Two constructions for adding an involution operator to residuated ordered monoids are investigated. One preserves integrality and the mingle axiom x 2x but fails to preserve the contraction property xx 2. The other has the opposite preservation properties. Both constructions preserve commutativity as well as existent nonempty meets and joins and self-dual order properties. Used in conjunction with either construction, a result of R.T. Brady can be seen to show that the equational theory of commutative distributive residuated lattices (without (...) involution) is decidable, settling a question implicitly posed by P. Jipsen and C. Tsinakis. The corresponding logical result is the (theorem-) decidability of the negation-free axioms and rules of the logic RW, formulated with fusion and the Ackermann constant t. This completes a result of S. Giambrone whose proof relied on the absence of t. (shrink)

This paper establishes several algebraic embedding theorems, each of which asserts that a certain kind of residuated structure can be embedded into a richer one. In almost all cases, the original structure has a compatible involution, which must be preserved by the embedding. The results, in conjunction with previous findings, yield separative axiomatizations of the deducibility relations of various substructural formal systems having double negation and contraposition axioms. The separation theorems go somewhat further than earlier ones in the literature, (...) which either treated fewer subsignatures or focussed on the conservation of theorems only. (shrink)

Excerpt from Involution The word "involution" implies a folding in towards the centre. It seems to bear a different sense to the word "concentration," inasmuch as the latter carries with it the suggestion of an organised movement directed from outside, while involution takes rather the form of an automatic convergence. The word is here used in the latter sense, and it has been chosen as the title of this book because it is the word which seems to (...) describe best the cosmic process at which a glimpse is taken in the following pages. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

A recent paper by Jakl, Jung and Pultr succeeded for the first time in establishing a very natural link between bilattice logic and the duality theory of d-frames and bitopological spaces. In this paper we further exploit, extend and investigate this link from an algebraic and a logical point of view. In particular, we introduce classes of algebras that extend bilattices, d-frames and N4-lattices to a setting in which the negation is not necessarily involutive, and we study corresponding logics. We (...) provide product representation theorems for these algebras, as well as completeness, algebraizability results for the corresponding logics. (shrink)

In [5] we study Nonassociative Lambek Calculus augmented with De Morgan negation, satisfying the double negation and contraposition laws. This logic, introduced by de Grooté and Lamarche [10], is called Classical Non-Associative Lambek Calculus. Here we study a weaker logic InNL, i.e. NL with two involutive negations. We present a one-sided sequent system for InNL, admitting cut elimination. We also prove that InNL is PTIME.

This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of Eilenberg-Moore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. A part of the so-called Gelfand–Naimark–Segal (GNS) construction is identified as an isomorphism of categories, relating states on involutive monoids and inner products. This correspondence exists in arbritrary involutive symmetric monoidal categories.

Involutive Nonassociative Lambek Calculus is a nonassociative version of Noncommutative Multiplicative Linear Logic, but the multiplicative constants are not admitted. InNL adds two linear negations to Nonassociative Lambek Calculus ; it is a strongly conservative extension of NL Logical aspects of computational linguistics. LNCS, vol 10054. Springer, Berlin, pp 68–84, 2016). Here we also add unary modalities satisfying the residuation law and De Morgan laws. For the resulting logic InNLm, we define and study phase spaces. We use them to prove (...) the cut elimination theorem for a one-sided sequent system for InNLm, introduced here. Phase spaces are also employed in studying auxiliary systems InNLm, assuming the k-cyclic law for negation. The latter behave similarly as Classical Nonassociative Lambek Calculus, studied in de Groote and Lamarche :355–388, 2002) and Buszkowski. We reduce the provability in InNLm to that in InNLm. This yields the equivalence of type grammars based on InNLm with -free) context-free grammars and the PTIME complexity of InNLm. (shrink)

We prove that there are two involutions defined by monadic terms that characterize Monadic Algebras. We further prove that the variety of Monadic Algebras is the smallest variety of Interior Algebras where these involutions give rise to an interpretation from the variety of Bounded Distributive Lattices into it.

