Logical Multilateralism

In this paper we will consider the existing notions of bilateralism in the context of proof-theoretic semantics and propose, based on our understanding of bilateralism, an extension to logical multilateralism. This approach differs from what has been proposed under this name before in that we do not consider multiple speech acts as the core of such a theory but rather multiple consequence relations. We will argue that for this aim the most beneficial proof-theoretical realization is to use sequent calculi with multiple sequent arrows satisfying some specific conditions, which we will lay out in this paper. We will unfold our ideas with the help of a case study in logical tetralateralism and present an extension of Almukdad and Nelson’s propositional constructive four-valued logic by unary operations of meaningfulness and nonsensicality. We will argue that in sequent calculi with multiple sequent arrows it is possible to maintain certain features that are desirable if we assume an understanding of the meaning of connectives in the spirit of proof-theoretic semantics. The use of multiple sequent arrows will be justified by the presence of congruentiality-breaking unary connectives.


Introduction
Proof-theoretic bilateralism is located in the area of proof-theoretic semantics (PTS), i.e., in an inferential theory of meaning of the logical connectives (see [52] for an extensive overview of this area, as well as [16]).In PTS it is argued that the meaning of logical connectives is determined by the rules of inference that govern their use in proofs.It is commonly agreed, however, that these rules cannot just be of any kind but that they have to meet certain conditions, among these harmony, uniqueness and purity are often listed.In standard intuitionistic natural deduction the rules display all these desirable properties, while in the classical calculus the rules for negation are not harmonious.That is why from a PTS point of view intuitionistic logic is often seen as preferable over classical logic.The term bilateralism was coined by Rumfitt in his seminal paper [50] in which he argues for a 'bilateral' proof system of natural deduction with introduction and elimination rules determining not only the assertion conditions for formulas containing the connective in question but also the denial conditions.Rumfitt's aim was to argue that with a proof system taking assertion and denial on a par, harmony could be retained in such a bilateral (classical) system and thus give a motivation why the rules of classical logic lay down the meaning of the connectives. 1Since then, bilateralism has been strongly associated with speech acts or speech act-related conditions.
However, we want to consider a different approach to bi-and ultimately multilateralism, which is n-lateral not on the level of formulas but on the level of the underlying finitary consequence relation in a proof system.Therefore, we do not use signed formulas, as e.g., Rumfitt does to indicate assertion or denial of this formula, but instead signed sequent arrows (in sequent calculus systems).On our understanding, proof systems are manipulating formulas or sequents interpreted as propositions, not as speech acts.Of course, inferences may be interpreted in terms of speech acts on certain readings but first and foremost we would not say that this is necessary for understanding inference rules as giving rise to derivations.Rather, we want to motivate proof systems which display more than the usual derivability relation capturing preservation of truth from premises to conclusion.This has been devised for bilateral inferential relations for the bi-intuitionistic logic 2Int in [63] and [3] with a natural deduction and a sequent calculus system respectively, or in [27] for Nelson's constructive logic with strong negation, N4.Also, in [11,12] a general theory about bilateral consequence relations is developed at the example of 2Int with a Hilbert-style calculus using both signed consequence relations and signed formulas, and in [8] an approach to studying logical consequence is given, which is based on bilateral consequence operators.
What we argue for in this paper is that from a multilateral point of view it may be desirable to extend our language with certain operators that are problematic in so far as they turn our system non-congruential.This means that in such a system it does not hold that whenever two formulas are interderivable, then also any two (complex) formulas containing them are interderivable.However, non-congruentiality poses a problem for the uniqueness condition of operators.If the condition of uniqueness is met, then it is ensured that the rules governing the inferential behavior of a connective characterize one and only one connective, i.e., a copy-cat connective characterized by the same rules cannot play a different role in inferences (cf.[5]).Hence, if uniqueness 1 For critical assessments of that paper see e.g.[19] or [29].The former points out that exchanging one of the rules for negation in Rumfitt's system would still meet the requirements of Rumfitt's own adequacy constraints while not yielding double negation elimination, which would be crucial for Rumfitt's argument.It is also mentioned [19, fn. 2] that the rules of Rumfitt's system without his so-called co-ordination principles (and which Rumfitt already describes as a formalisation of classical logic [50, p. 803]) actually yield the logic N4 and not classical logic.In [29] it is shown that the meaning-theoretical requirements, that Rumfitt imposes on his system and which he uses to argue for the preferability of classical logic over intuitionistic logic in a bilateral setting, can also be maintained in a bilateral formulation of intuitionistic logic.fails, i.e., the rules do not necessarily define one connective, it seems that we could not claim from a PTS point of view that the meaning of the connectives is indeed given by their rules of inference.After all, if we would have a set of rules for a single logical operation characterizing several connectives, how could the rules be said to give the meaning of those connectives?If it is the same rules for two connectives, for a PTS theorist these must have the same meaning.Failure of uniqueness, though, tells us that they behave differently in inferences, which in PTS must mean to be different in meaning.Note that this is not to say that uniqueness depends solely on the rules governing a connective since these have always to be considered against the background of the conditions on the consequence relation assumed in the proof system.
The problem with congruentiality-breaking operators is that in order to maintain uniqueness in such a system, uniqueness of the other operators would have to be specified in terms of the congruentiality-breaking operators (for details, see Section 4).With the solution we will propose, however, namely to consider finitary multiple consequence relations, we can show that these meaning-theoretical features can be retained in such a system.We will provide a definition of congruentiality that states it as a purely structural property, i.e., without the need to mention any specific operator and thereby, uniqueness can be characterized in such a system independently of any operators, as well.Thus, in our approach to what constitutes a multilateral logic, a proof-theoretic semantic understanding of the connectives is possible.
In Section 2 we will give an overview of what has been conceived of as 'bilateralism' in the literature (2.1) and also of tentative approaches to extend this to multilateralism (2.2).Then, we will also present our approach to multilateralism (2.3) and how this differs from and extends what has been proposed so far.In Section 3 we will present a case study showing as a specific instantiation of our general approach a tetralateral calculus for the logic N4 expanded with operators for meaningfulness and nonsensicality. 2Finally, in Section 4 we will give a general definition of the notion of an (n+1)lateral sequent calculus and show how the problems with non-congruentiality and uniqueness can be resolved in such a calculus.

