Sequent Calculus Research Articles

The Structure of Paradoxes in a Logic of Sentential Operators

AbstractAny language $$\mathcal {L}$$ L of classical logic, of first- or higher-order, is expanded with sentential quantifiers and operators. The resulting language $$\mathcal {L}^+\!$$ L + , capable of self-reference without arithmetic or syntax encoding, can serve as its own metalanguage. The syntax of $$\mathcal {L}^+$$ L + is represented by directed graphs, and its semantics, which coincides with the classical one on $$\mathcal {L}$$ L , uses the graph-theoretic concepts of kernels and semikernels. Kernels provide an explosive semantics, while semikernels generalize this to situations where paradoxes do not lead to explosion, thus distinguishing them from contradictions. Paradoxes arise only at the metalevel due to specific interpretations of the operators, but they can be avoided: $$\mathcal {L}^+$$ L + can express paradoxes but remains free from them. For an expansion $$\mathcal {L}^+$$ L + of any FOL language $$\mathcal {L}$$ L , with the non-explosive semantics, a complete reasoning system is obtained by extending Gentzen’s classical sequent calculus with two rules for the sentential quantifiers. Adding (cut) yields a complete system for the explosive semantics. The novel semantics and self-referential capabilities seem promising for a further extension of classical logic towards one capable also of consistently expressing its own syntax and truth theory.

Adaptive human–machine theorem proving system

This paper presents a novel approach to constructing human–machine theorem proving systems. These systems integrate machine learning capabilities, human expert knowledge, and rigorous logical control for the effective construction and verification of proofs. The innovation of this approach lies in its openness: the user is given a tool for building such systems. Users can create theorem proving systems by selecting existing logical inference strategies or adding new ones in accordance with the provided interfaces or relationship agreements. The systems being built have a significant human--machine adaptive nature. During their operation, a meta-strategy is developed and trained based on the foundational elements of the existing strategies. The system is designed as a universal framework for managing various strategies, with the provision of a basic architecture and a library of strategies with the possibility of adding new ones. During the learning process, a system of structural characteristics is also accumulated and trained, on the basis of which a decision is made on the use of specific strategy elements at the next step. The presentation is conducted for the sequential calculus of minimal positive predicate logic as the most suitable for the deductive synthesis of programs and algorithms, for which it is proposed to use this approach.

On structural proof theory of the modal logic K+ extended with infinitary derivations

Abstract We consider an extension of the modal logic of transitive closure $\textsf{K}^{+}$ with certain infinitary derivations and present a sequent calculus for this extension, which allows non-well-founded proofs. We establish continuous cut-elimination for the given calculus using fixed-point theorems for contractive mappings. The infinitary derivations mentioned above are well founded and countably branching, while the non-well-founded proofs of the sequent calculus can only be finitely branching. The ordinary derivations in $\textsf{K}^{+}$, as we show additionally, correspond to the non-well-founded proofs of the calculus that are regular and cut-free. Therefore, in this article, we explore the relationship between deductive systems for $\textsf{K}^{+}$ with well-founded infinitely branching derivations and (regular) non-well-founded finitely branching proofs.

Quantum modal logic

Abstract A modal logic based on quantum logic is formalized in its simplest possible form. Specifically, a relational semantics and a sequent calculus are provided, and the soundness and the completeness theorems connecting both notions are demonstrated. This framework is intended to serve as a basis for formalizing various modal logics over quantum logic, such as quantum alethic logic, quantum temporal logic, quantum epistemic logic, and quantum dynamic logic.

Strong Kleene Logics as a Tool for Modelling Formal Epistemic Norms

In this paper, we present two ways of modelling every epistemic formal conditional commitment that involves (at most) three key epistemic attitudes: acceptance, rejection and neither acceptance nor rejection. The first one consists of adopting the plurality of every mixed Strong Kleene logic (along with an epistemic reading of the truth-values), and the second one involves the use of a unified system of six-sided inferences, named 6SK, that recovers the validities of each mixed Strong Kleene logic. We also introduce a sequent calculus that is sound and complete with respect to both approaches. We compare both accounts, and finally, we suggest that the plurality of Strong Kleene logic as well as the general framework 6SK are linked to formal epistemic norms via bridge principles.

