Sequent-Type Calculi for Three-Valued and Disjunctive Default Logic

Default logic is one of the basic formalisms for nonmonotonic reasoning, a well-established area from logic-based artificial intelligence dealing with the representation of rational conclusions, which are characterised by the feature that the inference process may require to retract prior conclusions given additional premisses. This nonmonotonic aspect is in contrast to valid inference relations, which are monotonic. Although nonmonotonic reasoning has been extensively studied in the literature, only few works exist dealing with a proper proof theory for specific logics. In this paper, we introduce sequent-type calculi for two variants of default logic, viz., on the one hand, for three-valued default logic due to Radzikowska, and on the other hand, for disjunctive default logic, due to Gelfond, Lifschitz, Przymusinska, and Truszczyński. The first variant of default logic employs Łukasiewicz’s three-valued logic as the underlying base logic and the second variant generalises defaults by allowing a selection of consequents in defaults. Both versions have been introduced to address certain representational shortcomings of standard default logic. The calculi we introduce axiomatise brave reasoning for these versions of default logic, which is the task of determining whether a given formula is contained in some extension of a given default theory. Our approach follows the sequent method first introduced in the context of nonmonotonic reasoning by Bonatti, which employs a rejection calculus for axiomatising invalid formulas, taking care of expressing the consistency condition of defaults.


Introduction
Most formal logics studied in the literature are monotonic in the sense that an increased set of premisses never yields a reduced set of conclusions. An important class of logics, closely related to the formalisation of human common-sense reasoning and important in the area of logic-based artificial intelligence (AI), however, do not enjoy this property-they are nonmonotonic. A central nonmonotonic-reasoning formalism is default logic, introduced by Raymond Reiter in 1980 [1]. In default logic, conclusions may be asserted on the basis of having no evidence, making such inferences unjustified. A typical argument schema along these lines is to assume a certain statement given no evidence to the contrary. Such nonmonotonic conclusions are defeasible as they may be invalidated from a systematic construction for many-valued logics as described by Zach [26] and by Tompits and Bogojeski [27].
For the case of disjunctive default logic, the calculus we define employs the well-known sequent-type calculus following Gentzen [23] and an anti-sequent calculus due to Bonatti [7].
Concerning rejection systems in general, its history goes back already to Aristotle who not only analysed correct reasoning in his system of syllogisms but also studied invalid arguments, where in particular he rejected arguments by reducing them to other already rejected ones. The first usage of the term "rejection" in modern logic was done by Jan Łukasiewicz in his 1921 paper Logika dwuwartościowa ("Two-valued logic") in which he states that by doing so he follows Brentano [28]. An axiomatic treatment of rejection was then discussed in Łukasiewicz's treatment of Aristotle's syllogistic [20,29] where he introduced a Hilbert-type rejection system. This was then further elaborated by his student Jerzy Słupecki [30] and eventually extended to a theory of rejected propositions [31][32][33][34][35].
The paper is organised as follows. In the next section, we present the background on the formalisms employed in our work, that is, on the underlying monotonic logics (Section 2.1) and the two variants of default logic (Section 2.2). Afterwards, in Section 3, we introduce our sequent calculus for three-valued default logic, and in Section 4, we discuss our calculus for disjunctive default logic. The paper concludes with Section 5, providing a brief summary and an outlook for future work.

Underlying Monotonic Logics
We start with setting down the basic definitions and notation for classical propositional logic and Łukasiewicz's three-valued logic [11], which are required for our subsequent elaborations.

