On the Application of the Disjunctive Syllogism in Paraconsistent Logics Based on Four States of Information

Home , Disjunctive syllogism

Proceedings of the Twelfth International Conference on the Principles of Knowledge Representation and Reasoning (KR 2010)

Ofer Arieli School of Computer Science, The Academic College of Tel-Aviv, Israel. [email protected]

Abstract conflicting indications, causing a contradictory information about it. We identify three classes of four-state paraconsistent In the sequel, we divide four-state paraconsistent logics logics according to their different approaches towards the disjunctive syllogism, and investigate three repre- by the way they apply the disjunctive syllogism (and so, in- sentatives of these approaches: Quasi-classical logic, directly, by their attitude to the introduction of disjunction). which always accepts this principle, Belnap’s logic, that Each approach is demonstrated by a different logic, rep- rejects the disjunctive syllogism altogether, and a logic resenting a particular attitude for tolerating inconsistency: of inconsistency minimization that restricts its appli- The basic logic (known as Belnap’s four-valued logic (Bel- cation to consistent fragments only. These logics are nap 1977a; 1977b)), in which the propositional connectives defined in a syntactic and a semantic style, which are have their standard lattice-theoretic interpretations, invali- linked by a simple transformation. It is shown that the dates the disjunctive syllogism altogether. On other ex- three formalisms accommodate knowledge minimiza- treme, this principle is a cornerstone behind quasi-classical tion, and that the most liberal formalism towards the logic (Besnard and Hunter 1995), which is another formal- disjunctive syllogism is also the strongest among the three, while the most cautious logic is the weakest one. ism that is considered here. In between, we have intermediate approaches for cautious applications of the disjunctive syllogism. This class is represented by the logic of inconsis- Introduction tency minimization (Arieli and Avron 1996), according to which the disjunctive syllogism is allowed only with respect Paraconsistent (‘inconsistency-tolerant’) logics are for- to consistent fragments of the premises. malisms that reason with inconsistent information in a con- trolled and discriminating way. Thus, while classical logic, Each one of the logics mentioned above is presented in a intuitionistic logic, and many other standard logics are ‘ex- syntactic and a semantic style, which are linked by a simple plosive’ in the sense that they admit the inference of any transformation. We also show that knowledge minimization, conclusion from an inconsistent set of premises, paraconsis- used for reducing the amount of the relevant models without tent logics avoid this approach. In order to do so, at least one affecting the inferences, is supported by all the three log- of the following two classically valid rules has to be rejected: ics. Interestingly, Belnap’s logic, which is the most cautious in its attitude towards the disjunctive syllogism, is also The Disjunctive Syllogism: from ψ and ¬ψ ∨ φ infer φ. the weakest logic among the three formalisms, and quasi- The Introduction of Disjunction: from ψ infer ψ ∨ φ. classical logic, which is the most liberal one, is the strongest formalism. Formalisms that validate both of the rules above are necessarily explosive, as ψ entails ψ∨φ and this together with ¬ψ implies φ, thus the inconsistent set of assumptions {ψ, ¬ψ} Consequence Relations Based on Four States yields, in the presence of the two rules above, a derivation of Information of an arbitrary φ. In this paper, we consider different approaches of weak- Reasoning with four states of information is extensively ening the rules above in order to gain paraconsistency. We studied in computer science and AI. Some examples for do so in a framework that is based on four possible states of its application are in the context of symbolic model check- information regarding the truth of an assertion ψ: in addi- ing (Chechik et al. 2003), semantics of logic pro- tion to the two standard states, in which ψ is either accepted grams (Fitting 2002), natural language processing (Nelken or rejected, two additional states are allowed, reflecting the and Francez 2002), and, of-course, inconsistency-tolerant two extreme cases of uncertainty about ψ: either there is systems. In the context of KR the latter has been investi- no sufficient information to determine its truth, or there are gated, e.g., in the framework of description logic (Ma, Lin, and Lin 2006), default logic (Yue, Ma, and Lin 2006), belief Copyright c 2010, Association for the Advancement of Artificial revision operators (Gabbay, Rodrigues, and Russo 2000), Intelligence (www.aaai.org). All rights reserved. and by circumscriptive-like formulas (Arieli and Denecker