‘Creative Involution’ posits something of a philosophical genealogy, a line of flight that has neither need for nor interest in the periodisation of Modernism, a line of which Beckett (even reluctantly) is part. Murphy, among others, is deterritorialised as much as Beckett's landscapes are, and so he/they become a ‘complexification’ of being that manifests itself in Beckett not as represented, representative or a representation, since so much of Beckett deals with that which cannot be uttered, known or represented, but (...) whose image the works (and its figures) have become, a thinking through of negativity, becoming, and multiplicity through non-Newtonian motion, of being as becoming, where every movement brings something new into the world, but in something of a reverse Darwinism that moves from complex to simple organism, from Murphy to Worm, or Watt to Pim, or among the nameless figures in the short prose, a ‘becoming-animal’, in something not so much of a creative devolution but rather a ‘neoevolution’, or to adopt another term from Deleuze and Guattari, an ‘involution’, which ‘is in no way confused with regression’ (1987: 238), becomings creating nothing less than new worlds. Writing casually to his post-war confidant Georges Duthuit on 26 July 1951, Beckett noted in the midst of gardening chores, ‘Never seen so many butterflies in such worm-state, this little central cylinder, the only flesh, is the worm’ (2011: 271). The observation comes after the writing but before the premiere performance of Waiting for Godot in which Gogo tells Didi, ‘You and your landscapes! Tell me about the worms!’ (1954: 67). Such ‘becoming-animal always involves a pack, a band, a population, a peopling, in short a multiplicity’ (1987: 239), Deleuze and Guattari tell us. For Beckett it defines creativity as well, only possible through such un-tethered selves or beings, amid the generation of varieties and differences, accessible through moments of deterritorialisation, characterised by, ‘A fearsome involution calling us toward unheard-of becomings’ (240). (shrink)

As we settle further into the era of digital media and globalized visual culture, it might be tempting to think that photography holds no more than historical interest. Yet it continues to feature in debates with considerable significance for the present.1 The terms by which it was negotiated in the twentieth century – the print, the negative and the mechanical-optical apparatus, the affective experience of a moment stilled, and any truth that its rendering promises – have been technically and culturally (...) displaced and expanded. New instabilities have become familiar and have distanced us from how photography was understood, even in the fairly recent past. The current historical conjuncture is marked by a widespread suspicion that existing theories – including those that turned, in the 1970s, to Marxism, feminist critique, semiotics or psychoanalysis so as to politicize and contest mainstream photographic culture – might no longer be adequate to photography’s contemporary situation. That photography still matters, however, can be evidenced, prosaically and contingently, by noting the increasing number of new scholarly journals and exhibitions devoted to its past, present and future in recent years.2 There is, in this – as Fred Ritchin is only the latest to note – a sense that the undoing of photography’s prior certainties constitutes an ending and an enlargement. The fate of photography provides an ‘expansive filter’ through which to chart the ‘chaos of possibilities that emerge and recede, back up and move forward, crisscrossing each other.’3 Expectations of the new and the old, the obsolete and as yet only anticipated, are thrown into temporal disarray as its openness to reformation gives the photographic past a futural slant. (shrink)

The paper deals with involutive FLe-monoids, that is, commutative residuated, partially-ordered monoids with an involutive negation. Involutive FLe-monoids over lattices are exactly involutive FLe-algebras, the algebraic counterparts of the substructural logic IUL. A cone representation is given for conic involutive FLe-monoids, along with a new construction method, called twin-rotation. Some classes of finite involutive FLe-chains are classified by using the notion of rank of involutive FLe-chains, and a kind of duality is developed between positive and non-positive rank algebras. As a (...) side effect, it is shown that the substructural logic IUL plus t ↔ f does not have the finite model property. (shrink)

This article argues that the Indian philosopher-mystic Sri Aurobindo espoused a sophisticated form of cosmopsychism that has great contemporary relevance. After first discussing Aurobindo’s prescient reflections on the “central problem of consciousness” and his arguments against materialist reductionism, I explain how he developed a panentheistic philosophy of “realistic Adwaita” on the basis of his own spiritual experiences and his intensive study of the Vedāntic scriptures. He derived from this realistic Advaita philosophy a highly original doctrine of evolutionary cosmopsychism, according to (...) which the Divine Saccidānanda is “involved” in everything in the universe and gradually manifests itself at each stage of the evolutionary process from matter to life to mind, and ultimately, to Supermind—the final stage that is yet to come, upon the attainment of which we will attain knowledge of our true divine nature as Saccidānanda. I then reconstruct Aurobindo’s novel solution to the individuation problem, according to which the Divine Saccidānanda individuates into various distinct consciousnesses by playfully limiting itself through a process of “exclusive concentration.” Finally, I highlight the continued relevance of Aurobindo’s evolutionary cosmopsychism by bringing him into conversation with Itay Shani, a contemporary proponent of cosmopsychism. (shrink)

