becomes worthy of attention considering expansions of
$\mathbf{FL}_{\\_}\mathbf{e}$ in which assertions on “plausibility”, as well
as the inductive pattern, are amenable of a formal treatment.

To this aim, and to motivate a formal account of the notion of plausibility,
one might rely on the following assumptions:

  * $\blacklozenge$1

If $\neg A$ is more credible, then $A$ is not;

  * $\blacklozenge$2

If $\neg A$ is false, then $A$ is more credible.

  * $\blacklozenge$3

If $A$ is equivalent to $B$, then $A$ is more credible if and only if $B$ is.

Note that $\blacklozenge$1 encodes a weak version of Polya’s principle that
“non-$A$ more credible” is equivalent to “$A$ less credible” (see [25, vol. 2,
p.23]).

Let us consider then the language
$\mathcal{L}=\\{\land,\lor,\cdot,\rightarrow,\blacklozenge,0,1\\}$, where
$\blacklozenge$ is a unary (modal) operator. We will denote the absolutely
free algebra over $\mathcal{L}$ generated by an infinite countable set of
variables by $\mathbf{\mbox{{Fm}}}_{\\_}{\mathcal{L}}$. The formula
$\blacklozenge\varphi$ will be read as “$\varphi$ is more
credible/plausible/likely to be true”. In order to include
$\blacklozenge$1-$\blacklozenge$3 in our formal system, we consider the
expansion
$\vdash_{\\_}{\mathbf{FL}_{\mathbf{e}}^{\blacklozenge}}\subseteq\wp(\mbox{{Fm}}_{\\_}{\mathcal{L}})\times\mbox{{Fm}}_{\\_}{\mathcal{L}}$
of $\vdash_{\\_}{\mathbf{FL}_{\\_}\mathbf{e}}$ by the following axioms and
inference schemas concerning the behavior of $\blacklozenge$:

  1. $\blacklozenge$1:

$\vdash_{\\_}{\mathbf{FL}_{\mathbf{e}}^{\blacklozenge}}\blacklozenge\neg\varphi\to\neg\blacklozenge\varphi$;

  2. $\blacklozenge$2:

$\vdash_{\\_}{\mathbf{FL}_{\mathbf{e}}^{\blacklozenge}}\neg\neg\varphi\to\blacklozenge\varphi$;

  3. $\blacklozenge$3:

$\varphi\leftrightarrow\psi\vdash_{\\_}{\mathbf{FL}_{\mathbf{e}}^{\blacklozenge}}\blacklozenge\varphi\leftrightarrow\blacklozenge\psi$.

Due to $\blacklozenge$3, it follows from general facts concerning
algebraizable logics (see e.g. [8, Proposition 3.31]) that
$\vdash_{\\_}{\mathbf{FL}_{\mathbf{e}}^{\blacklozenge}}$ is algebraizable. Its
equivalent algebraic semantics is the variety
$\mathsf{FL}_{\mathsf{e}}^{\blacklozenge}$ of FL${}_{\mbox{\scriptsize
e}}^{\blacklozenge}$-algebras whose members are structures of the form
$(A,\land,\lor,\cdot,\to,\blacklozenge,0,1)$ where
$(A,\land,\lor,\cdot,\to,0,1)$ is an FL${}_{\mbox{\scriptsize e}}$-algebra,
and the following identities hold:

  * A1:

$\blacklozenge\neg x\leq\neg\blacklozenge x$;

  * A2:

$\neg\neg x\leq\blacklozenge x.$

As a consequence, we have the following

######  Proposition 4.1.

Any FL${}_{\mbox{\scriptsize e}}^{\blacklozenge}$-algebra satisfies $\neg\neg
x\approx\blacklozenge x$.

###### Proof.

One has that $\neg x\geq\neg\blacklozenge x\geq\blacklozenge\neg
x\geq\neg\neg\neg x=\neg x$. We conclude that $\neg x=\neg\blacklozenge x$.
Consequently, we have $\neg\neg x=\neg\neg\blacklozenge x\geq\blacklozenge x$
and $\neg\neg x=\blacklozenge x$. ∎

Now, we observe that, in general, the following _does not hold_ :

$\varphi\to\psi,\psi\vdash_{\\_}{\mathbf{FL}_{\mathbf{e}}^{\blacklozenge}}\blacklozenge\varphi,$
(4.1)

i.e., $\to$ does not satisfy the inductive pattern. However, it is easy to see
that ${\Rightarrow}_{\wedge}$ is _the weakest_ term-definable implication-like
connective $\leadsto$ satisfying, for any formulas
$\varphi,\psi\in\mbox{{Fm}}_{\\_}{\mathcal{L}}$, the following (stronger)
axiomatic renderings of _modus ponens_ , and the heuristic syllogism

  1. S1:

$\vdash_{\\_}{\mathbf{FL}_{\mathbf{e}}^{\blacklozenge}}(\varphi\leadsto\psi)\to(\varphi\to\psi)$;

  2. S2:

$\vdash_{\\_}{\mathbf{FL}_{\mathbf{e}}^{\blacklozenge}}(\varphi\leadsto\psi)\to(\psi\to\blacklozenge\varphi)$,

where “weakest” here means that, for any term-definable binary connective
$\leadsto$ satisfying S1-S2, we have that, for any
$\varphi,\psi\in\mbox{{Fm}}_{\\_}{\mathcal{L}}$:

