Willard Van Orman Quine

276 joseph s. ullian
a special hypothesis for the purpose at hand. Typical theorems thus
have their own comprehension premises, letting us see clearly just
what is required for their proof. Even the existence of infinite sets
needs hypothesization, since the battery of adopted axioms does not
deliver it. Quine has called the book’s treatment “pedagogically my
preferred approach to the subject,” identifying his “expository objectives:
clarity, elegance, and congenial philosophical perspective” (RW
645). The book’s explicitness lies at the other end of the spectrum
from the usual presentation of second-order logic, with its covert
assumptions of existence.
The final third of Set Theory and Its Logic surveys and compares
five set theories, all but MK of the theories mentioned together earlier.
Quine makes comparisons in terms of predicativity, safety, and
strength; issues of relative consistency are treated masterfully.
Quine wrote some twenty books, six of which are treatises entirely
devoted to symbolic logic and set theory and another of which
is a collection of logic papers culled from his enormous output of
technical articles. There are scores of innovations and insights: an
ingenious inversion trick for defining ‘natural number’, a definition
of ordered pair on which everything is one, a penetrating analysis of
ω-consistency, reduction of logic’s basis to inclusion and abstraction,
a host of proof procedures, and a rendering of Gödel’s Incompleteness
Theorem by way of concatenation theory. This only scratches the
surface. Quine’s contribution to the field is not to be measured in
terms of proved theorems, proposed theories, or striking simplifications,
however many there were of each. Much of what he taught us
has come to be taken for granted, as he has been one of the dominant
influences on students of logic for more than half a century. He
helped to shape the field of mathematical logic as much as anyone.
He showed us patiently and caringly how to see the subject, how to
express ourselves in it, and how to connect it with the rest of what
we would know.
II
Quine wrote that “the pedagogical motive has dominated my work
in logic” (RW 644). He had found no suitable text for teaching formal
logic in 1936, when he began his instructorship at Harvard. This
started him on a “project...ofpedagogical engineering” that led first
to his Elementary Logic (1941) and subsequently to the four editions
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