A reduction rule for Peirce formula

A reduction rule is introduced as a transformation of proof figures in implicational classical logic. Proof figures are represented as typed terms in a -calculus with a new constant P ⁽⁽⁾⁾. It is shown that all terms with the same type are equivalent with respect to -reduction augmented by this P-reduction rule. Hence all the proofs of the same implicational formula are equivalent. It is also shown that strong normalization fails for P-reduction. Weak normalization is shown for P-reduction with another reduction rule which simplifies of (( ) ) into an atomic type.This work was partially supported by a Grant-in-Aid for General Scientific Research No. 05680276 of the Ministry of Education, Science and Culture, Japan and by Japan Society for the Promotion of Science. Hiroakira Ono     

Dowód aksjomatyzowalności pewnychn-wartościowych rachunków zdań

Praca ta zostaa wykonana pod kierunkiem prof. drJerzego Supeckiego. Jej szczególny przypadek dotyczcyn-wartociowego rachunku zda o jednej wartoci wyrónionej by referowany w Warszawie na zjedzie Grupy Logiki Instytutu Filozofii i Socjologii PAN. Streszczenie tego referatu [5] zostao wydrukowane w Sprawozdaniach Wydziau Nauk Spoecznych PAN.Allatum est die 19 Aprilis 1963     

Interpretations of the first-order theory of diagonalizable algebras in peano arithmetic

For every sequence |p _n } _n of formulas of Peano ArithmeticPA with, every formulaA of the first-order theory diagonalizable algebras, we associate a formula ⁰ A, called the value ofA inPA with respect to the interpretation. We show that, ifA is true in every diagonalizable algebra, then, for every, ⁰ A is a theorem ofPA.     

Intuitionistic uniformity principles for propositions and some applications

This note deals with the prepositional uniformity principlep-UP: p x N A (p, x) x N p A (p, x) ( species of all propositions) in intuitionistic mathematics.p-UP is implied by WC and KS. But there are interestingp-UP-cases which require weak KS resp. WC only. UP for number species follows fromp-UP by extended bar-induction (ranging over propositions) and suitable weak continuity. As corollaries we have the disjunction property and the existential definability w.r.t. concrete objects. Other consequences are: there is no non-trivial countable partition of;id is the only injective function from to; there are no many-place injective prepositional functions; card () is incomparable with the cardinality of all metric spaces containing at least three elements.     

Eliminativism and the ambiguity of ‘belief’

It has recently been claimed (1) that mental states such as beliefs are theoretical entities and (2) that they are therefore, in principle, subject to theoretical elimination if intentional psychology were to be supplanted by a psychology not employing mentalistic notions. Debate over these two issues is seriously hampered by the fact that the key terms theoretical and belief are ambiguous. This article argues that there is only one sense of theoretical that is of use to the eliminativist, and in this sense some kinds of belief (dispositional states, infra-conscious states and the Freudian unconscious) are indeed theoretical and hence possible candidates for elimination, while others (consciously occurring thoughts like judgements and perceptualGestalten) are not theoretical and hence not candidates for elimination.     

Modal Logics of Succession for 2-Dimensional Integral Spacetime

We consider the problem of axiomatizing various natural successor logics for 2-dimensional integral spacetime. We provide axiomatizations in monomodal and multimodal languages, and prove completeness theorems. We also establish that the irreflexive successor logic in the standard modal language (i.e. the language containing and ) is not finitely axiomatizable.     

On a problem of P(α, δ, π) concerning generalized Alexandroff s cube

Universality of generalized Alexandroff's cube plays essential role in theory of absolute retracts for the category of , -closure spaces. Alexandroff's cube. is an , -closure space generated by the family of all complete filters. in a lattice of all subsets of a set of power .Condition P(, , ) says that is a closure space of all , -filters in the lattice ( ), .Assuming that P (, , ) holds, in the paper [2], there are given sufficient conditions saying when an , -closure space is an absolute retract for the category of , -closure spaces (see Theorems 2.1 and 3.4 in [2]).It seems that, under assumption that P (, , ) holds, it will be possible to givean uniform characterization of absolute retracts for the category of , -closure-spaces.Except Lemma 3.1 from [1], there is no information when the condition P (, , ) holds or when it does not hold.The main result of this paper says, that there are examples of cardinal numbers, , , such that P (, , ) is not satisfied.Namely it is proved, using elementary properties of Lebesgue measure on the real line, that the condition P (, ₁, 2 ) is not satisfied.Moreover it is shown that fulfillment of the condition is essential assumption in, Theorems 2.1 and 3.4 from [1] i.e. it cannot be eliminated.     

