Automated theorem proving for Łukasiewicz logics

This paper is concerned with decision proceedures for the ₀-valued ukasiewicz logics,. It is shown how linear algebra can be used to construct an automated theorem checker. Two decision proceedures are described which depend on a linear programming package. An algorithm is given for the verification of consequence relations in, and a connection is made between theorem checking in two-valued logic and theorem checking in which implies that determing of a -free formula whether it takes the value one is NP-complete problem.     

Proper n-valued Łukasiewicz algebras as S-algebras of Łukasiewicz n-valued prepositional calculi

Proper n-valued ukasiewicz algebras are obtained by adding some binary operators, fulfilling some simple equations, to the fundamental operations of n-valued ukasiewicz algebras. They are the s-algebras corresponding to an axiomatization of ukasiewicz n-valued propositional calculus that is an extention of the intuitionistic calculus.Dedicated to the memory of Gregorius C. Moisil     

Tableaux for Łukasiewicz Infinite-valued Logic

In this work we propose a labelled tableau method for ukasiewicz infinite-valued logic L . The method is based on the Kripke semantics of this logic developed by Urquhart [25] and Scott [24]. On the one hand, our method falls under the general paradigm of labelled deduction [8] and it is rather close to the tableau systems for sub-structural logics proposed in [4]. On the other hand, it provides a CoNP decision procedure for L validity by reducing the check of branch closure to linear programming     

Learning Łukasiewicz logic

The integration between connectionist learning and logic-based reasoning is a longstanding foundational question in artificial intelligence, cognitive systems, and computer science in general. Research into neural-symbolic integration aims to tackle this challenge, developing approaches bridging the gap between sub-symbolic and symbolic representation and computation. In this line of work the core method has been suggested as a way of translating logic programs into a multilayer perceptron computing least models of the programs. In particular, a variant of the core method for three valued Łukasiewicz logic has proven to be applicable to cognitive modelling among others in the context of Byrne’s suppression task. Building on the underlying formal results and the corresponding computational framework, the present article provides a modified core method suitable for the supervised learning of Łukasiewicz logic (and of a closely-related variant thereof), implements and executes the corresponding supervised learning with the backpropagation algorithm and, finally, constructs a rule extraction method in order to close the neural-symbolic cycle. The resulting system is then evaluated in several empirical test cases, and recommendations for future developments are derived.     

A finite model theorem for the propositional μ-calculus

We prove a finite model theorem and infinitary completeness result for the propositional -calculus. The construction establishes a link between finite model theorems for propositional program logics and the theory of well-quasi-orders.Supported by NSF grant DCR-8602663     

Does the deduction theorem fail for modal logic?

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified in a way that restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction- and cut-free sequent calculus equivalent to the Hilbert system for basic modal logic shows the standard failure argument untenable by proving the underivability of ${\square\,A}$ from A.     

Forcing in Łukasiewicz Predicate Logic

In this paper we study the notion of forcing for Łukasiewicz predicate logic (Ł∀, for short), along the lines of Robinson’s forcing in classical model theory. We deal with both finite and infinite forcing. As regard to the former we prove a Generic Model Theorem for Ł∀, while for the latter, we study the generic and existentially complete standard models of Ł∀. Presented by Jacek Malinowski     

Łukasiewicz and Popper on Induction

We compare Jan ?ukasiewicz's and Karl Popper's views on induction. The English translation of the two ?ukasiewicz's papers is included in the Appendix.     

A non-commutative generalization of Łukasiewicz rings

The goal of the present article is to extend the study of commutative rings whose ideals form an MV-algebra as carried out by Belluce and Di Nola [1] to non-commutative rings. We study and characterize all rings whose ideals form a pseudo MV-algebra, which shall be called here generalized Łukasiewicz rings. We obtain that these are (up to isomorphism) exactly the direct sums of unitary special primary rings.     

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).     

Averaging the truth-value in Łukasiewicz logic

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.     

A Generalization of the Łukasiewicz Algebras

We introduce the variety _n ^m , m 1 and n 2, of m-generalized ukasiewicz algebras of order n and characterize its subdirectly irreducible algebras. The variety _n ^m is semisimple, locally finite and has equationally definable principal congruences. Furthermore, the variety _n ^m contains the variety of ukasiewicz algebras of order n.     

Construction of monadic three-valued Łukasiewicz algebras

The notion of monadic three-valued ukasiewicz algebras was introduced by L. Monteiro ([12], [14]) as a generalization of monadic Boolean algebras. A. Monteiro ([9], [10]) and later L. Monteiro and L. Gonzalez Coppola [17] obtained a method for the construction of a three-valued ukasiewicz algebra from a monadic Boolea algebra. In this note we give the construction of a monadic three-valued ukasiewicz algebra from a Boolean algebra B where we have defined two quantification operations and ^* such that ^*x=^*x (where ^*x=-^*-x). In this case we shall say that and ^* commutes. If B is finite and is an existential quantifier over B, we shall show how to obtain all the existential quantifiers ^* which commute with .Taking into account R. Mayet [3] we also construct a monadic three-valued ukasiewicz algebra from a monadic Boolean algebra B and a monadic ideal I of B. The most essential results of the present paper will be submitted to the XXXIX Annual Meeting of the Unión Matemática Argentina (October 1989, Rosario, Argentina).     

Dual Equivalent Two-valued Under-determined and Over-determined Interpretations for Łukasiewicz’s 3-valued Logic Ł3

?ukasiewicz three-valued logic ?3 is often understood as the set of all 3-valued valid formulas according to ?ukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of ?3: “well-determined” ?3a and “truth-preserving” ?3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide (by using Dunn semantics) dual equivalent two-valued under-determined and over-determined interpretations for ?3, ?3a and ?3b. The logic ?3 is axiomatized as an extension of Routley and Meyer’s basic positive logic following Brady’s strategy for axiomatizing many-valued logics by employing two-valued under-determined or over-determined interpretations. Finally, it is proved that “well determined” ?ukasiewicz logics are paraconsistent.     

An axiomatization of the finite-valued Łukasiewicz calculus

In this paper the completeness theorems for the finite-valued ukasiewicz logics are proved with the use of the Lindenbaum algebra.The research was sponsored by the grant C.P.B.P. 08-15.I wish to thank Dr hab. Piotr Wojtylak for ideas and suggestions which enabled me to write this paper.     

A Generalization of Monadic n-Valued Łukasiewicz Algebras

Studia Logica - $${{\mathcal {M}} L}^{m}_n$$ of monadic m-generalized ?ukasiewicz algebras of order n (or $$M L^{m}_n$$ -algebras), namely a generalization of monadic n-valued ?ukasiewicz...     

Łukasiewicz logic and the foundations of measurement

The logic of inexactness, presented in this paper, is a version of the Łukasiewicz logic with predicates valued in [0, ∞). We axiomatize multi-valued models of equality and ordering in this logic guaranteeing their imbeddibility in the real line. Our axioms of equality and ordering, when interpreted as axioms of proximity and dominance, can be applied to the foundations of measurement (especially in the social sciences). In two-valued logic they provide theories of ratio scale measurement. In multivalued logic they enable us to treat formally errors arising in nominal and ordinal measurements.