Double Negation Law Research Articles

Routley Star in Information-Based Semantics

It is common in various non-classical logics, especially in relevant logics, to characterize negation semantically via the operation known as Routley star. This operation works well within relational semantic frameworks based on prime theories. We study this operation in the context of "information-based" semantics for which it is characteristic that sets of formulas supported by individual information states are theories that do not have to be prime. We will show that, somewhat surprisingly, the incorporation of Routley star into the information-based semantics does not lead to a collapse or a trivialization of the whole semantic system. On the contrary, it leads to a technically elegant though quite restricted semantic framework that determines a particular logic. We study some basic properties of this semantics. For example, we show that within this framework double negation law is valid only in involutive linear frames. We characterize axiomatically the logic of all linear frames and show that the logic of involutive linear frames coincides with a system that Mike Dunn coined Kalman logic. This logic is the fragment (for the language restricted to conjunction, disjunction and negation) of the "semi-relevant" logic known as R-mingle. Finally, we characterize by a deductive system the logic of all information frames equipped with Routley star.

Algebraizability of the Logic of Quasi-N4-Lattices

The class of quasi-N4-lattices (QN4-lattices) was introduced as a common generalization of quasi-Nelson algebras and N4-lattices, in such a way that N4-lattices are precisely the QN4-lattices satisfying the double negation law (~~x = x) and quasi-Nelson algebras are the QN4-lattices satisfying the explosive law (x ^ ~x) -> y = ((x ^ ~x) -> y) -> ((x ^ ~x) -> y). In this paper we introduce, via a Hilbert-style presentation, a logic (L_QN4) whose algebraic semantics is a class of algebras that we show to be term-equivalent to QN4-lattices. The result is obtained by showing that the calculus introduced by us is algebraizable in the sense of Blok and Pigozzi, and its equivalent algebraic semantics is term-equivalent to the class of QN4-lattices. As a prospect for future investigation, we consider the question of how one could place L_QN4 within the family of relevance logics.

Quasi-N4-lattices

Within the Nelson family, two mutually incomparable generalizations of Nelson constructive logic with strong negation have been proposed so far. The first and more well-known, Nelson paraconsistent logic, results from dropping the explosion axiom of Nelson logic; a more recent series of papers considers the logic (dubbed quasi-Nelson logic) obtained by rejecting the double negation law, which is thus also weaker than intuitionistic logic. The algebraic counterparts of these logical calculi are the varieties of N4-lattices and quasi-Nelson algebras, respectively. In the present paper we propose the class of quasi-N4-lattices as a common generalization of both. We show that a number of key results, including the twist-structure representation of N4-lattices and quasi-Nelson algebras, can be uniformly established in this more general setting; our new representation employs twist-structures defined over Brouwerian algebras enriched with a nucleus operator. We further show that quasi-N4-lattices form a variety that is arithmetical, possesses a ternary as well as a quaternary deductive term, and enjoys EDPC and the strong congruence extension property.

Eight Laws of Logic for the Forensic Construction Expert

Forensic experts’ primary and additional professional education does not include such a subject as logic. At the same time, knowledge of logic is crucial at all stages of the examination process. Filling this gap, the authors of this article reveal the content of the fundamental laws of logic and demonstrate their role in the cognitive activity of knowledgeable persons who implement their specialized (primarily construction and technical) knowledge in court proceedings. This paper deals successively with such laws of logic as the law of identity, the law of contradiction, the law of the excluded middle, the law of sufficient reason, the law of double negation, Clavius’s law, the law of contraposition, and laws of division (the dichotomy of logic) concerning various investigative and forensic situations. The projection of these laws of logic on the intellectual operations performed by experts will allow, from the authors’ point of view, to give the process of forensic examinations greater clarity and consistency, which, ultimately, should ensure an increase in its efficiency and quality of results. Logically verified approaches to work will also reduce time and financial costs.

