J. Lambert

The expression ‘free logic,’ coined by the author in 1960, is an abbreviation for ‘logic free of existence assumptions with respect to its terms, singular and general, but whose quantifiers are treated exactly as in standard quantifier logic.’ In more traditional language, such logics do not presume that either singular or general terms — the two distinct categories of terms emphasized in modern logical grammar — have existential import. A singular term ‘t’ has existential import just in case t exists (or, equivalently, there exists an object the same as t) and a general term (or predicate) ‘G’ has existential import just in case G exist (or, equivalently, there exists an object that is G).1 Examples from colloquial English customarily taken to be singular terms are expressions such as ‘Socrates’, ‘the planet causing perturbations in the orbit of Mercury’, ‘5’, ‘5/0’, ‘the square of 3’ and ‘having a heart’. Some of these do not have existential import — in particular, ‘5/0’ and ‘the planet causing perturbations in the orbit of Mercury’. Examples from colloquial English customarily taken to be general terms are expressions such as ‘is a philosopher’, ‘is a planet causing perturbations in the orbit of Mercury’, ‘number’, ‘is divisible by 0’, and ‘has a heart’. Some of these general terms do not have existential import — in particular, ‘is a planet causing perturbations in the orbit of Mercury’ and ‘is divisible by 0’. To say that the quantifers are treated exactly as in standard quantifier logic is to say, roughly, that the operator symbol ‘∃’ (the existential quantifier) reads: ‘There exists an object’, and the operator symbol ‘∀’ (the universal quantifier) reads: ‘Every existent object’.

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.

It is an historical fact that one of Russell's greatest philosophical contributions was to highlight the role that premises about logical form play in ontological arguments. A pair of quotations will introduce his point that great metaphysical systems are often not only based on, but are debased by, the belief that certain statements of philosophical discourse are logically subject-predicate in form.Speaking of Hegel's Absolute Idealism, Russell wrote in Our Knowledge of The Extemal World:Mr. Bradley has worked out a theory according to which, in all judgment, we are ascribing a predicate to Reality as a whole; and this theory is derived from Hegel. Now the traditional logic holds that every proposition ascribes a predicate to a subject, and from this it easily follows that there can be only one subject, the Absolute, for if there were two, the proposition that there were two would not ascribe a predicate to either. Thus Hegel's doctrine, that philosophical propositions must be of the form, “the Absolute is such and such,” depends on the traditional belief in the universality of the subject-predicate form. This belief, being traditionaL scarcely self-conscious, and not supposed to be important, operates underground, and is assumed in arguments which, like the refutation of relations, appear at first sight to e;tablish its truth.

This essay lays out the leading principles of the theories of definite descriptions advocated by Frege, Russell, and Hilbert and Bernays, and discusses various difficulties, philosophical and otherwise, with each treatment, fixing especially on the treatment of singular existence claims. Then the leading principles of free (definite) description theory are presented and it is shown how it resolves difficulties confronting the more traditional approaches. Finally, a pair of technical problems in free (definite) description theory are addressed. They help to show the fecundity of this treatment of definite descriptions

Those who want to interpret the quantifier ‘ (3 x) (...x...)'as having no existence commitment often fail to distinguish between this objective and that of merely changing the values of the variables. The confusion vitiates solutions of the singular existence anomalies which purport to be based on a non‐existential interpretation of the quantifier. An example of one who makes the distinction but still interprets the particular quantifier non‐existentially is offered by Czeslaw Lejewski. Objection to the classical interpretation of the quantifiers often runs hand in hand with aversion to extensional logic. However, it is at least arguable that such an aversion is the result of underestimating the resources of extensional logic. These points arc discussed in the wake of Professor Marcus's recent paper in this journal “Interpreting Quantification”.

This paper deals with the Meinong-Russell controversy on nonsubsistent objects. The first part notes the similarity of certain contemporary semantical developments to Meinonj;'s theory of nonsubsistent objects. Then it lays out the major features of Meinong's famous theory, considers Russell's objections to same and Meinong's counter-objections to Russell, and argues that Russell's well-known argument fails. However, it is possible to augment Russell's argument against Meinong with sound Russellian principles in such a way that it presents at least a strong inclining reason against Meinong's theory of impossible objects.

The essays in this volume are based on addresses presented during a colloquium on free logic, modal logic and related areas held at the University of California at Irvine, in May of 1968. With the single exception of Dagfinn F011esdal, whose revised address is included in a recent issue of Synthese honoring W. V. Quine, all of the speakers at the Irvine colloquium are contributors to this volume. Thanks are due to Professor A. I. Melden, Chairman of the Department of Philosophy at Irvine, for his enthusiastic support of the colloquium, and to Drs. Gordon Brittan and Daniel Dennett for their help in the administration of the colloquium. Finally. I should also like to thank Professor Ralph W. Gerard, Dean of the Graduate Division of the University of California at Irvine, for the financial support which made the colloquium possible. KAREL LAMBERT Laguna Beach, California, 1969 TABLE OF CONTENTS PREFACE V KAREL LAMBERT and BAS C. VAN FRAASSEN/ Meaning Relations, Possible Objects, and Possible Worlds 1 JAAKKO HINTIKKA / Existential Presuppositions and Uniqueness Presuppositions 20 RICHMOND H. THOMASON / Some Completeness Results for Modal Predicate Calculi 56 H. LEBLANC and R. K. MEYER / Truth-Value Semantics for the Theory of Types 77 J. M. VICKERS / Probability and Non Standard Logics 102 PETER W. WOODRUFF / Logic and Truth Value Gaps 121 DANA SCOTT / Advice on Modal Logic 143 INDEX OF NAMES 175 KAREL LAMBER T AND BAS C.