Perfect MV-algebras are categorically equivalent to abelianl-groups

In this paper we prove that the category of abelianl-groups is equivalent to the category of perfect MV-algebras. Furthermore, we give a finite equational axiomatization of the variety generated by perfect MV-algebras.Presented byDaniele Mundici     

Monadic MV-algebras are Equivalent to Monadic ?-groups with Strong Unit

In this paper we extend Mundici??s functor ?? to the category of monadic MV-algebras. More precisely, we define monadic ?-groups and we establish a natural equivalence between the category of monadic MV-algebras and the category of monadic ?-groups with strong unit. Some applications are given thereof.     

A Note on Bosbach’s Cone Algebras

In 2002, Dvure?enskij extended Mundici’s equivalence between unital abelian l-groups and MV-algebras to the non-commutative case. We analyse the relationship to Bosbach’s cone algebras and clarify the rôle of the corresponding pair of L-algebras. As a consequence, it follows that one of the two L-algebra axioms can be dropped.     

On Categorical Equivalences of Commutative BCK-algebras

A commutative BCK-algebra with the relative cancellation property is a commutative BCK-algebra (X;*,0) which satisfies the condition: if a ≤ x, a ≤ y and x * a = y * a, then x = y. Such BCK-algebras form a variety, and the category of these BCK-algebras is categorically equivalent to the category of Abelian ℓ-groups whose objects are pairs (G, G ₀), where G is an Abelian ℓ-group, G ₀ is a subset of the positive cone generating G ⁺ such that if u, v ∈ G ₀, then 0 ∨ (u - v) ∈ G ₀, and morphisms are ℓ-group homomorphisms h: (G, G ₀) → (G′,G′₀) with f(G ₀) ⫅ G′₀. Our methods in particular cases give known categorical equivalences of Cornish for conical BCK-algebras and of Mundici for bounded commutative BCK-algebras (= MV-algebras). This revised version was published online in June 2006 with corrections to the Cover Date.     

AI,ME AND LEWIS (ABELIAN IMPLICATION,MATERIAL EQUIVALENCE AND C I LEWIS 1920)

C I Lewis showed up Down Under in 2005, in e-mails initiated by Allen Hazen of Melbourne. Their topic was the system Hazen called FL (a Funny Logic), axiomatized in passing in Lewis 1921. I show that FL is the system MEN of material equivalence with negation. But negation plays no special role in MEN. Symbolizing equivalence with → and defining ∼A inferentially as A→f, the theorems of MEN are just those of the underlying theory ME of pure material equivalence. This accords with the treatment of negation in the Abelian l-group logic A of Meyer and Slaney (Abelian logic. Abstract, Journal of Symbolic Logic 46, 425–426, 1981), which also defines ∼A inferentially with no special conditions on f. The paper then concentrates on the pure implicational part AI of A, the simple logic of Abelian groups. The integers Z were known to be characteristic for AI, with every non-theorem B refutable mod some Zn for finite n. Noted here is that AI is pre-tabular, having the Scroggs property that every proper extension SI of AI, closed under substitution and detachment, has some finite Zn as its characteristic matrix. In particular FL is the extension for which n = 2 (Lewis, The structure of logic and its relation to other systems. The Journal of Philosophy 18, 505–516, 1921; Meyer and Slaney, Abelian logic. Abstract. Journal of Symbolic Logic 46, 425–426, 1981; This is an abstract of the much longer paper finally published in 1989 in G. G. Priest, R. Routley and J. Norman, eds., Paraconsistent logic: essays on the inconsistent, Philosophica Verlag, Munich, pp. 245–288, 1989). Meyer was supported in this work as a Visiting Fellow in the College of Engineering and Computer Science, ANU.     

A simplified duality for implicative lattices and l-groups

A topological duality is presented for a wide class of lattice-ordered structures including lattice-ordered groups. In this new approach, which simplifies considerably previous results of the author, the dual space is obtained by endowing the Priestley space of the underlying lattice with two binary functions, linked by set-theoretical complement and acting as symmetrical partners. In the particular case of l-groups, one of these functions is the usual product of sets and the axiomatization of the dual space is given by very simple first-order sentences, saying essentially that both functions are associative and that the space is a residuated semigroup with respect to each of them.The author is supported at the Mathematical Institute of Oxford by a grant of the Argentinian Consejo de Investigations Cientificas y Tecnicas (CONICET). The author wishes to acknowledge the CONICET and the kind hospitality of the Mathematical Institute.     

