Beyond Logic: Hypothetical Reasoning in Philosophy of Science, Informatics, and Law
Final Report Abstract
In two previous projects we investigated hypothetical reasoning from a logical point of view. This project targeted hypothetical reasoning in extra-logical areas. Our leading viewpoint was that logical investigations must prove useful outside logic, and conversely that logical investigations must take into account the way logic is applied in the ‘real world’. Thus we switched from an internal to an external perspective by considering what hypothetical reasoning is all about when we go beyond logic. As such extra-logical areas we chose philosophy of science as an application within philosophy, informatics as an application within the formal sciences, and law as an application within the field of social interaction. In each of these areas the idea of hypothetical reasoning plays a prominent role which can be used as a test case for our logical theories. We were particularly interested in how far the prooftheoretic perspective we had advocated and advanced in our logical investigations in the previous projects pertained to these subjects. In the area of logic in informatics we studied second-order logic, which is prominently applied in the computational interpretation of proofs and developed translations between first- and second-order logic that preserve fundamental aspects of proofs. We also studied the role of classical logic in relation to constructive logic and the role it may play in this context. A “calculus of natural calculation” that we developed allows one to intertwine deductive reasoning with calculation steps in a single system. By applying proof-theoretic means we also obtained novel results about computational complexity that may prove significant in the future. In the area of logic in relation to physics we studied the notion of explanation in the context of a general notion of “grounding”. In contradistinction to the metaphysical discussions on this concept within current philosophy, we used logical and in particular proof-theoretical methods to explicate this fundamental concept, which in natural language is expressed by the word “because”, and which in traditional logic had not received proper attention. Furthermore, we investigated physical computation models which go beyond the standard ones considered in informatics. We studied in particular the idea of analogous computation by physical means and what it implies for the interplay between philosophy of science, informatics and computational logic. More general investigations concerned “logic beyond logic”, that is, the logical treatment of aspects which obtain when logic is applied outside logic. In particular we analysed the influence of extra-logical bases for logical systems and the aspect of rule-based reasoning with associated reasoning principles that can capture logical and extra-logical reasoning in the same framework. A most surprising result was that a standard assumption normally made in deductive logic, its deductive adequacy or “completeness”, fails for constructive logic, at least when a kind of proof-theoretic semantics often employed in applications is used This brings up the question of the appropriate formalism for deductive reasoning in the constructive realm and therefore in informatics. Unfortunately, the project could not make proper advances in the area of legal reasoning and law in general. Apart from the institutional problem that a research group with whom we wanted to pursue this topic was unexpectedly no longer available, the idea of tackling legal reasoning with proof-theoretic tools turned out to be way too complex for the limited capacities of our project. The proof theory of legal reasoning continues to be a desideratum. It needs a project of its own, for which the ANR-DFG format would be an ideal framework, as the competences are available in France and Germany. Our project was the final project in an ANR-DFG cooperation which spread over a decade (2009-2019). This cooperation has proved extremely fruitful for both sides and has led to a variety of further research initiatives which we are jointly pursuing in Paris and Tübingen.
Publications
- Advances in Proof-Theoretic Semantics, Trends in Logic 43, Springer 2016
Thomas Piecha & Peter Schroeder-Heister (eds)
(See online at https://doi.org/10.1007/978-3-319-22686-6) - Completeness in Proof-Theoretic Semantics. In: T. Piecha & P. Schroeder-Heister (eds), Advances in Proof-Theoretic Semantics, Trends in Logic 43, Springer 2016, pp. 231-251
Thomas Piecha
(See online at https://doi.org/10.1007/978-3-319-22686-6_15) - Open problems in proof-theoretic semantics. In: T. Piecha & P. Schroeder- Heister (eds), Advances in Proof-Theoretic Semantics, Trends in Logic 43, Springer 2016, pp. 253- 283
Peter Schroeder-Heister
(See online at https://doi.org/10.1007/978-3-319-22686-6_16) - Ekman's Paradox. Notre Dame Journal of Formal Logic 58 (2017), pp. 567-581
Peter Schroeder-Heister & Luca Tranchini
(See online at https://doi.org/10.1215/00294527-2017-0017) - Popper’s Notion of Duality and His Theory of Negations. History and Philosophy of Logic 38 (2), 2017, pp. 154-189
David Binder & Thomas Piecha
(See online at https://doi.org/10.1080/01445340.2016.1278517) - How to Ekman a Crabbé-Tennant. Synthese (2018), pp. 1–23
Peter Schroeder-Heister & Luca Tranchini
(See online at https://doi.org/10.1007/s11229-018-02018-3) - Incompleteness of Intuitionistic Propositional Logic with Respect to Proof-Theoretic Semantics. Studia Logica 107(1), 233-246, 2019, pp. 233-246
Thomas Piecha & Peter Schroeder-Heister
(See online at https://doi.org/10.1007/s11225-018-9823-7) - Special issue on General Proof Theory. Studia Logica 107(1), 2019
Thomas Piecha & Peter Schroeder-Heister (eds)
(See online at https://doi.org/10.1007/s11225-018-9818-4)