Classical Propositional Logic
The alphabet of classical propositional logic, PL, consists of (i) a countable set P of propositional constants, (ii) the truth constants " " ("truth") and "⊥" ("falsehood"), (iii) the primitive logical connectives "¬" ("negation") and " ⊃ " ("implication"), and (iv) the punctuation symbols "("and")". The class of formulas is built from elements of the alphabet of PL in the usual inductive fashion, whereby the propositional constants and truth constants constitute the atomic formulas. Formulas which are non-atomic are referred to as composite formulas.
Besides the primitive connectives ¬ and ⊃ , we also make use of the standard connectives " ∨ " ("disjunction"), " ∧ " ("conjunction"), and " ≡ " ("equivalence"), defined in the usual way: In what follows, we will use the letters "P", "Q", "R", . . . (possibly appended with subscripts and/or with primes) or words from everyday English to refer to propositional constants, and use the letters "A", "B", "C", . . . (again possibly appended with subscripts and/or with primes) to refer to arbitrary formulas (distinct such letters need not represent distinct formulas).
A (two-valued) interpretation is a mapping I assigning each propositional constant from P an element from the set {t, f}, whose elements are referred to as truth values, where t represents truth and f represents falsity. The truth value of a composite formula A under an interpretation I, denoted by V I (A), is defined in terms of the usual truth-table conditions of classical propositional logic. Accordingly, a formula A is true under I iff V I (A) = t, and false under I if V I (A) = f. If A is true under I, then I is said to be a model of A, and if A is false under I, then I is a countermodel of A. If I is a countermodel of A, then we also say that I refutes A. We call A satisfiable (in PL) if it has some model, and falsifiable (in PL), or refutable (in PL), if it has some countermodel. Moreover, A is unsatisfiable (in PL) if it has no model. Finally, A is a tautology, symbolically |= 2 A, if it is true in every interpretation, and refutable (in PL), symbolically |= 2 A, otherwise.
A set of formulas is also referred to as a theory. An interpretation I is a model of a theory T if I is a model of all elements of T, otherwise I is a countermodel of T. If a theory T has a model, then T is satisfiable, and if T has a countermodel, then T is falsifiable. A theory is unsatisfiable if it has no model.
A formula A is a valid consequence of a theory T (in PL), or T entails A (in PL), in symbols T |= 2 A, iff A is true in any model of T. Two formulas, A and B, are (logically) equivalent (in PL) iff |= 2 (A ≡ B). In general, two theories are (logically) equivalent iff they have the same models.
As customary, we will write expressions like "T ∪ {A} |= 2 B" as "T, A |= 2 B", and similarly for finite sets of form {A 1 , . . . , A n } instead of a singleton set {A}.
We denote by 2 the usual derivability operator of PL with respect to some fixed sound and complete Hilbert-type system. The deductive closure operator of PL is given by: where T is a theory. A theory T is deductively closed iff T = Th 2 (T). As well known, the operator Th 2 (·) enjoys the following properties (for any theory T and T ):
T ⊆ T implies Th 2 (T) ⊆ Th 2 (T ). ("Monotonicity".) If A is not derivable from T, then we indicate this by writing T 2 A. Later on, we will define proof systems axiomatising formulas that are not derivable from a given theory. Such axiom systems are accordingly also referred to as complementary calculi as they axiomatise the complement of the provable formulas of a logic.
We say that a theory T is consistent iff there is a formula A such that T 2 A. Clearly, T is consistent iff it is satisfiable. Moreover, a formula A is consistent with T iff T 2 ¬A.

Łukasiewicz's Three-Valued Logic
We now turn to the three-valued logic of Łukasiewicz [11] for the propositional case, henceforth denoted by Ł 3 . Our presentation follows the one given by Radzikowska [9].
The alphabet of Ł 3 consists of the alphabet of PL along with the additional truth constant ("undetermined"). Again, we assume P as a countable set of propositional constants. The class of formulas of Ł 3 is built similarly to the formulas of PL, except that is counted as an additional atomic formula.
A difference to the syntax of the logic PL concerns the defined connectives; while conjunction, ∧ , and material equivalence, ≡ , are defined as in propositional logic, disjunction in Ł 3 is defined differently: Furthermore, there are also additional unary defined operators, viz.
Intuitively, LA expresses that A is certain, whilst MA means that A is possible. These operators will be used subsequently to distinguish between certain knowledge and defeasible conclusions. Furthermore, IA expresses that A is contingent or modally indifferent.
A (three-valued) interpretation is a mapping m assigning to each propositional constant from P an element from {t, f, u}. Here, besides the truth values t and f, the symbol u represents a truth value standing for "undetermined" or "indeterminacy". As usual, m(P) is the truth value of P under m, where now P is true under an interpretation m if m(P) = t, false under m if m(P) = f, and has undetermined truth value if m(P) = u.
The truth value, V m (A), of an arbitrary formula A under an interpretation m is given subject to the following conditions: Clearly, the classically valid principle of tertium non datur, i.e., the law of excluded middle, A ∨ ¬A, as well as the corresponding law of non-contradiction, ¬(A ∧ ¬A), are not valid in Ł 3 . However, their three-valued pendants, viz., the principle of quartum non datur, i.e., the law of excluded fourth, A ∨ IA ∨ ¬A, and the corresponding extended non-contradiction principle, ¬(A ∧ ¬IA ∧ ¬A), are valid in Ł 3 .
In classical logic, two formulas are logically equivalent if and only if, they have the same models, where logical equivalence between formulas A and B is defined by the condition that |= 2 (A ≡ B) holds. However, such a relation between logical equivalence and equality of models does not hold in general in the three-valued logic case. Indeed, following Radzikowska [9], let us define that two formulas A and B are strongly equivalent, symbolically A ⇔ s B, iff |= 3 (A ≡ B). That is, A and B are strongly equivalent iff, for any three-valued interpretation m, V m (A) = V m (B). Furthermore, let us call A and B equivalent (in Ł 3 ), symbolically A ⇔ B, iff A and B have the same models. Clearly, strong equivalence implies equivalence, but in general not vice versa. For instance, P and LP, for an atom P, are equivalent but not strongly equivalent. In addition, strong equivalence is an equivalence relation (i.e., reflexive, symmetric, and transitive) and enjoys a substitution principle, similar to the one of classical logic, i.e., if a formula C A contains a subformula A, and C B is the result of substituting at least one occurrence of A in C A by a formula B, then A ⇔ s B implies C A ⇔ s C B .
Let us also note some strong equivalences which hold in Ł 3 : ¬LA ⇔ s M¬A. 6.
¬MA ⇔ s L¬A. 7. ( The notion of a theory in Ł 3 is defined as in PL, i.e., a theory is a set of formulas. Likewise, the notion of a model or of a countermodel of a theory, and of a theory being satisfiable, falsifiable (or refutable), or unsatisfiable are defined in Ł 3 mutatis mutandis as in PL. A theory T is said to entail a formula A (in Ł 3 Sound and complete Hilbert-style axiomatisations of the logic Ł 3 can be readily found in the literature [47,48]; the first one was introduced by Wajsberg in 1931 [49]. We write T 3 A if A has a derivation (in some fixed Hilbert-style calculus) from T in Ł 3 . As well, the deductive closure operator of Ł 3 is given by where T is a theory. The notions of a theory being deductively closed and of being consistent, as well as of a formula being consistent with a theory, are defined similarly as in PL. Moreover, the properties of inflationaryness, idempotency, and monotonicity hold for Th 3 (·) like for Th 2 (·), and consistency of a theory T in Ł 3 is equivalent to the satisfiability of T in Ł 3 . While in PL we have the well-known properties that (i) T 2 A iff T ∪ {¬A} is inconsistent and (ii) T, A 2 B iff T 2 (A ⊃ B) (the "only if" part of the latter is generally referred to as the deduction theorem), for a theory T and formulas A and B, in Ł 3 sight variations thereof hold: Proposition 1. Let T be a theory, and A and B formulas.
Note that, as a consequence, the consistency of a formula A with a theory T implies the consistency of the theory T ∪ {MA}, but it does not necessarily imply the consistency of T ∪ {A}. For instance, ¬P is consistent with {MP}, for an atomic formula P, so {MP, M¬P} is consistent, but {MP, ¬P} is not.
Furthermore, although in Ł 3 it always holds that T 3 A ⊃ B implies T, A 3 B, it is the converse direction (i.e., the classical version of the deduction theorem) that fails in general.

