Proof Theory for Intuitionistic Stable Theories

In this paper we first introduce a new neighbourhood semantics for propositional intuitionistic logic. We then naturally extend this semantics to first-order intuitionistic logic. We also study bisimulation between neighbourhood models and prove some of their basic properties for both propositional and first-order intuitionistic logic.

Intuitionistic completeness of first-order logic

A Lindström theorem for intuitionistic first-order logic

Prawitz suggested expanding a natural deduction system for intuitionistic logic to include rules for classical logic constructors, allowing both intuitionistic and classical elements to coexist without losing their inherent characteristics. Looking at the added rules from the point of view of the Gödel-Gentzen translation, led us to propose a general method for the coexistent combination of two logics when a conservative translation exists from one logic (the source) to another (the host). Then we prove that the combined logic is a conservative extension of the original logics, thereby preserving the unique characteristics of each component logic. In this way there is no collapse of one logic into the other in the combination. We also demonstrate that a Gentzen calculus for the combined logic can be induced from a Gentzen calculus for the host logic by considering the translation. This approach applies to semantics as well. We then establish a general sufficient condition for ensuring that the combined logic is both sound and complete. We apply these principles by combining classical and intuitionistic logics capitalizing on the Gödel-Gentzen conservative translation, intuitionistic and S4 modal logics relying on the Gödel-McKinsey-Tarski conservative translation, and classical and Jaśkowski’s paraconsistent logics taking into account the existence of a conservative translation.

From Strategies to Derivations and Back: An Easy Completeness Proof for First-Order Intuitionistic Dialogical Logic

The paper investigates interpretations of propositional and first-order logic in which validity is defined in terms of partial indices; sometimes called possibilities but here understood as non-empty subsets of a set W of possible worlds. Truth at a set of worlds is understood to be truth at every world in the set. If all subsets of W are permitted the logic so determined is classical first-order predicate logic. Restricting allowable subsets and then imposing certain closure conditions provides a modelling for intuitionistic predicate logic. The same semantic interpretation rules are used in both logics for all the operators.

Classical logic is symmetric with respect to True and False but intuitionistic logic is not. We introduce and study a conservative extension of first-order intuitionistic logic that is symmetric with respect to True and False.

A multi-succedent sequent calculus for intuitionistic epistemic logic (IEL) is introduced in the paper. It is proved that the structural rules of weakening and contraction and the rule of cut are admissible in the calculus. It is also proved that any sequent with at most one formula in succedent is derivable in the calculus, iff it is derivable in the standard non-multi-succedent sequent calculus of IEL.

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."

Automatic reasoning tools play an important role when developing provably correct software. Both main approaches, program verification and program synthesis employ automated reasoning tools, more specifically, automated theorem provers. Besides classical logic, non-classical logics are particularly relevant in this field. This chapter presents calculi to automate theorem proving in classical and some important non-classical logics, namely first-order intuitionistic and first-order modal logics. These calculi are based on the connection method, which permits a goal-oriented and, hence, a more efficient proof search. The connection calculi for these non-classical logics extend the calculus for classical logic in an elegant and uniform way by adding so-called prefixes to atomic formulae. The leanCoP theorem prover is a very compact PROLOG implementation of the connection calculus for classical logics. We present details of the implementation and describe some basic techniques to improve its efficiency. leanCoP is adapted to non-classical logics by integrating a prefix unification algorithm, which depends on the specific logic. This results in leading theorem provers for the aforementioned non-classical logics.

In this paper, we share our experiences concerning the introduction of the Android-based self-study app AXolotl within the first-semester logic course offered at our university. This course is mandatory for students majoring in Computer Science and Artificial Intelligence. AXolotl was used as part of an optional lab assignment bridging clausal reasoning and SAT solving with classical reasoning, proof construction, and first-order logic. The app provides an intuitive interface for proof construction in various logical calculi and aids the students through rule application. The goal of the lab assignment was to help students make a smoother transition from clausal and decompositional reasoning used earlier in the course to inferential and contextual reasoning required for proof construction and first-order logic. We observed that the lab had a positive influence on students' understanding and end the paper with a discussion of these results.

In the domain of ontology design as well as in Conceptual Modeling, representing part-whole relations is a long-standing challenging problem. However, in most papers the focus has been on properties of the part-whole relation, rather than on its semantics. In the last decades, most approaches which have addressed the formal specification of the part-whole relation (i) rely on First Order Logic (FOL) which is unable to address multiple levels of granularity and (ii) do not support any typing mechanism useful for the extensional side of concepts and then, many difficulties remain especially about expressiveness. In mathematical logic and program checking, type theories have proved to be appealing but so far, they have not been applied in the formalization of ontological relations. To bridge this gap, we present an axiomatization of the part-whole relation which hold between typed terms. Relation structures in the dependently-typed framework rely on a constructive logic. We define in a precise way what relation structures and their meta-properties, are in term of type classes using the Coq language.

We formalize in Isabelle/HOL soundness and completeness of a one-sided sequent calculus for first-order logic. The completeness is shown via a translation from a semantic tableau calculus, whose completeness proof we base on the theory entry ‘First-Order Logic According to Fitting’ by Berghofer in the Archive of Formal Proofs. The calculi and proof techniques are taken from Ben-Ari’s textbook Mathematical Logic for Computer Science (Springer, 2012). We thereby demonstrate that Berghofer’s approach works not only for natural deduction but also constitutes a framework for mechanically checked completeness proofs for a range of proof systems.

Classical first-order logic is in many ways central to work in mathematics, linguistics, computer science and artificial intelligence, so it is worthwhile to define it in full detail. We present soundness and completeness proofs of a sequent calculus for first-order logic, formalized in the interactive proof assistant Isabelle/HOL. Our formalization is based on work by Stefan Berghofer, which we have since updated to use Isabelle's declarative proof style Isar (Archive of Formal Proofs, Entry FOL-Fitting, August 2007 / July 2018). We represent variables with de Bruijn indices; this makes substitution under quantifiers less intuitive for a human reader. However, the nature of natural numbers yields an elegant solution when compared to implementations of substitution using variables represented by strings. The sequent calculus considered has the special property of an always empty antecedent and a list of formulas in the succedent. We obtain the proofs of soundness and completeness for the sequent calculus as a derived result of the inverse duality of its tableau counterpart. We strive to not only present the results of the proofs of soundness and completeness, but also to provide a deep dive into a programming-like approach to the formalization of first-order logic syntax, semantics and the sequent calculus. We use the formalization in a bachelor course on logic for computer science and discuss our experiences.

: With better natural language semantic representations, computers can do more applications more efficiently as a result of better understanding of natural text. However, no single semantic representation at this time fulfills all requirements needed for a satisfactory representation. Logic-based representations like first-order logic capture many of the linguistic phenomena using logical constructs, and they come with standardized inference mechanisms, but standard first-order logic fails to capture the graded aspect of meaning in languages. Distributional models use contextual similarity to predict the graded semantic similarity of words and phrases but they do not adequately capture logical structure. In addition there are a few recent attempts to combine both representations either on the logic side (still, not a graded representation), or in the distribution side (not full logic). We propose using probabilistic logic to represent natural language semantics combining the expressivity and the automated inference of logic, and the gradedness of distributional representations. We evaluate this semantic representation on two tasks, Recognizing Textual Entailment (RTE) and Semantic Textual Similarity (STS). Doing RTE and STS better is an indication of a better semantic understanding. Our system has three main components, 1. Parsing and Task Representation, 2. Knowledge Base Construction, and 3. Inference. The input natural sentences of the RTE/STS task are mapped to logical form using Boxer which is a rule based system built on top of a CCG parser, then they are used to formulate the RTE/STS problem in probabilistic logic. Then, a knowledge base is represented as weighted inference rules collected from different sources like WordNet and on-the-fly lexical rules from distributional semantics.