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

Sources

• 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (next): Appendix $\text{B}$: The Algebra of Classes