Two Variants of Default Logic
We continue with the basic elements of three-valued default logic, due to Radzikowska [9], and of disjunctive default logic, introduced by Gelfond, Lifschitz, Przymusinska, and Truszczyński [10]. Note that we deal here with propositional versions of the formalisms as our subsequent calculi are defined for the propositional case only, similar to the undertaking of Bonatti [7,37] and of Bonatti and Olivetti [8].

Three-Valued Default Logic
Radzikowska's three-valued default logic [9], which in what follows we will denote by DL 3 , differs from Reiter's standard default logic [1] (henceforth referred to as DL) in two aspects; not only is in DL 3 the deductive machinery of classical logic replaced with Ł 3 , but there is also a modified consistency check for default rules employed in which the consequent of a default is taken into account as well. The latter feature is somewhat reminiscent to the consistency checks used in justified default logic [12] and in constrained default logic [13,14], where a default may only be applied if it does not lead to a contradiction a posteriori.
Formally, a default rule, or simply a default, d, is an expression of the form where A is the prerequisite, B 1 , . . . , B n are the justifications, and C is the consequent of d. The intuitive meaning of such a default is: if A is believed, and B 1 , . . . , B n and LC are consistent with what is believed, then MC is asserted.
Note that under this reading, by applying a default of the above form, it is assumed that C cannot be false, but it is not assumed that C is true in all situations. It is only assumed that C must be true in at least one such situation. This reflects the intuition that accepting a default conclusion, we are prepared to rule out all situations where it is false, but we can imagine at least one such situation in which it is true. As a consequence, we cannot conclude both MC and M¬C simultaneously.
In what follows, formulas of the form MC obtained by applying defaults will be referred to as default assumptions. For simplicity, defaults will also be written in the form (A : B 1 , . . . , B n /C).
A default theory, T, is a pair W, D , where W is a set of formulas (i.e., a theory in Ł 3 ), called the premisses of T, and D is a set of defaults. An extension of a default theory T = W, D in the three-valued default logic DL 3 is defined thus: For a set S of formulas, let Γ T (S) be the smallest set K of formulas obeying the following conditions: If (A : B 1 , . . . , B n /C) ∈ D, A ∈ K, ¬B 1 ∈ S, . . . , ¬B n ∈ S, and ¬LC / ∈ S, then MC ∈ K.
Then, E is an extension of T iff Γ T (E) = E. Note that the criterion of the applicability of a default in DL 3 makes the two defaults: Note further that, for obtaining extensions in the sense of Reiter [1], in the above definition, instead of Th 3 (·) we use Th 2 (·), and the condition 3 is replaced by: If (A : B 1 , . . . , B n /C) ∈ D, A ∈ K, and ¬B 1 ∈ S, . . . , ¬B n ∈ S, then C ∈ K.
There are two basic reasoning tasks in the context of default logic, viz., brave reasoning and skeptical reasoning. The former task is the problem of checking whether a formula A belongs to at least one extension of a given default theory T, whilst the latter task examines whether A belongs to all extensions of T. Our aim is to give a sequent-type axiomatisation of brave default reasoning, following the approach of Bonatti [7] for standard default logic.
To conclude our review of three-valued default logic, we give two examples, as discussed by Radzikowska [9], showing the representational advantages of DL 3 . The only default of this theory is inapplicable since W 3 ¬LSun_Shining holds. Hence, T has a single extension, viz. E = Th 3 (W). Note that T has no extension in Reiter's default logic due to the weaker consistency check which results in a vicious circle where the application of the default violates its justification for applying it.

