Completeness of quantum logic

This paper is based on a semantic foundation of quantum logic which makes use of dialog-games. In the first part of the paper the dialogic method is introduced and under the conditions of quantum mechanical measurements the rules of a dialog-game about quantum mechanical propositions are established. In the second part of the paper the quantum mechanical dialog-game is replaced by a calculus of quantum logic. As the main part of the paper we show that the calculus of quantum logic is complete and consistent with respect to the dialogic semantics. Since the dialog-game does not involve the excluded middle the calculus represents a calculus of effective (intuitionistic) quantum logic.In a forthcoming paper it is shown that this calculus is equivalent to a calculus of sequents and more interestingly to a calculus of propositions. With the addition of the excluded middle the latter calculus is a model for the lattice of subspaces of a Hilbert space.On leave of absence from the Institut für Theoretische Physik der Universität zu Köln, W.-Germany.     

Kenneth Strike on Liberalism,Citizenship and the Private Interest in Schooling

After indicating a number of points of agreement with the argument 0eveloped by Kenneth Strike in his article Liberalism, Citizenship and the Private Interest in Schooling, this article identifies and explores a number of queries and criticisms which arise in relation to that argument. These queries and criticisms relate especially to the nature and extent of the expansiveness involved in Strike's conception of public or common educational influence, and to the implications and justification of the claim that private educational interests enjoy a greater salience and recognition on Strike's view of public or common educational influence than on some alternative views.     

An abstract setting for Henkin proofs

A general result is proved about the existence of maximally consistent theories satisfying prescribed closure conditions. The principle is then used to give streamlined proofs of completeness and omitting-types theorems, in which inductive Henkin-style constructions are replaced by a demonstration that a certain theory respects a certain class of inference rules.     

Theorem counting

Consider the set of tautologies of the classical propositional calculus containing no connective other than and, or, and not. Consider the subset of this set containing tautologies in exactlyn propositional variables. This paper provides a method for determining the number of equivalence classes of each such subset modulo equivalence in the infinite-valued Lukasiewicz propositional calculus.     

Family-of-Origin Predictors of Hostile Conflict in Early Marriage*

This study used data from 367 married couples to examine the relationship between select family-of-origin variables measured during engagement and hostile marital conflict measured 4–7 years later. Each spouse completed family-of-origin measures from the PREParation for Marriage questionnaire during their engagement. Four to seven years later, they completed a measure of hostile marital conflict involving insults, name calling, unwillingness to listen, and lack of emotional involvement. Wives mother–child relationship and the quality of parental discipline, as well as the husbands perceived quality of their parents marriage, were predictive of wives perception of hostile conflict in their marriages. Only wives quality of parental discipline was predictive of husbands perception of hostile marital conflict. Implications for premarital counseling and marital therapy are discussed.This is a revised version of a poster presented at the Annual Conference of the National Council on Family Relations, Irvine, California, November, 1999.     

On two arguments for the indeterminacy of personal identity

Both arguments are based on the breakdown of normal criteria of identity in certain science-fictional circumstances. In one case, normal criteria would support the identity of person A with each of two other persons, B and C; and it is argued that, in the imagined circumstances, A=B and A=C have no truth value. In the other, a series or spectrum of cases is tailored to a sorites argument. At one end of the spectrum, persons A and B are such that A=B is clearly true; at the other end, A and B are such that the identity is clearly false. In between, normal criteria of identity leave the truth or falsehood of A=B undecided, and it is argued that in these circumstances A=B has no truth value.These arguments are to be understood counterfactually. My claim is that, so understood, neither establishes its conclusion. The first involves a pair of counterfactual situations that are equally possible or tied. If A=B and A=C have no truth value, a counterfactual conditional with one of them as consequent and an antecedent that is true in circumstances in which either is true should have no truth value. Intuitively, however, any such counterfactual is false. The second argument can be seen to invite an analogous response. If this is right, however, there is an important disanalogy between this and the classical paradox of the heap. If the disanalogy is only apparent, the argument shows at most that the existence of persons can be indeterminate.     

A partially ordered extention of the integers

This paper presents a monotonic system of Post algebras of order +* whose chain of Post constans is isomorphic with 012 ... -3-2-1. Besides monotonic operations, other unary operations are considered; namely, disjoint operations, the quasi-complement, succesor, and predecessor operations. The successor and predecessor operations are basic for number theory.The editing of this unified version of two previous papers [Epstein, Rasiowa 1990, 1991] by the authors was completed, except for footnotes, while the first author visited at Warsaw University during June–July, 1994. The footnotes were added by the first author in September, 1994.Presented byRyszard Wójcicki     

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     

A game-based formal system for Ł∞

A formal system for , based on a game-theoretic analysis of the ukasiewicz prepositional connectives, is defined and proved to be complete. An Herbrand theorem for the predicate calculus (a variant of some work of Mostowski) and some corollaries relating to its axiomatizability are proved. The predicate calculus with equality is also considered.     

