Richard L. Mendelsohn

Quantified Modal Logic

This is a book on first-order modal logic. Adding quantifier machinery to classical propositional logic yields first-order classical logic, fully formed and ready to go. For modal logic, however, adding quantifiers is far from the end of the story, as we will soon see. But certainly adding quantifiers is the place to start. As it happens, even this step presents complications that do not arise classically. We say what some of these are after we get language syntax matters out of the way.

Existence and Actualist Quantification

We saw, in Chapter 4, that two basic kinds of quantification were natural in first-order modal logics: possibilist and actualist. Possibilist quantifiers range over what might exist. This corresponds semantically to constant domain models where the common domain is, intuitively, the set of things that could exist. We also saw that in the possibilist approach we could introduce an existence primitive (Section 4.8) and relativize quantifiers to it, permitting the possibilist approach to paraphrase the actualist version.

Terms and Predicate Abstraction

Most presentations of first-order classical logic allow constant and function symbols to appear in formulas. This is the appropriate point for us to introduce them into our treatment of first-order modal logic. But doing so brings some unexpected complications with it. On the other hand, the rewards are great. The material that follows is, perhaps, the central part of this book.

Tableau Proof Systems

Informally, a proof is an argument that convinces. Formally, a proof is of a formula, and is a finite object constructed according to fixed syntactic rules that refer only to the structure of formulas and not to their intended meaning. The syntactic rules that define proofs are said to specify a proof procedure. A proof procedure is sound for a particular logic if any formula that has a proof must be a valid formula of the logic. A proof procedure is complete for a logic if every valid formula has a proof. Then a sound and complete proof procedure allows us to produce “witnesses,” namely proofs, that formulas are valid.

First-Order Axiom Systems

In order to keep this book to a reasonable size we have decided not to give an extensive treatment of subjects that can be found elsewhere and that are somewhat peripheral to our primary interests. That policy begins to have an effect now. Proving completeness of first-order axiomatizations can be quite complex—see (Garson, 1984) for a full discussion of the issues involved. Indeed, a common completeness proof that can cover constant domains, varying domains, and models meeting other conditions, does not seem available, so a thorough treatment would have to cover things separately for each version. Instead we simply present some appropriate axiomatizations, make a few pertinent remarks, and provide references. For machinery presented in later chapters we omit an axiomatic treatment altogether.

First-Order Tableaus

In Section 2.2 we gave prefixed tableau rules for several propositional modal logics. Now we extend these to deal with quantifiers. But we have considered two versions of quantifier semantics: varying domain and constant domain. Not surprisingly, these correspond to different versions of tableau rules for quantifiers. Also not surprisingly, the rules corresponding to constant domain semantics are somewhat simpler, so we will start with them.

Propositional Modal Logic

For analytic philosophy, formalization is a fundamental tool for clarifying language, leading to better understanding of thoughts expressed through language. Formalization involves abstraction and idealization. This is true in the sciences as well as in philosophy. Consider physics as a representative example. Newton’s laws of motion formalize certain basic aspects of the physical universe. Mathematical abstractions are introduced that strip away irrelevant details of the real universe, but which lead to a better understanding of its “deep” structure. Later Einstein and others proposed better models than that of Newton, reflecting deeper understanding made possible by the experience gained through years of working with Newton’s model.