Fundamental unit structures: A theory of ratio scalability

A theory of nonassociative measurement structures is developed which produces a natural generalization of associative measurement (i.e., extensive structures), and representation and uniqueness theorems are established for these generalized structures, and it is shown that in many cases these representations are ratio scales. The methods of proof strongly relate the structure of the automorphism group of the nonassociative structure to its underlying concatenation operation.     

First order logic with empty structures

For first order languages with no individual constants, empty structures and truth values (for sentences) in them are defined. The first order theories of the empty structures and of all structures (the empty ones included) are axiomatized with modus ponens as the only rule of inference. Compactness is proved and decidability is discussed. Furthermore, some well known theorems of model theory are reconsidered under this new situation. Finally, a word is said on other approaches to the whole problem.     

Causal inference

Conclusion We have examined only a few of the basic questions about causal inference that result from Reichenbach's two principles. We have not considered what happens when the probability distribution is a mixture of distributions from different causal structures, or how unmeasured common causes can be detected, or what inferences can reliably be drawn about causal relations among unmeasured variables, or the exact advantages that experimental control offers. A good deal is known about these questions, and there is a good deal more to find out.We thank the Office of Naval Research and the Navy Personnel Research and Development Center for supporting this research. Proofs of theorems will be found in Spirtes, Glymour, and Scheines (1991).     

Utility-uncertainty trade-off structures

A qualitative axiomatization of a generalization of expected utility theory is given in which the expected value of simple gambles is not necessarily the product of subjective probability and utility. Representation and uniqueness theorems for these generalized structures are derived for both Archimedean and nonarchimedean cases. It is also shown that a simple condition called distributivity is necessary and sufficient in the case of simple gambles for one of these generalized expected utility structures to have simultaneously an additive subjective probability function and a multiplicative combining rule for expected values.     

Some Limitations to the Psychological Orientation in Semantic Theory

The psychological orientation treats semantics as a matter of idealized computation over symbolic structures, and semantic relations like denotation as relations between linguistic expressions and these structures. I argue that results similar to Gödel’s incompleteness theorems and Tarski’s theorem on truth create foundational difficulties for this view of semantics.     

Goodness of fit tests for probabilistic measurement models

General goodness of fit tests for probabilistic response models are developed. The tests are applicable in psychophysics, in the theory of choice behavior and in mathematical learning theories. The necessary and sufficient constraints that a measurement model puts on the response probabilities are used for testing this model. In addition, representation theorems for some models are proved and the goodness of fit to experimental data is considered.     

Rosser-Type Undecidable Sentences Based on Yablo’s Paradox

It is widely considered that Gödel’s and Rosser’s proofs of the incompleteness theorems are related to the Liar Paradox. Yablo’s paradox, a Liar-like paradox without self-reference, can also be used to prove Gödel’s first and second incompleteness theorems. We show that the situation with the formalization of Yablo’s paradox using Rosser’s provability predicate is different from that of Rosser’s proof. Namely, by using the technique of Guaspari and Solovay, we prove that the undecidability of each instance of Rosser-type formalizations of Yablo’s paradox for each consistent but not Σ₁-sound theory is dependent on the choice of a standard proof predicate.     

What is the axiomatic method?

The modern notion of the axiomatic method developed as a part of the conceptualization of mathematics starting in the nineteenth century. The basic idea of the method is the capture of a class of structures as the models of an axiomatic system. The mathematical study of such classes of structures is not exhausted by the derivation of theorems from the axioms but includes normally the metatheory of the axiom system. This conception of axiomatization satisfies the crucial requirement that the derivation of theorems from axioms does not produce new information in the usual sense of the term called depth information. It can produce new information in a different sense of information called surface information. It is argued in this paper that the derivation should be based on a model-theoretical relation of logical consequence rather than derivability by means of mechanical (recursive) rules. Likewise completeness must be understood by reference to a model-theoretical consequence relation. A correctly understood notion of axiomatization does not apply to purely logical theories. In the latter the only relevant kind of axiomatization amounts to recursive enumeration of logical truths. First-order “axiomatic” set theories are not genuine axiomatizations. The main reason is that their models are structures of particulars, not of sets. Axiomatization cannot usually be motivated epistemologically, but it is related to the idea of explanation.     

Double threshold orders: a new axiomatization

The paper presents some new results concerning the axiomatization of double threshold preference structures. Such structures, which have been introduced in order to model a situation of hesitation between the strict preference and the indifference, were not axiomatized through the use of a single characteristic relation. We give two theorems for this purpose, exploiting a four valued logic recently introduced by the authors as a preference modeling language under hesitation. © 1998 John Wiley & Sons, Ltd.     

