During the first two years of the realization of the project, the team of researchers working in the project has been actively developing a proof theory of complex terms. The main effort was to construct well-behaving proof systems, like sequent calculi, tableau systems and natural deduction systems for several approaches to definite descriptions. Not only well-established theories, like the one by Russell or Frege, were dealt with, but also some new theories were proposed. All these attempts were analysed both in the context of standard formal languages and of richer languages, like the language of hybrid temporal logic. The theories of definite descriptions were investigated in different logics weaker than classical logic, like several variants of free logic where terms are not required to denote. For some of them, like neutral free logic which admits sentences which are neither true nor false, some non-standard proof systems were developed.
Most of the solutions provided so far are based on the formalisation of definite descriptions in terms of the iota-operator, sometimes accompanied with other operators like the lambda-operator. The basics of proof systems for the general theory of term-forming operators were presented and published in the proceedings of the conference TABLEAUX in Prague 2023. The approach developed in this way has recently been successfully applied to the construction of proof systems for set theories with set abstracts. This leads to a fruitful investigation of alternative set theories, like Quine’s system NF. Philosophical applications of systems with term-forming operators were also investigated, for example in the formalisation of Anzelm’s argument for the existence of God.
In addition to the investigation of systems based on the application of term-forming operators, some alternative approaches are also being developed. One, which uses a special kind of binary quantifier to represent definite descriptions provides significantly different solutions. Another alternative approach, originally proposed by Leśniewski as a calculus of names, called ontology, was formalised in terms of well-behaved sequent calculus. Both approaches were successfully developed and presented at several conferences and in publications.
In the works devoted to the presentation of these approaches to the proof theory of complex terms important metatheoretical problems were also investigated. In particular, interpolation theorems which are important indicators of the expressivity of theories, were proved for many systems, including the Russellian theory of definite descriptions, temporal hybrid logic with definite descriptions and Leśniewski’s ontology. A separate line of research resulted in a comparison of expressivity and computational complexity of systems with definite descriptions and without them. Investigation into decidability problems for restricted fragments are ongoing.