The Concept of Bilateralism
There are different conceptions captured under the notion of 'bilateralism', although the differences have been kept rather concealed in the literature.The only place, we are aware of, where a distinction between different 'kinds' of bilateralism is made explicit is in [48,49].Thus, it might be helpful to give a brief overview of how this notion is conceived before going on to the extension of this concept to a kind of multilateralism, in our case study tetralateralism.Although Rumfitt's paper [50] is the origin of bilateralism in the sense that the concrete term and idea are introduced therein and spelled out thoroughly, there are some predecessors to the general idea that are frequently cited, like [41], [57], and [22].The most frequent characterization that is used for bilateralism is that it is a theory of meaning displaying a symmetry between certain notions (or often rather: conditions governing these notions), which have not been considered being on a par by 'conventional' theories of meaning. 3The relevant notions are most often 'assertion' and 'denial', or 'assertibility' and 'deniability', sometimes also 'acceptance' and 'rejection'.While the former are usually taken to describe speech acts, the latter are usually -though not always (see [49] for a thorough distinction) -considered to describe the corresponding internal cognitive states or attitudes.'Assertibility' and 'deniability', on the other hand, are of a third kind, since they can be seen to describe something like properties of propositions.The symmetry between these respective concepts is often described with expressions like "both being primitive", "not reducible to each other", "being on a par", and "of equal importance".Another point to characterize bilateralism, which is often mentioned, though not as frequent or central as the former point, 4 is that in a bilateral approach the denial of A is not interpreted in terms of, or as the assertion of the negation of A but that it is the other way around: In bilateralism rejection and/or denial often are considered as conceptually prior to negation.
Ripley [48,49] distinguishes two camps of bilateral theories of meaning in terms of "what kinds of condition on assertion and denial they appeal to" [49]: a warrant-based approach and a coherence-based approach, for the latter of which he himself argues [47] and which was firstly devised by Restall [42,43]. 5As references for the first camp, which Ripley calls the 'orthodox' bilateralism, Price [41], Smiley [57], and Rumfitt [50] are given.Warrant-based bilateralism takes the relevant conditions to be the ones under which propositions can be warrantedly asserted or denied.Coherence-based bilateralism, on the other hand, takes the relevant conditions to be the conditions under which collections of propositions can be coherently asserted and/or denied together.The two approaches also differ in their design and interpretations of proof systems.Rumfitt uses these philosophical considerations on speech acts determining the sense of sentences to motivate a proof system of natural deduction with signed formulas for assertion and denial.In this system rules do not apply to propositions but to speech acts.On the other hand, the coherence-based approach rather suggests a specific kind of reading of our proof systems.Here, usually sequent calculi are of interest (see Section 2.2), although in a recent paper Restall considers an approach from this realm using a natural deduction system, which does not employ signed formulas but rather uses different positions for certain commitments from which the inference is drawn to the conclusion, namely from assumptions, which are ruled in, and alternatives, which are ruled out [44,45].The advantage of this system compared to Rumfitt's is that the pragmatic status of discharged formulas is much clearer in that they are taken as temporary suppositions for the sake of argument and not simply as assertions or denials.The latter has been criticized about Rumfitt's system because it seems to cause problems if we have to say about, e.g., a (discharged) formula signed with +, that an assertion is assumed (discharged). 6Firstly, making an assumption seems to be a speech act, as well; however, it is generally agreed upon (by Rumfitt, too) that speech acts cannot be embedded into one another.Secondly, it is not clear what it would mean to discharge a speech act since this is an action that is carried out and thus, cannot simply be 'taken back'.

Extensions of Bilateralism
The literature on a concept like multilateralism, understood as an extension of the concept of bilateralism in PTS, is relatively scarce.One example is a paper by Hjortland [21], in which he relates inference rules with truth conditions in order to determine the nature of the 'semantic content' of proof conditions in PTS.Therefore, he considers firstly, Carnap's categoricity condition, which is supposed to capture whether inference rules actually uniquely determine truth conditions, and secondly, multiple conclusion sequent calculus systems in a reading motivated by Restall [42], in which sequents can be read as statements containing the speech acts of assertion and denial.Restall argues that taking an argument to be valid (represented by the sequent sign ) means to commit to taking the assertion of the premises to stand against the denial of the conclusion. 7Logical consequence governs positions involving statements that are asserted and denied in the following way: If Γ , then the position of asserting each of the members of Γ while simultaneously denying each of the members of clashes or is incoherent.Hjortland extends these sequent systems in considering n-sided sequents with positions for more speech acts that go beyond assertion and denial, e.g., doubting or being indifferent.With these multilateral systems he shows that categoricity can be proven for arbitrary finite many-valued logics.
Another paper in which concepts extending bilateralism in terms of tri-or multilateralism are explicitly mentioned is by Francez [17].Similar to Hjortland's approach, Francez wants to extend Restall's bilateral understanding of multiple conclusion sequent systems by considering more positions representing more speech acts.However, instead of sequents he wants to consider poly-sequents having positions (in the bilateral case) for both assertions and denials on the assumption side as well as on the denial side, before he also goes on to extend this bilateral version to multilateral versions represented by n-sided poly-sequents connecting these with many-valued logics.Thus, he claims, the asymmetry that he says is inherent in Restall's understanding in that assumptions are only associated with assertions and conclusions only with denials is restored in his system.However, his criticism of Restall's approach seems misdirected.Such a criticism of Restall's system being asymmetric in the treatment of statements in the antecedent position as assertions and the ones in conclusion position as denials is already voiced by Steinberger [58, p. 351]. 8But this is crucially different from how Francez [17, p. 247f.]depicts Restall's position, namely that here assertions are assumed, while denials are concluded.Restall does not say that if A B, then if you assume the assertion of A, then you cannot conclude the denial of B, but that you would be incoherent in both asserting A and denying B. Also, another problem is that Francez adds an, in his words, equivalent reading of these positions, which would yield that if Γ , then if you assert every member in Γ , then you ought to assert some member of , which is what Restall [42, p. 4] explicitly says is different from his reading in that it is too strong a requirement.This is not to say that Restall is right in everything he says and that therefore Francez' approach is wrong.But considering that Francez claims that his approach is an extension of Restall's, these essential misrepresentations of Restall's position seem slightly problematic.
So, there are some attempts at extending the concept of bilateralism to something like tri-or multilateralism. 9Our approach to multilateralism will go in a slightly different direction, as we will sketch out below.

