2.1 Algebras, languages, and logics

A propositional signature is a family $\Sigma \,{:=}\, \{\Sigma _k\}_{k \in \omega }$ , where each $\Sigma _k$ is a collection of $k$ -ary connectives. A $\Sigma$ -multialgebra is a structure $\mathbf{A} \,{:=}\, \langle A, \cdot ^{\mathbf{A}} \rangle$ , where $A$ is a non-empty set called the carrier of $\mathbf{A}$ and, for each $\unicode{x00A9} \in \Sigma _k$ , the multioperation $\unicode{x00A9}^{\mathbf{A}} : A^k \to \wp (A)$ is the interpretation of $\unicode{x00A9}$ in $\mathbf{A}$ . We say that $\mathbf{A}$ is a $\Sigma$ -algebra (or deterministic $\Sigma$ -multialgebra) when $\unicode{x00A9}^{\mathbf{A}}(a_1,\ldots, a_k)$ is a singleton for every $a_1,\ldots, a_k \in A$ , $\unicode{x00A9} \in \Sigma _k$ and $k \in \omega$ . If $\unicode{x00A9}^{\mathbf{A}}(a_1,\ldots, a_k) \neq \varnothing$ for every $a_1,\ldots, a_k \in A$ , $\unicode{x00A9} \in \Sigma _k$ and $k \in \omega$ , we say that $\mathbf{A}$ is total. Note that the notion of $\Sigma$ -algebra matches the usual notion of algebra in Universal Algebra (cf. Sankappanavar Reference Sankappanavar1987). Given $\Sigma ^\prime \subseteq \Sigma$ (i.e., $\Sigma ^\prime _k \subseteq \Sigma _k$ for all $k \in \omega$ ), the $\Sigma '$ -reduct of a $\Sigma$ -multialgebra $\mathbf{A}$ is the $\Sigma '$ -multialgebra over the same carrier of $\mathbf{A}$ that agrees with $\mathbf{A}$ on the interpretation of the connectives in $\Sigma '$ .

A $\Sigma$ -homomorphism between $\Sigma$ -multialgebras $\mathbf{A}$ and $\mathbf{B}$ is a mapping $h : A \to B$ such that $h(\unicode{x00A9}^{\mathbf{A}}(a_1,\ldots, a_k)) \in \unicode{x00A9}^{\mathbf{B}}(h(a_1),\ldots, h(a_k))$ for all $a_1,\ldots, a_k \in A$ , $\unicode{x00A9} \in \Sigma _k$ and $k \in \omega$ . Note that for $\Sigma$ -algebras this definition coincides with the usual notion of homomorphism. The collection of all $\Sigma$ -homomorphisms between two $\Sigma$ -multialgebras $\mathbf{A}$ and $\mathbf{B}$ is denoted by $\textsf{Hom}(\mathbf{A},\mathbf{B})$ .

Given a denumerable set $P \supseteq \{p, q, r, s, x, y\}$ of propositional variables, the absolutely free $\Sigma$ -algebra freely generated by $P$ , or simply the language over $\Sigma$ (generated by $P$ ), is denoted by $\mathbf{L}_{\Sigma }(P)$ , and its members are called $\Sigma$ -formulas. The collection of all subformulas of a formula $\varphi$ is denoted by $\textsf{sub}(\varphi )$ and we let $\textsf{sub}(\Phi ) \,{:=}\, \bigcup _{\varphi \in \Phi }\textsf{sub}(\varphi )$ , for all $\Phi \subseteq L_{\Sigma }(P)$ . Similarly, the collection of all propositional variables occurring in $\varphi \in L_{\Sigma }(P)$ is denoted by $\textsf{props}(\varphi )$ , and we let $\textsf{props}(\Phi ) \,{:=}\, \bigcup _{\varphi \in \Phi }\textsf{props}(\varphi )$ , for all $\Phi \subseteq L_{\Sigma }(P)$ . The elements of $\textsf{Hom}(\mathbf{L}_{\Sigma }(P), \mathbf{A})$ will sometimes be referred to as valuations on $\mathbf{A}$ . When $\mathbf{A}$ is $\mathbf{L}_{\Sigma }(P)$ , valuations are endomorphisms on $\mathbf{L}_{\Sigma }(P)$ and are usually dubbed substitutions. The set of all substitutions is denoted by $\textsf{End}(\mathbf{L}_{\Sigma }(P))$ . Given $h,h' \in \textsf{Hom}(\mathbf{L}_{\Sigma }(P), \mathbf{A})$ , we shall say that $h'$ agrees with $h$ on $\Phi \subseteq L_{\Sigma }(P)$ provided that $h'(\varphi )=h(\varphi )$ for all $\varphi \in \Phi$ . In case $p_1,\ldots, p_k$ are the only propositional variables occurring in $\varphi \in L_{\Sigma }(P)$ , we say that $\varphi$ is $k$ -ary and denote by $\varphi ^{\mathbf{A}}$ the $k$ -ary multioperation on $A$ such that $\varphi ^{\mathbf{A}}(a_1,\ldots, a_k) \,{:=}\, \{ h(\varphi ) : h \in \textsf{Hom}(\mathbf{L}_{\Sigma }(P), \mathbf{A}) \text{ with } h(p_i) = a_i \text{ for each } 1 \leq i \leq k \}$ , for all $a_1,\ldots, a_k \in A$ . Also, if $\psi _1,\ldots, \psi _k \in L_{\Sigma }(P)$ , we let $\varphi (\psi _1,\ldots, \psi _k)$ denote the formula resulting from substituting each $p_i$ in $\varphi$ by $\psi _i$ , for all $1 \leq i \leq k$ . For a set $\Theta$ of formulas $\varphi (p_1,\ldots, p_k)$ , we let $\Theta (\psi _1,\ldots, \psi _k) \,{:=}\, \{ \varphi (\psi _1,\ldots, \psi _k) : \varphi \in \Theta \}$ .

A $\Sigma$ -equation is a pair $(\varphi, \psi )$ of $\Sigma$ -formulas that we denote by $\varphi \approx \psi$ , and a $\Sigma$ -multialgebra $\mathbf{A}$ is said to satisfy $\varphi \approx \psi$ if $h(\varphi )=h(\psi )$ for every $h \in \textsf{Hom}(\mathbf{L}_{\Sigma }(P), \mathbf{A})$ . For any given collection of $\Sigma$ -equations, the class of all $\Sigma$ -algebras that satisfy such equations is called a $\Sigma$ -variety. An equation is said to be valid in a given variety if it is satisfied by each algebra in the variety. The variety generated by a class $\textsf{K}$ of $\Sigma$ -algebras, denoted by $\mathbb{V}(\textsf{K})$ , is the closure of $\textsf{K}$ under homomorphic images, subalgebras and direct products. We denote the latter operations, respectively, by $\mathbb{H}$ , $\mathbb{S}$ and $\mathbb{P}$ . We write $\textsf{Cng}\mathbf{A}$ to refer to the collection of all congruence relations on $\mathbf{A}$ , which is known to form a complete lattice under inclusion.

