# Set theory and logic and their dual

(1) Set theory and logic, (2) number and algebra, and (3) space and time are three foundational topics that each have duals. Let us begin with the standard approaches to these three topics, and then define duals to each of them. To some extent, the original and the dual may be used together.

Set theory and logic

A set is defined by its elements or members. Its properties may also be known or specified, but what is essential to a set is its members, not its properties. The notation for “x is an element of set S” is “x ∈ S”. A subset is a set whose members are all within another set: “s is a subset of S” is “s ⊆ S”. If subset s does not (or cannot) equal S, then it is a proper subset: “s ⊂ S”.

The null set (∅) is a unique set defined as having no members. That is paradoxical but not contradictory. A universal set (Ω) is defined as having all members within a particular universe. An unrestricted universal set is not defined because it would lead to contradictions.

The complement of a set (c) is the set of all elements within a particular universe that are not in the set. A union (∪) of sets is the set containing all members of the referenced sets. An intersection (∩) of sets is defined as the set whose members are contained in every referenced set.

Set theory has a well-known correspondence with logic: negation (¬) corresponds to complement, disjunction (OR, ∨) corresponds to union, and conjunction (AND, ∧) corresponds to intersection. Material implication (→) corresponds to “is a subset of”. Contradiction corresponds to the null set, and tautology corresponds to the universal set.

Class theory

The dual to set theory is what I call “class theory”. A class is defined by the attributes or properties it includes. Its instances may also by known or specified, but what is essential to a class is its attributes. Generally, the more attributes, the fewer instances, and the fewer attributes, the more instances.

The notation for “x is an instance of class C” is “x ⋉ C”. A subclass is a class whose instances are all within another class: “c is a subclass of class C” is “c ⊑ C”. If subclass c does not (or cannot) equal C, then it is a proper subclass: “c ⊏ C”.

The universal class (Ω) is a unique class defined as having all instances. That is paradoxical but not contradictory. A null class (∅) is defined as having no instances within a particular universe. An unrestricted null class is not defined because it would lead to contradictions.

The complement of a class (c) is the class of all instances within a particular universe that are not in the class. A union (∪) of classes is the class containing all instances of the referenced classes. An intersection (∩) of classes is defined as the class whose instances are contained in every referenced class.

Class theory also has a correspondence with logic: negation (¬) corresponds to complement, disjunction (OR, ∨) corresponds to intersection, and conjunction (AND, ∧) corresponds to union. Material implication (→) corresponds to “is a subclass of” (⊑). Contradiction corresponds to the universal class, and tautology corresponds to the null class.

Together

Set theory and class theory can be used together within a particular universal set and null class. Each set has its associated class of properties, and each class has its associated set of members. There is an inverse correspondence between set intersection and class union, set union and class intersection.

For example, the more properties a class has, the fewer members of its associated set. The fewer properties a class has, the more members of its associated set.

For (2) see here and for (3) see here. Revised 2023-06-29.