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 [Reference Galatos, Jipsen, Kowalski and Ono9].

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 $x,y,z\in R$ ,

(residuation) $$ \begin{align} x\cdot y \leq z \iff x\leq y\to z, \end{align} $$

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., , with the convention that concatenation binds tighter than the remaining connectives in the signature. We also use a bi-implication abbreviation . 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.

An FL ${}_{\scriptsize\mbox{e}}$ -algebra is simply a $0$ -pointed CRL, i.e., a CRL whose signature is expanded by a constant $0$ , and the variety of FL ${}_{\scriptsize\mbox{e}}$ -algebras is denoted by ${\mathsf {FL}_{\mathsf {e}}}$ . An FL ${}_{\scriptsize\mbox{e}}$ -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 ${}_{\scriptsize\mbox{e}}$ -algebras satisfying integrality (weakening) by $\mathsf {FL}_{\mathsf {ei}}$ ( $\mathsf {FL}_{\mathsf {ew}}$ ).

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

(pc) $$ \begin{align} x\wedge \neg x\leq 0. \end{align} $$

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

(spc) $$ \begin{align} \neg x \wedge \neg(x{\to} y) \leq 0. \end{align} $$

Clearly (spc) entails (pc) as an instance; however, the converse need not hold as is witnessed by Figure 1.

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

It is easy to see that, for FL ${}_{\scriptsize\mbox{e}}$ -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 ${}_{\scriptsize\mbox{e}}$ -algebra ${\mathbf A}$ , if $\gamma \colon A\to A$ is a nucleus then is also an FL ${}_{\scriptsize\mbox{e}}$ -algebra, where and [e.g., see Chapter 3.4.11 in [Reference Galatos, Jipsen, Kowalski and Ono9]].

For simplicity, we use the notation 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.

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

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) ${\vdash }\subseteq \wp (\textit {Fm}_{\mathcal {L}})\times \textit {Fm}_{\mathcal {L}}$ . The interested reader is referred to [Reference Font8] for details.

The name “FL ${}_{\scriptsize\mbox{e}}$ ” comes from the fact that FL ${}_{\scriptsize\mbox{e}}$ -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 $\Phi \cup \psi \subseteq \textit {Fm}_{\mathcal {L}}$ of formulas over the language $\mathcal {L}=\{\wedge ,\vee ,\cdot ,{\to },0,1 \}$ ,

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

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

• • $\Phi \vdash _{\mathbf {FL}_{\mathbf {e}}}\psi $ if and only if $\tau (\Phi )\models _{{\mathsf {FL}_{\mathsf {e}}}}\tau (\psi )$ ,

• • ,

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. [Reference Galatos and Ono10] 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,Footnote ² is dually isomorphic to the lattice of subvarieties of ${\mathsf {FL}_{\mathsf {e}}}$ . Indeed, for every class $\mathsf {K}$ of FL ${}_{\scriptsize\mbox{e}}$ -algebras and for every set $\Phi \subseteq {\textit {Fm}}_{\mathcal {L}}$ of formulas over $\mathcal {L}$ , let $\mathbf {L}(\mathsf {K}):= \{\phi \in {\textit {Fm}}_{\mathcal {L}}: \mathsf {K}\ \models\ 1\leq \phi \}$ and $\mathsf {V}(\Phi )$ the subvariety of ${\mathsf {FL}_{\mathsf {e}}}$ axiomatized by $\{ 1\leq \phi :\phi \in \Phi \}$ , we state the well-known algebraization theorem (e.g., Theorem 2.29 in [Reference Galatos, Jipsen, Kowalski and Ono9]):

2.3 Boolean algebras and their Glivenko varieties

ItFootnote ³ is well known that Boolean algebras are term-equivalent to FL ${}_{\scriptsize\mbox{e}}$ -algebras satisfying $x\cdot y \approx x\wedge y$ and $x {\to } y \approx \neg x \vee y$ (e.g., Lemma 8.32 in [Reference Galatos, Jipsen, Kowalski and Ono9]). By $\mathsf {BA}$ we will denote the variety of FL ${}_{\scriptsize\mbox{e}}$ -algebras which are term-equivalent to Boolean algebras. It is well known that Boolean algebras are the equivalent algebraic semantics of classical logic $\mathbf {CPL}$ , i.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.

