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Ivan Chajda - One of the best experts on this subject based on the ideXlab platform.
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a representation of residuated lattices satisfying the Double Negation law
Soft Computing, 2018Co-Authors: Ivan ChajdaAbstract:Every residuated lattice can be considered as an idempotent semiring. Conversely, if an idempotent semiring is finite, then it can be organized into a residuated lattice. Unfortunately, this does not hold in general. We show that if an idempotent semiring is equipped with an involution which satisfies certain conditions, then it can be organized into a residuated lattice satisfying the Double Negation law. Also conversely, every residuated lattice satisfying the Double Negation law can be considered as an idempotent semiring with an involution satisfying the mentioned conditions.
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Residuation in orthomodular lattices
Topological Algebra and its Applications, 2017Co-Authors: Ivan Chajda, Helmut LängerAbstract:Abstract We show that every idempotent weakly divisible residuated lattice satisfying the Double Negation law can be transformed into an orthomodular lattice. The converse holds if adjointness is replaced by conditional adjointness. Moreover, we show that every positive right residuated lattice satisfying the Double Negation law and two further simple identities can be converted into an orthomodular lattice. In this case, also the converse statement is true and the corresponence is nearly one-to-one.
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General coupled semirings of residuated lattices
Fuzzy Sets and Systems, 2016Co-Authors: Ivan Chajda, Helmut LängerAbstract:Abstract Di Nola and Gerla showed that MV-algebras and coupled semirings are in a natural one-to-one correspondence. We generalize this correspondence to residuated lattices satisfying the Double Negation law.
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Set Representation of Partial Dynamic De Morgan Algebras
2016 IEEE 46th International Symposium on Multiple-Valued Logic (ISMVL), 2016Co-Authors: Ivan Chajda, Jan PasekaAbstract:By a De Morgan algebra is meant a bounded poset equipped with an antitone involution considered as Negation. Such an algebra can be considered as an algebraic axiomatization of a propositional logic satisfying the Double Negation law. Our aim is to introduce the so-called tense operators in every De Morgan algebra for to get an algebraic counterpart of a tense logic with Negation satisfying the Double Negation law which need not be Boolean. Following the standard construction of tense operators G and H by a frame we solve the following question: if a dynamic De Morgan algebra is given, how to find a frame such that its tense operators G and H can be reached by this construction.
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ISMVL - Set Representation of Partial Dynamic De Morgan Algebras
2016 IEEE 46th International Symposium on Multiple-Valued Logic (ISMVL), 2016Co-Authors: Ivan Chajda, Jan PasekaAbstract:By a De Morgan algebra is meant a bounded poset equipped with an antitone involution considered as Negation. Such an algebra can be considered as an algebraic axiomatization of a propositional logic satisfying the Double Negation law. Our aim is to introduce the so-called tense operators in every De Morgan algebra for to get an algebraic counterpart of a tense logic with Negation satisfying the Double Negation law which need not beBoolean. Following the standard construction of tense operatorsG and H by a frame we solve the following question: if adynamic De Morgan algebra is given, how to find a framesuch that its tense operators G and H can be reached bythis construction.
Jan Paseka - One of the best experts on this subject based on the ideXlab platform.
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Set Representation of Partial Dynamic De Morgan Algebras
2016 IEEE 46th International Symposium on Multiple-Valued Logic (ISMVL), 2016Co-Authors: Ivan Chajda, Jan PasekaAbstract:By a De Morgan algebra is meant a bounded poset equipped with an antitone involution considered as Negation. Such an algebra can be considered as an algebraic axiomatization of a propositional logic satisfying the Double Negation law. Our aim is to introduce the so-called tense operators in every De Morgan algebra for to get an algebraic counterpart of a tense logic with Negation satisfying the Double Negation law which need not be Boolean. Following the standard construction of tense operators G and H by a frame we solve the following question: if a dynamic De Morgan algebra is given, how to find a frame such that its tense operators G and H can be reached by this construction.
