IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic (...) interpretation to this construction, we generalize the concepts of perfect, bipartite and local algebra used in the classification of MV-algebras to the wider variety of IMTL-algebras and we prove that perfect algebras are exactly those algebras obtained from a prelinear semihoop by Jenei's disconnected rotation. We also prove that the variety generated by all perfect IMTL-algebras is the variety of the IMTL-algebras that are bipartite by every maximal filter and we give equational axiomatizations for it. (shrink)

BL-algebras rise as Lindenbaum algebras from many valued logic introduced by Hájek [2]. In this paper Boolean ds and implicative ds of BL-algebras are defined and studied. The following is proved to be equivalent: (i) a ds D is implicative, (ii) D is Boolean, (iii) L/D is a Boolean algebra. Moreover, a BL-algebra L contains a proper Boolean ds iff L is bipartite. Local BL-algebras, too, are characterized. These results generalize some theorems presented in [4], (...) [5], [6] for MV-algebras which are BL-algebras fulfiling an additional double negation law x = x **. (shrink)

In this paper we study the set of MV-algebras with given prime spectrum and we introduce the class of spectral MV-algebras. An MV-algebra is spectral if it is generated by the union of all its prime ideals (or proper ideals, or principal ideals, or maximal ideals). Among spectral MV-algebras, special attention is devoted to bipartite MV-algebras. An MV-algebra is bipartite if it admits an homomorphism onto the MV-algebra of two elements. We prove that both (...) bipartite MV-algebras and spectral MV-algebras can be finitely axiomatized in first order logic. We also prove that there is only, up to isomorphism, a set of MV-algebras with given prime spectrum. A further part of the paper is devoted to some relations between bipartite MV-algebras and their states. Recall that a state on an MV-algebra is a generalization of a probability measure on a Boolean algebra. Particular states are the states with Bayes’ property. We show that an MV-algebra admits a state with the Bayes’ property if and only if it is bipartite. (shrink)

The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...) of the reasons why CDs form a good notation is that they are highly mathematically tractable: experts can obtain valid results by ‘calculating’ with CDs. These calculations, take the form of ‘diagram chases’. In order to draw inferences, experts move algebraic elements around the diagrams. It will be argued that these diagrams are dynamic. It is thanks to their dynamicity that CDs can externalize the relevant reasoning and allow experts to draw conclusions directly by manipulating them. Lastly, it will be shown that CDs play essential roles in the context of proof as well as in other phases of the mathematical enterprise, such as discovery and conjecture formation. (shrink)

The study of perfect, local and bipartite IMTL-algebras presented in [29] is generalized in this paper to the general non-involutive case, i.e. to MTL-algebras. To this end we describe the radical of MTL-algebras and characterize perfect MTL-algebras as those for which the quotient by the radical is isomorphic to the two-element Boolean algebra, and a special class of bipartite MTL-algebras,.

Some mathematical theories in physics justify their explanatory superiority over earlier formalisms by the clarity of their postulates. In particular, axiomatic reconstructions drive home the importance of the composition rule and the continuity assumption as two pillars of quantum theory. Our approach sits on these pillars and combines new mathematics with a testable prediction. If the observer is defined by a limit on string complexity, information dynamics leads to an emergent continuous model in the critical regime. Restricting it to a (...) family of binary codes describing ‘bipartite systems,’ we find strong evidence of an upper bound on bipartite correlations equal to 2.82537. This is measurably different from the Tsirelson bound. The Hilbert space formalism emerges from this mathematical investigation as an effective description of a fundamental discrete theory in the critical regime. (shrink)

It is an old idea, lately out of fashion but now experiencing a revival, that quantum mechanics may best be understood, not as a physical theory with a problematic probabilistic interpretation, but as something closer to a probability calculus per se. However, from this angle, the rather special C *-algebraic apparatus of quantum probability theory stands in need of further motivation. One would like to find additional principles, having clear physical and/or probabilistic content, on the basis of which this apparatus (...) can be reconstructed. In this paper, I explore one route to such a derivation of finite-dimensional quantum mechanics, by means of a set of strong, but probabilistically intelligible, axioms. Stated very informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of (basic, discrete) measurements as orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space – in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short. (shrink)

