Efficient Reasoning with Inconsistent Information Using C-systems

Efficient Reasoning with Inconsistent Information Using C-systems Arnon Avron∗1 , Beata Konikowska†2 and Anna Zamansky‡3 1

School of Computer Science, Tel Aviv University, P.O.B. 39040, Ramat Aviv, Israel Institute of Computer Science, Polish Academy of Sciences, ul. Jana Kazimierza 5, 01-248 Warsaw, Poland 3 Department of Information Systems, University of Haifa, 199 Abba Hushi Blvd., Mount Carmel, Haifa, Israel 2

Abstract The paper subsumes and extends in a radical way a line of research aimed at automation of reasoning with inconsistent information using paraconsistent logics. We provide a new method for uniform, modular construction of analytic calculi for all major logics in the crucial class of paraconsistent logics known as C-systems. The method is based on semantic characterization of those logics via non-deterministic matrices (Nmatrices), and — unlike that developed previously — is also applicable to C-systems which can only be characterized by infinite Nmatrices. What is more, we show that the results obtained in this paper for infinite semantics imply our earlier results for finite semantics.

Keywords: inconsistent information; analytic proof systems; Gentzen-style calculi; paraconsistent logics; C-systems; many-valued logics; non-deterministic logical matrices

1

Introduction

This paper continues a line of research motivated by the need for efficient reasoning with inconsistent information, which is ubiquitous in the present day world. Such reasoning can be carried out using socalled paraconsistent logics, which, unlike classical logic, allow for non-trivial reasoning in the presence of contradictions. One of the oldest and most famous approachesto paraconsistency is that of da Costa ([22, 23, 24]). It involves two key ideas. The first is that propositions should be divided into two sorts: the “normal” (or “well-behaved”, or consistent) ones, and the “abnormal” (or inconsistent) ones. While classical logic can be applied freely to normal propositions, its application to the abnormal ones is restricted. The second idea is to reflect this classification within the language used. In da Costa’s papers this was done by using a certain formula of the classical language, denoted shortly by ◦ϕ, to represent the statement “ϕ is well-behaved”.1 ∗ [email protected] † [email protected] ‡ [email protected]

In his main system C1 [22] da Costa used ϕ◦ as an abbreviation for ¬(ϕ ∧ ¬ϕ). An alternative, equally justified, choice could have been ¬(¬ϕ ∧ ϕ), which is not equivalent to ¬(ϕ ∧ ¬ϕ) in C1 . 1

1

A radical generalization of da Costa’s original systems are the so-called Logics of Formal (In)consistency (LFIs) proposed in [18, 17]. In that family, the notion of consistency is treated like an independent operator, rather than just as an abbreviation for “well-behavedness”. In the most important class of LFIs known as C-systems ([18]), consistency is represented by a special unary connective ◦ (either primitive or defined), where ◦ϕ means “ϕ is consistent”. This allows for ample generalizations of the concept of consistency and its relations to other logical operators. Consequently, we can separate contradictions from inconsistencies by forbidding consistent contradictions (see the axiom b below). Until recently, despite extensive research on this subject ([13, 14, 16, 17, 18, 19, 20, 25, 27, 28, 29]), the main problem with C-systems was the lack of analytic proof systems for most of them2 — while the availability of such systems is a must for efficient reasoning. This drawback was partially remedied in our previous work [8]. In that paper, we succeeded to provide in a uniform way quasi-canonical (see Definition 4 below) cut-free Gentzen-type systems for all important C-systems without the axiom l, which reflects da Costa’s definition of the consistency operator, and the axiom d, which reflects its obvious alternative (see Footnote 1). We achieved this using the three-valued non-deterministic semantics provided for C-systems in [2, 3, 4, 5], and employing the general method for generating cut-free sequent calculi for logics induced by finite nondeterministic matrices (Nmatrices) given in [6]. However, this method is not applicable to C-systems with either l or d, since — as shown in [5] — such systems do not have finite characteristic Nmatrices. Indeed, by [1], such C-systems cannot have quasi-canonical Gentzen-type systems. Unfortunately, some of the most important and well-known C-systems (including da Costa’s original system C1 ) do include at least one of these two axioms. In the present paper we overcome this problem by presenting a completely new method for uniform construction of analytic calculi for C-systems. The resulting systems are weakly quasi-canonical (Definition 4), which suffices for providing terminating proof-search procedures and for guaranteeing decidability. This new method works for all major C-systems, including those that do not possess finite semantics (as well as those handled in [8], of course).3

2

Preliminaries

In this section we review all the background material needed to understand the rest of the paper. Though much of it can be found in [8], we repeat it here to make this paper self-contained. In what follows, L is a propositional language, and F rmL is its set of wffs. The metavariables ϕ, ψ range over L-formulas, p, q — over atomic formulas, T, S — over sets of L-formulas, and Γ, ∆ — over finite sets of L-formulas. The set of atomic formulas which occur in a formula ϕ is denoted by F v(ϕ). Notation 1 Let Lcl be the classical propositional language with the set of connectives {∧, ∨, ⊃, ¬}, and let L+ cl be its positive fragment, i.e. the propositional language with the set of connectives {∧, ∨, ⊃}. Definition 2 A (Tarskian) consequence relation (tcr) for a language L is a binary relation ` between sets of L-formulas and L-formulas, satisfying the following three conditions: Reflexivity:

if ψ ∈ T then T ` ψ.

Monotonicity:

if T ` ψ and T ⊆ T 0 then T 0 ` ψ.

Transitivity:

if T ` ψ and T, ψ ` ϕ then T ` ϕ.

2 An “analytic proof system” for a logic L is usually understood as a proof system P such that F ` ϕ for some finite L theory F iff there is a proof of this fact in P which uses solely sub-formulas of the formulas in {ϕ} ∪ F , or negations of such sub-formulas. Certain useful extensions of that notion exist (see e.g. [12, 26]), but the version quoted above is sufficient here. 3 The main results of this paper were first announced in [7]. However, that extended abstract did not include their proofs, and contained a number of mistakes, which are corrected here.

2

Definition 3 A propositional logic is a pair L = hL, `i, where ` is a tcr for L which satisfies the following two conditions: Structurality:

if T ` ϕ then σ(T ) ` σ(ϕ) for any substitution σ in L.

Non-triviality:

p 6` q for any distinct propositional variables p, q.

The main proof-theoretical tool for characterizing propositional logics applied in this paper are (cutfree) Gentzen-type systems. In fact, we use the following special class of such systems, particularly convenient for proof search: Definition 4 1. A weakly quasi-canonical rule of arity n is an expression of the form {Πi ⇒ Σi }1≤i≤m C where m ≥ 0, Πi , Σi ⊆ {p1 , ¬p1 , p2 , ¬p2 , . . . , pn , ¬pn } for 1 ≤ i ≤ m, and C is a singleton sequent (i.e. for some formula ϕ, C is either ϕ ⇒ or ⇒ ϕ) such that F v(C) = {p1 , . . . , pn }. 2. A weakly quasi-canonical rule as in 1. above is quasi-canonical if in addition C has one of the following forms: (p1 , p2 , . . . , pn ) ⇒

⇒ (p1 , p2 , . . . , pn )

¬  (p1 , p2 , . . . , pn ) ⇒

⇒ ¬  (p1 , p2 , . . . , pn )

3. An application of a weakly quasi-canonical rule {Πi ⇒ Σi }1≤i≤m / ϕ ⇒ as in 1. above to formulas ψ1 , . . . , ψn is an inference step of the form: {Γ, Π∗i ⇒ ∆, Σ∗i }1≤i≤m Γ, ϕ∗ ⇒ ∆ where Π∗i , Σ∗i , and ϕ∗ are obtained from Πi , Σi , and ϕ (respectively) by substituting ψj for pj (for all 1 ≤ j ≤ n), and Γ, ∆ are any sets of formulas. An application of a weakly quasi-canonical rule of the form {Πi ⇒ Σi }1≤i≤m / ⇒ ϕ is defined similarly. 4. A Gentzen-type system in which all rules except Weakening and Cut4 are (weakly) quasi-canonical is called (weakly) quasi-canonical. Remark 5 Obviously, if G is a weakly quasi-canonical Gentzen-type system, and P is a cut-free proof of a sequent Γ ⇒ ∆ in G, then every formula occurring in P is either a subformula of some ϕ ∈ Γ ∪ ∆, or a negation of a proper subformula of some ϕ ∈ Γ ∪ ∆. Hence if the cut-elimination theorem obtains for G then G is analytic (in the sense explained in Footnote 2). Moreover: Any backward proof search in such a G is guaranteed to terminate without entering loops, and so the corresponding logic is decidable. 5 4 For convenience, we assume here that the two sides of a sequent consists of finite sets, so there is no need for the rules of Contraction and Permutation. 5 Examples of Gentzen-type systems (in the form of tableaux) which are not weakly quasi-canonical, and where proof search may indeed involve coping with loops, can e.g. be found in [20], which treats three out of the hundreds of logics given cut-free quasi-canonical system in [8]. Following [15], [20] claims that such loops do not interfere with decidability issues. Even so, their existence certainly makes the task of proof search more complicated.

3

2.1

Taxonomy of LFIs

The notion of paraconsistency (with respect to ¬) and its stronger variant, known as strong paraconsistency, are usually defined as follows: Definition 6 Let L be a language which has a unary connective ¬. A propositional logic L = hL, `i is: • paraconsistent (with respect to ¬) if there are ψ, ϕ ∈ F rmL such that ψ, ¬ψ 6` ϕ. • strongly paraconsistent (with respect to ¬) if there are ψ, ϕ ∈ F rmL such that ψ, ¬ψ 6` ¬ϕ.

6

Logics of Formal (In)consistency (LFIs) represent a large family of paraconsistent logics in which consistency can be expressed in the language of the logic itself. Namely, a paraconsistent logic L = hL, `i is an LFI if there is a propositional variable p and a set X(p) of L-formulas containing no variable different from p such that ψ, ¬ψ, X{ψ/p} ` ϕ for every ψ, ϕ ∈ F rmL .7 The best known and particularly useful subfamily of LFIs are C-systems, in which X(p) is a singleton: Definition 7 A logic L = hL, `i with Lcl ⊆ L is said to be a C-system if the following conditions hold: 1. L contains the L+ cl -fragment of classical logic, 2. L is paraconsistent, 3. L has a (primitive or defined) unary connective ◦ obeying the following axioms in L: (t) ¬ϕ ∨ ϕ

(b) ◦ ϕ ⊃ (ϕ ∧ ¬ϕ ⊃ ψ)

(k) ◦ ϕ ∨ (ϕ ∧ ¬ϕ)

Remark 8 Like in [8], our notion of a “C-system” is somewhat narrower than in [17, 18], where the validity of (k) was not required. A detailed general justification for our choice to include (k) in our basic system was given in [8].8 Notation 9 Until stated otherwise, we shall assume that ◦ is a primitive connective of the language, and leave handling of C-systems without such a connective until Section 5. We shall denote by LC the propositional language with the set of connectives {∧, ∨, ⊃, ¬, ◦}.

Definition 10 Let HCL+ be some standard Hilbert-style system with Modus Ponens as the only inference rule which is sound and strongly complete for the positive fragment (i.e., the L+ cl -fragment) of classical propositional logic. 1. By B we shall denote the system for LC obtained by adding (t) and (b) to HCL+ . 2. By BK we shall denote the system (for LC ) obtained by adding (k) to B . 6 As ` is structural, it is enough to require that there are propositional variables p, q such that p, ¬p 6` q for paraconsistency, and p, ¬p 6` ¬q for strong paraconsistency. 7 As noted in [8], this is a slight generalization of the standard definition of LFIs ([17, 18]), where a certain additional requirement on X is added. However, that requirement can be easily shown to be satisfied by all the logics considered in this paper anyway. 8 One argument of particular importance for the present paper is the simple fact (shown in [8]) that (k) is anyway derivable from (t), (b) and each of the two main axioms investigated in this paper: (l) and (d) (Definition 11).

4

Next we provide a more or less extensive list of axioms considered in the literature (see, e.g., [17, 18, 22, 24, 23]) that BK can be extended with. They may be divided into two main subsets: axioms dealing with combinations of negation with classical connectives, and axioms handling the connective ◦. Definition 11 Let A be the following set of axioms, where ] ranges over {∧, ∨, ⊃}: (c) (nl∧ ) (nl∨ ) (nl⊃ ) (i) (o1] ) (a] ) (l)

¬¬ϕ ⊃ ϕ ¬(ϕ ∧ ψ) ⊃ (¬ϕ ∨ ¬ψ) ¬(ϕ ∨ ψ) ⊃ (¬ϕ ∧ ¬ψ) ¬(ϕ ⊃ ψ) ⊃ (ϕ ∧ ¬ψ) ¬◦ϕ ⊃ (ϕ ∧ ¬ϕ) ◦ϕ ⊃ ◦(ϕ]ψ) (◦ϕ ∧ ◦ψ) ⊃ ◦(ϕ]ψ) ¬(ϕ ∧ ¬ϕ) ⊃ ◦ϕ

(e) ϕ ⊃ ¬¬ϕ (nr∧ ) (¬ϕ ∨ ¬ψ) ⊃ ¬(ϕ ∧ ψ) (nr∨ ) (¬ϕ ∧ ¬ψ) ⊃ ¬(ϕ ∨ ψ) (nr⊃ ) (ϕ ∧ ¬ψ) ⊃ ¬(ϕ ⊃ ψ) (o2] )

◦ψ ⊃ ◦(ϕ]ψ)

(d)

¬(¬ϕ ∧ ϕ) ⊃ ◦ϕ

Let A0 = A \ {(l), (d)}. Definition 12 For any AX ⊆ A, by BK[AX] we shall denote the system obtained out of BK by adding to it all the axioms from AX. Notation 13 In what follows we shall usually omit various brackets and sometimes also decorations on the symbols, writing, e.g., BKco1∧ instead of BK[{(c), (o1∧ )}] and BKa instead of BK[{(a∧ ), (a∨ ), (a⊃ )}]. Remark 14 The system names used in this paper sometimes differ from those previously used in the literature on LFIs (see [17, 18]). Most of the differences are due to the fact that in B the axiom (k) follows from (i), and that (c) is usually included in the systems studied in [17, 18]. As a result, in the names of all the extensions of B including (c) and (i) considered in [17, 18], the combination “BKc” is replaced by “C”. The full table matching the names we use here against their previously used alternatives is provided below for the reader’s convenience (see also [8] for further discussion). Our notation B BKci BKcia BKcie BKcio BKciae BKcil BKcila BKcilo BKcile BKcilae BKcida BKcidae

Alternative mbC Ci Cia Cie Cio Ciae Cil Cila Cilo Cile Cilae Cida Cidae

Table 1: Alternative system names

5

Remark 15 In the list above, we have omitted the axiom (a¬ ) : ◦ϕ ⊃ ◦¬ϕ, since it is equivalent to (c) in BK (which was in principle discovered already by da Costa). Actually, also (a∧ ) is equivalent to (nl∧ ) in BK (see [8]), but for the reader’s convenience we have retained it in our list of major axioms for C-systems. Remark 16 da Costa’s original system C1 ([22, 23]) is known to be equivalent to the ◦-free fragment of BKcila. By our results (see Corollary 74 below), it is also equivalent to the ◦-free fragment of BKcla. Remark 17 An even more modular treatment of the axioms from A becomes possible if we split some of them. Thus (i) is equivalent to the conjunction of (i1 ) ¬◦ϕ ⊃ ϕ and (i2 ) ¬◦ϕ ⊃ ¬ϕ, which was the way of its presenting in [8]. Analogously, (nr∧ ) is equivalent to the conjunction of (nr,1 ∧ ) ¬ϕ ⊃ ¬(ϕ ∧ ψ) l l ) ¬ψ ⊃ ¬(ϕ ∧ ψ), and similar splitting can be done for (n ) and (n ). However, for the sake of and (nr,2 ∧ ∨ ⊃ brevity, we shall stick to the current presentation. Remark 18 It is easy to see that the “converse” of (i) (i.e., ϕ ∧ ¬ϕ ⊃ ¬ ◦ ϕ) and the converse of (l) (i.e., ◦ϕ ⊃ ¬(ϕ ∧ ¬ϕ)) are theorems of B. Together, the four implications intuitively imply that ◦ϕ and ¬(ϕ ∧ ¬ϕ) “have the same meaning”. On the other hand, (d), its converse (which is a theorem of B, too) and (i) taken together imply that ◦ϕ and ¬(¬ϕ ∧ ϕ) “have the same meaning”. This intuition will be exploited in the sequel to provide Gentzen-type rules corresponding to (l) and (d). It should be noted that (l) and (d) are not equivalent in BK (which follows directly from the semantics provided below).

2.2

Non-deterministic Matrices

The semantic tool underlying the results of this paper are non-deterministic multi-valued matrices (Nmatrices), introduced in [9] (a comprehensive survey on non-deterministic matrices can be found in [11]). These structures are a natural generalization of the concept of a many-valued matrix, in which the truthvalue assigned to a complex formula is chosen non-deterministically out of a given non-empty set of allowed options. Definition 19 1. A non-deterministic matrix (Nmatrix) for a language L is a tuple M = hV, D, Oi, where: V is a non-empty set of truth values, D (the set of designated truth values) is a non-empty proper subset of V, and O includes an interpretation function ˜M : V n → P + (V) for every n-ary connective  (where P + (V) is the set of nonempty subsets of V). We say that M is finite if so is V. 2. Let M = hV, D, Oi be an Nmatrix. Let F be some set of L-formulas closed under subformulas. A legal valuation in M (shortly M-valuation) on F is a function v : F → V which satisfies the following condition for every n-ary connective  of L and every ψ1 , . . . , ψn ∈ F such that (ψ1 , . . . , ψn ) ∈ F : v((ψ1 , . . . , ψn )) ∈ ˜M (v(ψ1 ), . . . , v(ψn )) A full M-valuation is an M-valuation on F rmL , i.e. the set of all formulas of L. 3. Let F be as above, and let ψ ∈ F . An M-valuation v on F satisfies ψ, denoted by v |=M ψ, if v(ψ) ∈ D. Further, v satisfies a set T ⊆ F of formulas, denoted by v |=M T , if it satisfies every formula of T . 4. Let F be as above, and let v be an M-valuation on F . A sequent Γ ⇒ ∆ such that Γ ∪ ∆ ⊆ F is satisfied by v if v |=M ψ for some ψ ∈ ∆, or v6|=M ψ for some ψ ∈ Γ. A sequent is valid in M if it is satisfied by every full M-valuation. 6

5. The consequence relation induced by M, denoted by `M , is defined by: T `M ψ if v |=M ψ for every full M-valuation v such that v |=M T . We shall identify an ordinary (deterministic) matrix with an Nmatrix whose functions in O always return singletons. Notation 20 Below we shall frequently write just  instead of ˜M , relying on the context to indicate whether we mean the connective itself or its interpretation in some Nmatrix M. Remark 21 In [9] it was shown that Nmatrices enjoy many of the crucial properties of usual (deterministic) matrices, like semantic analyticity, and in the finite case compactness and decidability. Definition 22 We say that an Nmatrix M is characteristic for a Gentzen-type system G if, for every Γ and ∆, `G Γ ⇒ ∆ holds iff Γ ⇒ ∆ is valid in M. Remark 23 If M is characteristic for G, then `G Γ ⇒ ψ iff Γ `M ψ. Consequently, if the compactness theorem holds for `M , then `M =`G .9 This is true for all the Nmatrices considered in this paper (including the infinite ones), since they all have (and were historically induced by) corresponding sound and complete Hilbert-type systems. We complete this section by discussing the important notion of refinement. Later in this paper, we shall use this notion to prove that the main result obtained for C-systems with infinite semantics implies the result obtained in [8] for C-systems with finite semantics. Definition 24 Let M = hV, D, Oi and M0 = hV 0 , D0 , O0 i be Nmatrices for a language L. • A reduction of M0 to M is a mapping F : V 0 → V such that: 1. For every a ∈ V 0 , a ∈ D0 iff F (a) ∈ D . ˜M (F (a1 ), . . . , F (an )) for every n-ary connective  of L and every a1 , . . . , an , b ∈ V 0 2. F (b) ∈  such that b ∈ ˜ M0 (a1 , . . . , an ). • M0 is a refinement of M if there is a reduction of M0 to M. The following Lemma can be easily proved: Lemma 25 Let M0 = hV 0 , D0 , O0 i be a refinement of M = hV, D, Oi with the underlying reduction F : V 0 → V satisfying the conditions of Definition 24. Then, for any legal valuation v 0 in M0 , the mapping v = F ◦ v 0 is a legal valuation in M0 . The above Lemma implies the following useful result, first proved in [5]: Proposition 26 If M0 is a refinement of M, then `M ⊆`M0 . In this paper we shall use two special types of refinements: simple refinements and expansions. Definition 27 An Nmatrix M0 = hV 0 , D0 , O0 i is called a simple refinement of Nmatrix M = hV, D, Oi if V 0 = V, D0 = D and ˜ M0 (a1 , . . . , an ) ⊆ ˜M (a1 , . . . , an ) for every n-ary connective  and every a1 , . . . , an ∈ V 0 . 9 The

consequence relation `G is defined, as usual, by: T `G ψ if there is a finite Γ ⊆ T such that `G Γ ⇒ ψ.

