DOCSLIB.ORG

Translating Non-Classical Logics Into the Classical Logic by Using Hidden Variables

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Translating Non-Classical Logics Into the Classical Logic by Using Hidden Variables Translating non-classical logics into the classical logic by using hidden variables Juan C. Agudelo-Agudelo Institute of Mathematics, University of Antioquia, Medellin, Colombia Abstract It is here proposed a method for constructing conservative translations of logics character- ized by `dyadic semantics' (a kind of non-truth-functional bivalued semantics) into the classical logic. The method particularly works for several finite many-valued logics and paraconsistent logics. The translation method uses `hidden variables', which are propositional variables used to represent the indeterminism that arises when non-classical logics are provided of bivalued semantics. Then, it is shown that intuitionistic logic, for instance, is not characterizable by dyadic semantics, then the translation method here proposed do not applied for this logic. Moreover, it is provided an alternative method (not based on dyadic semantics) for construct- ing conservative translations of any finite many-valued logic into the classical logic. In this translation method `hidden variables' are also used, but in this case to represent the degree of true or falsehood of propositions. Introduction Suszko's thesis, as it is nowadays known, states that there are only two logical values, true and false, and that any many-valued logic can be provided with a bivalent semantics (cf. [12]). In [2], it is pointed out that Suszko's original presentation of this reductive result is quite non-constructive and it is provided a method for effectively construct a kind of bivalent semantics, which the authors call gentzenian semantics, for logics that has a truth-functional finite-valued semantics and a ‘sufficiently expressive language'. Then, the dyadic semantics is defined as a specialization of gentzenian semantics where `quasi-tabular' decision procedures can be stablished, \in a deliberate intent to capture the computable class of such semantics". Moreover, it is shown that some logics that are not characterizable by truth-functional finite-valued semantics are characterizable by dyadic semantics, as it is the case of da Costa's paraconsistet logic C1 (in [6], dyadic semantics for other paraconsistent logics are also defined). It is important to mention that dyadic semantics are generally non-truth-functional, but they are usefull, for instance, to define tableau decision procedures (cf. [3] and [4]). The non-truth-functional feature of dyadic semantics can be viewed as a kind of indeterminism. In [5], `hidden variables' are introduced for defining translations of paraconsistent logics, charac- terized by dyadic semantics, into polynomials with coefficients in Z2, providing in this way a proof method for these logics based on polynomial reduction rules. In such translations, hidden variables 1 are (algebraic) variables included in polynomials for representing the indeterminism of dyadic se- mantics. Here, in Section 1, instead of translating non-classical logics into polynomials, we will take advantage of the idea of hidden variables for constructing translations of non-classical logics characterized by dyadic semantics into the Classical Propositional Logic (CPL), showing in this way that non-classical reasoning can be interpreted in classical terms, but at the cost of introducing new (hidden) propositional variables and possibly increasing the complexity of formulas. Hidden variables are, in this context, used to represent the indeterminism that arises when non-classical logics are provided of bivalued semantics. The following question naturally arises, >is any propositional logic characterizable by a dyadic semantics and translatable into CPL by the method here proposed? A negative answer for this question is presented in Section 2, by showing that intuitionistic logic, for instance, is not char- acterizable by dyadic semantics and consequently the method here proposed for constructing con- servative translations into CPL does not applied for this logic. However, the scope of dyadic semantics is really wide, including many finite-valued logics (cf. [2]), logics not characterizable by a two-valued Avron's non-deterministic matrix (as it is the case of several paraconsistent logics in the hierarchy of logics of formal inconsistency introduced in [6], that are only characterizable by non-deterministic matrices with more than two values, cf. [1]), and even logics not characterizable by any finite non-deterministic matrix (as it is the case of C1, cf. [1]). As it was mentioned above, the method introduced in [2] for constructing a dyadic semantics for a finite-valued logic L demands that the logic has a ‘sufficiently expressive language'. This condition consists in requiring that, for any pair