De Morgan's Laws Research Articles

Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore -- 721102, India

In this paper, soft matrices are defined based on soft set. Cartesian product of two soft sets is defined here. The very common operations of soft matrices are defined like AND, OR, union, intersection. commutative, associative, distributive, De Morgan's laws and convergent property for soft matrices are investigated. Key words: Soft set, soft relation, soft matrices, AND, OR operations, complement, union and intersection of soft matrices, commutative, associative, distributive laws, De Morgan's laws, convergence.

Universal models and definability

AbstractWe establish some general results on universal models in Topos Theory and show that the investigation of such models can shed light on problems of definability in Logic as well as on De Morgan's law and the law of excluded middle for Grothendieck toposes.

Uncovering Knowledge of Core Syntactic and Semantic Principles in Individuals With Williams Syndrome

We investigate knowledge of core syntactic and semantic principles in individuals with Williams Syndrome (WS). Our study focuses on the logico-syntactic properties of negation and disjunction (or) and tests knowledge of (a) core syntactic relations (scope and c-command), (b) core semantic relations (entailment relations and DeMorgan's laws of propositional logic), and (c) the relationship between (a) and (b). We examine the performance of individuals with WS, children matched for mental age (MA), and typical adult native speakers of English. Performance on all conditions suggests that knowledge of (a-c) is present and engaged in all three groups. Results also indicate slightly depressed performance on (c) for the WS group, compared to MA, consistent with limitation in processing resources. Implications of these results for competing accounts of language development in WS, as well as for the relevance of WS to the study of cognitive architecture and development are discussed.

De Morgan's law and the theory of fields

We show that the classifying topos for the theory of fields does not satisfy De Morgan's law, and we identify its largest dense De Morgan subtopos as the classifying topos for the theory of fields of nonzero characteristic which are algebraic over their prime fields.

De Morgan classifying toposes

We present a general method for deciding whether a Grothendieck topos satisfies De Morgan's law (resp. the law of excluded middle) or not; applications to the theory of classifying toposes follow. Specifically, we obtain a syntactic characterization of the class of geometric theories whose classifying toposes satisfy De Morgan's law (resp. are Boolean), as well as model-theoretic criteria for theories whose classifying toposes arise as localizations of a given presheaf topos.

On Fuzzy Soft Sets

We further contribute to the properties of fuzzy soft sets as defined and studied in the work of Maji et al. ( 2001), Roy and Maji (2007), and Yang et al. (2007) and support them with examples and counterexamples. We improve Proposition 3.3 by Maji et al., (2001). Finally we define arbitrary fuzzy soft union and fuzzy soft intersection and prove DeMorgan Inclusions and DeMorgan Laws in Fuzzy Soft Set Theory.

Technology Tips: Visualizing DeMorgan's Laws Using Logic Simulation Software

This month's tips illustrate an interactive, inquiry-based approach for exploring logic operators and equivalence of various Boolean expressions using technology. Specifically, I illustrate the use of several logic simulation tools to perform the following tasks: (1) build and “run” a virtual logic circuit illustrating x ∧ y and (2) verify the validity of DeMorgan's laws.

Basic intuitionistic principles in fuzzy set theories and its extensions (A terminological debate on Atanassov IFS)

In this paper we contribute to the terminological debate about Atanassov's use of the term “Intuitionistic” in defining his structure based on ortho-pairs of fuzzy sets. In particular, we stress that it is defined as “intuitionistic” a negation which from one side does not satisfy a standard property of the intuitionistic Brouwer negation (contradiction law) and on the contrary asserts some principles rejected by intuitionism (strong double negation law and one of the de Morgan laws). An algebraic Brouwer negation is studied in the context of IFS showing that it can be induced from a Heyting implication. A similar situation occurs in the case of standard fuzzy sets (FS). Some conditions which allow one to distinguish from the algebraic point of view FS from IFS are treated. Finally, a particular subclass of IFS consisting of ortho-pairs of crisp sets (denoted by ICS) is studied, showing that shadowed sets can be algebraically identified with ICSs.