Universal proof theory: Feasible admissibility in intuitionistic modal logics

We introduce a general and syntactically defined family of sequent-style calculi over the propositional language with the modalities {□,◇} and its fragments as a formalization for constructively acceptable systems. Calling these calculi constructive, we show that any strong enough constructive sequent calculus, satisfying a mild technical condition, feasibly admits all Visser's rules. This means that there exists a polynomial-time algorithm that, given a proof of the premise of a Visser's rule, provides a proof for its conclusion. As a positive application, we establish the feasible admissibility of Visser's rules in sequent calculi for several intuitionistic modal logics, including CK, IK, their extensions by the modal axioms T, B, 4, 5, and the axioms for bounded width and depth and their fragments CK□, propositional lax logic and IPC. On the negative side, we show that if a strong enough intuitionistic modal logic (satisfying a mild technical condition) does not admit at least one of Visser's rules, it cannot have a constructive sequent calculus. Consequently, no intermediate logic other than IPC has a constructive sequent calculus.

Bilateralism, coherence, and incoherence

AbstractBilateralism is the view that the speech act of denial is as primitive as that of assertion. Bilateralism has proved helpful in providing an intuitive interpretation of formalisms that, prima facie, look counterintuitive, namely, multiple‐conclusion sequent calculi. Under this interpretation, a sequent of the form is regarded as the statement that it is incoherent, according to our conversational norms, to occupy the position of asserting all the sentences in and denying all the sentences in . Some have argued, based on this interpretation, that the notion of invalidity is as conceptually primitive and important as the notion of validity: whereas the latter is couched in terms of what positions are incoherent and hence untenable, the former is couched in terms of what positions are coherent and hence tenable. My ultimate goal in this paper is to contest this view. Based on a novel technical account of the two notions—one that I find more accurate than the existing accounts in the literature—I shall argue that the notion of incoherence takes conceptual priority over the notion of coherence.

Coherence via focusing for symmetric skew monoidal and symmetric skew closed categories

Abstract The symmetric skew monoidal categories of Bourke and Lack are a weakening of Mac Lane’s symmetric monoidal categories where (i) the three structural laws of left and right unitality and associativity are not required to be invertible, they are merely natural transformations with a specific orientation; (ii) the structural law of symmetry is a natural isomorphism involving three objects rather than two. In a similar fashion, symmetric skew closed categories are a weakening of de Schipper’s symmetric closed categories with non-invertible structural laws of left and right unitality. In this paper, we study the structural proof theory of symmetric skew monoidal and symmetric skew closed categories, progressing the project initiated by Uustalu et al. on deductive systems for categories with skew structure. We discuss three equivalent presentations of the free symmetric skew monoidal (resp. closed) category on a set of generating objects: a Hilbert-style categorical calculus; a cut-free sequent calculus; a focused subsystem of derivations, corresponding to a sound and complete goal-directed proof search strategy for the cut-free sequent calculus. Focusing defines an effective normalization procedure for maps in the free symmetric skew monoidal (resp. closed) category, as such solving the coherence problem for symmetric skew monoidal (resp. closed) categories.

The higher dimensional propositional calculus

Abstract In recent research, some of the present authors introduced the concept of an $n$-dimensional Boolean algebra and its corresponding propositional logic $n\textrm{CL}$, generalizing the Boolean propositional calculus to $n\geq 2$ perfectly symmetric truth values. This paper presents a sound and complete sequent calculus for $n\textrm{CL}$, named $n\textrm{LK}$. We provide two proofs of completeness: one syntactic and one semantic. The former implies as a corollary that $n\textrm{LK}$ enjoys the cut admissibility property. The latter relies on the generalization to the $n$-ary case of the classical proof based on the Lindenbaum algebra of formulas and Boolean ultrafilters.

Universal proof theory: Semi-analytic rules and Craig interpolation

We provide a general and syntactically defined family of sequent calculi, called semi-analytic, to formalize the informal notion of a “nice” sequent calculus. We show that any sufficiently strong (multimodal) substructural logic with a semi-analytic sequent calculus enjoys the Craig Interpolation Property, CIP. As a positive application, our theorem provides a uniform and modular method to prove the CIP for several multimodal substructural logics, including many fragments and variants of linear logic. More interestingly, on the negative side, it employs the lack of the CIP in almost all substructural, superintuitionistic and modal logics to provide a formal proof for the well-known intuition that almost all logics do not have a “nice” sequent calculus. More precisely, we show that many substructural logics including UL−, MTL, R, Łn (for n⩾3), Gn (for n⩾4), and almost all extensions of IMTL, Ł, BL, RMe, IPC, S4, and Grz (except for at most 1, 1, 3, 8, 7, 37, and 6 of them, respectively) do not have a semi-analytic calculus.

Grokking the Sequent Calculus (Functional Pearl)

The sequent calculus is a proof system which was designed as a more symmetric alternative to natural deduction. The 𝜆𝜇𝜇-calculus is a term assignment system for the sequent calculus and a great foundation for compiler intermediate languages due to its first-class representation of evaluation contexts. Unfortunately, only experts of the sequent calculus can appreciate its beauty. To remedy this, we present the first introduction to the 𝜆𝜇𝜇-calculus which is not directed at type theorists or logicians but at compiler hackers and programming-language enthusiasts. We do this by writing a compiler from a small but interesting surface language to the 𝜆𝜇𝜇-calculus as a compiler intermediate language.

Uniform Cut-Free Bisequent Calculi for Three-Valued Logics

We present a uniform characterisation of three-valued logics by means of a bisequent calculus (BSC). It is a generalised form of a sequent calculus (SC) where rules operate on the ordered pairs of ordinary sequents. BSC may be treated as the weakest kind of system in the rich family of generalised SC operating on items being some collections of ordinary sequents, like hypersequent and nested sequent calculi. It seems that for many non-classical logics, including some many-valued, paraconsistent and modal logics, the reasonably modest generalisation of standard SC offered by BSC is sufficient. In this paper, we examine a variety of three-valued logics and show how they can be formalised in the framework of BSC. We present a constructive syntactic proof that these systems are cut-free, satisfy the subformula property, and allow one to prove the interpolation theorem in many cases.

Decidability of topological quasi-Boolean algebras

A sequent calculus S for the variety tqBa of all topological quasi-Boolean algebras is established. Using a construction of syntactic finite algebraic model, the finite model property of S is shown, and thus the decidability of S is obtained. We also introduce two non-distributive variants of topological quasi-Boolean algebras. For the variety TDM 5 of all topological De Morgan lattices with the axiom 5, we establish a sequent calculus S 5 and prove that the cut elimination holds for it. Consequently the decidability of S 5 is established. Furthermore, this proof-theoretic method is applied to some subvarieties of TDM 5 .

Sequent Calculi for First-order ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ST}$$\\end{document}

Strict-Tolerant Logic (ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ST}$$\\end{document}) underpins naïve theories of truth and vagueness (respectively including a fully disquotational truth predicate and an unrestricted tolerance principle) without jettisoning any classically valid laws. The classical sequent calculus without Cut is sometimes advocated as an appropriate proof-theoretic presentation of ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ST}$$\\end{document}. Unfortunately, there is only a partial correspondence between its derivability relation and the relation of local metainferential ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ST}$$\\end{document}-validity – these relations coincide only upon the addition of elimination rules and only within the propositional fragment of the calculus, due to the non-invertibility of the quantifier rules. In this paper, we present two calculi for first-order ST\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{ST}$$\\end{document} with an eye to recapturing this correspondence in full. The first calculus is close in spirit to the Epsilon calculus. The other calculus includes rules for the discharge of sequent-assumptions; moreover, it is normalisable and admits interpolation.

Verifying a Sequent Calculus Prover for First-Order Logic with Functions in Isabelle/HOL

We describe the design, implementation and verification of an automated theorem prover for first-order logic with functions. The proof search procedure is based on sequent calculus and we formally verify its soundness and completeness in Isabelle/HOL using an existing abstract framework for coinductive proof trees. Our analytic completeness proof covers both open and closed formulas. Since our deterministic prover considers only the subset of terms relevant to proving a given sequent, we do the same when building a countermodel from a failed proof. Finally, we formally connect our prover with the proof system and semantics of the existing SeCaV system. In particular, the prover can generate human-readable SeCaV proofs which are also machine-verifiable proof certificates. The abstract framework we rely on requires us to fix a stream of proof rules in advance, independently of the formula we are trying to prove. We discuss the efficiency implications of this and the difficulties in mitigating them.

