Intuitionistic Systems Research Articles

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

Nested sequents for intermediate logics: the case of Gödel-Dummett logics

We present nested sequent systems for propositional Gödel-Dummett logic and its first-order extensions with non-constant and constant domains, built atop nested calculi for intuitionistic logics. To obtain nested systems for these Gödel-Dummett logics, we introduce a new structural rule, called the linearity rule, which (bottom-up) operates by linearising branching structure in a given nested sequent. In addition, an interesting feature of our calculi is the inclusion of reachability rules, which are special logical rules that operate by propagating data and/or checking if data exists along certain paths within a nested sequent. Such rules require us to generalise our nested sequents to include signatures (i.e. finite collections of variables) in the first-order cases, thus giving rise to a generalisation of the usual nested sequent formalism. Our calculi exhibit favourable properties, admitting the height-preserving invertibility of every logical rule and the (height-preserving) admissibility of a large collection of structural and reachability rules. We prove all of our systems sound and cut-free complete, and show that syntactic cut-elimination obtains for the intuitionistic systems. We conclude the paper by discussing possible extensions and modifications, putting forth an array of structural rules that could be used to provide a sizable class of intermediate logics with cut-free nested sequent systems.

Maria Hämeen-Anttila and Jan von Plato, eds,Kurt Gödel: The Princeton Lectures on Intuitionism

This book publishes for the first time notes from two notebooks of Gödel which formed the basis of a course on intuitionism Gödel delivered at Princeton in the spring of 1941. These notes contain by far the most detailed treatment by Gödel (and anyone prior to Luckhardt [1973] and Troelstra [1973]) of his now famous functional (‘Dialectica’) interpretation which was published by Gödel himself only much later in the very brief paper [Gödel, 1958]. The essence of the Princeton notes is close to that of Gödel’s lecture at Yale from April 15, 1941, which was posthumously published in [Gödel, 1995], pp. 189–200. (See [Troelstra, 1995] for insightful comments on these notes.) However, the notes for the course given at Princeton are much more detailed covering in total nine lectures and contain interesting additional material and insights into what Gödel aimed to achieve by his interpretation. After Gödel’s functional interpretation was made public first by Kreisel in [1959] and then by Gödel himself in the aforementioned paper [1958], it received some constant interest until the early 70s, but then this interest declined for a while as Gentzen-style methods were considered to be superior being ‘ordinal informative’ and in 1978 the so-called |$A$|-translation [Friedman, 1978; Dragalin, 1980] provided a simple proof of the closure of intuitionistic systems such as Heyting arithmetic (HA) under Markov’s rule which by that time was considered to be a main application of the Dialectica interpretation. Beginning in the 90s (starting with [Kohlenbach, 1992]), Gödel’s functional interpretation saw a revival as a major tool in the emerging proof mining paradigm which aims at extracting new effective data from prima facie noneffective proofs in many areas of mathematics, most prominently in (nonlinear) analysis (see, e.g., [Kohlenbach, 2008]). This led to the development of many variants and extensions of Gödel’s original interpretation and numerous applications to core mathematics. By now, Gödel’s Dialectica paper [1958] is his second most cited (both according to Zentralblatt and MathSciNet) work on mathematical logic next to his celebrated incompleteness paper. Given this massive renewed interest in the Dialectica interpretation in the past thirty years, the publication of Gödel’s extensive course notes on the subject is of course highly relevant although some of its content had been already commented on in [Troelstra, 1995] (while certain parts of the notes were only discovered in 2017 according to the preface1). As these notes were written at the time when Gödel conceived his interpretation and contain many comments for himself on how to present things best in the lectures, they provide an insight into the way Gödel worked allowing the reader to, in a sense, look over his shoulder while he is working. That Gödel is still struggling himself with the material is illustrated by the fact that in one lecture he gives a wrong definition for the interpretation of ‘|$\vee$|’ (as in the notes to his Yale lecture) which he then acknowledges as false in the next lecture presenting a second attempt which is the final solution as known today.

One-Step Modal Logics, Intuitionistic and Classical, Part 2

Hodes (2021) “looked under the hood” of the familiar versions of the classical propositional modal logic K and its intuitionistic counterpart (see Plotkin & Sterling 1986). This paper continues that project, addressing some familiar classical strengthenings of K (D, T, K4, KB, K5, Dio (the Diodorian strengthening of K) and GL), and their intuitionistic counterparts (see Plotkin & Sterling 1986 for some of these counterparts). Section 9 associates two intuitionistic one-step proof-theoretic systems to each of the just mentioned intuitionistic logics, this by adding for each a new rule to those which generated IK in Hodes (2021). For the systems associated with the intuitionistic counterparts of D and T, these rules are “pure one-step”: their schematic formulations does not use □ or ♢. For the systems associated with the intuitionistic counterparts of K4, etc., these rules meet these conditions: neither □ nor ♢ is iterated; none use both □ and ♢. The join of the two systems associated with each of these familiar logics is the full one-step system for that intuitionistic logic. And further “blended” intuitionistic systems arise from joining these systems in various ways. Adding the 0-version of Excluded Middle to their intuitionistic counterparts yields the one-step systems corresponding to the familiar classical logics. Each proof-theoretic system defines a consequence relation in the obvious way. Section 10 examines inclusions between these consequence relations. Section 11 associates each of the above consequence relations with an appropriate class of models, and proves them sound with respect to their appropriate class. This allows proofs of some failures of inclusion between consequence relations. (Sections 10 and 11 provide an exhaustive study of a variety of intuitionistic modal logics.) Section 12 proves that the each consequence relation is complete or (for those corresponding to GL) weakly complete, that relative to its appropriate class of models. The Appendix presents three further results about some of the intuitionistic consequence relations discussed in the body of the paper.

König's lemma, weak König's lemma, and the decidable fan theorem

AbstractWe provide a fine‐grained analysis on the relation between König's lemma, weak König's lemma, and the decidable fan theorem in the context of constructive reverse mathematics. In particular, we show that double negated variants of König's lemma and weak König's lemma are equivalent to double negated variants of the general decidable fan theorem and the binary decidable fan theorem, respectively, over a nearly intuitionistic system containing a weak countable choice only. This implies that the general decidable fan theorem is not equivalent to the binary decidable fan theorem in the absence of a strong countable choice. In addition, we introduce a principle which fills the gap between König's lemma and weak König's lemma, and show that König's lemma is equivalent to weak König's lemma plus this principle. We also solve several problems arising from the investigation.

An ecumenical notion of entailment

Much has been said about intuitionistic and classical logical systems since Gentzen’s seminal work. Recently, Prawitz and others have been discussing how to put together Gentzen’s systems for classical and intuitionistic logic in a single unified system. We call Prawitz’ proposal the Ecumenical System, following the terminology introduced by Pereira and Rodriguez. In this work we present an Ecumenical sequent calculus, as opposed to the original natural deduction version, and state some proof theoretical properties of the system. We reason that sequent calculi are more amenable to extensive investigation using the tools of proof theory, such as cut-elimination and rule invertibility, hence allowing a full analysis of the notion of Ecumenical entailment. We then present some extensions of the Ecumenical sequent system and show that interesting systems arise when restricting such calculi to specific fragments. This approach of a unified system enabling both classical and intuitionistic features sheds some light not only on the logics themselves, but also on their semantical interpretations as well as on the proof theoretical properties that can arise from combining logical systems.

Monoidal logics: completeness and classical systems

ABSTRACTMonoidal logics were introduced as a foundational framework to analyze the proof theory of logical systems. Inspired by Lambek's seminal work in categorical logic, the objective is to define logical systems in order to make explicit their categorical (monoidal) structure. In this setting, logical connectives can be proven to be functors with specific properties. Accordingly, monoidal logics allow a classification of logical systems in function of their categorical structure and the functorial properties of their connectives. As they stand, however, strong parallels can be made between monoidal logics and the broader proof-theoretical framework of display logics. In this paper, we extend the results presented in Peterson ((2016). A comparison between monoidal and substructural logics. Journal of Applied Non-Classical Logics, 26(2), 126–159) and we show that monoidal logics are sound and complete with respect to associative display logics, thus providing a completeness result with regards to the algebraic semantics of display and substructural logics. In addition, we discuss the notions of classical and intuitionistic systems. Starting from Lambek's and Grishin's analyses, we explore the role played by partial De Morgan dualities and discuss the necessary and sufficient conditions required for the definitions of classical and intuitionistic deductive systems.

INTUITIONISTIC ANALYSIS AT THE END OF TIME

AbstractKripke recently suggested viewing the intuitionistic continuum as an expansion in time of a definite classical continuum. We prove the classical consistency of a three-sorted intuitionistic formal system IC, simultaneously extending Kleene’s intuitionistic analysis I and a negative copy C° of the classically correct part of I, with an “end of time” axiom ET asserting that no choice sequence can be guaranteed not to be pointwise equal to a definite (classical or lawlike) sequence. “Not every sequence is pointwise equal to a definite sequence” is independent of IC. The proofs are by Crealizability interpretations based on classical ω-models ${\cal M}$ = $\left( {\omega ,{\cal C}} \right)$ of C°.

The strength of countable saturation

In earlier work we introduced two systems for nonstandard analysis, one based on classical and one based on intuitionistic logic; these systems were conservative extensions of first-order Peano and Heyting arithmetic, respectively. In this paper we study how adding the principle of countable saturation to these systems affects their proof-theoretic strength. We will show that adding countable saturation to our intuitionistic system does not increase its proof-theoretic strength, while adding it to the classical system increases the strength from first- to full second-order arithmetic.

An Interpretability-Accuracy Tradeoff in Learning Parameters of Intuitionistic Fuzzy Rule-Based Systems

Parameter learning of Intuitionistic Fuzzy Rule-Based Systems (IFRBSs) is discussed and applied to medical diagnosis with intent of establishing a sound tradeoff between interpretability and accuracy. This study aims to improve the accuracy of IFRBSs without sacrificing its interpretability. This paper proposes an Objective Programming Method with an Interpretability-Accuracy tradeoff (OPMIA) to learn the parameters of IFRBSs by tuning the types of membership and non-membership functions and by adjusting adaptive factors and rule weights. The proposed method has been validated in the context of a medical diagnosis problem and a well-known publicly available auto-mpg data set. Furthermore, the proposed method is compared to Objective Programming Method not considering the interpretability (OPMNI) and Objective Programming Method based on Similarity Measure (OPMSM). The OPMIA helps achieve a sound a tradeoff between accuracy and interpretability and demonstrates its advantages over the other two methods.

Multi-focused Proofs with Different Polarity Assignments

In this work, we will reason on how a given focused proof, where atoms are assigned with some polarity, can be transformed into another focused proof, where the polarity assignment to atoms is changed. This will allow, in principle, transforming a proof obtained using one proof system into a proof using another proof system. More specifically, using the intuitionistic focused system LJF restricted to Harrop formulas, we define a procedure, introducing cuts, for transforming a focused proof where an atom is assigned with positive polarity into another focused proof where the same atom is assigned negative polarity and vice-versa. Then we show how to eliminate these cuts, obtaining a very interesting result: while the process of eliminating a cut on a positive atom gives rise to a proof with one smaller cut, in the negative case the number of introduced cuts grows exponentially. We end the paper by showing how to use maximal multi-focusing identify proofs in LJF, giving rise to a 1-1 translation between maximal proofs in LJF and proofs in the natural deduction system for intuitionistic logic NJ, restricted to Harrop formulas.

A MODAL TRANSLATION FOR DUAL-INTUITIONISTIC LOGIC

AbstractWe construct four binary consequence systems axiomatizing entailment relations between formulas of classical, intuitionistic, dual-intuitionistic and modal (S4) logics, respectively. It is shown that the intuitionistic consequence system is embeddable in the modal (S4) one by the usual modal translation prefixing □ to every subformula of the translated formula. An analogous modal translation of dual-intuitionistic formulas then consists of prefixing ◊ to every subformula of the translated formula. The philosophical importance of this result is briefly discussed.

Classical Consequences of Continuous Choice Principles from Intuitionistic Analysis

The sequential form of a statement ∀ξ(B(ξ)→∃ζA(ξ,ζ))(†) is the statement ∀ξ(∀nB(ξn)→∃ζ∀nA(ξn,ζn)). There are many classically true statements of the form (†) whose proofs lack uniformity, and therefore the corresponding sequential form is not provable in weak classical systems. The main culprit for this lack of uniformity is of course the law of excluded middle. Continuing along the lines of Hirst and Mummert, we show that if a statement of the form (†) satisfying certain syntactic requirements is provable in some weak intuitionistic system, then the proof is necessarily sufficiently uniform that the corresponding sequential form is provable in a corresponding weak classical system. Our results depend on Kleene’s realizability with functions and the Lifschitz variant thereof.

Degree Dominance Interval Relation-Based RSM in Intuitionistic Fuzzy Decision Systems

By introducing a degree dominance relation to dominance interval intuitionistic fuzzy decision systems, we establish a degree dominance interval rough set model (RSM), which is mainly based on replacing the indiscernibility relation in classical rough set theory with the degree dominance interval relation. To simplify knowledge representation and extract some nontrivial simpler degree dominance interval intuitionistic fuzzy decision rules, we propose two attribute reductions of the degree dominance interval intuitionistic fuzzy decision systems that eliminate the redundant condition attributes that are not essential from the viewpoint of degree dominance interval intuitionistic fuzzy decision rules.

Labelled natural deduction for a bundled branching temporal logic

We give a sound and complete labelled natural deduction system for a bundled branching temporal logic, namely the until-free version of BCTL*. The logic BCTL* is obtained by referring to a more general semantics than that of CTL*, where we only require that the set of paths in a model is closed under taking suffixes (i.e. is suffix-closed) and is closed under putting together a finite prefix of one path with the suffix of any other path beginning at the same state where the prefix ends (i.e. is fusion-closed). In other words, this logic does not enjoy the so-called limit-closure property of the standard CTL* validity semantics. We give both a classical and an intuitionistic version of our labelled natural deduction system for the until-free version of BCTL*, and carry out a proof-theoretical analysis of the intuitionistic system: we prove that derivations reduce to a normal form, which allows us to give a purely syntactical proof of consistency (for both the intuitionistic and classical versions) of the deduction system.

Intuitionistic Frege systems are polynomially equivalent

In a paper by Cook and Reckhow (1979), it is shown that any two classical Frege systems polynomially simulate each other. The same proof does not work for intuitionistic Frege systems, since they can have nonderivable admissible rules. (The rule A/B is derivable if the formula A → B is derivable. The rule A/B is admissible if for all substitutions σ, if σ(A) is derivable, then σ(B) is derivable.) In this paper, we polynomially simulate a single admissible rule. Therefore any two intuitionistic Frege systems polynomially simulate each other. Bibliography: 20 titles.

Notes on Constructive Negation

We put together several observations on constructive negation. First, Russell anticipated intuitionistic logic by clearly distinguishing propositional principles implying the law of the excluded middle from remaining valid principles. He stated what was later called Peirce’s law. This is important in connection with the method used later by Heyting for developing his axiomatization of intuitionistic logic. Second, a work by Dragalin and his students provides easy embeddings of classical arithmetic and analysis into intuitionistic negationless systems. In the last section, we present in some detail a stepwise construction of negation which essentially concluded the formation of the logical base of the Russian constructivist school. Markov’s own proof of Markov’s principle (different from later proofs by Friedman and Dragalin) is described.

Paraconsistent logics and translations

In 1999, da Silva, D'Ottaviano and Sette proposed a general definition for the term translation between logics and presented an initial segment of its theory. Logics are characterized, in the most general sense, as sets with consequence relations and translations between logics as consequence-relation preserving maps. In a previous paper the authors introduced the concept of conservative translation between logics and studied some general properties of the co-complete category constituted by logics and conservative translations between them. In this paper we present some conservative translations involving classical logic, Lukasiewicz three-valued system L 3, the intuitionistic system I 1 and several paraconsistent logics, as for instance Sette's system P 1, the D'Ottaviano and da Costa system J 3 and da Costa's systems C n, 1≤ n≤ω.

A common axiom set for classical and intuitionistic plane geometry

We describe a first order axiom set which yields the classical first order Euclidean geometry of Tarski when used with classical logic, and yields an intuitionistic (or constructive) Euclidean geometry when used with intuitionistic logic. The first order language has a single six place atomic predicate and no function symbols. The intuitionistic system has a computational interpretation in recursive function theory, that is, a realizability interpretation analogous to those given by Kleene for intuitionistic arithmetic and analysis. This interpretation shows the unprovability in the intuitionistic theory of certain “nonconstructive” theorems of the classical geometry.

Intuitionistic and classical natural deduction systems with the catch and the throw rules

Intuitionistic and classical natural deduction systems with the catch and the throw rules