Periodic Reporting for period 1 - ExtenDD (Coming to Terms: Proof Theory Extended to Definite Descriptions and other Terms)
Période du rapport: 2022-10-01 au 2025-03-31
Logic is concerned with an analysis of reasoning. The optimal forms of reasoning are proofs, and in modern logic, a special branch, called proof theory, was developed as a formal tool for analysing proofs. After the development of computer science, the results and tools of proof theory were successfully applied in the area of automated deduction.
One of the main differences between natural languages and formal languages applied in logic and computer science is the way naming phrases, briefly called terms, are treated. In natural languages there is a variety of such phrases: simple and complex, in singular and plural form. Complex terms exhibit a structure that contributes to their meaning. The most important examples of such expressions in natural languages are definite and indefinite descriptions, expressions of the form ‘the so and so’, and ‘a so and so’. They are used not only to refer to objects (sometimes there is nothing to refer to) but also to convey information. On the other hand, in formal languages, the role of terms is strongly restricted. They are mainly used to refer to objects, but not to convey information; the latter is performed by predicates. In fact, complex terms are very helpful in formal sciences as tools for communication. Set abstracts of the form `the set of all objects such that ...’, may serve as an example. However, proof theory and automated deduction developed tools suitable only for dealing with simple referring terms; complex terms are just translated away. The investigation into complex terms was neglected in this area and left to informal analyses by linguists and philosophers of language.
The project ExtenDD aims to develop the apparatus of proof theory to handle various forms of complex terms. So far, only a small effort has been put into an adequate treatment of these phenomena. ExtenDD aims to fill this gap by providing novel, proof-theoretic tools for several term-forming operators that allow us to formalise complex terms. Applying the methods of proof theory to complex terms is profitable to all sides. Competing theories of definite descriptions, their advantages and shortcomings, are shown in a new light. The behaviour of complex terms needs a subtle syntactical analysis and requires a significant enrichment of the toolkit of proof theory.
One of the main differences between natural languages and formal languages applied in logic and computer science is the way naming phrases, briefly called terms, are treated. In natural languages there is a variety of such phrases: simple and complex, in singular and plural form. Complex terms exhibit a structure that contributes to their meaning. The most important examples of such expressions in natural languages are definite and indefinite descriptions, expressions of the form ‘the so and so’, and ‘a so and so’. They are used not only to refer to objects (sometimes there is nothing to refer to) but also to convey information. On the other hand, in formal languages, the role of terms is strongly restricted. They are mainly used to refer to objects, but not to convey information; the latter is performed by predicates. In fact, complex terms are very helpful in formal sciences as tools for communication. Set abstracts of the form `the set of all objects such that ...’, may serve as an example. However, proof theory and automated deduction developed tools suitable only for dealing with simple referring terms; complex terms are just translated away. The investigation into complex terms was neglected in this area and left to informal analyses by linguists and philosophers of language.
The project ExtenDD aims to develop the apparatus of proof theory to handle various forms of complex terms. So far, only a small effort has been put into an adequate treatment of these phenomena. ExtenDD aims to fill this gap by providing novel, proof-theoretic tools for several term-forming operators that allow us to formalise complex terms. Applying the methods of proof theory to complex terms is profitable to all sides. Competing theories of definite descriptions, their advantages and shortcomings, are shown in a new light. The behaviour of complex terms needs a subtle syntactical analysis and requires a significant enrichment of the toolkit of proof theory.
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.
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.
The results of ExtenDD were presented at several conferences, including CADE (Rome 2023), TABLEAUX (Prague 2023), JELIA (Dresden 2023), AWPL (Sapporo 2024), LC (Gothenburg 2024), IJCAR (Nancy 2024), KR (Hanoi 2024). Presentations at several workshops and seminars (e.g. in Bochum 2022, London 2023, Sendai 2024, Tokio 2024, Dagstuhl 2024, Stirling 2023 and 2024) were also important occasions for dissemination of the results. In many cases the talks gave rise to publications in conference proceedings. Other papers were published in top-tier journals, including Synthese and Review of Symbolic Logic. So far eight papers were published and at least four are accepted for publication. A special online seminar is organised on a bi-weekly basis for regular presentations of the work of team members but also guest speakers. Some of them visited Łódź to present talks and have consultations. We prepared in cooperation with Bochum University two workshops and the 12th edition of the conference Non-classical Logics in Łódź.
We are planning to go beyond strictly theoretical research and focus on the construction of proof assistants and automated provers. It makes it possible to more efficiently realise reasoning tasks in natural languages and compare them with the standard approaches. To this aim the team of ExtenDD was extended with a specialist in programming.
Let us summarize the outcomes of ExtenDD after two years of realization. Eight papers were published in top-tier journals and proceedings of the leading conferences in logic and computer science. The results were presented at several conferences, workshops and seminars all over the world. Special workshops as a joined activity of Bochum and Łódź were organised in Bochum and in Łódź, as well as one edition of the conference Non-classical Logics: Theory and Applications in Łódź. The online seminar is dedicated to the ongoing presentation of the research of team members and guest speakers working on topics related to ExtenDD. Moreover, as a result of the established collaborations, some guests visited Łódź University for consultations and talks.
The full record of ExtenDD activities can be found on the project website: EXTENDD
We are planning to go beyond strictly theoretical research and focus on the construction of proof assistants and automated provers. It makes it possible to more efficiently realise reasoning tasks in natural languages and compare them with the standard approaches. To this aim the team of ExtenDD was extended with a specialist in programming.
Let us summarize the outcomes of ExtenDD after two years of realization. Eight papers were published in top-tier journals and proceedings of the leading conferences in logic and computer science. The results were presented at several conferences, workshops and seminars all over the world. Special workshops as a joined activity of Bochum and Łódź were organised in Bochum and in Łódź, as well as one edition of the conference Non-classical Logics: Theory and Applications in Łódź. The online seminar is dedicated to the ongoing presentation of the research of team members and guest speakers working on topics related to ExtenDD. Moreover, as a result of the established collaborations, some guests visited Łódź University for consultations and talks.
The full record of ExtenDD activities can be found on the project website: EXTENDD