An algebraic proof is presented for the finite strong standard completeness of the Involutive Uninorm Logic with Fixed Point ($${{\mathbf {IUL}}^{fp}}$$ IUL fp ). It may provide a first step towards settling the standard completeness problem for the Involutive Uninorm Logic ($${\mathbf {IUL}}$$ IUL, posed in G. Metcalfe, F. Montagna. (J Symb Log 72:834–864, 2007)) in an algebraic manner. The result is proved via an embedding theorem which is based on the structural description of the class of odd involutive FL$$_e$$ (...) e -chains which have finitely many positive idempotent elements. (shrink)

A new logic, quantized intuitionistic linear logic, is introduced, and is closely related to the logic which corresponds to Mulvey and Pelletier's involutive quantales. Some cut-free sequent calculi with a new property quantization principle and some complete semantics such as an involutive quantale model and a quantale model are obtained for QILL. The relationship between QILL and Wansing's extended intuitionistic linear logic with strong negation is also observed using such syntactical and semantical frameworks.

We establish the existence uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice.

The logic just corresponding to involutive quantales, which was introduced by Wendy MacCaull, is reconsidered in order to obtain a cut-free sequent calculus formulation, and the completeness theorem for this logic is proved using a new admissible rule.

We establish a duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the representation of involutive bisemilattices as Płonka sum of Boolean algebras, from the other. Furthermore, we show that the dual space of an involutive bisemilattice can be viewed as a GR space with involution, a generalization of the spaces introduced by Gierz and Romanowska equipped with an involution as additional operation.

We discuss Einstein’s knowledge of projective geometry. We show that two pages of Einstein’s Scratch Notebook from around 1912 with geometrical sketches can directly be associated with similar sketches in manuscript pages dating from his Princeton years. By this correspondence, we show that the sketches are all related to a common theme, the discussion of involution in a projective geometry setting with particular emphasis on the infinite point. We offer a conjecture as to the probable purpose of these geometric (...) considerations. (shrink)

One tolerably clear statement of Determinism has it that all events are caused. Expanded upon, this thesis has been taken as the claim that the existence of any event E1, has a set of events, E2 … En which antedate E1, and which are causally sufficient for the occurrence of E1. That is, given the occurrence of E2 … En, E1 is causally necessary. I would hardly wish to claim that this is the only plausible statement of the doctrine of (...) Determinism; nonetheless it is a common one, and the one that I’ll use as definitive. Some have believed both that this premise is true and that it inevitably leads to a denial of free actions. While remaining neutral as to the merits of the Hard Determinist position, what I wish to show is that the consequences of the argumentation that has led them to such a position have been neither fully nor adequately understood. (shrink)

Motivated by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras we define the functor K • relating integral residuated lattices with 0 with certain involutive residuated lattices. Our work is also based on the results obtained by Cignoli about an adjunction between Heyting and Nelson algebras, which is an enrichment of the basic adjunction between lattices and Kleene algebras. The lifting of the functor to the category of residuated lattices leads us to study (...) other adjunctions and equivalences. For example, we treat the functor C whose domain is cuRL, the category of involutive residuated lattices M whose unit is fixed by the involution and has a Boolean complement c. If we restrict to the full subcategory NRL of cuRL of those objects that have a nilpotent c, then C is an equivalence. In fact, C M is isomorphic to C e M, and C e is adjoint to, where assigns to an object A of IRL0 the product A × A 0 which is an object of NRL. (shrink)

