Superintuitionistic Companions of Classical Modal Logics

This paper investigates partitions of lattices of modal logics based on superintuitionistic logics which are defined by forming, for each superintuitionistic logic L and classical modal logic , the set L[] of L-companions of . Here L[] consists of those modal logics whose non-modal fragments coincide with L and which axiomatize if the law of excluded middle p V p is added. Questions addressed are, for instance, whether there exist logics with the disjunction property in L[], whether L[] contains a smallest element, and whether L[] contains lower covers of . Positive solutions as concerns the last question show that there are (uncountably many) superclean modal logics based on intuitionistic logic in the sense of Vakarelov [28]. Thus a number of problems stated in [28] are solved. As a technical tool the paper develops the splitting technique for lattices of modal logics based on superintuitionistic logics and ap plies duality theory from [34].     

ANOVA and ANCOVA of pre- and post-test,ordinal data

With random assignment to treatments and standard assumptions, either a one-way ANOVA of post-test scores or a two-way, repeated measures ANOVA of pre- and post-test scores provides a legitimate test of the equal treatment effect null hypothesis for latent variable . In an ANCOVA for pre- and post-test variablesX andY which are ordinal measures of and , respectively, random assignment and standard assumptions ensure the legitimacy of inferences about the equality of treatment effects on latent variable . Sample estimates of adjustedY treatment means are ordinal estimators of adjusted post-test means on latent variable .     

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).     

Parallel action: Concurrent dynamic logic with independent modalities

Regular dynamic logic is extended by the program construct, meaning and executed in parallel. In a semantics due to Peleg, each command is interpreted as a set of pairs (s,T), withT being the set of states reachable froms by a single execution of, possibly involving several processes acting in parallel. The modalities << and [] are given the interpretations<>A is true ats iff there existsT withsRT andA true throughoutT, and[]A is true ats iff for allT, ifsRT thenA is true throughoutT, which make <> and [] no longer interdefinable via negation, as they are in the regular case.We prove that the logic defined by this modelling is finitely axiomatisable and has the finite model property, hence is decidable. This requires the development a new theory of canonical models and filtrations for reachability relations.     

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     

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.     

On superintuitionistic logics as fragments of proof logic extensions

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). The result of placing in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and the decoding of is meant), by virtue of which the diagram is constructedIn this diagram the maps, x and are isomorphisms, thereforex ^–1 = ; and the maps and are the semilattice epimorphisms that are not commutative with lattice operation +. Besides, the given diagram is commutative, and the next equalities take place: ^–1 = ^–1 and = ^–1 x. The latter implies in particular that any superintuitionistic logic is a superintuitionistic fragment of some proof logic extension.     

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.     

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.     

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).     

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.     

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.     

Three prepositional calculi of probability

Attempts are made to transform the basis of elementary probability theory into the logical calculus.We obtain the propositional calculus NP by a naive approach. As rules of transformation, NP has rules of the classical propositional logic (for events), rules of the ukasiewicz logic ₀ (for probabilities) and axioms of probability theory, in the form of rules of inference. We prove equivalence of NP with a fragmentary probability theory, in which one may only add and subtract probabilities.The second calculus MP is a usual modal propositional calculus. It has the modal rules x x, x y x y, x x, x y (y x), (y x), in addition to the rules of classical propositional logic. One may read x as x is probable. Imbeddings of NP and of ₀ into MP are given.The third calculus P is a modal extension of ₀. It may be obtained by adding the rule ((xy)y) xy to the modal logic of quantum mechanics Q [5]. One may read x in P as x is observed. An imbedding of NP into P is given.     

Modal logic with names

We investigate an enrichment of the propositional modal language with a universal modality having semanticsx iff y(y ), and a countable set of names — a special kind of propositional variables ranging over singleton sets of worlds. The obtained language _c proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment () of, where is an additional modality with the semanticsx iff y(y x y ). Model-theoretic characterizations of modal definability in these languages are obtained. Further we consider deductive systems in _c. Strong completeness of the normal _c-logics is proved with respect to models in which all worlds are named. Every _c-logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties ([in]completeness, filtration, finite model property etc.) from to _c are discussed. Finally, further perspectives for names in multimodal environment are briefly sketched.     

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.     

Perceptual scission of surface-lightness and illumination: An examination of the gelb effect

Summary An attempt was made to examine how the photometric equation: luminance (L)=albedo (A)×illuminance (I) could be solved perceptually when a test field (TF) was not seen as figure, but as ground. A gray disk with two black or white patches was used as the TF. Illuminance of the TF was changed over 2.3 log units and TF albedo was varied from 2.5 to 8.0 in Munsell value. Albedos of the black- and white-appearing patches were 1.5 and 9.5 in Munsell values, respectively. Two types of category judgments for apparent TF lightness (A) and apparent overall illumination (I) were made on the total of 40 TFs (5 illuminances×4 TF-albedos×2 patch-albedos). The results indicated that when the black patches were added to the TF, A was indistinguishable from I and when the white patches were placed on the TF, A and I could be distinguished from each other. The Gelb effect was interpreted as a manifestation of such A–I scission. It was concluded, therefore, that as far as the Gelb effect was observed, the perceptual system could solve the equation, L=A×I, in the sense that for a fixed L, the product of A and I would be constant.     

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.     

Rules in relevant logic — II: Formula representation

This paper surveys the various forms of Deduction Theorem for a broad range of relevant logics. The logics range from the basic system B of Routley-Meyer through to the system R of relevant implication, and the forms of Deduction Theorem are characterized by the various formula representations of rules that are either unrestricted or restricted in certain ways. The formula representations cover the iterated form,A ₁ .A ₂ . ... .A _n B, the conjunctive form,A ₁&A ₂ & ...A _n B, the combined conjunctive and iterated form, enthymematic version of these three forms, and the classical implicational form,A ₁&A ₂& ...A _n B. The concept of general enthymeme is introduced and the Deduction Theorem is shown to apply for rules essentially derived using Modus Ponens and Adjunction only, with logics containing either (A B)&(B C) .A C orA B .B C .A C.I acknowledge help from anonymous referees for guidance in preparing Part II, and especially for the suggestion that Theorem 9 could be expanded to fully contraction-less logics.     

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.     

On a statistic arising in testing correlation

This paper is devoted to the study of a certain statistic,u, defined on samples from a bivariate population with variances ₁₁, ₂₂ and correlation . Let the parameter corresponding tou be . Under binormality assumptions the following is demonstrated. (i) If ₁₁ = ₂₂, then the distribution ofu can be obtained rapidly from theF distribution. Statistical inferences about = may be based onF. (ii) In the general case, allowing for _{11 22}, a certain quantity involvingu,r (sample correlation between the variables) and follows at distribution. Statistical inferences about may be based ont. (iii) In the general case a quantityt may be constructed which involves only the statisticu and only the parameter . If treated like at distributed magnitude,t admits conservative statistical inferences. (iv) TheF distributed quantity mentioned in (i) is equivalent to a certaint distributed quantity as follows from an appropriate transformation of the variable. (v) Three test statistics are given which can be utilized in making statistical inferences about = in the case ₁₁ = ₂₂. A comparison of expected lengths of confidence intervals for obtained from the three test statistics is made. (vi) The use of the formulas derived is illustrated by means of an application to coefficient alpha.This research was supported by the National Institute of Child Health and Human Development, under Research Grant 1 PO1 HD01762.