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.     

Characterization of prime numbers in Łukasiewicz's logical matrix

In this paper we define n+1-valued matrix logic K_n+1 whose class of tautologies is non-empty iff n is a prime number. This result amounts to a new definition of a prime number. We prove that if n is prime, then the functional properties of K_n+1 are the same as those of ukasiewicz's n +1-valued matrix logic _n+1. In an indirect way, the proof we provide reflects the complexity of the distribution of prime numbers in the natural series. Further, we introduce a generalization K _n+1 ^* of K_n+1 such that the set of tautologies of K_n+1 is not empty iff n is of the form p , where p is prime and is natural. Also in this case we prove the equivalence of functional properties of the introduced logic and those of _n+1. In the concluding part, we discuss briefly a partition of the natural series into equivalence classes such that each class contains exactly one prime number. We conjecture that for each prime number the corresponding equivalence class is finite.To the memory of Jerzy Supecki     

Bounded contraction and Gentzen-style formulation of Łukasiewicz logics

In this paper, we consider multiplicative-additive fragments of affine propositional classical linear logic extended with n-contraction. To be specific, n-contraction (n 2) is a version of the contraction rule where (n+ 1) occurrences of a formula may be contracted to n occurrences. We show that expansions of the linear models for (n + 1)- valued ukasiewicz logic are models for the multiplicative-additive classical linear logic, its affine version and their extensions with n-contraction. We prove the finite axiomatizability for the classes of finite models, as well as for the class of infinite linear models based on the set of rational numbers in the interval [0, 1]. The axiomatizations obtained in a Gentzen-style formulation are equivalent to finite and infinite-valued ukasiewicz logics.Presented by Jan Zygmunt     

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.     

What is the upper part of the lattice of bimodal logics?

We define an embedding from the lattice of extensions ofT into the lattice of extensions of the bimodal logic with two monomodal operators ₁ and ₂, whose ₂-fragment isS5 and ₁-fragment is the logic of a two-element chain. This embedding reflects the fmp, decidability, completenes and compactness. It follows that the lattice of extension of a bimodal logic can be rather complicated even if the monomodal fragments of the logic belong to the upper part of the lattice of monomodal logics.Presented byWolfgang Rautenberg     

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.     

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.     

The role of attack and defense semantics in skilled players’ memory for chess positions

There is much evidence that chess skill is based on chunks in memory that represent parts of positions from previously encountered games. However, the content of these chunks is a matter for debate. According to one view, (1) the closer two pieces are to each other on a board (proximity), the more likely they are to be in the same chunk, and (2) skilled players encode the precise locations of pieces. An alternative view is that what information is encoded in a chess chunk is determined more by processing of the attack/defense relations during evaluation. In three experiments, participants evaluated positions and completed recognition tests. Experiment 1 supported the view that expert players make more use of attack/defense relations than of locations of pieces in a recognition test. Experiments 2 and 3 demonstrated that, for both long and short presentation times, expert players' recognition for a piece within a position was primed more by a piece related by attack or defense than by a piece merely proximal. These findings challenge theories of expertise for chess that assume a primary role for proximity and location in determining which pieces are grouped together in memory.     

The semantics of personality in Russian “subjective sociology”

The article is devoted to the “subjective method” and the role of value preferences, as underscored in Russian proto-sociology, developed by the populists in discussions with the “ethical Marxism,” on the one hand, and with positivists, on the other. The main issue—how was the apologia of individual in these studies connected to the ideals of social development?—leads to the question, whether such ideals could be based on an inborn moral law or “universal good” in the spirit of empirical-positivist theories. Is the use of ethical categories in scientific knowledge admissible in principle, and if so, to what degree? How should the measure of subjectivity in historical or sociological studies be “regulated,” and are there universally binding solutions to these problems?