Example 2 ([51]
). Consider the default rules where P, Q, and R stand for the following propositions: • P: "Tony recites passages from Shakespeare"; • Q: "Tony can read and write"; • R: "Tony is over seven years old".
Obviously, common sense suggests that, given P, there are perfect reasons to apply both defaults to infer that Tony is over seven years old. Suppose now that we add the default rule where S stands for "Tony is a child prodigy". Given S, it is reasonable to infer that Tony can read and write, but the inference of R that Tony is over seven years old seems to be unjustified. In standard default logic DL, a common way of suppressing R in the latter scenario would be to employ a default rule with exceptions of the form However, this remedy is somewhat unsatisfactory as it requires that every default may possess a potentially large number of conceivable exceptions which, each time a new default is added, the previous ones must be revised, which is arguably ad hoc. In DL 3 , on the other hand, this can easily be accommodated by using the defaults P : LQ LQ and Q : LR LR instead of d 1 and d 2 , as well as Actually, the last example illustrates the difference between causal rules ("expectation-evoking rules") and evidential rules ("explanation-evoking rules") [51]. An example of the first kind of rules is "fire usually causes smoke" whilst "smoke usually suggests fire" is an instance of the second kind. As argued by Pearl [51], an evidential rule should not be applied if its prerequisite is derived by applying at least one causal rule. In

Disjunctive Default Logic
We now turn to the basics of disjunctive default logic [10], henceforth referred to as DL D .
The main motivation for introducing disjunctive default logics was to address a difficulty encountered when using defaults in the presence of disjunctive information, a problem which was first observed by David Poole [52]. More specifically, the difficulty lies in the difference between a default theory having two extensions, one containing a formula A and the other a formula B, and a theory with a single extension, containing the disjunction A ∨ B. This problem was also noted by Lin and Shoham [53], who gave an example of a theory in a modal-logic language, containing disjunctive information, and observed that no default theory exists which corresponds to this theory.
Another nice feature of disjunctive default logic is that it provides a one-to-one correspondence between answer-sets of disjunctive logic programs [4] and extensions of a corresponding disjunctive default theory. Such a correspondence does likewise not directly hold for standard default logic-and again the key problem lies in the presence of disjunctive information. More specifically, viewing P ∨ Q as a rule in a logic program under the answer-set semantics, the default naturally corresponding to this rule would be the default rule Now, while the program consisting of the single rule P ∨ Q has two answer sets, viz. {P} and {Q}, the default theory ∅, {d} has only one extension, Th 2 ({P ∨ Q}). As long as only programs without disjunctions are considered, such a natural translation of program rules into defaults gives rise to a one-to-one correspondence between answer sets of the given program and the extensions of its translation.
To formally introduce DL D , by a disjunctive default rule, or simply a disjunctive default, d, we understand an expression of the form where A, B 1 , . . . ,B n , and C 1 , . . . ,C m are formulas from PL. Similar to DL 3 , we call A the prerequisite, B 1 , . . . , B n the justifications, and C 1 , . . . , C n the consequents of d. Furthermore, following Baumgartner and Gottlob [54], we refer to the symbol "|" as effective disjunction.
The intuitive meaning of such a default is: if A is believed and B 1 , . . . , B n are consistent with what is believed, then one of C 1 , . . . , C m is asserted.
Similar to conventions in standard default logic, if the prerequisite of a default d is , then we will omit it from d. If, additionally, d has no justifications, then d is simply written as where C 1 , . . . , C m are the consequents of d. For convenience, disjunctive defaults will also be written in the form (A : B 1 , . . . , B n / C 1 | · · · |C m ).
A disjunctive default theory, T, is a pair W, D , where W is a set of formulas of PL (again referred to as the premisses of T) and D is a set of disjunctive defaults.
For defining extensions of disjunctive default theories, we need some further notation: Let us call a set S of formulas closed under propositional consequence if, whenever S 2 A, then A ∈ S. Clearly, the deductive closure of a set S, Th 2 (S), is the smallest set of formulas closed under propositional consequence containing S. Moreover, for a family F of sets, let min(F) denote the minimal elements of F, where minimality is defined with respect to set inclusion, i.e., Consider now a disjunctive default theory T = W, D . Given a set S of formulas of PL, let Cl T (S) be the collection of all sets K satisfying the following conditions: Moreover, let ∆ T (S) = min(Cl T (S)), i.e., ∆ T (S) consists of all minimal sets obeying conditions 1-3.
The notion of a brave and a skeptical consequence given a disjunctive default theory is defined as before mutatis mutandis.
Let us now discuss some examples showing the differences between disjunctive default logic and standard default logic, following Gelfond, Lifschitz, Przymusinska, and Truszczyński [10].
where P, Q, R, and S are atomic formulas. Intuitively, given the disjunctive information P ∨ Q, we would expect to derive R ∨ S, because, in case P holds, we could apply the first default, and in case Q holds, we could accordingly apply the second default. However, in DL, neither of the two defaults is applicable and the single extension of T is Th 2 (W). Now, in disjunctive default logic, we can represent the information expressed by T in terms of a disjunctive default theory T containing the three defaults P|Q, P : R R , and Q : S S .
In contrast to the situation in DL, T possesses two extensions in DL D , viz. Th 2 ({P, R}) and Th 2 ({Q, S}), and R ∨ S is contained in both, which is in accordance to our expectations. We next discuss the example by Poole [52].

