Early Career Researcher Workshop

Abstract: In a recent talk, we discussed the underlying common structure of semantic paradoxes which we called the Recipe for Paradox. In this talk, we will briefly sketch the Recipe for Paradox and focus on the possible uniform solutions the Recipe suggests. We will then provide our own revenge-immune uniform solution to semantic paradoxes. The upshot of our solution is to restrict the Cut rule to grounded sentences only. That is, if the Cut formula is ungrounded, then the Cut move is blocked. Since our solution depends on the notion of groundedness, we will present how we define “grounded” and “ungrounded” formally in a syntactic fashion. We will conclude the talk by discussing why restricting Cut is more appealing as opposed to getting rid of Cut completely.

Abstract: What are ‘good’ reduction procedures and why is it important to distinguish these from ‘bad’ ones? It has been argued that from a philosophical point of view, or more specifically a standpoint of proof-theoretic semantics, reduction procedures are closely connected to the question about identity of proofs and that accepting certain reductions would lead to a trivialization of identity of proofs in the sense that every derivation of the same conclusion would have to be identified [1]. Therefore, we need to be careful: We cannot just accept any reduction procedure, i.e. any procedure eliminating some kind of detour in a derivation. I agree with this conclusion, however, I will argue that the question, which reductions we accept in our system, is not only important if we see them as generating a theory of proof identity but is also decisive for the more general question whether a proof has meaningful content.

By annotating derivations and reductions with λ-terms in accordance with the Curry- Howard-correspondence, it becomes much clearer what may be wrong with certain reductions. I will give examples of such reductions and show that allowing these would not only trivialize identity of proofs of the same conclusion but that it would allow to reduce a term of one type to the term of an arbitrary other. The λ -calculus and some well-known properties thereof can provide us with directions as to why this happens in these cases but not in the cases of ‘well- behaved’ reductions. If we take reductions as inducing an identity relation then that would force us to identify proofs of different arbitrary formulas. But even if we reject this assumption about proof identity, I will argue that allowing such reductions would render derivations in such a system meaningless.

[1] Schroeder-Heister, P. & Tranchini, L. (2017). Ekman’s Paradox. Notre Dame Journal of Formal Logic, 58(4), 567-581.

Abstract: I would like to talk about my Master Research Project. This MRP is focused on the philosophy of Mathematics of Gödel. It has been noted and studied the relationship between Gödel's view of philosophy in mathematics and Husserl's phenomenology and how it fits and does not fit Gödel's view of intuition. On the other hand, Husserl's phenomenology has similarities with the works of Bolzano and his Philosophy of Science. In this project, I want to show the relationship between Bolzano and Gödel and how this is key to Gödel's work. Bolzano's ideas of intuition and representations are better suited for Gödel than those of Husserl.

Bolzano's work in philosophy responds to Kant's theory of the possibility of pure mathematics, his notions about intuition, the a priori, and how we obtain knowledge from experience. Bolzano refuted this theory, showing Kant's a priori intuition does not work and Kant's dismissal of semantics. Bolzano also affirmed his view that there are 'things' that exist independently of our consciousness. Among them, of course, are mathematical objects. Husserl is instead not interested in how things come to be but how we perceive them—making him neither a Realist nor an Idealist and having solid opinions about these two positions. Despite Husserl's view, Intuitionists and Platonists used phenomenology for their purposes. Gödel was not the exception. Gödel used Husserl's categoric intuition of ideal objects to base his mathematical intuition, even though these intuitions do not work the same way. I want to show that Bolzano's idea of intuition as an idea in oneself adapts better to Gödel's mathematical intuition.

Abstract: The analysis of knowledge, belief or information by means of modal logic's possible worlds semantics is widely generalised. However, one major complaint against this approach is that this kind of semantics is committed to an excessively idealized picture of human reasoning. This view has come to be known as the problem of logical omniscience. One possible way to address such issues is via the techniques of non-classical logics with non-normal worlds semantics. In this talk, we will focus on the ternary relational semantics, a semantics which was initially developed by Routley and Meyer to model relevant logics but has been shown to be a versatile tool, capable of modelling different kinds of logics (such as many-valued, paraconsistent or intuitionist logics). The aim of this talk is to examine the possibilities of modelling (relevant) epistemic logics by means of a ternary relational semantics while avoiding problems of logical omniscience.

Abstract: Classical bivalent logic arrived in Africa via colonialism. It mediated thought, theory and method in ways that allowed indigenous African episteme alien and useless. This system of logic has been construed by its originators as not only universal but absolute. The implication that this commands is that the African way of codifying reality, though unique in its own right, was disregarded and branded subaltern. Perhaps these are pertinent justifications to reinforce colonialism, this logic system continues to wield global influence in the face of its inadequacies across several fronts. In other words, the problems with the logic system are most times, overlooked in spite of the cautions of prominent scholars like Karl Marx, Gottlob Frege and Alfred N. Whitehead, to name a few. In this discussion, I invoke the method of conversational philosophy and limit my grouse with classical logic to two: rigidity and contexts. Though one may not be divorced from the other, I am convinced that they undergird the failure in articulating indigenous African ideas better and clearly. The classical laws of thought are too rigid to take note of the contexts which make truth-values volatile. It is this capacity to not be flexible and admit contexts that a three-valued logic proposal has been proposed for not only mediating thought, theory and method within the African place and beyond but to also fill the lacuna that informs the urgency of decolonising logic for the sake of Africa’s intellectual ingenuity.

