HYPER-REF: A General Model of Reference for First-Order Logic and First-Order Arithmetic

In this article I present HYPER-REF, a model to determine the referent of any given expression in First-Order Logic (FOL). I also explain how this model can be used to determine the referent of a first-order theory such as First-Order Arithmetic (FOA). By reference or referent I mean the non-empty set of objects that the syntactical terms of a well-formed formula (wff) pick out given a particular interpretation of the language. To do so, I will first draw on previous work to make explicit the notion of reference and its hyperintensional features. Then I present HYPER-REF and offer a heuristic method for determining the reference of any formula. Then I discuss some of the benefits and most salient features of HYPER-REF, including some remarks on the nature of self-reference in formal languages.

Why Can the Brain (and Not a Computer) Make Sense of the Liar Paradox?

Ordinary computing machines prohibit self-reference because it leads to logical inconsistencies and undecidability. In contrast, the human mind can understand self-referential statements without necessitating physically impossible brain states. Why can the brain make sense of self-reference? Here, we address this question by defining the Strange Loop Model, which features causal feedback between two brain modules, and circumvents the paradoxes of self-reference and negation by unfolding the inconsistency in time. We also argue that the metastable dynamics of the brain inhibit and terminate unhalting inferences. Finally, we show that the representation of logical inconsistencies in the Strange Loop Model leads to causal incongruence between brain subsystems in Integrated Information Theory.

CONTINUOUS PROFESSIONAL DEVELOPMENT AS A COMPONENT OF TEACHER'S SELF-REFERENCE

The article deals with the new opportunities for teachers’ creative self-realization and self-improvement in the context of educational reform with the application of the Framework for Continuous Professional Development of Teachers. The relevance of this topic is caused by the search for mechanisms of self-reflection, self-actualization, and self-development of the teacher. This is encouraged by the Teacher Framework for Continuing Professional Development, which is a guideline for creating innovative models of professional development of teachers. The purpose of the article is to reveal new opportunities for teachers’ creative self-realization and self-improvement in terms of educational reform with the application of the Framework for Continuous Professional Development of Teachers. We have used the analysis of scientific developments and the sources on the problem of continuous professional development of a teacher, psychological and pedagogical, educational and methodological literature, information sources on the problem of research. As a result of the study we have found out that supporting the continuous professional development of teachers using the Framework for Continuing Professional Development of Teachers is an effective tool for reflective analysis and reconstruction of their professional path, and therefore − a necessary condition for professional growth. So, an important component of the roadmap of professional development of the teacher is awareness of the teachers of his\her professional role as a facilitator of learning. A motivated teacher, who uses the Framework for Continuing Professional Development, can anticipate and design the trajectory of their growth, success in teaching, opportunities and obstacles to achieving goals The analysis of scientific literature, information sources on the research problem shows that self-reference acquires special significance for teachers of all levels from a novice teacher who has just chosen this profession to a creative professional.