Example 4.
Let us assume the following commonsense information: By default, a person's left arm is usable, the exception being when it is broken, and similarly for the right arm.
In standard default logic, we can express this by the following two defaults: where "U l " and "U r " stand for that the left arm is usable and that the right arm is usable, respectively, and, similarly, "B l " and "B r " mean that the left arm or the right arm is broken.
If there is no further information about one's hands, then one can conclude that both hands are usable. Indeed, the default theory T = ∅, {d 1 , d 2 } has a single extension in DL, containing both U l and U r .
However, if it is now known that the left arm is broken, i.e., B l is asserted, then the application of d 1 is blocked and the extended default theory has again one extension, containing U r .
But let us assume now that we only know that one arm is broken, but we do not remember exactly which one. So, what we can assert now is the formula Considering now the extensions of the default theory this default theory has still one extension, but unfortunately it contains both U l ∨ U r , which is contrary to our intuition.
Using DL D , on the other hand, we can represent the information of T by a disjunctive default theory containing B l |B r together with the two defaults d 1 and d 2 . The resulting theory has two extensions, viz.
which corresponds with our intuition. Note that the difference between a formula A ∨ B and a disjunctive default A|B amounts to the difference between the assertions "A or B is known" and "A is known or B is known".

A Sequent Calculus for Three-Valued Default Logic
We now introduce our sequent calculus B 3 for brave reasoning in DL 3 . Following the general design of the approach of Bonatti [7,55], B 3 involves three kinds of sequents, viz. assertional sequents for axiomatising validity in Ł 3 , anti-sequents for axiomatising non-tautologies of Ł 3 , and special default sequents representing brave reasoning in DL.
We start with laying down the postulates of B 3 and then, in Section 3.2, we show soundness and completeness.

Postulates of the Calculus
As far as sequent-type calculi for three-valued logics are concerned,-or, more generally, many-valued logics-different techniques have been discussed in the literature [21,24,26,[56][57][58]. Here, we use an approach due to Rousseau [25], which is a natural generalisation for many-valued logics of the classical two-sided sequent formulation as pioneered by Gentzen [23]. In Rousseau's approach, a sequent for a three-valued logic is a triple of sets of formulas where each component of the sequent represents one of the three truth values.

A Sequent Calculus for Ł 3
Formally, we introduce sequents for Ł 3 as follows: , is a finite set of formulas, called a component of the sequent.
For a (three-valued) interpretation m, a sequent Γ 1 | Γ 2 | Γ 3 is true under m if, for at least one i ∈ {1, 2, 3}, Note that a standard classical sequent Γ ∆ in the sense of Gentzen [23] corresponds to a pair Γ | ∆ under the usual two-valued semantics of PL.
As customary for sequents, we write sequent components comprised of a singleton set {A} simply as "A", and likewise we write Γ ∪ {A} as "Γ, A".
For obtaining the postulates of a many-valued logic in Rousseau's approach, the conditions of the logical connectives of a given logic are encoded in two-valued logic by means of a so-called partial normal form [47] and expressed by suitable inference rules.
The calculus we employ for Ł 3 , which we denote by SŁ 3 , is taken from Zach [26], which is obtained from a systematic construction of sequent-style calculi for many-valued logics and by applying some optimisations of the corresponding partial normal forms. Definition 2. The postulates of SŁ 3 are as follows: • axioms of SŁ 3 are sequents of the form the inference rules of SŁ 3 are comprised of the rules depicted in Figure 2.
Note that from the inference rules of SŁ 3 , we can easily obtain derived rules for the defined connectives of Ł 3 . Furthermore, the last three rules in Figure 2 are also referred to as weakening rules. Soundness and completeness of SŁ 3 follows directly from the method AS described by Zach [26]: Note that sequents in the style of Rousseau are truth functional rather than formalising entailment directly, but, by a general result for many-valued logics as shown by Zach [26], the latter can be expressed simply as follows: Proposition 3. For a theory T and a formula A, T 3 A iff the sequent T | T | A is provable in SŁ 3 .

An Anti-Sequent Calculus for Ł 3
As for axiomatising non-theorems of Ł 3 , a systematic construction of rejection calculi for many-valued logics has been developed by Bogojeski and Tompits [27], based on adapting the approach of Zach [26]. The refutation calculus we describe now for axiomatising invalid sequents in Ł 3 , denoted by RŁ 3 , is obtained from the method of Bogojeski and Tompits [27].

Definition 3.
A (three-valued) anti-sequent is a triple of form Γ 1 Γ 2 Γ 3 , where each Γ i , for i ∈ {1, 2, 3}, is a finite set of formulas, called a component of the anti-sequent.
Note that, in contrast to SŁ 3 , the inference rules of RŁ 3 have only single premisses. Indeed, this is a general pattern in sequent-style rejection calculi: If an inference rule for standard (assertional) sequents for a connective has n premisses, then there are usually n corresponding unary inference rules in the associated rejection calculus. Intuitively, what is an exhaustive search in a standard sequent calculus becomes nondeterminism in a rejection calculus. Again, soundness and completeness of RŁ 3 follow from the systematic construction as described by Bogojeski and Tompits [27]. Likewise, non-entailment in Ł 3 is expressed similarly as for SŁ 3 .