On models with variable universe

In this paper some parts of the model theory for logics based on generalised Kripke semantics are developed. Löwenheim-Skolem theorems and some applications of ultraproduct constructions for generalised Kripke models with variable universe are investigated using similar theorems of the model theory for classical logic. The results are generalizations of the theorems of [4].     

Sequences in countable nonstandard models of the natural numbers

Two different equivalence relations on countable nonstandard models of the natural numbers are considered. Properties of a standard sequence A are correlated with topological properties of the equivalence classes of the transfer of A. This provides a method for translating results from analysis into theorems about sequences of natural numbers.Most of the results in this paper appeared in the author's Ph.D. dissertation at the University of Colorado, 1985.     

SOME EXPLORATIONS IN INITIAL INTERACTION AND BEYOND: TOWARD A DEVELOPMENTAL THEORY OF INTERPERSONAL COMMUNICATION

This paper provides a theoretical perspective for dealing with the initial entry stage of interpersonal interaction. The seven axioms and 21 theorems presented suggest a set of research priorities for studying the development of interpersonal relationships. The paper concludes with a discussion of some of the problems to be considered if the theory is to be extended beyond the initial stages of interaction.     

Indirect-deduction theorems

Summary By indirect-deduction theorems introduced in the present paper we mean the theorems that allow to formalize indirect reasonings occurring in deductive practice in general and in mathematics in particular. We discuss the relationship between the introduced theorems and some logical calculi being virtually confined to propositional calculi with implication and negation. It is worth to notice that the above theorems are very handy and effective in proving logical theses.     

Group Differences in Regression Intercepts: Implications for Factorial Invariance

Studies of differential prediction typically examine group differences in linear regression slopes or intercepts for predicting criterion scores from one or more test scores. When there are no group differences in slopes, what are the implications of differences in regression intercepts for the measurement equivalence of the tests or criterion across groups? Measurement equivalence is here defined as factorial invariance under a single-factor model for the tests and criterion. Two theorems are given that describe conditions under which intercept differences can exist under factorial invariance. In such cases, intercept differences do not result from measurement bias in either the tests or criterion. The conditions of the theorems are testable using multiple-group confirmatory factor analysis. These test procedures are illustrated in real data. The implications of the theorems and the test procedures for studies of differential prediction are discussed.     

实践推理:从信念和愿望应该形成什么目标

这篇文章探讨实践推理中目标推理的问题,试图发现一个理性主体从信念和愿望形成目标的基本逻辑标准。文中的分析以循序渐进的方式展开,从只包含信念和愿望的基础模型开始,到增加了数值和排序方式表示愿望偏好的混合拓展模型结束,都给出了相应的目标形成标准,并构造出了形式化的目标形成理论。最后,我们探讨了不同模型形成的目标之间的关系,并证明了混合拓展模型中目标的存在性等基本定理。     

Proof Systems Combining Classical and Paraconsistent Negations

New propositional and first-order paraconsistent logics (called L _ω and FL _ω , respectively) are introduced as Gentzen-type sequent calculi with classical and paraconsistent negations. The embedding theorems of L _ω and FL _ω into propositional (first-order, respectively) classical logic are shown, and the completeness theorems with respect to simple semantics for L _ω and FL _ω are proved. The cut-elimination theorems for L _ω and FL _ω are shown using both syntactical ways via the embedding theorems and semantical ways via the completeness theorems. Presented by Yaroslav Shramko and Heinrich Wansing     

Two theorems in colour vision

Exact definitions and statements are given of some widely accepted concepts and assumptions in colour vision. Two theorems are stated, and one of them is proved (the other being self-evident). Two applications of the theorems are briefly discussed.     

Bunched sequential information

It is known that the logic BI of bunched implications is a logic of resources. Many studies have reported on the applications of BI to computer science. In this paper, an extension BIS of BI by adding a sequence modal operator is introduced and studied in order to formalize more fine-grained resource-sensitive reasoning. By the sequence modal operator of BIS, we can appropriately express “sequential information” in resource-sensitive reasoning. A Gentzen-type sequent calculus SBIS for BIS is introduced, and the cut-elimination and decidability theorems for SBIS are proved. An extension of the Grothendieck topological semantics for BI is introduced for BIS, and the completeness theorem with respect to this semantics is proved. The cut-elimination, decidability and completeness theorems for SBIS and BIS are proved using some theorems for embedding BIS into BI.     

A Conceptual Construction of Complexity Levels Theory in Spacetime Categorical Ontology: Non-Abelian Algebraic Topology, Many-Valued Logics and Dynamic Systems

A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Super-complex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item’s essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures that emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super- and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry’ might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, ?ukasiewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic ‘asymmetry’ of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics—which is presently unfolding in physics, mathematics, life and cognitive sciences—may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics.