Periodic Reporting for period 1 - REMODEL (Structures for modal and deontic logics)
Okres sprawozdawczy: 2024-06-01 do 2026-05-31
The aim of the REMODEL project is to provide a logical framework to describe, analyze and explain normative reasoning. At its core, the project is concerned with advancing logical methods for the study of deontic logics—a class of non-classical logics designed to formalize notions such as obligation, permission, and related normative concepts. These systems are useful to represent legal reasoning and ethical dilemmas. From a semantic point of view, they can be grouped in two categories: preference-based systems and norm-based ones. The main technical tool to represent formal reasoning is proof theory.
Proof theory, a branch of mathematical logic, investigates the structure of proofs (or derivations) to gain insights into the properties of formal systems. Traditionally, deontic logics are presented using Hilbert-style—or axiomatic—calculi. These systems consist of a set of axioms and a small number of inference rules, offering a compact and elegant formulation of non-classical logics. Moreover, they are modular: new systems can be constructed from base systems by simply adding axioms. However, while axiomatic systems are elegant, they are not well-suited for studying the internal structure of proofs or for practical reasoning tasks, as constructing derivations in them can be particularly challenging.
To foster applicability of non-classical logics it is desirable to develop analytic calculi. In contrast to axiomatic calculi, analytic systems allow for a bottom-up analysis of statements or arguments which are decomposed through the rules of the calculus. Therefore, analytic calculi allow for backward reasoning and they are key to design automated proof methods. The most prominent analytic proof methods are sequent calculi and their generalizations. While standard sequent calculi manipulate multisets of formulas, their extensions—such as hypersequents, nested sequents, and labelled sequents—operate on more complex structures like multisets of sequents, trees, or graphs of sequents, respectively.
The REMODEL project will provide uniform analytic calculi for deontic logics. A well-designed analytic calculus for deontic logics will be able to perform a twofold task. On the one hand, when a given formula or argument (suitably written in the formal Language) is valid, the calculus must produce a derivation which certifies its validity. On the other hand, whenever it turns out to be false, the calculus has to halt the search for a proof after a finite number of steps and then exhibit a finite countermodel, i.e. a semantic counterexample which gives a reason as to why the formula does not hold.
The project will begin by addressing preference-based deontic logics, before moving on to study non-monotonic forms of deontic reasoning. A key methodological approach of REMODEL is the transfer of results from modal logic to deontic logic. This will be achieved by establishing formal translations between the two, thereby creating bridges that allow for the reuse of well-established techniques and results from modal logic. This not only facilitates the development of analytic calculi for deontic logics but also opens up new perspectives in the theory of modal logics themselves.
Proof theory, a branch of mathematical logic, investigates the structure of proofs (or derivations) to gain insights into the properties of formal systems. Traditionally, deontic logics are presented using Hilbert-style—or axiomatic—calculi. These systems consist of a set of axioms and a small number of inference rules, offering a compact and elegant formulation of non-classical logics. Moreover, they are modular: new systems can be constructed from base systems by simply adding axioms. However, while axiomatic systems are elegant, they are not well-suited for studying the internal structure of proofs or for practical reasoning tasks, as constructing derivations in them can be particularly challenging.
To foster applicability of non-classical logics it is desirable to develop analytic calculi. In contrast to axiomatic calculi, analytic systems allow for a bottom-up analysis of statements or arguments which are decomposed through the rules of the calculus. Therefore, analytic calculi allow for backward reasoning and they are key to design automated proof methods. The most prominent analytic proof methods are sequent calculi and their generalizations. While standard sequent calculi manipulate multisets of formulas, their extensions—such as hypersequents, nested sequents, and labelled sequents—operate on more complex structures like multisets of sequents, trees, or graphs of sequents, respectively.
The REMODEL project will provide uniform analytic calculi for deontic logics. A well-designed analytic calculus for deontic logics will be able to perform a twofold task. On the one hand, when a given formula or argument (suitably written in the formal Language) is valid, the calculus must produce a derivation which certifies its validity. On the other hand, whenever it turns out to be false, the calculus has to halt the search for a proof after a finite number of steps and then exhibit a finite countermodel, i.e. a semantic counterexample which gives a reason as to why the formula does not hold.
The project will begin by addressing preference-based deontic logics, before moving on to study non-monotonic forms of deontic reasoning. A key methodological approach of REMODEL is the transfer of results from modal logic to deontic logic. This will be achieved by establishing formal translations between the two, thereby creating bridges that allow for the reuse of well-established techniques and results from modal logic. This not only facilitates the development of analytic calculi for deontic logics but also opens up new perspectives in the theory of modal logics themselves.
In its initial phase, the project concentrated on developing analytic calculi for preference-based deontic logics within the Åqvist family, focusing in particular on the logics F + (CM) and G. To achieve this, the researcher, in collaboration with the project supervisor Agata Ciabattoni, employed both sequent calculi extended with analytic cuts—which permit the application of the cut rule on subformulas—and hypersequent calculi. The resulting systems exhibited desirable structural properties, including the admissibility of the (non-analytic) cut rule, and were proved to be sound and complete. In addition, they supported countermodel extraction in cases where proof search failed, thus enhancing their applicability to automated reasoning.
Subsequently, the researcher collaborated with Han Gao, Emiliano Lorini, and Nicola Olivetti to develop an analytic calculus for the logic of evaluation—a system that integrates three distinct modal operators to formalize ethical dilemmas. The complex interaction between these modalities—two normal and one non-normal—was effectively managed using hypersequents with blocks, which captured the underlying connections between preference, knowledge, and values.
In parallel, the researcher also conducted independent work on the proof theory of modal logics using nested sequents, and collaborated with Mario Piazza on topics in non-monotonic reasoning. These efforts laid the grounds for the project's upcoming focus on non-monotonic deontic systems.
Finally, in joint work with Agata Ciabattoni and Dmitry Rozplokhas, the project introduced a novel family of calculi for Åqvist logics, leveraging their connections with the provability logic GL. As a significant byproduct, the project extended beyond its original objectives by identifying new syntactic methods relevant to the field of conditional logics.
Subsequently, the researcher collaborated with Han Gao, Emiliano Lorini, and Nicola Olivetti to develop an analytic calculus for the logic of evaluation—a system that integrates three distinct modal operators to formalize ethical dilemmas. The complex interaction between these modalities—two normal and one non-normal—was effectively managed using hypersequents with blocks, which captured the underlying connections between preference, knowledge, and values.
In parallel, the researcher also conducted independent work on the proof theory of modal logics using nested sequents, and collaborated with Mario Piazza on topics in non-monotonic reasoning. These efforts laid the grounds for the project's upcoming focus on non-monotonic deontic systems.
Finally, in joint work with Agata Ciabattoni and Dmitry Rozplokhas, the project introduced a novel family of calculi for Åqvist logics, leveraging their connections with the provability logic GL. As a significant byproduct, the project extended beyond its original objectives by identifying new syntactic methods relevant to the field of conditional logics.
The REMODEL project has delivered several contributions that advance the state of the art. To begin with, it introduced new proof systems for the Åqvist family of deontic logics, thereby completing their proof-theoretic characterization. These systems also support effective reasoning and countermodel construction.
Moreover, the project established new conceptual bridges between modal and deontic logics, particularly by leveraging the connection between deontic and provability logics. Building on a semantic framework based on finite models, the truth conditions for deontic operators were reformulated using an internal modality from the family of provability logics. This led to a uniform and modular presentation of the Åqvist systems up to F + (CM).
Finally, the project achieved a novel result in the area of conditional logics. Specifically, it developed a simple sequent calculus for PCL (Preferential Conditional Logic), a logic for which a satisfactory proof-theoretic framework had previously been lacking.
Moreover, the project established new conceptual bridges between modal and deontic logics, particularly by leveraging the connection between deontic and provability logics. Building on a semantic framework based on finite models, the truth conditions for deontic operators were reformulated using an internal modality from the family of provability logics. This led to a uniform and modular presentation of the Åqvist systems up to F + (CM).
Finally, the project achieved a novel result in the area of conditional logics. Specifically, it developed a simple sequent calculus for PCL (Preferential Conditional Logic), a logic for which a satisfactory proof-theoretic framework had previously been lacking.