The Default-Sequent Calculus B 3
We are now in a position to specify our calculus B 3 for brave reasoning in DL 3 .

Definition 5.
A (brave) default sequent is an ordered quadruple of the form Γ; ∆ ⇒ Σ; Θ, where Γ, Σ, and Θ are finite sets of formulas and ∆ is a finite set of defaults.
A default sequent Γ; ∆ ⇒ Σ; Θ is true if there is an extension E of the default theory Γ, ∆ such that Σ ⊆ E and Θ ∩ E = ∅; E is called a witness of Γ; ∆ ⇒ Σ; Θ.
The default sequent calculus B 3 consists of three-valued sequents, anti-sequents, and default sequents. It incorporates the systems SŁ 3 for three-valued sequents and RŁ 3 for anti-sequents, as well as additional axioms and inference rules for default sequents, described as follows: Definition 6. The postulates of B 3 comprise the following items: • all axioms and inference rules of SŁ 3 and RŁ 3 ; • axioms of the form Γ; ∅ ⇒ ∅; ∅, where Γ is a finite set of formulas of Ł 3 ; and • the inference rules depicted in Figure 4.  The informal meaning of the inference rules for the default sequents is the following: (i) rules (l 1 ) and (l 2 ) combine three-valued sequents and anti-sequents with default sequents, respectively; (ii) rule (mu) is the rule of "monotonic union"-it allows the joining of information in case that no default is present; and (iii) rules (d 1 )-(d 4 ) are the default introduction rules, where rules (d 1 ), (d 2 ), and (d 3 ) take care of introducing non-active defaults, whilst rule (d 4 ) allows to introduce an active default.
Let us give an example to illustrate the functioning of the calculus. As we saw, the single default of this theory is inapplicable since W 3 ¬LSun -Shining and E = Th 3 (W) is therefore the only extension of T. Consequently, Sun -Shining / ∈ E also holds. Hence, the default sequent Summer, ¬Sun -Shining; (Summer : ¬Rain/Sun -Shining) ⇒ ¬LSun -Shining; Sun -Shining is true. We will give a proof of (1) in B 3 . The proof of (1), depicted below and denoted by β, uses the proof α as subproof. For brevity, we will use "S" for "Summer", "R" for "Rain", and "H" for Sun -Shining".

Adequacy of the Calculus
We now show soundness and completeness of B 3 . To this end, we need some auxiliary results first, dealing with alternative characterisations and properties of extensions.

Preparatory Characterisations: Residues and Extensions
We start with some properties of extensions concerning adding defaults to default theories which provide the groundwork on which our adequacy proofs are built. In doing so, we first introduce an alternative formulation of DL 3 extensions, adapting a proof-theoretical characterisation as described by Marek and Truszczyński [59] for standard default logic, and afterwards we provide results concerning so-called residues, which are inference rules resulting from defaults satisfying their consistency conditions. The latter endeavour generalises the approach of Bonatti [7] to the three-valued case. Whenever it is clear from the context, we will allow ourselves to drop the prefix "DL 3 -" in "DL 3 -active", "DL 3 -reduct", and "DL 3 -residue" to ease notation.
For a set R of inference rules, let R 3 be the inference relation obtained from 3 by augmenting the postulates of the Hilbert-type calculus for Ł 3 underlying the relation 3 with the inference rules from R. Let the corresponding deductive closure operator for R 3 be given by Clearly, Th ∅ 3 (W) = Th 3 (W). We then obtain the following characterisation of the operator Γ T , mirroring the analogous property for standard default logic as discussed by Marek and Truszczyński [59]: Theorem 1. Let T = W, D be a three-valued default theory, E a set of formulas of Ł 3 , and D E the DL 3 -reduct of D with respect to E. Then, Proof. The result follows by a straightforward adaption of the proof of the analogous result for the case of standard default logic as given by Marek and Truszczyński [59].
By the definition of an extension, we thus obtain: Corollary 1. Let T = W, D be a three-valued default theory and E a set of formulas. Then, Next, we give some properties of extensions with respect to active and non-active defaults which underlay the construction of the default inference rules of B 3 . We start with two lemmata whose proofs are obvious. Lemma 1. Let R and R be sets of inference rules, and let W and W be sets of formulas. Then, the following properties hold: (W). We thus get, in view of part 2 of Lemma 2, and therefore, By observing that the assumption ¬j(d)