$\vdash_{\\_}{\mathbf{FL}_{\mathbf{e}}^{\blacklozenge}}(\varphi\leadsto\psi)\to(\varphi{\Rightarrow}_{\wedge}\psi).$

In light of the above discussion, it can be argued that, under Polya’s
_desiderata_ [24], ${\Rightarrow}_{\wedge}$ might be considered as a
reasonable candidate for formalizing the kind of conditionals which express a
connection between a conjecture and one of its consequences in a monotonic
framework. In fact, given that in any FL${}_{\mbox{\scriptsize e}}$-algebra
$\mathbf{A}$ it holds that

$\begin{array}[]{r c l }x{\Rightarrow}_{\wedge}y&=&(x\to
y)\land(y\to{\diamond}x)\\\ &=&(x\to y)\land({\diamond}y\to{\diamond}x)\\\
&=&(x\to y)\land({\diamond}x\to{\diamond}y)\land({\diamond}y\to{\diamond}x)\\\
&=&(x\to y)\land({\diamond}x\leftrightarrow{\diamond}y),\end{array}$

$x{\Rightarrow}_{\wedge}y$ can be read as “$x$ implies $y$ and $x$ is
plausible/more credible/likely to be true if and only if $y$ is”, whenever “it
is plausible/more credible/likely to be true that $x$” is meant to satisfy
$\blacklozenge 1-\blacklozenge 3$, and so it can be formalized by
${\diamond}$. Note that, under such interpretation, the identity

$x\leq{\diamond}1$ (4.2)

seems to be a reasonable assumption once $1$ is meant as an absolutely true
statement and so ${\diamond}1$ can be read as “an absolutely true statement is
plausible”. Therefore, if we confine ourselves to consider
FL${}_{\mbox{\scriptsize e}}$-algebras satisfying (4.2), then Corollary 3.19
states a full equivalence between the connexivity of ${\Rightarrow}_{\wedge}$
and ${\Rightarrow}_{\circ}$ (cf. p. 3.19).

### 4.2 On the strength of ${\Rightarrow}_{\wedge}$

Beside having particularly fair motivations, ${\Rightarrow}_{\wedge}$ plays a
special role for our discussion, since it enjoys interesting properties which
are not shared by its product-variant ${\Rightarrow}_{\circ}$. Indeed, the
connexivity of ${\Rightarrow}_{\wedge}$ can be formulated in a stronger form.

######  Proposition 4.2.

Let $\mathbf{A}$ be an FL${}_{\mbox{\scriptsize e}}$-algebra. Then
$(\mathbf{A},{\Rightarrow}_{\wedge})$ satisfies _( BT)_ if and only if it
satisfies:

$1\leq(x{\Rightarrow}y){\Rightarrow}[(y{\Rightarrow}z){\Rightarrow}\neg(x{\Rightarrow}\neg
z)].$

###### Proof.

Set ${\Rightarrow}\coloneq{\Rightarrow}_{\wedge}$. Concerning the left-to-
right direction, observe that (4.2) has the form $1\leq
r{\Rightarrow}(s{\Rightarrow}t)$, for suitable terms $r,s,t$, which holds iff
(i) $r\leq s{\Rightarrow}t$ and (ii) $s{\Rightarrow}t\leq{\diamond}r$ by the
definition of ${\Rightarrow}$. Clearly (i) holds iff $rs\leq t$ and
$rt\leq{\diamond}s$ hold, which are verified by simply using Proposition 2.1,
the fact that ${\mathsf{FL}_{\mathsf{e}}}\models a\to{\diamond}b\approx\neg
b\to\neg a$, and an application of (P3), by Lemma 3.8, via
$x{\Rightarrow}z\leq\neg(x{\Rightarrow}\neg z)$. Moreover, since
$(\mathbf{A},{\Rightarrow})\models\mbox{\eqref{BT}}$, we have
$\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$ by Theorem
3.17. The reader can easily verify that $\mathsf{BA}\models x\leftrightarrow
y\approx(x\leftrightarrow z)\wedge(z\leftrightarrow y)$. Therefore, using the
Glivenko property, Lemma 3.16, and the fact that ${\diamond}$ is a nucleus, it
follows that property (ii) holds as well. Conversely, setting $y\coloneq x$ in
(4.2), we derive
$x{\Rightarrow}_{\wedge}x\leq(x{\Rightarrow}_{\wedge}z){\Rightarrow}_{\wedge}\neg(x{\Rightarrow}_{\wedge}\neg
z)$. Thus (BT) holds since $1\leq x{\Rightarrow}_{\wedge}x$. ∎

We show the above proposition does not hold for ${\Rightarrow}_{\circ}$ with
the following example.