In what follows, we assume the reader is familiar with basic notations and terminology of lattice theory (Davey and Priestley Reference Davey and Priestley2002). We denote by $\Sigma ^{\textsf{bL}}$ the signature containing but two binary connectives, $\land$ and $\lor$ , and two nullary connectives $\top$ and $\bot$ , and by $\Sigma ^{\textsf{DM}}$ the extension of the latter signature by the addition of a unary connective $\sim$ . Moreover, by adding the unary connective $\nabla$ to $\Sigma ^{\textsf{DM}}$ we obtain a signature we shall call $\Sigma ^{\textsf{IS}}$ , and by adding instead the unary connective $\circ$ we obtain a signature we shall call $\Sigma ^{\textsf{PP}}$ . We provide below the definitions and some examples of De Morgan algebras and of involutive Stone algebras, in order to illustrate some of the notions introduced above.

Definition 1. Given a $\Sigma ^{\textsf{DM}}$ -algebra whose $\Sigma ^{\textsf{bL}}$ -reduct is a bounded distributive lattice, we say that it constitutes a De Morgan algebra if it satisfies the equations:

\begin{align*} \mathbf{(DM1)} \; {\sim }{\sim } x \approx x \qquad \mathbf{(DM2)} \; {\sim } (x \land y) \approx{\sim } x \lor{\sim } y \end{align*}

Definition 2. Given a $\Sigma ^{\textsf{IS}}$ -algebra whose $\Sigma ^{\textsf{DM}}$ -reduct is a De Morgan algebra, we say that it constitutes an involutive Stone algebra (IS-algebra) if it satisfies the equations:

\begin{align*} \textbf{(IS1)} \; \nabla \bot \approx \bot \quad \textbf{(IS2)} \; x \land \nabla x \approx x\quad \textbf{(IS3)} \; \nabla (x \land y) \approx \nabla x \land \nabla y\quad \textbf{(IS4)} \; {\sim }\nabla x \land \nabla x \approx \bot \end{align*}

Example 2. Let $\mathcal{V}_6 \,{:=}\, \mathcal{V}_4 \cup \{{\hat{\mathbf{f}}},{\hat{\mathbf{t}}}\}$ and let $\mathbf{IS}_6 \,{:=}\, \langle \mathcal{V}_6, \cdot ^{\mathbf{IS}_6} \rangle$ be the $\Sigma ^{\textsf{IS}}$ -algebra whose lattice structure is depicted in Fig. 1(b) and interprets $\sim$ and $\nabla$ as per the following:

\begin{equation*} {\sim }^{\mathbf {IS}_6} a \,{:=}\, \begin {cases} {\sim }^{\mathbf {DM}_4} a & a \in \mathcal {V}_4\\ {\hat {\mathbf {f}}} & a = {\hat {\mathbf {t}}}\\ {\hat {\mathbf {t}}} & a = {\hat {\mathbf {f}}}\\ \end {cases}\qquad \nabla ^{\mathbf {IS}_6} a \,{:=}\, \begin {cases} {\hat {\mathbf {t}}} & a \in \mathcal {V}_6\setminus \{{\hat {\mathbf {f}}}\}\\ {\hat {\mathbf {f}}} & a = {\hat {\mathbf {f}}}\\ \end {cases} \end{equation*}

The subalgebras of $\mathbf{IS}_6$ exhibited in Fig. 1(b) constitute the only subdirectly irreducible IS-algebras (Cignoli and Sagastume Reference Cignoli and Sagastume1983).

The notion of residuation in $\Sigma$ -algebras will be essential for us here, as it is tightly connected to the intuitionistic implication. Let $\textsf{K}$ be a class of $\Sigma$ -algebras, with $\Sigma \supseteq \Sigma ^{\textsf{bL}}$ , whose $\{ \land, \top \}$ -reducts are meet-semilattices with a greatest element assigned to $\top$ . Given $\mathbf{A} \in \textsf{K}$ , we say that $\unicode{x00A9}$ is the residuum of $\land$ in $\mathbf{A}$ provided that $a \,\land ^{\mathbf{A}}\, b \leq c$ if, and only if, $a \leq b \,\unicode{x00A9}^{\mathbf{A}}\, c$ for all $a,b,c \in A$ . It is well-known that in classical logic and in intuitionistic logic the implication plays the role of residuum of conjunction. We will also consider the notion of pseudo-complement of $a \in A$ : this will be defined as the greatest element $\neg a \in A$ such that $a \land ^{\mathbf{A}} \neg a = \bot ^{\mathbf{A}}$ . When every element of $\mathbf{A}$ has a pseudo-complement (unique by definition), we may think of $\neg ^{\mathbf{A}}$ as a unary operation, known as the pseudo-complement operation of $\mathbf{A}$ .

Figure 1. The subdirectly irreducible De Morgan (a) and the subdirectly irreducible IS-algebras (b).

We move now to the logical preliminaries. A logic (over $\Sigma$ ) is a consequence relation $\vdash$ on $L_{\Sigma }(P)$ and a Set-Set logic (over $\Sigma$ ) is a generalized consequence relation $\rhd$ on $L_{\Sigma }(P)$ (Humberstone Reference Humberstone2011). The companion of a given Set-Set logic $\rhd$ is the logic $\vdash _{\rhd }$ such that $\Phi \vdash _{\rhd } \psi$ if, and only if, $\Phi \rhd \{\psi \}$ . We will write $\Phi \;{\lhd \rhd } \; \Psi$ when $\Phi \; \rhd \; \Psi$ and $\Psi \; \rhd \; \Phi$ ; that being the case, we say these sets are logically equivalent. We will adopt the convention of omitting curly braces when writing sets of formulas in statements involving (generalized) consequence relations. The complement of a given Set-Set logic $\rhd$ , that is, $(\wp L_{\Sigma }(P) \times \wp L_{\Sigma }(P)){\setminus }\rhd$ , will be denoted by $\blacktriangleright _{}$ .