The artist as transgressor in Mandel'štam's poetry

In Mandel'tam's writing, artistic creativity is described as based on the indispensable yet contradictory modes of compliance and deviation. The artist, by his artistic nature, must be an obedient disciple to the tradition that inspires him, and, at the same time, a violator who renders what inspires him in an individual form. Thus, art implies iterability through novelty. In the totalitarian state, this double nature of art acquires a sinister context and brings the artist to an unavoidable conflict with the state. He has a choice between a servile compliance with the state's command and artistic independence. If the artist complies, he loses his ingenuity; if, on the other hand, he has the courage to break away from the established order, his fate is martyrdom. The criteria of truth and falsehood, the issue of loyalty, of compromise and collaboration or resistance become most relevant. Such words as outcast or non-contemporary acquire the meaning of non-collaborationist or enemy of the people. In the totalitarian state a genuine artist is viewed as a law-breaker, and his art leads him to crime. The notions of compliance and deviation cease being merely aesthetic terms and assume in Mandel'tam's poetry complex, subtle and tragic overtones.I would like to express my gratitude to John Barnstead, Thomas Blakeley, James Doull, Elena Glazov-Corrigan, Gregory Glazov, Leonard Kazdan, Peter Kussmaul, Malcolm Ross, Detlev Steffen, and Ieva Vitins for their valuable criticism, suggestions, and revisions of my English.     

Third Men: The Logic of the Sophisms at Arist. SE 22, 178b36–170a10

This article aims at elucidating the logic of Arist. SE 22, 178b36–179a10 and, in particular, of the sophism labelled "Third Man" discussed in it. I suggest that neither the sophistic Walking Man argument, proposed by ancient commentators, nor the Aristotelian Third Man of the , suggested by modern interpreters, can be identified with the fallacious argument Aristotle presents and solves in the passage. I propose an alternative reconstruction of the Third Man sophism and argue that an explanation of the lines regarding the identity of Coriscus and Coriscus the musician (178b39–179a3) is indispensable for its correct understanding, since they hint at another sophism in some important aspects analogous. Finally, I show that two contradictions concerning spotted by scholars in the passage are only apparent and can be dissolved once the assumption that the anti-Platonic Third Man argument is at stake here is discarded, and once the passage is read in the light of its agonistic context.     

The responsive order: A new empiricism

The uniqueness of logic is upheld and contrasted with twenty roles of a wider responsive order that includes us and our procedures. Empirical responses are precise, but different in different approaches. Procedures and findings are independent of (not separable from) their concepts. Two-way feedback obviates a top-down derivation of findings from assumptions, hypotheses, history, or language. The postmodern problems of interpretation, conditions of appearances and relativism involve the ancient error of making perception the model-instance of experience. Instead, bodily interaction functions in language and precedes perception and interpretation. Logic, space time locations and individuated referents involve positional relations derived from comparing. Beyond Kuhn, Feyerabend, Newton and Einstein, if we can give interaction priority over comparing, the responsive objectivity of both can be upheld. A new empiricism, neither naive nor constructivist, uses the words order, explication, truth, and exactly to build on Wittgenstein and on Dilthey's hermeneutic. Natural language is metaphor-like, originally crossed. Logic must ignore its assumptions. It must render everything as a machine and drop humans and animals out. A new discipline is proposed, to move between the logical and the responsive orders, to deal with the machine/human interface and the social uses of science such as bioengineering.     

A longitudinal study of the marriage role expectations of college women: 1961–1984

The marriage role expectations of female university students enrolled in an introductory marriage and family course at the University of Kentucky in 1961 (N=72), 1972 (N=58), 1978 (N=115), and 1984 (N=81) were compared. The 1972 group was found to be significantly more egalitarian than the 1961 group in overall marriage role expectations and on all subscales except authority. The 1978 and 1984 groups were significantly more egalitarian than the 1961 group on overall marriage role expectations and on all seven of the subscales: authority, homemaking, child care, personal characteristics, social participation, education, and employment and support. The 1978 group was significantly more egalitarian than the 1972 group only on authority and homemaking, with no significant differences on the other five subscales or on overall role expectation score. The 1984 group was significantly more egalitarian than the 1972 group only on authority, homemaking, and child care, and was slightly (not satistically significant) more traditional on personal characteristics and social participation. There were no significant differences between the 1978 and 1984 groups on overall expectation or on any of the subscales. In fact, the 1984 group was slightly more traditional on homemaking, personal characteristics, social participation, employment and support, and overall expectations. The data from this study showed the greatest gain in egalitarian trends to be between 1961 and 1972, with a continuation of the trends at a somewhat slower rate of change between 1972 and 1978, and a discontinuation of the trend toward more egalitarian expectations between 1978 and 1984.The data were collected in February 1961, January 1972, September 1978, and September 1984. Special acknowledgement and appreciation are expressed to Jessie Ringo (retired), University of Kentucky, for making the 1961 data available for this study.     