In any probabilistic theory, we say that a bipartite state ω on a composite system AB steers its marginal state ω B if, for any decomposition of ω B as a mixture ω B =∑ i p i β i of states β i on B, there exists an observable {a i } on A such that the conditional states $\omega_{B|a_{i}}$ are exactly the states β i . This is always so for pure bipartite states in quantum mechanics, (...) a fact first observed by Schrödinger in 1935. Here, we show that, for weakly self-dual state spaces (those isomorphic, but perhaps not canonically isomorphic, to their dual spaces), the assumption that every state of a system A is steered by some bipartite state of a composite AA consisting of two copies of A, amounts to the homogeneity of the state cone. If the state space is actually self-dual, and not just weakly so, this implies (via the Koecher-Vinberg Theorem) that it is the self-adjoint part of a formally real Jordan algebra, and hence, quite close to being quantum mechanical. (shrink)

Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$ -valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$, which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$. The main (...) aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$, we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$, giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$, and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras. (shrink)

Assertion is a speech act that stands at the intersection of the philosophy of language and social epistemology. It is a phenomenon that bears on such wide-ranging topics as testimony, truth, meaning, knowledge and trust. It is thus no surprise that analytic philosophers have devoted innumerable pages to assertion, trying to give the norms that govern it, its role in the transmission of knowledge, and most importantly, what assertion is, or how assertion is to be defined. In this thesis I (...) attempt to show that all previous answers to the question “What is assertion?” are flawed. There are four major traditions in the literature: constitutive norm theories of assertion, accounts that treat assertion as the expression of speaker attitudes, accounts that treat assertion as a proposal to add some proposition to the common ground, and accounts that treat assertion as the taking of responsibility for some claim. Each tradition is explored here, the leading theories within the tradition developed, and then placed under scrutiny to demonstrate flaws within the positions surveyed. I follow the work of G.E. Moore and William P. Alston, whilst drawing on the work of Robert Brandom in order to give a new bipartite theory of assertion. I argue that assertion consists in the explicit presentation of a proposition, along with a taking of responsibility for that proposition. Taking Alston's explicit presentation condition and repairing it in order to deal with problems it faces, whilst combining it with Brandom's responsibility condition, provides, I believe, the best account of assertion. (shrink)

Assertion is a speech act that stands at the intersection of the philosophy of language and social epistemology. It is a phenomenon that bears on such wide-ranging topics as testimony, truth, meaning, knowledge and trust. It is thus no surprise that analytic philosophers have devoted innumerable pages to assertion, trying to give the norms that govern it, its role in the transmission of knowledge, and most importantly, what assertion is, or how assertion is to be defined. -/- In this thesis (...) I attempt to show that all previous answers to the question “What is assertion?” are flawed. There are four major traditions in the literature: constitutive norm theories of assertion, accounts that treat assertion as the expression of speaker attitudes, accounts that treat assertion as a proposal to add some proposition to the common ground, and accounts that treat assertion as the taking of responsibility for some claim. Each tradition is explored here, the leading theories within the tradition developed, and then placed under scrutiny to demonstrate flaws within the positions surveyed. -/- I follow the work of G.E. Moore and William P. Alston, whilst drawing on the work of Robert Brandom in order to give a new bipartite theory of assertion. I argue that assertion consists in the explicit presentation of a proposition, along with a taking of responsibility for that proposition. Taking Alston’s explicit presentation condition and repairing it in order to deal with problems it faces, whilst combining it with Brandom’s responsibility condition, provides, I believe, the best account of assertion. (shrink)

In this paper, the fixed-time practical bipartite tracking problem for the networked robotic systems with parametric uncertainties, input disturbances, and directed signed graphs is investigated. A new fixed-time estimator-based control algorithm for the NRSs is presented to address the abovementioned problem. By applying a sliding surface and the time base generator approach, a new stability analysis method is proposed to achieve the fixed-time practical bipartite tracking for the NRSs. We also derive the upper bound of the convergence time (...) for employing the presented control algorithm to solve the practical bipartite tracking problem and further demonstrate that the convergence time is independent of the initial value. Finally, the simulation examples are given to verify the effectiveness of the presented algorithms. (shrink)

This comprehensive text shows how various notions of logic can be viewed as notions of universal algebra providing more advanced concepts for those who have an introductory knowledge of algebraic logic, as well as those wishing to delve into more theoretical aspects.