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 [Reference Galatos, Jipsen, Kowalski and Ono9, Reference Galatos and Ono10]), 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 is equivalently stated: for any FL ${}_{\scriptsize\mbox{e}}$ -term t, $\mathsf {V}(\mathbf {L})\models 1\leq t$ iff $\mathsf {V}(\mathbf {K})\models 1\leq \neg \neg t$ .

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

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

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 ${}_{\scriptsize\mbox{ew}}$ -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 $\mathbf {L}$ .

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.

Proof. (1) implies (2) is obvious. Since ${\diamond } \mathbf{A}$ is involutive by fiat, (2) implies (3) follows by 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 ${\mathbf A}$ :

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

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

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

from which it follows that

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

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

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

Thus we obtain

$$ \begin{align*}\begin{array}[b]{r c l l} 1 &\leq& {\diamond}[{\diamond}1{\to} 1] & \mbox{By } (\star)\\ & \leq &{\diamond}[xy {\to} (x\wedge y)] &\mbox{By the above}\\ & =& xy {\to} {\diamond}(x\wedge y) &\mbox{By } (\star\star)\\ &=& {\diamond}(xy) {\to} {\diamond}(x\wedge y). \end{array}\\[-32pt] \end{align*} $$

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

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

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

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

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

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

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. Thus,

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

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

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

In light of the Algebraization Theorem for substructural logics [cf., Theorem 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 ${}_{\scriptsize\mbox{e}}$ -algebras.

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

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

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 satisfying, to some respect, connexive theses.

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

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

(P1) $$ \begin{align} 1 &\leq {x {\Rightarrow} \neg\neg x}, \end{align} $$

(P2) $$ \begin{align} \neg\neg(x {\Rightarrow} y) &\approx \neg(x {\Rightarrow} \neg y ), \end{align} $$

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 ${}_{\scriptsize\mbox{e}}$ -algebras 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):

$$ \begin{align*} 1\leq (x{\Rightarrow} y){\Rightarrow} \neg\neg(x{\Rightarrow} y) \approx (x{\Rightarrow} y){\Rightarrow} \neg(x{\Rightarrow} \neg y), \end{align*} $$

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.

It is worth noticing that an FL ${}_{\scriptsize\mbox{e}}$ -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 from the valuation $x\mapsto \neg z$ and $y\mapsto 1$ . Therefore, the variety of FL ${}_{\scriptsize\mbox{e}}$ -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 $\mathsf {V}_\emptyset = {\mathsf {FL}_{\mathsf {e}}} + (x\approx y)$ . In fact:

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., [Reference Arieli, Avron and Zamansky1, Reference Marcos17, Reference Wansing, Odintsov, Andreas and Verdée31]).

In the light of the above discussion, it seems reasonable to wonder if there exists other term-definable binary operations on FL ${}_{\scriptsize\mbox{e}}$ -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 [Reference Fazio, Ledda and Paoli6] as follows. Given a map $\delta \colon A\to A$ on ${\mathbf A}$ , we define the two following operations:

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., and .

It is worth noticing that, for ${\Rightarrow }\in \{{\Rightarrow }_{\hspace {-.2em}\circ },{\Rightarrow }_{\hspace {-.2em}\wedge }\}$ , the identity $1\leq x{\Rightarrow } x$ holds in all FL ${}_{\scriptsize\mbox{e}}$ -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 ${\Rightarrow }\in \{{\Rightarrow }_{\hspace {-.2em}\circ },{\Rightarrow }_{\hspace {-.2em}\wedge } \}$ :

$$ \begin{align*} x\leq y ~\mathsf{and}~ {\diamond} x \approx {\diamond} y &\quad\mathsf{implies}\quad z{\Rightarrow} x \leq z{\Rightarrow} y,\\ x\leq y ~\mathsf{and}~ {\diamond} x \approx {\diamond} y &\quad\mathsf{implies}\quad y{\Rightarrow} z \leq x{\Rightarrow} z. \end{align*} $$

