Abstract
The characterizing properties of a prooftheoretical presentation of a given logic may hang on the choice of proof formalism, on the shape of the logical rules and of the sequents manipulated by a given proof system, on the underlying notion of consequence, and even on the expressiveness of its linguistic resources and on the logical framework into which it is embedded. Standard (onedimensional) logics determined by (nondeterministic) logical matrices are known to be axiomatizable by analytic and possibly finite proof systems as soon as they turn out to satisfy a certain constraint of sufficient expressiveness. In this paper we introduce a recipe for cooking up a twodimensional logical matrix (or Bmatrix) by the combination of two (possibly partial) nondeterministic logical matrices. We will show that such a combination may result in Bmatrices satisfying the property of sufficient expressiveness, even when the input matrices are not sufficiently expressive in isolation, and we will use this result to show that onedimensional logics that are not finitely axiomatizable may inhabit finitely axiomatizable twodimensional logics, becoming, thus, finitely axiomatizable by the addition of an extra dimension. We will illustrate the said construction using a wellknown logic of formal inconsistency called mCi. We will first prove that this logic is not finitely axiomatizable by a onedimensional (generalized) Hilbertstyle system. Then, taking advantage of a known 5valued nondeterministic logical matrix for this logic, we will combine it with another one, conveniently chosen so as to give rise to a Bmatrix that is axiomatized by a twodimensional Hilbertstyle system that is both finite and analytic.
Keywords
 Hilbertstyle proof systems
 finite axiomatizability
 consequence relations
 nondeterministic semantics
 paraconsistency
V. Greati acknowledges support from CAPES—Finance Code 001 and from the FWF project P 33548. J. Marcos acknowledges support from CNPq.
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1 Introduction
A logic is commonly defined nowadays as a relation that connects collections of formulas from a formal language and satisfies some closure properties. The established connections are called consecutions and each of them has two parts, an antecedent and a succedent, the latter often being said to ‘follow from’ (or to be a consequence of) the former. A logic may be manufactured in a number of ways, in particular as being induced by the set of derivations justified by the rules of inference of a given proof system. There are different kinds of proof systems, the differences between them residing mainly in the shapes of their rules of inference and on the way derivations are built. We will be interested here in Hilbertstyle proof systems (‘Hsystems’, for short), whose rules of inference have the same shape of the consecutions of the logic they canonically induce and whose associated derivations consist in expanding a given antecedent by applications of rules of inference until the desired succedent is produced. A remarkable property of an Hsystem is that the logic induced by it is the least logic containing the rules of inference of the system; in the words of [24], the system constitutes a ‘logical basis’ for the said logic.
Conventional Hsystems, which we here dub ‘SetFmla Hsystems’, do not allow for more than one formula in the succedents of the consecutions that they manipulate. Since [23], however, we have learned that the simple elimination of this restriction on Hsystems —that is, allowing for sets of formulas rather than single formulas in the succedents— brings numerous advantages, among which we mention: modularity (correspondence between rules of inference and properties satisfied by a semantical structure), analyticity (control over the resources demanded to produce a derivation), and the automatic generation of analytic proof systems for a wide class of logics specified by sufficiently expressive nondeterministics semantics, with an associated straightforward proofsearch procedure [13, 18]. Such generalized systems, here dubbed ‘SetSet Hsystems’, induce logics whose consecutions involve succedents consisting in a collection of formulas, intuitively understood as ‘alternative conclusions’.
An Hsystem \(\mathcal {H}\) is said to be an axiomatization for a given logic \(\mathcal {L}\) when the logic induced by \(\mathcal {H}\) coincides with \(\mathcal {L}\). A desirable property for an axiomatization is finiteness, namely the property of consisting on a finite collection of schematic axioms and rules of inference. A logic having a finite axiomatization is said to be ‘finitely based’. In the literature, one may find examples of logics having a quite simple, finite semantic presentation, being, in contrast, not finitely based in terms of SetFmla Hsystems [21]. These very logics, however, when seen as companions of logics with multiple formulas in the succedent, turn out to be finitely based in terms of SetSet Hsystems [18]. In other words, by updating the underlying prooftheoretical and the logical formalisms, we are able to obtain a finite axiomatization for logics which in a more restricted setting could not be said to be finitely based. We may compare the above mentioned movement to the common mathematical practice of adding dimensions in order to provide better insight on some phenomenon. A wellknown example of that is given by the Fundamental Theorem of Algebra, which provides an elegant solution to the problem of determining the roots of polynomials over a single variable, demanding only that real coefficients should be replaced by complex coefficients. Another example, from Machine Learning, is the ‘kernel trick’ employed in support vector machines: by increasing the dimensionality of the input space, the transformed data points become more easily separable by hyperplanes, making it possible to achieve better results in classification tasks.
It is worth noting that there are logics that fail to be finitely based in terms of SetSet Hsystems. An example of a logic designed with the sole purpose of illustrating this possibility was provided in [18]. One of the goals of the present work is to show that an important logic from the literature of logics of formal inconsistency (LFIs) called mCi is also an example of this phenomenon. This logic results from adding infinitelymany axiom schemas to the logic mbC, a logic that is obtained by extending positive classical logic with two axiom schemas. Incidentally, along the proof of this result, we will show that mCi is the limit of a strictly increasing chain of LFIs extending mbC (comparable to the case of \(\mathcal {C}_{\mathrm {Lim}}\) in da Costa’s hierarchy of increasingly weaker paraconsistent calculi [16]). A natural question, then, is whether we can enrich our technology, in the same vein, in order to provide finite axiomatizations for all these logics. We answer that in the affirmative by means of the twodimensional frameworks developed in [11, 17]. Logics, in this case, connect pairs of collections of formulas. A consecution, in this setting, may be read as involving formulas that are accepted and those that are not, as well as formulas that are rejected and those that are not. ‘Acceptance’ and ‘rejection’ are seen, thus, as two orthogonal dimensions that may interact, making it possible, thus, to express more complex consecutions than those expressible in onedimensional logics. Twodimensional Hsystems, which we call ‘\(\textsc {Set}^{2}\text {}\textsc {Set}^{2}\) Hsystems’, generalize SetSet Hsystems so as to manipulate pairs of collections of formulas, canonically inducing twodimensional logics and constituting logical bases for them. Another goal of the present work is, therefore, to show how to obtain a twodimensional logic inhabited by a (possibly not finitely based) onedimensional logic of interest. More than that, the logic we obtain will be finitely axiomatizable in terms of a \(\textsc {Set}^{2}\text {}\textsc {Set}^{2}\) analytic Hsystem. The only requirements is that the onedimensional logic of interest must have an associated semantics in terms of a finite nondeterministic logical matrix and that this matrix can be combined with another one through a novel procedure that we will introduce, resulting in a twodimensional nondeterministic matrix (a Bmatrix [9]) satisfying a certain condition of sufficient expressiveness [17]. An application of this approach will be provided here in order to produce the first finite and analytic axiomatization of mCi.
