Standard Gödel Modal Logics

We prove strong completeness of the □-version and the ?-version of a Gödel modal logic based on Kripke models where propositions at each world and the accessibility relation are both infinitely valued in the standard Gödel algebra [0,1]. Some asymmetries are revealed: validity in the first logic is reducible to the class of frames having two-valued accessibility relation and this logic does not enjoy the finite model property, while validity in the second logic requires truly fuzzy accessibility relations and this logic has the finite model property. Analogues of the classical modal systems D, T, S4 and S5 are considered also, and the completeness results are extended to languages enriched with a discrete well ordered set of truth constants.     

Dialektik Und Formalisierung

Dialectics and Formalization. In an appendix to his article Can the concept of dialectic be made clear? (JGPS 27: 131–165) the author critically examined an attempt by D. Wandschneider of reconstructing Hegelian dialectical logic by displaying antinomic structures. The main steps of this reconstruction were formalized within the framework of second order predicate logic. In his reply (JGPS 27: 347–352) Wandschneider argues that formalization is the wrong way of interpreting and judging his reconstruction project on the ground that the understanding of predication is already presupposed by predicate logic and, therefore, cannot be formalized. In the present contribution it is shown that Wandschneider's criticism is based on a fundamentally mistaken conception of formalization and formal logic and that the problems posed by his project remain completely unresolved.     

The ontological foundation of Russell's theory of modality

Prominent thinkers such as Kripke and Rescher hold that Russell has no modal logic, even that Russell was indisposed toward modal logic. In Part I, I show that Russell had a modal logic which he repeatedly described and that Russell repeatedly endorsed Leibniz's multiplicity of possible worlds. In Part II, I describe Russell's theory as having three ontological levels. In Part III, I describe six Parmenidean theories of being Russell held, including: literal in 1903; universal in 1912; timeless in 1914; transcendental in 1918–1948. The transcendental theory underlies the primary level of Russell's modal logic. In Part IV, I examine Rescher's view that Russell and modal logic did not mix.The United States Naval Academy Research Council kindly provided a summer 1988 research grant for work on this essay. This essay was presented at the Bertrand Russell Society Meeting during the December 1988 Eastern Division Meeting of the American Philosophical Association in Washington, D.C.     

Systems of Quantum Logic

Logical implications are closely related to modal operators. Lattice-valued logic LL and quantum logic QL were formulated in Titani S (1999) Lattice Valued Set Theory. Arch Math Logic 38:395–421, Titani S (2009) A Completeness Theorem of Quantum Set Theory. In: Engesser K, Gabbay DM, Lehmann D (eds) Handbook of Quantum Logic and Quantum Structures: Quantum Logic. Elsevier Science Ltd., pp. 661–702, by introducing the basic implication → which represents the lattice order. In this paper, we fomulate a predicate orthologic provided with the basic implication, which corresponds to complete ortholattices, and then formulate a quantum logic which is equivalent to QL, by using a modal operator instead of the basic implication.     

REVIEWS

Book reviewed in this article: Theology and Dialogue: Essays in Conversation with George Lindbeck, Edited by Bruce D. Marshall Biblical Criticism in Crisis? The Impact of the Canonical Approach on Old Testament Studies, by Mark G. Brett An Aplolgy for Apologetics: A Study in the logic of Interreligious Dialogue, by Paul J. Griffiths Between the Two Fires: The Anglican Church and Apartheid 1948 to 1957, by Michael E. Worsnip     

A Propositional Dynamic Logic with Qualitative Probabilities

This paper presents an -completeness theorem for a new propositional probabilistic logic, namely, the dynamic propositional logic of qualitative probabilities (D Q P), which has been introduced by the author as a dynamic extension of the logic of qualitative probabilities (Q P) introduced by Segerberg.     

Extending Montague's system: a three valued intensional logic

In this note we present a three-valued intensional logic, which is an extension of both Montague's intensional logic and ukasiewicz three-valued logic. Our system is obtained by adapting Gallin's version of intensional logic (see Gallin, D., Intensional and Higher-order Modal Logic). Here we give only the necessary modifications to the latter. An acquaintance with Gallin's work is pressuposed.     