302 2003) or quantified Boolean formulas (Arieli 2007). In what follows we shall write Γ,φ4 ψ for abbreviating The basic idea behind our framework is to acknowledge Γ ∪{φ} 4 ψ. four states of information for a given assertion: either the Proposition 3 4 is a Tarskian consequence relation assertion is known to be true, known to be false, no infor- (Tarski 1941), i.e., it has the following properties, for ev- mation about it is available, or there are contradictory indi- ery theory Γ and formulas ψ, φ ∈N. cations about its truth. Two common ways of representing Reflexivity: Γ,ψ 4 ψ. those states are by incorporating a related (meta-) language Monotonicity:ifΓ 4 ψ and Γ ⊆ Γ ,thenΓ 4 ψ. for describing the reasoner’s information (we call these ap- Γ φ Γ,φ ψ Γ, Γ ψ proaches syntactical, as they do not involve truth functions), Transitivity:if 4 and 4 ,then 4 . or by using truth values for the same purpose (such ap- Proof. Reflexivity immediately follows from Notation 2. proaches are called semantical). In this section we define Monotonicity follows from the fact that if Γ ⊆ Γ then Mod (Γ) ⊆ Mod (Γ) these approaches and describe an obvious link between them =4 =4 . For transitivity, suppose that Γ φ Γ,φ ψ I ∈ Mod (Γ ∪ Γ) by a simple transformation. 4 and 4 ,andlet =4 .Since I ∈ Mod (Γ) Γ φ I ∈ Mod ({φ}) =4 and 4 , =4 . Thus, A Syntactic Approach I ∈ Mod (Γ) I ∈ Mod (Γ ∪{φ}) as =4 ,wehavethat =4 . L Γ,φ ψ I ∈ Mod ({ψ}) Γ, Γ Let be a propositional language consisting of an alpha- Now, since 4 , =4 ,andso 4 bet A of propositional variables (atomic formulas), and the ψ. 2 logical symbols ¬, ∧, ∨. We denote the elements in A by Proposition 4 4 is paraconsistent: there are ψ, φ ∈N p, q, r, literals (i.e., atomic formulas or their negations) by such that ψ, ¬ψ 4 φ. li (i =1, 2,...), clauses (∨-disjunction of literals) by Ci + − (i =1, 2,...), and formulas in a conjunctive normal form Proof. Let I = {p ,p } for some p ∈A.Foreveryq ∈ (CNF; ∧-conjunction of clauses) by ψ, φ.ThesetofCNF- A\{p} we have that I =4 p and I =4 ¬p but I =4 q, formulas of L is denoted N . Theories are sets of CNF- thus p, ¬p 4 q. 2 formulas, and are denoted by Γ, Δ.1 The set of all atoms occurring in a formula ψ is denoted by A(ψ) and the set of A Semantic Counterpart all atoms occurring in a theory Γ is denoted by A(Γ) (that The paraconsistent consequence relation 4 considered in ± is, A(Γ) = ∪ψ∈ΓA(ψ)). A signed alphabet A is a set that the previous section is constructed in a syntactical manner, consists of symbols p+,p− for each atom p ∈A. without using truth values. Yet, it is evident (e.g., by the An information state I is an element in P(A±),the intuitive meaning of information states) that it may be as- power-set of A±. Intuitively, p+ ∈ I means that in I there sociated with a four-valued semantics. The corresponding − FOUR is a reason for accepting p,andp ∈ I means that in I there four-valued algebraic structure, denoted , was intro- is a reason for accepting ¬p. Clearly, an information state duced by Belnap (1977a; 1977b) (see Figure 1). This struc- + − {t, f, ⊥, } may contain both of p and p , or neither of them. The ture is composed of four elements . Intuitively, t f ⊥ former situation reflects inconsistent data about p and the and have their usual classical meanings, represents latter situation corresponds to incomplete data about it. This incomplete information and represents inconsistent infor- is formalized in the following definition, which also extends mation. These values are arranged in two lattice structures. atomic information to CNF-formulas and theories: One, represented along the horizontal axis of Figure 1, is the standard logical partial order, ≤t, which intuitively reflects Definition 1 Denote by =4 the binary relation on ± differences in the ‘measure of truth’ that every value repre- P(A )×N, inductively defined as follows: sents. We shall denote by ∧ and by ∨ the meet and the join + I =4 p if p ∈ I, (respectively) of the lattice obtained by ≤t, and by ¬ its or- − I =4 ¬p if p ∈ I, der reversing involution, for which ¬ = and ¬⊥ = ⊥. I =4 l1 ∨ ...∨ ln if I =4 li for some 1 ≤ i ≤ n, The other partial order of FOUR, denoted ≤k, is un- I =4 C1 ∧ ...∧ Cn if I =4 Ci for every 1 ≤ i ≤ n. derstood (again, intuitively) as reflecting differences in the amount of knowledge or information that each truth value Given an information state I and a theory Γ, we denote exhibits. This partial order is represented along the vertical I =4 Γ if I =4 ψ for every ψ ∈ Γ. axis of Figure 1, and it will also have a great importance in Definition 1 induces a corresponding relation on P(N )× what follows. N . The next step in using FOUR for reasoning is to choose its set of designated elements. The obvious choice is the Γ ψ ∈N Notation 2 Given a theory and a formula ,we set D = {t, }, since both of the values in D intuitively denote: represent formulas that are ‘known to be true’. The set D Mod (Γ) = {I ∈P(A±) | I = Γ} a ∧ b ∈D a b D =4 4 . has the property that iff both and are in , while a ∨ b ∈Diff at least one of a or b is in D. Γ 4 ψ if Mod= (Γ) ⊆ Mod= (ψ). 4 4 The semantical analogue of information states are valua- 1The concentration on (sets of) CNF-formulas is justified by tions, which are functions that assign a truth value to each the fact that, just as in the classical semantics, in the four-valued atomic formula. For a valuation ν we shall sometimes de- semantics for L, every well-formed formula is logically equivalent note ν = {pi : xi | i =1, 2,...} instead of ν(pi)=xi.The to a CNF-formula (see (Arieli and Avron 1996)). space of valuations is denoted Λ.