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 Proposition 2.1(2,3), while the latter implies $x{\to } {\diamond } z = y{\to } {\diamond } z$ , by Proposition 2.3(6). Since $*\in \{\cdot ,\wedge \}$ is order-preserving in both arguments, we obtain

$$ \begin{align*}\begin{array}[b]{l} z{\Rightarrow} x =(z{\to} x)*(x{\to} {\diamond} z)= (z{\to} x)*(y{\to} {\diamond} z)\leq (z{\to} y)*( y{\to} {\diamond} z) = z{\Rightarrow} y, \\ y{\Rightarrow} z =(y{\to} z)*(z{\to} {\diamond} y)=(y{\to} z)*(z{\to} {\diamond} x) \leq (x{\to} z)*(z{\to} {\diamond} x) =x{\Rightarrow} z. \end{array}\\[-35pt]\end{align*} $$

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

In fact, even more can be said for the connective ${\Rightarrow }_{\hspace {-.2em}\wedge }$ , as we see below.

Furthermore, we obtain a (stronger) converse of Lemma 3.6 for ${\Rightarrow }_{\hspace {-.2em}\wedge }$ .

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

3.3 Characterizing connexivity for ${\Rightarrow }_{\hspace {-.2em}\wedge }$ and ${\Rightarrow }_{\hspace {-.2em}\circ }$

We will now investigate the varieties of FL ${}_{\scriptsize\mbox{e}}$ -algebras satisfying the connexive laws for both ${\Rightarrow }_{\hspace {-.2em}\wedge }$ and ${\Rightarrow }_{\hspace {-.2em}\circ }$ , and their relationship to the Glivenko variety relative to Boolean Algebras, which is, at least implicitly, hinted at in [Reference Fazio, Ledda and Paoli6]. We provide a characterization for ${\Rightarrow }_{\hspace {-.2em}\wedge }$ in the general case, and a restricted one for ${\Rightarrow }_{\hspace {-.2em}\circ }$ . To that aim, let us start with the following technical lemmas.

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).Footnote ⁴ Furthermore, note that ${\diamond } 1$ being the greatest element implies ${\diamond }({\diamond } x{\Rightarrow }_{\hspace {-.2em}\wedge } x) \approx {\diamond }({\diamond } x {\to } x)$ .

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

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

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

$$ \begin{align*}x{\Rightarrow}_{\hspace{-.2em}\circ} \neg y=(x{\to} \neg y)\cdot (\neg y{\to} {\diamond} x)\leq(x{\to} \neg y)\wedge (\neg y{\to} {\diamond} x)=x{\Rightarrow}_{\scriptscriptstyle\wedge} \neg y.\end{align*} $$

For the other identity, observe that ${\diamond } x{\Rightarrow }_{\hspace {-.2em}\circ } x \leq ({\diamond } x {\to } x)\cdot {\diamond } 1\leq {\diamond }({\diamond } x {\to } x)={\diamond }({\diamond } x {\Rightarrow }_{\hspace {-.2em}\wedge } x)$ .

For (3), let ${\Rightarrow }\in \{{\Rightarrow }_{\hspace {-.2em}\circ },{\Rightarrow }_{\hspace {-.2em}\wedge }\}$ . Towards (a), set . 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

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

$$ \begin{align*}\begin{array}[b]{rcll} 1 &\leq& \neg({\diamond} x{\Rightarrow} \neg x) &\mbox{By (AT')}\\ &=&{\diamond}({\diamond} x{\Rightarrow} x) &\mbox{By (P2)} \\ &\leq& {\diamond}({\diamond} x {\Rightarrow}_{\scriptscriptstyle\wedge} x) & \mbox{By (2)}\\ &=& {\diamond}({\diamond} x {\to} x). \end{array}\\[-33pt] \end{align*} $$

