One’s exposure to formal logic usually begins with the study of classical truth-functional propositional logic, in which well-formed formulas (wffs) are built from basic propositions using a small set of connectives (and, or, if…then…, and not), and an assignment of the values true and false to the constituents of a wff determines its value recursively from the truth values of the basic propositions. The rules defining this assignment are called truth tables for the connectives. It is common for students to be uncomfortable about the truth table for if…then. The notion that, for given propositions A and B, “if A, then B” is false only when A is true and B is false creates a number of problems. A simple example is that anything implies a false proposition, an assertion that flies in the face of the sense that there should be some connection between the premise and the consequence.
Another connective that creates discomfort in a truth-functional form is not. At one level, the simple reversal of values its truth table decrees is fairly cut and dried. However, it creates difficulties for those trying to extend a constructive orientation to mathematics beyond the finite. For details, a primer on intuitionism is a good start.
Of course, none of these connectives as used in everyday language is truth-functional, and the classical conception that two truth values suffice is widely contested. While the success of the classical two-valued model in mainstream mathematics has marginalized such considerations for mathematicians, there has been active research among philosophers that takes these discomforts seriously.
As a result, several modifications of classical propositional logic exist. They can be defined initially via formal systems or via semantics. A formal system typically defines proof rules, and then a wff A is a syntactic consequence of a set S of wffs if it follows members of the set via the proof rules. For semantics, one defines the notion of a model (classically, a collection of assignments of true and false to the wffs that is consistent with the truth tables for the connectives), and then A is a syntactic consequence of S if it is true in every model in which every wff in S is true. For nonclassical logics, widely varying schemes of semantics have been devised by varying the set of truth values, or even the notion of model in general.
Epstein presents a single semantic framework that maintains two-valuedness, while introducing a notion of the contents of a wff (reflected formally as a mapping from wffs to some set S of potential contents, and an assignment of relations on subsets of S to connectives). The truth value of a wff now depends on the truth values of its constituents as well as on whether the contents of the constituent wffs stand in the relevant relation. For example, one could now reflect the idea that whether “if A, then B” is true depends on whether the subject matter of A relates to that of B.
What is remarkable about this framework is that, with suitable specialization, it allows one to provide semantics for a wide variety of logics, without abandoning the foundation of allowing propositions only two truth values. Key examples include modal logics, multivalued logics, and intuitionistic logic.
The book is well written and includes extensive motivation, though it is intended primarily for specialists in logic. After some review of classical logic, the author introduces the framework, and applies it in separate chapters (some with co-authors) to a series of nonclassical logics, in each case showing the equivalence of his formulation to an alternative preexisting definition.
A formal semantic framework that is so technically successful raises the question of how it corresponds to actual human practice in argument, and Epstein includes a short discussion of this question at the end of the book. The discussion addresses the questions of how we understand and fail to understand one another (especially in argument), and how this may relate to underlying assumptions. Through all of this, he argues, “always present is the Yes-No, Accept-Reject dichotomy that we impose on (or is imposed on us by) experience.”
A second volume by the same author extends his work to predicate logic [1].
