Review of “Logic, Logic and Logic”
A Biannual Journal
Vol. 1 No. 2
Essays in Philosophy
Cancer robbed our community of an outstanding philosopher. One is tempted to say “philosopher and logician,” but as Richard Cartwright remarked in his eulogy for George Boolos, “he would have not been altogether happy with the description: accurate, no doubt, but faintly redundant-a little like describing someone as 'mathematician and algebraist.'” While the title of this book is redundant, its content is not. Completed by colleagues, students and friends, this posthumous collection is not only a testament to Boolos' legacy but also to his logical virtuosity. George Boolos made significant contributions in every area of logic in which he worked. The volume is a treasure trove of insightful observations and elegant formal contributions to central questions in the philosophy of logic. The book is divided into three sections: Studies on Set Theory and the Nature of Logic, Frege Studies, and Various Logical Studies and Lighter Papers. The first section contains Boolos' seminal essay “The Iterative Conception of Set,” (article 1) published in the early 1970s, which introduced philosophers to the underlying intuitive conception of set articulated by Zermelo's axioms. Most philosophers had regarded these axioms as ad hoc attempts to avoid Russell's, Burali-Forti's, and other set-theoretical paradoxes, but Boolos demonstrates how a core of Zermelo's axioms (excluding the Axioms of Choice and Replacement) can be logically deduced from a set of axioms characterizing the idea of the formation of sets in stages over time. Article 2 contains Boolos' defense of Fraenkel's, in contrast to Zermelo's, position that first-order but not second-order logic is applicable to set theory. Boolos criticizes the view of Charles Parsons (and D.
-
A. Martin) that it makes sense to use second-order quantifiers when first-order quantifiers range over
entities that do not form a set. Boolos’ answer to the title of article 8, “Must We Believe in Set Theory?”
is ‘no’: the phenomenological argument (due to Gödel) does not imply that the axioms of set theory
correspond to something real, and the indispensability argument (due to Carnap) that mathematics is
required by our best physical theory, is dismissed as “rubbish.” Boolos laments not having time to
discuss the predicative set theory of Solomon Feferman and cabalistic views of Penelope Maddy (and D.
A. Martin, John Steel, Robert Solovay, and W. Hugh Woodin).
The first section also contains Boolos’ papers on the logic of plurals and second-order logic.
Secondorder, unlike first-order, logic can express such mathematically useful notions as coextensiveness,
equinumerousity, and the ancestral. Nevertheless, many logicians argued that second-order logic should
not be regarded logic because the failure of the completeness theorem cast doubts on the claim that it
could serve as a model of deductive reasoning. The question of what is and is not logic aside, Boolos
examines the issues of whether second-order logic is of use in formalizing many natural language
sentences that cannot be captured in first-order logic such as the following:
(1) Some relative of each villager and some relative of each townsperson hate each other (Hintikka).
(2) The richer the country, the more powerful is one of its officials (Barwise).
(3) Some critics admire only one another (Kaplan).
Boolos’ plural quantifiers: ‘some things, the U’s are such that…” adds to the expressive vocabulary of
logic. Plural quantifiers provide an alternative interpretation of second-order logic in which second-order
entities are subsets of the universe. In his excellent introductory remarks to the various sections, John
Burgess points out that plural quantification is limited by the fact that plurals provide a monadic, but not
dyadic, second-order quantification.
Section II contains Boolos’s work in forging a new direction in Frege studies. Boolos claims that when
Frege was confronted with the derivation of Russell’s paradox from his Basic Law V, he “grievously
undervalued his actual achievement” and mistakenly regarded the paradox as invalidating the whole of
his formal work. Instead, Boolos’ work vindicates Frege. Boolos claims that Frege deserves a place in
the philosophical pantheon beside Descartes, Leibniz, and Kant: “What is sad is not so much that Frege’s
system turned out to be vulnerable to Russell’s paradox as that both he and we failed to realize how
valuable his actual accomplishment was. Frege proved the first great theorem of logic: arithmetic can be
derived from the number principle.”
Boolos (with student Richard Heck) shows that arithmetic can be logically derived from Hume’s
Principle. Frege’s Basic Law V, which leads to the inconsistency of Russell’s paradox,
(Vb) "F "G(#F = #G « "x(Fx « Gx)),
is formalistically similar to the consistent number principle, dubbed “Hume’s Principle” because Frege
cited a passage from Hume’s Treatise (Book I, Part III, Section I) (...truncated)