Ranks in positive logic (KIRBYJ2_U26EMP)
Key Details
- Application deadline
- 31 January 2026 for International, 31 March 2026 for Home
- Location
- UEA
- Funding type
- Self-funded
- Start date
- 1 June 2026
- Mode of study
- Full-time
- Programme type
- PhD
)
Welcome to Norwich
According to the Sunday Times, this city is one of the best places to live in the UK.
Project description
Primary supervisor - Dr Jonathan Kirby
Model theory is traditionally done with “classical first-order logic”, the logic which allows unlimited use of the operators AND, OR, NOT, with the EXISTS and FOR ALL quantifiers. More recently, positive logic has emerged as a useful generalisation. For a theory in positive logic, one can specific how much you are allowed to use the NOT operator and the FOR ALL quantifier: either without restriction (to get the classical case) or much less. Positive logic is more appropriate to use directly for some applications in algebra, such as for modules, and for theories which do not admit quantifier elimination.
Theories in classical logic can be classified according to their combinatorial complexity, via a number of dividing lines which are mostly due to Shelah. A map of this classification appears at http://forkinganddividing.com.
Many of these dividing lines are related to the existence of a suitable rank notion on formulas such as Morley rank, U-rank, dp-rank. These are ordinal-valued ranks which explain the complexity of a formula, not as a piece of syntax but in terms of how it relates to other formulas within the context of the particular theory. For example, in vector spaces, any formula has an associated vector subspace and all these ranks will give the dimension of that subspace. These ranks are then often the most useful model-theoretic tool to apply to the theories.
We know how to extend some of these dividing lines to positive logic, but so far ranks have not been studied in positive logic. This PhD project will aim to develop suitable analogues of Morley rank, U-rank and dp-rank in positive theories, and will explore new applications of the results obtained.
Students should have some knowledge of mathematical logic and preferably also model theory, and are advised to contact Dr Kirby directly to discuss their application.
Entry requirements
The standard minimum entry requirement is 2:1 in Mathematics.
Funding
This project is offered on a self-funding basis. It is open to applicants with funding or those applying to funding sources. Details of tuition fees can be found here.
A bench fee is also payable in addition to the tuition fee to cover specialist equipment or laboratory costs required for the research. Applicants should contact the primary supervisor for further information about the fee associated with the project.
UEA Alumni 10% Scholarships - A scholarship of a 10% fee reduction is available to UEA Alumni looking to return for postgraduate study at UEA, Terms and conditions apply. For a postgraduate master’s loan, visit our Postgraduate Student Loans page for more information.
References
i) Gabriel Conant: http://forkinganddividing.com.
ii) Anna Dmitrieva, Francesco Gallinaro, Marks Kamsma. DIVIDING LINES BETWEEN POSITIVE THEORIES. The Journal of Symbolic Logic. 2023, doi:10.1017/jsl.2023.89
iii) Katrin Tent and Martin Ziegler, A Course in Model Theory , Cambridge University Press, Cambridge, 2012.