Insights from Bernard Bolzano – IBS

The current project is composed of three main objectives: the first concerning conceptual grounding, the second grounding and mathematical, and more generally non-causal, explanations, the third grounding and causation. These three objectives will be studied and analyzed drawing inspiration and guidance from Bolzano’s work. Given various issues treated, the methodological maxim of the project is that of cooperation and exchange between project members who are all experts in the different notions treated. This will be done as rigorously as possible in the following way.
First of all, the scientific coordinator will coordinate and be responsible of each objective of the project. Moreover, members of the project, both the internal as well as the external collaborators, will be organized in four teams. Team 1: {Carrara, Kriegel, Skiles, Textor} is concerned with Objective 1 (grounding in the conceptual sciences); Team 2: {Arana, Huneman, Lange, Panza}s concerned with Objective 2 (grounding and mathematical explanation); Team 3 : {Baumgartner, Kistler, Gebharter, Schnieder} is concerned with Objective 3 (grounding and causation); Team 4: {Benoist, Benis-Sinaceur, de Rouilhan, Sebestik, Roski}. is concerned with Bolzano’s theory of grounding.
Each team is composed of four members (except Team 4 who has five), experts of the main notion involved in the team’s objective and will help to develop the tasks linked to the Objective. Team 4 will supervise the whole project helping with links and references with Bolzano's theory of grounding.

Objective 1: Research in Objective 1 has proven to be fruitful and rich of results, in part even unexpected; indeed several new areas of investigation have naturally arise
T1.1. In order to elaborate an account of grounding in the conceptual sciences, we have started by extending Poggiolesi’s approach to logical grounding into two different and meaningful ways for the conceptual sciences themselves. On the one hand, Poggiolesi’s approach has been used to study the grounds of conditionals sentences, sentences of the form “if A, then B.” On the other hand, Poggiolesi’s approach has been used to study the grounds of quantifiers.
T1.2. Poggiolesi’s approach to the notion of complete and immediate grounding has been extended to the kindred notions of “complete and mediate,” “partial and immediate” and “partial and mediate grounding.” A proof-theoretical framework for several notions of grounding has thus been developed. This framework has served as the background framework for a formal and conceptual comparison with the logic of metaphysical grounding introduced by Kit Fine.
Objective 2: Research in Objective 2 is also turned out be fruitful, although it just started because of the time dedicated to Objective 1.
T2.1 The SC and F. Genco have deeply analysed the relation between the notion of conceptual grounding and the notion of conceptual explanation, which include mathematical explanation. In a nutshell, the SC and the post-doc have argued in favor of a distinction between conceptual grounding and conceptual explanation: In a nutshell, the SC and the post-doc have argued in favor of a distinction between conceptual grounding and conceptual explanation. Moreover, in order to support their claim, the SC and the post-doc have used the notion of proof and shaped conceptual explanations as special type of proofs.
Objective 3: research in Objective 3 has just started, however progress should be made in the continuation of the project.

What is the adequate notion of explanation in logic and conceptual sciences and how does it vary from one logic to another? The approach developed in this project, which is based on the SC’s former results and is mainly inspired by important Bolzano’s intuitions, turns out to be the best answer to these questions. Such an approach relies on two key-ingredients, namely derivability and complexity. Thanks to these two ingredients that can be easily varied from one logic to another, whilst remaining conceptually the same, Poggiolesi’s approach seems to represent the only account which is flexible enough to be adapted to several different frameworks. This is what comes out from the research, developed in the Objective 1 of the project, on quantifiers, conditionals, and connectives other from the classical ones.


What is the model for mathematical explanations and how can we account for the asymmetry that characterizes them? In the realm of explanations, conceptual explanations and more precisely mathematical explanations are those that seem to be more in need of investigation and models. The one proposed by the SC and F. Genco is based on the notion of proof and exploit and modernize insights coming from Bolzano. Moreover it makes a link with the kindred notion of grounding in the conceptual science. It presents several different advantages. First of all it explains the asymmetry of explanations via the notion of complexity. Secondly it revitalizes the Bolzanian tradition that has its illustrious roots in Aristotle. Thirdly it makes a bridge between the literature on grounding and that on explanations, which is in the case of conceptual sciences lacking. Finally it also makes a connection with proof-theory, a thriving area of research, whose results can be imported and used in the literature on explanations.

1. F. Poggiolesi, « Grounding principles for (relevant) implication », Synthese, forthcoming.
2. F. Poggiolesi and N. Francez, « Towards a generalization of the logic of grounding », Theoria, forthcoming.
3. F. Poggiolesi, « Grounding rules and (hyper-)isomorphic formulas », Australasian Journal of Logic, 17 : 70-80, 2020.
4. F. Poggiolesi, «Bolzano, the appropriate relevant logic and grounding rules for implication », in Bolzano and Grounding, S. Roski and B. Schnieder (eds.), Oxford University Press, forthcoming.
5. F. Poggiolesi, «Logics of grounding », in Routledge Handbook for Metaphysical Grounding, M. Raven (ed.), pp. 213-227, New York: Routledge, 2020.
6. F. Aschieri, F. Genco. «Par Means Parallel: Multiplicative Linear Logic Proofs as Concurrent Functional Programs », POPL 2020, Jan 2020, New Orleans, Louisiana, United States
7. Francesca Poggiolesi. Conditionnels. Précis de philosophie de la logique, F. Poggiolesi et P. Wagner (eds.), Editions de la Sorbonne, forthcoming.

One of the most under-appreciated thinkers of the XIX century is the great mathematician and philosopher Bernard Bolzano. Amongst the deep philosophical heritages that Bolzano left us with, there is the concept of 'Abfolge', or 'grounding', which was mainly ignored by his successors, but it is nowadays the centre of a renew and spread interest.
The goal of this project is to develop a neo-Bolzanian theory of grounding using contemporary formal and conceptual tools. The development of such a theory seems to be an interesting and valuable line of research from at least two different perspectives. First of all, from a conceptual perspective, because we firmly believe that the Bolzanian conception still has much to teach us. Secondly, from a historical perspective, because in developing a neo-Bolzanian approach to grounding, we will give new life to Bolzano and his philosophical contributions.