Preorder-Based Triangle: A Modified Version of Bilattice-Based Triangle for Belief Revision in Nonmonotonic Reasoning

09/19/2016 ∙ by Kumar Sankar Ray, et al. ∙ 0

Bilattice-based triangle provides an elegant algebraic structure for reasoning with vague and uncertain information. But the truth and knowledge ordering of intervals in bilattice-based triangle can not handle repetitive belief revisions which is an essential characteristic of nonmonotonic reasoning. Moreover the ordering induced over the intervals by the bilattice-based triangle is not sometimes intuitive. In this work, we construct an alternative algebraic structure, namely preorder-based triangle and we formulate proper logical connectives for this. It is also demonstrated that Preorder-based triangle serves to be a better alternative to the bilattice-based triangle for reasoning in application areas, that involve nonmonotonic fuzzy reasoning with uncertain information.



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1 Introduction:

In many application domains, decision making and reasoning deal with imprecise and incomplete information. Fuzzy set theory is a formalism for representation of imprecise, linguistic information. A vague concept is described by a membership function, attributing to all members of a given universe X a degree of membership from the interval [0,1]. The graded membership value refers to many-valued propositions in presence of complete information. But this ’one-dimensional’ measurement cannot capture the uncertainty present in information. In absence of complete information the membership degree may not be assigned precisely. This uncertainty with respect to the assignment of membership degrees is captured by assigning a range of possible membership values, i.e. by assigning an interval. Interval-valued Fuzzy Sets (IVFSs) deal with vagueness and uncertainty simultaneously by replacing the crisp [0,1]-valued membership degree by intervals in [0,1]. The intuition is that the actual membership would be a value within this interval. The intervals can be ordered with respect to their degree of truth as well as with respect to their degree of certainty by means of a bilattice-based algebraic structure, namely Triangle Arieli . (2004, 2005); Cornelis . (2007). This algebraic structure serves as an elegant framework for reasoning with uncertain and imprecise information.

The truth and knowledge ordering of intervals as induced by the bilattice-based triangle are inadequate for capturing the repetitive revision and modification of belief in nonmonotonic reasoning and are not always intuitive. In this paper we address this issue and attempt to propose an alternate algebraic structure to eliminate the shortcomings of bilattice-based triangle. The major contributions of this paper are as follows:

We demonstrate, with the help of proper examples (in section 3), that bilattice-based triangle is incapable of handling belief revision associated with nonmonotonic reasoning. In nonmonotonic reasoning, inferences are modified as more and more information is gathered. The prototypical example is inferring a particular individual can fly from the fact that it is a bird, but retracting that inference when an additional fact is added, that the individual is a penguin. Such continuous belief revision is not properly represented in bilattice-based triangle.

We point that the truth ordering is unintuitive regarding the ordering of intervals when one interval lies completely within the other (section 3)and hence not suitable for some practical applications.

Exploiting the discrepancies mentioned, we propose modifications for knowledge ordering and truth ordering of intervals so that the aforementioned shortcomings are removed (in section 4).

Using the modified knowledge and truth ordering we construct an alternate algebraic structure, namely preorder-based triangle (in section 5). This structure can be thought of as a unification of bilattice-based triangle and default bilattice Ginsberg (1988). With this we come out of the realm of bilattice-based structures and explore a new algebraic structure based on simple linear pre-ordering.

The proposed algebraic structure is then equipped with appropriate logical operators, i.e. negation, t-norms, t-conorms, implicators, in section 6. Most of the operators are in unison with those used for the bilattice-based structure. But the modified orderings offer additional flexibility.

The proposed algebraic structure is shown to be capable of handling commonsense reasoning problems that could not be handled by the bilattice-based triangle (in section 7). Moreover, it is demonstrated that the preorder-based triangle can be employed to construct an answer set programming paradigm suitable for nonmonotonic reasoning with vague and uncertain information.

2 Intervals as degree of belief:

This section addresses some of the basic definitions and notions that will ease the discussion in the forthcoming sections.

Uncertainty and incompleteness of information is unavoidable in real life reasoning. Hence, sometimes it becomes difficult and misleading, if not impossible, to assign a precise degree of membership to some fuzzy attribute or to assert a precise degree of truth to a proposition. Therefore, assigning an interval of possible truth values is the natural solution. Intervals are appropriate to describe experts’ degrees of belief, which may not be precise Nguyen . (1997). If an expert chooses a value, say 0.8, as his degree of belief for a proposition, actually we can only specify vaguely that his chosen value is around 0.8 and can be represented by an interval, say . Otherwise an interval can be thought of as a collection of possible truth values that a single or multiple rational experts would assign to a proposition in a scenario. Due to lack of complete knowledge the assertions made by different experts will be different and this lack of unanimity can be reflected by appropriate interval. The natural ordering of degree of memberships can be extended to the set of intervals and that gives rise to IVFS Sambuc (1975).

An IVFS can be viewed as an L-fuzzy set Goguen (1967) and the corresponding lattice can be defined as Deschrijver . (2007):

Definition 2.1.

,  where  and   and  iff    and  .

Figure 1: The grey area is

In the definition, is the set of all closed subintervals in [0,1].Figure 1 shows the set .

Bilattice-based Triangle: Bilattices are ordered sets where elements are partially ordered with respect to two orderings, typically one depicts the degree of vagueness or truth (namely, truth ordering) and the other one depicting the degree of certainty (namely, knowledge ordering) Arieli . (2004); Cornelis . (2007). A bilattice-based triangle, or simply Triangle, can be defined as follows:

Definition 2.2.

Let    be a complete lattice and let   and . A (bilattice-based) triangle is defined as a structure , where, for every in :

1. and .

2. and .

This triangle B(L) is not a bilattice, since, though the substructure is a complete lattice but is a complete semilattice.

When L is the unit interval [0,1], then I(L) describes membership of IVFS, , and the lattice becomes . In knowledge ordering, the intervals are ordered by set inclusion, as was suggested by Sandewall Sandewall (1989). The knowledge inherent in an interval is greater than another interval if .

Figure 2: Triangle B({0,0.5,1})

Triangle is shown in Figure 2.

3 Inadequacy of Bilattice-based Triangle:

Intervals are used to approximate degree of truth of propositions in absence of complete knowledge. All values within an interval are considered to be equally probable to be the actual truth value of the underlying proposition. Thus considering intervals as truth status or epistemic state of propositions enables efficient representation of vagueness and uncertainty of information and reasoning. However, the Triangle structure suffers from the following shortcomings that must be eliminated.

3.1 Inadequacy in modeling belief revision in nonmonotonic reasoning:

One important aspect of human commonsense reasoning is that it is nonmonotonic in nature Brewka (1991). In many cases conclusions are drawn in absence of complete information and we have to draw plausible conclusions based on the assumption that the world in which the reasoning is performed is normal and as expected. This is the best that can be done in contexts where the acquired knowledge is incomplete. But, these conclusions may have to be given up in light of further information. A proposition that was assumed to be true, may turn out to be false when new information is gathered. Such repetitive alterations of believes is an essential part of nonmonotonic reasoning. This type of belief revision may not be adequately represented by Triangle. The following discussion will illuminate this issue.

3.1.1 An intuitive explanation:

Example 1:Suppose the following information is given:


, [Birds Fly]

, [Penguin doesn’t Fly]


Bird (Tweety) [Tweety is a bird]

Given this information, suppose, multiple experts are trying to assess the degree of truth of the proposition ”Tweety Flies” [Fly (Tweety)]. The rule ”Birds Fly” is not a universally true fact, rather it’s a general assumption that has several exceptions. Thus, being a Bird is not sufficient to infer that it will fly, since it may be a Penguin, an ostrich or some other non-flying bird. Since, nothing is specified about Tweety except for it is a bird, it is natural in human commonsense reasoning to ”assume” that Tweety is not an exception and it will fly. Now, the confidence about this ”asumption” will be different for different experts. An expert may bestow his complete faith on the fact that Tweety is not an exceptional bird and he will assign truth value 1 to ”Tweety flies”. Another expert may remain indecisive as he cannot ignore the chances that Tweety may be a non-flying bird and he will assign 0.5 (neither true nor false) to the proposition ”Tweety flies”. Others’ assignments may be at some intermediate level depending on their perception about the world. Thus, the experts’ truth assignments collectively construct an interval as the epistemic state of the rule ”Birds fly” as well as of the fact ” Fly(Tweety)”.