Adjoint Operations in Twist-Products of Lattices

Given an integral commutative residuated lattices L=(L,∨,∧), its full twist-product (L2,⊔,⊓) can be endowed with two binary operations ⊙ and ⇒ introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Wille such that it becomes a commutative residuated lattice. For every a∈L we define a certain subset Pa(L) of L2. We characterize when Pa(L) is a sublattice of the full twist-product (L2,⊔,⊓). In this case Pa(L) together with some natural antitone involution ′ becomes a pseudo-Kleene lattice. If L is distributive then (Pa(L),⊔,⊓,′) becomes a Kleene lattice. We present sufficient conditions for Pa(L) being a subalgebra of (L2,⊔,⊓,⊙,⇒) and thus for ⊙ and ⇒ being a pair of adjoint operations on Pa(L). Finally, we introduce another pair ⊙ and ⇒ of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law, and we investigate when Pa(L) is closed under these new operations.

An Application of Classical Logic’s Laws in Formulas of Fuzzy Implications

The crucial role that fuzzy implications play in many applicable areas was our motivation to revisit the topic of them. In this paper, we apply classical logic’s laws such as De Morgan’s laws and the classical law of double negation in known formulas of fuzzy implications. These applications lead to new families of fuzzy implications. Although a duality in properties of the preliminary and induced families is expected, we will prove that this does not hold, in general. Moreover, we will prove that it is not ensured that these applications lead us to fuzzy implications, in general, without restrictions. We generate and study three induced families, the so-called D ′ -implications, QL ′ -implications, and R ′ -implications. Each family is the “closest” to its preliminary-“creator” family, and they both are simulating the same (or a similar) way of classical thinking.

On Bijective Correspondence between Fuzzy Reflexive Approximation Spaces and Fuzzy Transformation Systems

The objective of this paper is to establish the relationship between fuzzy approximation operators and fuzzy transformation systems. We show that for each upper fuzzy transformation system there exists a fuzzy reflexive approximation space and vice-versa. We further establish such relationship between lower fuzzy transformation systems and fuzzy reflexive approximation spaces under the condition that the underline lattice structure satisfies double negation law.

Residuation in non-associative MV-algebras

Abstract It is well known that every MV-algebra can be converted into a residuated lattice satisfying divisibility and the double negation law. In a previous paper the first author and J. Kühr introduced the concept of an NMV-algebra which is a non-associative modification of an MV-algebra. The natural question arises if an NMV-algebra can be converted into a residuated structure, too. Contrary to MV-algebras, NMV-algebras are not based on lattices but only on directed posets and the binary operation need not be associative and hence we cannot expect to obtain a residuated lattice but only an essentially weaker structure called a conditionally residuated poset. Considering several additional natural conditions we show that every NMV-algebra can be converted in such a structure. Also conversely, every such structure can be organized into an NMV-algebra. Further, we study an a bit more stronger version of an algebra where the binary operation is even monotone. We show that such an algebra can be organized into a residuated poset and, conversely, every residuated poset can be converted in this structure.

Separation axioms in \(L\)-fuzzifying supra-topology

In this paper, we define and investigate the notions of \(L\)-separation axioms in \(L\)-fuzzifying supra-topology. Also, some of their characterizations and a systematic discussion on the relationship among these notions is gave in \(L\)-fuzzifying supra-topology where \(L\) is a complete residuated lattice. Sometimes we need more conditions on \(L\) such as the completely distributive law or that the "\(\wedge\)" is distributive over arbitrary joins or the double negation law as we illustrate through this paper. As applications of our work the corresponding results (see \cite{2, 13}) are generalized and new consequences are obtained.

A representation of residuated lattices satisfying the double negation law

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.

Residuation in orthomodular lattices

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.

Orthomodular lattices can be converted into left residuated l-groupoids

We show that every orthomodular lattice can be considered as a left residuated l-groupoid satisfying divisibility, antitony, the double negation law and three more additional conditions expressed in the language of residuated structures. Also conversely, every left residuated l-groupoid satisfying the mentioned conditions can be organized into an orthomodular lattice.

When does a semiring become a residuated lattice?

It is an easy observation that every residuated lattice is in fact a semiring because multiplication distributes over join and the other axioms of a semiring are satisfied trivially. This semiring is commutative, idempotent and simple. The natural question arises if the converse assertion is also true. We show that the conversion is possible provided the given semiring is, moreover, completely distributive. We characterize semirings associated to complete residuated lattices satisfying the double negation law where the assumption of complete distributivity can be omitted. A similar result is obtained for idempotent residuated lattices.