7

Clearly, a simple refinement is a refinement where the reduction of M0 to M is just the identity function. In what follows, for any mapping F : X → Y and any A ⊆ X, F [A] shall denote the image of A under the mapping F . Definition 28 Let M = hV, D, Oi and M0 = hV 0 , D0 , O0 i be Nmatrices for a language L. M0 is called an expansion of M if there is a function F from M0 to M such that: 1. F [V 0 ] = V, F [D0 ] = D. 2. For every a1 , . . . , an ∈ V 0 and every n-ary connective , ˜ M (F (a1 ), . . . , F (an )) = F [˜M0 (a1 , . . . , an )] Intuitively, to define an expansion M0 = hV 0 , D0 , O0 i of an Nmatrix M = hV, D, Oi, for each element of V, we introduce a set of copies which behave like the original element, and then use at least one copy of each original element in the expansions of the original operations. Clearly, all expansions of an Nmatrix are its refinements. The importance of this type of refinements follows from the fact that they preserve the consequence relation of the original Nmatrix. To show this, we need to prove the following converse of Lemma 25: Lemma 29 Let M0 = hV 0 , D0 , O0 i be an expansion of M = hV, D, Oi with the underlying reduction F : V 0 → V satisfying the conditions of Definition 28. Then, for any legal valuation v in M, there is a legal valuation v 0 in M0 such that v = F ◦ v 0 . Proof: Let M, M0 , F and v meet the assumptions of the Lemma. We need to define a valuation v 0 : F rmL → V 0 such that (*) v(ϕ) = F (v 0 (ϕ)) for any formula ϕ. We proceed by induction on the complexity of ϕ. Suppose first ϕ = p is an atom. Then, as v(p) ∈ V and F (V 0 ) = V, there is an a ∈ V 0 such that F (a) = v(p). We put v 0 (p) = a. Obviously, we have F (v 0 (p)) = v(p), so (*) holds for ϕ . Assume now we have defined a partial valuation v 0 (ϕ) legal in M0 and satisfying (*) for all formulas ϕ of complexity up to k, and let ϕ be any formula of complexity k + 1. Then ϕ = (ϕ1 , . . . , ϕn ), where ϕ1 , . . . , ϕn are of complexity at most k. Since v is legal in M, we have v(ϕ) ∈ e M (v(ϕ1 ), . . . , v(ϕn )). As by inductive assumption v(ϕi ) = F (v 0 (ϕi )) for i = 1, . . . , n, then v(ϕ) ∈ e M (F (v 0 (ϕ1 )), . . . , F (v 0 (ϕn ))). However, by Point 2 of Definition 28, this implies there is b ∈ e M0 (v 0 (ϕ1 ), . . . , v 0 (ϕn )) with v(ϕ) = F (b). 0 0 Taking v (ϕ) = b, we have F (v (ϕ)) = v(ϕ), so (*) holds for ϕ. Since v 0 (ϕ) ∈ e M0 (v 0 (ϕ1 ), . . . , v 0 (ϕn )), 0 0 then the partial valuation v remains legal in M after its extension to formulas of rank k + 1. Hence the valuation v 0 defined inductively as above is legal in M0 and v = F ◦ v 0 . Proposition 30 If M0 is an expansion of M, then `M =`M0 . Proof: Assume M0 is an expansion of M, with an underlying reduction F : V 0 → V. Then M0 is a refinement of M , whence `M ⊆`M0 by Proposition 26. To prove the converse implication `M0 ⊆`M , we argue by contradiction. So assume that T ⊆ F rmL , ϕ ∈ F rmL and T `M0 ϕ but T 6`M ϕ. Then there is a valuation v legal in M such that v |= T and v 6|= ϕ. Hence v(ψ) ∈ D for every ψ ∈ T and v(ϕ) 6∈ D. By Lemma 29, there is a legal valuation v 0 in M0 such that v = F ◦ v 0 . Then, by assumption on v, we have F (v 0 (ψ)) ∈ D for every ψ ∈ T and F (v 0 (ϕ)) 6∈ D. However, by Item 1 of the definition of reduction, this implies v 0 (ψ) ∈ D0 for every ψ ∈ T and v 0 (ϕ) 6∈ D0 . As v 0 is a legal valuation in M0 , this contradicts the assumption that T `M0 ϕ. Based on this proposition, we shall show later that the results obtained here imply those of [8]. 8

3

C-systems without the Axioms l and d

As we have already said, systems without l and d were treated in [8]. In that paper we presented a method for systematic construction of cut-free quasi-canonical Gentzen-type calculi for those systems which was based on their semantic characterizations in terms of finite-valued (in fact, three-valued) Nmatrices. The said method, in turn, used the general algorithm given for this purpose in [6]. For the sake of completeness, and as a preparation for the more complicated proof to be given later for all major C-systems (including those with l or d), we shall now review the main results of [8].

3.1

Non-deterministic Three-valued Semantics

In this subsection we determine for which AX ⊆ A0 the system BK[AX] is strongly paraconsistent, and provide non-deterministic three-valued semantics for each such system. This semantics will be then used to derive first the semantic conditions corresponding to the axioms in BK[AX], and following this — the Gentzen-type rules corresponding to those semantic conditions. The non-deterministic semantics we use basically employs the following four truth values: t = h1, 0i, f = h0, 1i, > = h1, 1i, ⊥ = h0, 0i The intuition behind them is that a formula ϕ is assigned a truth value of the form hx, yi, where x = 1 iff ϕ is “true”, and y = 1 iff ¬ϕ is “true”. However, the axiom (t) ϕ ∨ ¬ϕ, included already in B, rules out the fourth truth value ⊥, since (t) implies that ϕ and ¬ϕ cannot be both “false”. Thus we are left with three truth values: t, f and >. As pointed out in [8], semantics for systems without the axiom (t) (which are obtained from the positive fragment of classical logic by adding some axioms from A0 ) can be provided in a similar way using the above four truth-values. We start by defining the Nmatrix M3 providing the semantics for BK: Definition 31 The Nmatrix M3 = ({t, f, >}, {t, >}, O) for LC is defined as follows: a ¬a t {f } > {t, >} f {t, >} ∨ t > f

t {t, >} {t, >} {t, >}

◦a {t, >} {f } {t, >} > {t, >} {t, >} {t, >}

f {t, >} {t, >} {f }

∧ t t {t, >} > {t, >} f {f }

> f {t, >} {f } {t, >} {f } {f } {f }

⊃ t t {t, >} > {t, >} f {t, >}

> {t, >} {t, >} {t, >}

f {f } {f } {t, >}

In [8] we provided non-deterministic three-valued semantics for the extensions of BK with axioms from A0 . Our semantics, quoted in this section, was modular in the following sense: each axiom ax ∈ A0 corresponds to some finite set C(ax) of semantic conditions. These conditions lead to simple refinements of the basic Nmatrix M3 (which amount to reducing the level of non-determinism in M3 ). The semantics of BK[AX] is then obtained by combining the semantic effects of all the axioms from AX in a straightforward way.

9

Tables 2 and 3 include the various semantic conditions that correspond to the axioms in A0 . We also include there a reformulation GC(ax) of each semantic condition C(ax) using the sets T = {t}, I = {>} and F = {f }. This reformulation can be applied to any expansion of M3 , and will be useful later in this paper for handling axioms (l) and (d), which must be treated using infinitely-valued semantics. Note that to ensure uniformity with GC(ax), we use inclusion instead of equality in the formulation of C(ax) (e.g., we write ◦f ⊆ {t} instead of ◦f = {t}). ˜, ∧ ˜ , and ⊃ ˜ in the Nmatrix M3 (and later in the basic Remark 32 Note that by using the definitions of ∨ infinite Nmatrix M0 introduced in Definition 49) we can simplify the conditions C(ax) and GC(ax) for ax ∈ {nr∧ , nr∨ , nr⊃ }. For example, GC(nr∧ ) from Table 2 can be simplified to: If a ∈ I and b ∈ T ∪ I, then a ∧ b ⊆ I and b ∧ a ⊆ I. Example 33 To see how the semantic conditions are derived, consider, e.g., the schema (a∨ ). To guarantee its validity, we must ensure that (∗) v(◦ϕ ∧ ◦ψ) ∈ {t, >} implies v(◦(ϕ ∨ ψ)) ∈ {t, >}. In any simple refinement of M3 , v(◦(ϕ ∨ ψ)) ∈ {t, >} iff v(ϕ ∨ ψ) ∈ {f, t}. Moreover, v(◦ϕ ∧ ◦ψ) ∈ {t, >} iff v(◦ϕ), v(◦ψ) ∈ {t, >} iff v(ϕ), v(ψ) ∈ {t, f }. Since v(ϕ ∨ ψ) ∈ {f, t} is already guaranteed if v(ϕ) = v(ψ) = f , this and the truth table of ∨ in M3 together entail the two conditions for b ∈ {t, f } which are given in Table 3: (i) t ∨ b ⊆ {t} and (ii) b ∨ t ⊆ {t}. Definition 34 A set AX ⊆ A0 is coherent if the semantic conditions C(ax) imposed by the axioms ax ∈ AX are consistent, i.e. if there is a simple refinement of M3 which satisfies those conditions. Based on the above definition, one can easily check that the coherent sets of axioms in A0 are exactly those given by the following proposition: Proposition 35 AX ⊆ A0 is coherent iff it does not contain any of the following pairs of axioms: (1) (o1∧ ), (nr∧ ); (2) (o2∧ ), (nr∧ ); (3) (o1∨ ), (nr∨ ); (4) (o2∨ ), (nr∨ ); (5) (o1⊃ ), (nr⊃ ). Proposition 36 For AX ⊆ A0 , BK[AX] is strongly paraconsistent iff AX is coherent10 . An implicit proof of Proposition 36 was contained in Remark 3.29 of [8]. Later in this paper we shall also give a generalization of this Proposition (see Proposition 64). Definition 37 For any coherent AX ⊆ A0 , let M3 [AX] be the weakest simple refinement of the Nmatrix M3 which satisfies all the semantic conditions C(ax) from Tables 2 and 3 corresponding to the axioms ax ∈ AX. Remark 38 Note that M3 [AX] is well-defined, because for a coherent AX the combination of the semantic conditions corresponding to axioms in AX is not contradictory, so their imposing on M3 in the way described above yields a correct three-valued Nmatrix, without empty entries. That Nmatrix is exactly the weakest simple refinement we have in mind. Example 39 Consider the set of axioms AX = {(nl∨ ), (nr∨ )}. Then the disjunction table in M3 [AX] is: ∨ t t {t} > {t} f {t}

> {t} {>} {>}

f {t} {>} {f }

10 An analogous proposition in [7] was erroneous because it used the notion of (standard) paraconsistency, which is insufficient here.