Now, suppose an additional information is acquired that:

Penguin(Tweety). [Tweety is a penguin]

Then all the experts will unanimously declare Tweety doesn’t fly and assign an interval as the revised epistemic state of the proposition ”Tweety flies”. The epistemic state of the proposition ” Tweety flies” was first asserted by an interval and later the experts retracted their previously drawn decision to assert another interval . From intuition it can be claimed that the interval makes a more confident and precise assertion than , since in the former case all the experts were unanimous. But this is not reflected in the bilattice-based triangle (Figure 2); since in Triangle and are incomparable in knowledge ordering. Thus, given the two intervals, based on the triangle structure, we remain clueless about which interval has higher degree of knowledge and which interval we should take up as final assertion of ” Tweety flies”. This is counter-intuitive and unwanted.

(a) Default Bilattice
(b) Multivalued Default Bilattice
Figure 3: Default bilattices for Nonmonotonic Reasoning

This type of scenario can be efficiently taken care of with the default bilattice Ginsberg (1988). The general rule ”Birds fly” will be assigned ’dt’, i.e. true by default. Hence, ’Tweety flies’ will also get dt. After acquiring the knowledge that Tweety is a penguin, ’Tweety flies’ is asserted definitely false, i.e. f. In the default bilattice (Figure 3.a) , expressing that the later conclusion is more certain than the earlier one.

The aforementioned example demonstrates that Triangle is incapable of depicting the continuous revision of decisions in absence of complete knowledge. Default bilattice is more appropriate than Triangle with respect to belief revision in nonmonotonic reasoning, but, vagueness or imprecision of information cannot be represented in Default bilattice.

3.1.2 Example from an application domain:

Bilattice-based structures are put to use for logical reasoning involving human detection and identity maintenance in visual surveillance systems by Shet et al. Shet (2007); Shet . (2007, 20061).

Multi-valued default bilattice (also known as prioritized default bilattice (Figure 3.b) has been used for identity maintenance and contextual reasoning in visual surveillance system Shet . (20061)

. However in practice, logical facts are generated from vision analytics, which rely upon machine learning and pattern recognition techniques and generally have noisy values. Thus, in practical applications it would be more realistic to attach arbitrary amount of beliefs to logical rules rather than values such as dt, df etc that are allowed in multivalued default bilattices. For instance, similarity of different persons based on their appearances is a fuzzy attribute and may attain any degree over the [0,1] scale. But this cannot be captured by the multivalued default bilattice.

Bilattice-based square Arieli . (2005) has been used for human detection in visual surveillance system. This algebraic structure is a better candidate than multivalued default bilattice as it provides continuous degrees of belief states.

The difference between bilattice-based square and bilattice-based triangle is that the former allows explicit representation of inconsistent information with different degrees of inconsistency. But it is pointed out by Dubois Dubois (2008) that square-like bilattices, where explicit representations of unknown and inconsistent epistemic states are allowed, can not preserve classical tautologies and sometimes give rise to unintuitive results.

Hence, bilattice-based triangle seems to be the most dependable and suitable algebraic structure to be used in the aforementioned applications.

Now let’s apply the bilattice-based triangle to a slightly modified version of an example demonstrated by Shet et. al. Shet . (20061, 20062) involving logical reasoning in identity maintenance.

Example 2: The example deals with determining whether two individuals observed in an image should be considered as being one and the same. The rules and facts along with the assigned epistemic states are as follows:








The specified set of facts depicts that individuals and are more similar than and . Rules r1 and r2 encode the judgments of two different information sources or different algorithms, none of which present a confident, full-proof answer. However rule r2 (which may be based on some more accurate and highly reliable information) gives greater assurance to the non-equality of two persons than the assertion of equality expressed by rule r1, which may have came from a simple appearance matching technique of low dependability.

Intuitively, from the given information, a rational agent would put more confidence to the fact that individuals are not equal than on their equality; since the degree of distinction is more than the degree of similarity.The inference mechanism is specified in Ginsberg (1988). The closure operator over the truth assignment function denotes the truth assignment that labels information entailed from the given set of rules and facts. The operator takes into account set of rules that entail q and considers set of rules that entail . Here the conjunctor, disjunctor and negator used are min, max and operators respectively.

Now the two intervals and are neither comparable with respect to in Triangle nor they have a in the Triangle structure. Thus the two intervals cannot be combined to get a single assertion for . Hence, using Triangle it is not possible to achieve the intended inference that and don’t seem to be equal.