On semi separation axioms in L-fuzzifying bitopological spaces

Abstract The main purpose of this paper is to introduce a concept of semi separation axioms in L-fuzzifying bitopological spaces (here L is a complete residuated lattice some times used double negation law) and discuss some of their basic properties and the structures. We shown that the category of semi-separation axioms in L- fuzzifying bitopological. Finlay we prove that its some a relation between a semi-separation axioms in L-fuzzifying bitopological spaces.

On semi separation axioms on \(L\)-fuzzifying topological spaces

In this paper we introduce and study the concepts of semi \(T_0\); semi 􀀀\(T_1\); semi 􀀀\(T_2\) (Hausdorff), semi 􀀀\(T_3\) (regularity), semi 􀀀\(T_4\) (normality), semi 􀀀\(R_0\); and semi 􀀀\(R_1\) separation axioms in \(L\)-fuzzifying topological space where \(L\) is a complete residuated lattice. Sometimes we add more conditions on \(L\) such as the completely distributive law or the double negation law.

Fixing the White Horse Discourse: Zhuangzi’s Proof of “A White Horse Is not a Horse”

This article proposes an interpretation of Zhuangzi’s comment, in “Discussion on Making all Things Equal” (“Qiwulun” 齊物論), on the “White Horse Discourse” (“Baimalun” 白馬論) that suggests his awareness of what Western logicians call intuitionist proof. According to logical intuitionism, one cannot rely on the Law of the Excluded Middle, and relatedly the Law of Double Negation, to facilitate formal reasoning. Consequent to this prohibition is an asymmetry in proof: one may indirectly prove his or her denial of a statement p by deriving an absurdity from p , but there is no similar way to prove p from the assumption of not- p . It is argued here that in this intuitionist framework, Zhuangzi’s strategy is in effect a recipe for silencing a debater who challenges the acceptability (可) of the opponent’s beliefs. An account shall also be given of how to reconcile Zhuangzi’s relativism with his dialectical reasoning based on reductio ad absurdum (bei 悖).

General coupled semirings of residuated lattices

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.

New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

Intuitionistic logic, in which the double negation law not-not-P = P fails, is dominant in categorical logic, notably in topos theory. This paper follows a different direction in which double negation does hold. The algebraic notions of effect algebra/module that emerged in theoretical physics form the cornerstone. It is shown that under mild conditions on a category, its maps of the form X -> 1+1 carry such effect module structure, and can be used as predicates. Predicates are identified in many different situations, and capture for instance ordinary subsets, fuzzy predicates in a probabilistic setting, idempotents in a ring, and effects (positive elements below the unit) in a C*-algebra or Hilbert space. In quantum foundations the duality between states and effects plays an important role. It appears here in the form of an adjunction, where we use maps 1 -> X as states. For such a state s and a predicate p, the validity probability s |= p is defined, as an abstract Born rule. It captures many forms of (Boolean or probabilistic) validity known from the literature. Measurement from quantum mechanics is formalised categorically in terms of `instruments', using L\"uders rule in the quantum case. These instruments are special maps associated with predicates (more generally, with tests), which perform the act of measurement and may have a side-effect that disturbs the system under observation. This abstract description of side-effects is one of the main achievements of the current approach. It is shown that in the special case of C*-algebras, side-effect appear exclusively in the non-commutative case. Also, these instruments are used for test operators in a dynamic logic that can be used for reasoning about quantum programs/protocols. The paper describes four successive assumptions, towards a categorical axiomatisation of quantitative logic for probabilistic and quantum systems.

