# What does $\alpha_r\supset\beta_r$ mean in the tolerance definition of System Z?

I'm reading up on System Z introduced by Judea Pearl (in System Z: a natural ordering of defaults with tractable applications to nonmonotonic reasoning). A central definition is that of tolerance of a subset $R'$ of a rule set $R$ for a rule $r = \alpha_r\to\beta_r$ (denoted by $T(r\ \vert\ R')$). Tolerance of $R'$ for $r$ is defined to be the set of satisfiable formulas $$(\alpha_r\land\beta_r)\bigcup_{r'\in R'}(\alpha_{r'}\supset\beta_{r'})$$ (see page 2 of the article by Pearl).

I don't understand what $\alpha_{r'}\supset\beta_{r'}$ is supposed to mean. Can anyone explain?

• What's the definition? – Raphael Oct 21 '15 at 16:28

## 1 Answer

In elementary logic, $A \supset B$ is notation for the formula $\neg A \lor B$ (for the implication "$A$ implies $B$"). In other words, $A \supset B$ is a formula; it is true if $A$ is false or if $B$ is true, and false otherwise. https://math.stackexchange.com/q/1106001/14578

How you could have figured this out on your own: Read the entire paper. For instance, a useful place would be to start would be reading the examples in Section 3, where from the examples one can reverse-engineer the probable meaning of the symbol.

• Thank you for the first part of your answer. I don't see why you want to lecture me on asking this question on cs.stackexchange rather than reading the whole paper and guessing the meaning of the notation. It's much better to ask someone who knows than guessing. – FK82 Oct 24 '15 at 9:51