The paper is organized as follows: Sect. 2 introduces basic terminology and definitions regarding algebras and languages. Section 3 presents the notions of onedimensional logics and SetSet Hsystems. Section 4 proves that mCi is not finitely axiomatizable by onedimensional Hsystems. Section 5 introduces twodimensional logics and Hsystems, and describes the approach to extending a logical matrix to a Bmatrix with the goal of finding a finite twodimensional axiomatization for the logic associated with the former. Section 6 presents a twodimensional finite analytic Hsystem for mCi. In the final remarks, we highlight some byproducts of our present approach and some features of the resulting proof systems, in addition to pointing to some directions for further research.^{Footnote 1}
2 Preliminaries
A propositional signature is a family \(\varSigma :=\{{\varSigma }_{k}\}_{k\in \omega }\), where each \({\varSigma }_{k}\) is a collection of \(k\)ary connectives. We say that \(\varSigma \) is finite when its base set \(\bigcup _{k\in \omega }{\varSigma }_{k}\) is finite. A nondeterministic algebra over \(\varSigma \), or simply \(\varSigma \) ndalgebra, is a structure \(\mathbf {A}:=\left\langle A, {\cdot }_{\mathbf {A}} \right\rangle \), such that \(A\) is a nonempty collection of values called the carrier of \(\mathbf {A}\), and, for each \(k\in \omega \) and \( {\copyright }\in {\varSigma }_{k}\), the multifunction \({ {\copyright }}_{\mathbf {A}} : A^k\rightarrow \mathcal {P}(A)\) is the interpretation of \( {\copyright }\) in \(\mathbf {A}\). When \(\varSigma \) and \(A\) are finite, we say that \(\mathbf {A}\) is finite. When the range of all interpretations of \(\mathbf {A}\) contains only singletons, \(\mathbf {A}\) is said to be a deterministic algebra over \(\varSigma \), or simply a \(\varSigma \) algebra, meeting the usual definition from Universal Algebra [12]. When \(\varnothing \) is not in the range of each \({ {\copyright }}_{\mathbf {A}}\), \(\mathbf {A}\) is said to be total. Given a \(\varSigma \)algebra \(\mathbf {A}\) and a \( {\copyright }\in {\varSigma }_{1}\), we let \({ {\copyright }}_{\mathbf {A}}^0(x) :=x\) and \({ {\copyright }}_{\mathbf {A}}^{i+1}(x) :={ {\copyright }}_{\mathbf {A}}({ {\copyright }}_{\mathbf {A}}^i(x))\). A mapping \(v: A\rightarrow B\) is a homomorphism from \(\mathbf {A}\) to \(\mathbf {B}\) when, for all \(k\in \omega \), \( {\copyright }\in {\varSigma }_{k}\) and \(x_1,\ldots ,x_k \in A\), we have \(f[{ {\copyright }}_{\mathbf {A}}(x_1,\ldots ,x_k)] \subseteq { {\copyright }}_{\mathbf {B}}(f(x_1),\ldots ,f(x_k))\). The set of all homomorphisms from \(\mathbf {A}\) to \(\mathbf {B}\) is denoted by \(\mathsf {Hom}_{\varSigma }(\mathbf {A},\mathbf {B})\). When \(\mathbf {B}= \mathbf {A}\), we write \(\mathsf {End}_{\varSigma }(\mathbf {A})\), rather than \(\mathsf {Hom}_{\varSigma }(\mathbf {A},\mathbf {A})\), for the set of endomorphisms on \(\mathbf {A}\).
Let \(P\) be a denumerable collection of \(propositional\ variables \) and \(\varSigma \) be a propositional signature. The absolutely free \(\varSigma \)algebra freely generated by \(P\) is denoted by \(\mathbf {L}_{\varSigma }(P)\) and called the \(\varSigma \) language generated by \(P\). The elements of \(L_{\varSigma }(P)\) are called \(\varSigma \) formulas, and those among them that are not propositional variables are called \(\varSigma \)compounds. Given \(\varPhi \subseteq L_{\varSigma }(P)\), we denote by \(\varPhi ^{\mathsf {c}}\) the set \({L_{\varSigma }(P)}{\setminus }{\varPhi }\). The homomorphisms from \(\mathbf {L}_{\varSigma }(P)\) to \(\mathbf {A}\) are called valuations on \(\mathbf {A}\), and we denote by \(\mathsf {Val}_{\varSigma }(\mathbf {A})\) the collection thereof. Additionally, endomorphisms on \(\mathbf {L}_{\varSigma }(P)\) are dubbed \(\varSigma \)substitutions, and we let \(\mathsf {Subs}_{\varSigma }^{P} :=\mathsf {End}_{\varSigma }(\mathbf {L}_{\varSigma }(P))\); when there is no risk of confusion, we may omit the superscript from this notation.
Given \(\varphi \in L_{\varSigma }(P)\), let \(\mathsf {props}(\varphi )\) be the set of propositional variables occurring in \(\varphi \). If \(\mathsf {props}(\varphi ) = \{p_1,\ldots ,p_k\}\), we say that \(\varphi \) is \(k\)ary (unary, for \(k=1\); binary, for \(k=2\)) and let \({\varphi }_{\mathbf {A}} : A^k\rightarrow \mathcal {P}(A)\) be the \(k\) ary multifunction on \(\mathbf {A}\) induced by \(\varphi \), where, for all \(x_1,\ldots ,x_k \in A\), we have \({\varphi }_{\mathbf {A}}(x_1,\ldots ,x_k) :=\{v(\varphi ) \mid v\in \mathsf {Val}_{\varSigma }(\mathbf {A}) \text { and } v(p_i) = x_i, \text { for } 1 \le i \le k\}\). Moreover, given \(\psi _1,\ldots ,\psi _k \in L_{\varSigma }(P)\), we write \(\varphi (\psi _1,\ldots ,\psi _k)\) for the \(\varSigma \)formula \({\varphi }_{\mathbf {L}_{\varSigma }(P)}(\psi _1,\ldots ,\psi _k)\), and, where \(\varPhi \subseteq L_{\varSigma }(P)\) is a set of \(k\)ary \(\varSigma \)formulas, we let \(\varPhi (\psi _1,\ldots ,\psi _k) :=\{\varphi (\psi _1,\ldots ,\psi _k) \mid \varphi \in \varPhi \}\). Given \(\varphi \in L_{\varSigma }(P)\), by \(\mathsf {subf}(\varphi )\) we refer to the set of subformulas of \(\varphi \). Where \(\theta \) is a unary \(\varSigma \)formula, we define the set \(\mathsf {subf}^{\theta }(\varphi )\) as \(\{\sigma (\theta )\mid \sigma ~:~P\rightarrow \mathsf {subf}(\varphi )\}\). Given a set \(\varTheta \supseteq \{p\}\) of unary \(\varSigma \)formulas, we set \(\mathsf {subf}^{\varTheta }(\varphi ) :=\bigcup _{\theta \in \varTheta }\mathsf {subf}^{\theta }(\varphi )\). For example, if \(\varTheta = \{p,{\lnot }p\}\), we will have \(\mathsf {subf}^{\varTheta }({\lnot }(q\vee r)) = \{q,r,q\vee r,{\lnot }(q\vee r)\}\cup \{ {\lnot }q,{\lnot }r,{\lnot }(q\vee r),{\lnot }{\lnot }(q\vee r)\}\). Such generalized notion of subformulas will be used in the next section to provide a more generous prooftheoretical concept of analyticity.
3 OneDimensional Consequence Relations
A SetSet statement (or sequent) is a pair \((\varPhi ,\varPsi ) \in \mathcal {P}(L_{\varSigma }(P)) \times \mathcal {P}(L_{\varSigma }(P))\), where \(\varPhi \) is dubbed the antecedent and \(\varPsi \) the succedent. A onedimensional consequence relation on \(L_{\varSigma }(P)\) is a collection of SetSet statements satisfying, for all \(\varPhi ,\varPsi ,\varPhi ^\prime ,\varPsi ^\prime \subseteq L_{\varSigma }(P)\),