Let $\rhd$ and $\rhd '$ be Set-Set logics over signatures $\Sigma$ and $\Sigma '$ , respectively, with $\Sigma \subseteq \Sigma '$ . We say that $\rhd '$ expands $\rhd$ when $\rhd ' \supseteq \rhd$ . When $\Sigma = \Sigma '$ , we say more simply that $\rhd '$ extends $\rhd$ . It is worth recalling that the collection of all extensions of a given logic forms a complete lattice under inclusion. We say that $\rhd '$ is a conservative expansion of $\rhd$ when $\rhd '$ expands $\rhd$ and, for all $\Phi, \Psi \subseteq L_{\Sigma }(P)$ , we have $\Phi \rhd ' \Psi$ if, and only if, $\Phi \rhd \Psi$ . The finitary companion of a Set-Set logic $\rhd$ is the Set-Set logic $\rhd _{\textsf{fin}}$ such that $\Phi \rhd _{\textsf{fin}} \Psi$ if, and only if, there are finite $\Phi ' \subseteq \Phi$ and $\Psi ' \subseteq \Psi$ such that $\Phi ' \rhd \Psi '$ . The latter concepts may be straightforwardly adapted to logics.

We call special attention to the notion of self-extensionality, as it will play an important role in the implicative extensions we consider in this paper:

A partial non-deterministic $\Sigma$ -matrix (or, more simply, $\Sigma$ -PNmatrix) $\mathfrak{M}$ is a structure $\langle \mathbf{A}, D \rangle$ where $\mathbf{A}$ is a $\Sigma$ -multialgebra and the members of $D \subseteq A$ are called designated values. We will write $\overline{D}$ to refer to $A{\setminus{}}D$ . When $\mathbf{A}$ is a $\Sigma$ -algebra, $\mathfrak{M}$ is called a $\Sigma$ -matrix, and coincides with the usual definition of logical matrix in the literature. A refinement of $\mathfrak{M}$ is a $\Sigma$ -PNmatrix obtained from $\mathbf{A}$ by deleting values from some entries of the interpretations of the connectives in $\mathbf{A}$ (the resulting interpretations are also said to be refinements of the ones in $\mathbf{A}$ ). The mappings in $\textsf{Hom}(\mathbf{L}_{\Sigma }(P), \mathbf{A})$ are called $\mathfrak{M}$ -valuations. Every $\Sigma$ -PNmatrix determines a logic $\rhd _{\mathfrak{M}}$ such that $\Phi\;{\rhd_{\mathfrak{M}}}\;\Psi$ iff $h(\Phi ) \cap \overline{D} \neq \varnothing$ or $h(\Psi ) \cap D \neq \varnothing$ for all $\mathfrak{M}$ -valuations $h$ , as well as a logic $\vdash _{\mathfrak{M}}$ with $\Phi \vdash _{\mathfrak{M}} \psi$ iff $h(\Phi ) \cap \overline{D} \neq \varnothing$ or $h(\psi ) \in D$ for all $\mathfrak{M}$ -valuations $h$ (notice that $\vdash _{\mathfrak{M}}$ is the companion of $\rhd _{\mathfrak{M}}$ ). Given a logic $\rhd$ (resp. a logic $\vdash$ ), if $\rhd \;\subseteq \;\rhd _{\mathfrak{M}}$ (resp. $\vdash \;\subseteq \; \vdash _{\mathfrak{M}}$ ), we shall say that $\mathfrak{M}$ is a matrix model of $\rhd$ (resp. $\vdash$ ), and if the converse also holds we shall say that $\mathfrak{M}$ determines $\rhd$ (resp. $\vdash$ ). The (resp. ) logic determined by a class $\mathcal{M}$ of $\Sigma$ -matrices is given by $\bigcap \{\rhd _{\mathfrak{M}} : \mathfrak{M} \in \mathcal{M}\}$ (resp. $\bigcap \{\vdash _{\mathfrak{M}} : \mathfrak{M} \in \mathcal{M}\}$ ).

Given a $\Sigma$ -matrix $\mathfrak{M} = \langle \mathbf{A}, D \rangle$ , a congruence $\theta \in \textsf{Cng} \mathbf{A}$ is said to be compatible with $\mathfrak{M}$ when $b \in D$ whenever both $a \in D$ and $a \theta b$ , for all $a,b \in A$ . We denote by $\Omega ^{\mathfrak{M}}$ the Leibniz congruence associated with $\mathfrak{M}$ , namely the greatest congruence of $\mathbf{A}$ compatible with $\mathfrak{M}$ . The matrix ${\mathfrak{M}}^{\ast } = \langle \mathbf{A}/\Omega ^{\mathfrak{M}}, D/\Omega ^{\mathfrak{M}} \rangle$ is the reduced version of $\mathfrak{M}$ . It is well-known that $\rhd _{\mathfrak{M}} = \rhd _{{\mathfrak{M}}^{\ast }}$ (and thus $\vdash _{\mathfrak{M}} \;=\; \vdash _{{\mathfrak{M}}^{\ast }}$ ) and, since every logic is determined by a class of matrix models, we have that every logic coincides with the logic determined by its reduced matrix models. As a shortcut, we call a matrix reduced when it coincides with its own reduced version (or, equivalently, when its Leibniz congruence is the identity relation on $A$ ).

When a $\Sigma$ -algebra $\mathbf{A}$ has a lattice structure with underlying order $\leq$ , for any $a\in A$ we write ${\uparrow }a$ to refer to the set $\{b \in A : a \leq b\}$ . For instance, over $\mathbf{IS_6}{}$ we may consider the set ${{\uparrow }{\mathbf{b}}} = \{{\mathbf{b}},{\mathbf{t}},{\hat{\mathbf{t}}}\}$ (see Fig. 1B). We may employ the same notation to write the designated set of a $\Sigma$ -matrix having a lattice as underlying algebra. A lattice filter of a meet-semilattice $\mathbf{A}$ with a top element $\top ^{\mathbf{A}}$ is a subset $D \subseteq A$ with $\top ^{\mathbf{A}} \in D$ and closed under $\land ^{\mathbf{A}}$ ; moreover, $D$ is a proper lattice filter of $\mathbf{A}$ when $D \neq A$ . A principal filter of $\mathbf{A}$ is a lattice filter of the form ${\uparrow } a$ , for some $a \in A$ . Note that, if $\mathbf{A}$ is finite, every lattice filter is principal. If $\mathbf{A}$ also has a join-semilattice structure, a prime filter of $\mathbf{A}$ is a proper lattice filter $D$ of $\mathbf{A}$ such that $a \lor b \in D$ iff $a \in D$ or $b \in D$ , for all $a,b \in A$ .

