Contradiction Support Detection

As has already been hinted in the introduction, it is useful and might even be crucial for some applications that a paraconsistent semantics not only provides information about which facts are inconsistent, but also about those which can only be derived by involving inconsistent knowledge. In this way, one knows when to be “suspicious” about a piece of information because an error might have occurred. Hence, approaches applying so- called suspicious reasoning are closer to the idea of belief revision as inconsistencies are rather regarded as errors, and detecting “suspicious” information can be viewed as a form of debugging. Suspicious reasoning can be achieved by implementing some way of propagating inconsistencies into a logic, which results in what Damásio and Pereira call Contradiction Support Detection [DP97]. The authors give a general formal definition of de- pendence on contradiction in Definition 8, which is independent of the specific semantics used. We adapt it to our notation here.

Definition 2.10(Dependence on contradiction [DP97]). Let Π be an extended logic pro- gram and C = {c, ¬c | c, ¬c ∈ Π}, the set of the contradictory facts in Π. We say that any objective literal L depends on a contradiction w.r.t. a semantics SEM if and only if there is

a set S ⊆ C such that

SEM (Π) ∩ {L} 6= SEM (Π − S) ∩ {L}.

This means, an objective literal depends on a contradiction if and only if its truth value changes after removing only contradictory facts. Note that the definition is limited to contradictory facts, so that contradictory atoms which are derivable, but not contained as facts in the program, are not considered. Damásio and Pereira additionally state that the notion of dependence on contradiction can be made stronger by referring to the para- consistent well-founded model of a program. In this case, an objective literal L is defined to depend on contradiction iff L and notL are both contained in the model [DP97].

Damásio and Pereira mention that in most work conducted on paraconsistent LP se- mantics the need for being able to detect support on contradictions has been “overlooked or not properly captured” [DP97]. Two approaches to paraconsistent LP semantics that integrate suspicious reasoning into their framework are those based respectively on the SMS [SI95] and the WFS [Sak92] which have been discussed in section2.3.1. However, Damásio and Pereira remark that both approaches suppose that the truth of a default- negated program atom can never depend on a contradiction, which is not the case ac- cording to the authors and hence, makes their approach to Contradiction Support De- tection incomplete [DP97]. They illustrate this issue by means of the following example.

Example 2.11(Support on contradiction through default negation [DP97]). Consider the following extended logic program.

a ← notb.

b ← notc, not¬c.

c ←

¬c ←

In this program, the program atom a clearly depends on a contradiction because it is not derivable anymore after removing one of the contradictory factscand ¬cfrom the program. It is desirable that a paraconsistent semantics detects this dependence on con- tradiction. This is not the case for the paraconsistent SMS of Sakama and Inoue [SI95] and the paraconsistent WFS of Sakama [Sak92] because they do not propagate inconsistencies

over default negation. ♦

In W F SXp, Contradiction Support Detection is already built-in due to the enforce-

ment of the Coherence Principle. For example, suppose that we add the program-rule

resolvedRisk(food) ← isMon(food).

is not part of the set Γs(I1)sinceisMon(food)is not derivable, i.e. it is also not in the set

of “true or undefined” objective literals. However, simultaneously it is derivable by the operator ΓΓsto be in the set of “true” objective literals sinceisMon(food)is contained in

it, too. Consequently, the paraconsistent model of the program is

W F Mp(Π) = {risk(food),certForw(food),isMon(food), ¬isMon(food), resolvedRisk(food), not¬risk(food), not¬certForw(food), notisMon(food), notresolvedRisk(food), not¬resolvedRisk(food), not¬isMon(food), notisLabelled(food)})

in this case andresolvedRisk(food)is evaluated to true with contradictory belief in the cor- responding nine-valued model [DP95].

Though W F SXp is capable of handling situations as described in Example2.11, it

fails to detect dependence on contradiction when the program atom that should be inter- preted to be true with contradictory belief occurs also in a rule with undefined body [DP97]. Damásio and Pereira provide the following example.

Example 2.12(Failure to detect support on contradiction [DP97]). Consider the following extended logic program Π.

a ← notb.

a ← c.

b ← notb.

c ←

¬c ←

In this program, the truth ofadepends on the contradictory program atomc. However, notais not contained in W F Mp(Π)becausebis undefined due to the third rule and hence,

the factais contained in the GL-transformation used by the operator Γs because of the

first rule. ♦

In contrast to W F SXp, this example is handled correctly by the approach of Sakama

and Inoue [SI95] and the approach by Sakama [Sak92]. Damásio and Pereira show that a stronger form of Contradiction Support Detection can be obtained by using a transfor- mation of the program in each step of the alternating fixpoint construction. For models computed by the resulting algorithm, Contradiction Support Detection is complete for arbitrary extended logic programs. In addition, Damásio and Pereira also demonstrate how the propagation of contradictions can be blocked in certain cases where the prop- agation is not desirable [DP97]. In our six-valued semantics, we implement the weaker form of Contradiction Support Detection and discuss an extension in the section about future work.