Equationally definable implication algebras for orthomodular lattices

The fact that it is possible to define three different material conditionals in orthomodular lattices suggests that there exist three different orthomodular logics whose conditionals are material conditionals and whose models are orthomodular lattices. The purpose of this paper is to provide equationally definable implication algebras for each of these material conditionals.     

On structural completeness of implicational logics

We consider the notion of structural completeness with respect to arbitrary (finitary and/or infinitary) inferential rules. Our main task is to characterize structurally complete intermediate logics. We prove that the structurally complete extension of any pure implicational in termediate logic C can be given as an extension of C with a certain family of schematically denned infinitary rules; the same rules are used for each C. The cardinality of the family is continuum and, in the case of (the pure implicational fragment of) intuitionistic logic, the family cannot be reduced to a countable one. It means that the structurally complete extension of the intuitionistic logic is not countably axiomatizable by schematic rules.This work was supported by the Polish Academy of Sciences, CPBP 08.15, Struktura logiczna rozumowa niesformalizowanych.     

Deduction theorems for weak implicational logics

The standard deduction theorem or introduction rule for implication, for classical logic is also valid for intuitionistic logic, but just as with predicate logic, other rules of inference have to be restricted if the theorem is to hold for weaker implicational logics.In this paper we look in detail at special cases of the Gentzen rule for and show that various subsets of these in effect constitute deduction theorems determining all the theorems of many well known as well as not well known implicational logics. In particular systems of rules are given which are equivalent to the relevance logics E,R, T, P-W and P-W-I.     

The shortest possible length of the longest implicational axiom

A four-valued matrix is presented which validates all theorems of the implicational fragment, IF, of the classical sentential calculus in which at most two distinct sentence letters occur. The Wajsberg/Diamond-McKinsley Theorem for IF follows as a corollary: every complete set of axioms (with substitution and detachment as rules) must include at least one containing occurrences of three or more distinct sentence letters.Additionally, the matrix validates all IF theses built from nine or fewer occurrences of connectives and letters. So the classic result of Jaskovski for the full sentential calculus —that every complete axiom set must contain either two axioms of length at least nine or else one of length at least eleven—can be improved in the implicational case: every complete axiom set for IF must contain at least one axiom eleven or more characters long.Both results are best possible, and both apply as well to most subsystems of IF, e.g., the implicational fragments of the standard relevance logics, modal logics, the relatives of implicational intutionism, and logics in the ukasiewicz family.Earlier proofs of these results, utilizing a five-valued matrix built from the product matrix of C2 with itself via the method of [8], were obtained in 1988 while the author was a Visiting Research Fellow at the Automated Reasoning Project, Research School of Social Sciences, Australian National University, and were presented in [9]. The author owes thanks to the RSSS for creating the Project, and to the members of the Project generally for the stimulating atmosphere they created in turn, but especially to Robert K. Meyer for making the visit possible, and for many discussions over the years.     

On the autological character of diagonalizable algebras

Summary Let be the first order theory of diagonalizable algebras. We define a bijection from the atomic formulas of (identities) to the open formulas of . is an algebraic analogous of ≒. We prove that , ^-1 preserve the validity.The algebraization of the theories which express TheorSee the precedent papers with the same subtitle.Allatum est die 21 Julii 1975     

On the lattice of quasivarieties of Sugihara algebras

Let S denote the variety of Sugihara algebras. We prove that the lattice (K) of subquasivarieties of a given quasivariety K S is finite if and only if K is generated by a finite set of finite algebras. This settles a conjecture by Tokarz [6]. We also show that the lattice (S) is not modular.     

Wajsberg algebras and post algebras

We give a presentation of Post algebras of ordern+1 (n1) asn+1 bounded Wajsberg algebras with an additional constant, and we show that a Wajsberg algebra admits a P-algebra reduct if and only if it isn+1 bounded.This autor is partially supported by Grant PB90-0465-C02-01 of D.G.I.C.Y.T. of SpainPresented byJan Zygmunt     

On yawning and its functions

The forms and behavioral correlates of yawning are described, and the phylogenetic and ontogenetic aspects of the act are examined with particular attention to its possible functions. Much evidence supports the view that yawning is an important mediator of behavioral arousal levels, a view that is further strengthened by a review of endocrine, neurotransmitter, and pharmacological mechanisms of yawning: A major function of yawning appears to involve maintenance or increase of arousal when environments provide relatively little stimulation.     

On minima of discrimination functions

A discrimination function ψ(x,y) assigns a measure of discriminability to stimulus pairs x,y (e.g., the probability with which they are judged to be different in a same-different judgment scheme). If for every x there is a single y least discriminable from x, then this y is called the point of subjective equality (PSE) for x, and the dependence h(x) of the PSE for x on x is called a PSE function. The PSE function g(y) is defined in a symmetrically opposite way. If the graphs of the two PSE functions coincide (i.e., g≡h⁻¹), the function is said to satisfy the Regular Minimality law. The minimum level functions are restrictions of ψ to the graphs of the PSE functions. The conjunction of two characteristics of ψ, (1) whether it complies with Regular Minimality, and (2) whether the minimum level functions are constant, has consequences for possible models of perceptual discrimination. By a series of simple theorems and counterexamples, we establish set-theoretic, topological, and analytic properties of ψ which allow one to relate to each other these two characteristics of ψ.