On the size of refutation Kripke models for some linear modal and tense logics

LetL be any modal or tense logic with the finite model property. For eachm, definer _L (m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr _L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr _L (m) for some linear modal and tense logicsL.     

Development and Evaluation of a Decision Aid About Prenatal Testing for Women of Advanced Maternal Age

Objective: To develop and evaluate a decision aid designed to prepare patients of advanced maternal age for counseling about prenatal diagnostic testing. Setting: A regional genetics center. Design: A before/after study. Interventions: Participants used an audioguided workbook to learn about options and outcomes and to clarify personal risks, values, questions, and predispositions. Subjects: 21 women of advanced maternal age and 17 spouses. Main outcome measures: Knowledge of prenatal testing alternatives, decisional conflict, level of anxiety, and acceptability of the decision aid. Results: After using the decision aid, participants had significantly reduced decisional conflict (uncertainty) and a significant increase in knowledge. There was no effect on state or trait anxiety. More than three-quarters of participants were satisfied with the length, clarity, balance, and acceptability of the decision aid. Conclusions: The decision aid shows promise as a useful aid for preparing couples for counseling.     

Proof theory and mathematical meaning of paraconsistent C-systems

A proof-theoretic analysis and new arithmetical semantics are proposed for some paraconsistent C-systems, which are a relevant sub-class of Logics of Formal Inconsistency (LFIs) introduced by W.A. Carnielli et al. (2002, 2005) [8] and [9]. The sequent versions BC, CI, CIL of the systems bC, Ci, Cil presented in Carnielli et al. (2002, 2005) [8] and [9] are introduced and examined. BC, CI, CIL admit the cut-elimination property and, in general, a weakened sub-formula property. Moreover, a formal notion of constructive paraconsistent system is given, and the constructivity of CI is proven. Further possible developments of proof theory and provability logic of CI-based arithmetical systems are sketched, and a possible weakened Hilbert?s program is discussed. As to the semantical aspects, arithmetical semantics interprets C-system formulas into Provability Logic sentences of classical Arithmetic PA (Artemov and Beklemishev (2004) [2], Japaridze and de Jongh (1998) [19], Gentilini (1999) [15], Smorynski (1991) [22]): thus, it links the notion of truth to the notion of provability inside a classical environment. It makes true infinitely many contradictions B∧¬B and falsifies many arbitrarily complex instances of non-contradiction principle ¬(A∧¬A). Moreover, arithmetical models falsify both classical logic LK and intuitionistic logic LJ, so that a kind of metalogical completeness property of LFI-paraconsistent logic w.r.t. arithmetical semantics is proven. As a work in progress, the possibility to interpret CI-based paraconsistent Arithmetic PACI into Provability Logic of classical Arithmetic PA is discussed, showing the role that PACIarithmetical models could have in establishing new meta-mathematical properties, e.g. in breaking classical equivalences between consistency statements and reflection principles.     

On the Closure Properties of the Class of Full G-models of a Deductive System

In this paper we consider the structure of the class FGModS of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the lattice-theoretical point of view; this set is represented by the lattice FACS_s A of all algebraic closed-set systems C on A such that (A, C) ε FGModS. We relate some properties of these structures with tipically logical properties of the sentential logic S. The main algebraic properties we consider are the closure of FGModS under substructures and under reduced products, and the property that for any A the lattice FACS_s A is a complete sublattice of the lattice of all algebraic closed-set systems over A. The logical properties are the existence of a fully adequate Gentzen system for S, the Local Deduction Theorem and the Deduction Theorem for S. Some of the results are established for arbitrary deductive systems, while some are found to hold only for deductive systems in more restricted classes like the protoalgebraic or the weakly algebraizable ones. The paper ends with a section on examples and counterexamples. Dedicated to the memory of Willem Johannes Blok     

Intermediate logics with the same disjunctionless fragment as intuitionistic logic

Given an intermediate prepositional logic L, denote by L ^–d its disjuctionless fragment. We introduce an infinite sequence {J _n}_n1 of propositional formulas, and prove:(1)For any L: L ^–d =I ^–d (I=intuitionistic logic) if and only if J _n L for every n 1.Since it turns out that L{J _{n} n}1 = Ø for any L having the disjunction property, we obtain as a corollary that L ^–d = I ^–d for every L with d.p. (cf. open problem 7.19 of [5]). Algebraic semantic is used in the proof of the if part of (1). In the last section of the paper we provide a characterization in Kripke's semantic for the logics J _n =I+ +J _n (n 1).     