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

$$ \begin{align*}\begin{array}{r c l l} x{\to} {\diamond}1&\geq& (x{\to} {\diamond}1) \wedge {\diamond} x \\ &=& (x{\to} {\diamond}1) \wedge ({\diamond} 1 {\to} {\diamond} x) \\ &=& x {\Rightarrow}_{\scriptscriptstyle\wedge} {\diamond} 1 \end{array} \end{align*} $$

$$ \begin{align*}\begin{array}{r c l l} &=&\neg (x{\Rightarrow}_{\scriptscriptstyle\wedge} 0)& \mbox{By (P3) since } {\diamond} 1 = \neg 0\\ &\geq & \neg (x{\to} 0) & \mbox{By def. of } {\Rightarrow}_{\scriptscriptstyle\wedge}\\ &\geq& x &\mbox{By Prop.~2.3(1)}, \end{array} \end{align*} $$

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.

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

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

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

$$ \begin{align*}(x{\to} \neg y)\wedge (\neg y {\to} \neg\neg x) \approx (x{\to} \neg y)\wedge (\neg y {\to} x) \approx \neg[(x{\to} y)\wedge (y{\to} x)],\end{align*} $$

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 $({\mathbf A},{\Rightarrow }_{\hspace {-.2em}\wedge })$ 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.

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.

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

(3.1) $$ \begin{align} \begin{array}{r c l l} 1 &\leq& a{\Rightarrow} \neg b \\ &\leq& {\diamond}[a{\Rightarrow} \neg b] & \mbox{By (ii)} \\ &\leq & {\diamond}[a{\Rightarrow}_{\scriptscriptstyle\wedge} \neg b] & \mbox{By assumption on } {\Rightarrow} \\ &=& a {\Rightarrow}_{\scriptscriptstyle\wedge} \neg b & \mbox{By Lem. 3.7(2)}. \end{array} \end{align} $$

Hence, by Lemma 3.7(1), $\neg \neg a = \neg b$ . So $({\mathbf A}, \Rightarrow )\ \models $ (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 $a:= x{\Rightarrow }_{\hspace {-.2em}\wedge } y$ and $b:=x{\Rightarrow }_{\hspace {-.2em}\wedge } \neg y$ ) we conclude $({\mathbf A},{\Rightarrow }_{\hspace {-.2em}\wedge })\ \models $ (P2), and the result follows from Lemma 3.8.

Now suppose (3) holds. On the one hand, by Lemma 3.8 $({\mathbf A},{\Rightarrow }_{\hspace {-.2em}\wedge })\ \models\ \mathrm {(P3)}$ , so ${\diamond }(x{\Rightarrow }_{\hspace {-.2em}\wedge } y)=x{\Rightarrow }_{\hspace {-.2em}\wedge } {\diamond } y$ . On the other hand, ${\mathbf A} \in \mathbf {G}_{{\mathsf {FL}_{\mathsf {e}}}}(\mathsf {BA})$ by Theorem 3.17, so ${\diamond }(x{\Rightarrow }_{\hspace {-.2em}\circ } y)= {\diamond }(x{\Rightarrow }_{\hspace {-.2em}\wedge } y)$ by Theorem 2.8(1a). Thus ${\diamond }(x{\Rightarrow } y)={\diamond }(x{\Rightarrow }_{\hspace {-.2em}\wedge } y)$ follows the assumption that ${\diamond }(x{\Rightarrow }_{\hspace {-.2em}\circ } y)\leq {\diamond }(x{\Rightarrow } y)\leq x{\Rightarrow }_{\hspace {-.2em}\wedge } {\diamond } y$ . Hence $({\mathbf A},{\Rightarrow })\ \models$ (P2) follows from the fact $({\mathbf A},{\Rightarrow }_{\hspace {-.2em}\wedge })$ does. Since $({\mathbf A},{\Rightarrow })\ \models$ (P1) by assumption, $({\mathbf A},{\Rightarrow })$ is proto-connexive by Proposition 3.2. This completes the equivalences.

