Bounded-analytic sequent calculi and embeddings for hypersequent logics

A sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics (specifically: those with a cut-free hypersequent calculus). We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory

Goal-directed proof theory

This report is the draft of a book about goal directed proof theoretical formulations of non-classical logics. It evolved from a response to the existence of two camps in the applied logic (computer science/artificial intelligence) community. There are those members who believe that the new non-classical logics are the most important ones for applications and that classical logic itself is now no longer the main workhorse of applied logic, and there are those who maintain that classical logic is the only logic worth considering and that within classical logic the Horn clause fragment is the most important one. The book presents a uniform Prolog-like formulation of the landscape of classical and non-classical logics, done in such away that the distinctions and movements from one logic to another seem simple and natural; and within it classical logic becomes just one among many. This should please the non-classical logic camp. It will also please the classical logic camp since the goal directed formulation makes it all look like an algorithmic extension of Logic Programming. The approach also seems to provide very good compuational complexity bounds across its landscape

Proof search in multi-succedent sequent calculi for intuitionistic logics (Theory and Applications of Proof and Computation)

In this note, terminating and bicomplete proof search procedures with respect to the Kripke semantics are given in multi-succedent sequent calculi for intuitionistic propositional logic and fragments of intuitionistic predicate logic. G. Mints [11, 12] in his later years investigated a proof search procedure in single-succedent sequent calculus for intuitionistic predicate logic

On Deriving Nested Calculi for Intuitionistic Logics from Semantic Systems

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

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

Self-referentiality in Constructive Semantics of Intuitionistic and Modal Logics

This thesis explores self-referentiality in the framework of justification logic. In this framework initialed by Artemov, the language has formulas of the form t:F, which means the term t is a justification of the formula F. Moreover, terms can occur inside formulas and hence it is legal to have t:F(t), which means the term t is a justification of the formula F about t itself. Expressions like this is not only interesting in the semantics of justification logic, but also, as we will see, necessary in applications of justification logic in formalizing constructive contents implicitly carried by modal and intuitionistic logics. Works initialed by Artemov and followed by Brezhnev and others have successfully extracted constructive contents packaged by modality in many modal logics. Roughly speaking, they offer methods of substituting modalities by terms in various justification logics, and then computing the exact structure of each term. After performing these methods, each formula prefixed by a modality becomes a formula prefixed by a term, which intuitively stands for the justification of the formula being prefixed. In terminology of this framework, we say that modal logics are realized in justification logics. Within the family of justification logics, the Logic of Proofs LP is perhaps the most important member. As Artemov showed, this logic is not only complete w.r.t. to arithmetical semantics about proofs, but also accommodates the modal logic S4 via realization. Combined with Godel\u27s modal embedding from intuitionistic propositional logic IPC to S4, the Logic of Proofs LP serves as an intermedium via which IPC receives its provability semantics, also known as Brouwer-Heyting-Kolmogorov semantics, or BHK semantics. This thesis presents the candidate\u27s works in two directions. (1) Following Kuznets\u27result that self-referentiality is necessary for the realization of several modal logics including S4, we show that it is also necessary for BHK semantics. (2) We find a necessary condition for a modal theorem to require self-referentiality in its realization, and using this condition to derive many interesting properties about self-referentiality