This paper deals with pretabularity of fuzzy logics. For this, we first introduce two systems NMnfp and NM½, which are expansions of the fuzzy system NM, and examine the relationships between NMnfp and the another known extended system NM—. Next, we show that NMnfp and NM½ are pretabular, whereas NM is not. We also discuss their algebraic completeness.

Inspired by the work done by Baaz et al. (Ann Pure Appl Log 147(1–2): 23–47, 2007; Lecture Notes in Computer Science, vol 4790/2007, pp 77–91, 2007) for first-order Gödel logics, we investigate Nilpotent Minimum logic NM. We study decidability and reciprocal inclusion of various sets of first-order tautologies of some subalgebras of the standard Nilpotent Minimum algebra, establishing also a connection between the validity in an NM-chain of certain first-order formulas and its order type. Furthermore, (...) we analyze axiomatizability, undecidability and the monadic fragments. (shrink)

In this paper we study finitary extensions of the nilpotent minimum logic or equivalently quasivarieties of NM-algebras. We first study structural completeness of NML, we prove that NML is hereditarily almost structurally complete and moreover NM\, the axiomatic extension of NML given by the axiom \^{2}\leftrightarrow ^{2})^{2}\), is hereditarily structurally complete. We use those results to obtain the full description of the lattice of all quasivarieties of NM-algebras which allow us to characterize and axiomatize all finitary extensions (...) of NML. (shrink)

In this paper we carry out an algebraic investigation of the weak nilpotent minimum logic and its t-norm based axiomatic extensions. We consider the algebraic counterpart of WNM, the variety of WNM-algebras and prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM-chains, in particular varieties generated by standard WNM-chains, or equivalently t-norm based axiomatic extensions of WNM, and we study (...) their standard completeness properties. We also characterize the generic WNM-chains, i. e. those that generate the variety [MATHEMATICAL DOUBLE-STRUCK CAPITAL W]ℕ[MATHEMATICAL DOUBLE-STRUCK CAPITAL M], and we give finite axiomatizations for some t-norm based extensions of WNM. (shrink)

In the present paper we give a description of the free algebra over an arbitrary set of generators in the variety of nilpotent minimum algebras. Such description is given in terms of a weak Boolean product of directly indecomposable algebras over the Boolean space corresponding to the Boolean subalgebra of the free NM-algebra.

We axiomatize the notion of state over finitely generated free NM-algebras, the Lindenbaum algebras of pure Nilpotent Minimum logic. We show that states over the free n -generated NM-algebra exactly correspond to integrals of elements of with respect to Borel probability measures.

This book is part of aaThe Minimum You Need to Knowaa family of books by Logikal Solutions. As the family expands they will cover an increasing variety of topics. This book is designed to be used as a text book for classes in logic from high school to college level. It should be one of the first courses you have on IT and this should be one of the first books you read when starting in IT. Not only (...) does this book cover flow charting and pseudocode, it teaches the reader to think before they start mapping out the logic to solve a problem. The author of this book is an industry veteran with nearly 20 years in the field. It has been his experience that recent graduates, from any country, are nearly useless at problem solving. If they cannot point, click, and drag, they cannot solve the problem. This book is an attempt to teach them how to solve the problem. An instructoraas guide is available for schools looking to make this book the basis of coursework. (shrink)

Let p be an odd prime. A method is described which given a structure M of finite similarity type produces a nilpotent group of class 2 and exponent p which is in the same stability class as M. Theorem. There are nilpotent groups of class 2 and exponent p in all stability classes. Theorem. The problem of characterizing a stability class is equivalent to characterizing the (nilpotent, class 2, exponent p) groups in that class.