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

Having established the proto-connexitivity of ${\Rightarrow }_{\hspace {-.2em}\wedge }$ and ${\Rightarrow }_{\hspace {-.2em}\circ }$ (and other related operations) in the Glivenko variety relative to Boolean algebras, we now investigate the principle of non-symmetry (NS).

Proof. Of course, if ${\mathbf A}$ is a Boolean algebra, then ${\Rightarrow }_{\hspace {-.2em}\circ } ={\leftrightarrow }={\Rightarrow }_{\hspace {-.2em}\wedge }$ 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 ${\Rightarrow }\in \{{\Rightarrow }_{\hspace {-.2em}\circ },{\Rightarrow }_{\hspace {-.2em}\wedge }\}$

$$ \begin{align*}{\diamond} 1 = (1\to{\diamond} 1)*({\diamond} 1 \to{\diamond} 1)=1{{\kern-1.5pt}\Rightarrow{\kern-1.5pt}}{\diamond} 1 {\kern-1pt}={\kern-1pt}{\diamond} 1{{\kern-1.5pt}\Rightarrow{\kern-1.5pt}} 1{\kern-1pt}={\kern-1pt} ({\diamond} 1{\kern-1pt}\to{\kern-1pt} 1)*(1{\kern-1pt}\to{\kern-1pt}{\diamond} 1)=({\diamond} 1{\kern-1pt}\to{\kern-1pt} 1)*{\diamond} 1.\end{align*} $$

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 $(\diamond 1\to 1)*\diamond 1 = \diamond1 \to 1$ , 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 Theorem 3.17, Corollary 3.19, Remark 2.9, and the lemma above:

3.4 Generalizations in the integral case

We now consider the special case when dealing with integral (and $0$ -bounded) FL ${}_{\scriptsize\mbox{e}}$ -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 $[{\Rightarrow }_{\hspace {-.2em}\circ },{\Rightarrow }_{\hspace {-.2em}\wedge }]$ 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-connexivity for subvarieties of $\mathsf {FL}_{\mathsf {ew}}$ .

Since integral FL ${}_{\scriptsize\mbox{e}}$ -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

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

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 $x\in A$ ,

$$ \begin{align*} 1{\Rightarrow}_{\hspace{-.2em}\circ} x = (1{\to} x)\cdot (x{\to} \delta{1})=x\cdot 1=x=x\wedge 1=(1{\to} x)\wedge (x{\to} \delta{1}) = 1{\Rightarrow}_{\scriptscriptstyle\wedge} x,\\ x{\Rightarrow}_{\hspace{-.2em}\circ} 1 = (x{\to} 1)\cdot( 1{\to} {\delta{x}}) = 1\cdot {\delta{x}} = {\delta{x}}= 1\cdot {\delta{x}}=(x{\to} 1)\wedge( 1{\to} {\delta{x}}) = x{\Rightarrow}_{\scriptscriptstyle\wedge} 1. \end{align*} $$

For (2), we follow essentially the same argument as Lemma 3.12(3a). Suppose $({\mathbf A},{\Rightarrow })$ satisfies (AT), and let $x\in A$ . Set 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

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

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 }_{\hspace {-.2em}\circ } \neg c \leq {\diamond } c {\Rightarrow } \neg c \leq 0 $ .

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

$$ \begin{align*} 1 = (1{\Rightarrow} x) {\Rightarrow} \neg (1 {\Rightarrow} \neg x) = x {\Rightarrow} \neg\neg x \leq x{\Rightarrow}_{\scriptscriptstyle\wedge}^{\delta} {\diamond} x =(x\to {\diamond} x) \wedge ({\diamond} x \to {\delta{x}}) = {\diamond} x {\to} {\delta{x}}, \end{align*} $$

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

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

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

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

Hence $\delta = {\diamond }$ , and so ${\Rightarrow }\in [{\Rightarrow }_{\hspace {-.2em}\circ },{\Rightarrow }_{\hspace {-.2em}\wedge }]$ . Since $x{\Rightarrow }_{\hspace {-.2em}\circ } y \leq x {\Rightarrow } y$ , ${\Rightarrow } $ clearly satisfies (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.