(O) if \(\varPhi \cap \varPsi \ne \varnothing \), then

(D) if , then

(C) if for all \(\varPi \subseteq L_{\varSigma }(P)\), then
Properties (O), (D) and (C) are called overlap, dilution and cut, respectively. The relation is called substitutioninvariant when it satisfies, for every \(\sigma \in \mathsf {Subs}_{\varSigma }^{}\),

(S) if , then
and it is called finitary when it satisfies

(F) if , then for some finite \({\varPhi }^{\mathrm f} \subseteq \varPhi \) and \({\varPsi }^{\mathrm f} \subseteq \varPsi \)
Onedimensional consequence relations will also be referred to as onedimensional logics. Substitutioninvariant finitary onedimensional logics will be called standard. We will denote by \(\blacktriangleright _{}^{}\) the complement of \(\rhd _{}^{}\), called the compatibility relation associated with \(\rhd _{}^{}\) [10].
A SetFmla statement is a sequent having a single formula as consequent. When we restrict standard consequence relations to collections of SetFmla statements, we define the socalled (substitutioninvariant finitary) Tarskian consequence relations. Every onedimensional consequence relation \(\rhd _{}^{}\) determines a Tarskian consequence relation , dubbed the SetFmla Tarskian companion of \(\rhd _{}^{}\), such that, for all \(\varPhi \cup \{\psi \} \subseteq L_{\varSigma }(P)\), if, and only if, \(\varPhi \rhd _{}^{} \{\psi \}\). It is wellknown that the collection of all Tarskian consequence relations over a fixed language constitutes a complete lattice under settheoretical inclusion [25]. Given a set C of such relations, we will denote by \(\bigsqcup C\) its supremum in the latter lattice.
We present in what follows two ways of obtaining onedimensional consequence relations: one semantical, via nondeterministic logical matrices [6], and the other prooftheoretical, via SetSet Hilbertstyle systems [18, 23].
A nondeterministic \(\varSigma \)matrix, or simply \(\varSigma \)ndmatrix, is a structure \(\mathbb {M}:=\left\langle \mathbf {A},D \right\rangle \), where \(\mathbf {A}\) is a \(\varSigma \)ndalgebra, whose carrier is the set of truthvalues, and \(D\subseteq A\) is the set of designated truthvalues. Such structures are also known in the literature as ‘PNmatrices’ [7]; they generalize the socalled ‘Nmatrices’ [5], which are \(\varSigma \)ndmatrices with the restriction that \(\mathbf {A}\) must be total. From now on, whenever \(X\subseteq A\), we denote \({A}{\setminus }{X}\) by \(\overline{X}\). In case \(\mathbf {A}\) is deterministic, we simply say that \(\mathbb {M}\) is a \(\varSigma \) matrix. Also, \(\mathbb {M}\) is said to be finite when \(\mathbf {A}\) is finite. Every \(\varSigma \)ndmatrix \(\mathbb {M}\) determines a substitutioninvariant onedimensional consequence relation over \(\varSigma \), denoted by \(\rhd _{\mathbb {M}}^{}{}\), such that \( \varPhi \rhd _{\mathbb {M}}^{}{} \varPsi \text { if, and only if, for all } v\in \mathsf {Val}_{\varSigma }(\mathbf {A}), v[\varPhi ]~\cap ~\overline{D} \ne \varnothing \text { or } v[\varPsi ] \cap D\ne \varnothing . \)
It is worth noting that \(\rhd _{\mathbb {M}}^{}{}\) is finitary whenever the carrier of \(\mathbf {A}\) is finite (the proof runs very similar to that of the same result for Nmatrices [5, Theorem 3.15]).
A strong homomorphism between \(\varSigma \)matrices \(\mathbb {M}_1 :=\left\langle \mathbf {A}_1,D_1 \right\rangle \) and \(\mathbb {M}_2 :=\left\langle \mathbf {A}_2, D_2 \right\rangle \) is a homomorphism h between \(\mathbf {A}_1\) and \(\mathbf {A}_2\) such that \(x\in D_1\) if, and only if, \(h(x) \in D_2\). When there is a surjective strong homomorphism between \(\mathbb {M}_1\) and \(\mathbb {M}_2\), we have that .
Now, to the Hilbertstyle systems. A (schematic) SetSet rule of inference \(R_\mathsf s\) is the collection of all substitution instances of the SetSet statement \(\mathsf s\), called the schema of \(R_\mathsf s\). Each \(r\in R_\mathsf s\) is called a rule instance of \(R_\mathsf s\). A (schematic) SetSet Hsystem \(\mathsf {R}\) is a collection of SetSet rules of inference. When we constrain the rule instances of \(\mathsf {R}\) to having only singletons as succedents, we obtain the conventional notion of Hilbertstyle system, called here SetFmla Hsystem.
An \(\mathsf {R}\)derivation in a SetSet Hsystem \(\mathsf {R}\) is a rooted directed tree \(\mathsf t\) such that every node is labelled with sets of formulas or with a discontinuation symbol \(*\), and in which every nonleaf node (that is, a node with child nodes) \(\mathsf n\) in \(t\) is an expansion of \(\mathsf n\) by a rule instance \(r\) of \(\mathsf {R}\). This means that the antecedent of \(r\) is contained in the label of \(\mathsf n\) and that \(\mathsf n\) has exactly one child node for each formula \(\psi \) in the succedent of \(r\). These child nodes are, in turn, labelled with the same formulas as those of \(\mathsf n\) plus the respective formula \(\psi \). In case \(r\) has an empty succedent, then \(\mathsf n\) has a single child node labelled with \(*\). Here we will consider only finitary SetSet Hsystems, in which each rule instance has finite antecedent and succedent. In such cases, we only need to consider finite derivations. Figure 1 illustrates how derivations using only finitary rules of inference may be graphically represented. We denote by \(\ell ^{\,\mathsf t}(\mathsf n)\) the label of the node \(\mathsf n\) in the tree \(\mathsf t\). It is worth observing that, for SetFmla Hsystems, derivations are linear trees (as rule instances have a single formula in their succedents), or, in other words, just sequences of formulas built by applications of the rule instances, matching thus the conventional definition of Hilbertstyle systems.
A node \(\mathsf n\) of an \(\mathsf {R}\)derivation \(\mathsf t\) is called \(\varDelta \) closed in case it is a leaf node with \(\ell ^{\,\mathsf t}(\mathsf n) = *\) or \(\ell ^{\,\mathsf t}(\mathsf n) \cap \varDelta \ne \varnothing \). A branch of \(\mathsf t\) is \(\varDelta \)closed when it ends in a \(\varDelta \)closed node. When every branch in \(\mathsf t\) is \(\varDelta \)closed, we say that \(\mathsf {R}\) is itself \(\varDelta \) closed. An \(\mathsf {R}\) proof of a SetSet statement \((\varPhi ,\varPsi )\) is a \(\varPsi \)closed \(\mathsf {R}\)derivation \(\mathsf t\) such that \(\ell ^{\,\mathsf t}(\mathsf {rt}(\mathsf t)) \subseteq \varPhi \).