Les tests dominos (D70 et D2000) : comment dépasser le constat du seul score total ? Propositions d’analyses des réponses

The domino-type tests (D48, D70 and D2000), measurements of the factor g and the fluid intelligence, are often regarded as resting exclusively on a numerical logic of resolution because of the type of support used (dominoes). A cognitive analysis of the items of the two most recent versions of this type of test moderates this consideration by showing that it is possible to distinguish several categories of items according to their logic from resolution (mainly the space items and the numerical items). We then propose leads of analysis of the performance of the subject which rest on the taking into account of these various categories of items.     

Narrative rationality and the logic of scientific discourse

This essay argues that scientific discourse is amenable to interpretation and assessment from the perspective of the narrative paradigm and its attendant logic, narrative rationality. It also contends that this logic entails a revised conception of knowledge, one that permits the possibility of wisdom. The text analyzed is James D. Watson and Francis H. Crick's proposal of the double helix model of DNA.     

Coherence and Computational Complexity of Quantifier-free Dependence Logic Formulas

We study the computational complexity of the model checking problem for quantifier-free dependence logic ${(\mathcal{D})}$ formulas. We characterize three thresholds in the complexity: logarithmic space (LOGSPACE), non-deterministic logarithmic space (NL) and non-deterministic polynomial time (NP).     

A Method of Generating Modal Logics Defining Jaśkowski’s Discussive Logic D<Subscript>2</Subscript>

Ja?kowski’s discussive logic D₂ was formulated with the help of the modal logic S5 as follows (see [7, 8]): \({A \in {D_{2}}}\) iff \({\ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\rm S}5}\), where (–)^? is a translation of discussive formulae from For^d into the modal language. We say that a modal logic L defines D₂ iff \({{\rm D}_{2} = \{A \in {\rm For^{\rm d}} : \ulcorner\diamond{{A}^{\bullet}}\urcorner \in {\it L}\}}\). In [14] and [10] were respectively presented the weakest normal and the weakest regular logic which (?): have the same theses beginning with ‘\({\diamond}\)’ as S5. Of course, all logics fulfilling the above condition, define D₂. In [10] it was prowed that in the cases of logics closed under congruence the following holds: defining D₂ is equivalent to having the property (?). In this paper we show that this equivalence holds also for all modal logics which are closed under replacement of tautological equivalents (rte-logics).We give a general method which, for any class of modal logics determined by a set of joint axioms and rules, generates in the given class the weakest logic having the property (?). Thus, for the class of all modal logics we obtain the weakest modal logic which owns this property. On the other hand, applying the method to various classes of modal logics: rte-logics, congruential, monotonic, regular and normal, we obtain the weakest in a given class logic defining D₂.     

Book Reviews

LOGIC AND SCIENCE

W. KNORR, The ancient tradition of geometric problems. Boston, Basel, Stuttgart:Birkhauser, 1986. ix+411 pp., 9 plts. SFr 128.

D. PEARCE, Roads to commensurability. Dordrecht, Boston, Lancaster, and Tokyo: D. Reidel Publishing Company, 1987. xi + 253 pp. £46/$59.

RAFFAELLA SIMILI [ed.], Logica, metodo e scienze in Gran Bretagna (1860-1930). Turin: Loescher, 1986. 272 pp. 16 000 Lire.

LOGIC AND SCIENCE

SEPPO SAJAMA and MATTI KAMPPINEN. A historical introduction to phenomenology. London, New York, Sydney: Croom Helm, 1987. vii + 131 pp. £22.50 (hardback)/£8.95 (paperback).

GRAHAM MACDONALD and CRISPIN WRIGHT (eds.), Fact, science and morality: Essays on A. J. Ayer's Language, truth and logic. Oxford: Basil Blackwell, 1986. 314 pp. £27.50.

BARRY GOWER (ed.). Logical positivism in perspective: Essays on Language, truth and logic. London Croom Helm, 1987. viii + 129 pp. £20.00.

CHRISTOPHER CHERNIAK, Minimal rationality. Cambridge, Mass.: Bradford and MIT Press, 1986. x+ 161 pp. £19.95.