2.2 Logics associated with classes of (ordered) algebras

Throughout this section, consider a propositional signature $\Sigma \supseteq \Sigma ^{\textsf{bL}}$ . In addition, for a set of $\Sigma$ -formulas $\Phi \,{:=}\, \{ \varphi _1,\ldots, \varphi _n \}$ , let $\bigwedge \Phi \,{:=}\, \varphi _1 \land \ldots \land \varphi _n$ and $\bigvee \Phi \,{:=}\, \varphi _1 \lor \ldots \lor \varphi _n$ , while, by convention, let $\bigwedge \varnothing \,{:=}\, \top$ and $\bigvee \varnothing \,{:=}\, \bot$ . Moreover, let the inequality $\varphi \leq \psi$ abbreviate the equation $\varphi \approx \varphi \land \psi$ .

For any order-preserving logic $\rhd _{\textsf{K}}^{\leq }$ associated with an appropriate class $\textsf{K}$ of algebras, the following characterization in terms of prime filters applies:

For a variety generated by a single finite distributive lattice, we have this simpler characterization in terms of a finite family of finite matrices:

Notice that $\vdash _{\textsf{K}}^{\leq }$ is the companion of $\rhd _{\textsf{K}}^{\leq }$ .

The above logics are particularly interesting for us in view of the following:

We close by defining the $\top$ -assertional logics (also known as 1-assertional logics) associated with a class of bounded lattices.

Notice that $\vdash ^{\top }_{\textsf{K}}$ is the companion of $\rhd ^\top _{\textsf{K}}$ .

2.3 Hilbert-style axiomatizations

Based on Shoesmith and Smiley (Reference Shoesmith and Smiley1978) and Caleiro and Marcelino (Reference Caleiro, Marcelino, Iemhoff, Moortgat and Queiroz2019), we define a symmetrical (Hilbert-style) calculus $\textsf{R}$ (or calculus, for short) as a collection of pairs $(\Phi, \Psi ) \in \wp{L_{\Sigma }(P)}\times \wp{L_{\Sigma }(P)}$ , denoted by $\frac{\Phi }{\Psi }$ and called (symmetrical or ) inference rules, where $\Phi$ is the antecedent and $\Psi$ is the succedent of the said rule. We will adopt the convention of omitting curly braces when writing sets of formulas and leaving a blank space instead of writing $\varnothing$ when presenting inference rules and statements involving (generalized) consequence relations. We proceed to define what constitutes a proof in such calculi.

A finite bounded rooted tree $t$ is a finite poset $\langle \textsf{nds}(t), {\leq }^t \rangle$ with a single minimal element $\textsf{rt}(t)$ , the root of $t$ , such that, for each node $n \in \textsf{nds}(t)$ , the set $\{n' \in \textsf{nds}(t) : n' {\leq }^t n \}$ of ancestors of $n$ (or the branch up to $n$ ) is linearly ordered under ${\leq }^t$ , and every branch of $t$ has a maximal element (a leaf of $t$ ). We may assign a label $l^{t}(n) \in \wp{L_{\Sigma }(P)} \cup \{\ast \}$ to each node $n$ of $t$ , in which case $t$ is said to be labeled. Given $\Psi \subseteq L_{\Sigma }(P)$ , a leaf $n$ is $\Psi$ -closed in $t$ when $l^{t}(n) \;=\; \ast$ or $l^{t}(n) \cap \Psi \neq \varnothing$ . The tree $t$ itself is $\Psi$ -closed when all of its leaves are $\Psi$ -closed. The immediate successors of a node $n$ with respect to ${\leq }^t$ are called the children of $n$ in $t$ .

Figure 2. Graphical representation of $\textsf{R}$ -derivations, where $\textsf{R}$ is a system. The dashed edges and blank circles represent other branches that may exist in the derivation. We usually omit the formulas inherited from the parent node, exhibiting only the ones introduced by the applied rule of inference. Recall that, in both cases, we must have $\Pi \subseteq \Phi$ .

Let $\textsf{R}$ be a symmetrical calculus. An $\textsf{R}$ -derivation is a finite labeled bounded rooted tree such that for every non-leaf node $n$ of $t$ there exists a rule of inference $\textsf{r} = \frac{\Pi }{\Theta } \in \textsf{R}$ and a substitution $\sigma$ such that $\sigma (\Pi ) \subseteq l^{t}(n)$ , and the set of children of $n$ is either (i) $\{n^\varphi : \varphi \in \sigma (\Theta )\}$ , in case $\Theta \neq \varnothing$ , where $n^\varphi$ is a node labeled with $l^{t}(n) \cup \{\varphi \}$ , or (ii) a singleton $\{n^\ast \}$ with $l^{t}(n^\ast )\;=\;\ast$ , in case $\Theta \;=\;\varnothing$ . Such node $n$ is said to be expanded by an application of $\textsf r$ . Fig. 2 illustrates the general shape of $\textsf{R}$ -derivations. We say that $\Phi \; \rhd _{\textsf{R}}\; \Psi$ whenever there is a $\Psi$ -closed derivation $t$ such that $\Phi \;\supseteq \;\textsf{rt}(t)$ ; such a tree consists in a proof that $\Psi$ follows from $\Phi$ in $\textsf{R}$ . As a matter of simplification when drawing such trees, we usually avoid copying the formulas inherited from the parent nodes (see Example 5 below, where we display some proof trees). The relation $\rhd _{\textsf{R}}$ so defined is a logic and, when $\rhd _{\textsf{R}} = \rhd$ , we say that $\textsf{R}$ axiomatizes $\rhd$ . It is worth emphasizing that we restrict ourselves to finite derivations because we are interested in finitary logics, but in general they could be infinite and cover also non-finitary logics (cf. Marcelino and Caleiro Reference Marcelino and Caleiro2021, Figure 1, for an example). A rule $\frac{\Phi }{\Psi }$ is sound in a PNmatrix $\mathfrak{M}$ when $\Phi \;\rhd _{\mathfrak{M}}\;\Psi$ . It should be pointed out that this proof formalism generalizes the conventional () Hilbert-style calculi: the latter corresponds to symmetrical calculi whose rules have, each, a finite antecedent and a singleton as succedent.