303 Proof. By induction on the structure of a CNF-formula ψ, ≤k 6 t it is easily veriﬁed that I =4 ψ iff σ(I)(ψ) ∈D,andso @ I = ψ σ(I) |= ψ I = Γ 4 iff 4 . Thisimpliesthat 4 iff @ σ(I) |=4 Γ. The second part is similar. 2 @ @ Proposition 10 For every theory Γ and formula ψ ∈N, t @t f t Γ 4 ψ iff Γ 4 ψ. @

@ Proof. If Γ 4 ψ then ν ∈ Mod|= (Γ) \ Mod|= (ψ) for 4 4 @ ν ∈ Λ Mod (Γ) = {σ−1(ν) | ν ∈ some . By Lemma 9, =4 @ Mod (Γ)} σ−1(ν) ∈ Mod (Γ) \ Mod (ψ) @ t |=4 , thus =4 =4 .It ⊥ follows that Γ 4 ψ. The proof of the converse is similar, ≤t Mod (Γ) = {σ(I) | I ∈ - using the fact that by Lemma 9, |=4 Mod (Γ)} 2 =4 . Figure 1: FOUR Proposition 10 shows that the two approaches above ac- tually coincide for CNF-formulas. This will be useful in what follows for showing correspondence among the differ- The following definition and notation should be compared ent derivatives of these consequence relations. with Definition 1 and Notation 2, respectively. Note 11 For relating 4 and 4 with respect to arbitrary sets Definition 5 Denote by |=4 the binary relation on Λ ×N, of formulas in L (not necessarily in a conjunctive normal inductively defined as follows: form), some generalizations of the basic definitions are re- ν |=4 p if ν(p) ∈D, quired. In the semantic approach this can be done rather di- ν |=4 ¬p if ¬ν(p) ∈D, rectly by extending Definition 5 so that for every L-formulas ν |=4 l1 ∨ ...∨ ln if ν |=4 li for some 1 ≤ i ≤ n, ψ, φ, ν |= C ∧ ...∧ C ν |= C 1 ≤ i ≤ n 4 1 n if 4 i for every . ν |=4 ¬ψ if ¬ν(ψ) ∈D, For a valuation ν and a theory Γ, we denote ν |=4 Γ if ν |=4 ν |=4 ψ ∨ φ if ν |=4 ψ or ν |=4 φ, ψ for every ψ ∈ Γ. ν |=4 ψ ∧ φ if ν |=4 ψ and ν |=4 φ. Notation 6 Given a theory Γ and a formula ψ ∈N,we The consequence relation that is obtained by extending Def- denote: inition 5 to this satisfiability relation coincides with Bel- nap’s well-known four-valued logic (Belnap 1977a; 1977b) Mod|= (Γ) = {ν ∈ Λ | ν |=4 Γ}. 4 and has strong ties to ‘first-degree entailments’ in relevance Γ ψ Mod (Γ) ⊆ Mod (ψ) 4 if |=4 |=4 . logic (Anderson and Belnap 1975; Dunn and Restall 2002). ψ ,...,ψ φ ψ ∧ ...∧ ψ → φ Proposition 7 4 is a paraconsistent Tarskian consequence Indeed, it holds that 1 n 4 iff 1 n relation. is a first-degree entailment. Adapting the syntactic approach to general formulas is Proof. A direct proof is obtained from the proofs of Proposi- somewhat more cumbersome, as extra conditions have to be tions 3 and 4, replacing information states by valuations and imposed on the satisfaction relation for assuring the well- Mod (Γ) Mod (Γ) =4 by |=4 . This proposition also follows behaving of the induced entailment with respect to general as a corollary of Propositions 3, 4, by the correspondence formulas (e.g., the double-negation rule, associating ¬¬ψ 2 between 4 and 4, shown in Proposition 10 below. with ψ, has to be explicitly required; see, e.g., (Hunter 2000, Section 5)). Relating the Two Approaches The syntactic and the semantic approaches described above Quasi-Classical Logic can be related by the following transformation: By their definition, paraconsistent logics are weaker than ± Definition 8 For an information state I ∈P(A ),theval- classical logic with respect to inconsistent sets of premises. uation σ(I) ∈ Λ is defined for every p ∈Aas follows: However, the consequence relations considered in the pre- ⎧ t p+ ∈ I p− ∈ I vious section seem to be too weak, as they exclude some ⎨⎪ if and , classically valid rules even when the premises are classically f if p+ ∈ I and p− ∈ I, σ(I)(p)= consistent. Thus, for instance, both of 4 and 4 reject the ⎪ ⊥ if p+ ∈ I and p− ∈ I, ⎩ disjunctive syllogism in any circumstance, as q is not 4- if p+ ∈ I and p− ∈ I. deducible (and, by Proposition 10, it is not 4-deducible ei- {p, ¬p∨q} 2 Clearly, σ is a one-to-one mapping from the space P(A±) ther) from . In order to overcome this shortcom- of the information states onto the space Λ of the four-valued ing, Besnard and Hunter (1995; 2000) introduced an alter- valuations. Given a valuation ν ∈ Λ, we shall denote by native formalism, called quasi-classical logic,inwhichthe σ−1(ν) the information state I for which σ(I)=ν. satisfaction relation of the syntactic-based approach (Defini- tion 1) is strengthen as follows: Lemma 9 It holds that I =4 Γ iff σ(I) |=4 Γ, and ν |=4 Γ −1 2 iff σ (ν) =4 Γ. Consider a valuation that assigns to p and f to q.