Soundness and Completeness of B 3
We are now in a position to prove soundness and completeness of B 3 .
Proof. We show that all axioms are true, and that the conclusions of all inference rules are true whenever its premisses are true (resp., valid or refutable in case of rules (l 1 ) and (l 2 )).
INDUCTION STEP. Assume |∆| > 0, and let the statement hold for all default sequents Γ ; ∆ ⇒ Σ ; Θ such that |∆ | < |∆|. We distinguish two cases: (i) There is some default in ∆ which is active in E, or (ii) none of the defaults in ∆ is active in E.
Since |∆ 0 | < |∆|, by induction hypothesis there is some proof α in B 3 of S . Furthermore, Γ 3 A, so there is some proof β of the sequent Γ | Γ | A in SŁ 3 . The following figure is a proof of S in B 3 (note that in this figure, the endsequents of α and β have been displayed explicitly for better clarity): Now assume that (ii) holds, i.e., no default in ∆ is active in E. Since |∆| > 0, there is some default d = (A : B 1 , . . . , B n /C) in ∆ such that ∆ = ∆ 0 ∪ {d} with ∆ 0 := ∆ \ {d}. Since d is not active in E, according to Theorem 2, E is an extension of Γ, ∆ 0 . Furthermore, either: Consequently, E is either a witness of: Since |∆ 0 | < |∆|, by induction hypothesis there is thus either: Therefore, one of the three figures below constitutes a proof of S (again, the respective endsequents of α, β, and γ are explicitly shown):

A Sequent Calculus for Disjunctive Default Logic
We now introduce our sequent calculus for brave reasoning for disjunctive default logic which we denote by B D . Again, the calculus comprises of three kinds of sequents: (i) sequents for expressing validity in PL; (ii) anti-sequents for expressing non-tautologies; and (iii) special default inference rules reflecting brave reasoning in DL D .
As sequents for propositional logic, we use standard two-sided sequents in the sense of Gentzen [23] and a corresponding calculus, LK, which is a slight simplification of the one originally introduced by Gentzen. As a calculus for anti-sequents, we use the one due to Bonatti [37] which he introduced in connection to his calculus for brave reasoning for standard default logic [7,55]; we will denote this calculus by LK r (note that, independently from Bonatti [37], Goranko [38] developed a similar calculus as part of his refutation systems for different modal logics).

Postulates of the Calculus
We start with defining the sequent calculus LK for classical sequents. Following customs, we write sequents of the form Γ ∪ {A} → ∆ simply as "Γ, A → ∆", and if the antecedent or succedent of a sequent is the empty set, then it is omitted from the sequent. the inference rules of LK are those given in Figure 5.
Note that the last two rules in Figure 5 are the weakening rules of LK. Moreover, from the rules of LK, we can easily obtain derived rules for the defined connectives ∧ , ∨ , and ≡ . For instance, the derived rules for ∧ are as follows: Soundness and completeness of LK is well known: Figure 5. Rules of the sequent calculus LK.
In particular, the following relation follows immediately: For every formula A,

The Anti-Sequent Calculus LK r
Now we introduce our complementary calculus LK r for axiomatising invalidity in propositional logic, following Bonatti [37] (and Goranko [38]). Definition 11. An anti-sequent is an ordered pair of the form Γ → Θ, where Γ and Θ are finite sequences of formulas.
For a two-valued interpretation I, an anti-sequent Γ → Θ is refuted by I, or I refutes Γ → Θ, if every formula in Γ is true under I and every formula in Θ is false under I. An anti-sequent Γ → Θ is refutable if there is at least one interpretation that refutes Γ → Θ.
Hence, the anti-sequent Γ → Θ is refutable iff the classical sequent Γ → Θ is invalid. Also, in accordance to the convention for classical sequents, we write " → Θ" and "Γ → " whenever Γ or Θ is the empty set. Definition 12. The postulates of LK r are as follows: • the axioms of LK r are anti-sequents of the form Φ → Ψ, where Φ and Ψ are disjoint finite sets of atomic formulas such that ⊥ / ∈ Φ and / ∈ Ψ; and • the inference rules of LK r are those depicted in Figure 6. Note that, following the general pattern of complementary calculi, the inference rules of LK r have only single premisses.
We again can obtain corresponding derived rules for the defined connectives. Below we give the ones for ∧ : Soundness and completeness for LK r was shown by Bonatti [37] (and, independently, by Goranko [38]): For formulas, we have then the following immediate corollary:

The Default-Sequent Calculus B D
We can now specify our calculus B D for brave reasoning in disjunctive default logic.
Definition 13. By a (brave) disjunctive default sequent we understand an ordered quadruple of the form Γ; ∆ ⇒ Σ; Θ, where Γ, Σ, and Θ are finite sets of formulas and ∆ is a finite set of disjunctive defaults.
A disjunctive default sequent Γ; ∆ ⇒ Σ; Θ is true iff there is an extension E of the disjunctive default theory Γ, ∆ such that Σ ⊆ E and Θ ∩ E = ∅; E is called a witness of Γ; ∆ ⇒ Σ; Θ.
The default sequent calculus B D consists of sequents, anti-sequents, and disjunctive default sequents. It incorporates the systems LK for sequents and LK r for anti-sequents, as well as additional axioms and inference rules for disjunctive default sequents, similar to the case of B 3 .

