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     

Decision problems for classes of diagonalizable algebras

We make use of a Theorem of Burris-McKenzie to prove that the only decidable variety of diagonalizable algebras is that defined by 0=1. Any variety containing an algebra in which 01 is hereditarily undecidable. Moreover, any variety of intuitionistic diagonalizable algebras is undecidable.     

The undecidability of the first-order theory of diagonalizable algebras

The undecidability of the first-order theory of diagonalizable algebras is shown here.     

Interpretations of the first-order theory of diagonalizable algebras in peano arithmetic

For every sequence |p _n } _n of formulas of Peano ArithmeticPA with, every formulaA of the first-order theory diagonalizable algebras, we associate a formula ⁰ A, called the value ofA inPA with respect to the interpretation. We show that, ifA is true in every diagonalizable algebra, then, for every, ⁰ A is a theorem ofPA.     

An effective fixed-point theorem in intuitionistic diagonalizable algebras

Summary Within the technical frame supplied by the algebraic variety of diagonalizable algebras, defined by R. Magari in [2], we prove the following:LetT be any first-order theory with a predicate Pr satisfying the canonical derivability conditions, including Löb's property. Then any formula inT built up from the propositional variables q, p₁, ..., p_n, using logical connectives and the predicate Pr, has the same fixed-points relative to q (that is, formulas (p₁ ..., p_n) for which for all p₁, ..., p_n T((p₁, ..., p_n), p₁, ..., p_n) (p₁, ..., p_n)) of a formula ^* of the same kind, obtained from in an effective way.Moreover, such ^* is provably equivalent to the formula obtained from substituting with ^* itself all the occurrences of q which are under Pr. In the particular case where q is always under Pr in , ^* is the unique (up to provable equivalence) fixedpoint of .Since this result is proved only assumingPr to be canonical, it can be deduced that Löb's property is, in a sense, equivalent to Gödel's diagonalization lemma.All the results are proved more generally in the intuitionistic case.The algebraization of the theories which express Theor, IXAllatum est die 19 Decembris 1975     

The uniqueness of the fixed-point in every diagonalizable algebra

Summary It is well known that, in Peano arithmetic, there exists a formulaTheor (x) which numerates the set of theorems. By Gödel's and Löb's results, we have that Therefore, the considered «equations» admit, up to provable equivalence, only one solution.In this paper we prove (Corollary 1), that, in general, ifP (x) is an arbitrary formula built fromTheor (x), then the fixed-point ofP (x) (which exists by the diagonalization lemma) is unique up to provable equivalence. This result is settled referring to the concept of diagonalizable algebra (see Introduction).The algebraization of the theories which express TheorAllatum est die 3 Augustii 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.     

The logic of algebraic rules as a generalization of equational logic

In this paper we start an investigation of a logic called the logic of algebraic rules. The relation of derivability of this logic is defined on universal closures of special disjunctions of equations extending the relation of derivability of the usual equational logic. The paper contains some simple theorems and examples given in justification for the introduction of our logic. A number of open questions is posed.     

On the perception of a class of bilaterally symmetric forms

Two experiments were performed to determine the manner in which people perceive a class of bilaterally symmetric forms. In the first experiment, paired, comparison dissimilarity judgments and unidimensional single stimulus ratings were collected from the Ss. The second experiment involved the collecting of semantic differential and paired comparison dissimilarity judgments from two groups of Ss. The judgments and ratings from both experiments were compared to each other and to a set of objective measures of the forms by a series of multivariate statistical techniques. It was found that three primary attributes, orientation, convexity-concavity, and curvature, accounted for the perceptual reports, but the manner in which the attributes were used differed for different response procedures and for the same response on different occasions.