Our Approach to Multilateralism
What Ripley [49] mentions in a footnote is that there are also other kinds of bilateralism, which do not fit into either camp because they do not consider speech acts (i.e., assertion and denial) as the primary notions to act upon in the context of PTS.An example of this can be found in [38] in which a sequent calculus is provided which is called 'bilateral' because it can prove all the valid formulas and refute all the invalid formulas considered in the setting.Another approach, in the realm of which we will argue in this paper, is to rather consider notions being on a par with proof, provability, or verification, i.e., refutation, refutability, or falsification, respectively (see, e.g., [62], [64]).The point of interest here is, thus, a duality between different inferential relationships.It can be argued then that this gives rise to accounting for more than one derivability relation that needs to be implemented in the proof-theoretic framework.Hence, this leads to yet another way to devise proof systems, which can be claimed to establish bilateralism on a very fundamental level.
The idea is, thus, to detach bilateralism, or more general multilateralism, from being a theory of speech acts and rather propose it being a theory of multiple kinds of inferential relationships.So, instead of a bilateral reading of ordinary sequents we will study sequent calculi which are multilateral in that they display different derivability relations, which can be understood as preserving different semantic values.The motivation comes from the viewpoint of PTS and the question of uniqueness 8 He says that with this, Restall's system fails to give the meaning of logical constants in the way that is usually expected in PTS.While traditionally in PTS we have to specify both assertibility conditions and consequences of asserting formulas containing the logical constant in question, a bilateralist approach would have to do the same, while also specifying this for denial.This, however, as Steinberger says, is not done with Restall's reading but here all we get is a view on the meaning of logical constants in light of their role in the interplay between sets of assertions and denials. 9Another approach that should be mentioned is by Incurvati and Schlöder [24,25] who give an account of a multilateral epistemic logic, which is based on a Rumfittian natural deduction system with signed formulas extended to modal logic with an operator which is explained in terms of the speech act of weak assertion.
therein.If we want the rules of the connectives to be understood as conveying the meaning of those connectives, then it seems like a desirable feature of these rules to uniquely define the connectives in question.However, there are certain features of logical systems which can cause problems for uniqueness and one among those is a system being non-congruential, which is the case for the multilateral systems we will outline.Our account of multilateral derivability relations will offer a solution to this problem, though, and thus, uniqueness will be secured in our setting.

A Case Study in Proof-Theoretic Tetralateralism
Proof-theoretic multilateralism may give rise to rather convoluted proof systems.Nevertheless we believe that there are convincing reasons to study multilateral sequent calculi.In this section we present a tetralateral sequent calculus for an expansion of the well-known constructive paraconsistent and paracomplete logic with strong negation nowadays called N4.This logic is an expansion of the basic four-valued paraconsistent and paracomplete Belnap-Dunn logic of first-degree entailment, FDE. 10 Usually N4 is associated with David Nelson, who, in a seminal paper from 1949 [34], introduced a constructive and paracomplete logic with strong negation nowadays referred to as N3.The logic N4 has been studied by Nelson in a joint paper with Ahmad Almukdad from 1984 [1] and it has been presented and discussed already by Dag Prawitz [40] with a reference to [34], and, independently of Nelson's work, by Franz von Kutschera [33].We will expand the language of propositional N4 by two unary connectives, [m] and [n].A formula [m]A is to be read as 'it is meaningful to say that A' or simply 'it is meaningful that A', and [n]A is to be understood as 'it is nonsensical to say that A' or simply 'it is nonsensical that A'.Just to avoid any misunderstanding, let us emphasize that by 'nonsensical' we do not mean 'ungrammatical'.The operators [m] and [n] precede formulas, and we are not dealing with predicates that apply to names of sentences.
The model-theoretically defined logic of the expanded language will be referred to as N4mn.We shall introduce and motivate the system N4mn through its Kripke semantics.In Section 3.1, we will thus first look at the Kripke semantics and then, in Section 3.2, we will take up the definition of a subformula sequent calculus and a dual sequent calculus for N4 in [27,28] and obtain a tetralateral sequent system SN4mn for N4mn.In Section 3.3 we highlight some properties of SN4mn and prove its completeness with respect to the class of all models for N4mn.The presentation of SN4mn will then serve as the motivation and background for our suggestion of a notion of an (n + 1)lateral sequent calculus in Section 4.

