The problem of the additive constant and eigenvalues in metric multidimensional scaling

This paper is concerned with the additive constant problem in metric multidimensional scaling. First the influence of the additive constant on eigenvalues of a scalar product matrix is discussed. The second part of this paper is devoted to the introduction of a new formulation of the additive constant problem. A solution is given for fixed dimensionality, by maximizing a normalized index of fit with a gradient method. An experimental computation has shown that the author's solution is accurate and easy to follow.     

Piecewise method of reciprocal averages for dual scaling of multiple-choice data

The proposed method handles the classical method of reciprocal averages (MRA) in a piecewise (item-by-item) mode, whereby one can deal with smaller matrices and attain faster convergence to a solution than the MRA. A new concept the principle of constant proportionality is introduced to provide an interesting interpretation for scaling multiple-choice data a la Guttman. A small example is presented for discussion of the technique.This study was supported by the Natural Sciences and Engineering Research Council Canada Grant (No. A4581) to S. Nishisato. The authors are indebted to reviewers for valuable comments.     

V.A. Smirnov’s Results in the Field of Modern Formal Logic

This paper is a survey of V.A. Smirnovs main results in modern logic.     

Construction of the World: Ontology in the Tractatus

Why does Wittgenstein say in the Tractatus Logico-Philosophicus that the world has as its members only facts, not things? Focusing on this question, I start with the problem, in its general form, “How is it possible to determine something as X?” and establish the excluding-allowing model for determination. From this model, I derive an argument for Wittgenstein’s aforementioned statement. The argument shows that a whole cannot be determined as consisting of components that are determined separately in a strong sense, whereas in a weak sense it can be. This thus demonstrates why the context principle holds. The recommended interpretation places suitable weight on the Tractarian notion of possibility. It provides new insights into Wittgenstein’s conception of logic, and his atomism about facts and states of affairs.     

FINDING THE SOMETHING IN NOTHING

Formless Infinity: Clinical Explorations of Matte Blanco and Bion. By Riccardo Lombardi. Translated by Karen Christenfeld, Gina Atkinson, Andrea Sabbadini, and Philip Slotkin. London/New York: Routledge, 2016. 282 pp.     

Cartographic systems and non-linguistic inference

It is often assumed that the capability to make inferences requires language. Against this assumption, I claim that inferential abilities do not necessarily require a language. On the contrary, certain cartographic systems could be used to explain some forms of inferences, and they are capable of warranting rational relations between contents they represent. By arguing that certain maps, as well as sentences, are adequate for inferential processes, I do not mean to neglect that there are important differences between maps and sentences. Instead, the purpose of this paper is to highlight interesting distinctions that might affect the way that cartographic thinking works.     

Protoalgebraic Gentzen Systems and the Cut Rule

In this paper we show that, in Gentzen systems, there is a close relation between two of the main characters in algebraic logic and proof theory respectively: protoalgebraicity and the cut rule. We give certain conditions under which a Gentzen system is protoalgebraic if and only if it possesses the cut rule. To obtain this equivalence, we limit our discussion to what we call regular sequent calculi, which are those comprising some of the structural rules and some logical rules, in a sense we make precise. We note that this restricted set of rules includes all the usual rules in the literature. We also stress the difference between the case of two-sided sequents and the case of many-sided sequents, in which more conditions are needed.     

A Note on Formality and Logical Consequence

Logic is formal in the sense that all arguments of the same form as logically valid arguments are also logically valid and hence truth-preserving. However, it is not known whether all arguments that are valid in the usual model-theoretic sense are truth-preserving. Tarski claimed that it could be proved that all arguments that are valid (in the sense of validity he contemplated in his 1936 paper on logical consequence) are truth-preserving. But he did not offer the proof. The question arises whether the usual model-theoretic sense of validity and Tarski's 1936 sense are the same. I argue in this paper that they probably are not, and that the proof Tarski had in mind, although unusable to prove that model-theoretically valid arguments are truth-preserving, can be used to prove that arguments valid in Tarski's 1936 sense are truth-preserving.     

The Logical Style of Confucius’ Analects

Considered from a logical point of view, Confucius’ Analects contain many implicit forms of reasoning and argumentation. This is shown first by analyzing the phenomenon of parallelism: direct parallelism is often a way of hinting at a general assertion, whereas anti-parallelism hides dilemmas, generalizations and modal notions of “moral preference.” The Analects also have various types of conditionals, ranging from material implications, to modalized implications, and counterfactual conditionals, which are the germs of implicit reasoning, concluding with a moral recommendation. Analogies are particularly abundant and a presentation of three examples suggests that, beyond their explicative role, they also involve moral recommendations. The implicit logic of The Analects requires an active, albeit unconscious participation of the reader, which could be an important element in explaining the enduring influence of the text.     

LOGICAL PLURALISM AND SEMANTIC INFORMATION

Up to now theories of semantic information have implicitly relied on logical monism, or the view that there is one true logic. The latter position has been explicitly challenged by logical pluralists. Adopting an unbiased attitude in the philosophy of information, we take a suggestion from Beall and Restall at heart and exploit logical pluralism to recognise another kind of pluralism. The latter is called informational pluralism, a thesis whose implications for a theory of semantic information we explore. Research Assistant of the Fund for Scientific Research – Flanders (Belgium)