Finitary Polyadic Algebras from Cylindric Algebras

It is known that every α-dimensional quasi polyadic equality algebra (QPEA _α ) can be considered as an α-dimensional cylindric algebra satisfying the merrygo- round properties . The converse of this proposition fails to be true. It is investigated in the paper how to get algebras in QPEA from algebras in CA. Instead of QPEA the class of the finitary polyadic equality algebras (FPEA) is investigated, this class is definitionally equivalent to QPEA. It is shown, among others, that from every algebra in a β-dimensional algebra can be obtained in QPEA _β where , moreover the algebra obtained is representable in a sense. Presented by Daniele Mundici Supported by the OTKA grants T0351192, T43242.     

Completeness theorems for some intermediate predicate calculi

We give completeness results — with respect to Kripke's semantic — for the negation-free intermediate predicate calculi: (1) $$\begin{gathered} BD = positive predicate calculus PQ + B:(\alpha \to \beta )v(\beta \to \alpha ) \hfill \\ + D:\forall x\left( {a\left( x \right)v\beta } \right) \to \forall xav\beta \hfill \\ \end{gathered}$$ (2) $$T_n D = PQ + T_n :\left( {a_0 \to a_1 } \right)v \ldots v\left( {a_n \to a_{n + 1} } \right) + D\left( {n \geqslant 0} \right)$$ and the superintuitionistic predicate calculus: (3) $$B^1 DH_2^ \urcorner = BD + intuitionistic negation + H_2^ \urcorner : \urcorner \forall xa \to \exists x \urcorner a.$$ The central point is the completeness proof for (1), which is obtained modifying Klemke's construction [3]. For a general account on negation-free intermediate predicate calculi — see Casari-Minari [1]; for an algebraic treatment of some superintuitionistic predicate calculi involving schemasB andD — see Horn [4] and Görnemann [2].     

An optimal property of least squares weights in prediction models

In predicting scores fromp > 1 observed scores in a sample of sizeñ, the optimal strategy (minimum expected loss), under certain assumptions, is shown to be based upon the least squares regression weights computed from a previous sample. Letting represent the correlation between and the predicted values , and letting represent the correlation between and a different set of predicted values , where w is any weighting system which is not a function of , it is shown that the probability of being less than cannot exceed .50. The relationship of this result to previous research and practical implications are discussed.     

Aspects of analytic deduction

Let ? be the ordinary deduction relation of classical first-order logic. We provide an “analytic” subrelation ?³ of ? which for propositional logic is defined by the usual “containment” criterion $$\Gamma \vdash ^a \varphi iff \Gamma \vdash \varphi and Atom(\varphi ) \subseteq Atom(\Gamma ),$$ whereas for predicate logic, ?^a is defined by the extended criterion $$\Gamma \vdash ^a \varphi iff \Gamma \vdash \varphi and Atom(\varphi ) \subseteq ' Atom(\Gamma ),$$ where Atom(?) $ \subseteq '$ Atom(Γ) means that every atomic formula occurring in ? “essentially occurs” also in Γ. If Γ, ? are quantifier-free, then the notions “occurs” and “essentially occurs” for atoms between Γ and ? coincide. If ? is formalized by Gentzen's calculus of sequents, then we show that ?^a is axiomatizable by a proper fragment of analytic inference rules. This is mainly due to cut elimination. By “analytic inference rule” we understand here a rule r such that, if the sequent over the line is analytic, then so is the sequent under the line. We also discuss the notion of semantic relevance as contrasted to the previous syntactic one. We show that when introducing semantic sequents as axioms, i.e. when extending the pure logical axioms and rules by mathematical ones, the property of syntactic relevance is lost, since cut elimination no longer holds. We conclude that no purely syntactic notion of analytic deduction can ever replace successfully the complex semantico-syntactic deduction we already possess.     

Comparison of ranks of cross-product and covariance solutions in component analysis

SupposeD is a data matrix forN persons andn variables, and is the matrix obtained fromD by expressing the variables in deviation-score form. It is shown that ifD has rankr, will always have rank (r−1) ifr=N<n, otherwise it will generally have rankr. If has ranks,D will always have ranks ifs=n, but ifs<n it will generally have rank (s+1). Thus two cases can arise, Case A in whichD has rank one greater than , and Case B in whichD has rank equal to . Implications of this distinction for analysis of cross products versus analysis of covariances are briefly indicated.     

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     

Probability Propagation in Generalized Inference Forms

Probabilistic inference forms lead from point probabilities of the premises to interval probabilities of the conclusion. The probabilistic version of Modus Ponens, for example, licenses the inference from \({P(A) = \alpha}\) and \({P(B|A) = \beta}\) to \({P(B)\in [\alpha\beta, \alpha\beta + 1 - \alpha]}\) . We study generalized inference forms with three or more premises. The generalized Modus Ponens, for example, leads from \({P(A_{1}) = \alpha_{1}, \ldots, P(A_{n})= \alpha_{n}}\) and \({P(B|A_{1} \wedge \cdots \wedge A_{n}) = \beta}\) to an according interval for P(B). We present the probability intervals for the conclusions of the generalized versions of Cut, Cautious Monotonicity, Modus Tollens, Bayes’ Theorem, and some SYSTEM O rules. Recently, Gilio has shown that generalized inference forms “degrade”—more premises lead to less precise conclusions, i.e., to wider probability intervals of the conclusion. We also study Adam’s probability preservation properties in generalized inference forms. Special attention is devoted to zero probabilities of the conditioning events. These zero probabilities often lead to different intervals in the coherence and the Kolmogorov approach.     

The resting membrane potential of cells are measures of electrical work,not of ionic currents

Living cells create electric potential force,E, between their various phases by at least three distinct mechanisms. Charge separation,

Positive modal logic

We give a set of postulates for the minimal normal modal logicK ₊ without negation or any kind of implication. The connectives are simply , , , . The postulates (and theorems) are all deducibility statements . The only postulates that might not be obvious are

A computer simulation/mathematical model of learning: Extension of DMOD from appetitive to aversive situations

The computer simulation/mathematical model called DMOD, which can simulate over 35 different phenomena in appetitive discrete-trial and simple free-operant situations, has been extended to include aversive discrete-trial situations. Learning (V) is calculated using a three-parameter equation \(\Delta V = \alpha \beta (\lambda - \bar V)\) (see Daly & Daly, 1982; Rescorla & Wagner, 1972). The equation is applied to three possible goal events in the appetitive (e.g., food) case and to three in the aversive (e.g., shock) case. The original goal event can be present, absent, or reintroduced; in the appetitive situation, these events condition approach (Vap), avoidance (Vav), and courage (Vcc), respectively. In the aversive situation, the events condition avoidance (Vav*), approach (Vap*), and cowardice (Vcc*), respectively. The model was developed in simple learning situations and subsequently was applied to complex situations. It can account for such diverse phenomena as contrast effects after reward shifts, greater persistence following partial than following continuous reinforcement, and a preference for predictable appetitive and predictable aversive events. Application of the aversive version of the model to “reward” shifts is described.     

Finite basis problems and results for quasivarieties

Let be a finite collection of finite algebras of finite signature such that SP( ) has meet semi-distributive congruence lattices. We prove that there exists a finite collection ₁ of finite algebras of the same signature, , such that SP( ₁) is finitely axiomatizable.We show also that if , then SP( ₁) is finitely axiomatizable. We offer new proofs of two important finite basis theorems of D. Pigozzi and R. Willard. Our actual results are somewhat more general than this abstract indicates.While working on this paper, the first author was partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant no. T37877 and the second author was supported by the US National Science Foundation grant no. DMS-0245622.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko     

Decidability of General Extensional Mereology

The signature of the formal language of mereology contains only one binary predicate P which stands for the relation “being a part of”. Traditionally, P must be a partial ordering, that is, ${\forall{x}Pxx, \forall{x}\forall{y}((Pxy\land Pyx)\to x=y)}$ and ${\forall{x}\forall{y}\forall{z}((Pxy\land Pyz)\to Pxz))}$ are three basic mereological axioms. The best-known mereological theory is “general extensional mereology”, which is axiomatized by the three basic axioms plus the following axiom and axiom schema: (Strong Supplementation) ${\forall{x}\forall{y}(\neg Pyx\to \exists z(Pzy\land \neg Ozx))}$ , where Oxy means ${\exists z(Pzx\land Pzy)}$ , and (Fusion) ${\exists x\alpha \to \exists z\forall y(Oyz\leftrightarrow \exists x(\alpha \land Oyx))}$ , for any formula α where z and y do not occur free. In this paper, I will show that general extensional mereology is decidable, and will also point out that the decidability of the first-order approximation of the theory of complete Boolean algebras can be shown in the same way.     

Linguistically invariant inductive logic

Summary Carnap's early system of inductive logic make degrees of confirmation depend on the languages in which they are expressed. They are sensitive to which predicates are, in the language, taken as primitive. Hence they fail to be linguistically invariant. His later systems, in which prior probabilities are assigned to elements of a model rather than sentences of a language, are sensitive to which properties in the model are called primitive. Critics have often protested against these features of his work. This paper shows how to make his systems independent of any choice of primitive predicates or primitive properties.The solution is related to another criticism of inductive logic. It has been noticed that Carnap's systems are too all-embracing. Hisc(h, e) is defined for all sentencesh ande. Yet for manyh ande, the evidencee does not warrant any assessment of the probability ofh. We need an inductive logic in whichc(h, e) is defined only whene really does bear onh. This paper sketches the measure theory of such a logic, and, within this measure theory, provides relativized versions of Carnap's systems which are linguistically invariant.     

