The inverse of the distance between two structures $\mathscr{A} \not\equiv \mathscr{B}$ of finite type $\tau$ is naturally measured by the smallest integer $q$ such that a sentence of quantifier rank $q - 1$ is satisfied by $\mathscr{A}$ but not by $\mathscr{B}$. In this way the space $\operatorname{Str}^\tau$ of structures of type $\tau$ is equipped with a pseudometric. The induced topology coincides with the elementary topology of $\operatorname{Str}^\tau$. Using the rudiments of the theory of uniform spaces, in this elementary note we (...) prove the convergence of every Cauchy net of structures, for any type $\tau$. (shrink)

The inverse of the distance between two structures $\mathscr{A} \not\equiv \mathscr{B}$ of finite type τ is naturally measured by the smallest integer q such that a sentence of quantifier rank q - 1 is satisfied by A but not by B. In this way the space $\operatorname{Str}^\tau$ of structures of type τ is equipped with a pseudometric. The induced topology coincides with the elementary topology of $\operatorname{Str}^\tau$ . Using the rudiments of the theory of uniform spaces, in this elementary note (...) we prove the convergence of every Cauchy net of structures, for any type τ. (shrink)

Ultrafilters play a significant role in model theory to characterize logics having various compactness and interpolation properties. They also provide a general method to construct extensions of first-order logic having these properties. A main result of this paper is that every class $\Omega $ of uniform ultrafilters generates a $\Delta $ -closed logic ${\mathcal {L}}_\Omega $. ${\mathcal {L}}_\Omega $ is $\omega $ -relatively compact iff some $D\in \Omega $ fails to be $\omega _1$ -complete iff ${\mathcal {L}}_\Omega $ does not (...) contain the quantifier “there are uncountably many.” If $\Omega $ is a set, or if it contains a countably incomplete ultrafilter, then ${\mathcal {L}}_\Omega $ is not generated by Mostowski cardinality quantifiers. Assuming $\neg 0^\sharp $ or $\neg L^{\mu }$, if $D\in \Omega $ is a uniform ultrafilter over a regular cardinal $\nu $, then every family $\Psi $ of formulas in ${\mathcal {L}}_\Omega $ with $|\Phi |\leq \nu $ satisfies the compactness theorem. In particular, if $\Omega $ is a proper class of uniform ultrafilters over regular cardinals, ${\mathcal {L}}_\Omega $ is compact. (shrink)

In 1999, da Silva, D'Ottaviano and Sette proposed a general definition for the term translation between logics and presented an initial segment of its theory. Logics are characterized, in the most general sense, as sets with consequence relations and translations between logics as consequence-relation preserving maps. In a previous paper the authors introduced the concept of conservative translation between logics and studied some general properties of the co-complete category constituted by logics and conservative translations between them. In this paper we (...) present some conservative translations involving classical logic, Lukasiewicz three-valued system L 3, the intuitionistic system I 1 and several paraconsistent logics, as for instance Sette's system P 1, the D'Ottaviano and da Costa system J 3 and da Costa's systems C n, 1 n. (shrink)

Syntax and semantics in Łukasiewicz infinite-valued sentential logic Ł are harmonized by revising the Bolzano-Tarski paradigm of “semantic consequence,” according to which, θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} follows from Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} iff every valuation v that satisfies all formulas in Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} also satisfies θ.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta.$$\end{document} For θ\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} to be a consequence of Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document}, we also require that any infinitesimal perturbation of v that preserves the truth of all formulas of Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document} also preserves the truth of θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document}. An elementary characterization of Łukasiewicz implication shows that the Łukasiewicz axiom →Y)→→X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \rightarrow Y ) \rightarrow \rightarrow X )$$\end{document} guarantees the continuity and the piecewise linearity of the implication operation →\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow $$\end{document}, an appropriate fault-tolerance property of any logic of [0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{[0,1]}\,}}$$\end{document}-valued observables. The directional derivability of the functions coded by all ψ∈Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi \in \Theta $$\end{document} and by θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} then provides a quantitative formulation of our refinement of Bolzano-Tarski consequence, which turns out to coincide with the time-honored syntactic Ł-consequence. (shrink)

Chang's MV algebras are the algebras of the infinite-valued sentential calculus of ukasiewicz. We introduce finitely additive measures (called states) on MV algebras with the intent of capturing the notion of average degree of truth of a proposition. Since Boolean algebras coincide with idempotent MV algebras, states yield a generalization of finitely additive measures. Since MV algebras stand to Boolean algebras as AFC*-algebras stand to commutative AFC*-algebras, states are naturally related to noncommutativeC*-algebraic measures.