Kripke Semantics and Semantic Embedding
The propositional language L of N4mn is defined in Backus-Naur form as follows: 10 For a survey see [35].
propositional variables Φ : p ∈ Φ formulas: A ∈ Form L (Φ) We use A ↔ B as an abbreviation of (A → B) ∧ (B → A).The language L of positive intuitionistic propositional logic (i.e., intuitionistic propositional logic without the falsum or a primitive negation), IPL + , is obtained from L by dropping the unary connectives, i.e., ∼, [m], and [n], and the language L of the propositional logic N4 is obtained from L by dropping [m] and [n].We sometimes omit outer brackets in formulas, and we write A ≡ B to express that A and B are the same formula.
In a first step, we will recall the Kripke semantics for IPL + because it will be used later and because it prepares for the semantic definition of N4mn in terms of its Kripke semantics.

Definition 2 A valuation | on a Kripke frame M, R is a mapping from the set Φ of propositional variables to the power set 2 M of M such that for any p ∈ Φ and any x, y ∈ M, if x ∈ | ( p) and x Ry, then y ∈ | ( p). We will write x | p for x ∈ | ( p). This valuation
| is extended to a mapping from the set of all L -formulas to 2 M by: The following heredity condition holds for | : For any L -formula A and any x, y ∈ M, if x | A and x Ry, then y | A. This is proved by induction on A.
We now turn to the language L and define four separate valuation functions | + , | − , | m , and | n .These mappings determine for a given propositional variable p, the set of states that support the truth, the falsity, the meaningfulness, and the nonsensicality (meaninglessness) of p, respectively.Support of truth, support of falsity, support of meaningfulness, and support of meaninglessness are seen as properties that are independent of each other.In particular, it is not excluded that an information state supports both the truth and the falsity of a given propositional variable or both its meaningfulness and its nonsensicality. 11Following Nuel Belnap's [6,7] reading of the four truth values in the many-valued semantics of FDE, our four dimensions of semantic evaluation stand for possible 'told values'.As a source of information, a state may tell a propositional variable true, false, meaningful, or nonsensical, where telling true, false, meaningful, or nonsensical is not a speech act but is to be understood as providing support of truth, falsity, meaningfulness, or meaninglessness, for example by someone's testimony, or by other pieces of information.An anonymous reviewer raised the question how something could be true without being priorly meaningful.It is important to highlight that support of truth, falsity, meaningfulness, or nonsensicality is different from truth, falsity, meaningfulness, respectively nonsensicality.We can imagine ourselves at a state that supports the nonsensicality of the sentence 'God exists' because at that state Rudolf Carnap tells the sentence meaningless, a state at which Michael Dummett tells 'God exists' meaningful and true, and a state at which both Carnap and Dummett are present, one telling 'God exists' meaningless, the other telling it true, so that the state supports both the truth and the nonsensicality of 'God exists'.If we think of compound sentences, the semantics we are about to present is such that every state in a model supports the truth of an implication A → A no matter whether A receives support of meaningfulness at that state in the model or not.
If we are at a state that supports the nonsensicality of the statement 'Something is the first negation of negation, as simple self-relation in the form of being', the state both supports the nonsensicality of 'If something is the first negation of negation, as simple self-relation in the form of being, then something is the first negation of negation, as simple self-relation in the form of being' and nevertheless supports the truth of that conditional "If something is the first negation of negation, as simple self-relation in the form of being, then something is the first negation of negation, as simple self-relation in the form of being" in virtue of the meaning of 'if . . .then'.x x Do the operators [m] and [n] really express meaningfulness, respectively nonsensicality?If one assumes that a sentence is true or false only if it is meaningful, then if an atomic statement A is true, [m]A should be true as well.However, truth and falsity are to be distinguished from support of truth and support of falsity.It may well be the case that a state supports the truth of A, i.e., tells A true, and nevertheless does not tell A meaningful, i.e., support the meaningfulness of A. But that A is told true does not give any reason to believe that [m]A is told true, if A is not told meaningful.Moreover, if A is told nonsensical and is part of a more complex statement, say B, then it seems natural to assume that thereby B is to be seen as told nonsensical.As to a motivation of the semantic clauses for [m] and [n], we may note that a compound formula is meaningful (nonsensical) iff all (some) of its immediate proper subformulas are; meaninglessness is 'infectious'. 12Thus, in particular, For the statement that A is nonsensical to be meaningful, A must be meaningful, although [n]A may well be false.Thus, if A is meaningful, one can meaningfully say that A is nonsensical.Moreover, a state supports the falsity of a formula A's meaningfulness, respectively nonsensicality, iff it supports A's nonsensicality, respectively meaningfulness, and a state supports the truth of a formula A's meaningfulness (nonsensicality) iff it supports A's meaningfulness (nonsensicality).Note also that All remaining clauses for compound formulas are justified by the motivation coming with the Kripke semantics of N4, which does not mean that none of these clauses can be debated. 13efinition 5 An L-formula A is said to be true in a Kripke model for N4mn M, R, Note that the above definition follows a standard approach to defining entailment and presenting a logic model-theoretically.The perspective of logical multilateralism suggests to consider in addition to entailment as support-of-truth preservation also the preservation of support of falsity, meaningfulness, and nonsensicality, i.e., to consider the four relations defined by: The logic N4mn could then be defined model-theoretically as the quintuple. 14

Proposition 1 Each of the unary connectives • ∈ {∼, [m],
[n]} is congruentialitybreaking in the sense that there are L-formulas A and B such that A | + B and B | + A but not: The mutual entailment of two L-formulas A and B thus does not in general license the replacement of B for A (or vice versa) without affecting mutual entailment.
We now define a translation from Form L (Φ) to the set of formulas of the language L based on an enlarged set of propositional variables.This translation will be shown to give rise to two kinds of embeddings of N4mn into IPL + , which turn out to be useful to transfer results from IPL + to N4mn.

