Wojciech Krysztofiak

The presentation of the formal conception of noemata is the main aim of the article. In the first section, three informal approaches to noemata are discussed. The goal of this chapter is specifying main controversies and their sources concerned with different ways of the understanding of noemata. In the second section, basic assumptions determining the proposed way of understanding noemata are presented. The third section is devoted to the formal set-theoretic construction needed for the formal comprehension of noemata. In the fourth section, definitions of noemata and their various kinds, as well as definitions of other phenomenological notions are formulated. In the last section, possibilities of further developing the proposed formal conception are indicated.

The paper presents the construction of a new class of logics, which are called logics of derivational trees. The presentation comprises three sections: (i) intuitive psychological motivations for these logics stemming from some results of developmental psycho-linguistics (Piaget, Chomsky), (ii) the construction of formal calculus with help of algorithmic tools, and (iii) the construction of set-theoretic semantic model for our logic. They determine criteria of valid deriving and transforming structures which are usually described in the literature as derivational trees. These structures are used in linguistics or in computational sciences as tools of modelling deep sentential structures or information-bases. Furthermore cognitive anthropologists notice that most of our ordinary taxonomies arranging the experienced world in our Lebenswelt possess various tree-structures. It seems that our abilities of applying tree-structures (without explicit knowledge concerned with algebraic mechanisms of construing tree-structures) in various segments of our life are mental and behavioral manifestation of some special logical disposal belonging to the machinery of logical competence in general. It is interesting to put the hypothesis according to which the competence of construing and applying tree-structures is even more primitive than the competence of applying logical rules of natural deduction. The presented calculus possesses some peculiar feature, namely its formal language is composed of expressions of three syntactic levels. In standard, formal languages all expressions may be divided as belonging to two levels: the level of formulas and the level of constituents of formulas. In the language of derivational trees there are distinguished the following levels: (i) the level of lexical expressions, (ii) the level of derivation-expressions, (iii) and finally the level of transformation-expressions. These last category fulfils the role of formulas. Proofs are appropriate sequences of derivation-expressions. However what is proved is not a derivation-expression but it is a transformation-expression. The peculiarity of our logic consists in that an expression which is proved, does not belong to the category of constituents of proofs

The main aim of the paper is paraphrasing Heidegger's category of being in the theoretic framework of Fregean phenomenological semantics. The choice of Fregean phenomenological semantics as the tool of the paraphrase is justified by the fact that philosophy articulated in Sein und Zeit may be interpreted as the modification of Husserl's project of phenomenology which is treated, in turn, as generalisation of Frege's theory of sense and nominatum. So in the paper it is defended that Heidegger's category of being stems from Frege's considerations devoted to basic semantic questions. In the first chapter there are specified in existentialistic language ten principles clarifying the notion of being. In the second part the principles are paraphrased in the phenomenological theory of noema (this category is understood in spirit of the so called Californian interpretation). This move allows us to show how the conception of being is the modification of Frege's and Husserl's semantics. In the last chapters there is reconstructed Heidegger's model of acts of reference. The formal tool of the reconstruction is Leśniewski's formal language (enriched by Ajdukiewicz) with indexes designating ways of existence and referential backgrounds

In the paper there is presented the semantic interpretation of idealism/ realism controversy which is one of the most essential issues in Ingarden’s phenomenological project of ontology. The procedure of semantic paraphrase which is contemporary developed by Wolen´ ski, is the main interpretative tool. In the central part of the paper, there is formulated the formal theory of the semantic framework underlying idealism/realism discourse. Finally, there are formulated some notes showing that intentional conception of negation may be used for defending various idealistic positions.

In the paper there are presented main assumptions underlying the construction of theoretic models of mental processes of numeral reference in mathematical practice which comprises such abilities as counting, solving story-tasks, estimating cardinalities and comparing magnitudes. Numerals are understood as any expressions which enable mind to refer to numbers, cardinalities and magnitudes. The main research question formulated in the article sounds: What cognitive processes do there occur in the mind during execution of various numeral acts of reference?

In this paper, representational structures of arithmetical thinking, encoded in human minds, are described. On the basis of empirical research, it is possible to distinguish four types of mental number lines: the shortest mental number line, summation mental number lines, point-place mental number lines and mental lines of exact numbers. These structures may be treated as generative mechanisms of forming arithmetical representations underlying our numerical acts of reference towards cardinalities, ordinals and magnitudes. In the paper, the theoretical framework for a formal model of mental arithmetical representations is constructed. Many competitive conceptions of the mental system responsible for our arithmetical thinking may be unified within the presented framework. The paradigm underlying our research may be interpreted philosophically as a neo-Kantian approach to modeling the mind’s representational structures.

The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the mind is based on the linear number representation. This classical conception is rejected and a competitive hypothesis is formulated according to which the basic mature representational system of cognitive arithmetic is a structure composed of many numerical axes which possess a common constituent, namely, the numeral zero. Arithmetic of indexed numbers is just a formal tool for modelling the basic mature arithmetic competence. The third task is to develop a standpoint called temporal pluralism, which is motivated by neo-Kantian philosophy of arithmetic