10

Table 2: A0 : axioms, the corresponding semantic conditions and Gentzen-type rules (Note that both here and in Table 3 below x ranges over {t, >, f }, and y over T ∪ I ∪ F.) ax

C(ax)

GC(ax)

(c)

¬¬ϕ ⊃ ϕ

¬f ⊆ {t}

for a ∈ F : ¬a ⊆ T

(e)

ϕ ⊃ ¬¬ϕ

¬> ⊆ {>}

for a ∈ I: ¬a ⊆ I

(i)

¬◦ϕ ⊃ (ϕ ∧ ¬ϕ)

◦f ⊆ {t} and ◦t ⊆ {t}

for a ∈ F ∪ T : ◦a ⊆ T

(nr∧ )

(¬ϕ ∨ ¬ψ) ⊃ ¬(ϕ ∧ ψ)

for a ∈ {f, >} :

for a ∈ F ∪ I :

Γ ⇒ ∆, ¬ψ, ¬ϕ

a ∧ x, x ∧ a ⊆ {f, >}

a ∧ y, y ∧ a ⊆ F ∪ I

Γ ⇒ ∆, ¬(ϕ ∧ ψ)

(nl∧ )

¬(ϕ ∧ ψ) ⊃ (¬ϕ ∨ ¬ψ)

t ∧ t ⊆ {t}

for a, b ∈ T : a ∧ b ⊆ T

(nr∨ )

(¬ϕ ∧ ¬ψ) ⊃ ¬(ϕ ∨ ψ)

for a, b ∈ {f, >} :

for a, b ∈ F ∪ I :

a ∨ b ⊆ {f, >}

a∨b⊆F ∪I

(nl∨ )

¬(ϕ ∨ ψ) ⊃ (¬ϕ ∧ ¬ψ)

(nr⊃ )

(ϕ ∧ ¬ψ) ⊃ ¬(ϕ ⊃ ψ)

(nl⊃ )

¬(ϕ ⊃ ψ) ⊃ (ϕ ∧ ¬ψ)

t ∨ x, x ∨ t ⊆ {t}

R(ax) Γ, ϕ ⇒ ∆ Γ, ¬¬ϕ ⇒ ∆

Γ ⇒ ∆, ϕ Γ ⇒ ∆, ¬¬ϕ

Γ, ϕ, ¬ϕ ⇒ ∆ Γ, ¬◦ ϕ ⇒ ∆

Γ, ¬ϕ ⇒ ∆

Γ, ¬ψ ⇒ ∆

Γ, ¬(ϕ ∧ ψ) ⇒ ∆

Γ ⇒ ∆, ¬ϕ

Γ ⇒ ∆, ¬ψ

Γ ⇒ ∆, ¬(ϕ ∨ ψ)

for a ∈ T :

Γ, ¬ϕ, ¬ψ ⇒ ∆

a ∨ y, y ∨ a ⊆ T

Γ, ¬(ϕ ∨ ψ) ⇒ ∆

Γ ⇒ ∆, ϕ

Γ ⇒ ∆, ¬ψ

for a ∈ {t, >}, b ∈ {f, >} :

for a ∈ D, b ∈ F ∪ I :

a ⊃ b ⊆ {f, >}

a⊃b∈F ∪I

Γ ⇒ ∆, ¬(ϕ ⊃ ψ)

for a ∈ F , b ∈ T :

Γ, ϕ, ¬ψ ⇒ ∆

a ⊃ y, y ⊃ b ⊆ T

Γ, ¬(ϕ ⊃ ψ) ⇒ ∆

f ⊃ x, x ⊃ t ⊆ {t}

11

Table 3: A0 : axioms, the corresponding semantic conditions, and Gentzen-type rules — continued

(a∧ )

(a∨ )

(a⊃ )

ax

C(ax)

GC(ax)

R(ax)

(◦ϕ ∧ ◦ψ) ⊃ ◦(ϕ ∧ ψ)

for a, b ∈ {t}, a ∧ b ⊆ {t}

for a, b ∈ T : a ∧ b ⊆ T

Γ, ¬ϕ ⇒ ∆ Γ, ¬ψ ⇒ ∆ Γ, ¬(ϕ ∧ ψ) ⇒ ∆

for b ∈ {t, f }, t ∨ b ⊆ {t}

for a ∈ T , b ∈ T ∪ F : a ∨ b ⊆ T

for b ∈ {t, f }, b ∨ t ⊆ {t}

for a ∈ T , b ∈ T ∪ F : b ∨ a ⊆ T

Γ, ¬ψ ⇒ ∆ Γ, ¬ϕ, ϕ ⇒ ∆ Γ, ¬(ϕ ∨ ψ) ⇒ ∆

for a ∈ {t, f }, f ⊃ a ⊆ {t}

for b ∈ F , a ∈ T ∪ F : b ⊃ a ⊆ T

Γ, ϕ ⇒ ∆ Γ, ¬ψ, ψ ⇒ ∆ Γ, ¬(ϕ ⊃ ψ) ⇒ ∆

for a ∈ {t, f }, a ⊃ t ⊆ {t}

for b ∈ T , a ∈ T ∪ F : a ⊃ b ⊆ T

Γ, ¬ϕ, ϕ ⇒ ∆ Γ, ¬ψ ⇒ ∆ Γ, ¬(ϕ ⊃ ψ) ⇒ ∆

Γ, ¬ϕ ⇒ ∆ Γ, ¬ψ, ψ ⇒ ∆ Γ, ¬(ϕ ∨ ψ) ⇒ ∆

(◦ϕ ∧ ◦ψ) ⊃ ◦(ϕ ∨ ψ)

(◦ϕ ∧ ◦ψ) ⊃ ◦(ϕ ⊃ ψ)

(o1 ∧)

◦ϕ ⊃ ◦(ϕ ∧ ψ)

for b ∈ {t, >} : t ∧ b ⊆ {t}

for a ∈ T and b ∈ D : a ∧ b ⊆ T

Γ, ¬ϕ ⇒ ∆ Γ ⇒ ∆, ψ Γ, ¬(ϕ ∧ ψ) ⇒ ∆

(o2 ∧)

◦ψ ⊃ ◦(ϕ ∧ ψ)

for b ∈ {t, >} : b ∧ t ⊆ {t}

for a ∈ T and b ∈ D : b ∧ a ⊆ T

Γ, ¬ψ ⇒ ∆ Γ ⇒ ∆, ϕ Γ, ¬(ϕ ∧ ψ) ⇒ ∆

t ∨ x ⊆ {t}

for a ∈ T : a ∨ y ⊆ T

Γ, ¬ϕ ⇒ ∆ Γ, ¬(ϕ ∨ ψ) ⇒ ∆

for b ∈ {t, >} : f ∨ b ⊆ {t}

for c ∈ F and b ∈ D : c ∨ b ⊆ T

Γ, ϕ ⇒ ∆ Γ ⇒ ∆, ψ Γ, ¬(ϕ ∨ ψ) ⇒ ∆

x ∨ t ⊆ {t}

for a ∈ T : y ∨ a ⊆ T

Γ, ¬ψ ⇒ ∆ Γ, ¬(ϕ ∨ ψ) ⇒ ∆

for b ∈ {t, >} : b ∨ f ⊆ {t}

for c ∈ F and b ∈ D : b ∨ c ⊆ T

Γ, ψ ⇒ ∆ Γ ⇒ ∆, ϕ Γ, ¬(ϕ ∨ ψ) ⇒ ∆

(o1 ∨)

(o2 ∨)

◦ϕ ⊃ ◦(ϕ ∨ ψ)

◦ψ ⊃ ◦(ϕ ∨ ψ)

for b ∈ {t, >} : t ⊃ b ⊆ {t} (o1 ⊃)

(o2 ⊃)

for a ∈ T and b ∈ D:

a⊃b⊆T

Γ, ¬ϕ ⇒ ∆ Γ ⇒ ∆, ψ Γ, ¬(ϕ ⊃ ψ) ⇒ ∆

◦ϕ ⊃ ◦(ϕ ⊃ ψ) f ⊃ x ⊆ {t}

for c ∈ F : c ⊃ y ⊆ T

Γ, ϕ ⇒ ∆ Γ, ¬(ϕ ⊃ ψ) ⇒ ∆

x ⊃ t ⊆ {t}

for a ∈ T : y ⊃ a ⊆ T

Γ, ¬ψ ⇒ ∆ Γ, ¬(ϕ ⊃ ψ) ⇒ ∆

f ⊃ f ⊆ {t}

for b, c ∈ F : b ⊃ c ⊆ T

Γ, ϕ ⇒ ∆ Γ, ψ ⇒ ∆ Γ, ¬(ϕ ⊃ ψ) ⇒ ∆

◦ψ ⊃ ◦(ϕ ⊃ ψ)

12

The truth tables for other connectives remain the same as in M3 , because the semantic effects of are limited to disjunction only.

(nl∨ ), (nr∨ )

Theorem 40 ([8]) For any coherent AX ⊆ A0 , T `M3 [AX] ψ iff T `BK[AX] ψ.

3.2

Corresponding Gentzen-type Systems

The development of Gentzen systems for the considered logics will be based on the following simple fact, expressing the basic properties of the non-deterministic semantics we have provided for them: Proposition 41 Let v be a full M-valuation, where M is any simple refinement of M3 . Then: • v(ψ) = t iff ¬ψ ⇒ is true under v. • v(ψ) = f iff ψ ⇒ is true under v. • v(ψ) = > iff ⇒ ψ and ⇒ ¬ψ are both true under v. • v(ψ) ∈ {f, >} iff ⇒ ¬ψ is true under v. • v(ψ) ∈ {t, >} iff ⇒ ψ is true under v. • v(ψ) ∈ {t, f } iff ψ, ¬ψ ⇒ is true under v. By translating the truth tables of M3 into the corresponding sequent rules using Proposition 41, we obtain the following Gentzen system for M3 : Table 4: Logical rules of GBK (∧ ⇒)

Γ, ϕ, ψ ⇒ ∆ Γ, ϕ ∧ ψ ⇒ ∆

(⇒ ∧)

Γ ⇒ ∆, ϕ Γ ⇒ ∆, ψ Γ ⇒ ∆, ϕ ∧ ψ

(∨ ⇒)

Γ, ϕ ⇒ ∆ Γ, ψ ⇒ ∆ Γ, ϕ ∨ ψ ⇒ ∆

(⇒ ∨)

Γ ⇒ ∆, ϕ, ψ Γ ⇒ ∆, ϕ ∨ ψ

(⊃⇒)

Γ ⇒ ϕ, ∆ Γ, ψ ⇒ ∆ Γ, ϕ ⊃ ψ ⇒ ∆

(⇒⊃)

Γ, ϕ ⇒ ψ, ∆ Γ ⇒ ϕ ⊃ ψ, ∆

(⇒ ¬)

Γ, ϕ ⇒ ∆ Γ ⇒ ∆, ¬ϕ

(⇒ ◦)

Γ, ϕ, ¬ϕ ⇒ ∆ Γ ⇒ ◦ϕ, ∆

(◦ ⇒)

Γ ⇒ ϕ, ∆ Γ ⇒ ¬ϕ, ∆ Γ, ◦ϕ ⇒ ∆

Definition 42 The system GBK consists of the basic axiom ψ ⇒ ψ, the structural rules of cut and weakening, and the logical rules given in Table 4. The system GB is obtained from GBK by deleting the rule (⇒ ◦).