Consider the binary relation \(\rhd _{\mathsf {R}}^{}\) on \(\mathcal {P}(L_{\varSigma }(P))\) such that \(\varPhi \rhd _{\mathsf {R}}^{} \varPsi \) if, and only if, there is an \(\mathsf {R}\)proof of \((\varPhi ,\varPsi )\). This relation is the smallest substitutioninvariant onedimensional consequence relation containing the rules of inference of \(\mathsf {R}\), and it is finitary when \(\mathsf {R}\) is finitary. Since SetSet (and SetFmla) Hsystems canonically induce onedimensional consequence relations, we may refer to them as onedimensional Hsystems or onedimensional axiomatizations. In case there is a proof of \((\varPhi ,\varPsi )\) whose nodes are labelled only with subsets of \(\mathsf {subf}^{\varTheta }[\varPhi \cup \varPsi ]\), we write \(\varPhi \rhd _{\mathsf {R}}^{\varTheta } \varPsi \) . In case \(\rhd _{\mathsf {R}}^{} = \rhd _{\mathsf {R}}^{\varTheta }\), we say that \(\mathsf {R}\) is \(\varTheta \)analytic. Note that the ordinary notion of analyticity obtains when \(\varTheta = \{p\}\). From now on, whenever we use the word “analytic” we will mean this extended notion of \(\varTheta \)analyticity, for some \(\varTheta \) implicit in the context. When the \(\varTheta \) happens to be important for us or we identify any risk of confusion, we will mention it explicitly.
In [13], based on the seminal results on axiomatizability via SetSet Hsystems by Shoesmith and Smiley [23], it was proved that any nondeterministic logical matrix \(\mathbb {M}\) satisfying a criterion of sufficient expressiveness is axiomatizable by a \(\varTheta \)analytic SetSet Hilbertstyle system, which is finite whenever \(\mathbb {M}\) is finite, where \(\varTheta \) is the set of separators for the pairs of truthvalues of \(\mathbb {M}\). According to such criterion, an ndmatrix is sufficiently expressive when, for every pair \((x,y)\) of distinct truthvalues, there is a unary formula \(\mathrm S\), called a separator for \((x,y)\), such that \({\mathrm S}_{\mathbf {A}}(x) \subseteq D\) and \({\mathrm S}_{\mathbf {A}}(y) \subseteq \overline{D}\), or viceversa; in other words, when every pair of distinct truthvalues is separable in \(\mathbb {M}\).
We emphasize that it is essential for the above result the adoption of SetSet Hsystems, instead of the more restricted SetFmla Hsystems. In fact, while twovalued matrices may always be finitely axiomatized by SetFmla Hsystems [22], there are sufficiently expressive threevalued deterministic matrices [21] and even quite simple twovalued nondeterministic matrices [19] that fail to be finitely axiomatized by SetFmla Hsystems. When the ndmatrix at hand is not sufficiently expressive, we may observe the same phenomenon of not having a finite axiomatization also in terms of SetSet Hsystems, even if the said ndmatrix is finite. The first example (and, to the best of our knowledge, the only one in the current literature) of this fact appeared in [13], which we reproduce here for later reference:
Example 1
Consider the signature \(\varSigma :=\{{\varSigma }_{k}\}_{k\in \omega }\) such that \(\varSigma _1 :=\{g, h\}\) and \(\varSigma _k :=\varnothing \) for all \(k \ne 1\). Let \(\mathbb {M}:=\left\langle \mathbf {A}, \{\mathbf {a}\} \right\rangle \) be a \(\varSigma \)ndmatrix, with \(A:=\{\mathbf {a},\mathbf {b},\mathbf {c}\}\) and
This matrix is not sufficiently expressive because there is no separator for the pair \((\mathbf {b},\mathbf {c})\), and [13] proved that it is not axiomatizable by a finite SetSet Hsystem, even though an infinite SetSet system that captures it has a quite simple description in terms of the following infinite collection of schemas:
In the next section, we reveal another example of this same phenomenon, this time of the known LFI [14] called mCi. In the path of proving that this logic is not axiomatizable by a finite SetSet Hsystem, we will show that there are infinitely many LFIs between mbC and mCi, organized in a strictly increasing chain whose limit is mCi itself.
Before continuing, it is worth emphasizing that any given nonsufficiently expressive ndmatrix may be conservatively extended to a sufficiently expressive ndmatrix provided new connectives are added to the language [18]. These new connectives have the sole purpose of separating the pairs of truthvalues for which no separator is available in the original language. The SetSet system produced from this extended ndmatrix can, then, be used to reason over the original logic, since the extension is conservative. However, these new connectives, which a priori have no meaning, are very likely to appear in derivations of consecutions of the original logic. This might not look like an attractive option to inferentialists who believe that purity of the schematic rules governing a given logical constant is essential for the meaning of the latter to be coherently fixed. In the subsequent sections, we will introduce and apply a potentially more expressive notion of logic in order to provide a finite and analytic Hsystem for logics that are not finitely axiomatizable in one dimension, while preserving their original languages.
4 The Logic mCi is Not Finitely Axiomatizable
A onedimensional logic \(\rhd _{}^{}\) over \(\varSigma \) is said to be \({\lnot }\)paraconsistent when we have \(p,{\lnot }p\;\blacktriangleright _{}^{}\; q\), for \(p,q\in P\). Moreover, \(\rhd _{}^{}\) is \({\lnot }\)gently explosive in case there is a collection \(\bigcirc (p) \subseteq L_{\varSigma }(P)\) of unary formulas such that, for some \(\varphi \in L_{\varSigma }(P)\), we have \(\bigcirc (\varphi ), \varphi \blacktriangleright _{}^{} \varnothing \); \(\bigcirc (\varphi ), {\lnot }\varphi \blacktriangleright _{}^{} \varnothing \), and, for all \(\varphi \in L_{\varSigma }(P)\), \(\bigcirc (\varphi ),\varphi ,{\lnot }\varphi \; \rhd _{}^{} \varnothing \). We say that \(\rhd _{}^{}\) is a logic of formal inconsistency (LFI) in case it is \({\lnot }\)paraconsistent yet \({\lnot }\)gently explosive. In case \(\bigcirc (p) = \{{\circ }p\}\), for \({\circ }\) a (primitive or composite) consistency connective, the logic is said also to be a Csystem. In what follows, let \(\varSigma ^{{\circ }}\) be the propositional signature such that \(\varSigma ^{{\circ }}_{1} :=\{{\lnot },{\circ }\}\), \(\varSigma ^{{\circ }}_{2} :=\{\wedge ,\vee ,{\,\supset \,}\}\), and \(\varSigma ^{{\circ }}_{k} :=\varnothing \) for all \(k \not \in \{1,2\}\).
One of the simplest Csystems is the logic mbC, which was first presented in terms of a SetFmla Hsystem over \(\varSigma ^{{\circ }}\) obtained by extending any SetFmla Hsystem for positive classical logic (\({\textbf {CPL}}^{+}\)) with the following pair of axiom schemas:

\((\mathsf {em})\) \(p\vee {\lnot }p\)

\((\mathsf {bc1})\) \({\circ }p{\,\supset \,}(p{\,\supset \,}({\lnot }p{\,\supset \,}q))\)
The logic mCi, in turn, is the Csystem resulting from extending the Hsystem for mbC with the following (infinitely many) axiom schemas [20] (the resulting SetFmla Hsystem is denoted here by \(\mathcal {H}_{{\textbf {mCi }}}\)):

\((\mathsf {ci})\) \({\lnot }{\circ }p{\,\supset \,}(p\wedge {\lnot }p)\)

\((\mathsf {ci})\) \(_j\) \({\circ }{\lnot }^j{\circ }p\) (for all \(0 \le j < \omega \))
A unary connective \( {\copyright }\) is said to constitute a classical negation in a onedimensional logic \(\rhd _{}^{}\) extending \({\textbf {CPL}}^{+}{}\) in case, for all \(\varphi ,\psi \in L_{\varSigma }(P)\), \(\varnothing \rhd _{}^{} \varphi \vee {\copyright }(\varphi )\) and \(\varnothing \rhd _{}^{} \varphi {\,\supset \,}( {\copyright }(\varphi ) {\,\supset \,}\psi )\). One of the main differences between mCi and mbC is that an inconsistency connective \(\bullet \) may be defined in the former using the paraconsistent negation, instead of a classical negation, by setting \(\bullet \varphi :={\lnot }{\circ }\varphi \) [20].
Both logics above were presented in [15] in ways other than Hsystems: via tableau systems, via bivaluation semantics and via possibletranslations semantics. In addition, while these logics are known not to be characterizable by a single finite deterministic matrix [20], a characteristic ndmatrix is available for mbC [1] and a 5valued nondeterministic logical matrix is available for mCi [2], witnessing the importance of nondeterministic semantics in the study of nonclassical logics. Such characterizations, moreover, allow for the extraction of sequentstyle systems for these logics by the methodologies developed in [3, 4]. Since mCi’s 5valued ndmatrix will be useful for us in future sections, we recall it below for ease of reference.
Definition 1
Let \(\mathcal {V}_5:=\{f,F, I, T, t\}\) and \({\mathsf {Y}}_{5}:=\{I, T, t\}\). Define the \(\varSigma ^{{\circ }}\)matrix \(\mathbb {M}_{{\textbf {mCi }}}:=\left\langle \mathbf {A}_{5}, {\mathsf {Y}}_{5} \right\rangle \) such that \(\mathbf {A}_{5}:=\left\langle \mathcal {V}_5, {\cdot }_{\mathbf {A}_{5}} \right\rangle \) interprets the connectives of \(\varSigma ^{{\circ }}\) according to the following:
One might be tempted to apply the axiomatization algorithm of [13] to the finite nondeterministic logical matrix defined above to obtain a finite and analytic SetSet system for mCi. However, it is not obvious, at first, whether this matrix is sufficiently expressive or not (we will, in fact, prove that it is not). In what follows, we will show now mCi is actually axiomatizable neither by a finite SetFmla Hsystem (first part), nor by a finite SetSet Hsystem (second part); it so happens, thus, that it was not by chance that \(\mathcal {H}_{{\textbf {mCi }}}\) has been originally presented with infinitely many rule schemas. For the first part, we rely on the following general result:
Theorem 1
([25], Theorem 2.2.8, adapted). Let be a standard Tarskian consequence relation. Then is axiomatizable by a finite SetFmla Hsystem if, and only if, there is no strictly increasing sequence of standard Tarskian consequence relations such that .
In order to apply the above theorem, we first present a family of finite SetFmla Hsystems that, in the sequel, will be used to provide an increasing sequence of standard Tarskian consequence relations whose supremum is precisely mCi. Next, we show that this sequence is stricly increasing, by employing the matrix methodology traditionally used for showing the independence of axioms in a proof system.
Definition 2
For each \(k \in \omega \), let \(\mathcal {H}_{{\textbf {mCi }}}^k\) be a SetFmla Hsystem for positive classical logic together with the schemas (em), (bc1), (ci) and (ci)\(_j\), for all \(0 \le j \le k\).
Since \(\mathcal {H}_{{\textbf {mCi }}}^k\) may be obtained from \(\mathcal {H}_{{\textbf {mCi }}}\) by deleting some (infinitely many) axioms, it is immediate that:
Proposition 1
For every \(k \in \omega \), .
The way we define the promised increasing sequence of consequence relations in the next result is by taking the systems \(\mathcal {H}_{{\textbf {mCi }}}^{k}\) with odd superscripts, namely, we will be working with the sequence Excluding the cases where k is even will facilitate, in particular, the proof of Lemma 3.
Lemma 1
For each \(1 \le k < \omega \), let . Then , and
Finally, we prove that the sequence outlined in the paragraph before Lemma 1 is strictly increasing. In order to achieve this, we define, for each \(1 \le k < \omega \), a \(\varSigma ^{{\circ }}\)matrix \(\mathbb {M}_k\) and prove that \(\mathcal {H}_{{\textbf {mCi }}}^{2k1}\) is sound with respect to such matrix. Then, in the second part of the proof (the “independence part”), we show that, for each \(1 \le k < \omega \), \(\mathbb {M}_k\) fails to validate the rule schema (ci)\(_j\), for \(j = 2k\), which is present in \(\mathcal {H}_{{\textbf {mCi }}}^{2(k+1)1}\). In this way, by the contrapositive of the soundness result proved in the first part, we will have (ci)\(_j\) provable in \(\mathcal {H}_{{\textbf {mCi }}}^{2(k+1)1}\) while unprovable in \(\mathcal {H}_{{\textbf {mCi }}}^{2k1}\). In what follows, for any \(k \in \omega \), we use \({k}^{*}\) to refer to the successor of k.
Definition 3
Let \(1 \le k < \omega \). Define the \(2{k}^{*}\)valued \(\varSigma ^{{\circ }}\)matrix \(\mathbb {M}_{k} :=\left\langle \mathbf {A}_k, D_k \right\rangle \) such that \(D_k :=\{{k}^{*} + 1, \ldots , 2{k}^{*}\}\) and \(\mathbf {A}_k :=\left\langle \{1,\ldots ,2{k}^{*}\}, {\cdot }_{\mathbf {A}_k} \right\rangle \), the interpretation of \(\varSigma ^{{\circ }}\) in \(\mathbf {A}_k\) given by the following operations:
Before continuing, we state results concerning this construction, which will be used in the remainder of the current line of argumentation. In what follows, when there is no risk of confusion, we omit the subscript ‘\(\mathbf {A}_k\)’ from the interpretations to simplify the notation.
Lemma 2
For all \(k \ge 1\) and \(1 \le m \le 2k\),
Lemma 3
For all \(1 \le k < \omega \), we have but .
Finally, Theorem 1, Lemma 1 and Lemma 3 give us the main result:
Theorem 2
mCi is not axiomatizable by a finite SetFmla Hsystem.
For the second part —namely, that no finite SetSet Hsystem axiomatizes mCi—, we make use of the following result:
Theorem 3
([23], Theorem 5.37, adapted). Let be a onedimensional consequence relation over a propositional signature containing the binary connective \(\vee \). Suppose that the SetFmla Tarskian companion of , denoted by , satisfies the following property:
If a SetSet Hsystem \(\mathsf {R}\) axiomatizes , then \(\mathsf {R}\) may be converted into a SetFmla Hsystem for that is finite whenever \(\mathsf {R}\) is finite.
It turns out that:
Lemma 4
mCi satisfies (Disj).
Proof
The nondeterministic semantics of mCi gives us that, for all \(\varphi ,\psi \in L_{\varSigma ^{{\circ }}}(P)\), ; , and , and such facts easily imply (Disj).
Theorem 4
mCi is not axiomatizable by a finite SetSet Hsystem.
Proof
If \(\mathsf {R}\) were a finite SetSet Hsystem for mCi, then, by Lemma 4 and Theorem 3, it could be turned into a finite SetFmla Hsystem for this very logic. This would contradict Theorem 2.
Finding a finite onedimensional Hsystem for mCi (analytic or not) over the same language, then, proved to be impossible. The previous result also tells us that there is no sufficiently expressive nondeterministic matrix that characterizes mCi (for otherwise the recipe in [13] would deliver a finite analytic SetSet Hsystem for it), and we may conclude, in particular, that:
Corollary 1
The ndmatrix \(\mathbb {M}_{{\textbf {mCi }}}\) is not sufficiently expressive.
The pairs of truthvalues of \(\mathbb {M}_{{\textbf {mCi }}}\) that seem not to be separable (at least one of these pairs must not be, in view of the above corollary) are \((t,T)\) and \((f,F)\). The insufficiency of expressive power to take these specific pairs of values apart, however, would be circumvented if we had considered instead the matrix defined below, obtained from \(\mathbb {M}_{{\textbf {mCi }}}\) by changing its set of designated values:
Definition 4
Let \(\mathbb {M}_{{\textbf {mCi }}}^{\mathsf {n}}:=\left\langle \mathbf {A}_{5}, {\mathsf {N}}_{5} \right\rangle \), where \({\mathsf {N}}_{5}:=\{f, I, T\}\).
Note that, in \(\mathbb {M}_{{\textbf {mCi }}}^{\mathsf {n}}\), we have \(t\not \in {\mathsf {N}}_{5}\), while \(T\in {\mathsf {N}}_{5}\), and we have that \(f\in {\mathsf {N}}_{5}\), while \(F\not \in {\mathsf {N}}_{5}\). Therefore, the single propositional variable \(p\) separates in \(\mathbb {M}_{{\textbf {mCi }}}^{\mathsf {n}}\) the pairs \((t,T)\) and \((f,F)\). On the other hand, it is not clear now whether the pairs \((t,F)\) and \((f,T)\) are separable in this new matrix. Nonetheless, we will see, in the next section, how we can take advantage of the semantics of nondeterministic Bmatrices in order to combine the expressiveness of \(\mathbb {M}_{{\textbf {mCi }}}\) and \(\mathbb {M}_{{\textbf {mCi }}}^{\mathsf {n}}\) in a very simple and intuitive manner, preserving the language and the algebra shared by these matrices. The notion of logic induced by the resulting structure will not be onedimensional, as the one presented before, but rather twodimensional, in a sense we shall detail in a moment. We identify two important aspects of this combination: first, the logics determined by the original matrices can be fully recovered from the combined logic; and, second, since the notions of Hsystems and sufficient expressiveness, as well as the axiomatization algorithm of [13], were generalized in [17], the resulting twodimensional logic may be algorithmically axiomatized by an analytic twodimensional Hsystem that is finite if the combining matrices are finite, provided the criterion of sufficient expressiveness is satisfied after the combination. This will be the case, in particular, when we combine \(\mathbb {M}_{{\textbf {mCi }}}\) and \(\mathbb {M}_{{\textbf {mCi }}}^{\mathsf {n}}\). Consequently, this novel way of combining logics provides a quite general approach for producing finite and analytic axiomatizations for logics determined by nondeterministic logical matrices that fail to be finitely axiomatizable in one dimension; this includes the logics from Example 1, and also mCi.
5 TwoDimensional Logics
From now on, we will employ the symbols \({\mathsf {Y}}\), , \({\mathsf {N}}\) and to informally refer to, respectively, the cognitive attitudes of acceptance, nonacceptance, rejection and nonrejection, collected in the set . Given a set \(\varPhi \subseteq L_{\varSigma }(P)\), we will write \(\varPhi _{\alpha }\) to intuitively mean that a given agent entertains the cognitive attitude \(\alpha \in \mathsf {Atts}\) with respect to the formulas in \(\varPhi \), that is: the formulas in \(\varPhi _{\mathsf {Y}}\) will be understood as being accepted by the agent; the ones in , as nonaccepted; the ones in \(\varPhi _{\mathsf {N}}\), as rejected; and the ones in , as nonrejected. Where \(\alpha \in \mathsf {Atts}\), we let \(\tilde{\alpha }\) be its flipped version, that is, , , and .
We refer to each as a B statement, where \((\varPhi _{{\mathsf {Y}}},\varPhi _{\mathsf {N}})\) is the antecedent and is the succedent. The sets in the latter pairs are called components. A B consequence relation is a collection of Bstatements satisfying:

(O2) if or , then

(D2) if and \({\varPsi }_{\alpha } \subseteq {\varPhi }_{\alpha }\) for every \(\alpha \in \mathsf {Atts}\), then

(C2) if for all and , then
A Bconsequence relation is called substitutioninvariant if, in addition, holds whenever, for every \(\sigma \in \mathsf {Subs}_{\varSigma }^{}\):

(S2) and \({\varPhi }_{\alpha } = {\sigma }({\varPsi }_{\alpha })\) for every \(\alpha \in \mathsf {Atts}\)
Moreover, a Bconsequence relation is called finitary when it enjoys the property

(F2) if , then , for some finite \({\varPhi }_{\alpha }^{\mathtt f} \subseteq {\varPhi }_{\alpha }\), and each \(\alpha \in \mathsf {Atts}\)
In what follows, Bconsequence relations will also be referred to as twodimensional logics. The complement of , sometimes called the compatibility relation associated with [10], will be denoted by . Every Bconsequence relation induces onedimensional consequence relations and \(\rhd _{\mathsf {f}}^{\mathsf {C}}\), such that iff , and iff . Given a onedimensional consequence relation , we say that it inhabits the \(\mathsf {t}\)aspect of \(\mathsf {C}\) if , and that it inhabits the \(\mathsf {f}\)aspect of \(\mathsf {C}\) if . Bconsequence relations actually induce many other (even nonTarskian) onedimensional notions of logics; the reader is referred to [9, 11] for a thorough presentation on this topic.
As we did for onedimensional consequence relations, we present now realizations of Bconsequence relations, first via the semantics of ndBmatrices, then by means of twodimensional Hsystems.
A nondeterministic B matrix over \(\varSigma \), or simply \(\varSigma \) ndB matrix, is a structure \(\mathfrak {M}:=\left\langle \mathbf {A}, {\mathsf {Y}}, {\mathsf {N}} \right\rangle \), where \(\mathbf {A}\) is a \(\varSigma \)ndalgebra, \({\mathsf {Y}} \subseteq A\) is the set of designated values and \({\mathsf {N}} \subseteq A\) is the set of antidesignated values of \(\mathfrak {M}\). For convenience, we define to be the set of nondesignated values, and to be the set of nonantidesignated values of \(\mathfrak {M}\). The elements of \(\mathsf {Val}_{\varSigma }(\mathbf {A})\) are dubbed \(\mathfrak {M}\) valuations. The B entailment relation determined by \(\mathfrak {M}\) is a collection of Bstatements such that

(B ent)
for every . Whenever , we say that the Bstatement holds in \(\mathfrak {M}\) or is valid in \(\mathfrak {M}\). An \(\mathfrak {M}\)valuation that bears witness to is called a countermodel for in \(\mathfrak {M}\). One may easily check that is a substitutioninvariant Bconsequence relation, that is finitary when \(A\) is finite. Taking \(\mathsf {C}\) as , we define \(\rhd _{\mathsf {t}}^{\mathfrak {M}} :=\rhd _{\mathsf {t}}^{\mathsf {C}}\) and \(\rhd _{\mathsf {f}}^{\mathfrak {M}} :=\rhd _{\mathsf {f}}^{\mathsf {C}}\).
We move now to twodimensional, or \(\textsc {Set}^{2}\text {}\textsc {Set}^{2}\), Hsystems, first introduced in [17]. A (schematic) \(\textsc {Set}^{2}\text {}\textsc {Set}^{2}\) rule of inference \(R_\mathfrak {s}\) is the collection of all substitution instances of the \(\textsc {Set}^{2}\text {}\textsc {Set}^{2}\) statement \(\mathfrak {s}\), called the schema of \(R_\mathfrak {s}\). Each \(r\in R_\mathfrak {s}\) is said to be a rule instance of \(R_\mathfrak {s}\). In a prooftheoretic context, rather than writing the Bstatement , we shall denote the corresponding rule by . A (schematic) \(\textsc {Set}^{2}\text {}\textsc {Set}^{2}\) Hsystem \(\mathfrak {R}\) is a collection of \(\textsc {Set}^{2}\text {}\textsc {Set}^{2}\) rules of inference. \(\textsc {Set}^{2}\text {}\textsc {Set}^{2}\) derivations are as in the SetSet Hsystems, but now the nodes are labelled with pairs of sets of formulas, instead of a single set. When applying a rule instance, each formula in the succedent produces a new branch as before, but now the formula goes to the same component in which it was found in the rule instance. See Fig. 2 for a general representation and compare it with Fig. 1.
Let \(\mathfrak {t}\) be an \(\mathfrak {R}\)derivation. A node \(\mathfrak {n}\) of \(\mathfrak {t}\) is closed in case it is discontinued (namely, labelled with \(*\)) or it is a leaf node with \(\ell ^{\,\mathfrak {t}}(\mathfrak {n}) = (\varPhi _{{\mathsf {Y}}},\varPhi _{\mathsf {N}})\) and either or . A branch of \(\mathfrak {t}\) is closed when it ends in a closed node. An \(\mathfrak {R}\)derivation \(\mathfrak {t}\) is said to be closed when all of its branches are closed. An \(\mathfrak {R}\) proof of is a closed \(\mathfrak {R}\)derivation \(\mathfrak {t}\) with \(\ell ^{\,\mathfrak {t}}(\mathsf {rt}(\mathfrak {t})) \subseteq (\varPhi _{{\mathsf {Y}}},\varPhi _{\mathsf {N}})\). The definitions of the (finitary) substitutioninvariant Bconsequence relation induced by a (finitary) \(\textsc {Set}^{2}\text {}\textsc {Set}^{2}\) Hsystem \(\mathfrak {R}\) and \(\varTheta \)analyticity are obvious generalizations of the corresponding SetSet definitions.
In [17], the notion of sufficient expressiveness was generalized to ndBmatrices. We reproduce here the main definitions for selfcontainment:
Definition 5
Let \(\mathfrak {M}:=\left\langle \mathbf {A}, {\mathsf {Y}}, {\mathsf {N}} \right\rangle \) be a \(\varSigma \)ndBmatrix.

Given \(X,Y\subseteq A\) and \(\alpha \in \{{\mathsf {Y}},{\mathsf {N}}\}\), we say that \(X\) and \(Y\) are \(\alpha \) separated, denoted by \({X}\#_{\alpha }{Y}\), if \(X\subseteq \alpha \) and \(Y\subseteq \tilde{\alpha }\), or viceversa.