Exploring complexity when diversity is limited: institutional complementarity in theories of rule of law and national systems revisited

AbstractCategories reflect particular theories about the world in the form of causal and performative claims. Unlike attributes that are the mainstay of statistical analysis, these discrete entities consist of the contradiction of being easy to understand and yet hard to analyze. An important obstacle to the exploration of causal claims about categories (e.g. nations) is the limited diversity of observed cases. We propose the use of methodologies that take greater exploratory account of causal complexity and that respects the importance of case narratives for theexplicitdecisions made to arrive at theoretical claims. One such methodology is qualitative comparative analysis developed by Charles Ragin. This method is applied to data provided by two independent lines of study (i.e. rule of law and governance and varieties of capitalism) to show how the identification, and adoption, ofprototypesis complex. Through the use of logic (e.g. De Morgan's law) and reductive inferences, we explore the space of observed and unobserved configurations, showing how the identification of institutional configurations relies upon logical assumptions that are rarely made explicit. The analysis rejects the hypothesis of rule of law and financial development and qualifies the institutional prototypes of corporatism and market as useful descriptions of the varieties of capitalism.

General Jurisprudence: A 25th Anniversary Essay

Law is a parochial business whose character and content depends on local institutions, events, and understandings. Thus it is that the most learned scholar of English law can be an ignoramus about Estonian law, let alone the law of remote societies or ancient times. To call law parochial may sound like a complaint. It is not-law would be of less value were it not firmly rooted in the societies it serves. But it does raise the question of how far we can hope for a theory of law. Particular facts can be explained; but to have a theory is to have an explanation at some level of generality. It is one thing to theorize doctrines within a legal system, to develop the branches of special jurisprudence; but how far can we work out a general jurisprudence, a theory of law as such, that will apply to contracts and to constitutions, and in Estonia as much as in England? Not every project that calls itself a 'general theory' is intelligible. Back when political scientists aspired to general theories of politics, Alasdair MacIntyre mocked them with a parable about an eccentric quest for the general theory of holes.1 It was a needed warning; but in law things are not yet that dire. Modern jurisprudence does not advance a unified theory of everything that is properly called 'law', including the law of England, the moral law, de Morgan's laws, and Boyle's law.2 All the same, it does aspire to great generality, and there is the question of how even that can be done. One answer was offered by Aristotle, who said that in addition to 'particular law', i.e. 'that which each community lays down and applies to its own members', there is also 'universal law', the principles of natural justice rationally binding on everyone apart from any local terms of association.3 That is the germ of the

Soft set theory

In this paper, the authors study the theory of soft sets initiated by Molodtsov. The authors define equality of two soft sets, subset and super set of a soft set, complement of a soft set, null soft set, and absolute soft set with examples. Soft binary operations like AND, OR and also the operations of union, intersection are defined. De Morgan's laws and a number of results are verified in soft set theory.

Elkan's theoretical argument, reconsidered

This paper takes part in the discussion motivated by Elkan's paper “The Paradoxical Success of Fuzzy Logic” printed in 1993, whose main theoretical point was that Fuzzy Logic does not properly deal with a specific Law of Classical Logic: ¬(p∧¬q)=q∨(¬p∧¬q). The given answer can be summarized, like in other previous cases, by the sentence “Yes it can but, of course, at some cost”. As it is shown this cost is, basically, duality. But without De Morgan laws there are uncountable many theories of Fuzzy Sets on which that classical law holds. On the way it is observed that, this equation, is not universally verified in De Morgan lattices and the solution given by Elkan, in a particular case, is incorrect.