######  Example 4.3.

Consider $A=\\{\bot,0,a,b,c,1\\}$ with the lattice order via $\bot<0<b,c<a<1$.
Let $\cdot$ be the commutative operation with unit $1$ described via: For all
$x\in A$, $\bot$ is absorbing; $x\cdot x=x$ for $x\neq 0$; $0\cdot x=\bot$ for
$x\neq 1$; $a\cdot x=x$ for $x=b,c$; and $b\cdot c=\bot$. It is not difficult
to show that $\cdot$ is associative and distributes over joins and therefore,
by general results (since the lattice is complete), has a residual operation
${\to}$. Hence $\mathbf{A}=\langle A,\wedge,\vee,\cdot,{\to},0,1\rangle$ is an
integral FL${}_{\mbox{\scriptsize e}}$-algebra, which is furthermore verified
to be a member of $\mathbf{G}_{\mathsf{FL}_{\mathsf{ei}}}(\mathsf{BA})$.
Therefore, by Corollary 3.19 one has that ${\Rightarrow}_{\circ}$ is proto-
connexive. However, noting that $\neg c=c{\to}b=b$, $\neg b=b{\to}c=c$, and
$0{\to}\bot=a$, the failure of (4.2) is computed:
$(0{\Rightarrow}_{\circ}1){\Rightarrow}_{\circ}((1{\Rightarrow}_{\circ}b){\Rightarrow}_{\circ}\neg(0{\Rightarrow}_{\circ}c))=0{\Rightarrow}_{\circ}(b{\Rightarrow}_{\circ}\neg\neg
c)=0{\Rightarrow}_{\circ}(b{\Rightarrow}_{\circ}c)=0{\Rightarrow}_{\circ}\bot=a\ngeq
1.$

Furthermore, we see that ${\Rightarrow}_{\wedge}$ has a privileged status
among arrows of the form ${\Rightarrow}_{\wedge}^{\delta}$ which can be
defined over an FL${}_{\mbox{\scriptsize e}}$-algebra by means of an
increasing map $\delta$. In fact, we show below that it is the _only_
operation of this form capable of satisfying connexive theses in the Glivenko
variety relative to Boolean algebras.

######  Proposition 4.4.

Let $\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$ and
$\delta$ be an increasing map on $\mathbf{A}$. If
$(\mathbf{A},{\Rightarrow}_{\wedge}^{\delta})\models\mbox{\eqref{BT}}$ then
$\delta={\diamond}$.

###### Proof.

By (BT) we have
$1\leq(1{\Rightarrow}_{\wedge}^{\delta}x){\Rightarrow}_{\wedge}^{\delta}\neg(1{\Rightarrow}_{\wedge}^{\delta}\neg
x)$, and from the definition of ${\Rightarrow}_{\wedge}^{\delta}$, it follows
that $1\leq\neg(1{\Rightarrow}_{\wedge}^{\delta}\neg
x){\to}\delta(1{\Rightarrow}_{\wedge}^{\delta}x)$. Expanding this, we obtain

$\displaystyle 1$ $\displaystyle\leq\neg[(1{\to}\neg x)\land(\neg
x{\to}\delta{1})]{\to}\delta[(1{\to}x)\land(x{\to}\delta{1})]$
$\displaystyle=\neg[\neg x\land(\neg
x{\to}\delta{1})]{\to}\delta[x\land(x{\to}\delta{1})]$ ($\ast$)

Setting $x\coloneq 1$, ($\ast$ ‣ 4.2) yields $1\leq\neg[\neg 1\land(\neg
1{\to}\delta{1})]{\to}\delta[1\land(1{\to}\delta{1})]=(0{\to}0){\to}\delta{1}$,
hence $0\to 0\leq\delta{1}$. Since
$\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$ and
$0{\to}0$ is its greatest element, it follows that $0{\to}0=\delta{1}$. So
$\delta{1}$ is the greatest element of $\mathbf{A}$, and thus the identity
$y\leq y{\to}\delta{1}$, in particular, holds in $\mathbf{A}$. Using this
fact, ($\ast$ ‣ 4.2) simplifies to $1\leq\neg\neg x{\to}\delta{x}$, i.e.
${\diamond}x\leq\delta{x}$. Moreover, using the fact that $\delta$ is
increasing and $\delta{1}$ is the greatest element, it is easily deduced that
${\Rightarrow}_{\wedge}^{\delta}$ also satisfies (AT) as a consequence
instantiating of $y\coloneq x$ in (BT). As
$\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$, from (AT)
we conclude

$\begin{array}[b]{r c l}1&\leq&\neg(x{\Rightarrow}_{\wedge}^{\delta}\neg x)\\\
&=&\neg((x{\to}\neg x)\land(\neg x{\to}\delta{x}))\\\ &=&\neg(\neg
x\land\delta{x})\\\ &=&\neg(\neg x\cdot\delta{x})\\\
&=&\delta{x}{\to}{\diamond}x.\end{array}\qed$

As a consequence of the above proposition, we have the following

######  Corollary 4.5.

Let $\mathbf{A}$ be an FL${}_{\mbox{\scriptsize e}}$-algebra. Then
$(\mathbf{A},{\Rightarrow}_{\wedge})$ is proto-connexive if and only if it is
the unique binary operation of the form ${\Rightarrow}_{\wedge}^{\delta}$ to
be so.

The example below shows neither Prop. 4.4 nor the corollary above hold for
${\Rightarrow}_{\circ}$.