13

Example 43 To see how the logical rules of GBK are derived, consider the semantic tables for ◦. First, we observe that (1) v(◦ϕ) = f iff v(ϕ) = >, and (2) v(◦ϕ) ∈ {t, >} iff v(ϕ) ∈ {t, f }. As by Proposition 41 v(ψ) = f iff ψ ⇒ is true under v, and v(ψ) = > iff both ⇒ ψ and ⇒ ¬ψ are true under v, then after adding contexts (1) translates to the rule (◦ ⇒) above. Further, since by Proposition 41 v(ψ) ∈ {t, >} iff ⇒ ψ is true under v, and v(ψ) ∈ {t, f } iff ψ, ¬ψ ⇒ is true under v, then (2) translates to the rule (⇒ ◦). Remark 44 Here we should comment on the relationship between the Gentzen-type systems given here and their corresponding Hilbert-style systems. Using a standard method based on cuts, one can show that each such Gentzen-type system G is equivalent to the corresponding Hilbert-type system H in the sense that T `H ψ iff T `G ψ. In particular, ψ is a theorem of H iff `G ⇒ ψ. The above can be even shown mechanically using the system described in [21], implemented in Prolog. The method we have just used for M3 can be applied to translate the semantic effect of each extra axiom into Gentzen rules. Adding those rules to the system GBK , we can obtain in a modular way a cut-free Gentzen-type formulation for each of the C-systems BK[AX] with AX ⊆ A0 . This process results in generating the rules R(ax) given in Tables 2 and 3. Example 45 To see how the Gentzen-type rules from Tables 2,3 are derived, let us consider the axiom (a∨ ). By Table 2, the validity of this axiom is equivalent to the combination of the following two conditions for b ∈ {t, f }: (i) t ∨ b ⊆ {t}, and (ii) b ∨ t ⊆ {t}. Now (i) can be reformulated as follows: if ¬ϕ ⇒ and ψ, ¬ψ ⇒ are true, then ¬(ϕ ∨ ψ) ⇒ is true. By adding context, we obtain: Γ, ¬ϕ ⇒ ∆ Γ, ψ, ¬ψ ⇒ ∆ Γ, ¬(ϕ ∨ ψ) ⇒ ∆ Similarly, (ii) can be reformulated as follows: if ϕ, ¬ϕ ⇒ and ¬ψ ⇒ are true, then ¬(ϕ ∨ ψ) ⇒ is true. By adding context, we obtain: Γ, ¬ψ ⇒ ∆ Γ, ϕ, ¬ϕ ⇒ ∆ Γ, ¬(ϕ ∨ ψ) ⇒ ∆ Definition 46 • For each ax ∈ A, the set of Gentzen-type rules R(ax) corresponding to ax is defined as in Tables 2,3. • For AX ⊆ A, GBK [AX] is the Gentzen-type system obtained by adding to GBK the set of rules R(ax) for every ax ∈ AX. Proposition 47 If AX ⊆ A0 then BK[AX] is equivalent to GBK [AX]. Proof: This can be easily proved directly (see Remark 44). For a coherent AX, it can also be derived semantically from Theorem 40 and Theorem 48 below. Theorem 48 ([8]) Let AX be a coherent subset of A0 . 1. M3 [AX] is a characteristic Nmatrix for GBK [AX] (and so for BK[AX]). 2. GBK [AX] enjoys cut-admissibility. In the sequel we shall show (Theorem 66) a general version of Theorem 48 for any set of axioms AX ⊆ A (which can also include the axioms (l), (d)), whose proof can easily be adapted (and simplified) to provide a new proof of this Theorem. 14

4

Cut-free Calculi for C-systems

The method for constructing cut-free quasi-canonical Gentzen calculi for C-systems without the axioms (l), (d) described in the foregoing does not apply as it is to the systems which include one of those axioms. This is because the method was based on the use of a three-valued Nmatrix M3 providing the semantics for the C-systems considered previously, which is no longer adequate for C-systems containing (l) or (d). What is more, it was shown in [5] that such systems cannot have finite-valued characteristic Nmatrices (and so, by [1], they cannot have quasi-canonical Gentzen calculi). However, it was also shown there that they do have infinitely-valued characterizations of this type (which still suffice for guaranteeing their decidability). Below we show that the those characterizations can also be exploited for a modular construction of cut-free sequent calculi. Those calculi are of course not quasi-canonical, but they are still weakly quasi-canonical — which is almost as good.

4.1

Non-deterministic Infinitely-valued Semantics

The treatment of axioms (l) and (d) is more complicated than that of the axioms from A0 , since (as noted above) the semantic effect of adding them to BK cannot be formulated as a condition on the three-valued truth-table of some connective, leading to a certain simple refinement of M3 . This is due to the fact that both (l) and (d) involve a conjunction of a formula with its negation. Accordingly, to handle them we must be able to distinguish between the conjunction of an “inconsistent” formula ψ with its negation and the conjunction of ψ with other formulas. This requires an infinite number of truth-values, corresponding to the infinitely many formulas of the language. Therefore, the finite Nmatrix M3 for BK was replaced in [5] by the infinite Nmatrix M0 defined below. Instead of t, > and f , M0 uses three sets of truth-values: T = {tji | i ≥ 0, j ≥ 0}, I = {>ji | i ≥ 0, j ≥ 0} and F = {f }, respectively. Intuitively, the set T contains infinitely many “copies” of the classical value t, the set F contains (one “copy” of) f , and the set I contains infinitely many “copies” of the inconsistent truth-value >. Definition 49 The Nmatrix M0 = hV, D, Oi for LC , where V = T ∪ I ∪ F and D = T ∪ I, is defined as follows:  D if either a ∈ D or b ∈ D, eb = a∨ F if a, b ∈ F  D if either a ∈ F or b ∈ D e = a⊃b F if a ∈ D and b ∈ F  F if either a ∈ F or b ∈ F eb = a∧ D otherwise  if a ∈ T  F D if a ∈ F ¬ ea =  j+1 {>j+1 , t } if a = >ji i i  D if a ∈ F ∪ T e ◦a = F if a ∈ I

Now we turn to providing non-deterministic semantics for the extensions of BK with axioms from A, including the problematic axioms (l) and (d), which (as noted above) do not allow finite non-deterministic semantics. The modularity of our semantics is preserved — each axiom corresponds to a certain semantic 15

condition on the basic Nmatrix M0 . Moreover, the semantic conditions induced by the axioms from A0 turn out to be identical with their general formulations GC(ax) presented in Tables 2 and 3. However, there is a difference in the meaning of what is written there: now we take T = {tji | i ≥ 0, j ≥ 0}, I = {>ji | i ≥ 0, j ≥ 0}, and F = {f }. Example 50 Let us consider again the axiom (a∨ ), considered earlier in the context of three-valued simple refinements of M3 . To guarantee its validity in the context of infinite-valued refinements of M0 , we must ensure that (∗) v((◦ϕ ∧ ◦ψ)) ∈ D implies v(◦(ϕ ∨ ψ)) ∈ D. In any simple refinement of M0 , v(◦(ϕ ∨ ψ)) ∈ D iff v(ϕ ∨ ψ) ∈ T ∪ F. Moreover, v((◦ϕ ∧ ◦ψ)) ∈ D iff v(◦ϕ), v(◦ψ) ∈ D iff v(ϕ), v(ψ) ∈ F ∪ T . Since v(ϕ ∨ ψ) ∈ T ∪ F is already guaranteed when v(ϕ), v(ψ) ∈ F, the truth table of ∨ in M0 implies that we only have to require the two conditions for (a∨ ) given in Table 3: if a ∈ T and b ∈ T ∪ F then: (i) a ∨ b ⊆ T and (ii) b ∨ a ⊆ T . Note the similarity of this derivation of semantic conditions to that in Example 33. Now we have to define the semantic conditions induced by (l) and (d): Definition 51 GC(l): For a = >ji and b ∈ {>j+1 , tj+1 }, a ∧ b ⊆ T . i i j+1 j+1 j GC(d): For b = >i and a ∈ {>i , ti }, a ∧ b ⊆ T . If the set of axioms contains (l) or (d), we need to modify the notion of coherence given in Definition 34 to one applicable to the Nmatrix M0 rather than to M3 . Definition 52 For AX ⊆ A, we say that AX is coherent if the semantic conditions GC(ax) imposed by the axioms ax ∈ AX are consistent, i.e. if there is a simple refinement of M0 which satisfies those conditions. Again one can easily check that the coherent sets of axioms in A are exactly those given by the following proposition: Proposition 53 For any AX ⊆ A, AX is coherent iff it does not contain any of the following pairs of axioms:11 1. (o1∧ ) and (nr∧ ); 2. (o2∧ ) and (nr∧ ); 3. (o1∨ ) and (nr∨ ); 4. (o2∨ ) and (nr∨ ); 5. (o1⊃ ) and (nr⊃ ). 6. (l) and (nr∧ ) 7. (d) and (nr∧ ). 11 Note that pairs 1-5 were already forbidden for AX ⊆ A in Proposition 35. Note also that in [7] the notion of coherence 0 was defined for AX ⊆ A by simply listing the forbidden pairs of axioms given in the above Proposition. However, that definition was tailored to the specific set of axioms considered in that paper, while the present one is general, and does not rely on any specific choice of axioms.

16

Definition 54 For any coherent AX ⊆ A, the Nmatrix M0 [AX] is the weakest simple refinement of M0 in which GC(ax) (from Tables 2,3 and Definition 51) holds for every ax ∈ AX. A crucial property of the infinite Nmatrix M0 is that M0 is an expansion of the 3-valued Nmatrix M3 underlying the semantics of C-systems without axioms (l), (d). What is more, an analogous relationship holds also for the weakest simple refinements of the above Nmatrices corresponding to any set of axioms AX ⊆ A0 : Proposition 55 For any coherent AX ⊆ A0 , the Nmatrix M0 [AX] is an expansion of the finite-valued Nmatrix M3 [AX]. Proof: We shall prove the basic case where AX = ∅. It implies the claim for any AX, since none of the semantic conditions given in Tables 2,3 which correspond to the axioms in AX distinguish between the individual elements of T , F or I. For the purposes of this proof, let us affix a subscript equal to either 3 or 0, respectively, to all components of the above Nmatrices, and denote additionally F3 = {f } and T3 = {t}. As D3 = {t, >}, then from the truth tables of M3 it immediately follows that the interpretations of all connectives in M3 save for negation can be written down in exactly the same way as in case of M0 , with D0 and F0 replaced by D3 and F3 , respectively. Accordingly, if we define F : V0 → V3 by F (f ) = f, F (tji ) = t, F (>ji ) = > for any i, j ≥ 0, then F [V0 ] = V3 , F [D0 ] = D3 , F [T0 ] = T3 , F [F0 ] = F3 , and for any n-ary connective  other than ¬ we have (∗)

F r[e M0 (a1 , . . . , an )] = e M3 (F (a1 ), . . . , F (an ))

for any a1 , . . . , an ∈ V3 . In case of ¬, this obviously holds for n = 1 and a ∈ F0 ∪ T0 . Consider now a ∈ I0 . Then a = >ji for some i, j, whence F (a) = > and ¬ e 3 F (a) = {t, >}. On the other hand, F (e ¬0 a) = F ({>j+1 , tj+1 }) = i i j+1 {F (>j+1 ), F (t )} = {>, t}, whence (*) holds in this case too. Hence the conditions of Definition 28 i i are satisfied, and M0 is an expansion of M3 . As a direct corollary of Proposition 55, Proposition 30 and Theorem 48 we obtain the following: Proposition 56 The following is true for any coherent AX ⊆ A0 : 1. `M0 [AX] =`M3 [AX] . 2. T `M0 [AX] ψ iff T `BK[AX] ψ. The second part of Proposition 56 can be generalized as follows: Proposition 57 T `M0 [AX] ψ iff T `BK[AX] ψ for every coherent AX ⊆ A. The proof of Proposition 57 can be given by modifying the proofs of the strongly related soundness and completeness theorems from [5] and [3]. However, later we shall provide an independent completeness proof for the corresponding Gentzen type systems. Since the equivalence between those systems and the original Hilbert type systems can be proved mechanically (e.g. using the system from [21]), this would yield a new proof of Proposition 57.

