If, not when

AbstractWe present a logic of verified and unverified assertions and prove it sound and complete with respect to its possible-worlds semantics. The logic, a constructive modal logic, is motivated by considerations of the interpretation of conditionals in natural language semantics, but, we claim, is of independent interest

Cut-Elimination for Full Intuitionistic Linear Logic

We describe in full detail a solution to the problem of proving the cut elimination theorem for FILL, a variant of (multiplicative and exponential-free) Linear Logicintroduced by Hyland and de Paiva. Hyland and de Paiva's work used a term assignmentsystem to describe FILL and barely sketched the proof of cut elimination. In this paper, as well as correcting a small mistake in their paper and extending thesystem to deal with exponentials, we introduce a different formal system describing the intuitionistic character of FILL and we provide a full proof of the cut eliminationtheorem. The formal system is based on a notion of dependence between formulae within a given proof and seems of independent interest. The procedure forcut elimination applies to (classical) multiplicative Linear Logic, and we can (with care) restrict our attention to the subsystem FILL. The proof, as usual with cutelimination proofs, is a little involved and we have not seen it published anywhere

Lineales

The first aim of this note is to describe an algebraic structure, more primitive than lattices and quantales, which corresponds to the intuitionistic flavour of Linear Logic we prefer. This part of the note is a total trivialisation of ideas from category theory and we play with a toy-structure a not distant cousin of a toy-language. The second goal of the note is to show a generic categorical construction, which builds models for Linear Logic, similar to categorical models GC of [deP1990], but more general. The ultimate aim is to relate different categorical models of linear logic

The Dialectica Categories

This thesis describes two classes of Dialectica categories. Chapter one introduces dialectica categories based on Goedel's Dialectica interpretation and shows that they constitute a model of Girard's Intuitionistic Linear Logic. Chapter two shows that, with extra assumptions, we can provide a comonad that interprets Girard's !-course modality. Chapter three presents the second class of Dialectica categories, a simplification suggested by Girard, that models (classical) Linear Logic and chapter four shows how to provide modalities ! and ? for this second class of construction

The ILLTP Library for Intuitionistic Linear Logic

Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of Kleene's intuitionistic theorems in the traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems