This paper is devoted to the study of some subvarieties of the variety Qof Q-Heyting algebras, that is, Heyting algebras with a quantifier. In particular, a deeper investigation is carried out in the variety Q 3 of three-valued Q-Heyting algebras to show that the structure of the lattice of subvarieties of Qis far more complicated that the lattice of subvarieties of Heyting algebras. We determine the simple and subdirectly irreducible algebras in Q 3 and we construct the lattice of subvarieties (...) (Q 3 ) of the variety Q 3. (shrink)

This paper is devoted to the study of some subvarieties of the variety Q of Q-Heyting algebras, that is, Heyting algebras with a quantifier. In particular, a deeper investigation is carried out in the variety Q subscript 3 of three-valued Q-Heyting algebras to show that the structure of the lattice of subvarieties of Q is far more complicated that the lattice of subvarieties of Heyting algebras. We determine the simple and subdirectly irreducible algebras in Q subscript 3 and we construct (...) the lattice of subvarieties lambda (Q subscript 3) of the variety Q subscript 3. (shrink)

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This (...) historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic. (shrink)

We isolate two abstract determinacy theorems for games of length $\omega_1$ from work of Neeman and use them to conclude, from large-cardinal assumptions and an iterability hypothesis in the region of measurable Woodin cardinals thatif the Continuum Hypothesis holds, then all games of length $\omega_1$ which are provably $\Delta_1$ -definable from a universally Baire parameter are determined;all games of length $\omega_1$ with payoff constructible relative to the play are determined; andif the Continuum Hypothesis holds, then there is a model of (...) ${\mathsf{ZFC}}$ containing all reals in which all games of length $\omega_1$ definable from real and ordinal parameters are determined. (shrink)

The history behind the ‘Madagascar’ example of Gareth Evans is traced, suggesting that the decisive reference-shift occurred in the 16th, not the 17, century. The difference between this example and the ‘Gödel’ example of Saul Kripke is explained in terms of the distinction between de re and de dicto beliefs and intentions.

In their thought-provoking article, Sedlakova and Trachsel (2023) defend the view that the status—both epistemic and ethical—of Conversational Artificial Intelligence (CAI) used in psychotherapy is complicated. While therapeutic CAI seems to be more than a mere tool implementing particular therapeutic techniques, it falls short of being a “digital therapist.” One of the main arguments supporting the latter claim is that even though “the interaction with CAI happens in the course of conversation… the conversation is profoundly different from a conversation with (...) a human therapist” (Sedlakova and Trachsel 2023, 8). In particular, unlike a human therapist, CAI cannot help its users gain new insight and self-understanding (Sedlakova and Trachsel 2023). We agree that currently available therapeutic CAI cannot be considered a “digital therapist,” however, we think that the issue surrounding the acquisition of new self-understanding in the interaction with therapeutic CAI is more complicated than Sedlakova and Trachsel suggest. (shrink)

Despite Vancil’s (1979) proclamation over twenty years ago that topoi have been abandoned in argument theory, this essay contends that topoi should have a vital role in contemporary argumentation theory. Four key areas are identified where topoi are (or can be) essential tools for argumentation: Locating argument, building argument, development of critical thinking, and argument pedagogy. As a result, teachers and students of argument can both benefit from a (re)discovery of topoi.

It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$. Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.

We characterize the determinacy of $F_\sigma $ games of length $\omega ^2$ in terms of determinacy assertions for short games. Specifically, we show that $F_\sigma $ games of length $\omega ^2$ are determined if, and only if, there is a transitive model of ${\mathsf {KP}}+{\mathsf {AD}}$ containing $\mathbb {R}$ and reflecting $\Pi _1$ facts about the next admissible set.As a consequence, one obtains that, over the base theory ${\mathsf {KP}} + {\mathsf {DC}} + ``\mathbb {R}$ exists,” determinacy for $F_\sigma $ (...) games of length $\omega ^2$ is stronger than ${\mathsf {AD}}$, but weaker than ${\mathsf {AD}} + \Sigma _1$ -separation. (shrink)

The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously. Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not literally). A (...) modal stage theory, MST, is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that MST interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo–Fraenkel set theory. (shrink)

Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism (which is now lacking) and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argumentative part, (...) an evaluation of arguments and counterarguments is presented. Results: The main result is that strict finitism is indeed a viable option, next to other constructive approaches, in (the foundations of) mathematics. Implications: Arguing for strict finitism is more complex than is usually thought. For future research, strict finitist mathematics itself needs to be written out in more detail to increase its credibility. In as far as strict finitism is a viable option, it will change our views on such “classics” as the platonist-constructivist discussion, the discovery-construction debate and the mysterious applicability problem (why is mathematics so successful in its applications?). Constructivist content: Strict finitism starts from the idea that counting is an act of labeling, hence the mathematician is an active subject right from the start. It differs from other constructivist views in that the finite limitations of the human subject are taken into account. (shrink)

Growing demand for broadly accessible mental health care, together with the rapid development of new technologies, trigger discussions about the feasibility of psychotherapeutic interventions based on interactions with Conversational Artificial Intelligence (CAI). Many authors argue that while currently available CAI can be a useful supplement for human-delivered psychotherapy, it is not yet capable of delivering fully fledged psychotherapy on its own. The goal of this paper is to investigate what are the most important obstacles on our way to developing CAI (...) systems capable of delivering psychotherapy in the future. To this end, we formulate and discuss three challenges central to this quest. Firstly, we might not be able to develop effective AI-based psychotherapy unless we deepen our understanding of what makes human-delivered psychotherapy effective. Secondly, assuming that it requires building a therapeutic relationship, it is not clear whether psychotherapy can be delivered by non-human agents. Thirdly, conducting psychotherapy might be a problem too complicated for narrow AI, i.e., AI proficient in dealing with only relatively simple and well-delineated tasks. If this is the case, we should not expect CAI to be capable of delivering fully-fledged psychotherapy until the so-called “general” or “human-like” AI is developed. While we believe that all these challenges can ultimately be overcome, we think that being mindful of them is crucial to ensure well-balanced and steady progress on our path to AI-based psychotherapy. (shrink)