It is introduced a new algebra$$(A, \otimes, \oplus, *, \rightharpoonup, 0, 1)$$(A,⊗,⊕,∗,⇀,0,1)called$$L_PG$$LPG-algebra if$$(A, \otimes, \oplus, *, 0, 1)$$(A,⊗,⊕,∗,0,1)is$$L_P$$LP-algebra (i.e. an algebra from the variety generated by perfectMV-algebras) and$$(A,\rightharpoonup, 0, 1)$$(A,⇀,0,1)is a Gödel algebra (i.e. Heyting algebra satisfying the identity$$(x \rightharpoonup y ) \vee (y \rightharpoonup x ) =1)$$(x⇀y)∨(y⇀x)=1). The lattice of congruences of an$$L_PG$$LPG-algebra$$(A, \otimes, \oplus, *, \rightharpoonup, 0, 1)$$(A,⊗,⊕,∗,⇀,0,1)is isomorphic to the lattice of Skolem filters (i.e. special type ofMV-filters) of theMV-algebra$$(A, \otimes, \oplus, *, 0, 1)$$(A,⊗,⊕,∗,0,1). The variety$$\mathbf {L_PG}$$LPGof$$L_PG$$LPG-algebras (...) is generated by the algebras$$(C, \otimes, \oplus, *, \rightharpoonup, 0, 1)$$(C,⊗,⊕,∗,⇀,0,1)where$$(C, \otimes, \oplus, *, 0, 1)$$(C,⊗,⊕,∗,0,1)is ChangMV-algebra. Any$$L_PG$$LPG-algebra is bi-Heyting algebra. The set of theorems of the logic$$L_PG$$LPGis recursively enumerable. Moreover, we describe finitely generated free$$L_PG$$LPG-algebras. (shrink)

Nous tentons dans cet article de proposer une thèse cohérente concernant la formation de la notion d’involution dans le Brouillon Project de Desargues. Pour cela, nous donnons une analyse détaillée des dix premières pages dudit Brouillon, comprenant les développements de cas particuliers qui aident à comprendre l’intention de Desargues. Nous mettons cette analyse en regard de la lecture qu’en fait Jean de Beaugrand et que l’on trouve dans les Advis Charitables.

My book is about the human creativity being a source of progress, and cycling of evolution caused by platitude and triviality of once high-reaching idealism. In essence the book presents an original perception of human history, based on Christian values as vital coordinates system. I hope this book will revive the interest to the Russian school of thoughts and to humanism in general.

In this paper, we try to understand what considerations and possible sources of inspiration Desargues used to formulate his concepts of involution and transversal, and to state the related theorems that are at the basis of his Brouillon project. To this end, we trace some clues which are found scattered throughout his works, we connect them together in the light of his experience and knowledge in the field of perspective, and we investigate what were his motivations within Mersenne’s academy. (...) As a result of our research, we can safely say that were his great geometrical insight and his projective vision of space which, guided by some classical theorems, led him to these completely new concepts in the panorama of the geometry of that time that were destined to remain misunderstood for about two centuries. (shrink)

In order to investigate the competitive psychology of college students in the current context of fierce social competition, this study compiled a competition psychology scale for college students [i.e., the Competition Psychology Scale for College Students ]. The scale was administered online to 628 university students in different regions of China. After item analysis, reliability analysis, and validity analysis, a 6-item scale was finally formed. CPS-CS contains four dimensions: hypercompetitive attitude, competitive motivation, personal development, competitive attitude, and competitive interpersonal relationships. (...) The reliability and validity of the CPS-CS developed in this research meet the requirements of psychometrics. (shrink)

Involutive Stone algebras were introduced by R. Cignoli and M. Sagastume in connection to the theory of $n$-valued Łukasiewicz–Moisil algebras. In this work we focus on the logic that preserves degrees of truth associated to S-algebras named Six. This follows a very general pattern that can be considered for any class of truth structure endowed with an ordering relation, and which intends to exploit many-valuedness focusing on the notion of inference that results from preserving lower bounds of truth values, and (...) hence not only preserving the value $1$. Among other things, we prove that Six is a many-valued logic that can be determined by a finite number of matrices. Besides, we show that Six is a paraconsistent logic. Moreover, we prove that it is a genuine Logic of Formal Inconsistency with a consistency operator that can be defined in terms of the original set of connectives. Finally, we study the proof theory of Six providing a Gentzen calculus for it, which is sound and complete with respect to the logic. (shrink)

Propositional fuzzy logics given by a combination of a continuous SBL t-norm with finitely many idempotents and of an involutive negation are investigated. A characterization of continuous t-norms which, in combination with different involutive negations, yield either isomorphic algebras or algebras with distinct and incomparable sets of propositional tautologies is presented.