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ISMVL - Set Representation of Partial Dynamic De Morgan Algebras
2016 IEEE 46th International Symposium on Multiple-Valued Logic (ISMVL), 2016Co-Authors: Ivan Chajda, Jan PasekaAbstract:By a De Morgan algebra is meant a bounded poset equipped with an antitone involution considered as Negation. Such an algebra can be considered as an algebraic axiomatization of a propositional logic satisfying the Double Negation law. Our aim is to introduce the so-called tense operators in every De Morgan algebra for to get an algebraic counterpart of a tense logic with Negation satisfying the Double Negation law which need not beBoolean. Following the standard construction of tense operatorsG and H by a frame we solve the following question: if adynamic De Morgan algebra is given, how to find a framesuch that its tense operators G and H can be reached bythis construction.
Henriëtte De Swart - One of the best experts on this subject based on the ideXlab platform.
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A Typology of Negation in a Constraint-Based Framework of Syntax and Semantics
2020Co-Authors: Henriëtte De SwartAbstract:Negation and negative indefinites raise problems for the principle of compositionality of meaning, because we find both Double and single Negation readings in natural languages. De Swart and Sag (2002) solve the compositionality problem in a polyadic quantifier framework. All negative quantifiers are collected into an N-store, and are interpreted by means of iteration (Double Negation) or resumption (negative concord) upon retrieval. This paper extends the earlier analysis with a typology of Negation and negative indefinites using bi-directional optimality theory (OT). The constraints defined are universal, but their ranking varies from one language to the next. In negative concord languages, the functional motivation for the marking of ‘negative variables’ wins out. Double Negation languages value first-order iteration. The bi-directional set-up is essential, for syntactic and semantic variation go hand in hand.
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Double Negation Readings
The Oxford Handbook of Negation, 2020Co-Authors: Henriëtte De SwartAbstract:This chapter is concerned with the linguistic environments in which Double Negation readings do and do not arise in Double Negation and negative concord languages. The theoretical background comes from other chapters in the Oxford Handbook of Negation. The chapter briefly surveys the experimental literature on the role of prosody in the comprehension of negative concord and Double Negation, and continues with a multilingual corpus investigation that focuses on language use. Under the assumption that all languages convey the same message in a specific context, production data in parallel corpora enable us to detect grammatical variation through translation. The examples are extracted from the parallel corpus EuroParl and the languages discussed are English, Dutch, German, Italian, French, and Spanish. Even though the set of languages is relatively small, the spread of grammars should be wide enough to shed light on the phenomenon of Double Negation in natural language.
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Double Negation in negative concord languages
2010Co-Authors: Henriëtte De SwartAbstract:This chapter investigates Double Negation readings in negative concord languages. This may look like a contradiction in terms. After all, negative concord is a system in which multiple expressions of Negation combine to convey a single Negation reading. If so, then how is it possible to express Double Negation in such languages? I will discuss three cases in which this arises.
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Sentential Negation and Negative Indefinites
Studies in Natural Language and Linguistic Theory, 2009Co-Authors: Henriëtte De SwartAbstract:This chapter integrates the results on sentential Negation (from Chapter 3) with the analysis of negative concord and Double Negation (from Chapter 4). Section 1 develops the classification of co-occurrence restrictions between sentential Negation and negative indefinites in negative concord and Double Negation languages. The grammar of negative spread supports the claims made by de Swart and Sag’s (2002) that in the presence of n-words, the marker of sentential Negation in negative concord languages is semantically redundant (Section 2).
Paulo Oliva - One of the best experts on this subject based on the ideXlab platform.
