Structural Proof Theory

ABSTRACT: This paper contends that Stoic logic (i.e. Stoic analysis) deserves more attention from contemporary logicians. It sets out how, compared with contemporary propositional calculi, Stoic analysis is closest to methods of backward proof search for Gentzen-inspired substructural sequent logics, as they have been developed in logic programming and structural proof theory, and produces its proof search calculus in tree form. It shows how multiple similarities to Gentzen sequent systems combine with intriguing dissimilarities that may enrich contemporary discussion. Much of Stoic logic appears surprisingly modern: a recursively formulated syntax with some truth-functional propositional operators; analogues to cut rules, axiom schemata and Gentzen's negation-introduction rules; an implicit variable-sharing principle and deliberate rejection of Thinning and avoidance of paradoxes of implication. These latter features mark the system out as a relevance logic, where the absence of duals for its left and right introduction rules puts it in the vicinity of McCall's connexive logic. Methodologically, the choice of meticulously formulated meta-logical rules in lieu of axiom and inference schemata absorbs some structural rules and results in an economical, precise and elegant system that values decidability over completeness.

Una presentazione molto sintetica e semplice dei problemi relativi alla traduzione tra i due sistemi di derivazione introdotti da Gentzen. Analizzo la proposta di traduzione avanzata originariamente da Gentzen e le modifiche proposte da Prawitz e Von Plato.

The present paper provides an analysis of the existing proof systems for dynamic epistemic logic from the viewpoint of proof-theoretic semantics. Dynamic epistemic logic is one of the best known members of a family of logical systems which have been successfully applied to diverse scientific disciplines, but the proof theoretic treatment of which presents many difficulties. After an illustration of the proof-theoretic semantic principles most relevant to the treatment of logical connectives, we turn to illustrating the main features of display calculi, a proof-theoretic paradigm which has been successfully employed to give a proof-theoretic semantic account of modal and substructural logics. Then, we review some of the most significant proposals of proof systems for dynamic epistemic logics, and we critically reflect on them in the light of the previously introduced proof-theoretic semantic principles. The contributions of the present paper include a generalisation of Belnap's cut elimination metatheorem for display calculi, and a revised version of the display-style calculus D.EAK [30]. We verify that the revised version satisfies the previously mentioned proof-theoretic semantic principles, and show that it enjoys cut elimination as a consequence of the generalised metatheorem.

In this thesis we consider generic tools and techniques for the proof-theoretic investigation of not necessarily normal modal logics based on minimal, intuitionistic or classical propositional logic. The underlying framework is that of ordinary symmetric or asymmetric two-sided sequent calculi without additional structural connectives, and the point of interest are the logical rules in such a system. We introduce the format of a sequent rule with context restrictions and the slightly weaker format of a shallow rule. The format of a rule with context restrictions is expressive enough to capture most normal modal logics in the S5 cube, standard systems for minimal, intuitionistic and classical propositional logic and a wide variety of non-normal modal logics. For systems given by such rules we provide sufficient criteria for cut elimination and decidability together with generic complexity results. We also explore the expressivity of such systems with the cut rule in terms of axioms in a Hilbert-style system by exhibiting a corresponding syntactically defined class of axioms along with automatic translations between axioms and rules. This enables us to show a number of limitative results concerning amongst others the modal logic S5. As a step towards a generic construction of cut free and tractable sequent calculi we then introduce the notion of cut trees as representations of rules constructed by absorbing cuts. With certain limitations this allows the automatic construction of a cut free and tractable sequent system from a finite number of rules. For cases where such a system is to be constructed by hand we introduce a graphical representation of rules with context restrictions which simplifies this process. Finally, we apply the developed tools and techniques and construct new cut free sequent systems for a number of Lewis' conditional logics extending the logic V. The systems yield purely syntactic decision procedures of optimal complexity and proofs of the Craig interpolation property for the logics at hand. 5

Which modal logics can be 'naturally' captured by a sequent system? Clearly, this question hinges on what one believes to be natural, i.e. which format of sequent rules one is willing to accept. This paper studies the relationship between the format of sequent rules and the corresponding syntactical shape of axioms in an equivalent Hilbert-system. We identify three different such formats, the most general of which captures most logics in the S5-cube. The format is based on restricting the context in rule premises and the correspondence is established by translating axioms into rules of our format and vice versa. As an application we show that there is no set of sequent rules of this format which is sound and cut-free complete for S5 and for which cut elimination can be shown by the standard permutation-of-rules argument.