This book presents an English translation of a classic Russian text on duality theory for Heyting algebras. Written by Georgian mathematician Leo Esakia, the text proved popular among Russian-speaking logicians. This translation helps make the ideas accessible to a wider audience and pays tribute to an influential mind in mathematical logic. The book discusses the theory of Heyting algebras and closure algebras, as well as the corresponding intuitionistic and modal logics. The author introduces the key notion of (...) a hybrid that “crossbreeds” topology and order, resulting in the structures now known as Esakia spaces. The main theorems include a duality between the categories of closure algebras and of hybrids, and a duality between the categories of Heyting algebras and of so-called strict hybrids. Esakia’s book was originally published in 1985. It was the first of a planned two-volume monograph on Heyting algebras. But after the collapse of the Soviet Union, the publishing house closed and the project died with it. Fortunately, this important work now lives on in this accessible translation. The Appendix of the book discusses the planned contents of the lost second volume. (shrink)

This paper investigates the bipartite consensus of linear discrete-time multiagent systems with exogenous disturbances. A discrete-time disturbance-observer- based technology is involved for attenuating the exogenous disturbances. And both the state feedback and observer-based output feedback bipartite consensus protocols are proposed by using the DTDO method. It turned out that bipartite consensus can be realized under the given protocols if the topology is connected and structurally balanced. Finally, numerical simulations are presented to illustrate the theoretical findings.

This volume is not restricted to papers presented at the 1988 Colloquium, but instead aims to provide the reader with a (relatively) coherent reading on Algebraic Logic, with an emphasis on current research. To help the non-specialist reader, the book contains an introduction to cylindric and relation algebras by Roger D. Maddux and an introduction to Boolean Algebras by Bjarni Joacute;nsson.

Let K = (p, q...; &, ∨, ~) be a zeroth-order formal language with sentence variables p, q..., two place connectives & (and), ∨ (or) and negation sign ~, and let F be the formula algebra (set of well-formed formulas in K defined in the standard way by induction from the sentence variables). If v is an assignment of truth values 1(true), 0(f alse) to the sentence variables p, q..., then classical propositional logic is characterized by extending v by induction (...) from p, q... to the formula algebra F in such a manner that the extension (called interpretation and also denoted by v) be a “homomorphism” from F into the two-element Boolean algebra B 2 ≡ (1, 0, ⋂, ⋃, ⊥), where “homomorphism” means that the following hold υ(q&p)=υ(q)∩υ(p)υ(q∨p)=υ(q)∪υ(p)υ(∼q)=υ(q)⊥. (shrink)

This paper deals with an (otherwise classical) two-(non-interacting) particle system immersed in a common stochastic zero-point radiation field. The treatment is an extension of the one-particle case for which it has been shown that the quantum properties of the particle emerge from its interaction with the background field under stationary and ergodic conditions. In the present case we show that non-classical correlations—describable only in terms of entanglement—arise between the (nearby) particles whenever both of them resonate to a common frequency of (...) the field. For identical particles the entanglement becomes maximum and must be described by totally (anti)symmetric states. (shrink)

This paper investigates a generalization of Boolean algebras which I call agglomerative algebras. It also outlines two conceptions of propositions according to which they form an agglomerative algebra but not a Boolean algebra with respect to conjunction and negation.

We investigate the effect of adding $\omega _2$ Cohen reals on graphs on $\omega _2$, in particular we show that $\omega _2 \to (\omega _2, \omega : \omega )^2$ holds after forcing with $\mathsf {Add}(\omega, \omega _2)$ in a model of $\mathsf {CH}$. We also prove that this result is in a certain sense optimal as $\mathsf {Add}(\omega, \omega _2)$ forces that $\omega _2 \not \to (\omega _2, \omega : \omega _1)^2$.

Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -- the theory of operator algebras, category theory, etc.. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT. This paper is a survey of AQFT, with an orientation towards foundational topics. In addition to covering the basics of the (...) theory, we discuss issues related to nonlocality, the particle concept, the field concept, and inequivalent representations. We also provide a detailed account of the analysis of superselection rules by Doplicher, Haag, and Roberts (DHR); and we give an alternative proof of Doplicher and Robert's reconstruction of fields and gauge group from the category of physical representations of the observable algebra. The latter is based on unpublished ideas due to J. E. Roberts and the abstract duality theorem for symmetric tensor *-categories, a self-contained proof of which is given in the appendix. (shrink)

Let $\Gamma$ be the random bipartite graph, a countable graph with two infinite sides, edges randomly distributed between the sides, but no edges within a side. In this paper, we investigate the reducts of $\Gamma$ that preserve sides. We classify the closed permutation subgroups containing the group $\operatorname {Aut}(\Gamma)^{\ast}$ , where $\operatorname {Aut}(\Gamma)^{\ast}$ is the group of all isomorphisms and anti-isomorphisms of $\Gamma$ preserving the two sides. Our results rely on a combinatorial theorem of Nešetřil and Rödl and a (...) strong finite submodel property for $\Gamma$. (shrink)

In classical logic, the proposition expressed by a sentence is construed as a set of possible worlds, capturing the informative content of the sentence. However, sentences in natural language are not only used to provide information, but also to request information. Thus, natural language semantics requires a logical framework whose notion of meaning does not only embody informative content, but also inquisitive content. This paper develops the algebraic foundations for such a framework. We argue that propositions, in order to embody (...) both informative and inquisitive content in a satisfactory way, should be defined as non-empty, downward closed sets of possibilities, where each possibility in turn is a set of possible worlds. We define a natural entailment order over such propositions, capturing when one proposition is at least as informative and inquisitive as another, and we show that this entailment order gives rise to a complete Heyting algebra, with meet, join, and relative pseudo-complement operators. Just as in classical logic, these semantic operators are then associated with the logical constants in a first-order language. We explore the logical properties of the resulting system and discuss its significance for natural language semantics. We show that the system essentially coincides with the simplest and most well-understood existing implementation of inquisitive semantics, and that its treatment of disjunction and existentials also concurs with recent work in alternative semantics. Thus, our algebraic considerations do not lead to a wholly new treatment of the logical constants, but rather provide more solid foundations for some of the existing proposals. (shrink)

Volume I provides a detailed analysis of cylindric algebras, starting with a formulation of their axioms and a development of their elementary properties, and proceeding to a deeper study of their interrelationships by means of general algebraic notions such as subalgebras, homomorphisms, direct products, free algebras, reducts and relativized algebras.

The Algebra of Revolution is the first book to study Marxist method as it has been developed by the main representatives of the classical Marxist tradition, namely Marx and Engels, Luxembourg, Lenin, Lukacs, Gramsci, and Trotsky. This book provides the only single volume study of major Marxist thinkers' views on the crucial question of the dialectic, connecting them with pressing contemporary, political and theoretical questions. This title available in eBook format. Click here for more information . Visit our eBookstore at: (...) www.ebookstore.tandf.co.uk. (shrink)

1 Cognitive Architectures 2 Multilayer Perceptrons 3 Relations between Variables 4 Structured Representations 5 Individuals 6 Where does the Machinery of Symbol Manipulation Come From? 7 Conclusions.

This unique textbook states and proves all the major theorems of many-valued propositional logic and provides the reader with the most recent developments and trends, including applications to adaptive error-correcting binary search. The book is suitable for self-study, making the basic tools of many-valued logic accessible to students and scientists with a basic mathematical knowledge who are interested in the mathematical treatment of uncertain information. Stressing the interplay between algebra and logic, the book contains material never before published, such as (...) a simple proof of the completeness theorem and of the equivalence between Chang's MV algebras and Abelian lattice-ordered groups with unit - a necessary prerequisite for the incorporation of a genuine addition operation into fuzzy logic. Readers interested in fuzzy control are provided with a rich deductive system in which one can define fuzzy partitions, just as Boolean partitions can be defined and computed in classical logic. Detailed bibliographic remarks at the end of each chapter and an extensive bibliography lead the reader on to further specialised topics. (shrink)

We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive (...) algebras determine Medvedev’s logic of finite problems. In addition to the algebraic characterization of H(B), we give a topological characterization of H(B) in terms of the recently introduced choice-free duality for Boolean algebras using so-called upper Vietoris spaces (UV-spaces). In particular, while a Boolean algebra B is realized as the Boolean algebra of compact regular open elements of a UV-space dual to B, we show that H(B) is realized as the algebra of compact open elements of this space. This connection yields a new topological semantics for inquisitive logic. (shrink)