304 Definition 12 Let ∼ l be the complement of a literal l (i.e., Lemma 18 I =QC l1 ∨ ...∨ ln iff ∼ l = ¬p if l = p,and∼ l = p if l = ¬p). Denote by + − ± • there is 1 ≤ i ≤ n, such that li ∈ I and li ∈ I,or =QC the binary relation on P(A )×N, defined just as the • 1 ≤ i ≤ n l+ ∈ I l− ∈ I relation =4 of Definition 1, with the only exception that for all , i and i . I =QC l1 ∨ ...∨ ln if the following two conditions hold: Definition 19 Denote by |=QC the binary relation on Λ×N , • I =4 l1 ∨ ...∨ ln,and inductively defined as follows: • for every 1 ≤ i ≤ n, ν |=QC p if ν(p) ∈D 3 ν |= ¬p ¬ν(p) ∈D if I =QC ∼li then I =QC l1 ∨ ...li−1 ∨ li+1 ∨ ...∨ ln. QC if + + − ν |=QC l1 ∨ ...∨ ln if ν(li)=t for some 1 ≤ i ≤ n I1 = {q } I2 = {p ,p } Example 13 Let and .Then or ν(li)= for all 1 ≤ i ≤ n I1 =QC p ∨ q I1 =QC ¬p ∨ q and . On the other hand, ν |=QC C1 ∧ ...∧ Cn if ν |=QC Ci for all 1 ≤ i ≤ n I2 =QC q,andsoI2 =QC p ∨ q and I2 =QC ¬p ∨ q. Given a valuation ν and a theory Γ, we denote ν |=QC Γ if ∨ Note that according to Definition 12, the disjunction ν |=QC ψ for every ψ ∈ Γ. cease to have its usual lattice-based meaning. The idea Γ ψ ∈N behind this definition is, generally speaking, to link be- Notation 20 Given a theory and a formula ,we tween a disjunct and its complement, and by this to preserve denote: Mod (Γ) = {ν ∈ Λ | ν |= Γ} the meaning of the resolution principle. This gives rise to |=QC QC . the following definition of entailments in the quasi-classical Γ ψ Mod (Γ) ⊆ Mod (ψ) QC iff |=QC |=4 . logic. Proposition 21 For every theory Γ and a formula ψ ∈N, Γ ψ ∈N Notation 14 Given a theory and a formula ,we Γ QC ψ iff Γ QC ψ. denote: Proof. For a literal l we have that I =QC l iff σ(I)(l) ∈D, Mod (Γ) = {I ∈P(A±) | I = Γ} =QC QC . thus I =QC l iff σ(I) |=QC l. Also, by Lemma 18, Γ ψ Mod (Γ) ⊆ Mod (ψ) I = l ∨ ...∨ l σ(I)=t 1 ≤ i ≤ n QC iff =QC =4 . QC 1 n iff for some ,or σ(I)= for every 1 ≤ i ≤ n,iffσ(I) |=QC l1 ∨ ...∨ ln. Example 15 For Γ={p ∨ q, ¬p ∨ q}, Mod= (Γ) = QC Thus, by induction on the structure of a CNF-formula ψ, {{p+,p−,q+}, {p+,q+}, {p−,q+}, {q+}} Γ q , thus QC . we have that I =QC ψ iff σ(I) |=QC ψ. It follows Γ q {p+,p−} Mod (Γ) −1 Note that 4 ,since is in =4 but not that Mod= (Γ) = {σ (ν) | ν ∈ Mod|= (Γ)} and Mod (q) QC QC in =4 . Mod (Γ) = {σ(I) | I ∈ Mod (Γ)} |=QC =QC . This implies Proposition 16 QC is paraconsistent. (as in the proof of Proposition 10) that QC and QC coin- 2 {p+,p−}∈Mod ({p, ¬p}) cide. Proof. It holds that =QC but {p+,p−} ∈ Mod ({q}) p, ¬p q 2 =4 , therefore QC . Logic of Minimal Inconsistency Note that, as demonstrated in Example 15, unlike 4,the As noted before, Belnap’s logic excludes some classically entailment QC admits the principle of resolution, so for ar- valid rules even for classically consistent theories, and so it bitrary clauses C1,C2,C3,C4 we have: may be considered as too weak. Quasi classical logic, on the C ∨ l ∨ C ,C∨∼l ∨ C C ∨ C ∨ C ∨ C . other hand, may be considered as too strong. This is shown 1 2 3 4 QC 1 2 3 4 in the following example. C ∨C ∨C ∨C C ∨l∨ The clause 1 2 3 4 is called a resolvent of 1 Example 22 Consider the theory Γ={p, ¬p, p ∨ q}. Here, C2 and C3∨∼l∨C4. By the monotonicity of quasi-classical Mod (Γ) = {{p : ,q : t}, {p : ,q : }} Γ |=QC , thus QC logic we have, then, the next result. q. This may look counter-intuitive, as there is no justification Proposition 17 Any resolvent of two clauses in a theory Γ to infer q from p ∨ q (the only assertion in Γ that mentions is QC-deducible from Γ. q) as long as one already ‘knows’ p.