of truth values v1 and v2, there exists a formula '(p) in the language of L such that '(p) takes a designated truth value when p takes value v1 and takes an undesignated truth value when p takes value v2 (or vice versa). A condition that is not too restrictive, because many (if not all) genuinely finite-valued logics satisfy this condition. Consequently, by using the method in [2] and the method described in Section 1 below, conservative translations for many finite-valued logics into CPL can be obtained. However, in Section 3, we present a direct method (not based on dyadic semantics) for constructing conservative translations from any finite-valued logic into CPL, independently of the expressive power of its language. Hidden variables are in this method used for representing the degree of truth or falsehood of atomic propositions. A brief discussion of the translation methods here proposed and some possible future works are included in Section 4. 1 From dyadic semantics to conservative translations into the classical logic Before introducing the method for constructing conservative translations for logics characterized by dyadic semantics into CPL, some definitions are in order. For a finite set of connectives (or signature) C, an arity function A: C ! N and a denumerable set of propositional variables V = fp0; p1; p2;:::g, the set of formulas F orC generated by V over C is defined, as usual, as the term algebra of type C generated by V (it is, F orC is the least set containing V and closed under connectives in C). Having defined a set of formulas F orC , a logic is a structure L = hF orC ; i, where ⊆ P(F orC ) × F orC is called a consequence relation. A gentzenian semantics for a logic L, as defined in [2], is a set of bivalutions for formulas of 2 L determined by a set of `conditional clauses'. The following definitions consider a logic L = hF orC ; i. Definition 1. A bivaluation is any function v: F orC ! fT;F g (where T represents true and F represents false). Definition 2. A conditional clause is an expression φ =) , where both φ and are metaformulas of the form: 1 1 n1 n1 1 1 nm nm v('1) = w1; : : : ; v('1 ) = w1 j ::: jv('m) = wm; : : : ; v('m ) = wm ; j j where every 'i is a formula scheme of F orC and every wi is a truth value in fT;F g. In a conditional clause, `=)' represents implication, `,' represents conjunction and `j' represents disjunction. The metalogic that governs conditional clauses is the first order classical logic. Definition 3. Given a set of conditional clauses B, the set of bivaluations that satisfy all conditions in B is called a gentzenian semantics and is denoted by Sem(B) (i.e. Sem(B) = fv: F orC ! fT;F g j v satisfies all conditions in Bg). A gentzenian semantics allows to define the notion of model and a consequence relation in the usual way: Definition 4. Let B be a set of conditional clauses: • A bivaluation v 2 Sem(B) is a model of (or satisfies) a formula ' 2 F orC if v(') = T . The set of models of ', with respect to B, will be denoted by ModB('). • A bivaluation v is a model of (or satifies) a set of formulas Γ ⊆ F orC if it is a model of every formula in Γ. The set of models of Γ, with respect to B, will be denoted by ModB(Γ). • The consequence relation B is defined as follows: a formula ' is a consequence of a set of formulas Γ in B, which is denoted by Γ B ', if and only if ModB(Γ) ⊆ ModB('). • The logic L is characterized by the gentzenian semantics Sem(B) if = B. In this case, it is also said that Sem(B) is an adequate semantics for L. In shake of simplicity, the concept of dyadic semantics is here defined in a slightly different way than in [2]. Before of that, other two definitions are presented. Definition 5. A complexity measure is a function l: F orC ! N, such that l(') < l(δ) whenever ' is a subformula of δ. A function l: F orC ! N, defined by l(pi) = 0; for each propositional variable pi; and l(∗('1;:::;'n)) = maxfl('1); : : : ; l('n)g + 1; if ∗ is an n-ary connective in C; is a particular case of a complexity measure and will be called the canonical complexity measure for F orC . 3 Definition 6. A conditional clause c will be called admissible, with respect to a complexity measure l, if all formulas in the antecedent of c have strictly lower complexity than the formulas in the consequent. Definition 7. A gentzenian semantics, defined by a set of conditional clauses B, will be called a dyadic semantics if: 1. B is finite. 2. The consequent of every conditional clause in B has only one term (i.e. consequent of conditional clauses have the form v(') = w). 3. Formula schemes on the consequent of conditional clauses in B cover all compound formulas of F orC (i.e. any compound formula of F orC can be obtained by uniform substitution on some formula scheme on the consequent of a conditional clause in B). 4. There exists a recursively definable complexity measure l with respect to which all conditional clauses in B are admissible.
Load more

Recommended publications
  • A Simple Logic for Comparisons and Vagueness
    THEODORE J. EVERETT A SIMPLE LOGIC FOR COMPARISONS AND VAGUENESS ABSTRACT. I provide an intuitive, semantic account of a new logic for comparisons (CL), in which atomic statements are assigned both a classical truth-value and a “how much” value or extension in the range [0, 1]. The truth-value of each comparison is determined by the extensions of its component sentences; the truth-value of each atomic depends on whether its extension matches a separate standard for its predicate; everything else is computed classically. CL is less radical than Casari’s comparative logics, in that it does not allow for the formation of comparative statements out of truth-functional molecules. I argue that CL provides a better analysis of comparisons and predicate vagueness than classical logic, fuzzy logic or supervaluation theory. CL provides a model for descriptions of the world in terms of comparisons only. The sorites paradox can be solved by the elimination of atomic sentences. In his recent book on vagueness, Timothy Williamson (1994) attacks accounts of vagueness arising from either fuzzy logic or supervaluation theory, both of which have well-known problems, stemming in part from their denial of classical bivalence. He then gives his own, epistemic ac- count of vagueness, according to which the vagueness of a predicate is fundamentally a matter of our not knowing whether or not it applies in every case. My main purpose in this paper is not to criticize William- son’s positive view, but to provide an alternative, non-epistemic account of predicate vagueness, based on a very simple logic for comparisons (CL), which preserves bivalence.
    [Show full text]
  • Does Informal Logic Have Anything to Learn from Fuzzy Logic?
    University of Windsor Scholarship at UWindsor OSSA Conference Archive OSSA 3 May 15th, 9:00 AM - May 17th, 5:00 PM Does informal logic have anything to learn from fuzzy logic? John Woods University of British Columbia Follow this and additional works at: https://scholar.uwindsor.ca/ossaarchive Part of the Philosophy Commons Woods, John, "Does informal logic have anything to learn from fuzzy logic?" (1999). OSSA Conference Archive. 62. https://scholar.uwindsor.ca/ossaarchive/OSSA3/papersandcommentaries/62 This Paper is brought to you for free and open access by the Conferences and Conference Proceedings at Scholarship at UWindsor. It has been accepted for inclusion in OSSA Conference Archive by an authorized conference organizer of Scholarship at UWindsor. For more information, please contact [email protected]. Title: Does informal logic have anything to learn from fuzzy logic? Author: John Woods Response to this paper by: Rolf George (c)2000 John Woods 1. Motivation Since antiquity, philosophers have been challenged by the apparent vagueness of everyday thought and experience. How, these philosophers have wanted to know, are the things we say and think compatible with logical laws such as Excluded Middle?1 A kindred attraction has been the question of how truth presents itself — relatively and in degrees? in approximations? in resemblances, or in bits and pieces? F.H. Bradley is a celebrated (or as the case may be, excoriated) champion of the degrees view of truth. There are, one may say, two main views of error, the absolute and relative. According to the former view there are perfect truths, and on the other side there are sheer errors..
    [Show full text]
  • Intransitivity and Vagueness
    Intransitivity and Vagueness Joseph Y. Halpern∗ Cornell University Computer Science Department Ithaca, NY 14853 [email protected] http://www.cs.cornell.edu/home/halpern Abstract preferences. Moreover, it is this intuition that forms the ba- sis of the partitional model for knowledge used in game the- There are many examples in the literature that suggest that indistinguishability is intransitive, despite the fact that the in- ory (see, e.g., [Aumann 1976]) and in the distributed sys- distinguishability relation is typically taken to be an equiv- tems community [Fagin, Halpern, Moses, and Vardi 1995]. alence relation (and thus transitive). It is shown that if the On the other hand, besides the obvious experimental obser- uncertainty perception and the question of when an agent vations, there have been arguments going back to at least reports that two things are indistinguishable are both care- Poincare´ [1902] that the physical world is not transitive in fully modeled, the problems disappear, and indistinguishabil- this sense. In this paper, I try to reconcile our intuitions ity can indeed be taken to be an equivalence relation. More- about indistinguishability with the experimental observa- over, this model also suggests a logic of vagueness that seems tions, in a way that seems (at least to me) both intuitively to solve many of the problems related to vagueness discussed appealing and psychologically plausible. I then go on to ap- in the philosophical literature. In particular, it is shown here ply the ideas developed to the problem of vagueness. how the logic can handle the Sorites Paradox. To understand the vagueness problem, consider the well- n+1 1 Introduction known Sorites Paradox: If grains of sand make a heap, then so do n.
