# Decision Rules

• Fábio Romeu de Carvalho
• Jair Minoro Abe
Chapter
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 87)

## Abstract

This chapter gives in outline some concepts regarding the lattice associated to the paraconsistent annotated evidential logic Eτ such as certainty degree and uncertainty degree. Also, it is introduced the concept of control values (requirement level) and decision rules.

## Keywords

Level Requirements Certainty Degree Uncertainty Degree Paraconsistent Annotated Logic Favorable Evidence
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## 3.1 General Considerations

A convenient division of lattice τ is seen in Fig. 3.1, in which the unit square is divided into twelve regions. Out of these, the four regions denominated extreme regions are highlighted, which will be object of more detailed analysis.
In this division of τ, we highlight the segments AB, called perfectly undefined line (PUL), and CD, called perfectly defined line (PDL). For a given annotation constant (a; b), we will define the
• degree of uncertainty, by the expression $${\mathbf{G}}\left( {\varvec{a};\varvec{b}} \right) = \varvec{a} + \varvec{b} - 1$$ (proportional to the distance from the point that represents it to the PDL); and also, the

• degree of certainty, by the expression $${\mathbf{H}}\left( {\varvec{a};\varvec{b}} \right) = \varvec{a} - \varvec{b}$$ (proportional to the distance from the point that represents it to the PUL).

For points X = (a; b) next to A, the values of the degree of favorable evidence (or degree of belief) (a) and of the degree of contrary evidence (or degree of disbelief) (b) are close to 0, characterizing a region of paracompleteness (AMN); next to B; on the contrary, the values of a and b are close to 1, characterizing a region of inconsistency (BRS); in the surroundings of C, the values of a are close to 1 and the values of b are close to 0, defining a region with high degree if certainty (close to 1), called truth region (CPQ); and, finally, in the proximities of D, the values of a are close to 0, and the values of b are close to 1, defining a region with low degree of certainty (close to −1), but high in module, called falsity region (DTU).

Also defined, are:
• Paracompleteness limit line: line MN, so that $$\text{G = } -\text{k}_{\text{1}} = - 0.70,\;{\text{where}}\; 0{\text{ < k}}_{1} , < 1;$$

• Inconsistency limit line: line RS, so that $${\text{G = k}}_{1} = 0.70,\;{\text{where}}\; 0{\text{ < k}}_{1} < 1;$$

• Falsity limit line: line analyses for decision making; for that reason, it, so that $${\text{H = }} -{\text{k}}_{2} = - 0.70,\;{\text{where}}\; 0{\text{ < k}}_{2} < 1;$$

• Truth limit line: line PQ, so that $${\text{H = k}}_{2} = 0.70,{\text{where}}\,\, 0{\text{ < k}}_{ 2} ,< 1.$$

Except for contrary reference, in this book, k1 = k2 = k will be adopted, giving symmetry to the chart, as in Fig. 3.1, in which k1 = k2 = k = 0.70. The value of k2 will be called requirement level (control value), as it will be seen further ahead.

As seen in the previous chapter, four extreme regions and one central region are highlighted in Fig. 3.1.
\begin{aligned} & {\mathbf{AMN}}\,{\text{Region}}\text{:} - 1.\text{0} \le \text{G} \le - \text{0}\text{.70} \Rightarrow {\text{paracompleteness}}\,{\text{region}} \\ & {\mathbf{BRS}}\,{\text{Region:}}\, 0. 7 0\le {\text{G}} \le 1. 0\Rightarrow \,{\text{inconsistency}}\,{\text{region}} \\ \end{aligned}
In these regions, we have situations of high indefinition (‘very’ paracomplete or ‘very’ inconsistent). Therefore, if point X = (a; b), which translates a generic situation in study, belongs to one of these regions, it will be said that the data present a high degree of uncertainty (paracompleteness or inconsistency).
\begin{aligned} & {\mathbf{CPQ}}\,{\text{Region:}}\, 0. 7 0\le {\text{H}} \le 1. 0\Rightarrow {\text{truth}}\,{\text{region}} \\ & {\mathbf{DTU}}\,{\text{Region:}}\, - 1.0 \le {\text{H}} \le - 0. 7 0\Rightarrow {\text{falsity}}\,{\text{region}} \\ \end{aligned}

In contrast to the previous ones, in these regions, we have situations of high definition (truth or falsity). Therefore, if point X = (a; b), which translates a generic situation in study, belongs to the region CPQ or DTU, it will be said that the situation presents a high degree of favorable certainty (truth) or contrary certainty (falsity), respectively.

The first one, CPQ, is called favorable decision region (or feasibility), as when the point that translates the result of the analysis belongs to it, it means that the result presents a high degree of favorable evidence (degree of belief) and low degree of contrary evidence (degree of disbelief). That results in a high degree of certainty (close to 1), which leads to a favorable decision, translating the feasibility of the enterprise.