On a Logical Principle Useful in Music Theory

ON A LOGICAL PRINCIPLE USEFUL IN MUSIC THEORY ROY WHELDEN Our life is frittered away by detail…Simplify, simplify. —H. D. Thoreau in Walden N 1860 THE BRITISH LOGICIAN Augustus De Morgan published a challenging puzzle in a small gazette. The solution to the puzzle, which four years later he was referring to as Theorem K, proves itself to be of interest to music theorists.1 (A few logicians still call it Theorem K. Most logicians, if they know it at all, call it Schroeder’s theorem.) Although De Morgan’s Theorem K holds in very general systems of relations, musicians may be most interested in the theorem’s applications within group theory, since many common musical structures— the classical twelve-tone operations of the Viennese school, Babbitt’s duration series, the operations on Forte’s pitch-class sets, the cubic symmetries used by Xenakis, the operations on pitch-class sets from expanded microtonal systems—can be studied as groups. Within the context of group theory, then, here is a brief introduction to the notation used. (As an example group to hold in mind, used here because of its connection to twelve-tone equal temperament, take the group of integers 0, 1, 2, . . ., 11 with addition modulo 12. I 190 Perspectives of New Music As is customary, we’ll let 0, 1, 2, . . ., 11 represent the equal-tempered pitch classes C, C#, D, . . ., B.) Letting x and y be arbitrary elements of some group, simple concatenation xy will denote the group operation. (Thus, in our example, xy denotes addition x + y mod 12.) Let S, T, and U represent arbitrary subsets of G. (These subsets can be thought of as sonorities or as chords or, more accurately, as pitchclass sets.) • Then S-1 will represent the set of inverses of every element in S. (In our example, the inverse of an element x is 12 − x.) • The complement of S, that is, set of group elements not in S, will be denoted by S′. • ST will denote all possible concatenations of elements taken from S and T, in that order. • Finally, we use the sign ⊂ as the inclusion relation “is contained in.” De Morgan’s Theorem K says that the following statements are equivalent : ST ⊂ U, S-1U′ ⊂ T′, and U′T-1 ⊂ S′. De Morgan suggested a handy mnemonic for this: “Given an inclusion of the form ST ⊂ U, an equivalent form can be obtained by inverting one of the ‘multipliers’ S or T, then interchanging the complements of the other two members.” Our application of Theorem K will often require the use of some well known logical and group laws, including the law of double negation S′′ = S, the law of double inverse (S-1)-1 = S, and the law of contraposition, which says that S ⊂ T and T′ ⊂ S′ are equivalent. Before showing the proof of Theorem K, let’s apply the theorem to the subject of combinatoriality, the well known concept introduced by Milton Babbitt in 1955.2 I hasten to add that combinatoriality has been well understood since its introduction. The composer Donald Martino in particular offered insightful discussions of combinatoriality.3 My purpose here is to show how De Morgan’s powerful Theorem K provides a direct, simple, and easily programable approach to combinatoriality , generalizing it as well as several related musical concepts. 194 Perspectives of New Music the Common Tone theorem. For example, let S be the (just discussed ) sonority 6-32. We know that S is prime combinatorial, using traditional tools, from the fact that its interval vector (143250) has a rightmost entry of 0. It’s the presence of the 0 entry which informs us immediately (or at least after computing or looking up in a table the sonority’s interval vector) that S has prime combinatoriality. The reasoning is as follows: the interval vector (143250) of S tells us that the interval class of the tritone (6) is lacking. The Common Tone theorem assures us that the number of tritones to be found in S is the same as the size of the intersection of {6}S and S. But since the number of tritones in S is 0, the...

Concord and Doubling Phenomena: an Introduction

Formal semanticists have difficulty facing the fact that the law of double negation (DN) does not appear to hold in a vast majority of the languages of the world. In many languages, multiple negations may, semantically, yield one negation, and this is difficult to understand if each of the negations involves a semantic negation. One can assert, even in English, ‘If you ain’t got nothing to hide, then you don't need no privacy’—while meaning ‘If you got nothing to hide, you don't need privacy’. A primary school teacher could say that this is bad English, and that it reflects illogical thinking. However, for a lot of languages, and for a lot of speakers of American English, something like this is just the right way to say these things. So we have to live with the linguistic fact that, for instance, a DN may imply a single negation. Or, at least, that what seems to be a DN turns out to be a single negation after all. Once this fact is acknowledged, there appear to be many more similar facts about language. Multiple modality expressions can be seen to contribute to a single modality. Multiple number- or person-markers in a sentence contribute to one number- or person-related interpretation. And then the following questions readily pops up: are these all instances of the same phenomenon? And how should they be analyzed?