On Combining Intuitionistic and S4 Modal Logic

We address the problem of combining intuitionistic and S4 modal logic in a non-collapsing way inspired by the recent works in combining intuitionistic and classical logic. The combined language includes the shared constructors of both logics namely conjunction, disjunction and falsum as well as the intuitionistic implication, the classical implication and the necessity modality. We present a Gentzen calculus for the combined logic defined over a Gentzen calculus for the host S4 modal logic. The semantics is provided by Kripke structures. The calculus is proved to be sound and complete with respect to this semantics. We also show that the combined logic is a conservative extension of each component. Finally we establish that the Gentzen calculus for the combined logic enjoys cut elimination.

On Verified Automated Reasoning in Propositional Logic: Teaching Sequent Calculus to Computer Science Students

As the complexity of software systems is ever increasing, so is the need for practical tools for formal verification. Among these are automatic theorem provers, capable of solving various reasoning problems automatically, and proof assistants, capable of deriving more complex results when guided by a mathematician/programmer. In this paper we consider using the latter to build the former. In the proof assistant Isabelle/HOL we combine functional programming and logical program verification to build a theorem prover for propositional logic. We also consider how such a prover can be used to solve a reasoning task without much mental labor. The development is extended with a formalized proof system for writing machine-checked sequent calculus proofs. We consider how this can be used to teach computer science students about logic, automated reasoning and proof assistants.

Attack principles in sequent-based argumentation theory

Abstract Attack principles have been introduced in semi-abstract argumentation frameworks and, in the present work, we interpret them in sequent-based argumentation frameworks. Thus, we investigate the role of minimality and consistency of the support set of an argument. Through the notion of preservation of strength, we introduce a formal criterion to sort out the attack principles; isolate the more “acceptable” ones, i.e. those easier to justify; and recover a new argumentative semantics for the non-classical logic that arises from dropping the rules $(\neg , r)$, $(\land , r)$ and $(\supset , l)$ from Gentzen’s classical sequent calculus for classical logic $\textsf{LK}$.

Sequent Calculi for Choice Logics

Choice logics constitute a family of propositional logics and are used for the representation of preferences, with especially qualitative choice logic (QCL) being an established formalism with numerous applications in artificial intelligence. While computational properties and applications of choice logics have been studied in the literature, only few results are known about the proof-theoretic aspects of their use. We propose a sound and complete sequent calculus for preferred model entailment in QCL, where a formula F is entailed by a QCL-theory T if F is true in all preferred models of T. The calculus is based on labeled sequent and refutation calculi, and can be easily adapted for different purposes. For instance, using the calculus as a cornerstone, calculi for other choice logics such as conjunctive choice logic (CCL) and lexicographic choice logic (LCL) can be obtained in a straightforward way.

ONE-VARIABLE FRAGMENTS OF FIRST-ORDER LOGICS

AbstractThe one-variable fragment of a first-order logic may be viewed as an “S5-like” modal logic, where the universal and existential quantifiers are replaced by box and diamond modalities, respectively. Axiomatizations of these modal logics have been obtained for special cases—notably, the modal counterparts $\mathrm {S5}$ and $\mathrm {MIPC}$ of the one-variable fragments of first-order classical logic and first-order intuitionistic logic, respectively—but a general approach, extending beyond first-order intermediate logics, has been lacking. To this end, a sufficient criterion is given in this paper for the one-variable fragment of a semantically defined first-order logic—spanning families of intermediate, substructural, many-valued, and modal logics—to admit a certain natural axiomatization. More precisely, an axiomatization is obtained for the one-variable fragment of any first-order logic based on a variety of algebraic structures with a lattice reduct that has the superamalgamation property, using a generalized version of a functional representation theorem for monadic Heyting algebras due to Bezhanishvili and Harding. An alternative proof-theoretic strategy for obtaining such axiomatization results is also developed for first-order substructural logics that have a cut-free sequent calculus and admit a certain interpolation property.