We establish the Robinson joint consistency theorem for the infinite-valued propositional logic of Łukasiewicz. As a corollary we easily obtain the amalgamation property for MV-algebras—the algebras of Łukasiewicz logic: all pre-existing proofs of this latter result make essential use of the Pierce amalgamation theorem for abelian lattice-ordered groups together with the categorical equivalence Γ between these groups and MV-algebras. Our main tools are elementary and geometric.

We give a constructive proof of McNaughton's theorem stating that every piecewise linear function with integral coefficients is representable by some sentence in the infinite-valued calculus of Lukasiewicz. For the proof we only use Minkowski's convex body theorem and the rudiments of piecewise linear topology.

We prove that de Finetti coherence is preserved under taking products of coherent books on two sets of independent events. This establishes a desirable closure property of coherence: were it not the case it would raise a question mark over the utility of de Finetti’s notion of coherence.

De Finetti gave a natural definition of “coherent probability assessment” β:E→[0,1] of a set E={X1,…,Xm} of “events” occurring in an arbitrary set of “possible worlds”. In the particular case of yes–no events, , Kolmogorov axioms can be derived from his criterion. While De Finetti’s approach to probability was logic-free, we construct a theory Θ in infinite-valued Łukasiewicz propositional logic, and show: a possible world of is a valuation satisfying Θ, β is coherent iff it is a convex combination of valuations (...) satisfying Θ, iff β agrees on E with a state of the Lindenbaum MV-algebra of Θ, iff for some Borel probability measure μ on . Thus Łukasiewicz semantics, MV-algebraic states, and Borel probability measures provide a universal representation of coherent assessments of events occurring in any conceivable set of possible worlds. (shrink)

If S ⊆{0,1}; * and S ′ = {0,1} * \sb S are both recognized within a certain nondeterministic time bound T then, in not much more time, one can write down tautologies A n → A′ n with unique interpolants I n that define S ∩{0,1} n ; hence, if one can rapidly find unique interpolants, then one can recognize S within deterministic time T p for some fixed p \s>0. In general, complexity measures for the problem of finding (...) unique interpolants in sentential logic yield new relations between circuit depth and nondeterministic Turing time, as well as between proof length and the complexity of decision procedures of logical theories. (shrink)

Let $\to $ be a continuous $\protect \operatorname {\mathrm {[0,1]}}$ -valued function defined on the unit square $\protect \operatorname {\mathrm {[0,1]}}^2$, having the following properties: $x\to = y\to $ and $x\to y=1 $ iff $x\leq y$. Let $\neg x=x\to 0$. Then the algebra $W=$ satisfies the time-honored Łukasiewicz axioms of his infinite-valued calculus. Let $x\to _{\text {\tiny \L }}y=\min $ and $\neg _{\text {\tiny \L }}x=x\to _{\text {\tiny \L }} 0 =1-x.$ Then there is precisely one isomorphism $\phi $ of (...) W onto the standard Wajsberg algebra $W_{\text {\tiny \L }}= $. Thus $x\to y= \phi ^{-1}+\phi ))$. (shrink)

This study described the relationships between academic class and student moral sensitivity and reasoning and between curriculum design components for ethics education and student moral sensitivity and reasoning. The data were collected from freshman (n = 506) and senior students (n = 440) in eight baccalaureate nursing programs in South Korea by survey; the survey consisted of the Korean Moral Sensitivity Questionnaire and the Korean Defining Issues Test. The results showed that moral sensitivity scores in patient-oriented care and conflict were (...) higher in senior students than in freshman students. Furthermore, more hours of ethics content were associated with higher principled thinking scores of senior students. Nursing education in South Korea may have an impact on developing student moral sensitivity. Planned ethics content in nursing curricula is necessary to improve moral sensitivity and moral reasoning of students. (shrink)

Building on Wójcicki’s work on infinite-valued Łukasiewicz logic Ł ∞ , we give a self-contained proof of the deductive interpolation theorem for Ł ∞ . This paper aims at introducing the reader to the geometry of Łukasiewicz logic.