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

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 ${\Rightarrow }_{\hspace {-.2em}\wedge }$ and ${\Rightarrow }_{\hspace {-.2em}\circ }$ “on the map.” First, as a consequence of Theorem 3.17, Lemma 3.18, and Corollary 3.22, we obtain the following theorem.

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

Interestingly enough, in the framework of $\mathbf {FL}_{\mathbf {ew}}$ , the above result “does justice” to the idea that “Aristotles Thesis is the cornerstone of the logics belonging to the family of so-called connexive logics” (cf. [Reference Pizzi24]).

Since the variety $\mathsf {HA}$ of Heyting algebras is pseudo-complemented, we recover the following result from [Reference Fazio, Ledda and Paoli6] as a particular instance of Theorem 3.29.

As the connectives $\cdot $ and $\wedge $ coincide in Heyting algebras, so too do the operations ${\Rightarrow }_{\hspace {-.2em}\wedge }$ and ${\Rightarrow }_{\hspace {-.2em}\circ }$ . We therefore conclude with the following corollary.

4.1 A focus on ${\Rightarrow }_{\hspace {-.2em}\wedge }$

In the second volume of his famous Mathematics and plausible reasoning [Reference Polya26], 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” [Reference Polya26, 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

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 (implicitly understood as more credible than before). In other words, we have applied the following fundamental inductive pattern (in brief “inductive pattern” [Reference Polya26, vol. 2, p. 4]) or heuristic syllogism:

$$ \begin{align*} \frac{A\ \textrm{implies}\ B \quad B}{A\ \textrm{is}\ \textrm{more}\ \textrm{credible}}. \end{align*} $$

As stated in [Reference Polya25, 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 more likely to be true than before”) 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:Footnote ⁵

• $\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 [Reference Polya26, 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 denote the absolutely free algebra over $\mathcal {L}$ generated by a denumerable set of variables by $\textit {Fm}_{\mathcal {L}}$ . The formula $\blacklozenge \varphi $ will be read as “ $\varphi $ is more credible/plausible/likely to be true than before.” In order to include $\blacklozenge $ 1- $\blacklozenge $ 3 in our formal system, we consider the expansion $\vdash _{\mathbf {FL}_{\mathbf {e}}^{\blacklozenge } }\subseteq \wp ({\textit {Fm}}_{\mathcal {L}})\times {\textit {Fm}}_{\mathcal {L}}$ of $\vdash _{\mathbf {FL}_{\mathbf {e}}}$ by the following axioms and inference schemas:

• $\blacklozenge $ 1: $\vdash _{\mathbf {FL}_{\mathbf {e}}^{\blacklozenge } } \blacklozenge \neg \varphi \to \neg \blacklozenge \varphi $ ;

• $\blacklozenge $ 2: $\vdash _{\mathbf {FL}_{\mathbf {e}}^{\blacklozenge } } \neg \neg \varphi \to \blacklozenge \varphi $ ;

• $\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., [Reference Font8, 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 ${}_{\scriptsize\mbox{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 ${}_{\scriptsize\mbox{e}}$ -algebra, and the following identities hold:

1. A1: $\blacklozenge \neg x\leq \neg \blacklozenge x$ ;

2. A2: $\neg \neg x\leq \blacklozenge x.$

As a consequence, we have the following.

Note that, in particular, any FL ${}_{\scriptsize\mbox{e}}^\blacklozenge $ -algebra satisfies the identity $\blacklozenge \blacklozenge x\approx \blacklozenge x.$ It is arguable that this last fact seems to have a rather intuitive flavor, since it just establishes that the statement “The proposition ‘A is more credible than before’ is more credible than before” is equivalent to “A is more credible than before.”

