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A Treatise on ManyValued Logics
 Studies in Logic and Computation
, 2001
"... The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with som ..."
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Cited by 53 (3 self)
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The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics. Key words: mathematical fuzzy logic, algebraic semantics, continuous tnorms, leftcontinuous tnorms, Pavelkastyle fuzzy logic, fuzzy set theory, nonmonotonic fuzzy reasoning 1 Basic ideas 1.1 From classical to manyvalued logic Logical systems in general are based on some formalized language which includes a notion of well formed formula, and then are determined either semantically or syntactically. That a logical system is semantically determined means that one has a notion of interpretation or model 1 in the sense that w.r.t. each such interpretation every well formed formula has some (truth) value or represents a function into
Proof Transformations in HigherOrder Logic
, 1987
"... We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, ..."
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Cited by 22 (5 self)
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We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, H, a form of the sequent calculus we define, nonanalytic proofs are represented by natural deductions. A nondeterministic translation algorithm between expansion proofs and Hdeductions is presented and its correctness is proven. We also present an algorithm for translation in the other direction and prove its correctness. A cutelimination algorithm for expansion proofs is given and its partial correctness is proven. Strong termination of this algorithm remains a conjecture for the full higherorder system, but is proven for the firstorder fragment. We extend the translations to a nonanalytic proof system which contains a primitive notion of equality, while leaving the notion of expansion proof unaltered. This is possible, since a nonextensional equality is definable in our system of type theory. Next we extend analytic and nonanalytic proof systems and the translations between them to include extensionality. Finally, we show how the methods and notions used so far apply to the problem of translating expansion proofs into natural deductions. Much care is taken to specify this translation in a
Local Possibilistic Logic
 Journal of Applied NonClassical Logic
, 1997
"... Possibilistic states of information are fuzzy sets of possible worlds. They constitute a complete lattice, which can be endowed with a monoidal operation (a tnorm) to produce a quantal. An algebraic semantics is presented which links possibilistic formulae with information states, and gives a natur ..."
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Cited by 21 (14 self)
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Possibilistic states of information are fuzzy sets of possible worlds. They constitute a complete lattice, which can be endowed with a monoidal operation (a tnorm) to produce a quantal. An algebraic semantics is presented which links possibilistic formulae with information states, and gives a natural interpretation of logical connectives as operations on fuzzy sets. Due to the quantal structure of information states, we obtain a system which shares several features with (exponentialfree) intuitionistic linear logic. Soundness and completeness are proved, parametrically on the choice of the tnorm operation.
Mathematical fuzzy logic as a tool for the treatment of vague information
 Information Sciences
, 2005
"... The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by ..."
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Cited by 11 (1 self)
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The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by a calculus for the derivation of formulas. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon these theoretical considerations. Key words: mathematical fuzzy logic, algebraic semantics, continuous tnorms, leftcontinuous tnorms, Pavelkastyle fuzzy logic, fuzzy set theory, nonmonotonic fuzzy reasoning 1
The Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."
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Cited by 8 (2 self)
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This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.
Truth functionality and measurebased logics
 Fuzzy Sets, Logics and Reasoning about Knowledge
, 1999
"... We present a truthfunctional semantics for necessityvalued logics, based on the forcing technique. We interpret possibility distributions (which correspond to necessity measures) as informational states, and introduce a suitable language (basically, an extension of classical logic, similar to Pave ..."
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Cited by 5 (4 self)
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We present a truthfunctional semantics for necessityvalued logics, based on the forcing technique. We interpret possibility distributions (which correspond to necessity measures) as informational states, and introduce a suitable language (basically, an extension of classical logic, similar to Pavelka’s language). Then we define the relation of “forcing ” between an informational state and a formula, meaning that the state contains enough information to support the validity of the formula. The subsequent step is the definition of a manyvalued truthfunctional semantics, by simply taking the truth value of a formula to be the set of all informational states that force the truth of the formula. A proof system in sequent calculus form is provided, and validity and completeness theorems are proved. 1
Coherent Functions in Autonomous Systems
, 2002
"... INTRODUCTION Advanced sensorimotor devices, like mobile robots, are often referred to as autonomous systems. The expression is used to intentionally remark on the difference between these systems and those of traditional industrial automation. Although no rigorous definition is easily available, a ..."
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Cited by 3 (2 self)
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INTRODUCTION Advanced sensorimotor devices, like mobile robots, are often referred to as autonomous systems. The expression is used to intentionally remark on the difference between these systems and those of traditional industrial automation. Although no rigorous definition is easily available, a general informal consensus seems to exist on the features that denote autonomy: a system is considered the more autonomous the more reliably it can survive and perform tasks in the real world, without the need for human intervention. 171 ____________________________ *email: claudio.sossai@isib.cnr.it J. of Mult.Valued Logic & Soft Computing., Vol. 9, pp. 171194 2003 Old City Publishing, Inc. Reprints available directly from the publisher Published by license under the OCP Science imprint, Photocopying permitted by license only a member of the Old City Publishing Group In the literature different ideas and techniques have been proposed and investigated to achieve these results, nonethe
3 rd Generation Partnership Project, “Stage 1 Service Requirements for the Open Service Access (OSA
 Spacetime Physics Research Trends, Horizons in World Physics, volume 248. Nova Science Publishers, 2006. Manuscript at http: //publish.uwo.ca/ ∼ jbell
, 2001
"... In the present paper the concept of a covering is presented and developed. The relationship between cover schemes, frames (complete Heyting algebras), Kripke models, and framevalued set theory is discussed. Finally cover schemes and framevalued set theory are applied in the context of Markopoulou’s ..."
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Cited by 3 (0 self)
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In the present paper the concept of a covering is presented and developed. The relationship between cover schemes, frames (complete Heyting algebras), Kripke models, and framevalued set theory is discussed. Finally cover schemes and framevalued set theory are applied in the context of Markopoulou’s account of discrete spacetime as sets “evolving ” over a causal set. We observe that Markopoulou’s proposal may be effectively realized by working within an appropriate framevalued model of set theory. We go on to show that, within this framework, cover schemes may be used to force certain conditions to prevail in the associated models: for example, rendering the universe timeless, obliterating a given event or forcing it to become the universe’s “beginning”. Preamble The concept of Grothendieck (pre)topology or covering issued from the efforts of algebraic geometers to study “sheaflike ” objects defined on categories more general than the lattice of open sets of a topological space (see, e.g. [4]). A Grothendieck pretopology on a category C with pullbacks is defined by specifying, for each object U of C, a set P(U) of arrows to U called covering families satisfying appropriate category 2 theoretic versions of the corresponding conditions for a family A of sets to cover a set U, namely: (i) {U} covers U, (ii) if A covers U and V ⊆ U, then