A COMMENT ON RCC: FROM RCC TO RCC<Superscript>++</Superscript>

The Region Connection Calculus (RCC theory) is a well-known spatial representation of topological relations between regions. It claims that the connection relation is primitive in the spatial domain. We argue that the connection relation is indeed primitive to the spatial relations, although in RCC theory there is no room for distance relations. We first analyze some aspects of the RCC theory, e.g. the two axioms in the RCC theory are not strong enough to govern the connection relation, regions in the RCC theory cannot be points, the uniqueness of the operation in the theory is not guaranteed, etc. To solve some of the problems, we propose an extension to the RCC theory by introducing the notion of region category and adding a new axiom which governs the characteristic property of the connection relation. The extended theory is named as RCC⁺⁺. We support the claim that the connection relation is primitive to spatial domain by showing how distance relations, size relations are developed in RCC⁺⁺. At last we revisit a sub-family of un-intended models in RCC theory, argue that RCC⁺⁺ is more suitable than RCC with regards to its original intended model, and discuss the representation limitation of the RCC, as well as RCC⁺⁺.     

An application of a Theorem of Ash to finite covers

The technique of covers is now well established in semigroup theory. The idea is, given a semigroup S, to find a semigroup having a better understood structure than that of S, and an onto morphism of a specific kind from to S. With the right conditions on , the behaviour of S is closely linked to that of . If S is finite one aims to choose a finite . The celebrated results for inverse semigroups of McAlister in the 1970s form the flagship of this theory.Weakly left quasi-ample semigroups form a quasivariety (of algebras of type(2, 1)), properly containing the classes of groups, and of inverse, left ample, and weakly left ample semigroups. We show how the existence of finite proper covers for semigroups in this quasivariety is a consequence of Ashs powerful theorem for pointlike sets. Our approach is to obtain a cover of a weakly left quasi-ample semigroup S as a subalgebra of S × G, where G is a group. It follows immediately from the fact that weakly left quasi-ample semigroups form a quasivariety, that is weakly left quasi-ample. We can then specialise our covering results to the quasivarieties of weakly left ample, and left ample semigroups. The latter have natural representations as (2, 1)-subalgebras of partial (one-one) transformations, where the unary operation takes a transformation to the identity map in the domain of . In the final part of this paper we consider representations of weakly left quasi-ample semigroups.This work was supported by the London Mathematical Society, Fundação para a Ciência e a Tecnologia, Centro de Álgebra da Universidade de Lisboa, Centro de Matemática da Universidade do Porto and projects POCTI/32440/MAT/2000 and POCTI/32817/Mat/2000 of FEDER.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko     

Fregean logics with the multiterm deduction theorem and their algebraization

A deductive system $\mathcal{S}$ (in the sense of Tarski) is Fregean if the relation of interderivability, relative to any given theory T, i.e., the binary relation between formulas $$\{ \left\langle {\alpha ,\beta } \right\rangle :T,\alpha \vdash s \beta and T,\beta \vdash s \alpha \} ,$$ is a congruence relation on the formula algebra. The multiterm deduction-detachment theorem is a natural generalization of the deduction theorem of the classical and intuitionistic propositional calculi (IPC) in which a finite system of possibly compound formulas collectively plays the role of the implication connective of IPC. We investigate the deductive structure of Fregean deductive systems with the multiterm deduction-detachment theorem within the framework of abstract algebraic logic. It is shown that each deductive system of this kind has a deductive structure very close to that of the implicational fragment of IPC. Moreover, it is algebraizable and the algebraic structure of its equivalent quasivariety is very close to that of the variety of Hilbert algebras. The equivalent quasivariety is however not in general a variety. This gives an example of a relatively point-regular, congruence-orderable, and congruence-distributive quasivariety that fails to be a variety, and provides what apparently is the first evidence of a significant difference between the multiterm deduction-detachment theorem and the more familiar form of the theorem where there is a single implication connective.     

On the lattice of quasivarieties of Sugihara algebras

Let S denote the variety of Sugihara algebras. We prove that the lattice (K) of subquasivarieties of a given quasivariety K S is finite if and only if K is generated by a finite set of finite algebras. This settles a conjecture by Tokarz [6]. We also show that the lattice (S) is not modular.     

The completeness of S

The subsystem S of Parry's AI [10] (obtained by omitting modus ponens for the material conditional) is axiomatized and shown to be strongly complete for a class of three valued Kripke style models. It is proved that S is weakly complete for the class of consistent models, and therefore that Ackermann's rule is admissible in S. It also happens that S is decidable and contains the Lewis system S4 on translation — though these results are not presented here. S is arguably the most relevant relevant logic known at this time to be decidable.I wish to thank Jill Pipher and Professora Nuel D. Belnap, Jr. and David Kaplan for their helpful comments on an earlier version of this paper.