Confirmation holism and semantic holism

Fodor and Lepore, in their recent book Holism, maintain that if an inference from semantic anatomism to semantic holism is allowed, certain fairly deleterious consequences follow. In Section 1 Fodor and Lepore's terminology is construed and amended where necessary with the result that the aforementioned deleterious consequences are neither so apparent nor straightforward as they had suggested. In Section 2 their Argument A is considered in some detail. In Section 3 their argument attributed to Quine is examined at length and a shorter and more perspicacious argument suggested which avoids their charge that the Quinean argument is guilty of an equivocation on the word statement.     

The subjective vertical under ‘dry’ and ‘wet’ conditions at clockwise and counterclockwise changed positions and the effect of a parallel-lined background field

Summary The dependence of the subjective vertical (SV, the angle between a subjective vertical line and body median plane) on the gravity vertical (body tilt position, angle ) and on the optical vertical (i.e., a field of parallel lines seen as background to the line to be adjusted) was investigated. The SV was measured under dry and wet conditions at different degrees of body tilt attained in either clockwise (CW) or counterclockwise (CCW) progression.The measured difference in between field-of-lines left and right of the line is smallest at the upright position (=O°) and largest at =150°/165°. All body positions show a -difference between CW and CCW attainment (hysteresis), this too being least at upright and greatest at inverted body positions.These results, and changes of with test time, are discussed relative to the hypothesis that efficiency of the statolith organs decreases with body tilt increase, favouring increase of interference of somatoreceptors and the optical reference.     

Before refraining: Concepts for agency

A structure is described that can serve as a foundation for a semantics for a modal agentive construction such as sees to it that Q ([ stit: Q]). The primitives are Tree,,Instant, Agent, choice. Eleven simple postulates governing this structure are set forth and motivated. Tree and encode a picture of branching time consisting of moments gathered into maximal chains called histories. Instant imposes a time-like ordering. Agent consists of agents, and choice assigns to each agent and each moment in Tree a set of possible choices, where each possible choice is a set of histories. All of these ingredients are referred to in the semantics suggested for [ stit: Q]. The most complex part of the discussion is the motivation for the definition of what it means for a typically non-terminating chain of moments jointly to witness the truth of [ stit: Q] at a moment.The paper begins with an informal account of the Refref conjecture, which says that the only way to refrain from refraining from seeing to something is to see to it. The paper ends with a consideration of an argument of Prior's that in a certain sense contemplation and action are inconsistent.     

Theological and Biblical Interpretations of Current Movements and Emerging Paradigms Within Bereavement Studies

Among the predominant themes in the last few years of bereavement studies have been those around disenfranchised grief, masculine grief, spirituality and grief, and the movement from medical/hierarchical models of grief work/counseling to a more non-hierarchical model that emerges internally, inherently, and intrinsically within the bereaved. In the latter model the grief counselor becomes a facilitator or, as Alan Wolfelt noted at the Chicago ADEC conference, a companioning presence. The intent of this article is to take this conversation into the realm of theology, and/or biblical theology. The following offers theological analysis of what is already good clinical theory. Integrating theories from anthropology and theology, this article will explore the concepts of voice, etic and emic theory, and incarnational theology as they relate to emerging patterns in bereavement studies. It will also relate a model of the Kingdom of God to disenfranchised grief, as well as offer support for the theory of masculine grief from biblical sources and complementary fields of study.     

New lower and upper bounds for communality in factor analysis

We derive several relationships between communalities and the eigenvalues for ap ×p correlation matrix under the usual factor analysis model. For suitable choices ofj, _j (), where _j () is thej-th largest eigenvalue of , provides either a lower or an upper bound to the communalities for some of the variables. We show that for at least one variable, 1 - _p () improves on the use of squared mulitiple correlation coefficient as a lower bound.This research was done while the second author was at Tokyo Institute of Technology.     

Algebraic aspects of deduction theorems

The first known statements of the deduction theorems for the first-order predicate calculus and the classical sentential logic are due to Herbrand [8] and Tarski [14], respectively. The present paper contains an analysis of closure spaces associated with those sentential logics which admit various deduction theorems. For purely algebraic reasons it is convenient to view deduction theorems in a more general form: given a sentential logic C (identified with a structural consequence operation) in a sentential language I, a quite arbitrary set P of formulas of I built up with at most two distinct sentential variables p and q is called a uniform deduction theorem scheme for C if it satisfies the following condition: for every set X of formulas of I and for any formulas and , C(X{{a}}) iff P(, ) AC(X). [P(, ) denotes the set of formulas which result by the simultaneous substitution of for p and for q in all formulas in P]. The above definition encompasses many particular formulations of theorems considered in the literature to be deduction theorems. Theorem 1.3 gives necessary and sufficient conditions for a logic to have a uniform deduction theorem scheme. Then, given a sentential logic C with a uniform deduction theorem scheme, the lattices of deductive filters on the algebras A similar to the language of C are investigated. It is shown that the join-semilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian.A part of this paper was presented in abstracted form in Bulletin of the Section of Logic, Vol. 12, No. 3 (1983), pp. 111–116, and in The Journal of Symbolic Logic.     