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the herbrand functional interpretation of the Double Negation shift
Journal of Symbolic Logic, 2017Co-Authors: Martin Hotzel Escardo, Paulo OlivaAbstract:This paper considers a generalisation of selection functions over an arbi- trary strong monad T, as functions of type J T RX = (X → R) → TX. It is assumed throughout that R is a T-algebra. We show that J T R is also a strong monad, and that it embeds into the continuation monad KRX = (X → R) → R. We use this to derive that the explicitly controlled product of T-selection functions is definable from the explicitly controlled product of quantifiers. We then prove several properties of this product in the special case when T is the finite power set monad Pf(·). These are used to show that when TX = Pf(X) the explicitly controlled product of T-selection functions calculates a witness to the Herbrand functional interpretation of the Double Negation shift, and hence countable choice.
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On Pocrims and Hoops.
arXiv: Logic, 2014Co-Authors: Rob Arthan, Paulo OlivaAbstract:Pocrims and suitable specialisations thereof are structures that provide the natural algebraic semantics for a minimal affine logic and its extensions. Hoops comprise a special class of pocrims that provide algebraic semantics for what we view as an intuitionistic analogue of the classical multi-valued {\L}ukasiewicz logic. We present some contributions to the theory of these algebraic structures. We give a new proof that the class of hoops is a variety. We use a new indirect method to establish several important identities in the theory of hoops: in particular, we prove that the Double Negation mapping in a hoop is a homormorphism. This leads to an investigation of algebraic analogues of the various Double Negation translations that are well-known from proof theory. We give an algebraic framework for studying the semantics of Double Negation translations and use it to prove new results about the applicability of the Double Negation translations due to Gentzen and Glivenko.
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what sequential games the tychonoff theorem and the Double Negation shift have in common
Proceedings of the third ACM SIGPLAN workshop on Mathematically structured functional programming, 2010Co-Authors: Martin Hotzel Escardo, Paulo OlivaAbstract:This is a tutorial for mathematically inclined functional programmers, based on previously published, peered reviewed theoretical work. We discuss a higher-type functional, written here in the functional programming language Haskell, which (1) optimally plays sequential games, (2) implements a computational version of the Tychonoff Theorem from topology, and (3) realizes the Double-Negation Shift from logic and proof theory. The functional makes sense for finite and infinite (lazy) lists, and in the binary case it amounts to an operation that is available in any (strong) monad. In fact, once we define this monad in Haskell, it turns out that this amazingly versatile functional is already available in Haskell, in the standard prelude, called sequence, which iterates this binary operation. Therefore Haskell proves that this functional is even more versatile than anticipated, as the function sequence was introduced for other purposes by the language designers, in particular the iteration of a list of monadic effects (but effects are not what we discuss here).
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MSFP@ICFP - What sequential games, the tychonoff theorem and the Double-Negation shift have in common
Proceedings of the third ACM SIGPLAN workshop on Mathematically structured functional programming - MSFP '10, 2010Co-Authors: Martin Hotzel Escardo, Paulo OlivaAbstract:This is a tutorial for mathematically inclined functional programmers, based on previously published, peered reviewed theoretical work. We discuss a higher-type functional, written here in the functional programming language Haskell, which (1) optimally plays sequential games, (2) implements a computational version of the Tychonoff Theorem from topology, and (3) realizes the Double-Negation Shift from logic and proof theory. The functional makes sense for finite and infinite (lazy) lists, and in the binary case it amounts to an operation that is available in any (strong) monad. In fact, once we define this monad in Haskell, it turns out that this amazingly versatile functional is already available in Haskell, in the standard prelude, called sequence, which iterates this binary operation. Therefore Haskell proves that this functional is even more versatile than anticipated, as the function sequence was introduced for other purposes by the language designers, in particular the iteration of a list of monadic effects (but effects are not what we discuss here).
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the peirce translation and the Double Negation shift
Conference on Computability in Europe, 2010Co-Authors: Martin Hotzel Escardo, Paulo OlivaAbstract:We develop applications of selection functions to proof theory and computational extraction of witnesses from proofs in classical analysis. The main novelty is a translation of classical minimal logic into minimal logic, which we refer to as the Peirce translation, and which we apply to interpret both a strengthening of the Double-Negation shift and the axioms of countable and dependent choice, via infinite products of selection functions.
