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.

A (normal) system of propositional modal logic is said to be complete iff it is characterized by a class of (Kripke) frames. When we move to modal predicate logic the question of completeness can again be raised. It is not hard to prove that if a predicate modal logic is complete then it is characterized by the class of all frames for the propositional logic on which it is based. Nor is it hard to prove that if a propositional modal (...) logic is incomplete then so is the predicate logic based on it. But the interesting question is whether a complete propositional modal logic can have an incomplete extension. In 1967 Kripke announced the incompleteness of a predicate extension of S4. The purpose of the present article is to present several such systems. In the first group it is the systems with the Barcan Formula which are incomplete, while those without are complete. In the second group it is those without the Barcan formula which are incomplete, while those with the Barcan Formula are complete. But all these are based on propositional systems which are characterized by frames satisfying in each case a single first-order sentence. (shrink)

The aim of this paper is to prove strong completeness theorems for several Anderson-like variants of Gödels theory wrt. classes of modal structures, in which: (i). 1st order terms order receive only rigid extensions in the constant objectual 1st order domain; (ii). 2nd order terms receive non-rigid extensions in preselected world-relative objectual domains of 2nd order and rigid intensions in the constant conceptual 2nd order domain.

In the paper we introduce a wide range of Anderson-like variants of Gödel's theory and prove for each of them strong completeness theorem wrt. corresponding class of modal structures.These theories — all formulated in the 2nd order modal language with a 2nd order unary predicate of positiveness — differ among themselves with respect of: properties of the necessity operator and of the predicate of positiveness, axioms characterizing identity between 1st sort terms, definitions of identity between 2nd sort terms, the treatment (...) of Godlikeness as a term of the 2nd order, the treatment of necessary existence as a term of the 2nd order if there exists the necessity of its presence, the treatment of Ix (Ix(y) is read: terms x, y are identical), for every 1st sort term x, as a term of the 2nd order if there exists the necessity of its presence.Concerning to semantical commitments, the following are essential: (i) 1st order terms order receive only rigid extensions in the constant objectual 1st order domain; (ii) 2nd order terms receive non-rigid extensions in the constant objectual domain of 2nd order and rigid intensions in the constant conceptual 2nd order domain; (iii) The behavior of the identity between 1st sort terms shifts from possible worlds to possible worlds. (shrink)

Gödel's modal ontological argument is the centerpiece of an extensive examination of intensional logic. First, classical type theory is presented semantically, tableau rules for it are introduced, and the Prawitz/Takahashi completeness proof is given. Then modal machinery is added to produce a modified version of Montague/Gallin intensional logic. Finally, various ontological proofs for the existence of God are discussed informally, and the Gödel argument is fully formalized. Parts of the book are mathematical, parts philosophical.

The essay outlines the character of free logic, and motivation for its construction and development. It details some technical achievements of high philosophical interest, but urges that the role of existence assumptions in logic is still not fully understood, that unresolved old problems, both technical and philosophical, abound, and presents some new problems of considerable philosophical import in free logic.

The author attacks the view that identity, Like largeness, Is a relative relation. The primary advocate of the view that identity is relative is p.T. Geach. It is argued that geach has not shown that the failure of the identity of indiscernibles principle, As a truth of logic, Forces us to stop taking indiscernibility within particular formal theories or languages as a sufficient condition for identity. The author also argues that the whole notion of relative identity, As explicated by geach, (...) Is very probably incoherent. (shrink)

Kurt Gödel’s version of the ontological argument was shown by J. Howard Sobel to be defective, but some plausible modifications in the argument result in a version which is immune to Sobel’s objection. A definition is suggested which permits the proof of some of Godel’s axioms.

First - order modal logic is very much under current development, with many different semantics proposed. The use of rigid objects goes back to Saul Kripke. More recently, several semantics based on counterparts have been examined, in a development that goes back to David Lewis. There is yet another line of research, using intensional objects, that traces back to Richard Montague. I have been involved with this line of development for some time. In the present paper, I briefly sketch several (...) of the approaches to first - order modal logic. Then I present one that I call FOIL in the Montague tradition that, I believe, is both expressive and natural. I briefly discuss in what sense it can be made to encompass the other approaches. Finally, I provide tableau rules to go with the FOIL semantics. (shrink)

In the first twelve chapters of this book, I am concerned with the Fregean notion of an object (the reference of a proper name) and its connection with the notion of identity. The rest of the book is devoted to a discussion of the problem of personal identity.

Free logic is an important field of philosophical logic that first appeared in the 1950s. J. Karel Lambert was one of its founders and coined the term itself. The essays in this collection explore the philosophical foundations of free logic and its application to areas as diverse as the philosophy of religion and computer science. Amongst the applications on offer are those to the analysis of existence statements, to definite descriptions and to partial functions. The volume contains a proof that (...) free logics of any kind are non-extensional and then uses that proof to show that Quine's theory of predication and referential transparency must fail. The purpose of this collection is to bring an important body of work to the attention of a new generation of professional philosophers, computer scientists and mathematicians. (shrink)