Disjunctive Quantum Logic in Dynamic Perspective

In Coecke (2002) we proposed the intuitionistic or disjunctive representation of quantum logic, i.e., a representation of the property lattice of physical systems as a complete Heyting algebra of logical propositions on these properties, where this complete Heyting algebra goes equipped with an additional operation, the operational resolution, which identifies the properties within the logic of propositions. This representation has an important application towards dynamic quantum logic, namely in describing the temporal indeterministic propagation of actual properties of physical systems. This paper can as such by conceived as an addendum to Quantum Logic in Intuitionistic Perspective that discusses spin-off and thus provides an additional motivation. We derive a quantaloidal semantics for dynamic disjunctive quantum logic and illustrate it for the particular case of a perfect (quantum) measurement.     

Boolean Skeletons of MV-algebras and ?-groups

Let ?? be Mundici??s functor from the category ${\mathcal{LG}}$ whose objects are the lattice-ordered abelian groups (?-groups for short) with a distinguished strong order unit and the morphisms are the unital homomorphisms, onto the category ${\mathcal{MV}}$ of MV-algebras and homomorphisms. It is shown that for each strong order unit u of an ?-group G, the Boolean skeleton of the MV-algebra ??(G, u) is isomorphic to the Boolean algebra of factor congruences of G.     

Duality Theory and Skeleta for Semisimple MV-Algebras

We start from Marra–Spada duality between semisimple MV-algebras and Tychonoff spaces, and we consider the particular cases when the \(\omega \)-skeleta of the MV-algebras are restricted in some way. In particular we consider antiskeletal MV-algebras, that is, the ones whose \(\omega \)-skeleton is trivial.     

Free Łukasiewicz and Hoop Residuation Algebras

Hoop residuation algebras are the {, 1}-subreducts of hoops; they include Hilbert algebras and the {, 1}-reducts of MV-algebras (also known as Wajsberg algebras). The paper investigates the structure and cardinality of finitely generated free algebras in varieties of k-potent hoop residuation algebras. The assumption of k-potency guarantees local finiteness of the varieties considered. It is shown that the free algebra on n generators in any of these varieties can be represented as a union of n subalgebras, each of which is a copy of the {, 1}-reduct of the same finite MV-algebra, i.e., of the same finite product of linearly ordered (simple) algebras. The cardinality of the product can be determined in principle, and an inclusion-exclusion type argument yields the cardinality of the free algebra. The methods are illustrated by applying them to various cases, both known (varieties generated by a finite linearly ordered Hilbert algebra) and new (residuation reducts of MV-algebras and of hoops).     

The Intimate Relationship Between the McNaughton and the Chinese Remainder Theorems for MV-algebras

We show the intimate relationship between McNaughton Theorem and the Chinese Remaindner Theorem for MV-algebras. We develop a very short and simple proof of McNaughton Theorem. The arguing is elementary and right out of the definitions. We exhibit the theorem as just an instance of the Chinese theorem. Since the variety of MV-algebras is arithmetic, the Chinese theorem holds for MV-algebras. However, to make this paper self-contained and entirely elementary, we include a simple proof of this theorem inspired in Ferraioli and Lettieri (Math Logic Q 1:27–43, 2011).     

Vagueness and Blurry Sets

This paper presents a new theory of vagueness, which is designed to retain the virtues of the fuzzy theory, while avoiding the problem of higher-order vagueness. The theory presented here accommodates the idea that for any statement S ₁ to the effect that Bob is bald is x true, for x in [0,1], there should be a further statement S ₂ which tells us how true S ₁ is, and so on – that is, it accommodates higher-order vagueness – without resorting to the claim that the metalanguage in which the semantics of vagueness is presented is itself vague, and without requiring us to abandon the idea that the logic – as opposed to the semantics – of vague discourse is classical. I model the extension of a vague predicate P as a blurry set, this being a function which assigns a degree of membership or degree function to each object o, where a degree function in turn assigns an element of [0,1] to each finite sequence of elements of [0,1]. The idea is that the assignment to the sequence 0.3,0.2, for example, represents the degree to which it is true to say that it is 0.2 true that o is P to degree 0.3. The philosophical merits of my theory are discussed in detail, and the theory is compared with other extensions and generalisations of fuzzy logic in the literature.     

Everything Else Being Equal: A Modal Logic for <Emphasis Type="Italic">Ceteris Paribus</Emphasis> Preferences

This paper presents a new modal logic for ceteris paribus preferences understood in the sense of “all other things being equal”. This reading goes back to the seminal work of Von Wright in the early 1960’s and has returned in computer science in the 1990’s and in more abstract “dependency logics” today. We show how it differs from ceteris paribus as “all other things being normal”, which is used in contexts with preference defeaters. We provide a semantic analysis and several completeness theorems. We show how our system links up with Von Wright’s work, and how it applies to game-theoretic solution concepts, to agenda setting in investigation, and to preference change. We finally consider its relation with infinitary modal logics.     