A modification of kendall's tau for measuring association in contingency tables

A coefficient of association is described for a contingency table containing data classified into two sets of ordered categories. Within each of the two sets the number of categories or the number of cases in each category need not be the same.=+1 for perfect positive association and has an expectation of 0 for chance association. In many cases also has –1 as a lower limit. The limitations of Kendall's _a and _b and Stuart's _c are discussed, as is the identity of these coefficients to' under certain conditions. Computational procedure for is given.     

Plantinga’s Reformed Epistemology: What’s the Question?

Alvin Plantingas Warranted Christian Belief is without questionone of the central texts of the Reformed epistemology movement. Critiques of Plantingas defence have been both multiple and varied. As varied as these responses are, however, it is my contention that many of them amount to the same thing. It is the purpose of this paper to offer an overview of the main lines of attack that have been directed as Plantingas project, and thereafter to show how many, if not most, of these objections can be understood as versions or aspects of the same criticism, what I call the Inadequacy Thesis.     

Foundations of a new system of probability theory

The aim of my book is to explain the content of the different notions of probability.Based on a concept of logical probability, which is modified as compared with Carnap, we succeed by means of the mathematical results of de Finetti in defining the concept of statistical probability.The starting point is the fundamental concept that certain phenomena are of the same kind, that certain occurrences can be repeated, that certain experiments are identical. We introduce for this idea the notion: concept K of similarity. From concept K of similarity we derive logically some probability-theoretic conclusions:If the events E() are similar —of the same kind - on the basis of such a concept K, it holds good that intersections of n of these events are equiprobable on the basis of K; in formulae: E(₁)...E( _{n K} E('₁)...E(' _n , _{i j} ,' _j ' _j for ij On the basis of some further axioms a partial comparative probability structure results from K, which forms the starting point of our further investigations and which we call logical probability on the basis of K.We investigate a metrisation of this partial comparative structure, i.e. normed -additive functions m _K, which are compatible with this structure; we call these functions m _K measure-functions in relation to K.The measure-functions may be interpreted as subjective probabilities of individuals, who accept the concept K.Now it holds good: For each measure-function there exists with measure one the limit of relative frequencies in a sequence of the E().In such an event, where all measure-functions coincide, we speak of a quantitative logical probability, which is the common measure of this event. In formulae we have: l _K (h _n lim h _n )=1 in words: There is the quantitative logical probability one that the limit of the relative frequencies exists. Another way of saying this is that the event ^* (h_n lim h _n) is a maximal element in the comparative structure resulting from K.Therefore we are entitled to introduce this limit and call it statistical probability P.With the aid of the measure-functions it is possible to calculate the velocity of this convergence. The analog of the Bernoulli inequation holds true: m _K (¦h _n –P¦)1–1/4n².It is further possible in the work to obtain relationships for the concept of statistical independence which are expressed in terms of the comparative probability.The theory has a special significance for quantum mechanics: The similarity of the phenomena in the domain of quantum mechanics explains the statistical behaviour of the phenomena.The usual mathematical statistics are explained in my book. But it seems more expedient on the basis of this new theory to use besides the notion of statistical probability also the notion of logical probability; the notion of subjective probability has only a heuristic function in my system.The following dualism is to be noted: The statistical behaviour of similar phenomena may be described on the one hand according to the model of the classical probability theory by means of a figure called statistical probability, on the other hand we may express all formulae by means of a function, called statistical probability function. This function is defined as the limit of the relative frequencies depending on the respective state of the universe. The statistical probability function is the primary notion, the notion of statistical probability is derived from it; it is defined as the value of the statistical probability function for the true unknown state of the universe.As far as the Hume problem, the problem of inductive inference, is concerned, the book seems to give an example of how to solve it.The developed notions such as concept, measure-function, logical probability, etc. seem to be important beyond the concept of similarity.The present work represents a summary of my book Grundzüge zu einem neuen Aufbau der Wahrscheinlich-keitstheorie [5], For this reason, I have frequently dispensed with providing proof and in this connection refer the interested reader to my book.