The process of building an $\textsf{R}$ -derivation, in principle, does not necessarily terminate, since there may be infinitely many rule instances that are applicable. We now introduce a notion of analyticity that allows a limit to be set on the search space in such a way that this process is guaranteed to finish. Given $\Lambda \subseteq L_{\Sigma }(P)$ , we write $\Phi \rhd _{\textsf{R}}^\Lambda \Psi$ whenever there is a proof of $\Psi$ from $\Phi$ using only formulas in $\Lambda$ . Given $\Theta, \Xi \subseteq L_{\Sigma }(P)$ , we define the set $\Upsilon ^\Xi$ of $\Xi$ -generalized subformulas of $\Theta$ as the set $\textsf{sub}(\Theta ) \cup \{ \sigma (\varphi ) : \varphi \in \Xi \mbox{ and } \sigma : P \to \textsf{sub}(\Theta )\}$ . We say that $\textsf{R}$ is $\Xi$ -analytic when, for all $\Phi, \Psi \subseteq L_{\Sigma }(P)$ , we always have $\Phi \rhd _{\textsf{R}}^{\Upsilon ^\Xi } \Psi$ whenever we have $\Phi \;\rhd _{\textsf{R}}\;\Psi$ ; intuitively, it means that proofs in $\textsf{R}$ that $\Psi$ follows from $\Phi$ , whenever they exist, do not need to use more material than subformulas of $\Phi \cup \Psi$ or substitution instances of the formulas in $\Xi$ built from the subformulas of $\Phi \cup \Psi$ . Searching for proofs in a finite $\Xi$ -analytic calculus is then guaranteed to terminate because (i) due to analyticity, there are finitely many rule instances to be considered when expanding a node and (ii) in view of Remark Reference Caleiro and Marcelino10, loops may always be averted.

A general method is introduced in Caleiro and Marcelino (Reference Caleiro, Marcelino, Iemhoff, Moortgat and Queiroz2019), Marcelino and Caleiro (Reference Marcelino and Caleiro2021) for obtaining analytic calculi (in the sense of analyticity introduced in the previous paragraph) for logics given by a $\Sigma$ -PNmatrix $\langle \mathbf{A}, D\rangle$ whenever a certain expressiveness requirement (called “monadicity” in Shoesmith and Smiley Reference Shoesmith and Smiley1978) is met: for every $a,b \in A$ , there is a single-variable formula $\textrm{S}$ such that $\textrm{S}^{\mathbf{A}}(a) \in D$ and $\textrm{S}^{\mathbf{A}}(b) \not \in D$ or vice-versa. We call such formula a separator (for $a$ and $b$ ).

The following example illustrates a symmetrical calculus for $\mathcal{B}$ generated by this method, as well as some proofs in this calculus.

Example 5. The matrix $\langle \mathbf{DM}_4,{\uparrow }{\mathbf{b}} \rangle$ fulfills the above-mentioned expressiveness requirement, with the following set of separators: $\mathcal{S} \,{:=}\, \{p,{\sim } p\}$ . We may therefore apply the method introduced in Marcelino and Caleiro (Reference Marcelino and Caleiro2021) to obtain for $\mathcal{B}$ the following $\mathcal{S}$ -analytic axiomatization we call $\textsf{R}_{\mathcal{B}}$ :

\begin{equation*} \frac {}{\top }\,\textsf {r}_1 \quad \frac {\sim \top }{} \, {\textsf {r}_2} \quad \frac {}{\sim \bot }\, \textsf {r}_3 \quad \frac {\bot }{}\, \textsf {r}_4 \quad \frac {p}{\sim \sim p}\, \textsf {r}_5 \quad \frac {\sim \sim p}{p}\, \textsf {r}_6 \end{equation*}

\begin{equation*} \frac {p \land q}{p}\, \textsf {r}_7 \quad \frac {p \land q}{q}\, \textsf {r}_8 \quad \frac {p, q}{p \land q}\, \textsf {r}_9 \quad \frac {\sim p}{\sim (p \land q)}\, \textsf {r}_{10} \quad \frac {\sim q}{\sim (p \land q)}\, \textsf {r}_{11} \quad \frac {\sim (p \land q)}{\sim p, \sim q}\, \textsf {r}_{12} \end{equation*}

\begin{equation*} \frac {p}{p \lor q}\, \textsf {r}_{13} \quad \frac {q}{p \lor q}\,\textsf {r}_{14} \quad \frac {p \lor q}{p, q}\, \textsf {r}_{15} \quad \frac {\sim p, \sim q}{\sim (p \lor q)}\, \textsf {r}_{16} \quad \frac {\sim (p \lor q)}{\sim p}\,\textsf {r}_{17} \quad \frac {\sim (p \lor q)}{\sim q}\, \textsf {r}_{18} \end{equation*}

Fig. 3 illustrates proofs in $\textsf{R}_{\mathcal{B}}$ .

2.4 Implication connectives and criteria for implicative expansions

In the present study, we will follow some very specific research paths with the goal of adding an implication to given logics; we will refer to the resulting logics as implicative expansions. First of all, it is important to define precisely what we mean by an implication connective on and logics. Throughout this subsection, let $\Sigma$ be an arbitrary signature with a binary connective $\lor$ and denote by $\Sigma _{\Rightarrow }$ the signature resulting from adding to $\Sigma$ the binary connective $\Rightarrow$ . In what follows, let $\bigvee \{ \psi _1,\ldots, \psi _m \} \,{:=}\, \psi _1 \lor (\psi _2 \lor \ldots (\ldots \lor \psi _n)\ldots )$ .

Figure 3. Proofs in $\textsf{R}_{\mathcal{B}}$ witnessing that ${\sim }(p \land q) \;{\lhd \rhd }_{\mathcal{B}}\;{\sim } p \lor{\sim } q$ and $p \lor \bot \;{\lhd \rhd }_{\mathcal{B}} \; p, r$ .

Definition 12. The connective $\Rightarrow$ is an implication in a logic $\rhd{}$ over $\Sigma _\Rightarrow$ provided, for all $\Phi, \{ \psi _1,\ldots, \psi _m \}, \{ \varphi, \psi \} \subseteq L_{\Sigma }(P)$ ,

\begin{equation*} \Phi, \varphi \rhd \psi, \psi _1,\ldots, \psi _m \text { if, and only if, } \Phi \rhd \varphi \Rightarrow \left (\bigvee \{\psi, \psi _1,\ldots, \psi _m \}\right ). \end{equation*}

Note that the above definition reduces in to the deduction-detachment theorem (DDT) that holds, for example, in intuitionistic and classical logics, and this is indeed what we will take to be an implication in , in this paper. That is, $\vdash$ has the DDT with respect to $\Rightarrow$ in case $\Phi, \varphi \vdash \psi$ if, and only if, $\Phi \vdash \varphi \Rightarrow \psi$ for all $\Phi, \{ \varphi, \psi \} \subseteq L_{\Sigma }(P)$ . What we did above was a convenient generalization of the DDT for logics, using it to abstractly characterize what we expect of an implication connective (namely, an internalization of the consequence relation that allows formulas to be discharged from the antecedent).