Abstract: We investigate the concept of definable, or inner, automorphism in the logical setting of partial Horn theories. The central technical result extends a syntactical characterization of the group of such automorphisms (called the covariant isotropy group) associated with an algebraic theory to the wider class of quasi-equational theories. We apply this characterization to prove that the isotropy group of a strict monoidal category is precisely its Picard group of invertible objects. Furthermore, we obtain an explicit description of the covariant isotropy group of a presheaf category.

Abstract: In contrast with the traditional view that suggests that logic and reasoning go hand in hand (Shapiro & Kouri Kissel, 2021), the normative role of logic towards reasoning has been both discussed and challenged (e.g. Steinberger, 2019; Russell, 2017); from the point of view of cognitive scientists, there is a widespread opinion that logic is irrelevant towards rationality (Stenning & Van Lambalgen, 2012: 23). Rationality is still a topic of interest for both cognitive scientists and logicians, but: is there link between the way cognitive scientist understand rational behavior and what logic tells us rationality should be?

According to some cognitive scientists (dual-process theorists), cognitive styles are considered a key aspect that distinguishes intelligence (as construed from the general theory of intelligence) from rationality (Stanovich, 2016); cognitive styles might be described as the differences among individuals while making decisions, establishing or calibrating beliefs (see Lucas-Stannard 2003: 1); they are an interesting topic for psychologists because they merge the study of two aspects of cognition: personality and rationality (Sternberg and Grigorenko, 1997: 701). Furthermore, among cognitive scientists, it is accepted that certain specific cognitive styles, labeled as self-government tendencies, favor rationality; unfortunately, so far, self-government theory has been self-explanatory, and hence, it has remained unclear why we consider those tendencies as distinctively rational. Here, I will contend that specific inferential features, taken from philosophical logic, can validate the self-government tendencies as rational behavior. I will proceed as follows:

First I will present the general features of cognitive styles and the self-government theory; second, I will speak about the distinction between intelligence and rationality, from cognitive scientists’ point of view; third, I will speak about the problem of the self-government theory validating itself; fourth, I will speak of certain inferential features, taken from philosophical logic (non-monotonicity, ampilative inferences and the intrinsic/extrinsic character of rationality) that might explain the rationality behind cognitive styles; finally, I’ll try to explain how those inferential features relate to specific self-government tendencies.

Abstract: Set-theoretic potentialism is the view that the universe of sets is never fully completed but is only given potentially. There are multiple varieties of this viewpoint, based on different views on just how this gradual unfolding of sets occurs. Tools from modal logic have been applied to more finely understand the commitments of and distinctions between different varieties of set-theoretic potentialism. In recent joint work with Neil Barton, we extended this analysis to study class-theoretic potentialism, the view that proper classes are given potentially (while the sets may or may not be fixed). In this talk I will survey some of the prior work on the mathematics of set-theoretic potentialism and present some of our work on class-theoretic potentialism.

Abstract: Prawitz (2007) frames the issue of formulating general logical laws in terms of Lewis Carroll’s dialogue between Achilles and the Tortoise: it is easier to agree on the truth of a particular instance than on the truth of the general law that might compel one to agree to the particular instance. Within the philosophy of science, the difficulty in formulating general laws is known as the induction problem: no matter the number of particular instances observed, one is never warranted in formulating a general law regarding such observances.

Lipton (1993) notes that there are two versions of the induction problem. The strong version leads to a wholesale Humean scepticism about induction, while the weak version leads only to theory underdetermination. Lipton claims that the method of Inference to the Best Explanation (IBE) manages to solve the issue of underdetermination via an account of scientific explanation. Logical abductivists attempt to do the same for logic.

Achilles and the Tortoise also motivates the adoption problem: “if a subject already infers in accordance with basic logical principles, no adoption is needed, and if the subject does not infer in accordance with them, no adoption is [...] possible” (Padro 2015, 18). That theory revision in logic is unwarranted seems rather similar to the strong version of the induction problem.

In discussing the impasse of Achilles and the Tortoise, Prawitz (2007) already argues (implicitly) both against the view which draws too close an analogy between logic and science and against what he calls logical nihilism. He suggests instead that logic be revised via reflective equilibrium. The present talk aims to frame the issue of revising logic following Prawitz (2007) and argues that those who take that no adoption of an alternative logic is possible find themselves in a stubborn equilibrium, while those who advocate for logical abductivism are in a very weak one.