    [Show full text]
  • On the Relationship Between Fuzzy Autoepistemic Logic and Fuzzy Modal Logics of Belief
    JID:FSS AID:6759 /FLA [m3SC+; v1.204; Prn:31/03/2015; 9:42] P.1(1-26) Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss On the relationship between fuzzy autoepistemic logic and fuzzy modal logics of belief ∗ Marjon Blondeel d, ,1, Tommaso Flaminio a,2, Steven Schockaert b, Lluís Godo c,3, Martine De Cock d,e a Università dell’Insubria, Department of Theorical and Applied Science, Via Mazzini 5, 21100 Varese, Italy b Cardiff University, School of Computer Science and Informatics, 5The Parade, Cardiff, CF24 3AA, UK c Artficial Intelligence Research Institute (IIIA) – CSIC, Campus UAB, 08193 Bellaterra, Spain d Ghent University, Department of Applied Mathematics, Computer Science and Statistics, Krijgslaan 281 (S9), 9000 Gent, Belgium e University of Washington Tacoma – Center for Web and Data Science, 1900 Commerce Street, Tacoma, WA 98402-3100, USA Received 12 February 2014; received in revised form 30 September 2014; accepted 26 February 2015 Abstract Autoepistemic logic is an important formalism for nonmonotonic reasoning originally intended to model an ideal rational agent reflecting upon his own beliefs. Fuzzy autoepistemic logic is a generalization of autoepistemic logic that allows to represent an agent’s rational beliefs on gradable propositions. It has recently been shown that, in the same way as autoepistemic logic generalizes answer set programming, fuzzy autoepistemic logic generalizes fuzzy answer set programming as well. Besides being related to answer set programming, autoepistemic logic is also closely related to several modal logics. To investigate whether a similar relationship holds in a fuzzy logical setting, we firstly generalize the main modal logics for belief to the setting of finitely-valued c Łukasiewicz logic with truth constants Łk, and secondly we relate them with fuzzy autoepistemic logics.
    [Show full text]
  • Consequence and Degrees of Truth in Many-Valued Logic
    Consequence and degrees of truth in many-valued logic Josep Maria Font Published as Chapter 6 (pp. 117–142) of: Peter Hájek on Mathematical Fuzzy Logic (F. Montagna, ed.) vol. 6 of Outstanding Contributions to Logic, Springer-Verlag, 2015 Abstract I argue that the definition of a logic by preservation of all degrees of truth is a better rendering of Bolzano’s idea of consequence as truth- preserving when “truth comes in degrees”, as is often said in many-valued contexts, than the usual scheme that preserves only one truth value. I review some results recently obtained in the investigation of this proposal by applying techniques of abstract algebraic logic in the framework of Łukasiewicz logics and in the broader framework of substructural logics, that is, logics defined by varieties of (commutative and integral) residuated lattices. I also review some scattered, early results, which have appeared since the 1970’s, and make some proposals for further research. Keywords: many-valued logic, mathematical fuzzy logic, truth val- ues, truth degrees, logics preserving degrees of truth, residuated lattices, substructural logics, abstract algebraic logic, Leibniz hierarchy, Frege hierarchy. Contents 6.1 Introduction .............................2 6.2 Some motivation and some history ...............3 6.3 The Łukasiewicz case ........................8 6.4 Widening the scope: fuzzy and substructural logics ....9 6.5 Abstract algebraic logic classification ............. 12 6.6 The Deduction Theorem ..................... 16 6.7 Axiomatizations ........................... 18 6.7.1 In the Gentzen style . 18 6.7.2 In the Hilbert style . 19 6.8 Conclusions .............................. 21 References .................................. 23 1 6.1 Introduction Let me begin by calling your attention to one of the main points made by Petr Hájek in the introductory, vindicating section of his influential book [34] (the italics are his): «Logic studies the notion(s) of consequence.
