Omega-consistency and the diamond

G is the result of adjoining the schema (qAA)qA to K; the axioms of G* are the theorems of G and the instances of the schema qAA and the sole rule of G* is modus ponens. A sentence is -provable if it is provable in P(eano) A(rithmetic) by one application of the -rule; equivalently, if its negation is -inconsistent in PA. Let -Bew(x) be the natural formalization of the notion of -provability. For any modal sentence A and function mapping sentence letters to sentences of PA, inductively define A by: p = (p) (p a sentence letter); = ; (AB)su}= (A B); and (qA)= -Bew(A )(S) is the numeral for the Gödel number of the sentence S). Then, applying techniques of Solovay (Israel Journal of Mathematics 25, pp. 287–304), we prove that for every modal sentence A, _G A iff for all , _PA A ; and for every modal sentence A, _G* A iff for all , A is true.I should like to thank David Auerbach and Rohit Parikh.     

Modal operators with probabilistic interpretations,I

We present a class of normal modal calculi P_FD, whose syntax is endowed with operators M _r (and their dual ones, L _r), one for each r [0,1]: if a is sentence, M _r is to he read the probability that a is true is strictly greater than r and to he evaluated as true or false in every world of a F-restricted probabilistic kripkean model. Every such a model is a kripkean model, enriched by a family of regular (see below) probability evaluations with range in a fixed finite subset F of [0,1]: there is one such a function for every world w, P _F(w,-), and this allows to evaluate M ^ra as true in the world w iff p _F(w, ) r.For every fixed F as before, suitable axioms and rules are displayed, so that the resulting system P _FD is complete and compact with respect to the class of all the F-restricted probabilistic kripkean models.     

The logic of linear tolerance

A nonempty sequence T₁,...,T_n of theories is tolerant, if there are consistent theories T ₁ ⁺ ,..., T _n ⁺ such that for each 1 i n, T _i ⁺ is an extension of T_i in the same language and, if i n, T _i ⁺ interprets T _i+1 ⁺ . We consider a propositional language with the modality , the arity of which is not fixed, and axiomatically define in this language the decidable logics TOL and TOL. It is shown that TOL (resp. TOL) yields exactly the schemata of PA-provable (resp. true) arithmetical sentences, if (A₁,..., A_n) is understood as (a formalization of) PA+A₁, ..., PA+A_n is tolerant.     

Varieties of Monadic Heyting Algebras. Part III

This paper is the concluding part of [1] and [2], and it investigates the inner structure of the lattice (MHA) of all varieties of monadic Heyting algebras. For every n , we introduce and investigate varieties of depth n and cluster n, and present two partitions of (MHA), into varieties of depth n, and into varieties of cluster n. We pay a special attention to the lower part of (MHA) and investigate finite and critical varieties of monadic Heyting algebras in detail. In particular, we prove that there exist exactly thirteen critical varieties in (MHA) and that it is decidable whether a given variety of monadic Heyting algebras is finite or not. The representation of (MHA) is also given. All these provide us with a satisfactory insight into (MHA). Since (MHA) is dual to the lattice NExtMIPC of all normal extensions of the intuitionistic modal logic MIPC, we also obtain a clearer picture of the lattice structure of intuitionistic modal logics over MIPC.     

Some modifications of Scott's theorem on injective spaces

D. Scott in his paper [5] on the mathematical models for the Church-Curry -calculus proved the following theorem.A topological space X. is an absolute extensor for the category of all topological spaces iff a contraction of X. is a topological space of Scott's open sets in a continuous lattice.In this paper we prove a generalization of this theorem for the category of , -closure spaces. The main theorem says that, for some cardinal numbers , , absolute extensors for the category of , -closure spaces are exactly , -closure spaces of , -filters in , >-semidistributive lattices (Theorem 3.5).If = and = we obtain Scott's Theorem (Corollary 2.1). If = 0 and = we obtain a characterization of closure spaces of filters in a complete Heyting lattice (Corollary 3.4). If = 0 and = we obtain a characterization of closure space of all principial filters in a completely distributive complete lattice (Corollary 3.3).     

