Hilbert's ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential (...) theorems obtained by this elimination procedure. (shrink)

This is a thorough treatment of first-order modal logic. The book covers such issues as quantification, equality (including a treatment of Frege's morning star/evening star puzzle), the notion of existence, non-rigid constants and function symbols, predicate abstraction, the distinction between nonexistence and nondesignation, and definite descriptions, borrowing from both Fregean and Russellian paradigms.

Herbrand’s theorem is a central fact about classical logic, [9, 10]. It provides a constructive method for associating, with each first-order formula X, a sequence of formulas X1, X2, X3, . . . , so that X has a first-order proof if and only if some Xi is a tautology. Herbrand’s theorem serves as a constructive alternative to..

First-order modal logics, as traditionally formulated, are not expressive enough. It is this that is behind the difficulties in formulating a good analog of Herbrand’s Theorem, as well as the well-known problems with equality, non-rigid designators, definite descriptions, and nondesignating terms. We show how all these problems disappear when modal language is made more expressive in a simple, natural way. We present a semantic tableaux system for the enhanced logic, and (very) briefly discuss implementation issues.

Modal logic is an enormous subject, and so any discussion of it must limit itself according to some set of principles. Modal logic is of interest to mathematicians, philosophers, linguists and computer scientists, for somewhat different reasons. Typically a philosopher may be interested in capturing some aspect of necessary truth, while a mathematician may be interested in characterizing a class of models having special structural features. For a computer scientist there is another criterion that is not as relevant for the (...) other disciplines: a logic should be ‘well-behaved.’ This is, admittedly, a vague notion, but some things are clear enough. A logic that can be axiomatized is better than one that can’t be; a logic with a simple axiomatization is better yet; and a logic with a reasonably implementable proof procedure is best of all. My current interests are largely centered in computer science, and so I will only discuss well-behaved modal logics. My talk is organized into three sections, depending on the expressiveness of the modal logics considered: propositional; first-order with rigid designators; first-order with non-rigid designators. Even limited as I have said, the subject is a big one, and this talk can be no more than an outline. For a general discussion of propositional modal logic see [2]; for this and first-order modal logic see [9], [10] and [8]; for tableau systems in modal logic see [6]. (shrink)