Adam Olszewski

After a brief reminder of the, so called, Linda problem and its solution by Kahneman & Tversky (KT) (the tame solution), I point out the implications of the solution adopted by the KTs. Among these implications, I emphasize the importance of the relation of probability between the sentences: ‘Linda is active in a feminist movement’ (F) and ‘Linda is a bank teller and active in a feminist movement’ (T∧F); while in KT’s paper the main emphasis was put on considering the relationship between the probability of sentences: ‘Linda is a bank teller’ (T) and ‘Linda is a bank teller and active in a feminist movement’ (T∧F). I offer a critical argument against the zero hypothesis H0 that ‘at least 85% of the respondents will choose the sentence F as more likely than the sentence (T∧F), and the opposite consequently will be selected at most by 15% of the respondents;’ being drawn from the assumptions made by Kahneman and Tversky. This hypothesis will be further partially refuted by means of results from the surveys N0.1. and N0.2. Then the reasoning supporting the result of surveys is presented and finally critical conclusions will be derived.

The present paper is devoted to a discussion of the role of Church’s thesis in setting limits to the cognitive possibilities of mathematics. The specific aim is to analyse the formalized theory of arithmetic as a fundamental mathematical structure related to the theory of computation. By introducing notional non-standard computational abilities into this theory, a non-trivial enlargement of the set of theorems is obtained. The paper also indicates the connection between the inclusion of new functions through the development of axioms and the potential modification of inference rules. In addition, the paper provides an explanation of the role of inclusion of a certain interpretation of the meaning of the axioms of the theory in that theory.

This paper presents a logical concept of probability which seems to be obvious, as it is, but the author is not aware of any elaboration of a developed studies on the issue or of any special philosophical application of it. Such a probability of the formula A, of the language of the propositional logic, is the quotient of the number of Boolean valuations of formula A of the classical propositional calculus, which takes the logical value 1, to the number of all Boolean valuations of such a formula A. An application of this concept of logical probability to the solution of the Linda problem is given.

This paper marks the first extensive attempt to articulate the view I have termed solideism. I begin by exploring various interpretations of solipsism, as solideism represents a particular version of it, and establish a schematic formulation for solipsism. Following a brief overview of absolute idealism, I reconstruct F. C. S. Schiller’s argument that absolute idealism inevitably leads to solipsism. Building on Schiller’s reasoning, along with my own argument against solipsism (developed in a separate paper), I derive an absolute form of solipsism – solideism. Lastly, I offer observations on the order structure of the set of all solipsisms.

Church's Thesis (CT) was first published by Alonzo Church in 1935. CT is a proposition that identifies two notions: an intuitive notion of an effectively computable function defined in natural numbers with the notion of a recursive function. Despite the many efforts of prominent scientists, Church's Thesis has never been disproven. There exists a vast literature concerning the thesis. The aim of this book is to provide a one volume summary of the state of research on Church's Thesis. These include the following: different formulations of CT, CT and intuitionism, CT and intensional mathematics, CT and physics, the epistemic status of CT, CT and philosophy of mind, provability of CT and CT and functional programming. Adam Olszewski, is assistant professor of the philosophy of mathematics and mathematical logic at the Department of Philosophy of Pontifical Academy of Theology, Cracow (Poland). Jan Wolenski is professor of philosophy, Institute of Philosphy, Jagiellonian University, Cracow (Poland). He is one of the most distinguished logicians in Poland. Robert Janusz is assistant professor in the Department of Philosophy of Ignatius College, Cracow (Poland).

The main purpose of this paper, which takes the form of an essay, is an attempt to answer the question of the limits of artificial intelligence (AI). In the introductory section, we present the key milestones in AI development, both historical and future projections, in which two terms – Artificial Human (AH) and Artificial ‘god’ (AG) – play a special role. In the second section, we clarify the question of the limits of AI by indicating the hypothetical goal of AI development. The third section develops the argument proposed by C. F. Weizsäcker, originally formulated for cybernetics. The conclusion of this argument is optimistic about limitations to the possibilities of cybernetic simulations. We apply this argument to AI and subject it to a critique which ultimately undermines the legitimacy of its conclusion. We base the critique on two well-known results: the theorem of the unsolvability of the halting problem and Gödel’s first incompleteness theorem, and we formulate two objections interpreted without adopting Church’s thesis. In the crucial fourth section, we present a third objection in the form of a hypothesis for which we argue that AI (AH), understood as a subject, will always be solipsistic.

