The Second Edition
With one exception, the changes to the second edition have been minimal. We have added a treatment of identity to chapters 3 and 4. In chapter 3 this required merely a slight modification to the definition of wff, some comments on translation, and the inclusion of introduction and elimination rules for identity. The changes made to chapter 4 are more extensive. In the first edition we avoided overt reference to the object language/metalanguage distinction and had no need to introduce into the specification of interpretations the extensions (denotations, referents) of names, but the inclusion of identity in the language necessitates them. To keep matters simple, when giving interpretations for sentences that involve identity we use italicized names in the metalanguage, and we recommend that no member of the universe of an interpretation be given more than one metalinguistic name. This makes it easy to specify whether or not two names of the object language have the same extension in an interpretation, for the same metalinguistic name will be used for names denoting the same object. Expansions now involve the use of italicized names, so that strictly speaking they are not wffs of the object language. This does not affect their use in determining truth values of quantified wffs in an interpretation, and facilitates their use in determining truth values of wffs involving identity. (We realize that italicization is not available for hand-written exercises, so we recommend that instructors adopt a convention such as underlining for blackboard presentations.) The addition of the material on identity is supplemented with new exercises in chapters 3 and 4. We have tried to organize the new material in such a way that an instructor who wishes to omit it can do so easily.
In chapter 1, a set of exercises has been inserted whose proofs do not require →I and RAA. That is, these proofs do not involve the discharge of assumptions. These exercises are intended to allow students to become comfortable with the remaining rules of proof before they are forced to learn the more complicated mechanics of →I and RAA.
In chapter 3 we have waited until after the section on translations to introduce the notions of a wff's universalization, existentialization, and instance. This change reduces the chance of the student's confusing the rules for constructing universally quantified wffs, where at least one occurrence of a name must be replaced by a variable, and universalization, where all occurrences of the name must be replaced.