Sheaves over Heyting lattices

For a complete Heyting lattice , we define a category Etale (). We show that the category Etale () is equivalent to the category of the sheaves over , Sh(), hence also with -valued sets, see [2], [1]. The category Etale() is a generalization of the category Etale (X), see [1], where X is a topological space.     

A semantical investigation into Leśniewski's axiom of his ontology

A structure A for the language L, which is the first-order language (without equality) whose only nonlogical symbol is the binary predicate symbol , is called a quasi -struoture iff (a) the universe A of A consists of sets and (b) a b is true in A ([p) a = {p } & p b] for every a and b in A, where a(b) is the name of a (b). A quasi -structure A is called an -structure iff (c) {p } A whenever p a A. Then a closed formula in L is derivable from Leniewski's axiom x, y[x y u (u x) u; v(u, v x u v) u(u x u y)] (from the axiom x, y(x y x x) x, y, z(x y z y x z)) iff is true in every -structure (in every quasi -structure).     

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.     

Equivalents of Mingle and Positive Paradox

Relevant logic is a proper subset of classical logic. It does not include among its theorems any ofpositive paradox A (B A)mingle A (A A)linear order (A B) (B A)unrelated extremes (A ) (B B¯)This article shows that those four formulas have different effects when added to relevant logic, and then lists many formulas that have the same effect as positive paradox or mingle.     

A note on the ω-incompleteness formalization

The paper studies two formal schemes related to -completeness.LetS be a suitable formal theory containing primitive recursive arithmetic and letT be a formal extension ofS. Denoted by (a), (b) and (c), respectively, are the following three propositions (where (x) is a formula with the only free variable x): (a) (for anyn) ( _T (n)), (b) _{T x} Pr _T (^–(x)^–) and (c) _T x(x) (the notational conventions are those of Smoryski [3]). The aim of this paper is to examine the meaning of the schemes which result from the formalizations, over the base theoryS, of the implications (b) (c) and (a) (b), where ranges over all formulae. The analysis yields two results overS : 1. the schema corresponding to (b) (c) is equivalent to ¬Cons _T and 2. the schema corresponding to (a) (b) is not consistent with 1-CON _T. The former result follows from a simple adaptation of the -incompleteness proof; the second is new and is based on a particular application of the diagonalization lemma.Presented byMelvin Fitting     

Parameter estimation in latent trait models

Latent trait models for binary responses to a set of test items are considered from the point of view of estimating latent trait parameters=( ₁, , _n ) and item parameters=( ₁, , _k ), where _j may be vector valued. With considered a random sample from a prior distribution with parameter, the estimation of (, ) is studied under the theory of the EM algorithm. An example and computational details are presented for the Rasch model.This work was supported by Contract No. N00014-81-K-0265, Modification No. P00002, from Personnel and Training Research Programs, Psychological Sciences Division, Office of Naval Research. The authors wish to thank an anonymous reviewer for several valuable suggestions.     

On a certain class ofm-address machines

Conclusion It follows from the proved theorems that ifM =Q, (whereQ={0,q ₁,q ₂,...,q }) is a machine of the classM _F then there exist machinesM _i such thatM _i(1,c)=M (q _i,c) andQ _i={0, 1, 2, ..., +1} (i=1, 2, ..., ).And thus, if the way in which to an initial function of content of memorycC a machine assigns a final onecC is regarded as the only essential property of the machine then we can deal with the machines of the formM ={0, 1, 2, ..., }, and processes (t) (wheret=1,c,cC) only.Such approach can simplify the problem of defining particular machines of the classM _F , composing and simplifying them.Allatum est die 19 Januarii 1970     