Based on the behavior of implication in classical logic, we also consider the following stronger notion of implication:

Definition 13. The connective $\Rightarrow$ is a classic-like implication in a logic $\rhd{}$ over $\Sigma _\Rightarrow$ provided, for all $\Phi, \Psi, \{ \varphi, \psi \} \subseteq L_{\Sigma }(P)$ ,

\begin{equation*} \Phi, \varphi \rhd \psi, \Psi \text { if, and only if, } \Phi \rhd \varphi \Rightarrow \psi, \Psi . \end{equation*}

It should be clear enough that, for logics $\rhd$ with $\lor$ satisfying $p \rhd p \lor q$ ; $q \rhd p \lor q$ ; and $p \lor q \rhd p,q$ , classic-like implications are implications in the sense of Definition 12. The converse, however, might not hold (see Remark Reference Monteiro32).

In this paper, we shall require two minimal criteria on implicative expansions:

1. $(\textsf{I}1)$ The connective being added must qualify as an implication (Definition 12);

2. $(\textsf{I}2)$ The expansion must be conservative.

We will prove below some facts regarding the connections among the notions of implication, residuation, and characterizability via a single PNmatrix.

Observe that the above result extends to $\vdash _{\textsf{K}}^{\leq }$ (recall Definition 7) since it is the companion of $\rhd _{\textsf{K}}^{\leq }$ . This also looks like the right moment to introduce intuitionistic-like implications:

Inspired by Avron (Reference Avron2020), we also consider the following additional criteria to guide our investigations:

1. $(\textsf{A}1)$ The expanded logic is determined by the single PNmatrix $\langle \mathbf A, D \rangle$ , and, for all $a,b \in A$ , we have $a \Rightarrow ^{\mathbf A} b \subseteq D$ if, and only if, either $a \not \in D$ or $b \in D$ .

2. $(\textsf{A}2)$ The expanded logic is self-extensional (Definition 3).

Note that, for logics, the criterion $(\textsf{A}1)$ is strong to the point of forcing the referred implication to be classic-like:

5.1 Characterizability by single finite PNmatrices

We saw in Theorem 34 that the logics we introduced in this section are not characterized by any single logical matrix. In this subsection, we will demonstrate the power of partial non-deterministic matrices by showing that both $\mathcal{PP}^{\rhd, \Rightarrow _{\textsf{H}}}_{\textsf{up}}$ and $\mathcal{PP}^{\rhd, \Rightarrow _{\textsf{H}}}_{\!\leq }$ are characterized by single finite PNmatrices. In consequence, $\mathcal{PP}^{\Rightarrow _{\textsf{H}}}_{\!\leq }$ will be characterized by either of these matrices. The essential idea is that the collections of matrices that characterize these logics can be packaged into a single structure using partiality.

The construction we will present makes use of the notion of total components of a $\Sigma$ -PNmatrix, which we now proceed to introduce. Let $\mathfrak{M} \,{:=}\, \langle \mathbf{A}, D \rangle$ be a $\Sigma$ -PNmatrix. For $X \subseteq A$ , denote by $\mathfrak{M}_X$ the $\Sigma$ -PNmatrix $\langle \mathbf{A}_X, D \cap X \rangle$ , where $\mathbf{A}_X \,{:=}\, \langle A \cap X, \cdot ^{\mathbf{A}_X} \rangle$ is a $\Sigma$ -multialgebra such that $\unicode{x00A9}^{\mathbf{A}_X}(a_1,\ldots, a_k) \,{:=}\, \unicode{x00A9}^{\mathbf{A}}(a_1,\ldots, a_k) \cap X$ for all $a_1,\ldots, a_k \in X$ , $k \in \omega$ and $\unicode{x00A9} \in \Sigma _k$ . This PNmatrix is called the restriction of $\mathfrak{M}$ to $X$ . We say that $X \neq \varnothing$ is a total component of $\mathfrak{M}$ whenever $\mathfrak{M}_X$ is total. A total component $X$ is maximal if adding any other value to $X$ leads to a component that is not total. Denote by $\mathbb{T}(\mathfrak{M})$ the collection of maximal total components of $\mathfrak{M}$ . Then we have that $\rhd _{\mathfrak{M}} ={\rhd _{\{ \mathfrak{M}_X : X \in \mathbb{T}(\mathfrak{M})\}}}$ (Caleiro and Marcelino Reference Caleiro and Marcelino2023). The latter observation is key to us: the matrices induced by the maximal total components of the PNmatrices we will construct are precisely the ones in the classes that determine the logics $\mathcal{PP}^{\rhd, \Rightarrow _{\textsf{H}}}_{\textsf{up}}$ and $\mathcal{PP}^{\rhd, \Rightarrow _{\textsf{H}}}_{\!\leq }$ .

We display below diagrams of the four matrices (see discussion after Proposition 33) that determine the logic $\mathcal{PP}^{\rhd, \Rightarrow _{\textsf{H}}}_{\textsf{up}}$ (in each case, the sets of designated elements are highlighted with an ellipse).

Below we depict the structure of the PNmatrix $\mathfrak{M}_{\textsf{up}}$ we propose for this logic, whose principle of construction is the combination of the above matrices, such that each of them consists of a total component of the PNmatrix. Note: There is nothing special about the dashed edge, it is pictured as dashed only because it crosses other edges.

We try to make the above idea clearer in the following pictures, which show how to identify each of the four matrices inside $\mathfrak{M}_{\textsf{up}}$ .

The PNmatrix for $\mathcal{PP}^{\rhd, \Rightarrow _{\textsf{H}}}_{\!\leq }$ , which we dub $\mathfrak{M}_{\leq }$ , is essentially the same, the only difference being the absence of the dashed line.

After this informal presentation of the general approach, we proceed to a precise definition of the PNmatrices $\mathfrak{M}_{\textsf{up}}$ and $\mathfrak{M}_{\leq }$ , and prove that they indeed determine, respectively, the logics $\mathcal{PP}^{\rhd, \Rightarrow _{\textsf{H}}}_{\textsf{up}}$ and $\mathcal{PP}^{\rhd, \Rightarrow _{\textsf{H}}}_{\!\leq }$ .