Now, we observe that, in general, the following does not hold:

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

i.e., $\to $ does not satisfy the inductive pattern. However, it is easy to see that ${\Rightarrow }_{\hspace {-.2em}\wedge }$ is the weakest term-definable implication-like connective $\leadsto $ satisfying, for any formulas $\varphi ,\psi \in {\textit {Fm}}_{\mathcal {L}}$ , 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 $\varphi ,\psi \in {\textit {Fm}}_{\mathcal {L}}$ :

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

In light of the above discussion, it can be argued that, under Polya’s desiderata [Reference Polya25], ${\Rightarrow }_{\hspace {-.2em}\wedge }$ 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 ${}_{\scriptsize\mbox{e}}$ -algebra ${\mathbf A}$ it holds that

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

$x{\Rightarrow }_{\hspace {-.2em}\wedge } y$ 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

(4.2) $$ \begin{align} x\leq{\diamond} 1 \end{align} $$

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 ${}_{\scriptsize\mbox{e}}$ -algebras satisfying (4.2), then Corollary 3.19 states a full equivalence between the connexivity of ${\Rightarrow }_{\hspace {-.2em}\wedge }$ and ${\Rightarrow }_{\hspace {-.2em}\circ }$ (cf. page 18).

4.2 On the strength of ${\Rightarrow }_{\hspace {-.2em}\wedge }$

Besides having particularly fair motivations, ${\Rightarrow }_{\hspace {-.2em}\wedge }$ plays a special role for our discussion since it enjoys interesting properties which are not shared by its product-variant ${\Rightarrow }_{\hspace {-.2em}\circ }$ . Indeed, the connexivity of ${\Rightarrow }_{\hspace {-.2em}\wedge }$ can be formulated in a stronger form.

Proposition 4.2. Let ${\mathbf A}$ be an FL ${}_{\scriptsize\mbox{e}}$ -algebra. Then $({\mathbf A},{\Rightarrow }_{\hspace {-.2em}\wedge })$ satisfies (BT) if and only if it satisfies

(BT^*) $$\begin{align} 1\leq(x{\Rightarrow} y){\Rightarrow}[(y{\Rightarrow} z){\Rightarrow}\neg(x{\Rightarrow}\neg z)]. \end{align}$$

We show the above proposition does not hold for ${\Rightarrow }_{\hspace {-.2em}\circ }$ with the following example.

Furthermore, we see that ${\Rightarrow }_{\hspace {-.2em}\wedge }$ has a privileged status among arrows of the form ${\Rightarrow }_{\wedge }^{\delta }$ which can be defined over an FL ${}_{\scriptsize\mbox{e}}$ -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.

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

(*) $$ \begin{align} 1&\leq \neg[(1{\to}\neg x)\land(\neg x{\to}{\delta{1}})]{\to}\delta[(1{\to} x)\land(x{\to}{\delta{1}})]\notag\\ &=\neg[\neg x\land(\neg x{\to}{\delta{1}})]{\to}\delta[x\land(x{\to}{\delta{1}})]. \end{align} $$

Setting , (*) 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, (*) 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 in (BT). As ${\mathbf A} \in \mathbf {G}_{{\mathsf {FL}_{\mathsf {e}}}}(\mathsf {BA})$ , from (AT) we conclude

$$ \begin{align*} \begin{array}[b]{r c l} 1&\leq& \neg(x{\Rightarrow}_{\scriptscriptstyle\wedge}^{\delta}\neg x)\\ &=& \neg((x{\to}\neg x)\land(\neg x{\to}{\delta{x}}))\\ &=&\neg(\neg x\land{\delta{x}})\\ &=&\neg(\neg x\cdot{\delta{x}})\\ &=&{\delta{x}}{\to}{\diamond} x. \end{array}\\[-34pt] \end{align*} $$

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

The example below shows neither Proposition 4.4 nor the corollary above hold for ${\Rightarrow }_{\hspace {-.2em}\circ }$ .