######  Example 4.6.

Consider the lattice $A=\\{\bot<1<a<0\\}$, where $\cdot$ is commutative and
idempotent with unit $1$, described via $\bot$ being absorbing and $0\cdot
x=0$ for any $x\neq\bot$. It is easily checked that $\cdot$ has a residual
$\to$ and hence $\mathbf{A}=\langle A,\wedge,\vee,\cdot,\to,0,1\rangle$ is an
FL${}_{\mbox{\scriptsize e}}$-algebra. By Proposition 3.11,
${\Rightarrow}_{\circ}^{\delta}$ is proto-connexive in $\mathbf{A}$ for any
map $\delta$, in particular for ${\diamond}$. Moreover, specifying $\delta$ to
be the map defined via $\delta{x}$ is $a$ if $x=a$, and $0$ otherwise, we see
that $\delta$ is increasing (and idempotent) which differs from ${\diamond}$.

A somewhat suggestive interpretation of the above result comes next. If one
wants to expand FL${}_{\mbox{\scriptsize e}}$-algebras with a modal operator
$\delta$ standing for “ being plausible/more credible/likely to be true”, then
a reasonable, minimal condition that $\delta$ should satisfy is being
extensive, since any true statement should be _a fortiori_ plausible/more
credible/likely to be true. Now, if one interprets
${\Rightarrow}_{\wedge}^{\delta}$ as the kind of implication which might
appear in Polya’s heuristic syllogism, then it seems reasonable to assume that
it satisfies connexive theses. In fact, one might notice that any _conjecture_
$A$ _whose truth or falsity is unknown_ should not have, among its
consequences, both a statement $B$ and its negation $\neg B$ on pain of being
_a priori_ false, a contradiction. Moreover, it is arguable that conditionals
involved in heuristic syllogisms have “epistemically possible” antecedents
(cf. p. 1 and [13]). But then, in the light of Proposition 4.4,
$\delta={\diamond}$, namely “being plausible/more credible/likely to be true”
must be _perforce_ expressed by double negation. Since inquiring into the
consequences of the above considerations is beyond the scope of the present
work, we postpone them to future investigations.

### 4.3 On weak connexitivity

In [32], a weaker notion of connexivity is formulated. A logic endowed with a
binary and a unary connective ${\Rightarrow}$ and ${\sim}$, respectively, is
called _weakly connexive_ if it satisfies Aristotle’s theses and the following
two _weak_ versions of Boethius theses:

$\displaystyle A{\Rightarrow}B$
$\displaystyle\vdash{\sim}(A{\Rightarrow}{\sim}B)$ $\displaystyle
A{\Rightarrow}{\sim}B$ $\displaystyle\vdash{\sim}(A{\Rightarrow}B)$

Clearly, in the light of Theorem 2.5, the above inference schemas hold in a
substructural logic w.r.t. (term defined) connectives ${\Rightarrow}$ and
${\sim}$ if and only if the following quasi-identities, which we call
_equational weak Boethius theses_ , hold:

$\displaystyle 1\leq x{\Rightarrow}y$ $\displaystyle\quad\mathsf{implies}\quad
1\leq\neg(x{\Rightarrow}\neg y)$ (BTw) $\displaystyle 1\leq x{\Rightarrow}\neg
y$ $\displaystyle\quad\mathsf{implies}\quad 1\leq\neg(x{\Rightarrow}y)$ (BTw’)

We will use the same naming convention for weak connexivity as in (and in the
same spirit of) Definition 3.1 for logics and (classes of) algebras.

######  Lemma 4.7.

For an FL${}_{\mbox{\scriptsize e}}$-algebra $\mathbf{A}$ and
${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$, the
following hold:

  1. 1.

$(\mathbf{A},{\Rightarrow})\models\eqref{BTw}$ iff
$(\mathbf{A},{\Rightarrow})\models\eqref{BTw'}$.

  2. 2.

If $(\mathbf{A},{\Rightarrow}_{\wedge})\models\eqref{BT}$ [or (BT’)] then
$(\mathbf{A},{\Rightarrow}_{\wedge})\models\eqref{BTw}$ and (BTw’).

  3. 3.

If ${\diamond}\mathbf{A}$ is integral, then
$(\mathbf{A},{\Rightarrow}_{\circ})\models\eqref{BT}$ [or (BT’)] implies
$(\mathbf{A},{\Rightarrow}_{\circ})\models\eqref{BTw}$ and (BTw’).

  4. 4.

If $(\mathbf{A},{\Rightarrow})\models\eqref{BTw}$ then
$(\mathbf{A},{\Rightarrow})\models\eqref{AT}$ and (AT’).

Consequently, ${\Rightarrow}$ is weakly connexive for a class $\mathsf{K}$ of
${\mathsf{FL}_{\mathsf{e}}}$-algebras if and only if at least one of the weak
Boethius theses holds for ${\Rightarrow}$ in $\mathsf{K}$.

