Logical Calculus Research Articles

Teaching a Formalized Logical Calculus

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.

A cut-free labelled sequent calculus for dynamic epistemic logic

Abstract Dynamic epistemic logic is a logic that is aimed at formally expressing how a person’s knowledge changes. We provide a cut-free labelled sequent calculus ($\textbf{GDEL}$) on the background of existing studies of Hilbert-style axiomatization $\textbf{HDEL}$ of dynamic epistemic logic and labelled calculi for public announcement logic. We first show that the $cut$ rule is admissible in $\textbf{GDEL}$ and show that $\textbf{GDEL}$ is sound and complete for Kripke semantics. Moreover, we show that the basis of $\textbf{GDEL}$ is extended from modal logic K to other familiar modal logics including S5 with keeping the admissibility of cut, soundness and completeness.

Influence mechanism of congestion degree of layout on domino effect in petrochemical plants

The study focused on the establishment of the congestion degree in describing the layout of process plants, and discovering the influence mechanism of congestion degree on the risk of domino effect, in view of the fact that the congestion degree has not been used to represent the characteristics of petrochemical plant layout. A characterization model was built for the congestion degree of process plant layouts applying the theory of fuzzy model that it took safety distance, the amount of materials involved and the hazard of these substances into consideration, and came out with a membership function through logical calculus, which made congestion degree a comparable data. Secondly, a systematic domino effect risk analysis method was generalized on account of domino effect quantitative risk analysis method which include the equipment damage probability, the domino effect occurrence probability, risk analysis based on the consequences, individual risk index and social risk index, the process plant natural risk index and real risk index, and then these indices were compared to select the most suitable ones to illustrate the risk of domino effect and to research about part of the storage tank field of a petrochemical plant. Finally, the influence mechanism of layout congestion degree on domino effect of regional accident was obtained through this quantitative analysis. Simultaneously, a range of the congestion degree was acquired according to the individual risk index which can be used as a reference in the design stage of the process plant. The result proved that high congestion degree in process plant greatly increased the risk of domino effect in most of the quantitative risk analysis index, such as the probability of resulting in secondary accidents, the harm on both human and the equipments caused by the initial accident, and the individual risk and social risk, etc. The characterization models for the congestion degree and the domino risk analysis method established and summarized in the study have been applied in the design stage of process plant layout to optimize the layout, and thus reduce the risk of domino effect caused by accidents.

Różne znaczenia definicji w logice. O krytyce Ajdukiewicza wobec klasyfikacji rozumowań Łukasiewicza, uwag kilka

Przedmiot badań: Artykuł analizuje rzadko poruszaną kwestię wyrażanej przez Kazimierza Aj­dukiewicza, polskiego logika, zwolennika logiki dwuwartościowej, krytyki klasyfikacji rozumo­wań autorstwa Jana Łukasiewicza, twórcy trójwartościowego rachunku logicznego.
 Cel badawczy: Celem artykułu jest próba wykazania, że podjęta przez Ajdukiewicza krytyka systemu rozumowań Łukasiewicza dotyczy problematyki roli i miejsca definicji w logice. Ajdu­kiewicz, który był uczniem Łukasiewicza, uznawanego na całym świcie za geniusza logiki, nie wdawał się w naukowy spór ze swoim mistrzem. Korzystając z płaszcza krytyki pewnego frag­mentu badań Łukasiewicza, wyraża jednak odmienne stanowisko wobec, jego zdaniem, błędnych sposobów definiowania przez Łukasiewicza terminów użytych w klasyfikacji rozumowań. Ajdukiewicz problematykę dotyczącą definicji uważał za bardzo ważną. W swoich pracach pod­kreśla, co zauważają współcześni badacze myśli Ajdukiewicza, że pewne aspekty teorii definicji wykraczają poza sformalizowane procedury definicyjne. Analiza jego poglądów świadczy o tym, że on sam aspirował do utworzenia nowej nauki o definicjach w logice. W krytyce, której się podejmuje wobec klasyfikacji rozumowań Jana Łukasiewicza, sam podkreśla, że dotyczy ona definicji, nie zaś procesu rozumowań. Zdaniem Ajdukiewicza definicje systemu logicznego muszą być sensowne i wyrażone w zdaniu danego języka logiki L, natomiast definicje Łukasiewicza prezentowane w klasyfikacji są sensu pozbawione oraz zawierają błąd ekwiwokacji. Fragmentaryczne ujęcie w krytyce problemu defi­niowania stanowi początek jego pogłębionych badań nad rozumieniem i rolą definicji w logice.
 Metoda badawcza: W artykule metodą analityczną wykazuje się odmienne podejścia logików do roli definicji w logice. Zestawienie poglądów Kazimierza Ajdukiewicza na temat klasyfikacji rozumowań z poglądami autora klasyfikacji – Jana Łukasiewicza wykazuje różnice w stanowi­skach logików, co do metodologii systemów logicznych. W ten sposób prezentowane jest rzadko podnoszone w nauce i dydaktyce zdanie Ajdukiewicza, iż metodologia Łukasiewicza jest niepra­widłowa, gdyż utworzona została w oparciu o definicje opatrzone błędem ekwiwokacji. Zdaniem Ajdukiewicza Łukasiewicz, wbrew zasadom metodologii logiki klasycznej, nadał definicjom ist­niejącym nowe znaczenie, tworząc przez to nieprawidłowy system logiki.
 Wyniki: Problematyka definicji podejmowana jest często zarówno w nauce, jak i procesie dydak­tycznym. Klasyfikacja rozumowań autorstwa Jana Łukasiewicza w dydaktyce podawana jest do dziś. Inaczej z jej krytyką autorstwa Kazimierza Ajdukiewicza, która jest podejmowana w lite­raturze niezwykle rzadko. Prezentowana w artykule analiza poglądów czołowych polskich logi­ków wobec klasyfikacji rozumowań stanowi cenny wkład w rozwój tematyki o rozumowaniach w logice. Prezentowana w artykule analiza krytyki Ajdukiewicza wskazuje na nowe oblicze jego poglądów dotyczące sporu o definicje w logice. Ta nowa perspektywa może zaowocować rozsze­rzeniem badań dotyczących znaczenia definicji w logice dwu- i trójwartościowej.