Definition 37. Let $\mathcal{V}_{10} \,{:=}\, \{{\hat{\mathbf{f}}},{\mathbf{f}}^-,{\mathbf{n}}^-,{\mathbf{b}}^-,{\mathbf{t}}^-,{\mathbf{f}}^+,{\mathbf{n}}^+,{\mathbf{b}}^+,{\mathbf{t}}^+,{\hat{\mathbf{t}}}\}$ and $D \,{:=}\, \{{\mathbf{f}}^+,{\mathbf{n}}^+,{\mathbf{b}}^+,{\mathbf{t}}^+,{\hat{\mathbf{t}}}\} \subseteq \mathcal{V}_{10}$ . Consider the predicates $\textsf{inc}_{\textsf{up}}$ and $\textsf{inc}_{\leq }$ over $\wp (\mathcal{V}_{10})$ , where $\textsf{inc}_{\textsf{up}}(X)$ (resp. $\textsf{inc}_{\leq }(X)$ ) means that $X$ is not contained in the image of valuations over $\mathfrak{M}_{\textsf{up}}$ (resp. $\mathfrak{M}_{\leq }$ ), and can be defined as follows:

\begin{align*} \textsf{inc}_{\textsf{up}}(X) & \text{ iff }\\ &X\supseteq Y\text{, for some }Y\in \{\{d^-,d^+\}:d\in \{{\mathbf{f}},{\mathbf{n}},{\mathbf{b}},{\mathbf{t}}\}\}\cup \\ &\qquad \{\{{\mathbf{b}}^-,{\mathbf{f}}^+\}, \{{\mathbf{n}}^-,{\mathbf{f}}^+\},\{{\mathbf{b}}^+,{\mathbf{t}}^-\}, \{{\mathbf{n}}^+,{\mathbf{t}}^-\}, \{{\mathbf{n}}^+,{\mathbf{b}}^+,{\mathbf{f}}^-\}\} \end{align*}

\begin{align*} {\textsf{inc}_{\leq }(X)}& \text{ iff }\\ &X\supseteq Y\text{, for some }Y\in \{\{d^-,d^+\}:d\in \{{\mathbf{f}},{\mathbf{n}},{\mathbf{b}},{\mathbf{t}}\}\}\cup \\ &\qquad \{ \{{\mathbf{b}}^-,{\mathbf{f}}^+\}, \{{\mathbf{n}}^-,{\mathbf{f}}^+\},\{{\mathbf{b}}^+,{\mathbf{t}}^-\}, \{{\mathbf{n}}^+,{\mathbf{t}}^-\}, \{{\mathbf{n}}^+,{\mathbf{b}}^+,{\mathbf{f}}^-\}, \{{\mathbf{n}}^-,{\mathbf{b}}^-,{\mathbf{t}}^+\}\} \end{align*}

Consider also the function $g:\mathcal{V}_{10} \to \{{\hat{\mathbf{f}}},{\mathbf{f}},{\mathbf{n}},{\mathbf{b}},{\mathbf{t}},{\hat{\mathbf{t}}}\}$ given by $g({\hat{\mathbf{f}}}) \,{:=}\,{\hat{\mathbf{f}}}$ , $g({\hat{\mathbf{t}}}) \,{:=}\,{\hat{\mathbf{t}}}$ , $g(a^i) \,{:=}\, a$ for $a\in \{{\mathbf{f}},{\mathbf{n}},{\mathbf{b}},{\mathbf{t}}\}$ and $i \in \{ +,- \}$ . We define the PNmatrices $\mathfrak{M}_{\textsf{up}} \,{:=}\,{\langle \mathbf{A}_{\textsf{up}},D\rangle }$ and $\mathfrak{M}_{\leq } \,{:=}\,{\langle \mathbf{A}_{\leq },D\rangle }$ , with $\mathbf{A}_{\textsf{up}} \,{:=}\, \langle \mathcal{V}_{10},\cdot _{\textsf{up}} \rangle$ and $\mathbf{A}_{\leq } \,{:=}\, \langle \mathcal{V}_{10},\cdot _{\leq } \rangle$ such that

• • for $\unicode{x00A9} \in \{{\sim },{\circ }\}$ ,

  \begin{align*} \unicode{x00A9}_{\textsf{up}}(a)& \,{:=}\, \{b\in \mathcal{V}_{10}:\unicode{x00A9}_{\mathbf{PP_6^{\Rightarrow _{\textsf{H}}}}}(g(a))=g(b) \text{ and not }{\textsf{inc}_{\textsf{up}}(\{ a,b \})}\}\\ \unicode{x00A9}_\leq (a)&\,{:=}\,\{b\in \mathcal{V}_{10}: \unicode{x00A9}_{\mathbf{PP_6^{\Rightarrow _{\textsf{H}}}}}(g(a))=g(b) \text{ and not }{\textsf{inc}_{\leq }(\{a,b\})} \} \end{align*}

• • for $\unicode{x00A9}\in \{\land, \lor, \Rightarrow \}$ ,

  \begin{align*} \unicode{x00A9}_{\textsf{up}}(a,b)& \,{:=}\, \{c\in \mathcal{V}_{10}: \unicode{x00A9}_{\mathbf{PP_6^{\Rightarrow _{\textsf{H}}}}}(g(a),g(b))=g(c) \text{ and not }{\textsf{inc}_{\textsf{up}}(\{a,b,c\})}\}\\ \unicode{x00A9}_{\leq }(a,b)&\,{:=}\,\{c\in \mathcal{V}_{10}: \unicode{x00A9}_{\mathbf{PP_6^{\Rightarrow _{\textsf{H}}}}}(g(a),g(b))=g(c)\text{ and not }{\textsf{inc}_{\leq }(\{a,b,c\})} \} \end{align*}

Proof. We have that if $\{a,b,c\} \subseteq X$ and $\textsf{inc}_{\textsf{up}}(\{a,b,c\})$ then $X$ is not contained in a total component of $\mathfrak{M}_{\textsf{up}}$ . In fact,

• • if $b_1,b_2\in X$ with

  \begin{equation*}\{b_1,b_2\}\in \{\{a^-,a^+\}:a\in \{{\mathbf {f}},{\mathbf {n}},{\mathbf {b}},{\mathbf {t}}\}\}\cup \{\{{\mathbf {b}}^-,{\mathbf {f}}^+\}, \{{\mathbf {n}}^-,{\mathbf {f}}^+\},\{{\mathbf {b}}^+,{\mathbf {t}}^-\}, \{{\mathbf {n}}^+,{\mathbf {t}}^-\}\}\end{equation*}
  then $b_1\land _{\textsf{up}} b_2=\varnothing$ .

• • if $\{{\mathbf{n}}^+,{\mathbf{b}}^+,{\mathbf{f}}^-\}\subseteq X$ , then ${\mathbf{n}}^+\land _{\textsf{up}}{\mathbf{b}}^+=\{{\mathbf{f}}^+\}$ and ${\mathbf{f}}^-\land _{\textsf{up}}{\mathbf{f}}^+=\varnothing$ .