The second one, DTU, is called unfavorable decision region (or unfeasibility), as, belonging to this region, the result presents a low degree of favorable evidence and high degree of contrary evidence. That results in a low degree of certainty (close to −1), which leads to an unfavorable decision, translating the unfeasibility of the enterprise.
\begin{aligned} & {\mathbf{MNTUSRQP}}\,{\text{Region}} \\ & \left| {\text{G}} \right| < 0.70 \Rightarrow - 0.70 < {\text{G < 0}} . 7 0\,{\text{and}}\,\left| {\text{H}} \right| < 0.70 \Rightarrow - 0.70 < {\text{H < 0}} . 7 0\\ \end{aligned}

This is a region that does not allow highlighted conclusions, that is, when the point that translates the result of the analysis belongs to this region, it is not possible to say that the result has a high degree of certainty or uncertainty. This region translates only the tendency of the analyzed situation, according to the considered decision states (see Sect.  and Table ).

This way, the favorable decision (feasibility) is made when the point that translates the analysis result belongs to the truth region (CPQ); and the unfavorable decision (unfeasibility), when the result belongs to the falsity region (DTU).

## 3.2 Requirement Level and the Decision Rule

As a result of the considerations made in the previous section, for the configuration in Fig. 3.1, we may enunciate the following decision rule [45]:

\begin{aligned} & {\mathbf{H}}{ \ge }{\mathbf{0}}.{\mathbf{70}} = {\mathbf{favorable}}\, \, {\mathbf{decision}} \, \left( {{\mathbf{the}}\, \, {\mathbf{enterprise}}\, \, {\mathbf{is}}\, \, {\mathbf{feasible}}} \right); \\ & {\mathbf{H}} \le {-}{\mathbf{0}}.{\mathbf{70}} = {\mathbf{unfavorable}}\, \, {\mathbf{decision}} \, \left( {{\mathbf{the}}\, \, {\mathbf{enterprise}} \, \,{\mathbf{is}} \, \,{\mathbf{unfeasible}}} \right); \\ & {-} \, {\mathbf{0}}.{\mathbf{70}} \, < \, {\mathbf{H}} \, < \, {\mathbf{0}}.{\mathbf{70}} = {\mathbf{inconclusive}} \, \,{\mathbf{analysis}}. \\ \end{aligned}

Observe that, for this configuration, the decision (favorable or unfavorable) is only made when |H| ≥ 0.70, or that is, when k2 = 0.70. Hence, this value (k2 = 0.70) represents the lowest value of the degree of certainty module for which a decision is made. For that reason, it is here denominated requirement level (NE) of the decision. With that, the decision rule, represented in the most generic form, is this way:

\begin{aligned} & {\mathbf{H}} \ge {\mathbf{NE}} = {\mathbf{favorable}}\,{\mathbf{decision}} \, \left( {{\mathbf{the}}\, \, {\mathbf{enterprise}} \, \,{\mathbf{is}} \, \,{\mathbf{feasible}}} \right); \\ & {\mathbf{H}} \le \, {-}{\mathbf{NE}} = {\mathbf{unfavorable}}\, \, {\mathbf{decision}} \, \left( {{\mathbf{the}} \, \,{\mathbf{enterprise}} \, \,{\mathbf{is}}\, \, {\mathbf{unfeasible}}} \right); \\ & {-} \, {\mathbf{NE}} < {\mathbf{H}} < {\mathbf{NE}} = {\mathbf{inconclusive}} \, \,{\mathbf{analysis}}. \\ \end{aligned}

It is appropriate to highlight that the requirement level depends on the safety, on the trust that you wish to have in the decision, which, in turn, depends on the responsibility it implicates, on the investment at stake, on the involvement or not of risk to human lives, etc.

If we want a stricter criterion for the decision making, that is, if we want safer, more reliable decisions, it is required to increase the requirement level, that is, we must approximate lines PQ and TU of points C and D, respectively.

Observe that, if the result belongs to the BRS region (inconsistency region), the analysis is inconclusive regarding the feasibility of the enterprise, but it points out to a high degree of inconsistency of the data (G ≥ 0.70).

Analogously, if the result belongs to the AMN region (paracompleteness), it means that the data present a high degree of paracompleteness or, equivalently, high lack of information about the data (G ≤ –0.70).

In these cases, therefore, the result does not enable a conclusion regarding the feasibility of the enterprise, but it allows us to conclude that the database, which will often be constituted by experts’ opinions, presents a high degree of uncertainty (paracompleteness or inconsistency). Therefore, it allowsus, at least, to have information regarding the degree of uncertainty of the elements contained in the database. This is a great advantage of using the paraconsistent annotated evidential logic Eτ, which manages to handle data, even if they are provided with paracompletenesses or inconsistencies (or contradictions).

As we may see, the application of the paraconsistent logic techniques enables us to determine possible inconsistencies of the database and verify to what extent they are acceptable or not in decision making.

The importance of the analysis of a real situation through MAX and MIN operators lies in the fact that it, even if the analyzed conditions present contradictory results, they are taken into account. That means that this method accepts databases that present contradictions, or paracompletenesses, that is, it manages to deal with situations provided with uncertainties, as long as they are not trivial. This is the great merit of the paraconsistent annotated logics.

It is verified, then, that lattice τ with the division into twelve logical decision states (Fig. 3.1) enables analyses for decision making; for that reason, it was considered to para-analyzer algorithm (or device) [37, 41].