Periodic Reporting for period 1 - STARTDIALOG (Towards a Systematic Theory of Aristotelian Diagrams in Logical Geometry)
Reporting period: 2022-09-01 to 2025-02-28
Aristotelian diagrams, such as the square of opposition, have been widely used throughout the history of philosophy and logic. Nowadays, they also have several applications in other disciplines that are concerned with logical reasoning, such as psychology, linguistics and computer science. However, many of the applications of Aristotelian diagrams suffer from substantial problems, often due to a lack of understanding of the intricate logical properties of these diagrams. Consequently, the tremendous heuristic potential of Aristotelian diagrams has remained vastly underappreciated thus far.
The overarching goal of the STARTDIALOG project is to develop a unified theory of Aristotelian diagrams. We will use a radically new research strategy to accomplish this goal, viz. developing a systematic typology of Aristotelian diagrams in close interplay with a comprehensive diagram database. This empirically informed typology will systematically organize all our (existing and new) knowledge about Aristotelian diagrams, similarly to the role of Mendeleev's periodic table of the elements in chemistry. This will enable us to clarify and solve many of the issues that currently surround the applications of these diagrams, and thereby lead to a more accurate understanding of their methodological importance.
This unified theory will allow us to move beyond the specific details of any given application, and to study Aristotelian diagrams as objects of independent interest. This will constitute a major breakthrough in logical geometry, i.e. the theoretical investigation of Aristotelian diagrams. However, because of the widely interdisciplinary use of these diagrams, the scientific impact of the STARTDIALOG project will reach far beyond the boundaries of logical geometry: its results will be relevant for other philosophical logicians as well, for philosophers in general, and ultimately, for all researchers who make use of Aristotelian diagrams in their research on logical reasoning.
The overarching goal of the STARTDIALOG project is to develop a unified theory of Aristotelian diagrams. We will use a radically new research strategy to accomplish this goal, viz. developing a systematic typology of Aristotelian diagrams in close interplay with a comprehensive diagram database. This empirically informed typology will systematically organize all our (existing and new) knowledge about Aristotelian diagrams, similarly to the role of Mendeleev's periodic table of the elements in chemistry. This will enable us to clarify and solve many of the issues that currently surround the applications of these diagrams, and thereby lead to a more accurate understanding of their methodological importance.
This unified theory will allow us to move beyond the specific details of any given application, and to study Aristotelian diagrams as objects of independent interest. This will constitute a major breakthrough in logical geometry, i.e. the theoretical investigation of Aristotelian diagrams. However, because of the widely interdisciplinary use of these diagrams, the scientific impact of the STARTDIALOG project will reach far beyond the boundaries of logical geometry: its results will be relevant for other philosophical logicians as well, for philosophers in general, and ultimately, for all researchers who make use of Aristotelian diagrams in their research on logical reasoning.
The overarching goal of the STARTDIALOG project leads to six more specific subgoals / questions:
1. Do the mathematical tools that are currently available suffice to accurately describe the entire range of logical behavior displayed by classical Aristotelian diagrams?
2. How exactly do non-classical Aristotelian diagrams differ from classical Aristotelian diagrams, and do they give rise to new types of logical behavior?
3. Which logical properties of Aristotelian diagrams can be discerned in the concrete applications of these diagrams that are found throughout the literature?
4. Does the small set of very well-studied Aristotelian diagrams suffice as a solid foundation for our theoretical investigations, or is it necessary to start from a broader empirical basis?
5. Can we establish a systematic typology of all Aristotelian diagrams found in the literature, and if so, what should be its underlying organizing principles?
6. What are the implications of such a new, empirically informed typology of Aristotelian diagrams with respect to the methodological importance of these diagrams?
Regarding 1 and 5, the STARTDIALOG team is developing a category-theoretical approach to logical geometry. This has already led to multiple publications in journals and conference proceedings, and we also have a book manuscript in preparation.
Regarding 2, the team has worked on Aristotelian diagrams for many-valued logics. We have also developed a modal logic of Aristotelian diagrams.
Regarding 3, the team has worked on Aristotelian diagrams for the proportional quantifier 'most' and for the late-ancient debate on privative negation.
Regarding 4, the Leonardi database has been expanded to well over 5000 diagrams (and still growing). The database is now reaching a sufficient size and coverage to facilitate empirical work on broad trends in diagram usage across time periods and across disciplines. Some pilot studies in this direction have proved to be very promising.
Regarding 6, the team has worked on the status of Aristotelian diagrams as logic diagrams. We are also developing an account on the heuristic value of Aristotelian diagrams (to facilitate analogical reasoning).
1. Do the mathematical tools that are currently available suffice to accurately describe the entire range of logical behavior displayed by classical Aristotelian diagrams?
2. How exactly do non-classical Aristotelian diagrams differ from classical Aristotelian diagrams, and do they give rise to new types of logical behavior?
3. Which logical properties of Aristotelian diagrams can be discerned in the concrete applications of these diagrams that are found throughout the literature?
4. Does the small set of very well-studied Aristotelian diagrams suffice as a solid foundation for our theoretical investigations, or is it necessary to start from a broader empirical basis?
5. Can we establish a systematic typology of all Aristotelian diagrams found in the literature, and if so, what should be its underlying organizing principles?
6. What are the implications of such a new, empirically informed typology of Aristotelian diagrams with respect to the methodological importance of these diagrams?
Regarding 1 and 5, the STARTDIALOG team is developing a category-theoretical approach to logical geometry. This has already led to multiple publications in journals and conference proceedings, and we also have a book manuscript in preparation.
Regarding 2, the team has worked on Aristotelian diagrams for many-valued logics. We have also developed a modal logic of Aristotelian diagrams.
Regarding 3, the team has worked on Aristotelian diagrams for the proportional quantifier 'most' and for the late-ancient debate on privative negation.
Regarding 4, the Leonardi database has been expanded to well over 5000 diagrams (and still growing). The database is now reaching a sufficient size and coverage to facilitate empirical work on broad trends in diagram usage across time periods and across disciplines. Some pilot studies in this direction have proved to be very promising.
Regarding 6, the team has worked on the status of Aristotelian diagrams as logic diagrams. We are also developing an account on the heuristic value of Aristotelian diagrams (to facilitate analogical reasoning).
The uses of category theory and of modal logic to study Aristotelian diagrams go significantly beyond the state of the art. The applicability of these methodologies clearly illustrates the rich and multi-faceted nature of Aristotelian diagrams.
The Leonardi database is starting to be appreciated -and used- as a valuable research tool within the broader research community.
The Leonardi database is starting to be appreciated -and used- as a valuable research tool within the broader research community.