In classical propositional logic, a theory T is prime iff it is complete. In Łukasiewicz infinite-valued logic the two notions split, completeness being stronger than primeness. Using toric desingularization algorithms and the fine structure of prime ideal spaces of free ℓ -groups, in this paper we shall characterize prime theories in infinite-valued logic. We will show that recursively enumerable prime theories over a finite number of variables are decidable, and we will exhibit an example of an undecidable r.e. prime theory (...) over countably many variables. (shrink)

Machines were introduced as calculating devices to simulate operations carried out by human computers following fixed algorithms. The mathematical study of (paper) machines is the topic of our essay. The first three sections provide necessary logical background, examine the analyses of effective calculability given in the thirties, and describe results that are central to recursion theory, reinforcing the conceptual analyses. In the final section we pursue our investigation in a quite different way and focus on principles that govern the operations (...) of physically realizable machines. (shrink)

Lexical Semantics is about the meaning of words. Although obviously a central concern of linguistics, the semantic behaviour of words has been unduly neglected in the current literature, which has tended to emphasize sentential semantics and its relation to formal systems of logic. In this textbook D. A. Cruse establishes in a principled and disciplined way the descriptive and generalizable facts about lexical relations that any formal theory of semantics will have to encompass. Among the topics covered in depth are (...) idiomaticity, lexical ambiguity, synonymy, hierarchical relations such as hyponymy and meronymy, and various types of oppositeness. Syntagmatic relations are also treated in some detail. The discussions are richly illustrated by examples drawn almost entirely from English. Although a familiarity with traditional grammar is assumed, readers with no technical linguistic background will find the exposition always accessible. All readers with an interest in semantics will find in this original text not only essential background but a stimulating new perspective on the field. (shrink)

The interpretation of propositions in Lukasiewicz's infinite-valued calculus as answers in Ulam's game with lies--the Boolean case corresponding to the traditional Twenty Questions game--gives added interest to the completeness theorem. The literature contains several different proofs, but they invariably require technical prerequisites from such areas as model-theory, algebraic geometry, or the theory of ordered groups. The aim of this paper is to provide a self-contained proof, only requiring the rudiments of algebra and convexity in finite-dimensional vector spaces.

Daniele Mundici and Wilfred Sieg. Computability Theory.

A record of the words and teachings of Confucius, _The Analects_ is considered the most reliable expression of Confucian thought. However, the original meaning of Confucius's teachings have been filtered and interpreted by the commentaries of Confucianists of later ages, particularly the Neo-Confucianists of the Song dynasty, not altogether without distortion.In this monumental translation by Professor D. C. Lau, an attempt has been made to interpret the sayings as they stand. The corpus of the sayings is taken as an organic (...) whole and the final test of the interpretation rests on the internal consistency it exhibits. In other words, _The Analects_ is read in the light of _The Analects._This results in a truer understanding of Confucius' thought than the traditional interpretation and paves the way for a re-assessment of its importance in the history of Chinese thought and its relevance to the present day world.This volume also contains an introduction to the life and teachings of Confucius, and three appendices on the events in the life of Confucius, on his disciples, and on the composition of _The Analects._. (shrink)