Definition 6 Given the set Φ of propositional variables, we define three more sets of propositional variables, namely Φ
We inductively define a mapping f from Form L (Φ) to the set of formulas of the language L of IPL + defined over Φ ∪ Φ − ∪ Φ m ∪ Φ n as follows: We write f (Γ ) to denote the result of replacing every formula A in Γ by f (A); thus, f (∅) = ∅.
Note that for L -formulas A defined over Φ (i.e., formulas of positive intuitionistic logic defined over Φ), f (A) = A.This fact will be used in the proof of Theorem 2.
R, | be the structure defined by setting for any x ∈ M and any p ∈ Φ, Clearly, the structure M is a Kripke model for IPL + .The claims (1)-( 4) can be proved by (simultaneous) induction on the complexity of A.
• Base step, A ≡ p ∈ Φ: For (1), we obtain: • Induction step.Case A ≡ (B ∧ C): (1)  2) (by the definition of f ).The claims (3) and ( 4) are dealt with in analogy to the claims (3) and ( 4) for the previous two cases.Case A ≡ ∼B: Proof Similar to the proof of Lemma 1.
Theorem 1 (semantic embedding) Let f be the mapping defined in Definition 6.For any set of L-formulas Γ ∪ {A},

A Tetralateral Sequent Calculus for N4mn
We now define a tetralateral sequent calculus SN4mn for N4mn that makes use of four different sequent arrows by generalizing a combination of the sequent calculi Sn4 and Dn4 from [27,28].As sequents are derivability statements, we thus assume four different kinds of derivability between sets of formulas and single formulas.Nevertheless, there is only one notion of derivability for a given sequent calculus and hence also for SN4mn.
A sequent is an expression of the form, where Γ 1 , . . ., Γ 4 are finite, possibly empty multisets of L-formulas, A is an Lformula, and * ∈ {+, −, m, n}. 15 For a singleton multiset {A} we usually write just A, and A, Γ as well as Γ, A ( , Γ as well as Γ, ) designates the union of the multisets Γ and {A} ( and Γ ).Intuitively, the four sequent arrows stand for four different relations of demonstrability, namely demonstrability of receiving support of truth, falsity, meaningfulness, and nonsensicality.
A sequent calculus is a non-empty set containing some axiomatic, initial sequents and rules of the form s 1 • • • s n s where s and all s i (1 ≤ i ≤ n, n ∈ N) are sequents.Derivations in a sequent calculus are inductively defined a usual.Every instance of an initial sequent is a derivation.Applications of sequent rules to instances of their schematic premise sequents as conclusions of derivations result in a derivation.If there is a derivation of a sequent s in a sequent calculus Cal, we say that s is provable in Cal and denote this as Cal s (or just as s if the sequent calculus in question is clear).A rule of inference R is admissible in a sequent calculus Cal if for any instance for any p ∈ Φ, where ∅ is the empty multiset.The structural rules of SN4mn are of the form: The introduction rules for unary connectives in succedent position of sequents are of the form: The introduction rules for unary connectives in antecedent position of sequents are of the form: The positive inference rules for the binary connectives of SN4mn are of the form: The negative inference rules for the binary connectives of SN4mn are of the form: The m-related inference rules for the binary connectives of SN4mn are of the form: The n-related inference rules for the binary connectives of SN4mn are of the form: Proposition 2 In SN4mn, for any L-formula A, Proof By induction on the structure of A. We here present the cases for the first and the fourth claim.1.:

4.: If
If A is of the form B C for ∈ {∧, ∨, →}, we have

Syntactical Embedding, Cut-Admissibility, Decidability, and Completeness
We will syntactically embed SN4mn into Gentzen's sequent calculus LJ + for positive intuitionistic logic.From this embedding we obtain the admissibility of SN4mn's cut-rules, the decidability of SN4mn, and its completeness with respect to the class of all models for N4mn.We first present LJ + .
A sequent of LJ + is an ordinary sequent, i.e., an expression of the form Γ ⇒ A where Γ is a finite multiset of L -formulas and A is an L -formula.We consider L defined over