NEW MATRIX BASED ALGORITHM FOR CALCULATION OF IMPORTANCE MEASURES

The system reliability/availability is complex term that is evaluated based on numerous indices and measures. There are different methods for the calculation of these indices and measures. Some of the most used are importance measures. These measures allow to evaluate the influence of fixed system components or set of components to the system reliability/availability. Importance measures are used to allow for various aspects of the impact of system elements on its failure or operability. Analysis of element importance is used in the system design, diagnosis, and optimization. In this paper new algorithm for the calculation some of importance measures is developed based on the matrix procedures. This paper goal is development of new algorithm to calculate importance measures of the system based on the matrix procedures that can be transformed in the parallel procedures/algorithm. This algorithm is developed based on the application of Logical Differential Calculus of Boolean logic for importance analysis of system. The application of parallel algorithm in importance analysis allows the evaluation of system of large dimension. Importance specific of the proposed matrix procedures for calculation of importance measures is the application of structure function for the mathematical representation of investigated system. This function defined the correlation of the system components states and system reliability/ availability. The structure function in this case is defined as truth vector to be used in the matrix transformation. The truth vector of Boolean function is column of the truth table of function if the values of the variables are lexicographically ordered

THE MATRIX PROCEDURES FOR CALCULATION OF IMPORTANCE MEASURES

System reliability/availability is a complex concept that is evaluated based on numerous indices and measures. There are different methods for the calculation of these indices and measures in reliability analysis. Some of the most used indices are important measures. These measures allow us to evaluate the influence of fixed system components or set of components to the system reliability/availability. Importance measures are used today to allow for various aspects of the impact of system elements on its failure or operability. Analysis of element importance is used in the system design, diagnosis, and optimization. In this paper new algorithms for the calculation, some of the important measures are developed based on the matrix procedures. This paper's goal is the development of a new algorithm to calculate importance measures of the system based on the matrix procedures that can be transformed in the parallel procedures/algorithms. These algorithms are developed based on the application of Logical Differential Calculus of Boolean logic for the important analysis of the system. The application of parallel algorithms in importance analysis allows the evaluation of the system of large dimensions. Importance specific of the proposed matrix procedures for calculation of importance measures is the application of structure-function for the mathematical representation of the investigated system. This function defined the correlation of the system components states and system reliability/ availability. The structure-function, in this case, is defined as a truth vector to be used in the matrix transformation. The truth vector of a Boolean function is a column of the truth table of function if the values of the variables are lexicographically ordered. Therefore, the structure-function of any system can be represented by the truth vector of 2n elements un-ambiguously.

On the system CB1 and a lattice of the paraconsistent calculi

In this paper, we present a calculus of paraconsistent logic. We propose an axiomatisation and a semantics for the calculus, and prove several important meta-theorems. The calculus, denoted as CB1, is an extension of systems PI, C min and B1, and a proper subsystem of Sette’s calculus P1. We also investigate the generalization of CB1 to the hierarchy of related calculi.

NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF

AbstractNatural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas.To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/.We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation.Hilbert 1918

A History Based Logic for Dynamic Preference Updates

History based models suggest a process-based approach to epistemic and temporal reasoning. In this work, we introduce preferences to history based models. Motivated by game theoretical observations, we discuss how preferences can dynamically be updated in history based models. Following, we consider arrow update logic and event calculus, and give history based models for these logics. This allows us to relate dynamic logics of history based models to a broader framework.

A resolution calculus for MinSAT

Abstract The logical calculus for SAT are not valid for MaxSAT and MinSAT because they preserve satisfiability but not the number of unsatisfied clauses. To overcome this drawback, a MaxSAT resolution rule preserving the number of unsatisfied clauses was defined in the literature. This rule is complete for MaxSAT when it is applied following a certain strategy. In this paper we first prove that the MaxSAT resolution rule also provides a complete calculus for MinSAT if it is applied following the strategy proposed here. We then describe an exact variable elimination algorithm for MinSAT based on that rule. Finally, we show how the results for Boolean MinSAT can be extended to solve the MinSAT problem of the multiple-valued clausal forms known as signed conjunctive normal form formulas.

CSE_E 1.0: An Integrated Automated Theorem Prover for First-Order Logic

First-order logic is an important part of mathematical logic, and automated theorem proving is an interdisciplinary field of mathematics and computer science. The paper presents an automated theorem prover for first-order logic, called C S E _ E 1.0, which is a combination of two provers contradiction separation extension (CSE) and E, where CSE is based on the recently-introduced multi-clause standard contradiction separation (S-CS) calculus for first-order logic and E is the well-known equational theorem prover for first-order logic based on superposition and rewriting. The motivation of the combined prover C S E _ E 1.0 is to (1) evaluate the capability, applicability and generality of C S E _ E , and (2) take advantage of novel multi-clause S-CS dynamic deduction of CSE and mature equality handling of E to solve more and harder problems. In contrast to other improvements of E, C S E _ E 1.0 optimizes E mainly from the inference mechanism aspect. The focus of the present work is given to the description of C S E _ E including its S-CS rule, heuristic strategies, and the S-CS dynamic deduction algorithm for implementation. In terms of combination, in order not to lose the capability of E and use C S E _ E to solve some hard problems which are unsolved by E, C S E _ E 1.0 schedules the running of the two provers in time. It runs plain E first, and if E does not find a proof, it runs plain CSE, then if it does not find a proof, some clauses inferred in the CSE run as lemmas are added to the original clause set and the combined clause set handed back to E for further proof search. C S E _ E 1.0 is evaluated through benchmarks, e.g., CASC-26 (2017) and CASC-J9 (2018) competition problems (FOFdivision). Experimental results show that C S E _ E 1.0 indeed enhances the performance of E to a certain extent.

Modal Resolution

Resolution-based provers for multimodal normal logics require pruning of the search space for a proof to ameliorate the inherent intractability of the satisfiability problem for such logics. We present a clausal modal-layered hyper-resolution calculus for the basic multimodal logic, which divides the clause set according to the modal level at which clauses occur to reduce the number of possible inferences. We show that the calculus is complete for the logics being considered. We also show that the calculus can be combined with other strategies. In particular, we discuss the completeness of combining modal layering with negative and ordered resolution and provide experimental results comparing the different refinements.

Two proofs of the algebraic completeness theorem for multilattice logic

Shramko [(2016). Truth, falsehood, information and beyond: The American plan generalized. In K. Bimbo (Ed.), J. Michael Dunn on information based logics, outstanding contributions to logic (pp. 191–212). Dordrecht: Springer] formulated multilattice logic and the algebraic completeness theorem for it. However, the proof has not been presented. In this paper, we consider Kamide and Shramko's multilattice logic [Kamide & Shramko (2017a). Embedding from multilattice logic into classical logic and vice versa. Journal of Logic and Computation, 27(5), 1549–1575] which is an extension of Shramko's original multilattice logic by several implications and coimplications. Using the technique of algebraic embedding, we show that Kamide and Shramko's sequent calculus for multilattice logic is sound and complete with respect to multilattices. Moreover, we introduce yet another algebraic semantics for this logic based on the notion of a De Morgan multilattice. Using Lindenbaum-Tarski algebras, we show that is sound and complete with respect to De Morgan multilattices. Besides, we modify Kamide and Shramko's notion of modal multilattice [Kamide & Shramko (2017b). Modal multilattice logic. Logica Universalis, 11(3), 317–343], i.e. we present the concept of De Morgan modal multilattice. We prove that Kamide and Shramko's modal multilattice logic (Kamide & Shramko, 2017b) is adequate with respect to De Morgan modal multilattices.