Given distinct truthvalues \(x, y\in A\), a unary formula \(\mathrm S\) is a separator for \((x,y)\) whenever \({{\mathrm S}_{\mathbf {A}}(x)}\#_{\alpha }{{\mathrm S}_{\mathbf {A}}(y)}\) for some \(\alpha \in \{{\mathsf {Y}},{\mathsf {N}}\}\). If there is a separator for each pair of distinct truthvalues in \(A\), then \(\mathfrak {M}\) is said to be sufficiently expressive.
In the same work [17], the axiomatization algorithm of [13] was also generalized, guaranteeing that every sufficiently expressive ndBmatrix \(\mathfrak {M}\) is axiomatizable by a \(\varTheta \)analytic \(\textsc {Set}^{2}\text {}\textsc {Set}^{2}\) Hsystem, which is finite whenever \(\mathfrak {M}\) is finite, where \(\varTheta \) is a set of separators for the pairs of truthvalues of \(\mathfrak {M}\). Note that, in the second bullet of the above definition, a unary formula is characterized as a separator whenever it separates a pair of truthvalues according to at least one of the distinguished sets of values. This means that having two of such sets may allow us to separate more pairs of truthvalues than having a single set, that is, the ndBmatrices are, in this sense, potentially more expressive than the (onedimensional) logical matrices.
Example 2
Let \(\mathbf {A}\) be the \(\varSigma \)ndalgebra from Example 1, and consider the ndBmatrix \(\mathfrak {M}:=\left\langle \mathbf {A},\{\mathbf {a}\},\{\mathbf {b}\} \right\rangle \). As we know, in this matrix the pair \((\mathbf {b},\mathbf {c})\) is not separable if we consider only the set of designated values \(\{\mathbf {a}\}\). However, as we have now the set \(\{\mathbf {b}\}\) of antidesignated truthvalues, the separation becomes evident: the propositional variable \(p\) is a separator for this pair now, since \(\mathbf {b}\in \{\mathbf {b}\}\) and \(\mathbf {c}\not \in \{\mathbf {b}\}\). The recipe from [17] produces the following \(\textsc {Set}^{2}\text {}\textsc {Set}^{2}\) axiomatization for \(\mathfrak {M}\), with only three very simple schematic rules of inference:
By construction, the onedimensional logic determined by the ndmatrix of Example 1 inhabits the \(\mathsf {t}\)aspect of , thus it can be seen as being axiomatized by this finite and analytic twodimensional system (contrast with the infinite SetSet axiomatization known for this logic provided in that same example).
We constructed above a \(\varSigma \)ndBmatrix from two \(\varSigma \)ndmatrices in such a way that the onedimensional logics determined by latter are fully recoverable from the former. We formalize this construction below:
Definition 6
Let \(\mathbb {M}:=\left\langle \mathbf {A},D \right\rangle \) and \(\mathbb {M}' :=\left\langle \mathbf {A},D' \right\rangle \) be \(\varSigma \)ndmatrices. The B product between \(\mathbb {M}\) and \(\mathbb {M}'\) is the \(\varSigma \)ndBmatrix \(\mathbb {M}\odot \mathbb {M}' :=\left\langle \mathbf {A},D,D' \right\rangle \).
Note that iff iff \(\varPhi \rhd _{\mathsf {t}}^{\mathbb {M}\odot \mathbb {M}'}\varPsi \), and iff iff \(\varPhi \rhd _{\mathsf {f}}^{\mathbb {M}\odot \mathbb {M}'}\varPsi \). Therefore, and are easily recoverable from , since they inhabit, respectively, the \(\mathsf {t}\)aspect and the \(\mathsf {f}\)aspect of the latter. One of the applications of this novel way of putting two distinct logics together was illustrated in that same Example 2 to produce a twodimensional analytic and finite axiomatization for a onedimensional logic characterized by a \(\varSigma \)ndmatrix. As we have shown, the latter onedimensional logic does not need to be finitely axiomatizable by a SetSet Hsystem. We present this application of Bproducts with more generality below:
Proposition 2
Let \(\mathbb {M}:=\left\langle \mathbf {A},D \right\rangle \) be a \(\varSigma \)ndmatrix and suppose that \(U \subseteq A\times A\) contains all and only the pairs of distinct truthvalues that fail to be separable in \(\mathbb {M}\). If, for some \(\mathbb {M}' :=\left\langle \mathbf {A},D' \right\rangle \), the pairs in U are separable in \(\mathbb {M}'\), then \(\mathbb {M}\odot \mathbb {M}'\) is sufficiently expressive (thus, axiomatizable by an analytic \(\textsc {Set}^{2}\text {}\textsc {Set}^{2}\) Hsystem, that is finite whenever \(\mathbf {A}\) is finite).
6 A Finite and Analytic Proof System for mCi
In the spirit of Proposition 2, we define below a ndBmatrix by combining the matrices \(\mathbb {M}_{{\textbf {mCi }}}:=\left\langle \mathbf {A}_{5}, {\mathsf {Y}}_{5} \right\rangle \) and \(\mathbb {M}_{{\textbf {mCi }}}^{\mathsf {n}}:=\left\langle \mathbf {A}_{5}, {\mathsf {N}}_{5} \right\rangle \) introduced in Sect. 4 (Definition 1 and Definition 4):
Definition 7
Let \(\mathfrak {M}_{{\textbf {mCi }}}:=\mathbb {M}_{{\textbf {mCi }}}\odot \mathbb {M}_{{\textbf {mCi }}}^{\mathsf {n}}= \left\langle \mathbf {A}_{5}, {\mathsf {Y}}_{5}, {\mathsf {N}}_{5} \right\rangle \), with \({\mathsf {Y}}_{5}:=\{I, T, t\}\) and \({\mathsf {N}}_{5}:=\{f,I,T\}\).
When we consider now both sets \({\mathsf {Y}}_{5}\) and \({\mathsf {N}}_{5}\) of designated and antidesignated truthvalues, the separation of all truthvalues of \(\mathbf {A}_{5}\) becomes possible, that is, \(\mathfrak {M}_{{\textbf {mCi }}}\) is sufficiently expressive, as guaranteed by Proposition 2. Furthermore, notice that we have two alternatives for separating the pairs \((I,t)\) and \((I,T)\): either using the formula \({\lnot }p\) or the formula \({\circ }p\). With this finite sufficiently expressive ndBmatrix in hand, producing a finite \(\{p,{\circ }p\}\)analytic twodimensional Hsystem for it is immediate by [17, Theorem 2]. Since mCi inhabits the \(\mathsf {t}\)aspect of , we may then conclude that:
Theorem 5
\({\textbf {mCi }}{}\) is axiomatizable by a finite and analytic twodimensional Hsystem.
Our axiomatization recipe delivers an Hsystem with about 300 rule schemas. When we simplify it using the streamlining procedures indicated in that paper, we obtain a much more succinct and insightful presentation, with 28 rule schemas, which we call \(\mathfrak {R}_{{\textbf {mCi }}}\). The full presentation of this system is given below:
Note that the set of rules \(\{{ {\copyright }}^{\textbf {mCi }}_{i} \mid {\copyright }\in \{\wedge ,\vee ,{\,\supset \,}\}, i \in \{1,2,3\}\}\) makes it clear that the \(\mathsf {t}\)aspect of the induced Bconsequence relation is inhabited by a logic extending positive classical logic, while the remaining rules for these connectives involve interactions between the two dimensions. Also, rule \({{\lnot }}^{\textbf {mCi }}_{2}\) indicates that \({\circ }\) satisfies one of the main conditions for being taken as a consistency connective in the logic inhabiting the \(\mathsf {t}\)aspect. In fact, all these observations are aligned with the fact that the logic inhabiting the \(\mathsf {t}\)aspect of is precisely mCi. See, in Fig. 3, \(\mathfrak {R}_{{\textbf {mCi }}}{}\)derivations showing that, in mCi, \({\lnot }{\circ }p\) and \(p{\wedge }{\lnot }p\) are logically equivalent and that \({\circ }{\lnot }{\circ }p\) is a theorem.
7 Concluding Remarks
In this work, we introduced a mechanism for combining two nondeterministic logical matrices into a nondeterministic Bmatrix, creating the possibility of producing finite and analytic twodimensional axiomatizations for onedimensional logics that may fail to be finitely axiomatizable in terms of onedimensional Hilbertstyle systems. It is worth mentioning that, as proved in [17], one may perform proof search and countermodel search over the resulting twodimensional systems in time at most exponential on the size of the Bstatement of interest through a straightforward proofsearch algorithm.