An Early Buddhist Text on Logic: Fang Bian Xin Lun

The Fang Bian Xin Lun is a text on Buddhist logic which is thought to be the earliest one still to be extant. It appears in Chinese only (T1632). The great Italian indologist Giuseppe Tucci, believing that the text was originally a Sanskrit text, translated it into Sanskrit and gave it the title Upāyahṛdaya. The paper provides the historical background of the development of logic in Classical India up to the time of this text, summarizes its content and translates its first section.

Dynamic epistemic logic with branching temporal structures

van Bentham et al. (Merging frameworks for interaction: DEL and ETL, 2007) provides a framework for generating the models of Epistemic Temporal Logic (ETL: Fagin et al., Reasoning about knowledge, 1995; Parikh and Ramanujam, Journal of Logic, Language, and Information, 2003) from the models of Dynamic Epistemic Logic (DEL: Baltag et al., in: Gilboa (ed.) Tark 1998, 1998; Gerbrandy, Bisimulations on Planet Kripke, 1999). We consider the logic TDEL on the merged semantic framework, and its extension with the labeled past-operator “P _ϵ” (“The event ϵ has happened before which. . .”). To axiomatize the extension, we introduce a method for transforming a given model into a normal form in a suitable sense. These logics suggest further applications of DEL in the theory of agency, the theory of learning, etc.     

Neat,Swine, Sheep,and Deer: Mill and Peirce on Natural Kinds

In the earliest phase of his logical investigations (1865–1870), Peirce adopts Mill's doctrine of real Kinds as discussed in the System of Logic and adapts it to the logical conceptions he was then developing. In Peirce's definition of natural class, a crucial role is played by the notion of information: a natural class is a class of which some non-analytical proposition is true. In Peirce's hands, Mill's distinction between connotative and non-connotative terms becomes a distinction between symbolic and informative and pseudo-symbolic and non-informative forms of representation. A symbol is for Peirce a representation which has information. Just as for Mill all names of Kind connote their being such, so for Peirce all symbols profess to correspond to a natural class.     

Constructive Logic with Strong Negation is a Substructural Logic. I

The goal of this two-part series of papers is to show that constructive logic with strong negation N is definitionally equivalent to a certain axiomatic extension NFL _ew of the substructural logic FL _ew . In this paper, it is shown that the equivalent variety semantics of N (namely, the variety of Nelson algebras) and the equivalent variety semantics of NFL _ew (namely, a certain variety of FL _ew -algebras) are term equivalent. This answers a longstanding question of Nelson [30]. Extensive use is made of the automated theorem-prover Prover9 in order to establish the result. The main result of this paper is exploited in Part II of this series [40] to show that the deductive systems N and NFL _ew are definitionally equivalent, and hence that constructive logic with strong negation is a substructural logic over FL _ew . Presented by Heinrich Wansing     

Don’t Ever Do That! Long-term Duties in <Emphasis Type="Italic">PD</Emphasis><Subscript><Emphasis Type="Italic">e</Emphasis></Subscript><Emphasis Type="Italic">L</Emphasis>

This paper studies long-term norms concerning actions. In Meyer’s Propositional Deontic Logic (PD _e L), only immediate duties can be expressed, however, often one has duties of longer durations such as: “Never do that”, or “Do this someday”. In this paper, we will investigate how to amend PD _e L so that such long-term duties can be expressed. This leads to the interesting and suprising consequence that the long-term prohibition and obligation are not interdefinable in our semantics, while there is a duality between these two notions. As a consequence, we have provided a new analysis of the long-term obligation by introducing a new atomic proposition I (indebtedness) to represent the condition that an agent has some unfulfilled obligation. Presented by Jacek Malinowski     

Henkin Quantifiers and the Definability of Truth

Henkin quantifiers have been introduced in Henkin (1961). Walkoe (1970) studied basic model-theoretical properties of an extension L _* ¹(H) of ordinary first-order languages in which every sentence is a first-order sentence prefixed with a Henkin quantifier. In this paper we consider a generalization of Walkoe's languages: we close L _* ¹(H) with respect to Boolean operations, and obtain the language L ¹(H). At the next level, we consider an extension L _* ²(H) of L ¹(H) in which every sentence is an L ¹(H)-sentence prefixed with a Henkin quantifier. We repeat this construction to infinity. Using the (un)-definability of truth – in – N for these languages, we show that this hierarchy does not collapse. In addition, we compare some of the present results to the ones obtained by Kripke (1975), McGee (1991), and Hintikka (1996).