Definition 8
The initial sequents of LJ + are of the form: p ⇒ p, for any p The structural inference rules of LJ + are of the form: The logical inference rules of LJ + are of the form: Theorem 2 (Syntactical embedding) Let f be the mapping from Definition 6.For any finite multiset of L-formulas Proof Since claim (b) follows from the proof of claim (a), we consider only (a).(We will appeal to both claims (a) and (b) in the proof of cut-admissibility in SN4mn.)Direction from left to right: By simultaneous induction on derivations π in SN4mn.
We distinguish the cases according to the last inference of π and present some cases for claims (a) ( 2) and (a) (4).Claim (a) (2).If the last inference of π is an axiomatic sequent p : ∅ : ∅ : ∅ ⇒ − p, then, by the definition of f , f (∼ p) = p − and LJ + p − ⇒ p − .The other cases of axiomatic sequents are similar.Case (∼r −): , by the induction hypothesis for (a) (1).Since f (∼∼A) = f (A), the claim follows.
Case ([m]r−): By the induction hypothesis for (a) (2), we have and may apply rule (∧l) in LJ + : which by the definition of f gives us Case (→ r−): By the induction hypothesis for (a) ( 2) and (a) (1), we have We can thus apply rule (∧r) in LJ + to obtain a derivation of which by the definition of f gives us Claim (a) (4).Case (∼rn): By the induction hypothesis for (a) (4), we may assume that Case ( ln): By the induction hypothesis for (a) (4), we have We can thus apply rule (∨l) in LJ + to obtain a derivation of Direction from right to left: By simultaneous induction on derivations π of the respective sequents under translation in LJ + .We distinguish the cases according to the last inference of π and present some cases again for claim (a) (2).Case (cut): The last inference of π is of the form: Since the L-formulas are defined over Φ, B is an L -formula defined over Φ.Therefore, B = f (B), by the induction hypotheses for (a) ( 1) and (a) (2), Γ 1 : Γ 2 : Γ 3 : . Hence, we can apply (cut+) in SN4mn: Case (∨l): We apply (∨l By the induction hypothesis for (a) (2), we have SN4mn Γ 1 , A : We assume that A and B are not both prefixed by the same unary connective and apply (∨l By the induction hypothesis for (a) (2), we have SN4mn Γ We apply (∨l By the induction hypothesis for (a) (2), we have SN4mn Γ 1 : By the induction hypothesis for (a) (2), we have SN4mn Γ 1 : Theorem 3 (Cut-admissibility) The rules (cut+), (cut−), (cutm), and (cutn) are admissible in cut-free SN4mn.
Proof Suppose that the sequent . By the cut-elimination theorem for LJ + , the sequent f is provable in cut-free LJ + , and by the syntactical embedding theorem, claim (b), Γ 1 : Γ 2 : Γ 3 : Γ 4 ⇒ + A is provable in cut-free SN4mn.The reasoning for sequents Γ 1 : The above indirect proof of cut-admissibility is one of the main advantages of the syntactical embedding method.The definition of a direct Gentzen-style cut-elimination procedure would give rise to very many case distinctions.A worked out example of a cut-elimination procedure for a bilateral sequent calculus with two kinds of sequent arrows can be found in [4].
Corollary 1 (Subformula property) SN4mn has the subformula property, i.e., if a sequent s is provable in SN4mn, then there is a derivation π of s such that all formulas appearing in π are subformulas of some formula in s.
Proof By the decidability of LJ + , for each L-formula A, it is possible to decide whether f (A) is provable in LJ + .Then, by the syntactical embedding theorems, SN4mn is decidable.

The Notion of an (n + 1)Lateral Sequent Calculus
If we think of logical multilateralism, it is appropriate to wonder whether there are philosophical or linguistic motivations for going beyond tetralateralism.One such motivation comes from many-valued logic.In [53], Shramko, Dunn, and Takenaka introduced a 16-element trilattice of semantical values that gives rise to the definition of three distinct entailment relations, see also [54,55].In a similar vein, starting from N4mn, in [66] a 65536-element pentalattice of generalized truth values is motivated.In addition to four entailment relations that preserve support of truth, falsity, meaningfulness, and nonsensicality, the subset relation on the set of values figures as a fifth relation of semantical consequence.The latter relation preserves informativeness of assigned values and is axiomatized by a formula-formula inference system.
In this section we consider proof-theoretic (n + 1)lateralism and want to define the notion of an (n + 1)lateral sequent calculus that makes use of n + 1 sequent arrows and sequents of the form Γ 1 : Γ 2 : . . .: Γ n : Γ n+1 ⇒ i A for 1 ≤ i ≤ n +1, n ∈ N, where the Γ i are finite possibly empty multisets of formulas.We say that formulas in Γ i occur at position i.In order to motivate such a definition, we will firstly lay down the desiderata that a system of rules should fulfill from the viewpoint of PTS.Then, we will show at the example of different sequent calculus representations for the logic N4 to what extent a bilateral sequent calculus displays advantageous features as opposed to a unilateral one.Since these considerations can be extended to multilateral features of a sequent calculus, our final definition will generalize our observations to the (n + 1)lateral case.
If the operational rules of a sequent calculus are meant to define the meaning of the logical operations in question, they must satisfy a number of adequacy conditions.For lists of such conditions and their motivation in the context of display and hypersequent calculi see, for example, [37,39,59].One format that has been suggested for ordinary Gentzen-style single-and multiple-succedent sequent calculi are the canonical sequent systems of Avron and Lev [2].A criterion that is shared by all these approaches is separation (or purity), namely that the left and right introduction rules for a given connective must display only and no other logical operation.Otherwise, the operational rules for would specify the meaning of in connection with the meaning of some other logical operations, and the proof-theoretic semantics of would be to some extent holistic.
Moreover, a crucial requirement for a definition is that definiens and definiendum are interreplaceable to within synonymy, as Lloyd Humberstone [23] puts it.Let be a syntactically or semantically defined Tarskian consequence relation for a propositional language L, i.e., a binary relation between sets of L-formulas and single L-formulas such that (i) {A} A, (ii) A and ⊆ Γ imply Γ A, and (iii) A and Γ ∪ {A} B imply ∪ Γ B. A Tarskian consequence relation is finitary if it is a relation between finite sets of formulas and single formulas.
Let us also assume that is structural (i.e., closed under uniform substitution) and non-trivial (i.e, there exists a non-empty set of L-formulas and an L-formula A such that it is not the case that A).The logic L = L, is said to be congruential (or self-extensional, or to satisfy the replacement property) if for all L-formulas A, B, and C and all propositional variables p, where C(A/ p) and C(B/ p) are the results of uniformly substituting A for p and B for p in C, respectively, and A B means that A B and B A. Equivalently, the congruentiality of L can be defined by requiring that for every n-place connective , the following holds: A connective is said to be congruential according to L if the congruence rule for it is admissible in L. The definition of congruentiality given in (1) is structural (or pure) insofar as it does not highlight any connective.If is a connective that violates (2), then (1) fails and the replacement property cannot be presented in terms of interderivability and substitutivity alone.Hence, if L = L, is not congruential, the replacement property required to be entitled to talk about synonymy is expressed by (1), and we consider a sequent calculus Cal with A B iff Cal A ⇒ B, then it is not justified to maintain that the operational sequent rules of Cal assign meaning to the connectives of L.
If the inference rules for a logical operation are meant to define the operation to within synonymy, then such a definition not only must be non-creative, 17 which is usually secured by the dispensability or the exclusion of cut, but in addition the rules must define the definiendum uniquely.The use of non-congruential connectives creates a problem with respect to this.Usually, in order to ensure that a connective is unique, we show that any formula containing this connective is interderivable with a formula built in the same way containing a copy-cat connective, i.e., a connective that is governed by the same rules as the connective in question.If the connective is noncongruential, however, then this would not ensure uniqueness to within synonymy, as Humberstone [23, §4.31] shows.To guarantee also this stronger notion of uniqueness, we must show that all compounds formed by that connective are interderivable as well, which is not the case in a non-congruential system.
With the example of N4, that is, a certain fragment of N4mn, we can show how these three aspects, purity of sequent rules, a purely structural definition of congruentiality, and the relation to uniqueness are highly relevant for choosing an adequate representation of rules from a PTS point of view.There is a kind of standard sequent calculus for N4, the system GN4, with non-separated left and right introduction rules for strong negation, ∼.

