[FOM] Concerning definition of formulas

B. Sidney Smith bsmith7 at radford.edu
Thu Oct 4 22:08:42 EDT 2007

None of the responses I've seen on this thread seem to me to do justice to
the central worry, which is the semantics of predicate logic.  If the
quantifiers are understood in the usual set-theoretic way, how are we
justified in formulating set theory in predicate logic?

Hilbert's (and Ackerman's) answer was that quantification should be
understood in terms of a primitive logical "choice" function, the so-called
epsilon operator.  Thus Exf(x) is understood to assert that a witness may be
chosen for f, and Axf(x) asserts that a witness may not be chosen for
neg(f).  This "natural" predicate logic, the logic that every cognitive
agent seems to use in practice, forms the intuitive basis for formal
(idealized) predicate logic as applied to idealized domains (such as pure

This is far from solving all metaphysical quandaries, but at least it
tackles the issue head-on.  It also suggests that we might not settle the
issue without addressing the role of the logical "agent" in the metaphysic.


Dr. B. Sidney Smith
Dept. of Math & Stats
Radford University

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