Algebraic proofs of the cut-elimination theorems for classical and intuitionistic logic are presented, and are used to show how one can sometimes extract a constructive proof and an algorithm from a proof that is nonconstructive. A variation of the double-negation translation is also discussed: if ϕ is provable classically, then ¬(¬ϕ)nf is provable in minimal logic, where θnf denotes the negation-normal form of θ. The translation is used to show that cut-elimination theorems for classical logic can be viewed as special (...) cases of the cut-elimination theorems for intuitionistic logic. (shrink)

The algebra of embeddings at the I3 level has been deeply analyzed, but nothing is known algebra-wise for embeddings above I3. In this article, we introduce an operation for embeddings at the level of I0 and above, and prove that they generate an LD-algebra that can be quite different from the one implied by I3.

Gödel algebras form the locally finite variety of Heyting algebras satisfying the prelinearity axiom =. In 1969, Horn proved that a Heyting algebra is a Gödel algebra if and only if its set of prime filters partially ordered by reverse inclusion–i.e. its prime spectrum–is a forest. Our main result characterizes Gödel algebras that are free over some finite distributive lattice by an intrisic property of their spectral forest.

An analysis of the structural properties of collections or pluralities, homogeneous objects like water, and the semantics and philosophy of events.

The aim of this paper is to give a description of the free algebras in some varieties of Glivenko MTL-algebras having the Boolean retraction property. This description is given (generalizing the results of [9]) in terms of weak Boolean products over Cantor spaces. We prove that in some cases the stalks can be obtained in a constructive way from free kernel DL-algebras, which are the maximal radical of directly indecomposable Glivenko MTL-algebras satisfying the equation in the (...) title. We include examples to show how we can apply the results to describe free algebras in some well known varieties of involutive MTL-algebras and of pseudocomplemented MTL-algebras. (shrink)

An aggregation procedure merges a list of objects into a representative object. This paper considers the problem of aggregating n rows in an n-by-m matrix into a summary row, where every entry is an element in an algebraic field. It focuses on consistent aggregators, which require each entry in the summary row to depend only on its column entries in the matrix and to be the same as the column entry if the column is constant. Consistent aggregators are related to (...) additive, linear and.. (shrink)

The logic DAI of demodalised analytic implication has been introduced by J.M. Dunn as a variation on a time-honoured logical system by C.I. Lewis’ student W.T. Parry. The main tenet underlying this logic is that no implication can be valid unless its consequent is “analytically contained” in its antecedent. DAI has been investigated both proof-theoretically and model-theoretically, but no study so far has focussed on DAI from the viewpoint of abstract algebraic logic. We provide several different algebraic semantics for DAI, (...) showing their equivalence with the known semantics by Dunn and Epstein. We also show that DAI is algebraisable and we identify its equivalent quasivariety semantics. This class turns out to be a linguistic and axiomatic expansion of involutive bisemilattices, a subquasivariety of which forms the algebraic counterpart of Paraconsistent Weak Kleene logic. This fact sheds further light on the relationship between containment logics and logics of nonsense. (shrink)

The main aim of the present paper is to explain a nature of relationships exist between Nelson and Heyting algebras. In the realization, a topological duality theory of Heyting and Nelson algebras based on the topological duality theory of Priestley for bounded distributive lattices are applied. The general method of construction of spaces dual to Nelson algebras from a given dual space to Heyting algebra is described. The algebraic counterpart of this construction being a generalization of the (...) Fidel-Vakarelov construction is also given. These results are applied to compare the equational category N of Nelson algebras and some its subcategories with the equational category H of Heyting algebras. It is proved that the category N is topological over the category H. (shrink)

Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.