Thus the knowledge ordering in bilattice-based triangle must be modified in order to remove the aforementioned discrepancy. The modified knowledge ordering must incorporate within Triangle the ability to perform reasoning in presence of nonmonotonicity and demonstrate the repetitive belief revision, as the default bilattice has.

3.2 Truth ordering is not always accurate:

An interval, taken as an epistemic state for a proposition, specifies the optimistic and pessimistic boundaries of the truth value of the proposition. In the bilattice-based triangle intervals are ordered with respect to ordering based on the boundaries of intervals; for two intervals and , iff and . Now with this ordering, any two intervals and are incomparable if is a proper sub-interval of or vice-versa, i.e. if one interval lies completely within the other with no common boundary. The justification behind this incomparability is that, if an interval, say , is a proper sub-interval of then the actual truth value approximated by interval may be greater or less than that of . For instance, if and then can be less than ( if ) or can be greater than (if ).

But similar situation may arise even when two intervals are not proper sub-interval of one another but just overlap, e.g. say and . Yet these intervals are t-comparable, i.e., . As the two intervals overlap, it is not ensured that the real truth value approximated by the lower interval will be smaller than the real truth value approximated by the higher interval (e.g. though but it may be the case that and ). In this respect the comparibility of these two intervals is not justified. Therefore, it is not always the most accurate ordering and can be regarded as a ” weak truth ordering” Esteva . (1994).

The intuitive justification in support for the truth ordering is given as Deschrijver (2009):

iff the probability that is larger than ”       

i.e. the basic intuition behind truth ordering of two intervals and lies in comparing the probabilities and , where, and are the actual truth values approximated by intervals and respectively.

Now, let us check whether this intuition holds good for the aforementioned pairs of intervals and eliminates the anomaly regarding their comparibility. Let us denote the three intervals , and by respectively. Now for the pair of intervals and , lets calculate the probabilities as specified in the right hand side of the iff condition in statement .

Since for an interval any value is equally probable to be equal to

(i.e. there is a uniform probability distribution over

) then for a sub-interval of we have, .

For intervals and , i.e. and respectively,





Now within the overlapped interval , and are equally probable,i.e.


So, .

For intervals and , i.e. and respectively,

and and



Again within the overlapped portion , and are equally probable,i.e.


So, .

Thus it can be seen for pairs of intervals and :


Thus we can see that from the probabilistic perspective the two pairs of intervals behave similarly but the truth ordering treats them differently. Following the intuition of truth ordering, as depicted in statement , both pairs of intervals should be comparable with respect to ordering and it should have been the case that and . However, surprisingly although we have (since and ) but intervals and are not comparable with respect to the truth ordering (since and ).

Thus the truth ordering in bilattice-based triangle is not intuitive when it compares partially or completely overlapped intervals. The incomparability of intervals and with respect to contradicts with the intuition of truth ordering. This may generate unintuitive and problematic results in application areas, specially when reasoning is done in absence of complete knowledge.

Application area where truth ordering fails to perform reasoning:

Artificial intelligence based systems are proposed to be used for medical diagnosis and artificial triage system in emergency wards Golding . (2008); Burke  Madison (1990); Wilkes . (2010). Use of possibilistic answer set programming for medical diagnosis has been reported in Bauters . (2014).

Possibilistic approach is suitable for capturing the uncertainty present in information. For instance, in the framework we can represent and reason with the possibility of a particular disease, given a set of symptoms. But sometimes the severity level of the particular disease,i.e. whether it is in primary stage or advanced stage, is also essential to know. For instance, only knowing a patient has coronary blockage is not sufficient, but whether there is any risk of heart attack or what type of surgery is to be done is dependent on the percentage of blockage of the coronary artery. Again if a patient comes in an emergency ward with stomach ache then the triage nurse would assess the urgency depending on the degree of pain the patient is experiencing. Sometimes the patient is asked to rate his/her pain on a scale of 1 to 10. These are not uncertain quantities but fuzzy quantities. Therefore the truth value of the statement ”Patient A is suffering from disease X” would not be bivalent rather would have to be chosen from a continuous range of values from . This information can not be represented in the Possibilistic ASP, which is essentially based on two valued logic.

