Formalization of functionally complete propositional calculus with the functor of implication as the only primitive term

The most difficult problem that Leniewski came across in constructing his system of the foundations of mathematics was the problem of defining definitions, as he used to put it. He solved it to his satisfaction only when he had completed the formalization of his protothetic and ontology. By formalization of a deductive system one ought to understand in this context the statement, as precise and unambiguous as possible, of the conditions an expression has to satisfy if it is added to the system as a new thesis. Now, some protothetical theses, and some ontological ones, included in the respective systems, happen to be definitions. In the present essay I employ Leniewski's method of terminological explanations for the purpose of formalizing ukasiewicz's system of implicational calculus of propositions, which system, without having recourse to quantification, I first extended some time ago into a functionally complete system. This I achieved by allowing for a rule of implicational definitions, which enabled me to define any propositionforming functor for any finite number of propositional arguments.To the memory of Jerzy Supecki     

Categorical and algebraic aspects of Martin-Löf Type Theory

In the paper there are introduced and discussed the concepts of an indexed category with quantifications and a higher level indexed category to present an algebraic characterization of some version of Martin-Löf Type Theory. This characterization is given by specifying an additional equational structure of those indexed categories which are models of Martin-Löf Type Theory. One can consider the presented characterization as an essentially algebraic theory of categorical models of Martin-Löf Type Theory. The paper contains a construction of an indexed category with quantifications from terms and types of the language of Martin-Löf Type Theory given in the manner of Troelstra [11]. The paper contains also an inductive definition of a valuation of these terms and types in an indexed category with quantifications.     

Repeated failure: a source of helplessness or a factor irrelevant to its emergence?

We tested the validity of the egotism model of human helplessness. In contrast to the original theoretical approach of Seligman and his associates, which points to response-outcome noncontingency as the main source of helplessness, the egotism alternative proposes that repeated failure itself is the critical determinant of helplessness symptoms. Repeated failure threatens the self-esteem of the subject, who supposedly engages in a least-effort strategy during the test phase of a typical learned helplessness study, which results in performance impairment. To examine the egotism explanation, we gave subjects noncontingent-feedback training with or without repeated failure on five consecutive discrimination problems. In two experiments, noncontingent-feedback preexposure produced helplessness deficits in performance on avoidance learning, whereas repeated failure appeared irrelevant to helplessness. This and our other findings from research are inconsistent with the egotism explanation and support instead Seligman's original proposal, in which helplessness is attributed to prolonged experience with noncontingency.     

On a problem of P(α, δ, π) concerning generalized Alexandroff s cube

Universality of generalized Alexandroff's cube plays essential role in theory of absolute retracts for the category of , -closure spaces. Alexandroff's cube. is an , -closure space generated by the family of all complete filters. in a lattice of all subsets of a set of power .Condition P(, , ) says that is a closure space of all , -filters in the lattice ( ), .Assuming that P (, , ) holds, in the paper [2], there are given sufficient conditions saying when an , -closure space is an absolute retract for the category of , -closure spaces (see Theorems 2.1 and 3.4 in [2]).It seems that, under assumption that P (, , ) holds, it will be possible to givean uniform characterization of absolute retracts for the category of , -closure-spaces.Except Lemma 3.1 from [1], there is no information when the condition P (, , ) holds or when it does not hold.The main result of this paper says, that there are examples of cardinal numbers, , , such that P (, , ) is not satisfied.Namely it is proved, using elementary properties of Lebesgue measure on the real line, that the condition P (, ₁, 2 ) is not satisfied.Moreover it is shown that fulfillment of the condition is essential assumption in, Theorems 2.1 and 3.4 from [1] i.e. it cannot be eliminated.     

On decidable consequence operators

The main theorem says that a consequence operator is an effective part of the consequence operator for the classical prepositional calculus iff it is a consequence operator for a logic satisfying the compactness theorem, and in which every finitely axiomatizable theory is decidable.     

The mutating preconscious archetype in present-day ecological conditions

Competition is an important force behind evolution in present-day ecological conditions, its intensity varying according to the organisms' expectations vis-à-vis resources available. On the other hand, the role of mutation in the evolutionary process can hardly be underestimated: Leading to the change of the preconsciously functioning archetype, mutation makes it impossible for the ego to realize its image. This is a consequence of the clash between the mutated, preconsciously functioning archetype and the system of ethical and moral norms functioning in the collective superego of a given population group. The type of mutation that results from the accelerated pace of evolution can be viewed as a continuity in the development of an organism's behavior. The preconsciously functioning archetype, resulting from these mutations, contains impulses and predispositions that differ markedly from the impulses and predispositions functioning in the genetically determined "pattern of behavior." In a situation where anxiety results from the inhibition of impulses and dispositions, hostility reactions are likely to occur. In its turn, reactive hostility can induce anxiety, thus creating a reaction cycle. Underlying this mechanism is the mutated, preconsciously functioning archetype, which itself is a result of the development of civilization in our contemporary world.     

When cognitive exertion does not yield cognitive gain: toward an informational explanation of learned helplessness

This study tested a new information-processing explanation of learned helplessness that proposes that an uncontrollable situation produces helplessness symptoms because it is a source of inconsistent, self-contradictory task information during problem-solving attempts. The flow of such information makes hypothesis-testing activity futile. Prolonged and inefficient activity of this kind leads in turn to the emergence of a state of cognitive exhaustion, with accompanying performance deficits. In 3 experiments, Ss underwent informational helplessness training (IHT): They were sequentially exposed to inconsistent task information during discrimination problems. As predicted, IHT was associated with subjective symptoms of irreducible uncertainty and resulted in (a) performance deterioration on subsequent avoidance learning, (b) heightened negative mood, and (c) subjective symptoms of cognitive exhaustion.