Definition of general aggregation operators through similarity relations

Various extensions of the original max and min aggregation operators in fuzzy set theory are successfully used in practical applications, but lack a clear conceptual model supporting them. Giving these operators a meaningful and simple interpretation is the topic of this paper. Aggregation operators are seen as different methods to measure distances to the essential reference points of the feature space, called Ideals. It has been proved that every general aggregation operator can be associated with a corresponding metric, in which the result of its application is the distance to the Ideal. Some widely used operators correspond to familiar l−p norms, and new operators can be defined by specifying different metrics. Heterogeneous combinations of ANDs and ORs are treated in such a way that the distributivity and De Morgan's laws hold. Applications to fuzzy constraint satisfaction problem and fuzzy control are discussed and interpreted geometrically. Classical operators are particular cases of the proposed semantic model, and several other examples are given.

Unification in intuitionistic logic

AbstractWe show that the variety of Heyting algebras has finitary unification type. We also show that the subvariety obtained by adding it De Morgan law is the biggest variety of Heyting algebras having unitary unification type. Proofs make essential use of suitable characterizations (both from the semantic and the syntactic side) of finitely presented projective algebras.

A Characterization of Finite Ternary Algebras

A ternary algebra is a De Morgan algebra (that is, a distributive lattice with 0 and 1 and a complement operation that satisfies De Morgan's laws) with an additional constant Φ satisfying [Formula: see text], [Formula: see text], and [Formula: see text]. We provide a characterization of finite ternary algebras in terms of "subset-pair algebras," whose elements are pairs (X, Y) of subsets of a given base set ℰ, which have the property X ∪ Y = ℰ, and whose operations are based on common set operations.

The Lee identities in Topoi, I

We investigate conditions under which various topoi satisfy equations, known as the Lee identities, that are similar to and weaker than De Morgan's law.

Negation trees: a unified approach to Boolean function complementation

Boolean function complementation is a basic operation of Boolean algebra. For functions given in SOP form, the complementation methods include the DeMorgan's Law, sharp, disjoint sharp, unate complementation, recursive, and Sasao's method. This paper accomplishes three purposes: (1) It exposes an underlying unification of the existing complementation algorithms. It is proven that (a) unate-complementation and sharp are the same as the DeMorgan Law algorithm; (b) Sasao's algorithm is the same as disjoint sharp; (c) disjoint sharp is a special case of the recursive method. (2) It proposes negation trees for the representation of functions and their complements. (3) It gives faster algorithms for finding complements of functions in SOP form based on negation trees.

Characterizing Belnap's Logic via De Morgan's Laws

AbstractThe aim of this paper is technically to study Belnap's four‐valued sentential logic (see [2]). First, we obtain a Gentzen‐style axiomatization of this logic that contains no structural rules while all they are still admissible in the Gentzen system what is proved with using some algebraic tools. Further, the mentioned logic is proved to be the least closure operator on the set of {Λ, V, ⌝}‐formulas satisfying Tarski's conditions for classical conjunction and disjunction together with De Morgan's laws for negation. It is also proved that Belnap's logic is the only sentential logic satisfying the above‐mentioned conditions together with Anderson‐Belnap's Variable‐Sharing Property. Finally, we obtain a finite Hilbert‐style axiomatization of this logic. As a consequence, we obtain a finite Hilbert‐style axiomatization of Priest's logic of paradox (see [12]).

An Algebra for Multiple Valued Logic Systems

An algebra for multiple valued logic systems is presented in this paper. This algebra uses r–1 new concurrent operators along with the traditional supremum and cycle operators. It is also shown that r + 1 operators are required to generate any r-valued logic function. Five new theorems, similar to DeMorgan's Laws, are then used to show that only one fundamental mapping is sufficient to represent any multi-valued logic function.

Fuzzy connectives via matrix logic

Abstract An approach to the definition of fuzzy logical connectives is proposed using matrix logic. This matrix representation of Boolean logical operators enables us to link together conjunction, disjunction and negation differently from the traditional De Morgan law. In this framework the Frank family appears as the only possibility. In addition, the parameters in conjuntion and in disjunction are reciprocal to each other. The usefulness of this case is illustrated by two applications.