An MV-algebra A=(A,0,¬,⊕) is an abelian monoid (A,0,⊕) equipped with a unary operation ¬ such that ¬¬x=x,x⊕¬0=¬0, and y⊕¬(y⊕¬x)=x⊕¬(x⊕¬y). Chang proved that the equational class of MV-algebras is generated by the real unit interval [0,1] equipped with the operations ¬x=1−x and x⊕y=min(1,x+y). Therefore, the free n-generated MV-algebra Free n is the algebra of [0,1]-valued functions over the n-cube [0,1] n generated by the coordinate functions ξ i ,i=1, . . . ,n, with pointwise operations. Any such function f is a (...) McNaughton function, i.e., f is continuous, piecewise linear, and each piece has integer coefficients. Conversely, McNaughton proved that all McNaughton functions f: [0,1] n →[0,1] are in Free n . The elements of Free n are logical equivalence classes of n-variable formulas in the infinite-valued calculus of Łukasiewicz. The aim of this paper is to provide an alternative, representation-free, characterization of Free n. (shrink)

Aim of this paper is to provide a self-contained presentation of the natural equivalence between MV-algebras and lattice-ordered abelian groups with strong unit.

We prove the following algebraic characterization of elementary equivalence: $\equiv$ restricted to countable structures of finite type is minimal among the equivalence relations, other than isomorphism, which are preserved under reduct and renaming and which have the Robinson property; the latter is a faithful adaptation for equivalence relations of the familiar model theoretical notion. We apply this result to Friedman's fourth problem by proving that if L = L ωω (Q i ) i ∈ ω 1 is an (ω 1 (...) , ω)-compact logic satisfying both the Robinson consistency theorem on countable structures of finite type and the Löwenheim-Skolem theorem for some $\lambda for theories having ω 1 many sentences, then $\equiv_L = \equiv$ on such structures. (shrink)

"By combining recent advances in the physical sciences with some of the novel ideas, techniques, and data of modern biology, this book attempts to achieve a new and different kind of evolutionary synthesis. I found it to be challenging, fascinating, infuriating, and provocative, but certainly not dull."--James H, Brown, University of New Mexico "This book is unquestionably mandatory reading not only for every living biologist but for generations of biologists to come."--Jack P. Hailman, Animal Behaviour , review of the first (...) edition "An important contribution to modern evolutionary thinking. It fortifies the place of Evolutionary Theory among the other well-established natural laws."--R.Gessink, TAXON. (shrink)

There is a natural map which assigns to every modelU of typeτ, (U ε Stτ) a groupG (U) in such a way that elementarily equivalent models are mapped into isomorphic groups.G(U) is a subset of a collection whose members are called Fraisse arrows (they are decreasing sequences of sets of partial isomorphisms) and which arise in connection with the Fraisse characterization of elementary equivalence. LetEC λ U be defined as {U εStr τ: ℬ ≡U and |ℬ|=λ; thenEG λ U can (...) be faithfully (i.e. 1-1) represented onto G(U) ×π *, whereπ *, is a collection of partitions over λ∪λ2∪.. (shrink)

In the algebraic treatment of quantum statistical systems, the claim “Nature does not have ideals” is sometimes used to convey the idea that the C*-algebras describing natural systems are simple, i.e., they do not have nontrivial homomorphic images. Using our interpretation of AF C*-algebras as algebras of Lukasiewicz calculus, in a previous paper the claim was shown to be incompatible with the existence of a G¨odel incomplete AF C*-algebra for a quantum physical system existing in nature. In this note we (...) survey recent developments on G¨odel incompleteness and decidability issues for AF C*-algebras. (shrink)

In the algebraic treatment of quantum statistical systems, the claim “Nature does not have ideals” is sometimes used to convey the idea that the C*-algebras describing natural systems are simple, i.e., they do not have nontrivial homomorphic images. Using our interpretation of AF C*-algebras as algebras of Lukasiewicz calculus, in a previous paper the claim was shown to be incompatible with the existence of a G¨odel incomplete AF C*-algebra for a quantum physical system existing in nature. In this note we (...) survey recent developments on G¨odel incompleteness and decidability issues for AF C*-algebras. (shrink)

In an earlier paper the present author proved that de Finetti coherence is preserved under taking products of coherent books on two finite sets of independent events. Conversely, in this note it is proved that product is the only coherence preserving operation on coherent books. Our proof shows that the traditional definition of stochastically independent classes of events actually follows from the combination of two more basic notions: boolean algebraic independence and de Finetti coherent betting system.