The following theorems are proved about the Frattini-free componentG Φ of a soluble stable ℜ-group: a) If it has a normal subgroupN with nilpotent quotientG Φ/N, then there is a nilpotent subgroupH ofG Φ withG Φ=NH. b) It has Carter subgroups; if the group is small, they are all conjugate. c) Nilpotency modulo a suitable Frattini-subgroup (to be defined) implies nilpotency. The last result makes use of a new structure theorem for the centre of the derivative of the (...) Frattini-free component of a centreless soluble ℜ-group. (shrink)

Do there exist real numbers d with d2 = 0? The question is formulated provocatively, to stress a formalist view about existence: existence is consistency, or better, coherence.Also, the provocation is meant to challenge the monopoly which the number system, invented by Dedekind et al., is claiming for itself as THE model of the geometric line. The Dedekind approach may be termed “arithmetization of geometry”.We know that one may construct a number system out of synthetic geometry, as Euclid and followers (...) did : “geometrization of arithmetic”..Starting from the geometric side, nilpotent elements are somewhat reasonable, although Euclid excluded them. The sophist Protagoras presented a picture of a circle and a tangent line; the apparent little line segment D which tangent and circle have in common, are, by Pythagoras' Theorem, precisely the points, whose abscissae d have d2 = 0. Protagoras wanted to use this argument for destructive reasons: to refute the science of geometry.A couple of millenia later, the Danish geometer Hjelmslev revived the Protagoras picture. His aim was more positive: he wanted to describe Nature as it was. According to him, the Real Line, the Line of Sensual Reality, had many nilpotent infinitesimals, which we can see with our naked eyes. (shrink)

It is well-known that an element of a commutative ring with identity is nilpotent if, and only if, it lies in every prime ideal of the ring. A modification of this fact is amenable to a very simple proof mining analysis. We formulate a quantitative version of this modification and obtain an explicit bound. We present an application. This proof mining analysis is the leitmotif for some comments and observations on the methodology of computational extraction. In particular, we emphasize (...) that the formulation of quantitative versions of ordinary mathematical theorems is of independent interest from proof mining metatheorems. (shrink)

We show that an infinite field is interpretable in a stable torsion-free nilpotent groupG of classk, k>1. Furthermore we prove thatG/Z k-1 (G) must be divisible. By generalising methods of Belegradek we classify some stable torsion-free nilpotent groups modulo isomorphism and elementary equivalence.

In this article I investigate the phenomenon of minimum and minimal models of second-order set theories, focusing on Kelley–Morse set theory KM, Gödel–Bernays set theory GB, and GB augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of ZFC has a minimum GBC-realization if and only if it admits a parametrically definable global well order. (2) Countable models of GBC admit minimal extensions with the same sets. (3) There is (...) no minimum transitive model of KM. (4) There is a minimum β-model of GB+ETR. The main question left unanswered by this article is whether there is a minimum transitive model of GB+ETR. (shrink)

We prove that the problem of determining the minimum propositional proof length is NP- hard to approximate within a factor of 2 log 1 - o(1) n . These results are very robust in that they hold for almost all natural proof systems, including: Frege systems, extended Frege systems, resolution, Horn resolution, the polynomial calculus, the sequent calculus, the cut-free sequent calculus, as well as the polynomial calculus. Our hardness of approximation results usually apply to proof length measured either (...) by number of symbols or by number of inferences, for tree-like or dag-like proofs. We introduce the Monotone Minimum (Circuit) Satisfying Assignment problem and reduce it to the problems of approximation of the length of proofs. (shrink)

While the Vienna Circle had virtually no impact on the Czech-speaking philosophical community during the 1930s, one can find a curious meeting point in the field of theory of architecture. There is now a growing literature on Otto Neurath as a theorist of architecture and urbanism, who emphasized the social aspects of modern building and approached architecture from his idiosyncratic viewpoint of Marxism interpreted as a physicalistic social science. It is less well known that a young Czech architecture critic and (...) theorist, Karel Teige, cultivated strikingly similar views during the same period—from 1920s to 1930s—albeit without any knowledge of Neurath’s thought in particular, or for that matter the Vienna Circle in general. The chapter reveals similarities as well as differences between Neurath and Teige on Marxism, science and architecture, and the Bauhaus, as well as a discussion of the relations of both to the contemporaries, most importantly Adolf Loos, Josef Frank and Hannes Meyer. (shrink)