17

4.2

The Corresponding Gentzen-type Systems

To construct cut-free Gentzen-type systems for logics with (l) and (d), we start with the obvious translation of (l) and (d) into Gentzen-type rules implied by the intuitive meaning of ◦ψ given in Remark 18. Namely, this translation is obtained by substituting in (◦ ⇒) the formulas ¬(ψ ∧ ¬ψ) and ¬(¬ψ ∧ ψ) (respectively) for ◦ψ (note that by applying the same procedure to (⇒ ◦), we get a rule which is derivable in BK). Definition 58 The Gentzen-type rules R(l) and R(d) are defined as follows: Γ ⇒ ϕ, ∆ Γ ⇒ ¬ϕ, ∆ R(d) Γ, ¬(¬ϕ ∧ ϕ) ⇒ ∆

Γ ⇒ ϕ, ∆ Γ ⇒ ¬ϕ, ∆ R(l) Γ, ¬(ϕ ∧ ¬ϕ) ⇒ ∆

As for the Gentzen-type rules which correspond to the axioms in A0 , luckily we need not start our search from scratch. Although we cannot construct a rule for each line of every truth-table like before (because of both the infinite number of truth-values, and the insufficient expressive power of our language, which does not allow for characterizing each of them), we can still perform the easier task of encoding the semantic effect of each axiom by a Gentzen-type rule. This can be done using the following analogue of Proposition 41: Proposition 59 Let v be a full M-valuation, where M is a simple refinement of M0 . Then: • v(ψ) ∈ T iff ¬ψ ⇒ is true under v. • v(ψ) ∈ F iff ψ ⇒ is true under v. • v(ψ) ∈ I iff ⇒ ψ and ⇒ ¬ψ are both true under v. • v(ψ) ∈ F ∪ I iff ⇒ ¬ψ is true under v. • v(ψ) ∈ T ∪ I iff ⇒ ψ is true under v. • v(ψ) ∈ F ∪ T iff ψ, ¬ψ ⇒ is true under v. Example 60 Revisiting Example 50, the semantic conditions GC(a∨ ) from Table 3 are: for a ∈ T , b ∈ T ∪ F: (i) a ∨ b ⊆ T and (ii) b ∨ a ⊆ T . The first of then can be reformulated as follows: if v(ϕ) ∈ T and v(ψ) ∈ T ∪ F, then v(ϕ ∨ ψ) ∈ T . Using Proposition 59, we rewrite this as: if ¬ϕ ⇒ and ¬ψ, ψ ⇒ are true, then ¬(ϕ ∨ ψ) ⇒ is true. By adding context we obtain the first corresponding rule from Table 3. The second rule is obtained similarly from condition (ii). We therefore retain our ability to provide cut-free Gentzen-type systems for BK[AX] with AX ⊆ A in a modular way. Moreover, as the semantic effects of the axioms in A0 remain the same, it is not surprising that the Gentzen-type rules corresponding to them are exactly those given in Tables 2,3: Definition 61 For each ax ∈ A, the set R(ax) of Gentzen-type rules corresponding to ax is defined as in Tables 2,3 and Definition 58. For AX ⊆ A, GBK [AX] is the Gentzen-type system obtained by augmenting GBK with the set of rules R(ax) for every ax ∈ AX. Example 62 Below we present a cut-free proof of (l) in GBK [{l}] (note that the rule R(l) defined above does not even mention the connective ◦): ϕ, ¬ϕ ⇒ ¬ϕ ϕ, ¬ϕ ⇒ ϕ ⇒ ¬ϕ, ◦ϕ (⇒ ◦) ⇒ ϕ, ◦ϕ (⇒ ◦) (R(l)) ¬(ϕ ∧ ¬ϕ) ⇒ ◦ϕ (⇒⊃) ⇒ ¬(ϕ ∧ ¬ϕ) ⊃ ◦ϕ 18

Proposition 63 If AX ⊆ A then BK[AX] is equivalent to GBK [AX]. Proof: Like in Proposition 47, this can easily be proved directly (Remark 44), and for coherent AX it can also be derived semantically from Theorem 66 below. Next we extend Proposition 36 to the full set A. Proposition 64 For AX ⊆ A, BK[AX] is strongly paraconsistent iff AX is coherent. Proof: (⇐) : Suppose AX is coherent. We note that for any such AX, ¬ ˜ M0 [AX] (x) ⊆ D for any x ∈ I, and ¬ ˜ M0 [AX] (y) = {f } for any y ∈ T Let p, q be some propositional variables. Then p, ¬p 6`M0 [AX] ¬q (as any M0 [AX]-valuation v such that v(p) ∈ I and v(q) ∈ T is a refuting one). By Proposition 57, p, ¬p 6`BK[AX] ¬q, and so BK[AX] is strongly paraconsistent. (⇒) : Suppose that AX is not coherent. By Proposition 53, it must contain at least one of the forbidden pairs listed in that Proposition. We show that this entails `GBK [AX] p, ¬p ⇒ ¬q, and so by Propositions 63, p, ¬p `BK[AX] ¬q for every two propositional variables p, q. By structurality of `BK[AX] , this implies that BK[AX] is not strongly paraconsistent. We consider the following two cases of forbidden pairs, leaving the rest to the reader: • AX contains both (o1∧ ) and (nr∧ ). Then the following is a derivation in GBK [AX]: ¬p ⇒ ¬p ¬p ⇒ ¬p, ¬q (weak) ¬q ⇒ ¬q p ⇒ p 1 R(nr∧ ) R(o∧ ) p, ¬(q ∧ p) ⇒ ¬q ¬p ⇒ ¬(q ∧ p) (cut) p, ¬p ⇒ ¬q • AX contains both (l) and (nr∧ ). Then the following is a derivation in GBK [AX]: p ⇒ p ¬p ⇒ ¬p ¬p ⇒ ¬p, ¬¬p R(l) R(nr∧ ) p, ¬p, ¬(p ∧ ¬p) ⇒ ¬p ⇒ ¬(p ∧ ¬p) (cut) p, ¬p ⇒ p, ¬p ⇒ ¬q (weak) We have thus shown that the induced logic is not strongly paraconsistent. It is interesting to note that when AX contains the first pair mentioned above, but not the second, the resulting logic is still paraconsistent (as shown in [8]). This, however, does not hold if AX contains the second pair referred to above, as p, ¬p ⇒ q is also derivable by replacing ¬q by q in the last line of the derivation. Remark 65 Proposition 64 implies that the cut-elimination theorem fails for GBK [AX] when AX is not coherent, because then there is no way to derive p, ¬p ⇒ ¬q in GBK [AX] without using cuts. As the next theorem shows, the situation is completely different when AX is coherent. Now we come to the main theorem of this paper, which applies to all coherent extensions of BK studied here. It includes direct proofs of completeness and cut-admissibility, without relying on the results from previous papers. Theorem 66 Let AX be a coherent subset of A. 19

1. M0 [AX] is a characteristic Nmatrix for GBK [AX]. 2. GBK [AX] enjoys cut-admissibility. Proof: We leave the easy proof of soundness to the reader. Below we prove completeness together with cut-admissibility. For AX ⊆ A, we call a sequent Γ ⇒ ∆ saturated with respect to AX if it is closed under the rules of GK [AX] applied backwards. More exactly, Γ ⇒ ∆ is saturated if it satisfies the properties (S1)-(S11) below: (S1) If ϕ ∧ ψ ∈ Γ, then ϕ, ψ ∈ Γ. If ϕ ∧ ψ ∈ ∆, then ϕ ∈ ∆ or ψ ∈ ∆. Similarly for ∨ and ⊃. (S2) If ¬ϕ ∈ ∆, then ϕ ∈ Γ. (S3) If ◦ϕ ∈ Γ, then either ϕ ∈ ∆ or ¬ϕ ∈ ∆. If ◦ϕ ∈ ∆, then ϕ ∈ Γ and ¬ϕ ∈ Γ. (S4) If (l) ∈ AX and ¬(ϕ ∧ ¬ϕ) ∈ Γ, then either ϕ ∈ ∆ or ¬ϕ ∈ ∆. (S5) If (d) ∈ AX and ¬(¬ϕ ∧ ϕ) ∈ Γ, then either ϕ ∈ ∆ or ¬ϕ ∈ ∆. (S6) If (c) ∈ AX and ¬¬ϕ ∈ Γ, then ϕ ∈ Γ. (S7) If (e) ∈ AX and ¬¬ϕ ∈ ∆, then ϕ ∈ ∆. (S8) If (i) ∈ AX and ¬ ◦ ϕ ∈ Γ, then ϕ, ¬ϕ ∈ Γ. (S9) If (o1∧ ) ∈ AX and ¬(ϕ ∧ ψ) ∈ Γ, then either ¬ϕ ∈ Γ or ψ ∈ ∆. If (o2∧ ) ∈ AX and ¬(ϕ ∧ ψ) ∈ Γ, then either ¬ψ ∈ Γ or ϕ ∈ ∆. If (o1∨ ) ∈ AX and ¬(ϕ ∨ ψ) ∈ Γ, then ¬ϕ ∈ Γ, and either ϕ ∈ Γ or ψ ∈ ∆. Similarly for (o2∨ ) and (oi⊃ ), where i ∈ {1, 2}. (S10) If (nr∧ ) ∈ AX and ¬(ϕ ∧ ψ) ∈ ∆, then ¬ϕ, ¬ψ ∈ ∆. If (nl∧ ) ∈ AX (or (a∧ ) ∈ AX) and ¬(ϕ ∧ ψ) ∈ Γ, then either ¬ϕ ∈ Γ or ¬ψ ∈ Γ. (S11) Similarly for the rest of the axioms from A. Now let AX be a coherent subset of A, and suppose that Γ0 ⇒ ∆0 has no cut-free proof in BK[AX]. It is a standard matter to show that Γ0 ⇒ ∆0 can be extended to a saturated (with respect to AX) sequent Γ ⇒ ∆ such that (i) Γ0 ⊆ Γ and ∆0 ⊆ ∆, and (ii) Γ ⇒ ∆ has no cut-free proof in GBK [AX]. Note that the latter means that we have: (∗) Γ ∩ ∆ = ∅ Let λi.αi be an enumeration of all the formulas in LC that do not begin with ¬. Then, for every formula ψ of LC , there are unique n(ψ) and k(ψ) such that ψ = ¬k(ψ) αn(ψ) , where ¬k ϕ is ϕ preceded by k negation symbols. Now we define a valuation v in M0 [AX] which refutes coherent). Below we write ˜  instead of ˜ M0 [AX] . If p is atomic, then   f v(p) = t0n(p)   0 >n(p)