A minimization algorithm for automata generated fault trees with priority gates

Fault tree analysis is still widely practiced in high-hazard industries. We propose in this article an algorithm for the reduction of fault tree expressions that are generated from automata representations of failure behaviors. Automata formalisms are increasingly being used to describe systems exhibiting sequence-dependent failures—i.e., the overall outcome like a total failure of the system can depend on the order in which events occur. A set of paths leading to a safety-relevant state is encoded as a standard sum of product canonical form, and without any loss of the significance of the sequencing of events. That is, the corresponding fault tree expression is basically a Boolean formula which is extended with the necessary temporal features (event occurrence priority). Such expressions can then be reduced into minimal canonical forms by using the Boolean methods together with the required temporal logic calculus. Since minimal failure sequences can be determined from the obtained reduced models, the proposed approach can improve the analysis of the dynamic effects of the sequencing of faults and propagated errors in such models. As a consequence, it can have a positive impact on the design of failure prevention measures. A fault tolerant example system exhibiting dynamic behavior is used to highlight the benefits of the approach.

Full Cut Elimination and Interpolation for Intuitionistic Logic with Existence Predicate

In previous work by Baaz and Iemhoff, a Gentzen calculus for intuitionistic logic with existence predicate is presented that satisfies partial cut elimination and Craig's interpolation property; it is also conjectured that interpolation fails for the implication-free fragment. In this paper an equivalent calculus is introduced that satisfies full cut elimination and allows a direct proof of interpolation via Maehara's lemma. In this way, it is possible to obtain much simpler interpolants and to better understand and (partly) overcome the failure of interpolation for the implication-free fragment.

Gentzen sequent calculi for some intuitionistic modal logics

Abstract Intuitionistic modal logics are extensions of intuitionistic propositional logic with modal axioms. We treat with two modal languages ${\mathscr{L}}_\Diamond $ and $\mathscr{L}_{\Diamond ,\Box }$ which extend the intuitionistic propositional language with $\Diamond $ and $\Diamond ,\Box $, respectively. Gentzen sequent calculi are established for several intuitionistic modal logics. In particular, we introduce a Gentzen sequent calculus for the well-known intuitionistic modal logic $\textsf{MIPC}$. These sequent calculi admit cut elimination and subformula property. They are decidable.

A bilateral Hilbert-style investigation of 2-intuitionistic logic

AbstractWe develop a bilateral Hilbert-style calculus for 2-intuitionistic logic of Heinrich Wansing. This calculus is defined over signed formulas of two types: formulas signed with plus correspond to assertions, while formulas signed with minus correspond to rejections. In this way, the provided system is a Hilbert-style calculus, which does take rejection seriously by considering it a primitive notion on par with assertion. We show that this presentation is not trivial and provide two equivalent axiomatizations obtained by extending intuitionistic and dual intuitionistic logics, respectively. Finally, we show that 2-intuitionistic logic is in some sense definitionally equivalent to a variant of Nelson’s logic with constructible falsity.

A Complete Axiomatisation for the Logic of Lattice Effect Algebras

In a recent work Foulis and Pulmannov\' a \cite{Foulis2012} studied the logical connectives in lattice effect algebras. In this paper we extend their study and investigate further the logical calculus for which the lattice effect algebras can serve as semantic models. We shall first focus on some properties of lattice effect algebras and will then give a complete axiomatisation of this logic.

Proof nets for multiplicative cyclic linear logic and Lambek calculus

AbstractThis paper presents a simple and intuitive syntax for proof nets of the multiplicative cyclic fragment (McyLL) of linear logic (LL). The main technical achievement of this work is to propose a correctness criterion that allows for sequentialization (recovering a proof from a proof net) for all McyLL proof nets, including those containing cut links. This is achieved by adapting the idea of contractibility (originally introduced by Danos to give a quadratic time procedure for proof nets correctness) to cyclic LL. This paper also gives a characterization of McyLL proof nets for Lambek Calculus and thus a geometrical (i.e., non-inductive) way to parse phrases or sentences by means of Lambek proof nets.

BUNDER’S PARADOX

AbstractSystems ofillative logicare logical calculi formulated in the untypedλ-calculus supplemented with certain logical constants.1In this short paper, I consider a paradox that arises in illative logic. I note two prima facie attractive ways of resolving the paradox. The first is well known to be consistent, and I briefly outline a now standard construction used by Scott and Aczel that establishes this. The second, however, has been thought to be inconsistent. I show that this isn’t so, by providing a nonempty class of models that establishes its consistency. I then provide an illative logic which is sound and complete for this class of models. I close by briefly noting some attractive features of the second resolution of this paradox.