Definition:Universal Negative/Set Theory

Definition
The universal negative $\forall x: S \left({x}\right) \implies \neg P \left({x}\right)$ can be expressed in set language as:


 * $\left\{{x: S \left({x}\right)}\right\} \implies \left\{{x: \neg P \left({x}\right)}\right\} = \varnothing$

or, more compactly:


 * $S \subseteq \complement \left({P}\right)$

Also defined as
Some sources give this rule as:


 * $S \cap P = \varnothing$


 * 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