Plural Logic: Second Edition, Revised and Enlarged

by Alex Oliver, Timothy Smiley

Alex Oliver and Timothy Smiley provide a natural point of entry to what for most readers will be a new subject. Plural logic deals with plural terms ('Whitehead and Russell', 'Henry VIII's wives', 'the real numbers', 'the square root of -1', 'they'), plural predicates ('surrounded the fort', 'are prime', 'are consistent', 'imply'), and plural quantification ('some things', 'any things'). Current logic is its terms stand for at most one thing. By contrast, the foundational thesis of this book is that a particular term may legitimately stand for several things at once; in other words, there is such a thing as genuinely plural denotation. The authors argue that plural phenomena need to be taken seriously and that the only viable response is to adopt a plural logic, a logic based on plural denotation. They expound a framework of ideas that includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions,
and lists. A formal system of plural logic is presented in three stages, before being applied to Cantorian set theory as an illustration.

Technicalities have been kept to a minimum, and anyone who is familiar with the classical predicate calculus should be able to follow it. The authors' approach is an attractive blend of no-nonsense argumentative directness and open-minded liberalism, and they convey the exciting and unexpected richness of their subject. Mathematicians and linguists, as well as logicians and philosophers, will find surprises in this book.

Genres: Logic

398 pages, Paperback

First published January 1, 2013

Reviews

[Krispy_Pages · Rating 5 out of 5]

This book is something of a tour de force of mathematical/linguistic analysis aimed at correcting the singularist orthodoxy by demonstrating how pervasive a feature of language and reasoning plurality is. Technicalities have been kept to a minimum, anyone familiar with the classical predicate calculus (∃y Ɐx (x ∈ y <-> x ∉ x)) should be able to follow along.

I found this rare gem while randomly browsing in the Kindle bookstore. It certainly brought back all the bitter sweet memories I had while taking PHIL 220. Plural quantification has become such a powerful tool in philosophy of logic, mathematics and linguistic. This book gives a healthy appreciation of just how pervasive and varied the phenomenon of plurality is in natural language, and how nominalism simply won’t do.

For example, the book is primarily concerned with the fact that these two sentences are not equivalent:

“Kris and Bob met.”
“Kris met and Bob met.”

Met who?
Met each other? Met different people? Met the same person? Met oneself?

When singularists attempt to address multiple things at once, they create this ambiguous interpretation. The sentences above contain collective predicates, which apply to their arguments collectively, not individually. As a result, the theory of plural quantification stated that plurals cannot be satisfactorily analyzed in terms of the singular. Thus, plural reference, plural quantification, and plural predication must be recognized as primitive. They can then form part of genuinely plural logics.

One may ask why is this even important? Are the authors making a mountain out of a molehill? Well first, semanticists would like to have the most adequate semantics for plurals. Second, plural logic became a popular tool in the philosophy of mathematics and metaphysics as some system of plural logic has the expressive and deductive power of monadic second-order logic. In addition, different from second-order logic, plural logic appears to be ontologically innocent.

The author’s approach is an attractive blend of no-nonsense argumentative directness and open-minded liberalism, and they convey the exciting and unexpected richness of their subject. Mathematician and linguists, as well as logicians and philosophers.

Personally I feel like reading chapters 5 to 10 was a delight as the author gradually establish grounds for the preferred approach to plural logic by evaluating manifestations of plurality in mathematical practice. The book examined various types of plural denoting terms, at plural predictions, at plural definite description and at multivalued functions. They reached a conclusion that plural logic is more philosophically respectable when taken at face value.

However, the authors appear to embrace a form of logical monism: the existence of a single topic-neutral logic that is preferable regardless of our purpose. Such a view is challenged frequently and I believe it deserves some motivation. Also, the authors clearly used the analysis of linguistic as the main source of evidence to motivate a choice of logic. Why should the need of linguistic be prioritized over the needs of other subjects (math/metaphysics etc.…) influenced by our choice of logic? It would have been nice if they had provided reasons why the precise connection between natural language and logic is of utmost importance. Regardless, I enjoyed the book and the find the ideas fairly interesting.