###### Proof.

Fix ${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$ and
let $*\in\\{\cdot,\wedge\\}$. First, note that, as a consequence of first
quasi-identity in Prop. 3.5, FL${}_{\mbox{\scriptsize e}}$-algebras satisfy
the following identity,

$x{\Rightarrow}y\leq x{\Rightarrow}\neg\neg y$ (4.2)

Now, for (1), suppose (BTw) holds and let $1\leq x{\Rightarrow}\neg y$. Then
by (BTw), $1\leq\neg(x{\Rightarrow}\neg\neg y)$. But
$\neg(x{\Rightarrow}\neg\neg y)\leq\neg(x{\Rightarrow}y)$ by Eq. (4.2), so
(BTw’) holds. Conversely, suppose (BTw’) holds. If $1\leq x{\Rightarrow}y$,
then again by Eq. (4.2), so too $1\leq x{\Rightarrow}\neg\neg y$. Hence by
(BTw’), $1\leq\neg(x{\Rightarrow}\neg y)$.

For (2) and (3), it is sufficient to verify that
$(\mathbf{A},{\Rightarrow}_{\wedge})\models\eqref{BT}$ implies
$(\mathbf{A},{\Rightarrow}_{\wedge})\models\eqref{BTw}$, in light of (1), Lem.
3.8 [for the case of (2)], and Lem. 3.18 [for the case of (3)]. Indeed,
suppose $1\leq a:=x{\Rightarrow}y$. Let $b:=\neg(x{\Rightarrow}\neg y)$. Then
by (BT), using antitonicity of the left argument for ${\to}$ and Prop. 2.3, we
find

$1\leq a{\Rightarrow}b=(a{\to}b)*(b{\to}{\diamond}a)\leq
b*(b{\to}{\diamond}a)={\diamond}b*{\diamond}(b{\to}{\diamond}a)\leq{\diamond}b\wedge(b{\to}{\diamond}a)\leq{\diamond}b=b,$

so (BTw) holds. This completes (2) and (3).

Lastly, (4) is immediate by (1) and the fact $1\leq x{\Rightarrow}x$ and
$1\leq\neg x{\Rightarrow}\neg x$. ∎

It naturally rises the question if a converse of Lemma 4.7(2) can be proven,
namely if weak connexivity is equivalent to connexivity. Unfortunately, the
next example shows that, even under the assumption of integrality, the former
concept is properly weaker than the latter.

######  Example 4.8.

Consider an arbitrary Heyting algebra $\mathbf{A}=(A,\land,\lor,\to,0,1)$
containing an element $a$ such that $b=\neg a\to a\neq a$, e.g. a three-
elements chain with operations defined in the expected way. Let
$\mathbf{A}^{*}$ be the algebra obtained from $\mathbf{A}$ by setting $0:=a$.
It is easily seen that $\mathbf{A}^{*}$ is still a pointed commutative
residuated lattice with a bottom element $\bot$ satisfying $\land=\cdot$.
Moreover, one has that $1\leq x{\Rightarrow}_{\wedge}y$ implies $x\leq y$ and
$\neg x=\neg y$. So $\neg(x{\Rightarrow}_{\wedge}\neg y)=\neg((x{\to}\neg
y)\land(\neg y{\to}{\diamond}x))=\neg(\neg x\land{\diamond}x)=\neg 0=1.$
Therefore, $\mathbf{A}^{*}$ satisfies (BTw) and, by Lemma 4.7, also (BTw’),
i.e. $\mathbf{A}^{*}$ is weakly connexive. However, one has also e.g.
$\neg(0{\Rightarrow}_{\wedge}\bot)=\neg((0{\to}\bot)\land(\bot{\to}{\diamond}0))=b\neq
a=((0{\to}1)\land
1{\Rightarrow}{\diamond}0)=0{\Rightarrow}_{\wedge}1=0{\Rightarrow}_{\wedge}\neg\bot$.
Therefore, since (P3) fails, by Lemma 3.8 we have that
$(\mathbf{A}^{*},{\Rightarrow}_{\wedge})$ is not proto-connexive.

However, with the assumption of weakening, Lemma 4.7 and Corollary 3.26 yield
the equivalence below. First, for a variety $\mathsf{V}$ of
FL${}_{\mbox{\scriptsize e}}$-algebras and term-definable connective
${\Rightarrow}$, let us denote by $Q_{\\_}w(\mathsf{V})$ the quasi-variety
axiomatized relative to $\mathsf{V}$ by the equational Aristotle’s and weak
Boethius’ theses.

######  Theorem 4.9.

If $\mathsf{K}$ is a class of FL${}_{\mbox{\scriptsize ew}}$-algebras and
${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$, then
${\Rightarrow}$ is weakly connexive for $\mathsf{K}$ if and only if
${\Rightarrow}$ is proto-connexive for $\mathsf{K}$. Moreover, for any
subvariety of $\mathsf{V}\subseteq\mathsf{FL}_{\mathsf{ew}}$,
$Q_{\\_}w(\mathsf{V})$ is a variety, namely the variety
$\mathsf{V}+\eqref{AT}$.