Cut-free tableau calculi for some propositional normal modal logics

We give sound and complete tableau and sequent calculi for the prepositional normal modal logics S4.04, K4B and G ⁰(these logics are the smallest normal modal logics containing K and the schemata A A, A A and A ( A); A A and AA; A A and ((A A) A) A resp.) with the following properties: the calculi for S4.04 and G ⁰are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of the cut-rule, the so-called analytical cut-rule.In addition we show that G ⁰is not compact (and therefore not canonical), and we proof with the tableau-method that G ⁰is characterised by the class of all finite, (transitive) trees of degenerate or simple clusters of worlds; therefore G ⁰is decidable and also characterised by the class of all frames for G ⁰.Research supported by Fonds zur Förderung der wissenschaftlichen Forschung, project number P8495-PHY.Presented by W. Rautenberg     

On a new property of partially balanced association schemes useful in psychometric structural analysis

In an earlier paper [Psychometrika,31, 1966, p. 147], Srivastava obtained a test for the HypothesisH ₀ : = ₀₀ + ... + _ll, where i are known matrices,i are unknown constants and is the unknown (p ×p) covariance matrix of a random variablex (withp components) having ap-variate normal distribution. The test therein was obtained under (p ×p) covariance matrix of a random variablex (withp components) the condition that ₀, ₁, ..., l form a commutative linear associative algebra and a certain vector, dependent on these, has non-negative elements. In this paper it is shown that this last condition is always satisfied in the special situation (of importance in structural analysis in psychometrics) where ₀, ₁, ..., l are the association matrices of a partially balanced association scheme.This research was partially supported by the U. S. Air Force under Grant No. AF33(615)-3231, monitored by the Aero Space Research Labs.Now at Colorado State University.     

Независимость одной системы аксиом и правила вывода пропозиционального исчисления

- Allatum est die 5 Martii 1970     

On defining necessity in terms of entailment

In their book Entailment, Anderson and Belnap investigate the consequences of defining Lp (it is necessary that p) in system E as (pp)p. Since not all theorems are equivalent in E, this raises the question of whether there are reasonable alternative definitions of necessity in E. In this paper, it is shown that a definition of necessity in E satisfies the conditions { _E Lpp, _EL(pq)(LpLq), _E pLp} if and only if its has the form C ₁.C₂ .... C_np, where each C _iis equivalent in E to either pp or ((pp)p)p.     

On a complexity-based way of constructivizing the recursive functions

Let g _E(m, n)=o mean that n is the Gödel-number of the shortest derivation from E of an equation of the form (m)=k. Hao Wang suggests that the condition for general recursiveness mn(g _E(m, n)=o) can be proved constructively if one can find a speedfunction _{s s}, with _s(m) bounding the number of steps for getting a value of (m), such that mn _s(m) s.t. g _E(m, n)=o. This idea, he thinks, yields a constructivist notion of an effectively computable function, one that doesn't get us into a vicious circle since we intuitively know, to begin with, that certain proofs are constructive and certain functions effectively computable. This paper gives a broad possibility proof for the existence of such classes of effectively computable functions, with Wang's idea of effective computability generalized along a number of dimensions.We are grateful to an anonymous referee for Studia Logica for valuable advice leading to substantial improvements in the presentation of the main definitions and theorem.     

Mel Gibson’s <Emphasis Type="Italic">The Passion of the Christ</Emphasis> Seen Through Jewish Eyes

Mel Gibsons The Passion of the Christ has stirred up long simmering tensions between Christian, and Jews. This article, by someone of the Jewish faith who has for many years worked at a Presbyterian seminary, discusses Gibsons morbid—almost pornographic—fascination with violence, Gibsons theology of sin and atonement, and the dangers of Gibsons disparaging and inflammatory portrayal of Jews through the use of despicable stereotypes. Gibsons violence drenched Christianity is contrasted with a more tolerant Christianity rooted in love and justice.     

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.     

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.