Sets vs. Boolean Algebra

Classical propositional logic and set theory are often considered to be two instances of Boolean algebra, with set union corresponding to logical or, and set intersection corresponding to logical and. However, this does not reflect the logical structure of set theory. Thus, any set may be considered as the union of one-element sets, the sense of this representation depending on interpretation:

• enumeration (strong union): a set is treated as element a and element b and element c and ...; this makes a simultaneous interpretation of set as actual integrity;

• sampling (weak union): a set is treated as element a or element b or element c or ...; this interpretation stresses the idea of potential integrity, referring to the operation of "probing" the set by random selection of one or another element.

On the other hand, enumeration is algorithmic, in the sense that one is supposed to be able to construct the set being given its elements; on the contrary, the sampling technique refers to the quality of the set, the properties of the elements that make them belong to the set. Compare:

• categorization by convention: let us refer to Mr. and Mrs. Jones as the Jones family;
• explanation by example: the vegetables are... well... the carrot, the cucumber, the onions, and like.

Similarly, the two types of definition:

• by construction (explicit): numeric data types include integer, real, and date-time;
• by function (implicit): let all the real numbers x < 0 be called negative.

8 Jan 2000