RADU J. BOGDAN (ed.) Jaakko Hintikka. (Projiles, volume 8.) Dordrecht: D. Reidel, 1987. 382 pp. US$89.

LOGIC AND SCIENCE

E. J. REULAND and A. G. B. TER MEULEN (eds.), The representation of (in) definiteness.Cambridge, (Mass.) and London: M.I.T. Press, 1987. xiv+ 344 pp. £27.50.

E. F. KITTAY, Metaphor: its cognitive force and linguistic structure. Oxford: Clarendon Press, 1987. x + 358 pp. £35.00.     

Algebraic Analysis of Demodalised Analytic Implication

Journal of Philosophical Logic - The logic DAI of demodalised analytic implication has been introduced by J.M. Dunn (and independently investigated by R.D. Epstein) as a variation on a...     

从主体、结构到资本逻辑的结构化——反思关于马克思思想之研究模式的主导逻辑

自第二国际时代以来,关于马克思思想的研究在主导逻辑上经历了两次重要转换:一是上个世纪20年代以卢卡奇等为标志的从传统的经济决定论构架转向以主体-客体为核心的历史辩证法构架,这一研究直接影响到后来的西方马克思主义者,并在20世纪80年代以来的国内马克思思想研究中得到回应;二是在上个世纪50—60年代关于人与结构的逻辑论争中,以主体-客体为核心的历史辩证     

Interpretations of intuitionist logic in non-normal modal logics

Historically, it was the interpretations of intuitionist logic in the modal logic S4 that inspired the standard Kripke semantics for intuitionist logic. The inspiration of this paper is the interpretation of intuitionist logic in the non-normal modal logic S3: an S3 model structure can be 'looked at' as an intuitionist model structure and the semantics for S3 can be 'cashed in' to obtain a non-normal semantics for intuitionist propositional logic. This non-normal semantics is then extended to intuitionist quantificational logic.     

Synchronization of Logics

Motivated by applications in software engineering, we propose two forms of combination of logics: synchronization on formulae and synchronization on models. We start by reviewing satisfaction systems, consequence systems, one-step derivation systems and theory spaces, as well as their functorial relationships. We define the synchronization on formulae of two consequence systems and provide a categorial characterization of the construction. For illustration we consider the synchronization of linear temporal logic and equational logic. We define the synchronization on models of two satisfaction systems and provide a categorial characterization of the construction. We illustrate the technique in two cases: linear temporal logic versus equational logic; and linear temporal logic versus branching temporal logic. Finally, we lift the synchronization on formulae to the category of logics over consequences systems.     

A Defense of Second-Order Logic

Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol Logic 45:464–482, 1980). Therefore, it is not clear how this logic should be interpreted. (3) Second-order logic also has strong ontological commitments: (a) it is ontologically committed to classes (Resnik, J Phil 85:75–87, 1988), and (b) according to Quine (Philosophy of logic, Prentice-Hall: Englewood Cliffs, 1970), it is nothing more than “set theory in sheep’s clothing”. (4) It is also not better than its first-order counterpart, in the following sense: if first-order logic does not characterize adequately mathematical systems, given the existence of non-isomorphic first-order interpretations, second-order logic does not characterize them either, given the existence of different interpretations of second-order theories (Melia, Analysis 55:127–134, 1995). (5) Finally, as opposed to what is claimed by defenders of second-order logic [such as Shapiro (J Symbol Logic 50:714–742, 1985)], this logic does not solve the problem of referential access to mathematical objects (Azzouni, Metaphysical myths, mathematical practice: the logic and epistemology of the exact sciences, Cambridge University Press, Cambridge, 1994). In this paper, I argue that the second-order theorist can solve each of these difficulties. As a result, second-order logic provides the benefits of a rich framework without the associated costs.     

BOOK REVIEWS

Book reviewed in this article:
Descubrimientos y refutaciones: La lógica de la indagación psicoanalítica [Discoveries and refutations: the logic of psychoanalytic enquiry]. By Jorge L. Ahumada
Kohut's Freudian Vision. By Philip F. D. Rubovits-Seitz in collaboration with Heinz Kohut.
Psychoanalysis on the Move: The Work of Joseph Sandler. Edited by Peter Fonagy, Arnold M.
La polarité de l'amour et de la mort [The polarity of love and death]. By Catherine Couvreur.
Being of Two Minds: The Vertical Split in Psychoanalysis and Psychotherapy. By Arnold Goldberg.
The Power of Feelings. By Nancy J. Chodorow
Nascita e costruzione della mente. La teoria del protomentale. [The birth and construction of the mind. The theory of the protomental]. By Antonio Imbasciati.     