In practice, several factors contribute to the severity of a particular ailment. Therefore sometimes it becomes difficult to diagnose a disease with a specific severity degree; rather it is more natural to ascribe a range of values.This may be due to the presence of subjective uncertainties, e.g. fluctuating blood pressure or body temperature or nonspecific rating of a patient’s pain. Another reason may be incomplete knowledge, based on which decisions are being taken. For example, in an emergency situation medical decisions have to be taken rapidly when the doctors can’t wait for all the test results and must make their decisions based on assumptions, rules of thumb and experience. Again, multiple agents (in this case doctors in medical board or different triage nurses) may not be unanimous about a judgment. Thus, the epistemic state of a statement like ’Patient A has disease d1’ would be comprised of an subinterval in , where each individual value in the interval represent an expert’s opinion regarding how true the statement is. The epistemic state of the statement ’Patient A has disease d1’ is same as the suspected severity degree of the disease. When the experts have complete knowledge and there isn’t any uncertainty, they provide unanimous opinion depicted by an exact interval of the form , where the value would signify the severity of the disease. More is the uncertainty wider would be the assigned interval. For example, the pain rated by a patient by the range on a scale of , can be represented by the interval . Therefore when a system is built to perform reasoning with these types of information subintervals of are assigned to simple propositions as their epistemic states denoting the vagueness and uncertainty. For example, the pain rated by a patient by the range on a scale of , can be represented by the interval . Decisions regarding the prescribed treatment would be based on the epistemic states and comparing them.

Bilattice-based triangle structure can be used to reason with such vague and uncertain information. The two natural orderings in bilattice-based triangle, namely knowledge ordering and truth ordering, can be employed to compare respectively certainty and vagueness about different propositions, e.g. severity degree of ailments. In medical decisions the truth ordering may play a more crucial role.

Example 3. Suppose an intelligent triage system has diagnosed a patient with two diseases (say and ). There may be situations where the doctor has to prioritize the treatment of the two diseases based on their severity. This situation may arise when medications for two diseases are mutually incompatible or both of them require surgery that can not be done simultaneously. Thus deciding which treatment is more urgent becomes crucial.

Suppose the artificial triage system has the following rules in its system regarding the two diseases di1 and di2 and its medications (dr1 and dr2):

1. not

2. not

These rules denote the mutual incompatibility of drugs for the two diseases. They suggest that if a patient is diagnosed with disease 1(2) and drug 2(1) is not being administered then drug 1(2) can be given to the patient. Hence when a patient is diagnosed with both of these diseases the triage system must calculate which of the two diseases is more severe and requires urgent medication. Based on the symptoms the severity of the diseases are specified by the intervals and respectively. If the assigned epistemic states, that denotes the level of severity, are exact intervals then simply comparing the assigned value would be enough. But when and are non-exact intervals then it must be decided which among the knowledge ordering and truth ordering has to be used for comparing the intervals to compare the severity. An example can help to clarify the intuition. Suppose, the intervals and are respectively and . Here, as a whole is suspected to be in a more severe condition than , because for each value , is greater than the upper limit of . Hence though the uncertainty regarding is more (since the interval is wider than that of ), but the chance that is more critical is higher than that of and is to be treated before . The truth ordering in the bilattice-based triangle is supposed to capture this intuition and indeed in this case we have (since, ). Thus the triage system must have the following rules to prioritize the treatment considering the urgency or severity of the diseases:



Now suppose. based on the symptoms and test results (which may not be complete) the severity of each disease are asserted and fed into the triage system’s database as follows:



Now, considering the input information the triage system would have to decide which of the drugs among and have to be administered based on the severity.

But now suppose and are respectively and . These two intervals cannot be compared with respect to . Therefore, in this situation, the artificial triage system cannot decide which disease is to be treated first. But this is not intuitive, since, following the similar analysis as demonstrated in the previous subsection it can be proved that it is more probable that the actual severity of (as approximated by the interval ) is higher than that of (as approximated by interval ). Thus, treatment of is more urgent and hence must be administered first.

The bilattice-based triangle is incapable of capturing this notion. Thus the truth ordering in bilattice-based triangle must be modified so that the new ordering behaves as when intervals are non-overlapping and captures the probabilistic essence stated in statement when intervals are overlapping.

4 Modification in Triangle structure:

Based on the discussions in the above two subsections the bilattice-based triangle is modified.