    [Show full text]
  • A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability and Statistics
    FLORENTIN SMARANDACHE A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS (fourth edition) NL(A1 A2) = ( T1 ({1}T2) T2 ({1}T1) T1T2 ({1}T1) ({1}T2), I1 ({1}I2) I2 ({1}I1) I1 I2 ({1}I1) ({1} I2), F1 ({1}F2) F2 ({1} F1) F1 F2 ({1}F1) ({1}F2) ). NL(A1 A2) = ( {1}T1T1T2, {1}I1I1I2, {1}F1F1F2 ). NL(A1 A2) = ( ({1}T1T1T2) ({1}T2T1T2), ({1} I1 I1 I2) ({1}I2 I1 I2), ({1}F1F1 F2) ({1}F2F1 F2) ). ISBN 978-1-59973-080-6 American Research Press Rehoboth 1998, 2000, 2003, 2005 FLORENTIN SMARANDACHE A UNIFYING FIELD IN LOGICS: NEUTROSOPHIC LOGIC. NEUTROSOPHY, NEUTROSOPHIC SET, NEUTROSOPHIC PROBABILITY AND STATISTICS (fourth edition) NL(A1 A2) = ( T1 ({1}T2) T2 ({1}T1) T1T2 ({1}T1) ({1}T2), I1 ({1}I2) I2 ({1}I1) I1 I2 ({1}I1) ({1} I2), F1 ({1}F2) F2 ({1} F1) F1 F2 ({1}F1) ({1}F2) ). NL(A1 A2) = ( {1}T1T1T2, {1}I1I1I2, {1}F1F1F2 ). NL(A1 A2) = ( ({1}T1T1T2) ({1}T2T1T2), ({1} I1 I1 I2) ({1}I2 I1 I2), ({1}F1F1 F2) ({1}F2F1 F2) ). ISBN 978-1-59973-080-6 American Research Press Rehoboth 1998, 2000, 2003, 2005 1 Contents: Preface by C. Le: 3 0. Introduction: 9 1. Neutrosophy - a new branch of philosophy: 15 2. Neutrosophic Logic - a unifying field in logics: 90 3. Neutrosophic Set - a unifying field in sets: 125 4. Neutrosophic Probability - a generalization of classical and imprecise probabilities - and Neutrosophic Statistics: 129 5. Addenda: Definitions derived from Neutrosophics: 133 2 Preface to Neutrosophy and Neutrosophic Logic by C.
    [Show full text]
  • Reading T. S. Eliot Reading Spinoza
    READING T. S. ELIOT READING SPINOZA A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Yoshiaki Mihara January 2013 © 2013 Yoshiaki Mihara READING T. S. ELIOT READING SPINOZA Yoshiaki Mihara, Ph. D. Cornell University 2013 “[The ordinary man] falls in love, or reads Spinoza, and these two experiences have nothing to do with each other” – a good reader of T. S. Eliot’s criticism probably knows this famous passage; a good researcher on Eliot’s apprenticeship in philosophy perhaps knows that he did actually read Spinoza; and yet, in the current Eliot studies, reading Eliot and reading Spinoza seem to have nothing to do with each other. In this dissertation, I attempt to reconstruct Eliot’s reading of Spinoza as faithfully and comprehensively as possible, by closely analyzing the marginalia in Eliot’s copy of Spinoza’s Opera, housed in the Archive Centre of King’s College, Cambridge. At the same time, the Spinozist context for Eliot’s apprenticeship at Harvard and Oxford (with the intermission of “a romantic year” in Paris) is also to be presented, which is, in fact, a glaring absence in the philosophical branch of Eliot studies. In addition to these positivistic contributions, I also take a theoretical approach so as to demonstrate how illuminative Eliot’s reading of Spinoza can be for understanding the characteristic style (or “ethology”) of Eliot’s reading in general (i.e., Theory), by way of extensively analyzing the unpublished as well as published materials of Eliot’s “academic philosophizing” that culminated in his doctoral dissertation on F.