In the 1950s Peter Strawson analyzed the works of Bertrand Russell regarding fundamental definitions of meaning, sentences and truth value. The debate between them uncovered many issues that Fregean, truth-functional logics have when defining concepts from natural language. To reconcile the Fregean paradigm with the reality of language use, Strawson proposed the concept of presuppositions  necessary preconditions for the truth of other sentences. We believe that his proposition stemmed primarily from the problem caused by the fact that Fregean, truth-functional logics are not sensitive to the contents of sentences and reduce them to their logical values. This is bound to produce a mismatch between the way logic models reasoning and the way language users reason since real-life reasoning is performed on the contents of sentences and not their logical values. Inspired by the ideas of Strawson and Roman Suszko, who initiated the paradigm of non-Fregean logics, we propose a new solution to the debate between Strawson and Russell. In our solution, the content implication connective is used to express content relations between sentences. We move away from truth and falsehood as the sole two semantic correlates of sentences and instead work in a system where the contents of sentences are their semantic correlates.

The starting point of this paper is the empirically determined ability to reason in natural language by employing probable sentences. A sentence is understood to be logically probable if its schema, expressed as a formula in the language of classical propositional calculus, takes the logical value of truth for the majority of Boolean valuations, i.e., as a logically probable formula. Then, the formal system P is developed to encode the set of these logically probable formulas. Based on natural semantics, a strong completeness theorem for P is proved. Alternative notions of consequence for logically probable sentences are also considered.

The aim of the paper is twofold. First, we want to recapture the genesis of the logics of order. The origin of this notion is traced back to the work of Jerzy Kotas, Roman Suszko, Richard Routley and Robert K. Meyer. A further development of the theory of logics of order is presented in the papers of Jacek K. Kabziński. Quite contemporarily, this notion gained in significance in the papers of Carles Noguera and Petr Cintula. Logics of order are named there _logics of weak implications_. They play a crucial role in their monograph (Noguera and Cintula _Logic and Implication. An Introduction to the General Algebraic Study of Non-Classical Logics_, Trends in Logic 57, Springer, Berlin, 2021). But, more importantly, the other goal is to define some subclasses of the logics of order in reference to later results of Jacek K. Kabziński and Michael Dunn. The original conception of implication is due to Kabziński. Implication is a stronger notion than the notion of the connective of order aka weak implication. As a result, the three subclasses of logics of order are isolated: logics of implication, logics of symmetry, and tonoidal logics. These notions are uniformly defined and investigated from various viewpoints in terms of consequence operations. The emphasis is put on their semantics.

The title question of the paper has its empirical origin in the form of an individual’s existential experience arising from the personal use of a computer, which we attempt to describe in the first section. The rest of the entire paper can be understood as a philosophical essay answering the question posed. First the connection between the main problem of the article and its “premonition” by mankind, which was expressed in the form of ancient myths and legends, is briefly suggested. After shortly discussing the problems that early considerations of AI focused on, i.e. whether machines can think at all, we move on to reformulate our title question, about the possibility of outsmarting AI. This outsmarting will be understood by us in a rather limited way as to prevent a machine from completing its implemented task. To achieve this objective, after softly clarifying the basic terms, an analogy is built between the “outsmarting” of a machine by a human (the target domain) and the playing of a mathematical game between two players (the base domain), where this outsmarting is assigned a “winning strategy” in the certain game. This mathematical model is formed by games similar to Banach-Mazur games. The strict theorems of such games are then proved and applied to the target of the analogy. We then draw conclusions and look for counter-examples to our findings. The answer to the title question posed is negative, and it is not clear how far it should be taken seriously.

The article analyses the role of Church’s Thesis (hereinafter CT) in the context of the development of hypercomputation research. The text begins by presenting various views on the essence of computer science and the limitations of its methods. Then CT and its importance in determining the limits of methods used by computer science is presented. Basing on the above explanations, the work goes on to characterize various proposals of hypercomputation showing their relative power in relation to the arithmetic hierarchy. The general theme of the article is the analysis of mutual relations between the content of CT and the theories of hypercomputation. In the main part of the paper the arguments for abolition of CT caused by the introduction of hypercomputable methods in computer science are presented and critique of these views is presented. The role of the efficiency condition contained in the formulation of CT is stressed. The discussion ends with a summary defending the current status of Church’s thesis within the framework of philosophy and computer science as an important point of reference for determining what the notion of effective calculability really is. The considerations included in this article seem to be quite up-to-date relative to the current state of affairs in computer science.1.

This work is interdisciplinary as it lies at the crossroads of three disciplines: philosophy, theology, and logic. The article attempts to discuss the way in which sentences (judgements) are accepted and rejected within theology. First, the role of the argument of authority in theology is discussed. Subsequently, the most important types of theorems with certain degrees of theological certainty were identified. Finally, censorship was presented as a means of rejecting theorems in theology. Finally, censorship was indicated as a model for philosophical-logical research on denial.