Let ${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$ and
$\mathbf{C}_{\\_}{\Rightarrow}^{wk}$ be the extension of
$\mathbf{FL}_{\\_}\mathbf{ew}$ by the weak Boethius’ theses for
${\Rightarrow}$. By algebraization and the theorem above, this logic is
deductively equivalent to the extension of $\mathbf{FL}_{\\_}\mathbf{ew}$ by
any weak Boethius’ thesis for ${\Rightarrow}$. Note that $\mathbf{IPL}$
satisfies the weak Boethius theses for ${\Rightarrow}$ since
$(\mathsf{HA},{\Rightarrow})\models\eqref{AT}$. By well known facts on
algebraization, Theorem 4.9 yields the following

######  Corollary 4.10.

Let ${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$.
Then, for any substructural logic $\mathbf{L}$ in the interval between
$\mathbf{FL}_{\\_}\mathbf{ew}$ and $\mathbf{IPL}$,
$(\mathbf{L},{\Rightarrow})$ is weakly connexive if and only if
$(\mathbf{L},{\Rightarrow})$ is connexive. Moreover, every logic between
$\mathbf{C}_{\\_}{\Rightarrow}^{wk}$ and $\mathbf{IPL}$ is connexive for
${\Rightarrow}$.

Consequently, $\mathbf{C}_{\\_}{\Rightarrow}^{wk}$ is deductively equivalent
to $\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{ew}}(\mathbf{CPL})$.

### 4.4 On strong connexivity

Lastly, we put our results in the context of what is often referred to in the
literature as strong connexivity [14, 30]. A logic is called strongly
connexive if it satisfies Aristotle’s and Boethius’ theses w.r.t. a non-
symmetric implication and, moreover, satisfies the requirements:

  1. (K1)

In no model, $A{\Rightarrow}\neg A$ is satisfiable (for any $A$), and in no
model, $\neg A{\Rightarrow}A$ is satisfiable (for any $A$);

  2. (K2)

In no model, $A{\Rightarrow}B$ and $A{\Rightarrow}\neg B$ are simultaneously
satisfiable (for any $A$ and $B$).

Accordingly, let us define a logic to be strongly connexive if it is connexive
and satisfies the principles (K1) and (K2). In this way, we also refer to a
connective ${\Rightarrow}$ being strongly connexive for some logic.

Now, if $\mathbf{A}$ is an FL${}_{\mbox{\scriptsize e}}$-algebra in which $0$
is the largest element, both ${\Rightarrow}_{\wedge}$ and
${\Rightarrow}_{\circ}$ are proto-connexive for $\mathbf{A}$ (see Proposition
3.11). Furthermore, if $\mathbf{A}$ is nontrivial, then (K1) and (K2) are
refuted. By the remarks above, we obtain the following

######  Proposition 4.11.

For the logics $\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{e}}(\mathbf{CPL})$
and $\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{ei}}(\mathbf{CPL})$, both
${\Rightarrow}_{\wedge}$ and ${\Rightarrow}_{\circ}$ are connexive but not
strongly connexive.

It is easily shown that if either (K1) or (K2) are refuted in an
FL${}_{\mbox{\scriptsize e}}$-algebra $\mathbf{A}$, for either
${\Rightarrow}_{\wedge}$ or ${\Rightarrow}_{\circ}$, then $\mathbf{A}\models
1\leq 0$.

######  Proposition 4.12.

Let $\mathbf{A}$ be an FL${}_{\mbox{\scriptsize e}}$-algebra.

  1. 1.

If $(\mathbf{A},{\Rightarrow}_{\wedge})$ is proto-connexive, then it refutes
_(K1)_ or _(K2)_ iff $\mathbf{A}\models x\leq 0$;

  2. 2.

If $(\mathbf{A},{\Rightarrow}_{\circ})$ is proto-connexive and it refutes
_(K1)_ or _(K2)_ , then $\mathbf{A}\models 1\leq 0$.

###### Proof.

(1). Concerning the right-to-left direction, note that $\mathbf{A}\models
x\leq 0$ implies that, for any $y\in A$, one has $1\leq y{\to}\neg y$ and
$1\leq\neg y{\to}{\diamond}y$. So we conclude that $1\leq
y{\Rightarrow}_{\wedge}\neg y$ and so ${\Rightarrow}_{\wedge}$ is not strongly
connexive over $A$. Conversely, let us distinguish the following cases:

  1. (i)

There exists $x\in A$ such that $1\leq x{\Rightarrow}_{\wedge}\neg x$. This
means by Lemma 3.7(1), $\neg x=\neg\neg x$. Since
$\mathbf{A}\in\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$, we have
that ${\diamond}\mathbf{A}$ is trivial and so we conclude $x\leq 0$.

  2. (ii)

If there exist $x,y\in A$ such that $1\leq x{\Rightarrow}_{\wedge}y$ and
$1\leq x{\Rightarrow}_{\wedge}\neg y$, then one has, again by Lemma 3.7(1),
$\neg x=\neg y$ and $\neg x=\neg\neg y$, i.e. $\neg y=\neg\neg y$. Reasoning
as in (1), the desired conclusion follows.