    [Show full text]
  • Modality and Factivity: One Perspective on the Meaning of the English Modal Auxiliaries
    MODALITY AND FACTIVITY: ONE PERSPECTIVE ON THE MEANING OF THE ENGLISH MODAL AUXILIARIES by NICOLA M,BREWER Submitted in accordance with the requirements for the degree of PhD The University of Leeds Department of Linguistics and Phonetics December 1987 ii ABSTRACT This study concentrates on modality as expressed by the set of modal auxiliaries and seeks to establish that these verbs share semantic as well as syntactic properties by identifying a single core meaning which they share. The relationship between modality and factivity is examined with the aim of gaining an insight into the former, more complex concept. When viewed from this perspective, the defining characteristic of all the modal auxiliary verbs in almost all of their uses is found to be nonfactivity. The meanings expressed by this set of verbs are classified according to a framework derived from modal logic consisting of three basic types of modality each of which relates to a different set of laws or principles; the relative factivity associated with the modal auxiliaries is seen to vary with the nature of modality as defined and classified by this framework. Within each of the three types of modality, a semantic scale is identified and modality is described as a gradable concept for which scalar analysis is appropriate, both within and beyond these three scales. Relative factivity is also shown to vary according to the degree of modality expressed by each of the modal verbs. The nature and degree of modality expressed interact with features of the linguistic (and pragmatic) context to determine the particular factive or a contrafactive interpretation conveyed by a given modal auxiliary token.
    [Show full text]
  • Irrationality and Human Reasoning Michael Aristidou Louisiana State University and Agricultural and Mechanical College
    Louisiana State University LSU Digital Commons LSU Master's Theses Graduate School 2004 Irrationality and human reasoning Michael Aristidou Louisiana State University and Agricultural and Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_theses Part of the Arts and Humanities Commons Recommended Citation Aristidou, Michael, "Irrationality and human reasoning" (2004). LSU Master's Theses. 2918. https://digitalcommons.lsu.edu/gradschool_theses/2918 This Thesis is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Master's Theses by an authorized graduate school editor of LSU Digital Commons. For more information, please contact [email protected]. IRRATIONALITY AND HUMAN REASONING A Thesis Submitted to the Graduate Faculty of the Louisiana State University in partial fulfillment of the requirements of the degree of Master of Arts in The Department of Philosophy by Michael Aristidou BSc, Aristotle University of Thessaloniki, 1999 MSc, Louisiana State University, 2001 May, 2004 To my parents Theodotos and Kristia, to my sister Eliza, and to my brother Konstantinos for all their love and support through all these years. ii ACKNOWLEDGMENTS I would like to thank my advisor Dr. Jon Cogburn for his great patience and valuable instruction on my thesis. I would also like to thank, Dr. Mary Sirridge for her advising and help through my graduate years in Philosophy, the members of my thesis committee Dr. Husain Sarkar and Dr. Ian Crystal for their cooperation, and Taneka Welch for her constant willingness to discuss issues related to my thesis, and her technical support.
    [Show full text]
  • Arxiv:1009.3391V3 [Cs.LO] 4 Nov 2010
    Fuzzy Ontology Representation using OWL 2 I Fernando Bobilloa, Umberto Stracciab aDepartment of Computer Science and Systems Engineering, University of Zaragoza, Spain bIstituto di Scienza e Tecnologie dell'Informazione (ISTI - CNR), Pisa, Italy Abstract The need to deal with vague information in Semantic Web languages is rising in importance and, thus, calls for a standard way to represent such information. We may address this issue by either extending current Semantic Web languages to cope with vagueness, or by providing a procedure to represent such informa- tion within current standard languages and tools. In this work, we follow the latter approach, by identifying the syntactic differences that a fuzzy ontology language has to cope with, and by proposing a concrete methodology to rep- resent fuzzy ontologies using OWL 2 annotation properties. We also report on the prototypical implementations. Key words: Fuzzy OWL 2, Fuzzy Ontologies, Fuzzy Languages for the Semantic Web, Fuzzy Description Logics 1. Introduction Today, there is a growing interest in the development of knowledge representa- tion formalisms able to deal with uncertainty, which is a very common require- ment in real world applications. Despite the undisputed success of ontologies, classical ontology languages are not appropriate to deal with vagueness or impre- cision in the knowledge, which is inherent to most of the real world application domains [29]. Since fuzzy set theory and fuzzy logic [30] are suitable formalisms to handle these types of knowledge, fuzzy ontologies emerge as useful in several applica- arXiv:1009.3391v3 [cs.LO] 4 Nov 2010 tions, ranging from (multimedia) information retrieval to image interpretation, ontology mapping, matchmaking, decision making, or the Semantic Web [19].
