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: the set is considered as element a and
element b and element c and ...
This is the simultaneous interpretation of set as an actual
integrity - "strong" union.
-
Exhausting: the set is considered 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 of the elements - "weak" union.
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; the probing technique, inversely, refers to the
quality of the set, the properties of the elements that
makes them belong to the set.
Compare:
-
the scheme of categorization by convention:
"Let us refer to Mr. and Mrs. Jones as the Jones family."
-
the scheme of explanation by example: "The
vegetables are... well... the carrot, the cucumber, the onions, and like."
Also the two types of definition:
-
by construction (explicit): "Numeric data types include integer,
real, and date-time."
-
by function (implicit): "Let all real numbers x<0 be called negative."
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