Audience with a Healer

Up From Scapegoating: Awakening Consciousness in Groups. 1995. By Arthur D. Colman Wilmette, IL: Chiron.

The Wounded Jung. (1996). By Robert C. Smith Evanston, IL: Northwestern University Press.

The Body: An Encyclopedia for Research in Archetypal Symbolism. 1996. By George R. Elder Boston: Shambhala.

Gifts of Suffering. 1996. By Polly Young-Eisendrath New York; Addison-Wesley Co.

The Religious Function of die Psyche. 1996. By Lionel Corbett London &; New York: Routledge.

Buddhism and the Art of Psychotherapy. 1996. By Hayao Kawai College Station, TX: Texas A &; M University Press.

The Soul's Code: In Search of Character and Calling. 1996. By James Hillman New York: Random House.

Kinds of Power A Guide to Its Intelligent Uses. 1995. By James Hillman New York: Doubleday.

Psychotherapy as a Mutual Process. 1996. By J. Marvin Spiegelman Tempe, AZ: New Falcon Publications.

Soul Cards. By Deborah Koff-Chapin 1995. Langley, WA: The Center for Touch Drawing.     

Book Reviews

Abstract

Beteendeterapi inom Psykiatrin L.G. Öst (Red). (1987). Stockholm: Natur &; Kultur. Pris: 330 sv kr. Reviewed by Arnold Torstenson.

“Till de Bildade bland dess Föraktare” Reviewed by Car-Erik Brattemo.

Beteendeterapi inom Psykiatrin Reviewed by Solveig og Carl Erik Grenness.

Schizofrenibehandling — Psykiatri på Liv och Död S. Belin (1987) Stockholm: Natur &; Kultur, 170 sidor. Pris: 205 SEK. Reviewed by Staffan Sohlberg.

Symtom — Diagnos G. Birke (1988) Stockholm: Almqvist &; Wiksell, 616 sidor Reviewed by Birgitta och Roberg Forsberg.     

A weak intuitionistic propositional logic with purely constructive implication

We introduce subsystems WLJ and SI of the intuitionistic propositional logic LJ, by weakening the intuitionistic implication. These systems are justifiable by purely constructive semantics. Then the intuitionistic implication with full strength is definable in the second order versions of these systems. We give a relationship between SI and a weak modal system WM. In Appendix the Kripke-type model theory for WM is given.This work was partially supported by NSF Grant DCR85-13417     

The existence of matrices strongly adequate for E,R and their fragments

A logic is a pair (P,Q) where P is a set of formulas of a fixed propositional language and Q is a set of rules. A formula is deducible from X in the logic (P, Q) if it is deducible from XP via Q. A matrix is strongly adequate to (P, Q) if for any , X, is deducible from X iff for every valuation in , is designated whenever all the formulas in X are. It is proved in the present paper that if Q = {modus ponens, adjunction } and P {E, R, E ⁺, R ⁺, E ^I, R ^I } then there exists a matrix strongly adequate to (P, Q).     

On the Existence of Continua of Logics Between Some Intermediate Predicate Logics

A method for constructing continua of logics squeezed between some intermediate predicate logics, developed by Suzuki [8], is modified and applied to intervals of the form [L, L+ ¬¬S], where Lis a predicate logic, Sis a closed predicate formula. This solves one of the problems from Suzuki's paper.     

