Conference Agenda

In this talk I will discuss the adequacy of prominent relations of theory equivalence as relations of theoretical synonymy.

A relation of theoretical synonymy should characterize the equivalence of formal theories with respect to their meaning or content. In logic and philosophy of science, the relation of definitional equivalence or its generalization, bi-interpretability, are predominantly used for this purpose. I will consider examples of theories that are bi-interpretable and argue that they should not be understood as synonymous. To do this, I will introduce five general principles of theoretical synonymy that seem intuitively plausible but are not satisfied by definitional equivalence or bi-interpretability. This gives rise to questioning the adequacy of bi-interpretability as a relation of theoretical synonymy.

Finally, I will briefly outline what interesting strengthenings of bi-interpretability might look like.

Bi-interpretability is a widely accepted standard of equivalence for first-order theories, understood both semantically via definable inner models and syntactically via mutually inverse (up to isomorphism) interpretations. However, in model-theoretic practice, interpretations frequently employ parameters, yielding bi-interpretability with parameters as an induced standard of equivalence. Despite their prevalence, interpretations with parameters remain understudied from both logical and philosophical perspectives, with Visser characterizing their category as "terra incognita."

This paper addresses this gap through three main contributions. First, I examine key examples including the Klein-Beltrami model and the geometrization of arithmetic. Second, I develop a systematic semantic analysis of interpretations with parameters using uniformly definable families of models, showing how they induce relation-valued profunctors between categories of models. This perspective yields a novel characterization of bi-interpretability with parameters and a new categorical invariant.

Finally, I analyze the semantic status of parameters, arguing against their treatment as additional constants, since this yields a coarser notion of equivalence. Instead, I propose understanding parameters as well-behaved Hilbertian ε-terms, developing a notion of interpretation in the ε-calculus. The main technical conjecture is that bi-interpretability with parameters coincides with bi-interpretability in the ε-calculus, providing another perspective on this notion of equivalence.

In this talk I consider the notion of logical duality with regard to some constructive logics. My approach is mainly semantic -- the dualization procedures are based on some important properties of truth values used in constructive logics. In the focus of consideration is an important logic of the constructivist family, constructive logic with strong negation, also known as Nelson's logic of constructible falsity. The logics in question are equipped with suitable proof systems in the form of binary consequence systems, in particular such a system is formulated for Nelson's explosive logic N3. It is shown how this logic can be dualized semantically and syntactically, and how Nelson's logic and its dual can be combined in the general framework of a bi-Nelson logic. In particular, I provide a definition of truth and falsity conditions for the connective of dual Nelson's implication and construct the respective binary consequence proof system in the FMLA-FMLA logical framework. A bi-Nelson logic in a language without implication and dual implication turns out to be the first-degree entailment fragment of the logic R-Mingle, whereas adding implication and dual implication leads to a system similar to Nelson's logic S.

Work in the field of feminist logic is still rather scarce and the field itself remains a contested area of study, but still, it is developing. One approach concentrates on analyzing logical systems with respect to structural features that may perpetuate discrimination or, on the other hand, features that may be helpful for resisting and opposing it. Upon this assumption, we want to investigate possible applications of queer feminist views on logic with respect to a very specific group, namely contradictory logics, i.e., logical systems containing contradictions in their set of theorems. We want to show that bringing together contradictory logics and reasoning about queer feminist issues may prove fruitful both as a ‘real-life’ motivation for these rather marginalized logical systems and as a formal basis for a philosophical field that is still characterized by a distrust of formalism.