20

Γ ⇒ ∆ (note that M0 [AX] exists because AX is

p∈∆ ¬p ∈ ∆ otherwise

For a sentence ϕ = (ψ1 , . . . , ψj ) (where j ∈ {1, 2}), define:   f    tk(ϕ) n(ϕ) v(ϕ) =     >k(ϕ)

n(ϕ)

˜ (v(ψ1 ), . . . , v(ψj )) = {f } k(ϕ) tn(ϕ) ∈ ˜ (v(ψ1 ), . . . , v(ψj )), and ˜(v(ψ1 ), . . . , v(ψj )) ⊆ T or ¬ϕ ∈ ∆ either  otherwise

Evidently, the valuation v above is well-defined by (∗) and (S2). Now we have to prove that v is a legal valuation in M0 [AX], that is, for every j-ary connective , v((ψ1 , . . . , ψj )) ∈ ˜(v(ψ1 ), . . . , v(ψj )). If k(ϕ) v(ϕ) = f or v(ϕ) = tn(ϕ) , this follows directly from the definition. It remains to prove the case for k(ϕ)

v(ϕ) = >n(ϕ) . Then, by the definition of v, we have: (i) ˜ (v(ψ1 , . . . , ψj )) 6= {f }

(ii) ˜(v(ψ1 , . . . , ψj )) 6⊆ T

We shall use this fact extensively in the proof below, remembering also that F = {f } ϕ = ¬ψ: Considering (i) above, by the truth table for negation in M0 , we have ¬ ˜ v(ψ) ⊆ D. Now, from Tables 2,3 it follows that the only axioms whose generalized semantic conditions can influence ¬ ˜ v(ψ) are (c) and (e). Since (c) could only imply ¬ ˜ v(ψ) ⊆ T , which is ruled out by (ii), the only relevant axiom is (e), which implies that ¬ ˜ v(ψ) ⊆ I if v(ψ) ∈ I. i+1 In view of the above, ¬ ˜ v(ψ) must be one of the following sets: (1) D — if v(ψ) ∈ F; (2) {ti+1 j , >j } i+1 — if v(ψ) ∈ I and (e) 6∈ AX; (3) {>j } — if v(ψ) ∈ I and (e) ∈ AX. If (1) holds, we are done. If (2) or (3) hold, then by the definition of v, i = k(ψ) and j = n(ψ), and since k(ψ) + 1 = k(¬ψ), the claim again holds.

ϕ = ◦ψ: The only axiom with generalized semantic condition influencing the interpretation of ◦ is (i). However, since the said condition could only imply that ˜◦v(ψ) ⊆ T , which is ruled out by condition (ii) above, axiom (i) has no influence on ˜◦v(ψ). Hence by (i) and the truth table for ◦ in M0 we k(ϕ) have ˜ ◦v(ψ) = D, and since >n(ϕ) ∈ D, the claim holds. ϕ = ψ1 ∧ ψ2 : In this case, the value of v(ϕ) can be influenced by all axioms whose generalized semantic conditions affect the interpretation of conjunction, namely: (nr∧ ), (nr∧ ), (a∧ ), (o1∧ ), (o2∧ ), (l), (d). ˜ (v(ψ1 ), v(ψ2 )) in view of (ii), However, after again eliminating the axioms which are irrelevant to ∧ we are only left with (nr∧ ). Hence, in view of (i) and the truth table for conjunction in M0 , we ˜ (v(ψ1 ), v(ψ2 )) must be either (1) D — if (nr∧ ) 6∈ AX, or (2) I — if (nr∧ ) ∈ AX. As conclude that ∧ k(ϕ) both of these sets contain >n(ϕ) . the claim holds once more. The easy cases of ⊃ and ∨ are left to the reader. It remains to show that v is a refuting valuation for Γ ⇒ ∆. First, it is easy to prove the following by induction on ϕ: k(ϕ) k(ϕ) (#) For every ϕ ∈ F rmL : v(ϕ) ∈ {f, tn(ϕ) , >n(ϕ) }. Next we prove (by induction on ϕ) the following four properties: 1. If ϕ ∈ ∆ then v(ϕ) = f . k(ϕ)

k(ϕ)

2. If ϕ ∈ Γ then v(ϕ) = tn(ϕ) or v(ϕ) = >n(ϕ) .

21

k(ϕ)

3. If ¬ϕ ∈ ∆ then v(ϕ) = tn(ϕ) . k(ϕ)

4. If ¬ϕ ∈ Γ then v(ϕ) = f or v(ϕ) = >n(ϕ) . If ϕ is atomic then 1-4 are immediate from the definition of v, the fact that if ϕ ∈ Γ then ϕ 6∈ ∆ (by (∗) above), and the fact that k(ϕ) = 0 in this case. • Suppose that ϕ = ¬ψ. k(ψ)

1. Suppose ϕ = ¬ψ ∈ ∆. By the induction hypothesis for ψ, we have v(ψ) = tn(ψ) . Since k(ψ)

¬ ˜ tn(ψ) = {f }, v(ϕ) = f by the definition of v. k(ψ)

2. Suppose ϕ = ¬ψ ∈ Γ. By the induction hypothesis for ψ, either v(ψ) = f or v(ψ) = >n(ψ) . k(ϕ)

Either way, ¬ ˜ v(ψ) 6= {f }. By the definition of v, v(ϕ) 6= f , and so by (#) either v(ϕ) = tn(ϕ) k(ϕ)

or v(ϕ) = >n(ϕ) . 3. Suppose ¬ϕ ∈ ∆. By property (S2), ¬ϕ = ¬¬ψ ∈ ∆ implies ¬ψ ∈ Γ. From the induck(ψ) tion hypothesis for ψ, it follows that either v(ψ) = f or v(ψ) = >n(ψ) . Now there are two possibilities: – (e) ∈ AX. Then ψ ∈ ∆ by (S7), and so v(ψ) = f by the induction hypothesis Hence in k(ϕ) this case tn(ϕ) ∈ ¬ e v(ψ) (because for every AX ⊆ A, ¬ e f can be either T or D). k(ψ)

k(ψ)+1

– (e)6∈AX. Then ¬ e >n(ψ) and ¬ e f can be either D, T or {>n(ψ)

k(ψ)+1

, tn(ψ)

¬ e v(ψ). Since k(ψ) + 1 = k(ϕ) and n(ψ) = n(ϕ), we again have k(ϕ)

k(ϕ) tn(ϕ)

k(ψ)+1

}, and so tn(ψ)

∈

∈¬ e v(ψ).

k(ϕ)

Thus in both cases tn(ϕ) ∈ ¬ e v(ψ). Since ¬ϕ ∈ ∆, v(ϕ) = tn(ϕ) by the definition of v. 4. Suppose ¬ϕ ∈ Γ. Then ¬ϕ 6∈ ∆ by (*) above, and so v(ϕ) ∈ T only if ¬ e v(ψ) ⊆ T . One of the following holds: – (c) ∈ A. Then by (S6) ¬ϕ = ¬¬ψ ∈ Γ implies ψ ∈ Γ. By the induction hypothesis, k(ψ) k(ψ) v(ψ) = tn(ψ) or v(ψ) = >n(ψ) . Thus we cannot have ¬ e v(ψ) ⊆ T (since ¬ e a = {f } for any a ∈ T , while we always have ¬ ea ∩I = 6 ∅ for any a ∈ I). – (c)6∈AX. Then ¬ e v(ψ)6⊆T even if v(ψ) = f (since in this case ¬ e f = D). k(ϕ)

Hence in both cases ¬ e v(ψ)6⊆T , and so either v(ϕ) = f or v(ϕ) = >n(ϕ) by the definition of v. • Suppose that ϕ = ◦ψ. 1. Suppose ϕ ∈ ∆. By (S3), this implies ψ, ¬ψ ∈ Γ. By the induction hypothesis, we have k(ψ) k(ψ) v(ψ) = >n(ψ) . As e ◦(>n(ψ) ) = {f }, v(ϕ) = f by the definition of v. 2. Suppose ϕ ∈ Γ. By (S3), either ψ ∈ ∆ or ¬ψ ∈ ∆. Hence v(ψ)6∈I by the induction hypothesis. k(ψ) k(ϕ) Then ˜ ◦(v(ψ)) 6= {f }, and so v(ϕ) = tn(ψ) or v(ϕ) = >n(ϕ) . 3. Suppose ¬ϕ ∈ ∆. By(S2), ϕ = ◦ψ ∈ Γ. By (S3), either ψ ∈ ∆ or ¬ψ ∈ ∆. Hence v(ψ)6∈I k(ϕ) by the induction hypothesis. Thus e ◦(v(ψ)) is either T or D, and so tn(ϕ) ∈ e ◦(v(ψ)). Since k(ϕ)

¬ϕ ∈ ∆, this and the definition of v imply that v(ϕ) = tn(ϕ) .