We generalize the notion of a monadic algebra to that of a pseudomonadic algebra. In the same way as monadic algebras serve as algebraic models of epistemic modal system S5, pseudomonadic algebras serve as algebraic models of doxastic modal system KD45. The main results of the paper are: Characterization of subdirectly irreducible and simple pseudomonadic algebras, as well as Tokarz's proper filter algebras; Ordertopological representation of pseudomonadic algebras; Complete description of the lattice of subvarieties of (...) the variety of pseudomonadic algebras. (shrink)

In this paper we study the relations between the fragment L of classical logic having just conjunction and disjunction and the variety D of distributive lattices, within the context of Algebraic Logic. We prove that these relations cannot be fully expressed either with the tools of Blok and Pigozzi's theory of algebraizable logics or with the use of reduced matrices for L. However, these relations can be naturally formulated when we introduce a new notion of model of a sequent calculus. (...) When applied to a certain natural calculus for L, the resulting models are equivalent to a class of abstract logics (in the sense of Brown and Suszko) which we call distributive. Among other results, we prove that D is exactly the class of the algebraic reducts of the reduced models of L, that there is an embedding of the theories of L into the theories of the equational consequence (in the sense of Blok and Pigozzi) relative to D, and that for any algebra A of type (2,2) there is an isomorphism between the D-congruences of A and the models of L over A. In the second part of this paper (which will be published separately) we will also apply some results to give proofs with a logical flavour for several new or well-known lattice-theoretical properties. (shrink)

Brown and Pooley’s ‘dynamical approach’ to physical theories asserts, in opposition to the orthodox position on physical geometry, that facts about physical geometry are grounded in, or explained by, facts about dynamical fields, not the other way round. John Norton has claimed that the proponent of the dynamical approach is illicitly committed to spatiotemporal presumptions in ‘constructing’ space-time from facts about dynamical symmetries. In this article, I present an abstract, algebraic formulation of field theories and demonstrate that the proponent of (...) the dynamical approach is not committed, in special relativity, to the illicit presumptions to which Norton refers. (shrink)

The admissibility of Ackermann's rule γ is one of the most important problems in relevant logics. The admissibility of γ was first proved by an algebraic method. However, the development of Routley-Meyer semantics and metavaluational techniques makes it possible to prove the admissibility of γ using the method of normal models or the method using metavaluations, and the use of such methods is preferred. This paper discusses an algebraic proof of the admissibility of γ in relevant modal logics based on (...) modern algebraic models. (shrink)

We introduce a generalization of MV algebras motivated by the investigations into the structure of quantum logical gates. After laying down the foundations of the structure theory for such quasi-MV algebras, we show that every quasi-MV algebra is embeddable into the direct product of an MV algebra and a “flat” quasi-MV algebra, and prove a completeness result w.r.t. a standard quasi-MV algebra over the complex numbers.

We investigate the polynomial time isomorphic type structure of (the class of tally, polynomial time computable sets). We partition P T into six parts: D −, D^ − , C, S, F, F^, and study their p-isomorphic properties separately. The structures of , , and are obvious, where F, F^, and C are the class of tally finite sets, the class of tally co-finite sets, and the class of tally bi-dense sets respectively. The following results for the structures of and (...) will be proved, where D^ is the class of tally, co-dense, polynomial time computable sets and S is the class of tally, scatted (i.e., neither dense nor co-dense), polynomial time computable sets. 1. is a countable distributive lattice with the greatest element. 2. Infinitely many intervals in can be distinguished by first order formulas. 3. There exist infinitely many nontrivial automorphisms for . 4. is not distributive, but any interval in is a countable distributive lattice. RID=""ID="" Mathematics Subject Classification (2000): 03D15, 03D25, 03D30, 03D35, 06A06, 06B20 RID=""ID="" Key words or phrases: Computational complexity – Polynomial time – Degree structure – Lattice – Isomorphism RID=""ID="" Part of this work was done when the author was a PhD student at the University of Heidelberg under the direction of Professor Ambos-Spies. (shrink)

A Souslin algebra is a complete Boolean algebra whose main features are ruled by a tight combination of an antichain condition with an infinite distributive law. The present article divides into two parts. In the first part a representation theory for the complete and atomless subalgebras of Souslin algebras is established (building on ideas of Jech and Jensen). With this we obtain some basic results on the possible types of subalgebras and their interrelation. The second part begins with a (...) review of some generalizations of results from descriptive set theory concerning Baire category which are then used in non-trivial Souslin tree constructions that yield Souslin algebras with a remarkable subalgebra structure. In particular, we use this method to prove that under the diamond principle there is a bi-embeddable though not isomorphic pair of homogeneous Souslin algebras. (shrink)