Kurt Gödel and the Foundations of Mathematics: Horizons of Truth

MEDIEVAL LOGICS

P. 0 . LEWRY (ed.), The rise of British logic. Acts of the sixth European Symposium on mediaeval logic and semantics, Balliol College, Oxford, 19-24 June 1983. Toronto: Pontifical Institute of Mediaeval Studies, 1985. xii+421 pp. $34.50.

L. M. ,DE RIJK, La philosophie au moyen cige. Translated from the Dutch by P.Swiggers, Leiden: E. J. Brill, 1985. xi + 244 pp. 50 Hfl.

RAIMUNDUS LULLUS, Die neue Logik. Logica nova. Textkritisch herausgegeben von Charles Lohr. Übersetzt von Vittorio Hösle und Walburga Büchel. Mit einer Einführung von Vittorio Hosle. (Lateinisch-deutsch.) Hamburg: Felix Meiner Verlag, 1985. xciv + 3 17s. 64DM.

PAULUS VENETUS, Logica parva. Translation of the 1472 edition, with introduction and notes by Alan R. Perreiah. Miinchen and Wein: Philosophia Verlag, 1984. 372 pp. DM 138.

E. J. ASHWORTH, Studies in post-medieval semantics. London: Variorum Reprints, 1985. x + 342 pp. £32.00.

D. GABBAY and F. GUENTHNER (eds.), Handbook of philosophical logic. Volume 2: Extensions of classical logic. Dordrecht, Boston and Lancaster: D. Reidel Publishing Company, 1984. xi + 779 pp. Dfl. 31 5/$12O.OO/£80.25.

THOMAS M. SEEBOHM, Philosophie der Logik. Freiburg: Alber, 1984. 364 pp. DM 74.

WERNER STELZNER, Epistemische Logik. Zur logischen Analyse von Akzeptationsformen.Berlin: Akademie-Verlag, 1984. 206 pp. No price stated.

JAMES BOGEN and JAMES E. MCGUIRE (eds.), How things are: studies in predication and the history and philosophy of science. (Philosophical studies series in philosophy, Volume 29.) Dordrecht: D. Reidel, 1985. 345 pp. US $46.00.

GEORG HENRIK VON WRIGHT, Truth, knowledge, and modality. (Philosophical papers, Volume 3.) Oxford: Basil Blackwell, 1984. ix + 155 pp. 15.00.

JEAN VAN HEIJENOORT, Selected essays. Naples: Bibliopolis, 1985. 166 pp. No price stated.

H. WANG, Beyond analytic philosophy. Doing justice to what we know. Cambridge (Massachusetts) and London: The MIT Press, 1986. xii + 273pp. No price stated.

TEXTBOOKS

R. L.VAUGHT, Set theory: an introduction. Boston, Basel, Stuttgart: Birkhäuser, 1985. x+141 pp. SFr. 76.

W. H. NEWTON-SMITH, Logic: An introductory course. London, Melbourne and Henley: Routledge and Kegan Paul, 1985. x+ 230 pp. £5.95 (paper).

MISCELLANEOUS

PIERRE DUHEM, Medieval cosmology. Edited and translated by Roger Ariew.Chicago and London: University of Chicago Press, 1985. xxxi+ 601 pp. £24.75.

DAVID A. WHITE, Logic and ontology in Heidegger. Columbus: Ohio State University Press, 1985. xiv + 245 pp. $22.50.

J. W. DEGEN, Systeme der kumulativen Logik. Miinchen und Wien: Philosophia Verlag (Reihe Analytica), 1983. 265pp. DM 128.

GABRIELE LOLLI, La ragioni jisiche e la dimonstrazioni matematiche. Bologna: I1 Mulino, 1985. 365 pp. 30,000 lire.