ion and general ideas.” Compare Meinong, Über Möglichkeit und Wahrscheinlichkeit, p. 178: “Besonders geeignet sind vielmehr Begriffsgegenstände, wie uns deren etwa durch Definitionen gegeben werden. Das Dreick z.B., darin hatte der sicher nicht überrationalistische Locke gegen Berkeley und gegen viele Spätere) am Ende doch recht, ist als solches weder gleichseitig noch gleichschenklig, weder rechtwinklig noch schiefwinklig, noch das Gegenteil davon: es ist in diesen Hinsichten und noch in vielen anderen eben unbestimmt. Gegenstände dieser Art stehen in deutlichen Gegensatz zu solchen, die, wie wir deren oben zuerst betrachtet haben, in bezug auf alle wie immer gearteten Gegenstände bestimmt sind. Man kann solche Gegenstände mit Recht vollständig bestimmte nennen, Blaues, Dreieck und ihresgleichen dagegen unvollständig bestimmte.” 15Hume, A Treatise of Human Nature, [1739-40], edited by Selby-Bigge, second edition revised by Nidditch (Oxford: The Clarendon Press, 1978), p. 17.

Handbook of the history of logic

The computer elicitation and analysis of personal construct systems has become a technique of great interest and wide application in recent years. This paper takes the current state of the art as a starting point and explores further developments that are natural extensions of it. The overall objective of the work described is to develop man-computer symbiotic systems in which the computer is a truly dialectical partner to the person in forming theories and making decisions. A logical model of constructs as predicates applying to elements is used to develop a logical analysis of construct structures and this is contrasted with various distance-based clustering techniques. A grid analysis program called ENTAIL is described based on these techniques which derives a network of entailments from a grid. This is compared and contrasted with various programs for repertory grid analysis such as INGRID, FOCUS and Q-Analysis. Entailment is discussed in relation to Kelly's superordination hierarchy over constructs and preference relations over elements. The entailment analysis is extended to rating-scale data using a fuzzy semantic model. The significance of Kelly's notion of the opposite to a construct as opposed to its negation is discussed and related to other epistemological models and the role of relevance. Finally, the interactive construct elicitation program PEGASUS is considered in terms of the psychological and philosophical importance of the dialectical processes of grid elicitation and analysis, and recommendations are made about its generalization and extension based on the logical foundations described. Links are established between the work on repertory grids and that on relational data bases and expert systems.

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New directions in the analysis and interactive elicitation of personal construct systems

Many-valued logic is a vast field with hundreds of published papers and numerous monographs devoted to it. I have attempted to keep this survey to manageable length by focusing on many-valued logic as an independent discipline. This means that such topics as the use of many-valued logics for proving the independence of axioms in propositional logic have been omitted.

Basic Many-Valued Logic

Fuzzy class theory

The semantical paradoxes are not a scientific subject like Inductive Definitions, Algebraic Geometry or Plasma Physics. At least not yet. On the other hand the paradoxes exert a strong fascination and many a philosopher or logician has spent some thought on them, mostly in relative isolation. The literature on the paradoxes is vast but scattered, repetitive and disconnected. This made it impossible to give a presentation in which all ideas in the literature receive their due.

Semantics and the Liar Paradox

It is natural to suggest that the paradoxes of naive set ,theory can be avoided not by asserting only special cases of the axiom of comprehension (,!?x)~)O, E x -A@)), as in set theories like ZF formulated in classical predicate logic, but rather by retaining the unrestricted comprehension axiom while weakening the underlying logic. Various experiments have been made with such “type-free” logics by Ackermann [l] , Fitch [3] , Schtitte [6] , and others. One of the most interesting proposals is Skolem’s. After showing that versions of Russell’s paradox can be produced from the unrestricted comprehension axiom in any finite-valued Lukasiewicz predicate logic, Skolem conjectured that the axiom can however be consistently added to the infinite-valued Lukasiewicz predicate logic [7]. He also suggested that it may be possible to derive a significant amount of mathematics in a set theory based on this logic. Skolem’s conjecture about the consistency of the axiom of comprehension in the infinite-valued logic has been partially confirmed by Skolem himself and by Chang and others [2]. In this paper I shall prove Skolem’s conjecture by using some simple modifications of methods from classical proof theory and model theory: normalization of proofs in a naturaldeduction calculus and the use of maximally consistent sets of formulas. It remains to investigate the mathematical strength of the system, and I conclude the paper with some remarks on tbis subject. In outline the consistency proof proceeds as follows. Louise Hay [4] has provided a complete axiomatization of the infinite-valued predicate logic, although her formalization is not an axiomatization in the strict sense since it requires an inftitary inference rule. (It follows from work by Scarpellini [S] that this is the most one can hope for.) The result of adding the comprehension axiom to Hay’s predicate logic, with only the dyadic predicate “e” is here called H, it is therefore sufficient to prove that H is consistent. For this purpose His enlarged by adding certain Hilbert r-terms to obtain a system Hr. Every theorem of H is a theorem of Hr. Hr is then

The consistency of the axiom of comprehension in the infinite-valued predicate logic of Łukasiewicz

This essay demonstrates proof-theoretically the consistency of a type-free theoryC with an unrestricted principle of comprehension and based on a predicate logic in which contraction (A → (A →B)) → (A →B), although it cannot holds in general, is provable for a wide range ofA's.C is presented as an axiomatic theoryCH (with a natural-deduction equivalentCS) as a finitary system, without formulas of infinite length. ThenCH is proved simply consistent by passing to a Gentzen-style natural-deduction systemCG that allows countably infinite conjunctions and in which all theorems ofCH are provable.CG is seen to be a consistent by a normalization argument. It also shown that in a senseC is highly non-extensional.

Richard B. White

Papers

The consistency of the axiom of comprehension in the infinite-valued predicate logic of Łukasiewicz

A consistent theory of attributes in a logic without contraction