Alternative Logics: A Book Review

A few years ago, Bill Gasarch, editor of the Book Review Column, sent me a copy of the book Alternative Logics: Do Sciences Need Them? [28], edited by Paul Weingartner, and asked if I was interested in reviewing it. The book is a collection of essays discussing whether there is a need for logics other than classical logic in various areas of science. Interestingly, I had just finished reading a copy of philosopher Susan Haack’s Deviant Logic [13], concerning the philosophical foundation of alternative logics. This was fortuitous, because Haack’s monograph provided a wonderful introduction to the essays collected in Weingartner’s volume. Let me start by giving a sense of the main question that underlies both books. Generally speaking, when one thinks of logic, one thinks of classical first-order predicate logic. (Throughout, I will take propositional logic as a sublanguage of first-order predicate logic.) Mathematicians sometimes need to go up to second-order, that is, allowing quantification over predicates, in order to express set theory (at least, Zermelo-Fraenkel set theory) and thereby most of modern mathematics. However, at various points in time, philosophers, mathematicians, and scientists have advocated using logics different from classical logic, arguing that the latter was not always appropriate. As a first example, take intuitionism. Intuitionism [16], a philosophical position about the meaning of mathematics, has profound implications on logic as a framework to express mathematics. Without describing intuitionism in depth, one common feature of intuitionism is that it is constructive. As a consequence, the law of excluded middle is often rejected in its full generality: intuitionism does not admit the validity of A∨¬A; rather, A∨¬A is true only when either A can be established explicitly, or ¬A can be established explicitly. Fortunately, much of mathematics survives such a restriction in proving power; but not all—some results are known only via a nonconstructive proof. Thus, the question of which logic is the right logic for mathematics impacts the mathematical results that can be proved. Around the same time as intuitionism was proposed, another view of classical logic was being questioned. Take propositional logic. There is a well known connection between propositional logic and sets, an instance of Stone Duality [27, 20]. This connection views propositional logic as being the logic of events, where an event is a set of states of the world. An elementary proposition A is identified with the set A of states where A is true. A formula A ∧ B is identified with the set of all states where both A and B are true, which is equivalent to A ∩ B, and similarly A ∨ B is identified with the set of all states where either A or B is true, which is equivalent to A ∪ B.