As regards (2), let us distinguish the following cases

  1. (i)

There exists $x\in A$ such that $1\leq x{\Rightarrow}_{\circ}\neg x$. Then, of
course, one has $1\leq x{\Rightarrow}\neg x\leq 0$;

  2. (ii)

If there exist $x,y\in A$ such that $1\leq x{\Rightarrow}_{\circ}y$ and $1\leq
x{\Rightarrow}_{\circ}\neg y$, then we have:

$\displaystyle 1\leq$
$\displaystyle(x{\Rightarrow}_{\circ}y)(x{\Rightarrow}_{\circ}\neg y)$
$\displaystyle=$ $\displaystyle(x{\to}y)(y{\to}{\diamond}x)(x{\to}\neg y)(\neg
y{\to}{\diamond}x)$ $\displaystyle\leq$ $\displaystyle(x{\to}\neg x)(\neg
x{\to}{\diamond}x)=(x{\Rightarrow}_{\circ}\neg x).$

By the previous case, our claim follows. ∎

Even more, the above results show that, for ${\Rightarrow}_{\wedge}$, the
failure of strong connexivity is witnessed _exactly_ by those
FL${}_{\mbox{\scriptsize e}}$-algebras having the equational Glivenko property
with respect to the trivial variety (cf. Section 3.2). As there is no non-
trivial algebra in $\mathsf{FL}_{\mathsf{ew}}$ in which $0=1$, Proposition
4.12 yields

######  Theorem 4.13.

Let ${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$.
Then, for any substructural logic $\mathbf{L}$ in the interval between
$\mathbf{FL}_{\\_}\mathbf{ew}$ and $\mathbf{IPL}$,
$(\mathbf{L},{\Rightarrow})$ is strongly connexive if and only if
$(\mathbf{L},{\Rightarrow})$ is connexive. Moreover, every logic between
$\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{ew}}(\mathbf{CPL})$ and
$\mathbf{IPL}$ is strongly connexive for ${\Rightarrow}$.

In light of Theorem 4.15 and the above Theorem 4.13, we are confronted with
the surprising fact:

######  Theorem 4.14.

For any logic between $\mathbf{FL}_{\\_}\mathbf{ew}$ and $\mathbf{IPL}$, the
conditions of strongly connexive, connexive, and weakly connexive are
equivalently satisfied for both ${\Rightarrow}_{\circ}$ and
${\Rightarrow}_{\wedge}$.

As already argued in [14, p. 4], a reasonable task to pursue is expressing
strong connexivity in the object language itself. To this aim, Kapsner
introduces the notion of _superconnexivity_ as the formal alter-ego of strong
connexivity, since it reflects the idea that violations of Aristotle’s and
Boethius’ theses should be regarded as genuine contradictions whose
satisfaction results into triviality. The latter fact can be codified by the
following axiom schemas

  * SA:

$(\varphi\to\neg\varphi){\to}\psi$;

  * SB:

$(\varphi\to\psi)\to((\varphi\to\neg\psi)\to\chi)$.

Unfortunately, SA and SB lead to triviality under a very narrow set of
assumptions (among the others, being closed under substitution). Therefore,
one might wonder whether, at least in some cases, strong connexivity is
conveyed by less demanding inference rules. And the answer is positive. In
fact, in what follows we show that, at least in our framework, strong
connexivity can be expressed by means of a simple (indeed very classical!)
schema: _ex falso quodlibet_.
Let us denote by $\mathsf{PC}^{\circ}$ and
$\mathsf{PC}^{\land}[=\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})]$
the varieties of FL${}_{\mbox{\scriptsize e}}$-algebras in which
${\Rightarrow}_{\circ}$ and ${\Rightarrow}_{\wedge}$ are proto-connexive,
respectively. In view of Proposition 4.12, the _largest_ sub-quasivariety
$\mathsf{M}^{\circ}$ ($\mathsf{M}^{\land}$) of $\mathsf{PC}^{\circ}$
($\mathsf{PC}^{\land}$) in whose $1$-assertional logic ${\Rightarrow}_{\circ}$
(${\Rightarrow}_{\wedge}$) is strongly connexive is axiomatized by the quasi-
identity

$1\leq 0\quad\mathsf{implies}\quad 1\leq x$

(or, equivalently, $1\leq 0\quad\mathsf{implies}\quad x=y$). Therefore, we
have that $\mathbf{L}(\mathsf{M}^{\circ})$ ($\mathbf{L}(\mathsf{M}^{\land})$)
can be obtained from $\vdash_{\\_}{\mathbf{L}(\mathsf{PC}^{\circ})}$
($\vdash_{\\_}{\mathbf{L}(\mathsf{PC}^{\land})}$) by adding the _explosion_
schema

$0\vdash\varphi.$ (EFQ)

Of course, $\mathbf{L}(\mathsf{M}^{\circ})$ ($\mathbf{L}(\mathsf{M}^{\land})$)
is the _least_ $1$-assertional sub-logic of
${\mathbf{L}(\mathsf{PC}^{\circ})}$ (${\mathbf{L}(\mathsf{PC}^{\land})}$) for
which ${\Rightarrow}_{\circ}$ (${\Rightarrow}_{\wedge}$) is strongly
connexive. The above considerations boil down to the following

######  Theorem 4.15.

Let ${\Rightarrow}\in\\{{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}\\}$ and
$\mathbf{L}$ be a substructural logic such $(\mathbf{L},{\Rightarrow})$ is
connexive. Then $(\mathbf{L},{\Rightarrow})$ is strongly connexive iff (EFQ)
holds in $\mathbf{L}$.

