Page 4 - Boundedness Revisited

The validity of a supposed or hypothesized inside can be
determined by analyzing its definition. If the principle of
boundedness is implicitly contained in that definition, the inside in
question is valid. If, on the other hand, that definition denies the
principle, the inside is invalid. Although validation of an inside by
boundary analysis provides no verification of or certainty about its
contents, the invalidation of an absolute nonfictional boundary
(see below) also invalidates the definition of the content of its
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inside or outside. The violations of boundedness fall into two
categories: insides not properly closed, and boundaries not
properly fictional.

1. Unclosed insides
Apart from insides improperly distinguished by virtue of “holes”
in their boundaries or by requiring non-finite extension in some
dimension, two other species of unclosed insides may be noted:
the paradoxical and the irrational. A definition of one of these
types, when superficially examined, provides an adequately closed
inside; complete analysis, however, reveals that the definition
destroys closure by “moving” the boundary in a way which
prevents the inside from being clearly distinguished from its
outside.

a. Paradoxical insides
This type of definition attempts to describe an inside, part or all
of which is outside itself. The result is a boundary appearing in
two places at once and an inside of varying inclusiveness. These
definitions reduce to “X or part of X is outside of X.” Once this
reduction has been made, the invalidity is obvious.
Two well-known paradoxes serve as examples. “This statement
is a lie” reduces to “‘this statement’ is both inside and outside ‘this
statement is a lie.’” “Who shaves the barber in a town where every
man shaves himself except those shaved by the barber?” reduces
to “a town contains two groups: those not shaving themselves and

dissolution, leaving the self-evident principle of boundedness or “law of the
excluded middle” as the only useful means of truth-evaluation.

7 See footnote 2, above.
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