We illustrated the abovementioned combination mechanism with two examples, one of them corresponding to a wellknown logic of formal inconsistency called mCi. We ended up proving not only that this logic is not finitely axiomatizable in one dimension, but also that it is the limit of a strictly increasing chain of LFIs extending the logic mbC. From the perspective of the study of Bconsequence relations, these examples allow us to eliminate the suspicion that a twodimensional Hsystem \(\mathfrak {R}\) may always be converted into SetSet Hsystems for the logics inhabiting the onedimensional aspects of without losing any desirable property (in this case, finiteness of the presentation).
At first sight, the formalism of twodimensional Hsystems may be confused with the formalism of nsided sequents [3, 4], in which the objects manipulated by rules of inference (the socalled nsequents) accommodate more than two sets of formulas in their structures. The reader interested in a comparison between these two different approaches is referred to the concluding remarks of [17].
We close with some observations regarding \(\mathfrak {M}_{{\textbf {mCi }}}\) and the twodimensional Hsystem \(\mathfrak {R}_{{\textbf {mCi }}}\). A onedimensional logic \(\rhd _{}^{}\) is said to be \({\lnot }\)consistent when \(\varphi ,{\lnot }\varphi \rhd _{}^{} \varnothing \) and \({\lnot }\)determined when \(\varnothing \rhd _{}^{} \varphi ,{\lnot }\varphi \) for all \(\varphi \in L_{\varSigma }(P)\). A Bconsequence relation is said to allow for gappy reasoning when and to allow for glutty reasoning when , for some \(\varphi \in L_{\varSigma }(P)\). Notice that \({\lnot }\)determinedness in the logic inhabiting the \(\mathsf {t}\)aspect of a Bconsequence relation by no means implies the disallowance of gappy reasoning in the twodimensional setting: we still have \(F\in \overline{{\mathsf {Y}}_{5}} \cap \overline{{\mathsf {N}}_{5}}\), so one may both nonaccept and nonreject a formula \(\varphi \) in , even though nonaccepting both \(\varphi \) and its negation in mCi is not possible, in view of rule \({{\lnot }}^{\textbf {mCi }}_{7}\). Similarly, the recovery of \({\lnot }\)consistency achieved via \({\circ }\) in such logic does not coincide with the gentle disallowance of glutty reasoning in , that is, we do not have, in general, or , even though for binary compounds both are derivable in view of rules \({ {\copyright }}^{\textbf {mCi }}_{5}\), for \( {\copyright }\in \{\wedge ,\vee ,{\,\supset \,}\}\), and \({\circ }_1^{{\textbf {mCi }}{}}\). With these observations we hope to call attention to the fact that Bconsequence relations open the doors for further developments concerning the study of paraconsistency (and, dually, of paracompleteness), as well as the study of recovery operators [8].
Notes
 1.
Detailed proofs of some results may be found in https://arxiv.org/abs/2205.08920.
References
Avron, A.: Nondeterministic matrices and modular semantics of rules. In: Beziau, J.Y. (ed.) Logica Universalis, pp. 149–167. Birkhäuser, Basel (2005). https://doi.org/10.1007/3764373040_9
Avron, A.: 5valued nondeterministic semantics for the basic paraconsistent logic mCi. Stud. Log. Grammar Rhetoric 127–136 (2008)
Avron, A., BenNaim, J., Konikowska, B.: Cutfree ordinary sequent calculi for logics having generalized finitevalued semantics. Log. Univers. 1, 41–70 (2007). https://doi.org/10.1007/s1178700600036
Avron, A., Konikowska, B.: Multivalued calculi for logics based on nondeterminism. Log. J. IGPL 13(4), 365–387 (2005). https://doi.org/10.1093/jigpal/jzi030
Avron, A., Lev, I.: Nondeterministic multiplevalued structures. J. Log. Comput. 15(3), 241–261 (2005). https://doi.org/10.1093/logcom/exi001
Avron, A., Zamansky, A.: Nondeterministic semantics for logical systems. In: Gabbay, D.M., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 16, pp. 227–304. Springer, Dordrecht (2011). https://doi.org/10.1007/9789400704794_4
Baaz, M., Lahav, O., Zamansky, A.: Finitevalued semantics for canonical labelled calculi. J. Autom. Reason. 51(4), 401–430 (2013). https://doi.org/10.1007/s108170139273x
Barrio, E.A., Carnielli, W.: Volume I: recovery operators in logics of formal inconsistency (special issue). Log. J. IGPL 28(5), 615–623 (2019). https://doi.org/10.1093/jigpal/jzy053
Blasio, C.: Revisitando a lógica de DunnBelnap. Manuscrito 40, 99–126 (2017). https://doi.org/10.1590/01006045.2017.v40n2.cb
Blasio, C., Caleiro, C., Marcos, J.: What is a logical theory? On theories containing assertions and denials. Synthese 198(22), 5481–5504 (2019). https://doi.org/10.1007/s1122901902183z
Blasio, C., Marcos, J., Wansing, H.: An inferentially manyvalued twodimensional notion of entailment. Bull. Sect. Log. 46(3/4), 233–262 (2017). https://doi.org/10.18778/01380680.46.3.4.05
Burris, S., Sankappanavar, H.: A Course in Universal Algebra, vol. 91 (1981)
Caleiro, C., Marcelino, S.: Analytic calculi for monadic PNmatrices. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds.) WoLLIC 2019. LNCS, vol. 11541, pp. 84–98. Springer, Heidelberg (2019). https://doi.org/10.1007/9783662595336_6
Carnielli, W., Marcos, J.: A taxonomy of Csystems. In: Paraconsistency: The logical way to the inconsistent. Taylor and Francis (2002). https://doi.org/10.1201/97802039101393
Carnielli, W.A., Coniglio, M.E., Marcos, J.: Logics of formal inconsistency. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 14, 2nd edn., pp. 1–93. Springer, Dordrecht (2007). https://doi.org/10.1007/9781402063244_1
Carnielli, W.A., Marcos, J.: Limits for paraconsistent calculi. Notre Dame J. Formal Log. 40(3), 375–390 (1999). https://doi.org/10.1305/ndjfl/1022615617
Greati, V., Marcelino, S., Marcos, J.: Proof search on bilateralist judgments over nondeterministic semantics. In: Das, A., Negri, S. (eds.) TABLEAUX 2021. LNCS (LNAI), vol. 12842, pp. 129–146. Springer, Cham (2021). https://doi.org/10.1007/9783030860592_8
Marcelino, S., Caleiro, C.: Axiomatizing nondeterministic manyvalued generalized consequence relations. Synthese 198(22), 5373–5390 (2019). https://doi.org/10.1007/s11229019021428
Marcelino, S.: An unexpected Boolean connective. Log. Univer. (2021). https://doi.org/10.1007/s11787021002807
Marcos, J.: Possibletranslations semantics for some weak classicallybased paraconsistent logics. J. Appl. NonClassical Log. 18(1), 7–28 (2008). https://doi.org/10.3166/jancl.18.728
Palasinska, K.: Deductive systems and finite axiomatization properties. Ph.D. thesis, Iowa State University (1994). https://doi.org/10.31274/rtd18081312680
Rautenberg, W.: 2element matrices. Stud. Log. 40(4), 315–353 (1981)
Shoesmith, D.J., Smiley, T.J.: MultipleConclusion Logic. Cambridge University Press, Cambridge (1978). https://doi.org/10.1017/CBO9780511565687
Wójcicki, R.: Some remarks on the consequence operation in sentential logics. Fundam. Math. 68, 269–279 (1970)
Wójcicki, R.: Theory of Logical Calculi. Synthese Library, 1 edn., Springer, Dordrecht (1988). https://doi.org/10.1007/9789401569422
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Greati, V., Marcos, J. (2022). Finite TwoDimensional Proof Systems for Nonfinitely Axiomatizable Logics. In: Blanchette, J., Kovács, L., Pattinson, D. (eds) Automated Reasoning. IJCAR 2022. Lecture Notes in Computer Science(), vol 13385. Springer, Cham. https://doi.org/10.1007/9783031107696_37
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