Meaning of "Monotone" in Monotone Disjunction

Question

I'm trying to understand and clarify what "Monotone" means in Monotone disjunctions.

A Disjunction is the result of a logical OR operation on a set. Monotone means "no negation". I'm interpreting that to mean that members of the set are booleans and have the values "is of class A" or "is not of class A". Wouldn't the latter constitute as negation since it asserts negation of the classification of A?

Answer 1

Disjunction is a monotone operator, because if you have a disjunction $A\lor B$ and change one of the inputs from one truth value to the other (while keeping the other one constant) then the truth value of the entire disjunction cannot possibly change in the opposite direction.

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The set of slides you link to speaks of "monotone disjunctions" in a particular AI context, and the terse woirding "no negations" makes sense only in this implicit context. It is common in this area to consider disjunctive clauses of the form $$ x \lor y \lor \neg z \lor w \lor \cdots \lor \neg u $$ -- that is, a disjunction of input variables and/or negations of input variables. The slides assume that the reader is used to working with such things and then tries to express something like

When we say we're trying to learn a "monotone disjunction" we mean an expression of the form that you're familiar with, but with the additional restriction that none of the variables can be negated.

Such an expression defines a truth function that is monotone in the above sense: changing one of the inputs cannot make the output change in the opposite direction.