Definition 14.
The postulates of B D comprise the following items: • all axioms and inference rules of LK and LK r ; • axioms of the form Γ; ∅ ⇒ ∅; ∅, where Γ is a finite set of formulas of PL; and • the inference rules are those depicted in Figure 7.
. . , ¬B n Γ; ∆, (A : B 1 , . . . , B n / C 1 | · · · |C i | · · · |C m ) ⇒ Σ; Θ (d 3 ) d The informal meaning of the nonmonotonic inference rules is similar to the meaning of the rules in B 3 : (i) rules (l 1 ) d and (l 2 ) d combine classical sequents and anti-sequents with disjunctive default sequents, respectively; (ii) rule (mu) d again allows the joining of information in case that no default is present; and (iii) rules (d 1  This disjunctive default theory has the two extensions: Th 2 ({B l , U r }) and Th 2 ({B r , U l }).
Accordingly, the following disjunctive default sequent is true: A proof, γ, of this sequent in B D is given below; it uses the two subproofs α and β: • Proof α:

Adequacy of the Calculus
Soundness and completeness of B D can be shown by similar arguments as in the case of B 3 . We sketch the relevant details.
We again need some preparatory characterisations of extensions, dealing with the introduction of active or non-active defaults.
We start with the notion of a reduct, adapted to the case of DL D , as introduced by Gelfond, Lifschitz, Przymusinska, and Truszczyński [10].
In what follows, we use the following terminology: By a disjunctive inference rule, or simply a disjunctive rule, r, we understand an expression of the form We say that a set S of formulas is closed under r if, whenever A ∈ S, then C i ∈ S, for some i ∈ {1, . . . , m}. Moreover, for a set R of disjunctive rules, we say that S is closed under R if S is closed under each r ∈ R.
Definition 15. Let D be a set of disjunctive defaults and E a set of formulas. The DL D -reduct of D with respect to E, denoted by D d E , is the set consisting of the following disjunctive inference rules: A disjunctive rule A C 1 | · · · |C m is called DL D -residue of a default A : B 1 , . . . , B n C 1 | · · · |C m .
We again allow ourselves to drop the prefix "DL D -" from "DL D -reduct" and "DL D -residue" if no ambiguity can arise.
Towards our characterisation of extensions of disjunctive default theories, we introduce the following notation: Definition 16. For a set W of formulas and a set R of disjunctive rules, let C R (W) be the collection of all sets which are closed under propositional consequence, and (iii) are closed under R.
Note that, for a disjunctive default theory T = W, D and a set E of formulas, we obviously have that: Cl T (E) = C D d E (W) and ∆ T (E) = Cn D d E (W).
From this, the following result is immediate: We again employ our notation p(d) as in case of DL 3 , but now we define and j(d) for a default d, . . , B n /C 1 | · · · |C m ). We obtain the following results corresponding to Lemma 2 and Theorems 2-4, respectively: Lemma 3. Let W and E be sets of formulas, R a set of disjunctive inference rules, and r = A B 1 | · · · |B n a disjunctive inference rule. Then:

1.
If E is an extension of W, D ∪ {d} and d is active in E, then E is an extension of W ∪ {C}, D , for some C ∈ c (d).

2.
If E is an extension of the disjunctive default theory W ∪ {C}, D , for some C ∈ c (d), W p(d), and ¬j (d) ∩ E = ∅, then E is an extension of W, D ∪ {d} .
From this, by similar arguments as in the case of B 3 , soundness and completeness of B D follows.

Theorem 10.
A disjunctive default sequent Γ; ∆ ⇒ Σ; Θ is provable in B D iff it is true.

Conclusions
In this paper, we introduced sequent-type calculi for brave reasoning for a three-valued version of default logic [9] and for disjunctive default logic [10], following the method of Bonatti [7]. This form of axiomatisation yielded a particular elegant formulation mainly due to their usage of anti-sequents. In addition, the approach was flexible and could be applied to formalise different versions of nonmonotonic reasoning. Indeed, other variants of default logic besides the versions studied here, including justified default logic [12] and constrained default logic [13,14], have also been axiomatised by this sequent method [60,61].
Related to the sequent approach discussed here are also works employing tableau methods. In particular, Niemelä [62] introduces a tableau calculus for inference under circumscription.
Variations of our calculi can be obtained by using different calculi for the underlying monotonic logics. As far as the three-valued case is concerned, we opted for the style of calculi as discussed by Rousseau [25] and Zach [26] because they naturally model the underlying semantic conditions of the considered logic. Alternatively, we could have also used two-sided sequent and anti-sequent calculi like the ones described by Avron [24] and Oetsch and Tompits [67], respectively. By employing such two-sided sequents, however, one then deals with calculi having also "non-standard" inference rules introducing two connectives simultaneously. Another prominent proof method for many-valued logics are hypersequent calculi [57], which are basically disjunctions of two-sided sequents. However, to the best of our knowledge, no rejection calculus based on hypersequents exist so far and establishing such a system in particular for Ł 3 would be worthwhile.
Another topic for future work is to develop calculi for sceptical reasoning for the considered versions of default logic as well as for other variants of default logic discussed in the literature [12][13][14], similar to the system for sceptical reasoning for standard default logic as introduced by Bonatti and Olivetti [8]. In that work, they also introduced a different version of a calculus for brave default reasoning-extending this calculus to DL 3 and DL D would provide an alternative to the calculi discussed here.