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant $\overline{0}$ , namely $\neg \varphi$ is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that $\neg$ is an involutive negation. However, (...) for t-norms without non-trivial zero divisors, $\neg$ is Gödel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation. (shrink)

The role played by the symmetric structure of a group of finite Morley rank without involutions in the proof by contradiction of Frécon 2018 was put in evidence in Poizat 2018; indeed, this proof consists in the construction of a symmetric space of dimension two (“a plane”), and then in showing that such a plane cannot exist.To a definable symmetric subset of such a group are associated symmetries and transvections, that we undertake here to study in the abstract, without mentioning (...) a group envelopping them. This leads us to consider axiomatically defined structures that we callsymmetrons(preferably todyadic symmetric setsas was done in Lawson & Lim 2004).Glauberman's$Z^*$-Theorem allows to elucidate completely the structure of the finite symmetrons: each of them is isomorphic to the set of symmetries associated to a symmetric subspace of a finite group without involutions, which is far from being uniquely determined. In fact, there exist non-isomorphic finite groups which have the same symmetries, and also finite symmetrons which are not isomorphic to the symmetries of a group.The situation is not so clear in the case of symmetrons of finite Morley rank, or even algebraic, which are the main objects of study of this paper. But in spite of the fact that a symmetron be a structure much weaker that a group, we can extend to symmetrons some well-known results concerning groups of finite Morley rank: chain condition, decomposition into connected components, characterisation of the generic definable subsets, elliptic generation, etc. These properties are new even in the case of a symmetric subspace of a group, and allow to bypass the computations made by Frécon during the construction of his paradoxical plane.Moreover, assuming the Algebricity Conjecture, we generalize Glauberman's Theorem to the finite Morley rank context. (shrink)

The focus of this article is on the Business Council of Australia, an association of the CEOs of the 100 or so largest companies operating in Australia. Since its inception the BCA has been an influential supporter of largely successful efforts to neoliberalize and internationalize the Australian economy. Running in parallel with these developments, however, the BCA has moved from being a “somewhat strong” to a relatively weak policy organization. This article argues these two trends are causally related. Neoliberal-inspired economic (...) restructuring and economic internationalization have weakened the “logic of membership” and the “logic of influence” of the BCA, leading to a process of organizational involution. Furthermore, potential offsets to what I describe as the organizational predations of neoliberalism and internationalization—especially via a willingness or capacity to forge supportive or mutualistic relations with the state—have not been realized. (shrink)

In general, there is only one fuzzy logic in which the standard interpretation of the strong conjunction is a strict triangular norm, namely, the product logic. We study several equations which are satisfied by some strict t-norms and their dual t-conorms. Adding an involutive negation, these equations allow us to generate countably many logics based on strict t-norms which are different from the product logic.

The COVID-19 pandemic has had a significant negative effect on university students’ mental health worldwide. The pandemic has resulted in individuals experiencing increased levels of anxiety and stress as well as intensified concerns about the future due to a rise in uncertainty. To eliminate the anxiety and stress caused by uncertainty, individuals who have high cognitive closure needs are strongly motivated to achieve certainty and seek answers, even if the decisions they make in the process are inappropriate or even irrational. (...) This study attempts to analyze the influence of anxiety and stress on university students’ academic involution behavior during the COVID-19 pandemic through the mediating effect of the need for cognitive closure. Analyzing the survey data collected from 402 university students from 3 different universities through the mediating effect model and path analysis with latent variable, our results indicate that: The COVID-19 pandemic has significantly increased the levels of anxiety and stress experienced by university students. The path coefficient of the perceived risk of contracting COVID-19 to perceived emotions was 0.352, anxiety and stress significantly and positively affect academic involution behavior. For each unit of increase in the scores of anxiety and stress, the scores of academic involution behavior increased by 0.408 and 0.398 units, respectively, and The need for cognitive closure had a complete and partial mediating effect on the relationship between the perceived risk of contracting COVID-19 and academic involution behavior, as well as anxiety and academic involution behavior, respectively. The results of PA-LV showed that the mediating effect values were 0.106 and 0.044, respectively. The impact of the COVID-19 pandemic has not yet come to an end and so clarifying the relationship between anxiety and academic involution behavior will assist university students to optimize the relationship in their own subjective situations, to establish good learning habits, and reduce psychological distress. (shrink)