We provide a direct method for proving Craig interpolation for a range of modal and intuitionistic logics, including those containing a "converse" modality. We demonstrate this method for classical tense logic, its extensions with path axioms, and for bi-intuitionistic logic. These logics do not have straightforward formalisations in the traditional Gentzen-style sequent calculus, but have all been shown to have cut-free nested sequent calculi. The proof of the interpolation theorem uses these calculi and is purely syntactic, without resorting to embeddings, semantic arguments, or interpreted connectives external to the underlying logical language. A novel feature of our proof includes an orthogonality condition for defining duality between interpolants.

Motivated by the fact that nearly all conditional logics are axiomatised by so-called shallow axioms (axioms with modal nesting depth ≤ 1) we investigate sequent calculi and cut elimination for modal logics of this type. We first provide a generic translation of shallow axioms to (one-sided, unlabelled) sequent rules. The resulting system is complete if we admit pseudo-analytic cut, i.e. cuts on modalised propositional combinations of subformulas, leading to a generic (but sub-optimal) decision procedure. In a next step, we show that, for finite sets of axioms, only a small number of cuts is needed between any two applications of modal rules. More precisely, completeness still holds if we restrict to cuts that form a tree of logarithmic height between any two modal rules. In other words, we obtain a small (Pspace-computable) representation of an extended rule set for which cut elimination holds. In particular, this entails Pspace decidability of the underlying logic if contraction is also admissible. This leads to (tight) Pspace bounds for various conditional logics.

In this paper, we define a multi-type calculus for inquisitive logic, which is sound, complete and enjoys Belnap-style cut-elimination and subfor-mula property. Inquisitive logic is the logic of inquisitive semantics, a semantic framework developed by Groenendijk, Roelofsen and Ciardelli which captures both assertions and questions in natural language. Inquisitive logic adopts the so-called support semantics (also known as team semantics). The Hilbert-style presentation of inquisitive logic is not closed under uniform substitution, and some axioms are sound only for a certain subclass of formulas, called flat formulas. This and other features make the quest for analytic calculi for this logic not straightforward. We develop a certain algebraic and order-theoretic analysis of the team semantics, which provides the guidelines for the design of a multi-type environment accounting for two domains of interpretation, for flat and for general formulas, as well as for their interaction. This multi-type environment in its turn provides the semantic environment for the multi-type calculus for inquisitive logic we introduce in this paper. Acknowledgements.

As part of a general research programme into the expressive power of different generalisations of the sequent framework we investigate hypersequent calculi given by rules of the newly introduced format of hypersequent rules with context restrictions. The introduced rule format is used to prove uniform syntactic cut elimination, decidability and complexity results. We also introduce transformations between hypersequent rules of this format and Hilbert axioms, entailing a result about the limits of such rules. As case studies, we apply our methods to several modal logics and obtain e.g. a complexity-optimal decision procedure for the logic S5 and new calculi for the logic K4.2 as well as combinations of modal logics in the form of simply dependent bimodal logics.

In the present paper, we introduce a multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus. The display-approach is suitable to modularly chart the space of dynamic epistemic logics on weaker-than-classical propositional base. The presence of types endows the language of the Dynamic Calculus with additional expressivity, allows for a smooth proof-theoretic treatment, and paves the way towards a general methodology for the design of proof systems for the generality of dynamic logics, and certainly beyond dynamic epistemic logic. We prove that the Dynamic Calculus adequately captures Baltag-Moss-Solecki's dynamic epistemic logic, and enjoys Belnap-style cut elimination.

We introduce a new Gentzen-style framework of grafted hypersequents that combines the formalism of nested sequents with that of hypersequents. To illustrate the potential of the framework, we present novel calculi for the modal logics K5 and KD5, as well as for extensions of the modal logics K and KD with the axiom for shift reflexivity. The latter of these extensions is also known as SDL+ in the context of deontic logic. All our calculi enjoy syntactic cut elimination and can be used in backwards proof search procedures of optimal complexity.

This thesis is mainly about Proof Theory.
It can be thought of as
Proof Theory
in the sense of Hilbert, Gentzen,
Sch\"utte, Buchholz, Rathjen, and in general
what could be called the
German school, but
it is also influenced
by many other branches,
of which the bibliography
might give an idea.
Intuitionism
and other
philosophical
approaches to mathematics are also
an important part of
what is studied, but
the Leitmotif of this thesis is 
Cut Elimination. 
The first part of
the thesis is concerned with
countable coded $\omega$-models of Bar Induction.
In this part we work from a reverse mathematics point of view.
A study for
an ordinal analysis of the theory of Bar
Induction BI is carried out,
and the equivalence between
the statement that every set is contained in an
$\omega$-model of this theory BI
and the well-ordering principle
which says that if X is a well-ordering,
then so is its Bachmann-Howard relativisation,
is proven.
In the second part of the thesis we
shift our viewpoint and
consider intuitionistic logic
and intuitionistic geometric theories.
We show that geometric
derivability in
classical infinitary logic
implies derivability in
intuitionistic
infinitary logic.
Again, our main
tool is Cut Elimination.
Next, we
present investigations regarding minimal logic and classical logical
principles, and give a precise classification of excluded middle, ex
falso, and double negation elimination.

The present paper provides an analysis of the existing proof systems for dynamic epistemic logic from the viewpoint of proof-theoretic semantics. Dynamic epistemic logic is one of the best known members of a family of logical systems which have been successfully applied to diverse scientific disciplines, but the proof theoretic treatment of which presents many difficulties. After an illustration of the proof-theoretic semantic principles most relevant to the treatment of logical connectives, we turn to illustrating the main features of display calculi, a proof-theoretic paradigm which has been successfully employed to give a proof-theoretic semantic account of modal and substructural logics. Then, we review some of the most significant proposals of proof systems for dynamic epistemic logics, and we critically reflect on them in the light of the previously introduced proof-theoretic semantic principles. The contributions of the present paper include a generalisation of Belnap's cut elimination metatheorem for display calculi, and a revised version of the display-style calculus D.EAK [30]. We verify that the revised version satisfies the previously mentioned proof-theoretic semantic principles, and show that it enjoys cut elimination as a consequence of the generalised metatheorem.

We present cut-free labelled sequent calculi for a central formalism in logics of agency: STIT logics with temporal operators. These include sequent systems for Ldm, Tstit and Xstit. All calculi presented possess essential structural properties such as contraction-and cut-admissibility. The labelled calculi G3Ldm and G3Tstit are shown sound and complete relative to irreflexive temporal frames. Additionally, we extend current results by showing that also Xstit can be characterized through relational frames, omitting the use of BT+AC frames.

The reduction of undecidable first-order logic to decidable propositional logic via Herbrand's theorem has long been of interest to theoretical computer science, with the notion of a Herbrand proof motivating the definition of expansion proofs. The problem of building a natural proof system around expansion proofs, with composition of proofs and cut-free completeness, has been approached from a variety of different angles. In this paper we construct a simple deep inference system for first-order logic, KSh2, based around the notion of expansion proofs, as a starting point to developing a rich proof theory around this foundation. Translations between proofs in this system and expansion proofs are given, retaining much of the structure in each direction.

University Of Aberdeen This paper explores how a semantics for Prior’s infamous connective tonk should be, a connective defined by inference rules that trivialize the logic of a deductive system if that logic is supposed to be transitive. To avoid triviality, one must reject transitivity and in a relatively recent paper, Roy Cook develops a semantics for tonk with non-transitive entailment. However, I show in this paper that a cut-free sequent calculus for tonk - the arguably most natural and simplest deductive system for a non-transitive logic - can neither be complete with respect to Cook’s semantics nor with respect to a semantics with non-transitive entailment based on a semantics for vagueness and transparent truth developed by Cobreros et al. It is argued that the failure to adequately represent tonk is connected with the fact that tonk is not uniquely defined in a cut-free sequent calculus system unless the logic is in addition non-reflexive. To remedy this, the paper develops a semantics with non-transitive and non-reflexive entailment based on the idea that complex formulae are true or false relative to them being assessed as premise or as conclusion.(Received August 05 2014)

We introduce transformations between hypersequent rules with context restrictions and Hilbert axioms extending classical (and intuitionistic) propositional logic and vice versa. The introduced rules are used to prove uniform cut elimination, decidability and complexity results as well as finite axiomatisations for many modal logics given by simple frame properties. Our work subsumes many logic-tailored results and allows for new results. As a case study we apply our methods to the logic of uniform deontic frames.

The aim of this paper is to introduce and explain display calculi for a variety of logics. We provide a survey of key results concerning such calculi, though we focus mainly on the global cut elimination theorem. Propositional, first-order, and modal display calculi are considered and their properties detailed.

In the present paper, we introduce a multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus. The displayapproach is suitable to modularly chart the space of dynamic epistemic logics on weaker-than-classical propositional base. The presence of types endows the language of the Dynamic Calculus with additional expressivity, allows for a smooth proof-theoretic treatment, and paves the way towards a general methodology for the design of proof systems for the generality of dynamic logics, and certainly beyond dynamic epistemic logic. We prove that the Dynamic Calculus adequately captures Baltag-Moss-Solecki's dynamic epistemic logic, and enjoys Belnap-style cut elimination.

This paper shows how to derive nested calculi from labelled calculi for propositional intuitionistic logic and first-order intuitionistic logic with constant domains, thus connecting the general results for labelled calculi with the more refined formalism of nested sequents. The extraction of nested calculi from labelled calculi obtains via considerations pertaining to the elimination of structural rules in labelled derivations. Each aspect of the extraction process is motivated and detailed, showing that each nested calculus inherits favorable proof-theoretic properties from its associated labelled calculus.

On the verge of the 20th century, Charles S. Peirce was convinced that his Existential Graphs were the best form of presenting every deductive argument. Between 1900 and 1909, Peirce chose the scroll as a basic sign in his Alpha system for Existential Graphs. According to a recent paper by Francesco Bellucci and Ahti-Veikko Pietarinen, the reason for this choice lies mainly in the non-analyzable nature of the scroll: Only one sign expresses the basic notion of illation. In this paper, some analogies between this early version of the Alpha system and Structural Reasoning (in the sense of Kosta Došen and Peter Schröder-Heister) are explored. From these analogies, it will be claimed that the system Alpha based on the scroll can be used as an accurate framework for (i) constructing basic structural deductions and (ii) accomplishing a diagrammatic interpretation of logical constants of First-Order Language. Moreover, EGs show cognitive advantages with respect to sequent systems. In this paper, the basic conception is outlined in an informal way, without making an exposition of the technical details.

We introduce a display calculus for the logic of Epistemic Actions and Knowledge (EAK) of Baltag-Moss-Solecki. This calculus is cut-free and complete w.r.t. the standard Hilbert-style presentation of EAK, of which it is a conservative extension, given that—as is common to display calculi—it is defined on an expanded language in which all logical operations have adjoints. The additional dynamic operators do not have an interpretation in the standard Kripke semantics of EAK, but do have a natural interpretation in the final coalgebra. This proof-theoretic motivation revives the interest in the global semantics for dynamic epistemic logics pursued among others by Baltag [4], Cˆırstea and Sadrzadeh [8].