The Geometry of Enhancement in Multiple Regression

In linear multiple regression, “enhancement” is said to occur when R ²=b′r>r′r, where b is a p×1 vector of standardized regression coefficients and r is a p×1 vector of correlations between a criterion y and a set of standardized regressors, x. When p=1 then b≡r and enhancement cannot occur. When p=2, for all full-rank R _xx≠I, R _xx=E[xx′]=V Λ V′ (where V Λ V′ denotes the eigen decomposition of R _xx; λ ₁>λ ₂), the set B₁:={b_i:R²=b_i￠r_i=r_i￠r_i;0 < R² ￡ 1}\boldsymbol{B}_{1}:=\{\boldsymbol{b}_{i}:R^{2}=\boldsymbol{b}_{i}'\boldsymbol{r}_{i}=\boldsymbol{r}_{i}'\boldsymbol{r}_{i};0R² ￡ 1;R²l_p ￡ r_i￠r_i < R²}0p≥3 (and λ ₁>λ ₂>⋯>λ _p ), both sets contain an uncountably infinite number of vectors. Geometrical arguments demonstrate that B ₁ occurs at the intersection of two hyper-ellipsoids in ℝ ^p . Equations are provided for populating the sets B ₁ and B ₂ and for demonstrating that maximum enhancement occurs when b is collinear with the eigenvector that is associated with λ _p (the smallest eigenvalue of the predictor correlation matrix). These equations are used to illustrate the logic and the underlying geometry of enhancement in population, multiple-regression models. R code for simulating population regression models that exhibit enhancement of any degree and any number of predictors is included in Appendices A and B.     

A Cut-Free Sequent System for the Smallest Interpretability Logic

The idea of interpretability logics arose in Visser [Vis90]. He introduced the logics as extensions of the provability logic GLwith a binary modality . The arithmetic realization of A B in a theory T will be that T plus the realization of B is interpretable in T plus the realization of A (T + A interprets T + B). More precisely, there exists a function f (the relative interpretation) on the formulas of the language of T such that T + B C implies T + A f(C).The interpretability logics were considered in several papers. An arithmetic completeness of the interpretability logic ILM, obtained by adding Montagna's axiom to the smallest interpretability logic IL, was proved in Berarducci [Ber90] and Shavrukov [Sha88] (see also Hájek and Montagna [HM90] and Hájek and Montagna [HM92]). [Vis90] proved that the interpretability logic ILP, an extension of IL, is also complete for another arithmetic interpretation. The completeness with respect to Kripke semantics due to Veltman was, for IL, ILMand ILP, proved in de Jongh and Veltman [JV90]. The fixed point theorem of GLcan be extended to ILand hence ILMand ILP(cf. de Jongh and Visser [JV91]). The unary pendant "T interprets T + A" is much less expressive and was studied in de Rijke [Rij92]. For an overview of interpretability logic, see Visser [Vis97], and Japaridze and de Jongh [JJ98].In this paper, we give a cut-free sequent system for IL. To begin with, we give a cut-free system for the sublogic IL4of IL, whose -free fragment is the modal logic K4. A cut-elimination theorem for ILis proved using the system for IK4and a property of Löb's axiom.     

Solitaire

Dancing In the Flames: The Dark Goddess in the Transformation of Consciousness. 1996. By Marion Woodman and Elinor Dickson. Boston: Shambala.

Urgings of the Heart: A Spirituality of Integration. 1995. By Wilkie Au and Noreen Cannon Mahwah, NJ: Paulist Press

Cast the First Stone: Ethics in Analytic Practice 1995. By Lena B. Ross and Manisha Roy Wilmette, IL: Chiron

Transforming Sexuality: The Archetypal World of Anima and Animus 1994. By Ann and Barry Ulanov Boston: Shambala

Fate, Love, and Ecstasy: Wisdom from the Lesser-Known Goddesses of the Greeks 1995. By John A. Sanford Wilmette, IL: Chiron

Jungian Perspectives on Clinical Supervision 1995. Edited by Paul Kugler Switzerland: Daimon

Synchronicity, Science and Soul-Making: Understanding Jungian Synchronicity through Physics, Buddhism, and Philosophy 1995. By Victor Mansfield Chicago: Open Court     

Book reviews

Preparing for the Twenty‐First Century, Paul Kennedy, Random House, New York, 1993

The Spirit of Community: Rights, Responsibilities, and the Communitarian Agenda, Amitai Etzione, Crown Publishers, New York, 1993

The Collapse of Chaos, Jack Cohen and Ian Stewart, New York: Viking, 1994.

