determine whether they can be expressed in first-order logic, existential secondorder logic or universal second-order logic.

9.Recalling the comment on page 142 that Alloy combines model checking M φ and validity checking Γ φ, can you discuss to what extent this is so?

2.9 Bibliographic notes

Many design decisions have been taken in the development of predicate logic in the form known today. The Greeks and the medievals had systems in which many of the examples and exercises in this book could be represented, but nothing that we would recognise as predicate logic emerged until the work of Gottlob Frege in 1879, printed in [Fre03]. An account of the contributions of the many other people involved in the development of logic can be found in the first few pages of W. Hodges’ chapter in [Hod83].

There are many books covering classical logic and its use in computer science; we give a few incomplete pointers to the literature. The books [SA91], [vD89] and [Gal87] cover more theoretical applications than those in this book, including type theory, logic programming, algebraic specification and term-rewriting systems. An approach focusing on automatic theorem proving is taken by [Fit96]. Books which study the mathematical aspects of predicate logic in greater detail, such as completeness of the proof systems and incompleteness of first-order arithmetic, include [Ham78] and [Hod83].

Most of these books present other proof systems besides natural deduction such as axiomatic systems and tableau systems. Although natural deduction has the advantages of elegance and simplicity over axiomatic methods, there are few expositions of it in logic books aimed at a computer science audience. One exception to this is the book [BEKV94], which is the first one to present the rules for quantifiers in the form we used here. A natural deduction theorem prover called Jape has been developed, in which one can vary the set of available rules and specify new ones6.

A standard reference for computability theory is [BJ80]. A proof for the undecidability of the Post correspondence problem can be found in the text book [Tay98]. The second instance of a Post correspondence problem is taken from [Sch92]. A text on the fundamentals of databases systems is [EN94]. The discussion of Section 2.6 is largely based on the text [Pap94] which we highly recommend if you mean to find out more about the intimate connections between logic and computational complexity.

6 www.comlab.ox.ac.uk/oucl/users/bernard.sufrin/jape.html