    [Show full text]
  • K3, Ł3, LP, RM3, A3, FDE, M: How to Make Many-Valued Logics Work for You
    K3, Ł3, LP, RM3, A3, FDE, M: How to Make Many-Valued Logics Work for You Allen P. Hazen Francis Jeffry Pelletier Dept. Philosophy Dept. Philosophy University of Alberta University of Alberta Abstract We investigate some well-known (and a few not-so-well-known) many-valued logics that have a small number (3 or 4) of truth values. For some of them we complain that they do not have any logical use (despite their perhaps having some intuitive semantic interest) and we look at ways to add features so as to make them useful, while retaining their intuitive appeal. At the end, we show some surprising results in the system FDE, and its relationships with features of other logics. We close with some new examples of “synonymous logics.” An Appendix contains a Fitch-style natural deduction system for our augmented FDE, and proofs of soundness and completeness. 1 Introduction This article is about some many-valued logics that have a relatively small number of truth values – either three or four. This topic has its modern (and Western) start with the work of (Post, 1921; Łukasiewicz, 1920; Łukasiewicz and Tarski, 1930), which slowly gained interest of other logicians and saw an increase in writings during the 1930s (Wajsberg, 1931; Bochvar, 1939, 1943; Słupecki, 1936, 1939a,b; Kleene, 1938). But it remained somewhat of a niche enterprise until the early 1950s, when two works brought the topic somewhat more into the “mainstream” of formal logic (Rosser and Turquette, 1952; Kleene, 1952). The general survey book, Rescher (1969), brought the topic of many-valued logics to the attention of a wider group within philosophy.
    [Show full text]
  • Hedges: a Study in Meaning Criteria and the Logic of Fuzzy Concepts
    UC Berkeley UC Berkeley Previously Published Works Title Hedges: A Study In Meaning Criteria And The Logic Of Fuzzy Concepts Permalink https://escholarship.org/uc/item/0x0010nv Author Lakoff, George Publication Date 1973 Peer reviewed eScholarship.org Powered by the California Digital Library University of California GEORGE LAKOFF HEDGES: A STUDY 1N MEANING CRITERIA AND THE LOGIC OF FUZZY CONCEPTS* 1. DEGREES OF TRUTH Logicians have, by and large, engaged in the convenient fiction that sen- tences of natural languages (at least declarative sentences) are either true or false or, at worst, lack a truth value, or have a third value often inter- preted as 'nonsense'. And most contemporary linguists who have thought seriously about semantics, especially formal semantics, have largely shared this fiction, primarily for lack of a sensible alternative. Yet students of language, especially psychologists and linguistic philosophers, have long been attuned to the fact that natural language concepts have vague boundaries and fuzzy edges and that, consequently, natural language sen- tences will very often be neither true, nor false, nor nonsensical, but rather true to a certain extent and false to a certain extent, true in certain re- spects and false in other respects. It is common for logicians to give truth conditions for predicates in terms of classical set theory. 'John is tall' (or 'TALL(j)') is defined to be true just in case the individual denoted by 'John' (or 'j') is in the set of tall men. Putting aside the problem that tallness is really a relative concept (tallness for a pygmy and tallness for a basketball player are obviously different)1, suppose we fix a population relative to which we want to define tallness.
    [Show full text]