The Problem of Monotonicity and the Skeleton

The premise p of a reasoning is usually a complex statement reflecting the information of departure and consisting in the conjunction of other statements, p = p1 · (p2 · (…(pn). Such p can be written without parenthesis provided conjunction is associative, and then with the possibility of placing the sub-indexes in any ordering if it is commutative; on the contrary neither parenthesis, nor ordering can be avoided.

Common Reasoning in a Computational Context

In the field of Computation Science, ‘Commonsense Reasoning’ usually expresses the formalization of Logic systems in order to efficiently automate replication of human performances. This is done by employing methodologies from Computational Learning, and deals with the construction of information of both deductive and inductive nature. The process often happens in an ecologic, natural context (i.e., in the real world, not in an artificial laboratory setting), and in presence of incomplete and imprecise information.

Quasi-transitivity

Seeing ordinary reasoning in the setting of the Poincaré continua by means of T-Indistinguishability Operators, opens a window towards the possibility of considering alternative types of transitivity. Let us concentrate on the so called quasi-transitive law.

Fuzziness, Cognition and Cybernetics: a historical perspective

In the present paper, we connect some old reflections about the relationships existing between the theory of fuzzy sets and cybernetics with modern, contemporary analyses of the crucial (better: unavoidable) role that fuzziness plays in the attempts at scientifically describing aspects of information sciences. The connection, which has a basic conceptual origin, has been triggered also by the recent 50th anniversary of Norbert Wiener’s death which has been instrumental in looking again at some crucial aspects of the birth of information sciences in the midst of the last century. Fuzzy sets are an essential part of this revolution and share all the innovations as well as the difficulties of …

A naïve way of looking at fuzzy sets

In this study, we consider the concept of a predicate (P) in a universe of discourse X from a specific viewpoint, i.e., the informational viewpoint with respect to its linguistic use. Its meaning and its different types are considered, particularly by considering the predicates that are "measurable" and designate a "collective" (P) in X, which is not always a classical subset of X. We show that the collective P manifests itself in different "states" or fuzzy sets, where knowledge and representation depend on the available information regarding the use of the predicate P in X. We also analyze the linguistic concept of a "collective" where the fuzzy sets are nothing other than informational s…

Introduction

In the intention of the authors, this booklet does not want to be a textbook; instead, it contains some reflections that aim to guide ‘Computing with Words and Percep- tions’, Zadeh’s final view on his Fuzzy Logic, towards its future as a new science of both Language and Reasoning. Towards an experimental and theoretical science concerning the Natural Phenomena of Language, Thinking and Reasoning, in rela- tionship with Neurosciences, whose possibilities are foreseen by the authors.

A Formal Skeleton of Commonsense Reasoning

After referring several times to Commonsense or Ordinary Reasoning, let’s devote a few pages to present a (minimal) mathematical model of it that can be seen as the ‘Skeleton’ of Reasoning, since it is defined by a set of few, simple laws appearing in the models of particular and specialized modes of reasoning like, for instance: Boolean Algebras for the reasoning with precise concepts; Orto-modular lattices for the reasoning with the concepts of Quantum Physics; and also in the so called Algebras of Fuzzy Sets for the reasoning with imprecise concepts, and among them De Morgan-Kleene algebras. All these models have interesting applications.

Conclusions for Part II

Human beings are animals endowed with a great curiosity. They continuously ask themselves how things are, where they come from, and where they go to. Questioning is at the origins of reasoning; and possibly, without the capability of self-questioning and guessing, neither directed thinking, nor reasoning, will exist. Their existence makes them a matter of study.

Fuzziness, Cognition and Cybernetics: an outlook on future

In the present paper, we connect some old reflections about the relationships existing between the theory of fuzzy sets and cybernetics with modern, contemporary analyses of the crucial (better: unavoidable) role that fuzziness plays in the attempts at scientifically describing aspects of information sciences. The connection, which has a basic conceptual origin, has been triggered also by the recent 50th anniversary of Norbert Wiener’ death which has been instrumental in looking again at some crucial aspects of the birth of information sciences in the midst of last Century. Fuzzy sets are an essential part of this revolution and share all the innovations as well as the difficulties of this …

Conclusions for Part I

Thanks to thinking, memory, and language, human beings can dedicate a non-minor part of their time to tell, themselves or the others, events, descriptions, true or imagined histories, etc. Part of the human conversation consists in telling, and most of the times reasoning starts from the self-telling of something.

An Overview of the Fuzzy Calculi

What has been discussed up to now allows us to assume that the roots of fuzzy sets are in Language and, thus, that Fuzzy Logic deals with both Language and Commonsense Reasoning. Fuzzy Logic’s main goal is the representation of statements whose meaning is not precise, but it can as well capture the case in which statements are precise.

Looking for Some Historical Roots

Knowledge comes from the observation that what surrounds us is provoked by the questions people poses when thinking on it; is reachecorrespondingd thanks to reasoning, and usually through establishing hypotheses and further testing them and their consequences against the reality. Speculation lays at the basic level of both ‘thinking on it’, and ‘establishing and testing hypotheses’.

Errata: On Fuzziness

Some algebraic clues towards a syntactic view on the Principles of Non-Contradiction and Excluded-Middle.

This short paper just considers the possibility of a new view for posing and proving the Aristotle’s Principles of Non-Contradiction and Excluded-Middle. It is done by means of their refutability, or deducibility, respectively, under Tarski’s Consequence Operators.

Introduction

‘Natural Reasoning’, that with which lay people decide their daily actions, also called Ordinary, Everyday, or Commonsense Reasoning, can be seen as ruled by the universal laws defining its formal ‘skeleton’.