The ‘price’ of this is that QC is not a consequence rela- The formalism that is considered in this section over- tion, since (although it is reflexive and monotonic) it is not comes this problem by applying classically valid rules (in transitive.4 Thus, proofs cannot be composed for more com- particular, the disjunctive syllogism) only to classically valid plicated proofs, and so inference in quasi-classical logic is fragments of the theory (see Corollary 29 and Note 30 be- not an iterative process, but rather a one-step procedure. We low). The idea here is to minimize the inconsistent states of refer to (Hunter 2000) for a more detailed discussion on the information of the reasoner, following the recognition that properties of QC and a corresponding sound and complete although inconsistency is unavoidable, it should be mini- proof theory. mized as much as possible. By the next lemma, shown in (Hunter 2000), we can give Definition 23 For ν, μ ∈ Λ we denote by ν ≤ μ that for a semantic counterpart to QC. every atom p ∈A, μ(p)= whenever ν(p)= .We ν< μ ν ≤ μ 3 = denote by that and there is an atomic In (Besnard and Hunter 1995; Hunter 2000) the relations QC formula p ∈A, such that μ(p)= while ν(p) = . and =4 are called, respectively, strong satisfiability and weak satisfiability. Notation 24 Given a theory Γ and a formula ψ ∈N,we 4 Indeed, p QC p ∨ q and ¬p, p ∨ q QC q,butp, ¬p QC q. denote:

305 Mod (Γ) = {ν ∈ Mod (Γ) | ν < ν ν μ A(Γ) |=MI |=4 Clearly, , and since is the same as on , μ< ν μ ∈ Mod (Γ)} ν Γ if then |=4 . is also a model of . Moreover, using the facts that both Γ and ψ are independent of Γ, it follows that ν ∈ Γ ψ Mod (Γ) ⊆ Mod (ψ) MI iff |=MI |=4 . Mod (Γ ) ν ∈ Mod (Γ) ν < ν |=4 . Thus, |=4 and ,which ν ∈ Mod (Γ) The logic MI was introduced in (Arieli and Avron implies that |=MI , but this is a contradiction to 1996; 1998) as a generalization to the four-valued case of the choice of ν. 2 Priest’s three-valued logic LPm of inconsistency minimiza- Corollary 29 Suppose that Γ and Γ are a partition of tion (Priest 1991). Γ and that Γ is classically consistent. Then any non- Proposition 25 MI is paraconsistent. tautological resolvent of two clauses in Γ is MI-deducible from Γ. Proof. Indeed, {p : ,q: f}∈Mod|= ({p, ¬p}) but {p : MI ψ Γ Γ ψ ,q:f} ∈ Mod|= ({q}),andsop, ¬p MI q. 2 Proof. If is a resolvent of two clauses in then . 4 Also, A(ψ) ⊆A(Γ), thus ψ is independent of Γ.By Example 26 Consider again the theory Γ={p, ¬p, p ∨ Proposition 28, then, Γ MI ψ. 2 q} of Example 22. As intuitively expected (and unlike Note 30 As Example 26 shows, Corollary 29 does not hold quasi-classical logic), q is not a MI-conclusion of Γ,as in general (i.e., for resolvents of clauses in any theory), so Mod|= (Γ) = {{p : ,q: t}, {p : ,q: f}, {p : ,q: ⊥}}. MI MI admits a restricted form of resolution (and so of the On the other hand, r, ¬r, ¬p, p ∨ q MI q. The latter is a disjunctive syllogism), adhering consistency (cf. Proposi- particular case of the next proposition that shows that every tion 17). non-tautological clause that can be inferred from a consistent fragment of a theory, can also be inferred by MI from The syntactical counterpart MI of MI is defined as fol- the whole theory, provided that the clause is independent of lows: ± the inconsistent part of the theory. Definition 31 For I,J ∈P(A ) we denote by I ≤ J that + − + − Definition 27 We say that Γ and Γ are independent,if {p ∈A | p ,p ∈ I}⊆{p ∈A | p ,p ∈ J}}.Ifthe A(Γ ) ∩A(Γ )=∅. Two independent theories Γ and Γ above containment is proper, we write I< J. are a partition of Γ,ifΓ=Γ ∪ Γ . Notation 32 Given a theory Γ and a formula ψ ∈N,we Proposition 28 Denote by the standard entailment of denote: Γ Γ Γ Mod (Γ) = {I ∈ Mod (Γ) | classical logic. Let and be a partition of . Sup- =MI =4 pose that Γ is classically consistent, and that ψ is a non- J< I J ∈ Mod (Γ)} if then =4 . tautological clause5 that is independent of Γ.IfΓ ψ 6 Γ ψ Mod (Γ) ⊆ Mod (ψ) then Γ MI ψ. MI iff =MI |=4 . I,J ∈ Proof. By (Arieli and Avron 1998, Proposition 64), if Γ Again, it is easy to see that for information states P(A±) I< J is classically consistent and ψ is as in the proposition, then it holds that (in the sense of Definition 31) iff σ(I) < σ(J) (in the sense of Definition 23). Simi- Γ ψ iff Γ MI ψ. It is enough to show, then, that under −1 Γ ψ Γ ψ larly, for valuations ν, μ ∈ Λ, ν< μ iff σ (ν) < the conditions of the proposition, if MI then MI . −1 −1 σ (μ). Thus, we have that Mod= (Γ) = {σ (ν) | Indeed, suppose otherwise that Γ MI ψ. Then there is a val- MI ν ∈ Mod (Γ)} Mod (Γ) = {σ(I) | I ∈ ν ∈ Mod (Γ) \ Mod (ψ) |=MI and that |=MI uation |=MI |=4 . Note that this in par- Mod (Γ)} ν ∈ Mod (Γ ) ν ∈ Mod (Γ ) =MI , which implies that MI and MI coincide. ticular implies that |=4 and |=4 . Now, consider the following valuation: Knowledge Minimization ν(p) if p ∈A(Γ ) ∪A(ψ), Mod (Γ) ⊆ Mod (Γ) Mod (Γ) ⊆ μ(p)= Since |=MI |=4 and |=QC t otherwise. Mod (Γ) |=4 , the set of models needed for drawing conclu-

As μ(φ)=ν(φ) for every formula φ ∈ Γ ∪{ψ},wehave sions by MI and by QC is never bigger than those needed μ ∈ Mod (Γ) μ ∈ Mod (ψ) for drawing conclusions by 4.Inthissectionwepresenta that |=4 and |=4 .Now,since Γ ψ μ ∈ Mod (Γ) natural approach for reducing the number of models even MI , |=MI , and so there is a valuation further, without changing the logic. The idea is to con- μ ∈ Mod|= (Γ ) such that μ < μ. Now, consider the 4 sider, in each logic, the ⊆-minimal information states of the following valuation: premises, or – equivalently – to take advantage of the knowl- μ(p) p ∈A(Γ) ∪A(ψ) edge order ≤k of FOUR and to consider the ≤k-minimal ν(p)= if , ν(p) otherwise. valuations among the models of the premises. The intuition behind this reﬁnement is that one should not assume any- 5That is, none of the disjunctions in ψ contains an atomic for- thing that is not really known. mula and its negation. Deﬁnition 33 For ν, μ ∈ Λ we denote by ν ≤k μ that for 6 The aspiration to preserve classically valid inferences when- every p ∈A, ν(p) ≤k μ(p). We denote by ν

306 Min⊆(S)={I ∈S| if J ⊂ I then J ∈S}. Sν has a minimal element, which is the required valuation in Min≤ (Mod|= (Γ)).Letμ be the following valuation: Min≤ (V)={ν ∈V| if μ

Proposition 37 All the logics considered above are pre- ⊆ ≤k b) Γ MI ψ iff Γ MI ψ iff Γ ψ iff Γ ψ. served under knowledge minimization: for every theory Γ MI MI ψ ⊆ ≤k (not necessarily ﬁnite) and a formula , c) Γ QC ψ iff Γ QC ψ iff Γ QC ψ iff Γ QC ψ. ≤k a) Γ 4 ψ iff Γ 4 ψ. Proof. Item (a) follows from Propositions 10, 36(a), and ≤ 37(a); Item (b) follows from the note at the end of the previ- Γ ψ Γ k ψ b) MI iff MI . ous section and from Propositions 36(b) and 37(b); Item (c) ≤k follows from Propositions 21, 36(c), 37(c). 2 c) Γ QC ψ iff Γ QC ψ. Proof. Parts (a) and (b) follow, respectively, from Corol- Comparing the Logics lary 32 and Proposition 59 of (Arieli and Avron 1998). For Part (c) we adjust a similar proof in (Arieli and Avron 1998) As noted before, an important difference among the three to quasi-classical logic. First, we show the following lemma: formalisms considered above is their attitude towards the disjunctive syllogism. The following example illustrates ν ∈ Mod (Γ) μ ∈ Lemma 38 For every |=QC there is a this. Min≤ (Mod|= (Γ)) such that μ ≤k ν. k QC Example 41 Let Γ={p, ¬p, q, ¬p ∨ r, ¬q ∨ s}. Belnap’s ν ∈ Mod (Γ) Proof. Indeed, for a valuation |=QC consider consequence relation excludes any application of the dis- S = {ν | ν ∈ Mod (Γ),ν ≤ ν} Γ r Γ s the set ν i i |=QC i k . Suppose junctive syllogism, thus 4 and 4 . On the other that C ⊆ Sν is a descending chain with respect to ≤k.We extreme, quasi-classical logic supports every application of shall show that C is bounded in Sν , so by Zorn’s lemma resolution in Γ, therefore Γ QC r and Γ QC s. The logic