##  5 Conclusion and future research

This work have been devoted to a preliminary investigation of term-definable
connexive implications in Substructural Logics and their semantic features.

We have shown that the Glivenko variety of ${\mathsf{FL}_{\mathsf{e}}}$
($\mathsf{FL}_{\mathsf{ei}}$, and $\mathsf{FL}_{\mathsf{ew}}$) relative to
Boolean algebras, provides a suitable semantical environment in which the
connectives ${\Rightarrow}_{\wedge}$ and ${\Rightarrow}_{\circ}$ serve as
bonafide connexive implications, and vice versa. Consequently, any axiomatic
extension of $\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{e}}(\mathbf{CPL})$
provides a full-fledged connexive logic once the
$\\{\land,\lor,\cdot,{\Rightarrow}_{\wedge},0,1\\}$-fragment (and, assuming
the axiom $\vdash\varphi\to\neg\neg 1$, the same holds replacing
${\Rightarrow}_{\wedge}$ by ${\Rightarrow}_{\circ}$) is considered.

We have also argued that these logics provide a suitable framework (resp.,
semantics) for the _logic of plausible inference_ à la G. Polya. As it has
been remarked in Section 3.3, focusing on ${\Rightarrow}_{\wedge}$ does not
result in a valuable loss of generality, since in most cases the connexivity
of ${\Rightarrow}_{\wedge}$ is equivalent to the connexivity of
${\Rightarrow}_{\circ}$ or, at least in the integral case, to the connexivity
of _any_ binary operation in the interval
$[{\Rightarrow}_{\circ},{\Rightarrow}_{\wedge}]$. See also Section 4.2.

Finally, we have shown that, in our framework, strong connexivity is codified
by (EFQ). Therefore, in some cases, it can be expressed in the object language
of a connexive logic by means of an anything but contra-classical inference
rule. We see this contribution as a stepping-stone for further investigations
of both the connective ${\Rightarrow}_{\wedge}$, in particular, and other
(term-defined) implications in residuated lattices and substructural logics in
general. For instance, it seems reasonable that the characterization results
presented in the work may generalize to the non-commutative case; while other
lines to be considered may be the following:

  * •

Obviously, for any sub-variety $\mathsf{V}$ of the variety
$\mathsf{In}{\mathsf{FL}_{\mathsf{e}}}$ of _involutive_ (i.e. satisfying
$\neg\neg x\approx x$) FL${}_{\mbox{\scriptsize e}}$-algebras, neither
$(\mathsf{V},{\Rightarrow}_{\wedge})$ nor $(\mathsf{V},{\Rightarrow}_{\circ})$
are connexive. Therefore, an interesting task would be investigating term-
definable connexive implications in involutive pointed commutative residuated
lattices.

  * •

Due to results obtained so far, ${\Rightarrow}_{\wedge}$ and
${\Rightarrow}_{\circ}$ fall short of behaving, in general, as suitable
connexive implications once an arbitrary FL${}_{\mbox{\scriptsize e}}$-algebra
is considered. However, extending the line of research [11, 22], it is
reasonable to wonder if there are _modal expansions_ of
FL${}_{\mbox{\scriptsize e}}$-algebras for which full-fledged connexive
implications can be term-defined.

  * •

Along the stream of research inaugurated by [28], expansions of residuated
lattices with connexive implications and/or unary operations $\sim$ which
satisfy connexive theses together with $/$ and $\backslash$ would be worth of
investigation.

  * •

As it has been pointed out in Section 4.3, in some cases (e.g. for extensions
of $\mathbf{FL}_{\\_}\mathbf{ew}$), weak connexivity is equivalent to
(strong!) proto-connexivity. However, as witnessed by Example 4.8, the former
concept is properly weaker in the general framework, even for the well
performing connective ${\Rightarrow}_{\wedge}$. Therefore, it naturally rises
the question if a characterization of algebras admitting weakly connexive
implications like the ones investigated in this paper can be provided.

And lastly there is, of course, the naturally risen question of if there is a
cut-free sequent calculus for at least the
$\\{\land,\lor,\cdot,{\Rightarrow}_{\wedge},0,1\\}$-fragment of
$\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{e}}(\mathbf{CPL})$
[$\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{ei}}(\mathbf{CPL})$ or
$\mathbf{G}_{\\_}{\mathbf{FL}_{\\_}\mathbf{ew}}(\mathbf{CPL})$]. The problem
is obviously strictly connected to finding a cut-free sequent calculus whose
equivalent algebraic semantics are
$\mathbf{G}_{{\mathsf{FL}_{\mathsf{e}}}}(\mathsf{BA})$,
$\mathbf{G}_{\mathsf{FL}_{\mathsf{ei}}}(\mathsf{BA})$ and
$\mathbf{G}_{\mathsf{FL}_{\mathsf{ew}}}(\mathsf{BA})$ which, to the best of
our knowledge, is still missing.

### Acknowledgements

The authors gratefully acknowledge Antonio Ledda, Hitoshi Omori, Francesco
Paoli, Adam Přenosil, and Sara Ugolini for the insightful discussions on the
subjects of the present work.

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