Rule complex

In this setting, a rule is type of knowledge (in the sense of epistemic logic (see Fagin, 2003)) formalized as a set of premises or conditions, a set of justifications, and a set of conclusions (this may be written as a triple, a rule ${\displaystyle r:=(X,Y,\alpha )}$). Elements of X should hold, and of Y may hold. If Y, the justifications, do not hold, then the rule cannot be applied. If X, the premises, obtain and the justifications are not known to not apply, then the rule is applied, and ${\displaystyle \alpha }$ is concluded. If X and Y are empty, then the rule is axiomatic (a "fact" or unconditional directive). Thus, rules can be seen as the basic objects of knowledge.

Formally, a rule complex is the class which contains all finite sets of rules, is closed under set-theoretical union and power set, and preserves inclusion:

1. Any finite set of rules is a rule complex;
2. If ${\displaystyle C_{1},C_{2}}$ are rule complexes, then ${\displaystyle C_{1}\cup C_{2}}$ and ${\displaystyle P(C_{1})}$ are rule complexes;
3. If ${\displaystyle C_{1}\subseteq C_{2}}$ and ${\displaystyle C_{2}}$ is a rule complex, then ${\displaystyle C_{1}}$ is a rule complex.

This means that for rule complexes ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$, ${\displaystyle C_{1}\cap C_{2},C_{1}-C_{2}}$ are also rule complexes. A complex ${\displaystyle B}$ is a subcomplex of the complex ${\displaystyle A}$ if ${\displaystyle B=A}$ or ${\displaystyle B}$ may be obtained from ${\displaystyle A}$ by deleting some rules from ${\displaystyle A}$ and/or redundant parentheses (Burns, 2005).