The maximal total components of $\mathfrak{M}_{\textsf{up}}$ are

\begin{equation*}\mathbb {T}({\mathfrak {M}_{\textsf {up}}})=\{ \{{\hat {\mathbf {f}}},{\mathbf {f}}^-,{\mathbf {n}}^-,{\mathbf {b}}^-,{\mathbf {t}}^-,{\hat {\mathbf {t}}}\}, \{{\hat {\mathbf {f}}},{\mathbf {f}}^-,{\mathbf {n}}^-,{\mathbf {b}}^-,{\mathbf {t}}^+,{\hat {\mathbf {t}}}\}, \{{\hat {\mathbf {f}}},{\mathbf {f}}^-,{\mathbf {n}}^-,{\mathbf {b}}^+,{\mathbf {t}}^+,{\hat {\mathbf {t}}}\}, \{{\hat {\mathbf {f}}},{\mathbf {f}}^+,{\mathbf {n}}^+,{\mathbf {b}}^+,{\mathbf {t}}^+,{\hat {\mathbf {t}}}\}\}\end{equation*}

since for every $X\in \mathbb{T}({\mathfrak{M}_{\textsf{up}}})$ and every $a,b,c\in X$ we have that $\textsf{inc}_{\textsf{up}}(\{a,b,c\})$ is not the case. Thus, the restriction of $\mathfrak{M}_{\textsf{up}}$ to $X$ is isomorphic to (that is, it is the same up to renaming of truth values) some matrix with set of designated values $D_X ={{\uparrow }a}$ for $a\in \mathcal{V}_6$ . The isomorphism is given by the restriction of $g$ to $X$ .

Similarly, the maximal total components of $\mathfrak{M}_{\leq }$ are

\begin{equation*}\mathbb {T}({\mathfrak {M}_{\leq }})=\{ \{{\hat {\mathbf {f}}},{\mathbf {f}}^-,{\mathbf {n}}^-,{\mathbf {b}}^-,{\mathbf {t}}^-,{\hat {\mathbf {t}}}\}, \{{\hat {\mathbf {f}}},{\mathbf {f}}^-,{\mathbf {n}}^-,{\mathbf {b}}^+,{\mathbf {t}}^+,{\hat {\mathbf {t}}}\}, \{{\hat {\mathbf {f}}},{\mathbf {f}}^+,{\mathbf {n}}^+,{\mathbf {b}}^+,{\mathbf {t}}^+,{\hat {\mathbf {t}}}\}\}\end{equation*}

This is so because $\textsf{inc}_{\leq }(\{a,b,c\})$ iff $\textsf{inc}_{\textsf{up}}(\{a,b,c\})$ or $\{{\mathbf{n}}^-,{\mathbf{b}}^-,{\mathbf{t}}^+\}\subseteq X$ , and the fact that if $\{{\mathbf{n}}^-,{\mathbf{b}}^-,{\mathbf{t}}^+\}\subseteq X$ then $X$ is not in a total component of $\mathfrak{M}_{\leq }$ , since ${\mathbf{n}}^-\lor _{\textsf{up}}{\mathbf{b}}^-=\{{\mathbf{t}}^-\}$ and ${\mathbf{t}}^-\land _{\textsf{up}}{\mathbf{t}}^+=\varnothing$ .

6.1 Analytic axiomatizations for $\mathcal{PP}^{\rhd, \Rightarrow _{\textsf{H}}}_{\textsf{up}}$ and $\mathcal{PP}^{\rhd, \Rightarrow _{\textsf{H}}}_{\!\leq }$

We begin by the analytic axiomatizations. What follows is a succession of definitions introducing groups of rules of inference that capture particular aspects of the collections of logical matrices determining $\mathcal{PP}^{\rhd, \Rightarrow _{\textsf{H}}}_{\textsf{up}}$ and $\mathcal{PP}^{\rhd, \Rightarrow _{\textsf{H}}}_{\!\leq }$ . We check the soundness of each of them and ultimately arrive at the desired completeness results. Throughout the proofs, we will make use of the following abbreviations:

By way of an example, the truth tables of the above-derived connectives interpreted in $\mathbf{PP_6^{\Rightarrow _{\textsf{H}}}}$ are given in Table 2. A key feature of these connectives is that they characterize the values $\mathbf{f}$ and $\mathbf{t}$ in the following sense: for each valuation $v$ over $\langle \mathbf{PP_6^{\Rightarrow _{\textsf{H}}}}, D\rangle$ , $v({\uparrow } p) \not \in D$ iff $v(p) ={\mathbf{f}}$ and $v({\downarrow } p) \not \in D$ iff $v(p) ={\mathbf{t}}$ , for $D$ containing $\hat{\mathbf{t}}$ and not containing $\hat{\mathbf{f}}$ .

Table 2. Truth tables of the connectives $\uparrow$ and $\downarrow$ interpreted in $\mathbf{PP_6^{\Rightarrow _{\textsf{H}}}}$

Note that the above definitions introduce new connectives by means of abbreviations, so we have ${\textsf{sub}}({\uparrow } p) = \{p,{\sim } p,{\sim } p\Rightarrow p,{\circ }({\sim } p\Rightarrow p)\}$ (the case of ${\textsf{sub}}({\downarrow } p)$ is similar).

Definition 41. Let $\textsf{R}_{\Diamond }$ be the calculus given by the following inference rules:

\begin{equation*} \frac {}{{\uparrow } p,{\downarrow } p}\, \textsf {r}^\Diamond _{{\uparrow }\textsf {or}{\downarrow }} \qquad \frac {}{\circ (p\Rightarrow p)}\, {\textsf {r}_{\textsf {id}}^{\Diamond }}\qquad \frac {\circ (p\Rightarrow q),\circ (q\Rightarrow r)}{\circ q,\circ (p\Rightarrow r)}\,\textsf {r}_{\textsf {trans}}^\Diamond \end{equation*}

\begin{equation*}\frac {}{{\downarrow } p,\circ q,\circ ( q\Rightarrow p)}\, \textsf {r}^\Diamond _{\leq \mathbf {t}} \qquad \frac {}{{\uparrow } p,\circ ( p\Rightarrow q)}\,\textsf {r}^\Diamond _{\geq \mathbf {f}} \end{equation*}

\begin{equation*}\frac {{\uparrow } p,\circ (p\Rightarrow q)}{\circ p,{\uparrow } q}\, \textsf {r}^\Diamond _{\textsf {incclass}_1} \qquad \frac {{\downarrow } q,\circ (p\Rightarrow q)}{\circ q,{\downarrow } p}\, \textsf {r}^\Diamond _{\textsf {incclass}_2} \qquad \frac {{\uparrow } p,{\downarrow } q,\circ (p\Rightarrow q)}{\circ q,\circ (q\Rightarrow p)}\, \textsf {r}^\Diamond _{\textsf {incclass}_3} \end{equation*}

\begin{equation*} \frac {{\downarrow } p,{\uparrow } r}{\circ p,\circ (p\Rightarrow q),\circ ( p\Rightarrow r),\circ (q\Rightarrow r)}\, \textsf {r}^\Diamond _{\textsf {just}_2} \end{equation*}