A somewhat suggestive interpretation of the above result comes next. If one wants to expand FL ${}_{\scriptsize\mbox{e}}$ -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 increasing, 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. page and [Reference Kapsner14]). 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 [Reference Wansing and Unterhuber33], 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:

$$ \begin{align*} A{\Rightarrow} B &\vdash {\sim}(A{\Rightarrow} {\sim} B),\\ A{\Rightarrow} {\sim} B &\vdash {\sim}(A{\Rightarrow} B). \end{align*} $$

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:

(BTw) $$ \begin{align} \ \ 1\leq x{\Rightarrow} y &\quad\ \ \,\mathsf{implies}\quad 1\leq \neg (x{\Rightarrow} \neg y), \end{align} $$

(BTw') $$ \begin{align} 1\leq x{\Rightarrow} \neg y &\quad\mathsf{implies}\quad1\leq \neg (x{\Rightarrow} y). \end{align} $$

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.

Proof. Fix ${\Rightarrow }\in \{{\Rightarrow }_{\hspace {-.2em}\circ },{\Rightarrow }_{\hspace {-.2em}\wedge } \}$ and let $*\in \{\cdot ,\wedge \}$ . First, note that, as a consequence of the first quasi-identity in Proposition 3.5, FL ${}_{\scriptsize\mbox{e}}$ -algebras satisfy the following identity:

(4.3) $$ \begin{align} x{\Rightarrow} y \leq x{\Rightarrow} \neg\neg y. \end{align} $$

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 Equation (4.3), so (BTw') holds. Conversely, suppose (BTw') holds. If $1\leq x{\Rightarrow } y$ , then again by Equation (4.3), 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 }_{\hspace {-.2em}\wedge }) \ \models $ (BT) implies $({\mathbf A},{\Rightarrow }_{\hspace {-.2em}\wedge })\ \models$ (BTw), in light of (1), Lemma 3.8 [for the case of (2)], and Lemma 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

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

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

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

The question of whether a converse of Lemma 4.7(2) can be proven naturally rises, 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.

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 ${}_{\scriptsize\mbox{e}}$ -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.

Let ${\Rightarrow }\in \{{\Rightarrow }_{\hspace {-.2em}\circ },{\Rightarrow }_{\hspace {-.2em}\wedge } \}$ 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 $(\mathsf {HA},{\Rightarrow })\ \models$ (AT). By well-known facts on algebraization, Theorem 4.9 yields the following.

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 [Reference Kapsner13, Reference Wansing and Omori32]. 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 (nontrivial) 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 (nontrivial) 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 ${}_{\scriptsize\mbox{e}}$ -algebra in which $0$ is the largest element, both ${\Rightarrow }_{\hspace {-.2em}\wedge }$ and ${\Rightarrow }_{\hspace {-.2em}\circ }$ 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.

It is easily shown that if either (K1) or (K2) is refuted in an FL ${}_{\scriptsize\mbox{e}}$ -algebra ${\mathbf A}$ , for either ${\Rightarrow }_{\hspace {-.2em}\wedge }$ or ${\Rightarrow }_{\hspace {-.2em}\circ }$ , then ${\mathbf A}\ \models\ 1\leq 0$ .

Even more, the above results show that, for ${\Rightarrow }_{\hspace {-.2em}\wedge }$ , the failure of strong connexivity is witnessed exactly by those FL ${}_{\scriptsize\mbox{e}}$ -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 the following result

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

As already argued in [Reference Kapsner13, 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:

1. SA: $(\varphi \to \neg \varphi ){\to }\psi $ .

2. 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 ${}_{\scriptsize\mbox{e}}$ -algebras in which ${\Rightarrow }_{\hspace {-.2em}\circ }$ and ${\Rightarrow }_{\hspace {-.2em}\wedge }$ 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 ${\Rightarrow }_{\hspace {-.2em}\circ }$ ( ${\Rightarrow }_{\hspace {-.2em}\wedge }$ ) is strongly connexive is axiomatized by the quasi-identity