For any Boolean Algebra A, let cmm be the smallest size of an infinite partition of unity in A. The relationship of this function to the 21 common functions described in Monk [4] is described, for the class of all Boolean algebras, and also for its most important subclasses. This description involves three main results: the existence of a rigid tree algebra in which cmm exceeds any preassigned number, a rigid interval algebra with that property, and the construction of an (...) interval algebra in which every well-ordered chain has size less than cmm. (shrink)

We study the problem of tiling a polyomino P with as few squares as possible such that every square in the tiling has a non-empty intersection with the boundary of P . Our main result is an algorithm which given a simply connected polyomino P computes such a tiling of P . We indicate how one can improve the running time of this algorithm for the more restricted row-column-convex polyominoes. Finally we show that a related decision problem is in NP (...) for rectangular polyominoes. (shrink)

`This book grew out of the conviction, not in itself strange or startling, that the ordinary person can and should think straight rather than crooked.' Patrick Shaw has written a commonsense introduction to the use of logic in everyday thought and argument. It explains some of the rules of good argument and some of the ways in which arguments can fail, drawing illustrations from a variety of contemporary and international sources, such as the press, radio, and television. Symbols and (...) technicalities are kept to a minimum in this thorough and provocative investigation of the rational approach to thought - and its limitations. Logic and Its Limits emphasizes the use of logic in helping to settle and clarify disputes. It will help the reader to avoid bad arguments, to detect them in others, and so to think and argue more effectively. A wide range of thought-provoking examples and exercises concerned with contemporary social and political issues make this a readable and stimulating guide for the student and general reader alike. (shrink)

Suppose that T is an equational theory of groups or of rings. If T is finitely axiomatizable, then there is a least number μ so that T can be axiomatized by μ equations. This μ can depend on the operation symbols that occur in T. In the 1960s, Tarski and Green completely determined the values of μ for arbitrary equational theories of groups and of rings. While Tarski and Green announced the results of their collaboration in 1970, the only fuller (...) publication of their work occurred as part of a seminar led by Tarski at Berkeley during the 1968–69 academic year. The present paper gives a full account of their findings and their proofs. (shrink)

We prove that for a finitely generated infinite nilpotent group $G$ with structure $(G,\cdot,\dots)$, the connected component ${G^*}^0$ of a sufficiently saturated extension $G^*$ of $G$ exists and equals \[ \bigcap_{n\in\N} \{g^n\colon g\in G^*\}. \] We construct an expansion of ${\mathbb Z}$ by a predicate $({\mathbb Z},+,P)$ such that the type-connected component ${{\mathbb Z}^*}^{00}_{\emptyset}$ is strictly smaller than ${{\mathbb Z}^*}^0$. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result (...) for the van der Waerden theorem for finite partitions of groups. (shrink)

In this paper we study the algebraic structure ofω-stable bilinear maps, arbitrary rings, and nilpotent groups. We will also provide rather complete structure theorems for the above structures in the finite Morley rank case.

We show constructively that every quasi-convex, uniformly continuous function \ with at most one minimum point has a minimum point, where C is a convex compact subset of a finite dimensional normed space. Applications include a result on strictly quasi-convex functions, a supporting hyperplane theorem, and a short proof of the constructive fundamental theorem of approximation theory.

A concise yet rigorous introduction to logic and discrete mathematics. This book features a unique combination of comprehensive coverage of logic with a solid exposition of the most important fields of discrete mathematics, presenting material that has been tested and refined by the authors in university courses taught over more than a decade. The chapters on logic - propositional and first-order - provide a robust toolkit for logical reasoning, emphasizing the conceptual understanding of the language and the (...) semantics of classical logic as well as practical applications through the easy to understand and use deductive systems of Semantic Tableaux and Resolution. The chapters on set theory, number theory, combinatorics and graph theory combine the necessary minimum of theory with numerous examples and selected applications. Written in a clear and reader-friendly style, each section ends with an extensive set of exercises, most of them provided with complete solutions which are available in the accompanying solutions manual. Key Features: Suitable for a variety of courses for students in both Mathematics and Computer Science. Extensive, in-depth coverage of classical logic, combined with a solid exposition of a selection of the most important fields of discrete mathematics Concise, clear and uncluttered presentation with numerous examples. Covers some applications including cryptographic systems, discrete probability and network algorithms. Logic and Discrete Mathematics: A Concise Introduction is aimed mainly at undergraduate courses for students in mathematics and computer science, but the book will also be a valuable resource for graduate modules and for self-study. (shrink)