307 Mod (Γ) ⊆ Mod (ψ) Γ ψ of inconsistency minimization is situated in an intermediate |=MI |=4 ,andso MI . level: as Γ can be partitioned to two independent fragments The proof for sets of CNF-formulas follows from the proof Γ = {q, ¬q ∨ s} Γ = {p, ¬p, ¬p ∨ r} and , the former of above by the fact that C1∧...∧Cn x ψ iff C1,...,Cn x ψ which is classically consistent, then by Proposition 28 the for both x = QC and x = MI. 2 disjunctive syllogism is supported only with respect to its A graphic representation of the results in Corollary 40 and consistent fragment, hence Γ MI s while Γ MI r. Proposition 42 is given in Figure 2. Next, we investigate the relative strength of the three for- As the next result shows (see Proposition 1–3 in (Coste- malisms. Marquis and Marquis 2005)), the formalisms considered Proposition 42 For every theory Γ and ψ ∈N, above also differ in their computational complexity. ψ ∈N a) if Γ 4 ψ then Γ MI ψ, Proposition 44 Deciding whether a formula follows from a theory Γ is in P for 4,iscoNP-complete for b) if Γ MI ψ then Γ QC ψ. p 8 QC, and is Π2-complete for MI. Note 43 The two items of Proposition 42 imply that Γ QC ψ whenever Γ 4 ψ,andso 4 ⊆ MI ⊆ QC. Note that Summary by Example 41 these containments are, in fact, strict (see Paraconsistent reasoning requires the weakening of either also Examples 22, 26). Moreover, unlike 4 and QC that the disjunctive syllogism or the law of disjunction introduc- are monotonic (see, respectively, (Arieli and Avron 1996, tion. In this paper, we have considered three different ap- Proposition 3.10) and (Hunter 2000, Proposition 4.32)), MI proaches for doing so in the context of four-states logics, is nonmonotonic. This is demonstrated in Example 26: for and investigated the relative expressive power of three log- Γ={¬p, p ∨ q} we have that Γ MI q but Γ ∪{p} MI q. ics, each one represents a different approach. More specifi- Proof Outline of Proposition 42. Part (a) simply follows cally, we compared quasi-classical logic – whose definition Mod (Γ) ⊆ Mod (Γ) is syntactical in nature and motivated by the need to pre- from the fact that |=MI |=4 . For Part (b), suppose first that Γ consists only of clauses. We consider serve the resolution principle – to a logic of inconsistency two operators that construct a theory from a theory and a minimization, that is based on purely semantical consider- valuation: ations, and which tolerates classically valid rules only with respect to classically consistent fragments of the premises. • T1(Γ,ν) is the theory that is obtained from Γ by removing Despite their different attitudes to the application of the all the clauses in which there is a literal l such that ν(l)= disjunctive syllogism, all the investigated formalisms share . the knowledge minimization property, which allows to re- duce the number of models without affecting the conclu- • T2(Γ,ν) is the theory that is obtained from Γ by removing sions. This also may be captured, in the context of four pos- from every clause in Γ the literals l such that ν(l)= .7 sible states of information, as a kind of consistency preserv- The following facts are easily verified: ing method: As long as one keeps the redundant information ν ∈ Mod (T (Γ,ν)) μ ∈ Mod (Γ) as minimal as possible the tendency of getting into conflicts Fact 1: If |=MI 1 there is |=MI such that μ ≤ ν and μ ≤k ν (either μ = ν or μ is obtained decreases. from ν by letting μ(pi) ∈{t, f, ⊥} for some pi ∈A(Γ) \ A(T1(Γ,ν)) such that ν(pi)= ). References ν ∈ Mod (Γ) ν ∈ Mod (T (Γ,ν)) Anderson, A., and Belnap, N. 1975. Entailment, volume I. Fact 2: If |=QC then |=QC 2 . Princton University Press. Also, by the definitions of T1 and T2, for every clause C1 in T1(Γ,ν) there is a clause C2 in T2(Γ,ν) such that C1 is a Arieli, O., and Avron, A. 1996. Reasoning with logical bilat- subformula of C2. Thus: tices. Journal of Logic, Language, and Information 5(1):25– 63. Fact 3: Mod|= (T2(Γ,ν)) ⊆ Mod|= (T1(Γ,ν)). 4 4 Arieli, O., and Avron, A. 1998. The value of the four values. Γ ψ ν ∈ Mod (Γ) \ Mod (ψ) Now, if QC ,thereis |=QC |=4 . Artificial Intelligence 102(1):97–141. ν ∈ Mod (T (Γ,ν)) \ Mod (ψ) By Fact 2, |=QC 2 |=4 , thus Arieli, O., and Denecker, M. 2003. Reducing preferential ν ∈ Mod (T (Γ,ν)) \ Mod (ψ) ν ∈ |=4 2 |=4 , and by Fact 3, paraconsistent reasoning to classical entailment. Journal of Mod (T (Γ,ν)) \ Mod (ψ) |=4 1 |=4 . Note that this implies, in Logic and Computation 13(4):557–580. p ∈A(T (Γ,ν))∪A(ψ) particular, that there is no 1 such that Arieli, O. 2007. Paraconsistent reasoning and preferential ν(p)= ν ∈ Λ . Consider a valuation that is identical to entailments by signed quantified boolean formulae. ACM ν A(T (Γ,ν)) ∪A(ψ) ν(p) = on 1 and elsewhere. Then Transactions on Computational Logic 8(3). ν ∈ Mod (T (Γ,ν)) \ Mod (ψ) |=MI 1 |=4 . Now, by Fact 1, Batens, D. 2000. A survey on inconsistency-adaptive logics. there is a valuation μ ∈ Mod|= (Γ) such that μ ≤k ν . MI In Frontiers of Paraconsistent Logic, volume 8 of Studies in ν ∈ Mod (ψ) μ ∈ Mod (ψ) Since |=4 , |=4 as well. Thus Logic and Computation. Research Studies Press. 69–73. 7Consider for instance the theory Γ={p, ¬p, p ∨ q} of Ex- 8When Γ and ψ are an arbitrary set of formulas and a formula amples 22 and 26, and let ν ∈ Λ such that ν(p)=.Then in L, the decision problem for 4 becomes coNP-complete, and p T1(Γ,ν)=∅ and T2(Γ,ν)={q}. remains Π2-complete for MI.

308 Figure 2: Classes of paraconsistent entailments in four-valued logics (Entailments within the same box are equivalent; Entail- ments in a higher box are weaker than those in a lower box).

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