Quantum Implications: Essays in Honour of David Bohm, B. J. Hiley and David Peat London: Rouledge, 1987.

The Undivided Universe, D. Bohm and B. J. Hiley London: Rouledge, 1993.     

Constructive Logic with Strong Negation is a Substructural Logic. I

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. This answers a longstanding question of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result. The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL _ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL _ew . Presented by Heinrich Wansing     

Book Reviews

Abstract

Learned Resourcefulness On Coping Skills, Self-Control, and Adaptive Behavior Rosenbaum, M. (Ed.) (1990) New York: Springer Publishing Company. Price: $38.95 Reviewed by Lars-Gunnar Lundh.

Nonverbal Communication in Depression Ellgring, H. (1989) Cambridge: Cambridge University Press. Price: $49.50 Reviewed by Lars-Gunnar Lundh.

Psykoanalys och kultur Rieland, H. och Ylander, F. (1991) Stockholm: Natur och Kultur. 250 sidor. Reviewed by Lars-Gunnar Lundh.     

Significance of a Preponderance of Human Movement Responses on the Rorschach in Children Below Age Ten

Abstract

Beck, Samuel J. Rorschach's Test III. Advance in Interpretation. New York: Grune &; Stratton, 1952. Reviewed by Mortimer M. Meyer

Brower, Daniel and Abt, Lawrence Progress in Clinical Psychology. Volume I. New York: Grune &; Stratton, 1952. Reviewed by Ohmer Milton

Buhler, C., Smitter, F., and Richardson, S. Childhood Problems and the Teacher. New York: Henry Holt and Co., 1952. Pp. 372, $3.75. Reviewed by Lois Barclay Murphy

Jolles, Isaac. A Catalogue for the Qualitative Interpretation of the H-T-P. Beverly Hills: Western Psychological Services, 1952, 97 pp. Reviewed by Wilson H. Guertin

Kinget, Marian G., The Drawing-Completion Test. New York: Grune and Stratton, Inc., 1952. Reviewed by Fred Brown

Meili, Richard. Lehrbuch der Psychologischen Diagnostik. Bern: Verlag Hans Huber, 1951, 372 pp. Reviewed by Walther Joel

Szondi, L. Experimental Diagnostics of Drives (Translated by Gertrude Aull). New York: Grune and Stratton, 1952, 220 pp. Reviewed by Henry P. David     

Democratic presumptions

At the Leading Edge: New Visions of Science, Spirituality, and Society. Michael Toms, 4936 State Route 414, Burdett, New York: Larson Publications, 1991. Softcover, 386 pp. U.S. $14.95.

A Dream Deferred: America's Discontent and the Search for a New Democratic Ideal, Philip Slater, Beacon, Boston, $19.95, 1991, ISBN 0–8070–4304–4

The American Replacement of Nature, William Irwin Thompson, Doubleday, New York, $20, 1991, ISBN 0,385–42025–0

Reimagination of the World, A Critique of the New Age, Science, and Popular Culture, David Spangler and William Irwin Thompson, Bear and Company, Santa Fe, New Mexico, $12.95, 1991, ISBN 0–939680–92–0

The New Evolutionary Paradigm, Ervin Laszlo, (Ed.), Gordon &; Breach, New York, $60 (libraries) $36 (individuals), 1991, ISBN 2–88124–375–4

The Evolutionary Approach for Social Scientists, Ignazio Masulli, Gordon &; Breach, New York, $50 (libraries) $30 (individuals), 1991, ISBN 2–88124–376–2

The Fifth Discipline. The Art and Practice of the Learning Organization, Peter M. Senge, Doubleday, New York, $19.95, 1990, ISBN 0–385–26094–6

Symbiosis as a Source of Evolutionary Innovation: Speciation and Innovation, Lynn Margulis and Rene Fester (Eds.), MIT Press, Cambridge, MA, $37.50, 454, Includes Index

One Earth, One Future: Our Changing Global Environment, Cheryl Simon Silver (with Ruth S. Defries for the National Academy of Sciences), National Academy Press, Washington, D.C., $14.95, 196, Includes Index

Toward a Transpersonal Ecology: Developing New Foundations for Environmentalism, Warwick Fox, Shambhala, Boston, MA, $16.95, 380, Includes Index

Gaia, A Way of Knowing: Political Implications of the New Biology, William Irwin Thompson (Ed.), The Lindisfarne Press, Great Barrington, MA, $10.95, 216

Gaia 2: Emergence, The New Science of Becoming, William Erwin Thompson, The Lindisfarne Press, Hudson, NY, $14.95, 272

A Replicative Perspective On Evolutionary Dynamics: The Organizing Process of the US Economy Elaborated Through Biological Metaphor, Mika Pantzar, Labour Institute for Economic Research, Helsinki, Finland, 274

Societal Systems: Planning, Policy and Complexity &; A Science of Generic Design: Managing Complexity through Systems Design, Vol I and II, John N. Warfield, Intersystems Publications, Salinas, CA, 1989 and 1990     

Brief eviews

Developments in humanistic psychology: Recent books by division members

Christopher M, Aanstoos (Ed.). Studies in Humanistic Psychology. West Georgia College Studies in the Social Sciences (Volume 29). Carrollton, GA: West Georgia College, 1991, 216 pp. $5.00.

Frank Barron. Creativity and Psychological Health: Origins of Personal Vitality and Creative Freedom. Buffalo: C.E.F. Press, 1990.

James F. T. Bugental. Intimate Journeys: Stories from Life‐Changing Therapy. San Francisco: Jossey‐Bass, 1990, 335 pp. $24.95.

Alvin R. Mahrer. How to Do Experiential Psychotherapy: A Manual for Practitioners. Ottawa: University of Ottawa Press, 1989. 153 pp. $12.95.

Alvin R. Mahrer. Dreamwork in Psychotherapy and Self‐Change. New York: W. W. Norton. 372 pp. $34.95.

Clark Moustakas. Heuristic Research: Design, Methodology, and Applications. Newbury Park, CA: Sage Publications, 1990. 130 pp. $14.95.

Clark Moustakas. Loneliness. New York: Prentice Hall Press, 1990, $6.95. and Clark Moustakas. Loneliness and Love. New York: Prentice Hall Press, 1990. 146 pp. $7.95.

Kirk J. Schneider. The Paradoxical Self: Toward an Understanding of our Contradictory Nature. New York: Plenum Press, 1990. 235 pp. $20.95.

M. Brewster Smith. Values, Self & Society: Toward a Humanistic Social Psychology. New Brunswick, NJ: Transaction Publishers, 1991, 289 pp. $34.95.

Richard E. Felder and Avrum Geurin Weiss. Experiential Psychotherapy: A Symphony of Selves. Lanham, MD: University Press of America, 1991. 164 pp.     

Some paraconsistent sentential calculi

In [8] Jakowski defined by means of an appropriate interpretation a paraconsistent calculusD ₂ . In [9] J. Kotas showed thatD ₂ is equivalent to the calculusM(S5) whose theses are exactly all formulasa such thatMa is a thesis ofS5. The papers [11], [7], [3], and [4] showed that interesting paraconsistent calculi could be obtained using modal systems other thanS5 and modalities other thanM. This paper generalises the above work. LetA be an arbitrary modality (i.e. string ofM's,L's and negation signs). Then theA-extension of a set of formulasX is {¦A X}}. Various properties ofA-extensions of normal modal systems are examined, including a problem of their axiomatizability