Definition:Universal Negative/Set Theory

Definition
The universal negative $\forall x: \map S x \implies \neg \map P x$ can be expressed in set language as:


 * $\set {x: \map S x} \implies \set {x: \map P x} = \O$

or, more compactly:


 * $S \subseteq \map \complement P$

Also defined as
Some sources give this rule as:


 * $S \cap P = \O$


 * There are no objects which are $S$ which are also $P$.

This is justified from Empty Intersection iff Subset of Complement.

The advantage to this approach is that it allows the complete set of categorical statements to be be defined using a combination of set intersection and set complement operators.

Also see

 * Equivalence of Definitions of Universal Negative


 * Definition:Square of Opposition


 * Definition:Universal Affirmative/Set Theory
 * Definition:Particular Affirmative/Set Theory
 * Definition:Particular Negative/Set Theory