Default Logic Rule

A Default Logic Rule is a production rule [math]\displaystyle{ delta }[/math] of the form [math]\displaystyle{ \frac{\alpha : \beta_1, ..., \beta_n}{\gamma} }[/math] where [math]\displaystyle{ \alpha, \beta_1, ..., \beta_n, \gamma }[/math] are first order formulae and [math]\displaystyle{ n \ge 1 }[/math], [math]\displaystyle{ alpha }[/math] is called the default prerequisite, [math]\displaystyle{ \beta_1, ..., \beta_n }[/math] are the default justifications, and [math]\displaystyle{ \gamma }[/math] is the default consequent.

• AKA: Assumption-based Default Rule.
• Context:
  • It can be interpreted to as: if [math]\displaystyle{ \alpha }[/math] is known, and if it is consistent to assume [math]\displaystyle{ \beta_1, ..., \beta_n }[/math] then conclude [math]\displaystyle{ \gamma }[/math].
  • It can range from being an Open Default Rule to being a Closed Default Rule, depending on whether [math]\displaystyle{ \alpha, \beta_1, ..., \beta_n, \gamma }[/math] have [[Free

Variable]]s.

• • It can be used to draw Plausible Conclusions in the face of Incomplete Information.
  • …
• Example(s):
  • a Default Subsumption Relation (which are only transitive by default).
• See: Default Theory.

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References

2012

• http://en.wikipedia.org/wiki/Default_logic#Syntax_of_default_logic
  • QUOTE: [math]\displaystyle{ D }[/math] is a set of default rules, each one being of the form: :[math]\displaystyle{ \frac{\text{Prerequisite : Justification}_1, \dots , \text{Justification}_n}{\text{Conclusion}} }[/math] According to this default, if we believe that [math]\displaystyle{ Prerequisite }[/math] is true, and each of [math]\displaystyle{ Justification_i }[/math] is consistent with our current beliefs, we are led to believe that [math]\displaystyle{ Conclusion }[/math] is true.

    The logical formulae in [math]\displaystyle{ W }[/math] and all formulae in a default were originally assumed to be first-order logic formulae, but they can potentially be formulae in an arbitrary formal logic. The case in which they are formulae in propositional logic is one of the most studied.

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