A focused framework for emulating modal proof systems

International audienceSeveral deductive formalisms (e.g., sequent, nested sequent, labeled sequent, hyperse-quent calculi) have been used in the literature for the treatment of modal logics, and some connections between these formalisms are already known. Here we propose a general framework, which is based on a focused version of the labeled sequent calculus by Negri, augmented with some parametric devices allowing to restrict the set of proofs. By properly defining such restrictions and by choosing an appropriate polarization of formulas, one can obtain different, concrete proof systems for the modal logic K and for its extensions by means of geometric axioms. In particular, we show how to use the expressiveness of the labeled approach and the control mechanisms of focusing in order to emulate in our framework the behavior of a range of existing formalisms and proof systems for modal logic

Another plan for negation

The paper  presents a plan for negation, proposing a paradigm shift from the Australian plan for negation,  leading to a family of contra-classical logics. The two main ideas are the following:  Instead of shifting points of evaluation (in a frame), shift the evaluated formula. Introduce an incompatibility set for every atomic formula, extended to any compound formula, and impose the condition on valuations that a formula evaluates to true iff all the formulas in its incompatibility set evaluate to false. Thus, atomic sentences are not independent in their truth-values.  The resulting negation, in addition to excluding the negated formula, provides a positive alternative to the negated formula. I  also present a sound and complete natural deduction proof systems for those logics. In addition, the kind of negation considered in this paper is shown to provide an innovative notion of grounding negation.  &nbsp

Beyond semantic pollution: Towards a practice-based philosophical analysis of labelled calculi

This paper challenges the negative attitudes towards labelled proof systems, usually referred to as semantic pollution, by arguing that such critiques overlook the full potential of labelled calculi. The overarching objective is to develop a practice-based philosophical analysis of labelled calculi to provide insightful considerations regarding their proof-theoretic and philosophical value. To achieve this, successful applications of labelled calculi and related results will be showcased, and comparisons with other relevant works will be discussed. The paper ends by advocating for a more practice-based approach towards the philosophical understanding of proof systems and their role in structural proof theory

Views from a peak:Generalisations and descriptive set theory

This dissertation has two major threads, one is mathematical, namely descriptive set theory, the other is philosophical, namely generalisation in mathematics. Descriptive set theory is the study of the behaviour of definable subsets of a given structure such as the real numbers. In the core mathematical chapters, we provide mathematical results connecting descriptive set theory and generalised descriptive set theory. Using these, we give a philosophical account of the motivations for, and the nature of, generalisation in mathematics.In Chapter 3, we stratify set theories based on this descriptive complexity. The axiom of countable choice for reals is one of the most basic fragments of the axiom of choice needed in many parts of mathematics. Descriptive choice principles are a further stratification of this fragment by the descriptive complexity of the sets. We provide a separation technique for descriptive choice principles based on Jensen forcing. Our results generalise a theorem by Kanovei.Chapter 4 gives the essentials of a generalised real analysis, that is a real analysis on generalisations of the real numbers to higher infinities. This builds on work by Galeotti and his coauthors. We generalise classical theorems of real analysis to certain sets of functions, strengthening continuity, and disprove other classical theorems. We also show that a certain cardinal property, the tree property, is equivalent to the Extreme Value Theorem for a set of functions which generalize the continuous functions.The question of Chapter 5 is whether a robust notion of infinite sums can be developed on generalisations of the real numbers to higher infinities. We state some incompatibility results, which suggest not. We analyse several candidate notions of infinite sum, both from the literature and more novel, and show which of the expected properties of a notion of sum they fail.In Chapter 6, we study the descriptive set theory arising from a generalization of topology, κ-topology, which is used in the previous two chapters. We show that the theory is quite different from that of the standard (full) topology. Differences include a collapsing Borel hierarchy, a lack of universal or complete sets, Lebesgue’s ‘great mistake’ holds (projections do not increase complexity), a strict hierarchy of notions of analyticity, and a failure of Suslin’s theorem.Lastly, in Chapter 7, we give a philosophical account of the nature of generalisation in mathematics, and describe the methodological reasons that mathematicians generalise. In so doing, we distinguish generalisation from other processes of change in mathematics, such as abstraction and domain expansion. We suggest a semantic account of generalisation, where two pieces of mathematics constitute a generalisation if they have a certain relation of content, along with an increased level of generality

Reasoning with Attitude:Foundations and Applications of Inferential Expressivism

Certain combinations of sounds or signs on paper are meaningful. What makes it the case that, unlike most combinations of sounds answers to these questions are based on the idea that words stand for something, but it is difficult to say what words such as good, if, or probable stand for. This book advances novel answers based on the idea that words get their meaning from the way they are used to express states of mind and what follows from them. It articulates a precise version of this idea, at a time when the shortcomings of the traditional answers are hotly discussed

Topics in the Proof Theory of Non-classical Logics. Philosophy and Applications

Chapter 1 constitutes an introduction to Gentzen calculi from two perspectives, logical and philosophical. It introduces the notion of generalisations of Gentzen sequent calculus and the discussion on properties that characterize good inferential systems. Among the variety of Gentzen-style sequent calculi, I divide them in two groups: syntactic and semantic generalisations. In the context of such a discussion, the inferentialist philosophy of the meaning of logical constants is introduced, and some potential objections – mainly concerning the choice of working with semantic generalizations – are addressed. Finally, I’ll introduce the case studies that I’ll be dealing with in part II. Chapter 2 is concerned with the origins and development of Jaśkowski’s discussive logic. The main idea of this chapter is to systematize the various stages of the development of discussive logic related researches from two different angles, i.e., its connections to modal logics and its proof theory, by highlighting virtues and vices. Chapter 3 focuses on the Gentzen-style proof theory of discussive logic, by providing a characterization of it in terms of labelled sequent calculi. Chapter 4 deals with the Gentzen-style proof theory of relevant logics, again by employing the methodology of labelled sequent calculi. This time, instead of working with a single logic, I’ll deal with a whole family of them. More precisely, I’ll study in terms of proof systems those relevant logics that can be characterised, at the semantic level, by reduced Routley-Meyer models, i.e., relational structures with a ternary relation between states and a unique base element. Chapter 5 investigates the proof theory of a modal expansion of intuitionistic propositional logic obtained by adding an ‘actuality’ operator to the connectives. This logic was introduced also using Gentzen sequents. Unfortunately, the original proof system is not cut-free. This chapter shows how to solve this problem by moving to hypersequents. Chapter 6 concludes the investigations and discusses the future of the research presented throughout the dissertation

Semantic pollution and syntactic purity

Logical inferentialism claims that the meaning of the logical constants should be given, not model-theoretically, but by the rules of inference of a suitable calculus. It has been claimed that certain proof-theoretical systems, most particularly, labelled deductive systems for modal logic, are unsuitable, on the grounds that they are semantically polluted and suffer from an untoward intrusion of semantics into syntax. The charge is shown to be mistaken. It is argued on inferentialist grounds that labelled deductive systems are as syntactically pure as any formal system in which the rules define the meanings of the logical constants.PostprintPeer reviewe