Frances Blanchette - One of the best experts on this subject based on the ideXlab platform.
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Negative Concord in English
Linguistic Variation, 2020Co-Authors: Frances BlanchetteAbstract:This paper argues that Negative Concord is generated by the grammars of all English varieties, but just not “realized” in the standardized variety, in the sense of Barbiers (2005, 2009). I show that Double Negation constructions, wherein two negative elements yield a doubly negated meaning, are formed identically by English varieties that realize Negative Concord and those that do not. Unlike previous Minimalist Agree approaches to English Negative Concord, this proposal accounts for the fact that English varieties generate both Double Negation and Negative Concord constructions. This paper employs Tortora’s (2009, in press) mechanism of feature spreading, and Lopez’s (2009) derivational assignment of the pragmatic feature [contrast], to successfully capture the facts of Negative Concord and Double Negation in English. In so doing, it contributes insight into the representation of sentential Negation, and supports the Barbiersian notion that not all grammatical structures are realized in a given variety.
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english negative concord and Double Negation the division of labor between syntax and pragmatics
Proceedings of the Linguistic Society of America, 2018Co-Authors: Frances Blanchette, Marianna Nadeu, Jeremy Yeaton, Viviane DeprezAbstract:Recent research demonstrates that prototypical negative concord (NC) languages allow Double Negation (DN) (Espinal & Prieto 2011; Prieto et al. 2013; Deprez et al. 2015; Espinal et al. 2016). In NC, two or more syntactic Negations yield a single semantic one (e.g., the ‘I ate nothing’ reading of “I didn’t eat nothing”), and in DN each Negation contributes to the semantics (e.g. ‘It is not the case that I ate nothing’). That NC and DN have been shown to coexist calls into question the hypothesis that grammars are either NC or DN (Zeijlstra 2004), and supports micro-parametric views of these phenomena (Deprez 2011; Blanchette 2017). Our study informs this debate with new experimental data from American English. We explore the role of syntax and speaker intent in shaping the perception and interpretation of English sentences with two negatives. Our results demonstrate that, like in prototypical NC languages (Espinal et al. 2016), English speakers reliably exploit syntactic, pragmatic, and acoustic cues to in selecting an NC or a DN interpretation.
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Micro-syntactic variation in American English Negative Concord
Glossa, 2017Co-Authors: Frances BlanchetteAbstract:This paper presents a series of quantitative gradient acceptability judgment studies of English negative sentences. Adult native speakers of American English recruited via Amazon’s Mechanical Turk were asked to rate sentences on a scale of 1 to 7 on the basis of their naturalness. The main study compares sentences with the marker n’t and either a negative object (e.g. ‘John didn’t eat nothing’) or a negative subject in canonical position (‘nobody didn’t eat’). Each sentence type has two possible interpretations, one in which the two negatives contribute a single semantic Negation, the so-called Negative Concord reading, and another in which the two Negations yield a semantic Double Negation logically equivalent to an affirmative. While mean acceptability ratings were below the median for all items, statistical analyses of the gradient data revealed that speakers prefer Negative Concord over Double Negation readings for sentences with negative objects. To rule out a processing explanation for the preference for negative objects over sentence initial negatives, a follow-up study tested the acceptability of sentences with a single negative subject or object and no negative marker. This revealed a preference for subjects, suggesting that the object preference in the two negatives study is a true grammatical effect. A third study revealed that Double Negation constructions are unacceptable overall even in explicit denial contexts, and a fourth study added Negative Auxiliary Inversion constructions (e.g. ‘Didn’t nobody eat’), to compare three types of Negative Concord. The results of all four studies are argued to reveal an English grammar that generates both Negative Concord and Double Negation, and in which Negative Concord is generated despite its unacceptability and reported absence in usage.