The relationship between perceived surface-lightness and perceived illumination

Summary An attempt was made to specify the relationship between perceived surface lightness and perceived illumination under the stimulus conditions where different combinations of albedo and illuminance gave 10 approximately equal-luminance levels of a test field (TF) on three different (black-, gray- and white-appearing) backgrounds. Two types of category judgments for TF-lightness and overall illumination were made on the total of 94 TFs by 5 Ss. The results indicated that under the condition where achromatic surface colors appear, the perceptual scission which produces two different perceptual dimensions (lightness and perceived illumination) from one sort of stimulation (luminance) was clearly observed. The relation between the two judgments was consistent with the lightness-illumination invariance hypothesis: As lightness judgments (A) changed from darker to lighter, illumination judgments (I) shifted form brighter to dimmer, the sum of A and I being kept nearly invariant for a given luminance of the TF; the psychological relationship between lightness and illumination, A+I, changed as a linear function of the photometric combination of albedo and illuminance, log A+log I. It was also found that the albedo of the background was an important factor determining the extent of perceptual scission between lightness and illumination.     

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.     

Stuffed Cabbage in the Old New School Cafeteria

The purpose of this lecture is to celebrate the memory of Aron Gurwitsch by examining and enlarging the domain of phenomenological clarification of some elements of what Gurwitsch called the logic of reality. Chief among those elements are the nature of the taken-for-grantedness of our existential belief, the difference between presentive and non-presentive indices of reality and the ground for the self-illumination of the world of working.     

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.     

‘Outdoing Lewis Carroll’: Judicial Rhetoric and Acceptable Fictions

This paper examines the functions of narrative within legal argumentation. My purposes are these: 1) to repudiate common assumptions that differentiate argumentation and storytelling in the law; 2) to begin to theorize anew how legal argumentation functions; 3) to explore the difficulties of evaluating the law's argumentative narratives; and 4) to trace some of the anxiety that judges themselves reveal about their roles as storytellers. I conclude that narrative is necessary to law's claims to authority, even as it complicates our understandings about how legislative policy decisions produce effects, and even as judges themselves seek to mask its importance.     

Set theory as modal logic

A logical systemBM ⁺ is proposed, which, is a prepositional calculus enlarged with prepositional quantifiers and with two modal signs, and These modalities are submitted to a finite number of axioms. is the usual sign of necessity, corresponds to transmutation of a property (to be white) into the abstract property (to be the whiteness). An imbedding of the usual theory of classesM intoBM ⁺ is constructed, such that a formulaA is provable inM if and only if(A) is provable inBM ⁺. There is also an inverse imbedding with an analogous property.     

Philosophie der Arrieregarde

Bulgarian philosophical life is particularly poor and fruitless. As well as in all communist countries the development of Bulgarian Marxism-Leninism was marked by some very sharp controversies between dogmatics and revisionists. Bulgarian Orthodoxes are particularly militant and intolerant and on the contrary, revisionists are in reality very moderate and rather conservative. Therefore the deviations are insignificant and the degree of the destalinization is very small.     

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.     

Conditionals and theory change: Revisions,expansions, and additions

This paper dwells upon formal models of changes of beliefs, or theories, which are expressed in languages containing a binary conditional connective. After defining the basic concept of a (non-trivial) belief revision model. I present a simple proof of Gärdenfors's (1986) triviality theorem. I claim that on a proper understanding of this theorem we must give up the thesis that consistent revisions (additions) are to be equated with logical expansions. If negated or might conditionals are interpreted on the basis of autoepistemic omniscience, or if autoepistemic modalities (Moore) are admitted, even more severe triviality results ensue. It is argued that additions cannot be philosophically construed as parasitic (Levi) on expansions. In conclusion I outline somed logical consequences of the fact that we must not expect monotonic revisions in languages including conditionals.I wish to thank Peter Gärdenfors for a number of helpful comments, André Fuhrmann and Wolfgang Spohn for extensive discussion of parts of this paper, and Winfred Klink for kindly checking my English.     

Proof-Theoretic Modal PA-Completeness I: A System-Sequent Metric

This paper is the first of a series of three articles that present the syntactic proof of the PA-completeness of the modal system G, by introducing suitable proof-theoretic objects, which also have an independent interest. We start from the syntactic PA-completeness of modal system GL-LIN, previously obtained in [7], [8], and so we assume to be working on modal sequents S which are GL-LIN-theorems. If S is not a G-theorem we define here a notion of syntactic metric d(S, G): we calculate a canonical characteristic fomula H of S (char(S)) so that _G H (S) and _GL-LIN H, and the complexity of H gives the distance d(S, G) of S from G. Then, in order to produce the whole completeness proof as an induction on this d(S, G), we introduce the tree-interpretation of a modal sequent Q into PA, that sends the letters of Q into PA-formulas describing the properties of a GL-LIN-proof P of Q: It is also a d(*, G)-metric linked interpretation, since it will be applied to a proof-tree T of H with H = char(S) and ( H) = d(S, G).     

Undecidability without Arithmetization

In the present paper the well-known Gödels – Churchs argument concerning the undecidability of logic (of the first order functional calculus) is exhibited in a way which seems to be philosophically interestingfi The natural numbers are not used. (Neither Chinese Theorem nor other specifically mathematical tricks are applied.) Only elementary logic and very simple set-theoretical constructions are put into the proof. Instead of the arithmetization I use the theory of concatenation (formalized by Alfred Tarski). This theory proves to be an appropriate tool. The decidability is defined directly as the property of graphical discernibility of formulas.