Franci Mangraviti

In logic, a theory is a deductively closed set of sentences. Theories containing all logical theorems are called regular, while irregular theories omit at least one logical theorem. We argue that the widespread assumption that theories must be regular is misguided. This is because (1) philosophical sense can be made of irregular theories when thinking of theories as modeling trasmission of commitment and (2) standard reasons for only considering regular theories are either question-begging or unnecessarily limit what theories can be used for.

Consistency appears to be an essential feature of mainstream mathematics: practices deemed inconsistent are either rejected as incorrect, or treated as outsiders. In this paper I analyze the role of consistency in mathematics by looking at deviations from consistency: I argue that attempts to ground the distinction between mainstream and inconsistent mathematics cannot succeed without incorporating some kind of practice-level commitment to inconsistency, and propose a notion of inconsistent practice which clarifies, by way of contrast, how consistency functions in the mainstream. This exemplifies a general methodology for better understanding core features of mainstream practices through contrast with deviating practices.

Philosophical accounts of disagreements within logic tend to focus on how logics themselves disagree with one another, such as how their theorems or (meta)inferences diverge, while neglecting the theoretical disagreements that lead to the advocacy of these logics. This has several shortcomings, amongst which it fails to explain why disagreements over the correct logic occur in the first place, as well as overlooking the possibility of substantive logical disagreements amongst those who advocate the same logic. In this paper, we move the debate away from this lawcentrism, highlighting the various ways in which logicians disagree with each other when engaged in theory choice. We propose a framework of logical research programs within which to understand the distinct commitments logicians take on when engaged in theory choice, and show how a range of existent disagreements amongst logicians can be fruitfully understood in terms of disagreements between elements of their respective research programs.

The relationship between relevant logic and mathematical practice has often been twofold: while some relevant logicians have focused on the descriptive project of capturing (in some sense) informal mathematical reasoning better than classical logic does, others have engaged in the normative project of suggesting how mathematics should be done as opposed to how it is done now. Both projects have received heavy criticism. In this paper, I argue that the intuitive idea of variable sharing does point to a real aspect of mainstream mathematical practices, and this fact can be used to make sense of both relevantist projects: on one hand, relevant logics may be used to provide a more accurate model of this particular aspect of current practices, while on the other, they can suggest new practices which deviate from the standard precisely in their approach to variable sharing.

I argue that the continued focus on the possibility question - whether feminist logic can exist as a respectable practice - has several harmful consequences. First, it invites the association of feminist logic with substantial positions in the philosophy of logic, which unnecessarily leaves room for dismissing the field a priori. Second, it invites a systematic reading of feminist logicians as arguing in isolation from their logical practice, which can hide some genuine possibilities for the field. To avoid these issues, I propose a very broad characterization of feminist logic as a kind of practice which addresses some harmful aspect of dominant practices by focusing on their interaction with logical practices. This characterization trivializes the possibility question, enforces no particular conception of logic to the exclusion of others, yet leaves room for both conservative and radical approaches.

In this paper, we aim to show that studies on the new epistemology of mathematics and other sciences are crucial, not only from a theoretical perspective but also in terms of practical aspects, which are particularly significant in the context of teaching and the development of new com petences among teachers. The first aspect addressed here is the age of AI and its potential impact, not just on scientists but on society as a whole. The second aspect focus es on mental imagery, which can be influenced by various factors and, in turn, shape scientific thinking. The third aspect, briefly mentioned, is the issue of the social responsibility of science. Our research approach is grounded in actor-network theory as well as the extended mind thesis. The main problem analysed here is the relationship between the diver sity of mental representations and critical concept kinds in the context of solving mathematical problems (e.g., constructing new structures). Our key conclusion is that this diversity in mental representations can be linked to the presence of critical concept kinds, thereby aiding in the effective construction of new concepts and problem-solving. The need for this diversity in representations can – and should – be cultivated and introduced at the educational level.

Axel A. Barceló takes issue with my discussion of the dominant gender conception—according to which “woman” is the classical negation of “man”—as an example of logic-based hermeneutical injustice. His arguments are embedded in a more general critique of revisionist projects within feminist logic. [...] Barceló’s particular response relies on a number of assumptions which I think are worth pushing back against. In particular, I will argue that feminist logical revisionism does not depend on giving up universality or proving classical logical truths wrong; that feminist logical revisionist arguments are not meant to be judged in isolation from the rest of the logic literature; and that exploratory work in non-classical feminist logic stands in no special need of justification.

The paper is structured as follows. First, I will single out three salient moments of trans philosophy, drawing on both my own experience as a trans person and the various attempts to theorize transness as laid out by Talia Mae Bettcher's "Trapped in the Wrong Theory". From there, I will extrapolate three ways to see the relationship between logic and trans philosophy, and provide for each some examples of both current and possible future work. Finally, in analogy with the literature on feminist logic, I will attempt to address the further question of what trans logic could be, as a discipline and a system.

Mathematics, even more than the other sciences, is often presented as essentially unique, as if it could not be any other way. And yet, prima facie alternative mathematics are all over the place, from non-Western mathematics to mathematics based on nonclassical logics. Taking inspiration from Robin Dembroff’s analysis of critical gender kinds, and from Andrew Aberdein and Stephen Read’s analysis of alternative logics, in this paper I will introduce a practice-centered framework for the study of alternative mathematics based on the notion of critical math kind. After sketching a model of mainstream mathematics, I will provide examples of how deviation along several distinct dimensions can occur, and how deviations can vary in their gravity. I will then discuss how the framework can be used to think of questions concerning the alternativeness status and philosophical implications of alleged alternative mathematics, and help us in identifying alternatives that suit our purposes.

While much has been said on the connection between dominant rationality standards and systemic oppression, the specific role of logic in supporting epistemic injustice has not received much explicit attention. In this paper I highlight several ways in which it is possible for logic – as a discipline, as a particular system and as a gloss for rational common sense – to be implicated in epistemic injustice. Concrete examples are given for testimonial, content-based, hermeneutical and contributory injustices. I conclude by elaborating on how the need to address these injustices affects the attitudes we should carry toward logic.

This dissertation has two main goals. The first is to provide a practice-based analysis of the field of inconsistent mathematics: what motivates it? what role does logic have in it? what distinguishes it from classical mathematics? is it alternative or revolutionary? The second goal is to introduce and defend a new conception of inconsistent mathematics - queer incomaths - as a particularly effective answer to feminist critiques of classical logic and mathematics. This sets the stage for a genuine revolution in mathematics, insofar as it suggests the need for a shift in mainstream attitudes about the role of logic and ethics in the practice of mathematics.

The relationship between logics with sets of theorems including contradictions (“inconsistent logics”) and theories closed under such logics is investigated. It is noted that if we take “theories” to be defined in terms of deductive closure understood in a way somewhat different from the standard, Tarskian, one, inconsistent logics can have consistent theories. That is, we can find some sets of formulas the closure of which under some inconsistent logic need not contain any contradictions. We prove this in a general setting for a family of relevant connexive logics, extract the essential features of the proof in order to obtain a sufficient condition for the consistency of a theory in arbitrary logics, and finally consider some concrete examples of consistent mathematical theories in Abelian logic. The upshot is that on this way of understanding deductive closure, common to relevant logics, there is a rich and interesting kind of interaction between inconsistent logics and their theories. We argue that this suggests an important avenue for investigation of inconsistent logics, from both a technical and a philosophical perspective.