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. The overarching goal of the STARTDIALOG project is to develop a unified theory of Aristotelian diagrams. In this PhD position, you will be responsible for carrying out an absolutely fundamental part of the project, viz., to study the behavior of Aristotelian diagrams in classical formal logic. A representative but non-exhaustive list of potential research topics looks like this:
- We currently have a well-defined notion of 'Aristotelian isomorphism', and some preliminary results on how to generalize this to a notion of 'Aristotelian homomorphism'. Based on these results, can we define one or more categories of Aristotelian diagrams, in the category-theoretical sense of the word? Do these categories enjoy interesting properties? (Demey and his current PhD student Leander Vignero already have some promising results in this direction.)
- Given the notion of Aristotelian isomorphism, it should be straightforward to define a notion of 'Aristotelian automorphism', and thus also of the automorphism group Aut(D) associated with an Aristotelian diagram D. Does the automorphism group provide any interesting information about the Aristotelian diagram it is associated with? For the mathematically inclined: can we prove a representation theorem in this context, i.e. for every group G there exists an Aristotelian diagram D such that G is isomorphic to Aut(D)?
- There are already some results on the informational optimality of Aristotelian diagrams (in relation to their so-called 'opposition' and 'implication' counterparts). We would like to expand on these results and to further unify them into a streamlined framework.
- Logic-sensitivity is an important and well-known phenomenon in logical geometry. There are already some basic results about this, but we would like to expand on them, for example by investigating their interaction with the notion of Boolean complexity. Typical example: it is well-known that there exist fragments of formulas F and logical systems S1, S2 such that the Aristotelian diagram for F relative to S1 is a classical square of opposition, but that for F relative to S2 is a degenerate square of opposition. Do there also exist fragments F and logical systems S1, S2 such that the Aristotelian diagrams for F relative to S1 and to S2 are two Boolean subtypes of one and the same Aristotelian family (e.g. strong vs weak Jacoby-Sesmat-Blanché hexagons)? And most importantly, can we e... For more information see https://www.kuleuven.be/personeel/jobsite/jobs/60140203
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