The Many Valued and Nonmonotonic Turn in Logic

Dov M. Gabbay, John Woods

• Publisher: North Holland
• Year: 2007
• ISBN:
  9780080549392
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

The present volume of the Handbook of the History of Logic brings together two of the most important developments in 20th century non-classical logic. These are many-valuedness and non-monotonicity. On the one approach, in deference to vagueness, temporal or quantum indeterminacy or reference-failure, sentences that are classically non-bivalent are allowed as inputs and outputs to consequence relations. Many-valued, dialetheic, fuzzy and quantum logics are, among other things, principled attempts to regulate the flow-through of sentences that are neither true nor false. On the second, or non-monotonic, approach, constraints are placed on inputs (and sometimes on outputs) of a classical consequence relation, with a view to producing a notion of consequence that serves in a more realistic way the requirements of real-life inference. Many-valued logics produce an interesting problem. Non-bivalent inputs produce classically valid consequence statements, for any choice of outputs. A major task of many-valued logics of all stripes is to fashion an appropriately non-classical relation of consequence.The chief preoccupation of non-monotonic (and default) logicians is how to constrain inputs and outputs of the consequence relation. In what is called "left non-monotonicity", it is forbidden to add new sentences to the inputs of true consequence-statements. The restriction takes notice of the fact that new information will sometimes override an antecedently (and reasonably) derived consequence. In what is called "right non-monotonicity", limitations are imposed on outputs of the consequence relation. Most notably, perhaps, is the requirement that the rule of or-introduction not be given free sway on outputs. Also prominent is the effort of paraconsistent logicians, both preservationist and dialetheic, to limit the outputs of inconsistent inputs, which in classical contexts are wholly unconstrained.In some instances, our two themes coincide. Dialetheic logics are a case in point. Dialetheic logics allow certain selected sentences to have, as a third truth value, the classical values of truth and falsity together. So such logics also admit classically inconsistent inputs. A central task is to construct a right non-monotonic consequence relation that allows for these many-valued, and inconsistent, inputs.The Many Valued and Non-Monotonic Turn in Logic is an indispensable research tool for anyone interested in the development of logic, including researchers, graduate and senior undergraduate students in logic, history of logic, mathematics, history of mathematics, computer science, AI, linguistics, cognitive science, argumentation theory, and the history of ideas.

• Detailed and comprehensive chapters covering the entire range of modal logic.
• Contains the latest scholarly discoveries and interprative insights that answers many questions in the field of logic.

Fuzzy-Set Based Logics — an History-Oriented Presentation of their Main Developments

Didier Dubois, Francesc Esteva, Lluís Godo and Henri Prade

1 Introduction: a Historical Perspective

The representation of human-originated information and the formalization of commonsense reasoning has motivated different schools of research in Artificial or Computational Intelligence in the second half of the 20th century. This new trend has also put formal logic, originally developed in connection with the foundations of mathematics, in a completely new perspective, as a tool for processing information on computers. Logic has traditionally put emphasis on symbolic processing at the syntactical level and binary truth-values at the semantical level. The idea of fuzzy sets introduced in the early sixties [Zadeh, 1965] and the development of fuzzy logic later on [Zadeh, 1975a] has brought forward a new formal framework for capturing graded imprecision in information representation and reasoning devices. Indeed, fuzzy sets membership grades can be interpreted in various ways which play a role in human reasoning, such as levels of intensity, similarity degrees, levels of uncertainty, and degrees of preference.

Of course, the development of fuzzy sets and fuzzy logic takes its roots in concerns already encountered in non-classical logics in the first half of the century, when the need for intermediary truth-values and modalities emerged. We start by briefly surveying some of the main issues raised by this research line before describing the historical development of fuzzy sets, fuzzy logic and related issues.

Jan Łukasiewicz (1878–1956) and his followers have developed three-valued logics, and other many-valued systems, since 1920 [Łukasiewicz, 1920]. This research was motivated by philosophical concerns as well as some technical problems in logic but not so much by issues in knowledge representation, leaving the interpretation of intermediate truth-values unclear. This issue can be related to a misunderstanding regarding the law of excluded middle and the law of non-contradiction, and the connections between many-valued logics and modal logics. The principle of bivalence,

Every proposition is either true or false,

formulated and strongly defended by Chrisippus and his school in antique Greece, was for instance questioned by Epicureans, and even rejected by them in the case of propositions referring to future contingencies.

Let us take an example considered already by Aristotle, namely the proposition:

“There will be a sea battle to-morrow (p) and there will not be a sea battle to-morrow (¬p)”

This proposition “p and ¬p” is ever false, because of the non-contradiction law and the proposition “p or ¬p” is ever true, because tertium non datur. But we may fail to know the truth of both propositions “there will be a sea battle to-morrow” and “there will not be a sea battle to-morrow”. In this case, at least intuitively, it seems reasonable to say that it is possible that there will be a sea battle to-morrow but at the same time, it is possible that there will not be a sea battle to-morrow. There has been a recurrent tendency, until the twentieth century many-valued logic tradition, to claim the failure of the bivalence principle on such grounds, and to consider the modality possible as a third truth value. This was apparently (unfortunately) the starting motivation of Łukasiewicz for introducing his three-valued logic. Indeed, the introduction of a third truth-value was interpreted by Łukasiewicz as standing for possible. However the proposition “possible p” is not the same as p, and “possible ¬p” is not the negation of “possible p”. Hence the fact that the proposition

may be true does not question the law of non-contradiction since “possible p” and “possible ¬p” are not mutually exclusive. This situation leads to interpretation problems for a fully truth-functional calculus of possibility, since even if p is “possible” and ¬p is “possible”, still p ∧¬p is ever false.

On the contrary, vague or fuzzy propositions are ones such that, due to the gradual boundary of their sets of models, proposition “p and ¬p” ...

Table of contents

• Cover image
• Title page
• Table of Contents
• Inside Front Cover
• Front Matter
• Copyright page
• Preface
• Contributors
• Many-Valued Logic and its Philosophy
• Preservationism: a Short History
• Paraconsistency and Dialetheism
• The History of Quantum Logic
• Logics of Vagueness
• Fuzzy-Set Based Logics — an History-Oriented Presentation of their Main Developments
• Nonmonotonic Logics: a Preferential Approach
• Default Logic
• Nonmonotonic Reasoning
• Free Logics
• Index