Definition 9
The initial sequents of GN4 are of the form p ⇒ p or ∼ p ⇒ ∼p, for any propositional variable p ∈ .In addition to the sequent rules of LJ + , GN4 consists of the following logical inference rules: Strong negation is not congruential in N4 and therefore it is not the case that in ( The left and right introduction rules for strong negation in GN4 are impure because they display other connectives in addition to strong negation.Moreover, the statement of the replacement property in (3) is not purely structural because it highlights ∼.The fact that ∼ is not congruential in N4 and the desideratum of having available a purely structural definition of replacement may be seen to justify the use of the bilateral sequent calculus SN4 for N4 in [65] that makes use of two different sequent arrows, ⇒ + and ⇒ − and sequents Γ : ⇒ + A and Γ : ⇒ − A. In SN4 the purely structural statement of the replacement property then takes the form The case C ≡ [n]D is analogous to the latter case.
Instead of assuming more than one derivability relation and using several sequent arrows, one could use a notational variant that makes use of only a single sequent arrow and in that way hides the idea of a multiplicity of derivability relations.In the case of SN4mn, instead of working with four single-succedent sequent arrows, one could use just one sequent arrow with four places on the right hand side of the sequent arrow.The sequents would thus be re-written as: respectively, so as to obtain eight-place sequents with a restriction to single formulas in succedent position. 18 Note that this is very different from the use of n-sided sequents in the proof theory of finitely-valued logics, where each of the n positions stands for one out of n truth values (for n ≥ 2).Note also that the above re-writing does not lead us to a Rumfittian multilateralism obtained by re-writing the above sequents as where Γ * = {A * | A ∈ Γ and * ∈ {+, −, m, n}}.This is so because on a Rumfittian conception, the superscripts would have to stand for speech acts, which is not the case and not intended at all here.The sequent arrows ⇒ + , ⇒ − , ⇒ m , and ⇒ n stand for inferential relationships that capture the meaning of the logical operations not in terms of pragmatic notions but with respect to the semantical concepts of demonstrating the receipt of support of truth, falsity, meaningfulness, and nonsensicality, as laid down by their left and right introduction rules against the background of the initial sequents and the given structural rules.
Moreover, the strength of our proposed use of multiple sequent arrows comes into play here, as well, which can also be shown at the example of GN4 and SN4.Since the rules of GN4 are impure, the uniqueness of strong negation could only be characterized as uniqueness in terms of the other connectives that are displayed in its rules.On the other hand, in order to get uniqueness to within synonymy it would not be enough to demand the interderivability of all formulas with their copy-cat connective formulas but one must also require the interderivability of all strongly negated formulas with their copy-cat connective formulas.This would mean, though, to define uniqueness of connectives in terms of negation, while the uniqueness of negation itself can only be characterized in terms of other connectives, which seems somehow circular.With a formulation of the proof rules as in SN4 this problem can be circumvented.The rules in SN4 are pure, i.e., it is not the case that we need to characterize one connective's uniqueness in terms of other connectives.As a condition for uniqueness (to within synonymy) all we have to demand is that the interderivability holds with respect to both consequence relations.
The same holds for our multilateral systems.The connectives we introduce make our system non-congruential, i.e., it is not enough to show that in a sequent calculus formulas are interderivable with respect to a single sequent arrow.The solution is simply to extend the definition of uniqueness by demanding interderivability with respect to all other finitary consequence relations that are represented in the respective system by different sequent arrows. 19Thus, in light of the above case study and considerations, we suggest a notion of an (n + 1)lateral sequent calculus.

Definition 10
Let L be a language with n primitive unary connectives, [• 1 ], . .., [• n ], where 1 ≤ n ∈ N, and let Cal be a set of sequent rules (including axiomatic sequents) manipulating sequents of the form Γ 1 : Γ 2 : . . .: Γ n : Γ n+1 ⇒ i A, for 1 ≤ i ≤ n + 1.Then the sequent system Cal is said to be an (n + 1)lateral sequent system.Definition 10 does not introduce the most general concept imaginable for the purpose of defining the notion of a multilateral sequent system as motivated by the above considerations and case study in tetralateralism.In particular, for the sake of simplicity and for accordance with notational conventions, the definition privileges one of the n + 1 sequent arrows insofar as it is not associated with any one-place connective.
If we follow standard usage and a common bias in favour of positive notions such as truth and meaningfulness over negative ones such as falsehood and nonsensicality, this privileged sequent arrow is one that represents the preservation of support of truth from the premises to the conclusion of an inference.
We assume that there is no unary connective in the formal language to express that a proposition has the feature the support of which is preserved by a modeltheoretic relation associated with the privileged sequent arrow.(If, for example, the preservation of support of falsity from the premises to the conclusion of an inference was the distinguished and predominant notion of entailment, then there would be no need for a negation sign and instead the use of an affirmation symbol would suggest itself.)However, we assume that there are unary connectives expressing that a formula has semantic features different from the one associated with the privileged sequent arrow such as, for example, being false, meaningful, or meaningless if the privileged sequent arrow is associated with the preservation of support of truth.In a more egalitarian approach to non-congruentiality in Cal, one would require that each of the unary operators [• 1 ], . .., [• n ] is non-congruential with respect to one and the same but not necessarily the privileged sequent arrow, or that each of the unary operators [• 1 ], . .., [• n ] is non-congruential with respect to at least one or all of the n + 1 sequent arrows.
Definition 10 has been motivated with a view on logics that can be presented as an (n + 1)lateral sequent system Cal in a language with unary connectives [• 1 ], . .., [• n ] that are non-congruential in Cal and where the replacement property (5) holds for Cal.Nevertheless, there are reasons not to build the non-congruentiality into the definition of an (n + 1)lateral sequent system.If we consider the {∼, ∧, ∨}-fragment of N4mn, for example, we obtain FDE.Since contraposition is an admissible rule in FDE, two formulas ∼A and ∼B are mutually derivable in a proof system for FDE if A and B are mutually derivable, and, as a result, ∼ is not congruentiality-breaking in FDE.The bilateral natural deduction proof systems in [36] for FDE and variants of FDE for which contraposition is an admissible rule are, however, well motivated already, though less strongly, by the purity of their introduction and elimination rules in comparison to the impurity of their unilateral counterparts.

Definition 4
The valuation functions | + , | − , | m , and | n on a Kripke frame M, R are mappings from the set Φ to the power set 2 M of M such that for any ∈ {+, −, m, n}, any p ∈ Φ and any x, y ∈ M, if x ∈ | ( p) and x Ry, then y ∈ | ( p).We will write x | p for x ∈ | ( p).The functions | + , | − , | m , and | n are extended to mappings from the set of all formulas to 2 M by: 1.
a Kripke model for N4mn based on F. It can easily be shown by induction on the complexity of L-formulas that the heredity condition holds for the extended functions | + , | − , | m , and | n , i.e., for any Lformula A and any x, y ∈ M, if x | * A and x Ry, then y | * A, for * ∈ {+, −, m, n}.
for any x ∈ M, and to be valid on a Kripke frame F = M, R if it is true for every Kripke model for N4mn based on F. An L-formula A is said to be N4mn-valid if A is valid on every Kripke frame.Let Γ ∪ {A} be a set of L-formulas.Entailment is defined in terms of support-of-truth preservation at each state: Γ | + A if for all Kripke models for N4mn M, R, | + , | − , | m , | n and for all x ∈ M, x | + A if x | + B for all B ∈ Γ .We write A | + B for {A} | + B. We define the logic N4mn model-theoretically as the pair L, { Γ, A | Γ | + A} and N4 is model-theoretically defined as the pair L , { Γ, A | Γ | + A} .

Lemma 1
Let f be the function from Definition 6.For any Kripke model for N4mn M, R, | + , | − , | m , | n , we can define a Kripke model for IPL + M, R, | such that for any formula A ∈ Form L (Φ) and any x ∈ M, by the definition of f ).For (3)-(4), we obtain: x | * p iff x | p * iff x | f ([ * ] p) for * ∈ {m, n} (by the definition of f ).
Case A ≡ (B ∨ C): analogous to the previous case.Case A ≡ (B → C): For (1), we obtain: x | + B → C iff ∀y ∈ M[x Ry and y | + B imply y | + C] iff ∀y ∈ M[x Ry and y | f (B) imply y | f (C)] (by the induction hypothesis for (1)) iff x | f (B) → f (C) iff x | f (B → C) (by the definition of f ).For (2), we get: x | − B → C iff x | + B and x | − C iff x | f (B) and x | f (∼C) (by the induction hypotheses for (1) and (

Lemma 2
Case A ≡ [n]B: analogous to the previous case.123 Let f be the function defined in Definition 6.For any Kripke model M, R, | for IPL + , we can construct a Kripke model M, R, | + , | − , | m , | n for N4mn such that for any L-formula A and any x ∈ M, 1

Proof 1 .
Direction from left to right: Suppose it is not the case that f (Γ ) | f (A) in IPL + .Then there is a model M, R, | for IPL + , and a state x ∈ M with x | f (B) for every B ∈ Γ , but x | f (A).By Lemma 2, there exists a model M, R, | + , | − , | m , | n for N4mn with x | + B for every B ∈ Γ , but x | + A. The direction from right to left is shown analogously by making use of Lemma 1. 2.-4.: analogous to the proof of the first claim.
GN4 A ⇒ B and B ⇒ A together imply C(A/ p) ⇒ C(B/ p) (and also C(B/ p) ⇒ C(A/ p)).One can, however, show that in GN4 A ⇒ B, B ⇒ A, ∼A ⇒ ∼B, and ∼B ⇒ ∼A together imply C(A/ p) ⇒ C(B/ p) and C(B/ p) ⇒ C(A/ p).
might seem as if that does not sound right, but upon reflection this equivalence reflects the intended understanding of [n], | + , and | n .We see that in general x | + A does not exclude x | n A. A state x support the nonsensicality of [n]A just in case x tells A nonsensical just in case x tells [n]A true.