22

4. Suppose ¬ϕ ∈ Γ. Then ¬ϕ 6∈ ∆ by (∗) above. e ◦(v(ψ)) ⊆ T can occur only if (i) ∈ AX and v(ψ) ∈ F ∪ T . However, this is impossible, as by (S8) ¬◦ ψ ∈ Γ implies ψ, ¬ψ ∈ Γ, and so v(ψ) ∈ I by the induction hypothesis. Hence e ◦(v(ψ))6⊆T , and since ¬ϕ6∈∆, by the definition k(ϕ) of v we have v(ϕ) = f or v(ϕ) = >n(ϕ) . • Suppose that ϕ = ψ1 ∧ ψ2 . 1. Suppose ϕ ∈ ∆. By (S1), ψ1 ∈ ∆ or ψ2 ∈ ∆. By the induction hypothesis, v(ψ1 ) = f or ˜ (v(ψ1 ), v(ψ2 )) = {f }, and v(ϕ) = f by the definition of v. v(ψ2 ) = f . Hence ∧ k(ψ )

2. Suppose ϕ ∈ Γ. By (S1), ψ1 , ψ2 ∈ Γ. By the induction hypothesis, v(ψi ) = tn(ψii ) or v(ψi ) = k(ψ ) e (v(ψ1 ), v(ψ2 )) 6= {f }, and by the definition of v, v(ϕ) = tk(ϕ) >n(ψii ) for i ∈ {1, 2}. Hence ∧ n(ϕ) or k(ϕ)

v(ϕ) = >n(ϕ) . 3. Suppose ¬ϕ ∈ ∆. By (S2), ϕ ∈ Γ. By (S1), ψ1 , ψ2 ∈ Γ. By the induction hypothesis, k(ϕ) ˜ (v(ψ1 ), v(ψ2 )), then v(ϕ) = tk(ϕ) v(ψ1 ), v(ψ2 ) ∈ D. If tn(ϕ) ∈ ∧ n(ϕ) by the definition of v, and we r are done. Otherwise, we must have (n∧ ) ∈ AX and either v(ψ1 ) ∈ I ∪ F or v(ψ2 ) ∈ I ∪ F. n(ψ ) n(ψ ) But then ¬ψ1 , ¬ψ2 ∈ ∆ by (S10), and so v(ψ1 ) = tk(ψ11) and v(ψ2 ) = tk(ψ22) by the induction hypothesis — which is a contradiction. ˜ (v(ψ1 ), v(ψ2 ))⊆T is applicable 4. Suppose ¬ϕ ∈ Γ. We show that none of the cases in which ∧ under this assumption: i+1 – Suppose (l) ∈ AX, v(ψ1 ) = >ij and v(ψ2 ) ∈ {ti+1 j , >j }. By the definition of v, we must have ψ2 = ¬ψ1 . Then by (S4) either ψ1 ∈ ∆ or ψ2 ∈ ∆. By the induction hypothesis, either v(ψ1 ) = f or v(ψ2 ) = f , which is impossible. i+1 – Suppose (d) ∈ AX, v(ψ2 ) = >ij and v(ψ1 ) ∈ {ti+1 j , >j }. Similarly to the previous case, this is impossible. – Suppose (nl∧ ) ∈ AX or (a∧ ) ∈ AX, and v(ψ1 ), v(ψ2 ) ∈ T . Then by (S10) either ¬ψ1 ∈ Γ or ¬ψ2 ∈ Γ. By the induction hypothesis, either v(ψ1 )6∈T or v(ψ2 )6∈T , which is impossible. – Suppose (o1∧ ) ∈ AX, v(ψ1 ) ∈ T and v(ψ2 ) ∈ D. By (S9), either ¬ψ1 ∈ Γ, or ψ2 ∈ ∆. By the induction hypothesis, either v(ψ1 )6∈T , or v(ψ2 )6∈D, which is impossible. – Suppose (o2∧ ) ∈ AX, v(ψ1 ) ∈ D and v(ψ2 ) ∈ T . Similarly to the previous case, this is impossible too. ˜ (v(ψ1 ), v(ψ2 ))6⊆T , and since by (∗) ¬ϕ6∈∆, the definition of v implies Hence It follows that ∧ k(ϕ)

that either v(ϕ) = f or v(ϕ) = >n(ϕ) . We leave the cases of ∨ and ⊃ to the reader. It follows that for every ψ ∈ Γ, v(ψ) ∈ D and for every ϕ ∈ ∆, v(ϕ) = f . Since Γ0 ⊆ Γ and ∆0 ⊆ ∆, Γ0 ⇒ ∆0 is not valid in M0 [AX]. Corollary 67 BK[AX] is decidable for every coherent AX ⊆ A. Proof: This follows easily from the facts that GBK [AX] is equivalent to BK[AX], and it enjoys cutadmissibility. Remark 68 The above corollary was first proved in [5] based on direct use of the characteristic Nmatrix M0 [AX] (which can be used for a semantic decision procedure despite its infinite nature).

23

Due to the results concerning the notion of expansion established earlier in this paper, from the above Theorem for the infinite semantics we can derive the corresponding result for finite semantics — namely, Theorem 48. Corollary 69 For coherent AX ⊆ A0 , M3 [AX] is a characteristic Nmatrix for GBK [AX], and GBK [AX] enjoys cut-admissibility. The proof follows directly from Theorem 66 and Propositions 30 and 55.

5

◦-free C-systems

All the C-systems treated above include ◦ as a primitive connective. However, many important C-systems, including da Costa’s historical C1 , are obtained by defining ◦ϕ using other connectives available in the language. The definitions usually employed for this purpose are: ¬(ϕ ∧ ¬ϕ) (like in C1 ), ¬(¬ϕ ∧ ϕ), and their disjunction. As shown by Proposition 73 below, these choices correspond to the ◦-free fragments of the C-systems considered in this paper that include (l) or (d). Therefore, it is important to extend our method to these fragments. This is a straightforward task, because the ◦-free fragments of all the paraconsistent extensions of BK by axioms from A are easily obtained from their corresponding Gentzentype systems. Proposition 70 Let AX ⊆ A. If AX is coherent, then the system obtained from GBK [AX] by discarding the rules for ◦ (i.e. (◦ ⇒), (⇒ ◦), and R(i) if (i) ∈ AX) is equivalent to the ◦-free fragment of BK[AX]. Proof: Denote by G0 the system obtained from GBK [AX] by discarding the rules for ◦. Let T ∪ {ψ} be a set of formulas over Lcl . Clearly, T `G0 ψ implies T `BK[AX] ψ. For the converse, suppose that T `BK[AX] ψ for some T ∪ {ψ} in Lcl . Then `GBK [AX] Γ ⇒ ψ for some finite Γ ⊆ T . By Theorem 66, Γ ⇒ ψ has a cut-free derivation P in GBK[AX] . Since Γ ⇒ ψ does not contain ◦, P does not contain applications of any rule which introduces ◦ (since that ◦ can only be eliminated using cuts). Hence P is also a derivation in G0 , and so T `G0 ψ. Corollary 71 Let AX ⊆ A \ {(i)} and AX 0 = AX ∪ {(i)}. If AX is coherent, then the ◦-free fragments of BK[AX] and BK[AX 0 ] are identical, that is: for any T ∪ {ψ} in Lcl , T `BK[AX] ψ iff T `BK[AX 0 ] ψ. In consequence, the inclusion of the axiom (i) in any of the systems studied in this paper does not affect the ◦-free fragment of that system. This is a somewhat surprising result, and we are not aware of any straightforward proof of it. Remark 72 The above results provide a straightforward way to obtain Hilbert-style axiomatizations for the ◦-free fragments of BK[AX] for all coherent AX ⊆ A. For this purpose, one can employ some standard method of translating Gentzen-type rules into corresponding axioms. For instance, we can obtain the following axioms, corresponding to the two Gentzen-type rules for (o1∨ ) in Table 3: ¬(ϕ ∨ ψ) ⊃ ¬ϕ and ¬(ϕ ∨ ψ) ⊃ (ψ ⊃ ϕ). Note that in this way we obtain a ◦-free equivalent for each axiom in Table 3. An alternative method to obtain Hilbert-style axiomatizations for the ◦-free fragments of BK[AX] for a set AX that includes either (l) or (d) is given in the next theorem: Theorem 73 Let AX ⊆ A be coherent. Given a formula σ(p), denote by BKσ [AX] the Hilbert-style system obtained from BK[AX] by deleting (k),(i), (l), and (d), and replacing ◦ϕ by σ(ϕ) in (b) as well as in all the (a)- and (o)-axioms of AX. Then BKσ [AX] is equivalent to the ◦-free fragment of BK[AX] whenever: 24

• AX contains (l), but not (d), and σ(p) = ¬(p ∧ ¬p). • AX contains (d), but not (l), and σ(p) = ¬(¬p ∧ p). • AX contains (l) and (d), and σ(p) = ¬(p ∧ ¬p) ∨ ¬(¬p ∧ p). Proof: By Corollary 71 we may assume that (i) 6∈ AX. Now it is easy to check that in all the three cases considered in the above theorem, both σ(ϕ) ⊃ ◦ϕ and ◦ϕ ⊃ σ(ϕ) are theorems of BK[AX]. Since BK[AX] is an extension of the classical positive logic, this easily implies that by replacing ◦ϕ by σ(ϕ) in any axiom of BK[AX] we obtain a theorem of BK[AX].12 Consequently, if T `BKσ [AX] ψ then T `BK[AX] ψ. For the converse, define a translation tr : F rmLC → F rmLcl as follows: tr(p) = p if p is atomic, tr(¬ϕ) = ¬tr(ϕ), tr(ϕ]ψ) = tr(ϕ)]tr(ψ) (for ] ∈ {∧, ∨, ⊃}), and tr(◦ϕ) = σ(tr(ϕ)). Using induction on the length of derivations in BK[AX] and our assumption that (i) 6∈ AX, we can easily show that if T `BK[AX] ψ, then {tr(ϕ) | ϕ ∈ T } `BKσ [AX] tr(ψ). Since tr(ϕ) = ϕ if ϕ is ◦-free, this in particular implies that if all formulas of T ∪ {ψ} are in Lcl and T `BK[AX] ψ, then T `BKσ [AX] ψ. Corollary 74 C1 is equivalent to the ◦-free fragments of BKcla, BKcila, Bcla, and Bcila. Proof: The identity of the ◦-free fragments of these four systems follows from Corollary 71, while the fact that (l) ⇒ (k) (i.e. ¬(ϕ ∧ ¬ϕ) ⊃ ◦ϕ ⇒ ◦ϕ ∨ (ϕ ∧ ¬ϕ)) was shown in [8] to be derivable in GB . Their equivalence to C1 follows from Theorem 73, because by applying the procedure described in that Theorem to BKcla (or BKcila) we obtain C1 . Corollary 75 The system obtained by discarding (◦ ⇒) and (⇒ ◦) from GBK [{(c), (l), (a∨ ), (a∧ ), (a⊃ )}] is equivalent to C1 .

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Summary and Further Research

The present paper subsumes and extends the results of our work on providing a uniform method for systematic construction of analytic calculi for C-systems, based on non-deterministic semantics defined for those systems. We believe that these results will help produce efficient tools for automated reasoning with inconsistency, eventually making LFIs a more appealing formalism for reasoning under uncertainty. Nevertheless, it is clear that for the purposes of building LFI-based theorem provers for real-life applications, the results of this paper need to be extended to the first-order case. To the best of our knowledge, currently no known analytic systems for LFIs are available on the first-order level.However, [10] provided non-deterministic modular semantics for first-order LFIs, which might hopefully be exploited along the lines of the approach presented in this paper.

Acknowledgements The first author is supported by The Israel Science Foundation under grant agreement no. 280-10. 12 Unless (c) ∈ AX, and in the third case also (nl ) ∈ AX, this is not true if (i) ∈ AX. Hence the possibility to assume ∨ that (i) 6∈ AX is crucial here!

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