Elio La Rosa MA
October 1, 2023 until March 31, 2024
Affiliation: Munich Center for Mathematical Philosophy
Research for a study about:
From Carnap's Definition of Theoretical Terms to the Definition of Contexts of Evaluation
“From Carnap's Definition of Theoretical Terms to the Definition of Contexts of Evaluation”
In his mature work on theoretical terms, Carnap was able to provide explicit definitions for them. Specifically, they were defined as unspecified witnesses of the Ramsified conjunction of scientific laws they occurred in, in accordance with their indeterminate nature and the synthetic/observational and analytic/theoretical content separation. Formally, this was made possible by epsilon calculus, a formalism originally developed by Hilbert in the context of his foundational program for mathematics. There, special ‘epsilon terms’ binding formulas denote non-deterministically an object (if any) satisfying them, resulting in an axiomatisable system more expressive than the predicate calculus. For Hilbert, these terms represented ‘ideal’ objects satisfying mathematical properties.
In this project, I aim at generalising both epsilon calculus and Carnap’s account at an intensional level, allowing for the reconstruction of theory structures more complex than Carnap’s advocated for in pragmatist approaches to philosophy of science. In particular, a more fine-grained analysis of theories inner structure can be achieved. To this end, I developed a new class of modal logics structurally analogous to epsilon calculus. These systems are based on ‘epsilon’ modalities indexed by formulas which select non-deterministically a related witness world (if any) satisfying them. Formulas under the scope of these modalities are thus evaluated in such a world. These modalities are more expressive than standard ones, and preserves some key properties of Hilbert’s epsilon calculus at the intensional level, and can embed them. As a result, Carnap’s explicit definition can be expressed there.
Currently, I am investigating a more direct connection between the formal framework of epsilon modalities and Carnap’s approach. In particular, I am working on the characterisation of a notion of ‘theoretical context’ acting as the intensional counterpart of theoretical terms. This would capture ‘ideal’ contexts of evaluation that, similarly to theoretical terms referents, are (only partially) determined by the laws of a scientific theory. Formally, these contexts would be represented as the scope of epsilon modalities indexed by the conjunction of theory laws. In this way, relations among different contexts and their consequences for the interpretation of theoretical language can be directly analysed in modal terms, and the definitions of theoretical terms simplified and characterised intensionally as well.
Lecture
From Carnap's Definition of Theoretical Terms to the Definition of Contexts of Evaluation
Logic Café Lecture
Date: November, 20
Time: 16h45 - 18h15
Meetings are usually held on Mondays from 16:45 to 18:15 in Raum 3F (D0313), 3. floor, Department of Philosophy
Abstract:
I develop a class of modal logics inspired by Hilbert's Epsilon Calculus and based on what I call 'epsilon modalities'. After presenting their formal properties, I show how these modalities can generalise Carnap's strategy of defining theoretical terms in scientific theories through epsilon terms. In this respect, epsilon modalities can be seen as denoting an arbitrary context of evaluation (if any) among those satisfying a certain formula. This means that a modal interpretation of Carnap's strategy similar to that of Andreas and Schiemer can be given at the level of the theory reconstruction. Finally, I show how the enhanced expressivity of the modal language allows for the representation of complex structures, with applications to the semantics of contemporary pragmatic reconstructions of scientific theories and to mathematical structuralism.
Final Report
The fellowship at the IVC was very helpful for developing several aspects of my research project, including some I did not foresee in my research plan.
In the first part of my visit, I worked on the representation in the epsilon modal framework of intensional account of Carnap’s definitions and on their application to mathematical structuralism already present in the literature. Mathematical structures can be explicitly defined by epsilon terms, allowing for simpler case studies and the investigation of specific (modal) structuralist interpretations. I presented these results in my talk at the Logik Café.
In the second part, I refined these reconstructions towards the characterisation of a notion of a theoretical context of evaluation. Epsilon modalities turned out to formally represent them as the worlds in which they evaluate formulas occurring in their scope, as anticipated in my research plan. More interestingly, they do so by preserving the analytic/synthetic and empiric/theoretic distinction in this scope and simplifying their formal representation. An equivalent ‘quantifier-free’ Ramsification is obtained by substituting occurrences of theoretical terms of theory laws by intensional variables, whose interpretation is world-dependent, and epsilon definitions by identities among such variables and theoretical terms. Since both components occur under epsilon modalities indexed by the same kind of Ramsification, intension variables preserve their intended interpretation. I presented part of these results at the 2024 WLD in Vienna, and I intend to submit them for publication soon.
In the final part of my stay, I worked on the interpretation of theoretical context of mathematical theories. Similar to Hilbert’s interpretation of epsilon terms as ideal elements of a mathematical theory, epsilon modalities represent ideal mathematical structures that could serve similar purposes at an intensional level. While working on the models of these theories, I also realised an analogy between the way worlds are singled out by epsilon modalities and the interpretation of antecedent in some conditional logics. This opens for the definition of conditionals preserving properties, including connexive ones, not represented in well-known accounts. I intend to further research both topics next.