Identity, existence, predication, necessity, and truth are fundamental philosophical concerns. Colin McGinn treats them both philosophically and logically, aiming for maximum clarity and minimum pointless formalism. He contends that there are real logical properties that challenge naturalistic metaphysical outlooks. These concepts are not definable, though we can say a good deal about how they work. The aim of Logical Properties is to bring philosophy back to philosophical logic.

In this paper, we give a complete algebraic description of groups elementarily equivalent to the P. Hall completion of a given free nilpotent group of finite rank over an arbitrary binomial domain. In particular, we characterize all groups elementarily equivalent to a free nilpotent group of finite rank.

We prove that in a nilpotent-by-finite stable group an equation that holds generically holds everywhere. Combining this result with results of Wagner and Bryant, we conclude that a soluble-by-finite stable group of generic exponent n has exponent n.

RésuméOn prouve que dans un groupe stable, nilpotent par fini, une équation générìquement satisfaite y est partout satisfaite. En combinant ce résultat avec des résultats de Wagner et Bryant, on déduit qu'un groupe stable résoluble-par-fini d'exposant généríquenest d'exposantn.

We show that it is consistent with MA + ¬CH that the Forcing Axiom fails for all forcing notions in the class of ωω–bounding forcing notions with norms of [17].

The terms of the upper and lower central series of a nilpotent computable group have computably enumerable Turing degree. We show that the Turing degrees of these terms are independent even when restricted to groups which admit computable orders.

'There is much food for thought in McGinn's discussions and each chapter is rich with a series of considerations for thinking that the currently received views on the various topics have some serious difficulties that need confronting... For those interested in metaphysics and the philosophy of logic, this book will stimulate much further thought' -Mind 'The sweep of the book is broad and the pace is brisk... There is much material here to provide the basis for many a deep (...) philosophical discussion' -Mind 'Lucid and provocative little book... clear, direct and well argued' -Times Higher Education SupplementIdentity, existence, predication, necessity, and truth are fundamental philosophical concerns. Colin McGinn treats them both philosophically and logically, aiming for maximum clarity and minimum pointless formalism. He contends that there are real logical properties that challenge naturalistic metaphysical outlooks. These concepts are not definable, though we can say a good deal about how they work. The aim of Logical Properties is to bring philosophy back to philosophical logic. (shrink)

This note is a continuation of our former paper ''Complexity of the r-query tautologies in the presence of a generic oracle.'' We give a very short direct proof of the nonexistence of t-generic oracles, a result obtained first by Dowd. We also reconstitute a proof of Dowd's result that the class of all r-generic oracles in his sense has Lebesgue measure one.

A notable early result of David Makinson establishes that every monotone modal logic can be extended to LI, LV, or LF, and every antitone logic can be extended to LN, LV, or LF, where LI, LN, LV, and LF are logics axiomatized, respectively, by the schemas □α↔α, □α↔¬α, □α↔⊤, and □α↔⊥. We investigate logics that are both monotone and antitone (hereafter amphitone). There are exactly three: LV, LF, and the minimum amphitone logic AM axiomatized by the (...) schema □α→□β. These logics, along with LI, LN, and a wider class of “extensional” logics, bear close affinities to classical propositional logic. Characterizing those affinities reveals differences among several accounts of equivalence between logics. Some results about amphitone logics do not carry over when logics are construed as consequence or generalized (“multiple-conclusion”) consequence relations on languages that may lack some or all of the nonmodal connectives. We close by discussing these divergences and conditions under which our results do carry over. (shrink)