4.1 Modification in knowledge ordering:

The knowledge ordering can be defined based on just the length of intervals and irrespective of the real truth values they attempt to approximate. Thus for two intervals and , where, is the set of sub-intervals of as shown in Figure 1


that is, wider the interval lesser is the knowledge content. Equality of the width of intervals is a necessary condition for , but not a sufficient condition; because two different intervals may have equal width, e.g. and .

Figure 4: I({0,0.25,0.5,0.75,1}) with modified knowledge ordering

The algebraic structure for is shown in Fig. 4.

4.2 Modification in Truth Ordering:

The truth ordering gives rise to certain discrepancies in ordering intervals, as discussed in section 3.2. Lets take statement as a starting point to revisit the truth ordering, especially in case when one interval is a proper sub-interval of the other. In this respect the following theorem is stipulated.

Theorem 4.1.

For two intervals and ,

where, stands for the actual truth value approximated by the interval ; and and are respectively the midpoints of intervals x and y.


The proof is constructed by considering several cases depending on how intervals and are situated on the scale. Without loss of generality it is assumed that for showing the proof. For the other case, i.e. similar proof can be constructed which is not shown here.

Since any is equally probable to be equal to (i.e. there is a uniform probability distribution over ) then for a sub-interval of we have, .

Case 1:

Figure 5: y is a proper sub-interval of x

Suppose, has as a proper sub-interval (Figure 5). For these intervals and , hence and can not be ordered using .

In this case,

1. iff or given ,

2. iff or given .

Within the smaller interval the and are equally probable,i.e.



or given or given

(since )

the midpoint of interval the midpoint of interval


Case 2:

Figure 6: x and y are overlapping

Suppose two intervals and are overlapping, as shown in Figure 6. In this case, and .


1. iff or and or given ,

2. iff given .



(since, )

(cancelling and and rearranging terms)

(since )

(since )

the midpoint of interval the midpoint of interval


Case 3:

Figure 7: x and y are disjoint and

We can have two subcases for disjoint intervals (Figure 7). For subcase a, the interval is lower than the interval , i.e. or in other words . Similarly, for subcase b, the interval is lower than the interval , i.e. or in other words .

In this case, since intervals are disjoint,

and if (Subcase a);

and if (Subcase b);



[since, ]



and and [since intervals are disjoint]



Thus .

Here the proof ends. ∎

Hence, it is proved that the straightforward way to compare the probabilities and for two intervals and is to compare their midpoints. Case 1 in the above proof is particularly interesting, where one interval is a proper sub-interval of the other. Though the chosen intervals and are not comparable with respect to ordering, but we can compare their midpoints and thus order the probabilities and . Thus following statement a truth ordering can be imposed on and based on the probabilistic comparison. The existing truth ordering as shown in Definition 2.2, doesn’t allow this comparability of and , and hence a new truth ordering is called for.

Now that we are able to estimate and order the probabilities, in light of statement

we are in a place to construct a generalised truth ordering as follows:


Figure 8: Intervals incomparable in t-ordering but not equal

The equality of midpoints of two intervals and , (i.e. ) is a necessary condition for , but not a sufficient condition; because two different intervals can have same midpoint, as shown in Figure 8.

Moreover, the discrepancy mentioned in section 3.2 is resolved, since cases where intervals are overlapped and when one interval is a proper sub-interval of the other are treated uniformly and in each case intervals are comparable with respect to .

Theorem 4.2.

For two intervals and , such that none is a proper subinterval of the other,



From the definition,



Thus, the probabilistic analysis gives a broader truth ordering of the intervals that can be achieved by comparing midpoints of intervals. For each pair of intervals if they are comparable with respect to they are also comparable with respect to the modified truth ordering and additionally can order intervals when one of them is a proper sub-interval of the other and hence are not comparable.

For instance, for two intervals and we have though and are not t-comparable w.r.t. .

Figure 9: I({0,0.25,0.5,0.75,1}) with modified truth ordering

The algebraic structure for is shown in Figure 9.

5 An alternative algebraic structure

Based on these modifications we propose a modified and more intuitive algebraic structure for ordering intervals with respect to degree of truth and knowledge (or certainty).

Notation: For an interval ; and will be used to denote the midpoint (or center) and the length of the interval respectively; i.e. and . The pair uniquely specifies an interval and hence may be used instead of the traditional representation .

Definition 5.1.

A preorder-based triangle is a structure , defined for every and as: