Propositional System Research Articles

An efficient propositional system for Abductive Logic Programming

Abductive logic programming (ALP) extends logic programming with hypothetical reasoning by means of abducibles, an extension able to handle interesting problems, such as diagnosis, planning, and verification with formal methods. Implementations of this extension have been using Prolog meta-interpreters and Prolog programs with Constraint Handling Rules (CHR). While the latter adds a clean and efficient interface to the host system, it still suffers in performance for large programs. Here, the concern is to obtain a more performant implementation of the SCIFF system following a compiled approach. This paper, as a first step in this long term goal, sets out a propositional ALP system following SCIFF, eliminating the need for CHR and achieving better performance.

La mediación en el ámbito de familia, niñez y adolescencia, como método efectivo para optimizar el sistema procesal en la ciudad de Milagro

Mediation as an alternative method of conflict resolution is recognised in Ecuadorian legislation, that is, in our Constitution of the Republic of Ecuador, in the Law of Arbitration and Mediation, as well as in the Organic Code of Childhood and Adolescence, and other regulations that refer to it. The objective of this study was to determine the effectiveness in the area of family, children and adolescents, as a mechanism to decongest the procedural system in the city of Milagro. The research focused on a mixed, descriptive, explanatory and propositional system that allowed us to understand the procedural phenomenon and to make a proposal with the aim of optimising the principle of procedural speed and economy. In the same way, empirical level methods, methods of analysis and documentary review, theoretical level methods and techniques and instruments such as interviews and surveys were used. For this reason, the proposal was made, which implies the obligatory nature of an act of impossibility of mediation that accompanies the lawsuit, prior to the initiation of a judicial process, which promotes the achievement of amicable, fast and satisfactory agreements, as well as the decongestion of the Ecuadorian justice system, based on factors such as speed, confidentiality and a lower amount of expenses for the processing of the process, and safeguards the welfare and interest of minors. A culture of peace prevails between parties, where an agreement is reached that is centred on their needs, contributing to a fair, equitable and collaborative society.

Base-extension semantics for modal logic

Abstract In proof-theoretic semantics, meaning is based on inference. It may seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems $K$, $KT$, $K4$ and $S4$, with $\square $ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between $\square $ and a natural presentation of $\lozenge $. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases.

Theory of ∆ conditional randomized truth degree in Gödel n-valued propositional logic system of adding ∆ operator

Theory of ∆ conditional randomized truth degree in Gödel n-valued propositional logic system of adding ∆ operator

An integrative control theory perspective on consciousness.

An integrative account of consciousness should have a number of properties. It should build upon a framework of nonconscious behavior in order to explain how and why consciousness contributes to, and addresses the limitations of, nonconscious processes. It should also encompass the primary (phenomenal), secondary (access), and tertiary (self-awareness) aspects of consciousness. A number of accounts have proposed a role for consciousness in the prediction of sensory input, yet these proposals do not address how organisms deal with multiple, unpredictable, disturbances to maintain control. According to perceptual control theory (PCT), purposiveness is the control of hierarchically organized perceptual variables via changes in output that counteract disturbances which would otherwise increase error between the current value and the reference value (goal state) of each perceptual variable. In PCT, reorganization is the process required for the adaptive modification of control systems in order to reduce the error in intrinsic systems that control essential, largely physiological, variables. The current article proposes that primary consciousness emerges from this system, and is sustained as secondary consciousness through a number of processes including the control of the integration rate of novel information via exploratory behavior, attention, imagination, and altering the mutation rate of reorganization. Tertiary consciousness arises when internally sustained perceptual information is associated with specific symbols that form a parallel, propositional system for the use of language, logic, and other symbolic systems. The hypotheses and initial research designs to test this account are provided. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Understanding the Axioms and Assumptions of Logical Mathematical Systems through Raster Images: Application to the Construction of a Likert Scale

This article presents different artistic raster images as a resource for correcting misconceptions about different laws and assumptions that underlie the propositional systems of binary logic, Łukasiewicz’s trivalent logic, Peirce’s trivalent logic, Post’s n-valent logic, and Black and Zadeh’s infinite-valent logic. Recognizing similarities and differences in how images are constructed allows us to deepen, through comparison, the laws of bivalence, non-contradiction, and excluded middle, as well as understanding other multivalent logic assumptions from another perspective, such as their number of truth values. Consequently, the first goal of this article is to illustrate how the use of visualization can be a powerful tool for better understanding some logic systems. To demonstrate the utility of this objective, we illustrate how a deeper understanding of logic systems helps us appreciate the necessity of employing Likert scales based on the logic of Post or Zadeh, which is the second goal of the article.

AN ALGEBRAIC PROOF OF COMPLETENESS FOR MONADIC FUZZY PREDICATE LOGIC

Abstract Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$ -valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$ , which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$ . The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$ , we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$ , giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$ , and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras.

A Comprehensive Formalization of Propositional Logic in Coq: Deduction Systems, Meta-Theorems, and Automation Tactics

The increasing significance of theorem proving-based formalization in mathematics and computer science highlights the necessity for formalizing foundational mathematical theories. In this work, we employ the Coq interactive theorem prover to methodically formalize the language, semantics, and syntax of propositional logic, a fundamental aspect of mathematical reasoning and proof construction. We construct four Hilbert-style axiom systems and a natural deduction system for propositional logic, and establish their equivalences through meticulous proofs. Moreover, we provide formal proofs for essential meta-theorems in propositional logic, including the Deduction Theorem, Soundness Theorem, Completeness Theorem, and Compactness Theorem. Importantly, we present an exhaustive formal proof of the Completeness Theorem in this paper. To bolster the proof of the Completeness Theorem, we also formalize concepts related to mappings and countability, and deliver a formal proof of the Cantor–Bernstein–Schröder theorem. Additionally, we devise automated Coq tactics explicitly designed for the propositional logic inference system delineated in this study, enabling the automatic verification of all tautologies, all internal theorems, and the majority of syntactic and semantic inferences within the system. This research contributes a versatile and reusable Coq library for propositional logic, presenting a solid foundation for numerous applications in mathematics, such as the accurate expression and verification of properties in software programs and digital circuits. This work holds particular importance in the domains of mathematical formalization, verification of software and hardware security, and in enhancing comprehension of the principles of logical reasoning.

Unified theory of integral true degrees in NM theory

Based on the NM propositional logic system of nilpotent minimum logic, the concept of the integral truth degree of the formula is firstly proposed by integrating the function induced by the formula, and the MP rule and HS rule of the integral truth degree are proved by means of the integral invariance. Secondly, in NM propositional logic the integral similarity and integral pseudo-distance are introduced into the set of general formulas of the system, and some good properties about the similarity and pseudo-distance are proved. Finally, in terms of the concept of divergence degrees and diameter, a new membership function for reflecting the consistency degrees of theories in NM propositional fuzzy logic is proposed, which is proved that of inconsistent theories are equal to 0 and that of completely consistent theories are equal to 1.

Neural correlates of performance monitoring vary as a function of competition between automatic and controlled processes: An ERP study

Dual process theories of attitude formation propose that an evolutionary old associative system automatically generates subjective judgments by processing mere spatiotemporal contiguity between paired objects, subjects, or events. These judgments can potentially contradict our well-reasoned evaluations and highjack decisional or behavioral outcomes. Contrary to this perspective, other models stress the exclusive work of a single propositional system that consciously process co-occurrences between environmental cues and produce propositions, i.e., mental statements that capture the specific manner through which stimuli are linked. We constructed an experiment on the premise that it would be possible, if the associative system does produce attitudes in a parallel non-conscious fashion, to condition two mutually exclusive attitudes (one implicit, the other explicit) toward a same stimulus. Through explicit ratings, inhibition performance, and neural correlates of performance monitoring, we assessed whether there was a discrepancy between stimuli that were conditioned with (1) the two systems working in harmony (i.e., producing congruent attitudes), or (2) the two systems working in competition (i.e., producing incongruent attitudes). Compared with congruent stimuli, incongruent stimuli consistently elicited more neutral liking scores, higher response times and error rates, as well as a diminished amplitudes in two well-studied neural correlates of automatic error detection (i.e., error-related negativity) and conscious appraisal of error commission (i.e., error-related positivity). Our findings are discussed in the light of evolutionary psychology, dual-process theories of attitude formation and theoretical frameworks on the functional significance of error-related neural markers.

Aspects of Language Monadic Predicate Logic System plus Identity (LMPL+I)

In this paper, we devoted to study the language monadic predicate logic system plus identity (LMPLS+I) as extension of the language of propositional logic system (LPLS). I.e., (LPLS)⊂ (LMPLS+I), which it contains all the hereditary traits (or features) of (LPLS) , furthermore that, we will add some new data information between relationship of object, subject and predicate. This is the task of monadic predicate logic system addressed. As mentioned in pervious papers, the main task of system of logic is classifying between valid and invalid arguments, moreover, the central role the system of logic how distinguishes between the conclusions which follow from their premises of the arguments and those do not follow from their premises. As a matter of fact, when we encounter some proofs that seem perceptually (or intuitively) sound, but we are -unable to prove their validity due to the inability language of propositional logic system (LPLS). Hence, it was necessary to uses the monadic predicate logic system (LMPLS+I) to overcome this problem. In this article, we study syntax, semantics and inference on language monadic predicate logic system plus identity (LMPLS+I) and investigation characteristics of arguments such valid \ invalid and types of formulas and relations between formulas like consistency and inconsistency sets.

A rewriting logic approach to specification, proof-search, and meta-proofs in sequent systems

This paper develops an algorithmic-based approach for proving inductive properties of propositional sequent systems such as admissibility, invertibility, cut-admissibility, and identity expansion. Although undecidable in general, these structural properties are crucial in proof theory because they can reduce the proof-search effort and further be used as scaffolding for obtaining other meta-results such as consistency. The algorithms –which take advantage of the rewriting logic meta-logical framework– are explained in detail and illustrated with examples throughout the paper. They have been fully mechanized in the L-Framework, thus offering both a formal specification language and off-the-shelf mechanization of the proof-search algorithms coming together with semi-decision procedures for proving theorems and meta-theorems of the object system. As illustrated with case studies in the paper, the L-Framework achieves a great degree of automation when used on several propositional sequent systems, including single conclusion and multi-conclusion intuitionistic logic, classical logic, classical linear logic and its dyadic system, intuitionistic linear logic, and normal modal logics.

Conditional probability measures on propositional systems

Conditional probability measures on propositional systems

AN ALGEBRAIC APPROACH TO INQUISITIVE AND -LOGICS

AbstractThis article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$ -logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$ -varieties. We prove that the lattice of $\mathtt {DNA}$ -logics is dually isomorphic to the lattice of $\mathtt {DNA}$ -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety theorems. We also introduce locally finite $\mathtt {DNA}$ -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of $\mathtt {InqB}$ is dually isomorphic to the ordinal $\omega +1$ and give an axiomatisation of these logics via Jankov $\mathtt {DNA}$ -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].1

Planning for an Efficient Implementation of Hypothetical Bousi∼Prolog

AbstractThis paper explores the integration of hypothetical reasoning into an efficient implementation of the fuzzy logic language Bousi∼Prolog. To this end, we first analyse what would be expected from a logic inference system, equipped with what is called embedded implication, to model solving goals with respect to assumptions. We start with a propositional system and incrementally build more complex systems and implementations to satisfy the requirements imposed by a system like Bousi∼Prolog. Finally, we propose an inference system, operational semantics and the translation function to generate efficient Prolog programmes from Bousi∼Prolog programmes.

Using linear algebra in decomposition of Farkas interpolants

The use of propositional logic and systems of linear inequalities over reals is a common means to model software for formal verification. Craig interpolants constitute a central building block in this setting for over-approximating reachable states, e.g. as candidates for inductive loop invariants. Interpolants for a linear system can be efficiently computed from a Simplex refutation by applying the Farkas’ lemma. However, these interpolants do not always suit the verification task—in the worst case, they can even prevent the verification algorithm from converging. This work introduces the decomposed interpolants, a fundamental extension of the Farkas interpolants, obtained by identifying and separating independent components from the interpolant structure, using methods from linear algebra. We also present an efficient polynomial algorithm to compute decomposed interpolants and analyse its properties. We experimentally show that the use of decomposed interpolants in model checking results in immediate convergence on instances where state-of-the-art approaches diverge. Moreover, since being based on the efficient Simplex method, the approach is very competitive in general.

Quantitative Method Based on Cotangent Similarity Degree in Three‐Valued Łukasiewicz Logic

The main purpose of this paper is to establish a type of quantitative model by using the contangent similarity function in the three-valued Ł ukasiewicz propositional logic system \begin{document}$\text{Ł}_{3}$\end{document} . We introduce the concepts of the cotangent similarity degree, cotangent pseudo-distance and cotangent truth degree of the propositions, together with their basic properties in \begin{document}$\text{Ł}_{3}$\end{document} . We investigate the relationship between the cotangent truth degree and contangent pseudo-distance, and prove the continuity of the logical connectives \begin{document}$\neg, \vee$\end{document} and \begin{document}$\rightarrow$\end{document} in the \begin{document}$\text{Ł}_{3}$\end{document} logical metric space. We propose a graded reduction method and three types of graded reasoning frameworks on the propositions set F(S), and provide several examples and basic properties of it.

Labelled Sequent Calculi for Lewis’ Non-normal Propositional Modal Logics

C. I. Lewis’ systems were the first axiomatisations of modal logics. However some of those systems are non-normal modal logics, since they do not admit a full rule of necessitation, but only a restricted version thereof. We provide G3-style labelled sequent calculi for Lewis’ non-normal propositional systems. The calculi enjoy good structural properties, namely admissibility of structural rules and admissibility of cut. Furthermore they allow for straightforward proofs of admissibility of the restricted versions of the necessitation rule. We establish completeness of the calculi and we discuss also related systems.

Biblical theology for oral-preference communities

Teaching in a non-individualist oral-preference culture in Indonesia, I have encountered the mismatch of theology developed for Western cultures and the needs of my students and their congregations that characterizes much postcolonial theological education. The use of proverbs and poetry as resources for theology addresses the need for theology to inspire and guide, but not the critical function of guarding responsible interpretation. Western theology has done the latter via abstraction of the biblical material into consistent propositional systems, an approach that also functions to inspire and guide for many of the highly literate people who developed Western theology. This connects with the deep Western philosophical tradition of finding order through abstraction, whereas oral and even literate Asian cultures create order through ritual. The Bible is not Western, however, and we show how the biblical story or theo-drama guides, guards, and inspires in a way that opens space for local theology. We then show how the identity of believers within this story is best developed for people in non-individualist cultures through ecclesial metaphors that in turn connect with divine metaphors. Three such “divine-ecclesial institutions” are sketched, namely family, kingdom, and cult. They shed light on missiological issues such as the priority of personal growth versus ritual observance. They provide a way of organizing the biblical material that offers a better starting point for the dialogue of contextualization than systematic Western theology.

Frege Systems for Quantified Boolean Logic

We define and investigate Frege systems for quantified Boolean formulas (QBF). For these new proof systems, we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class C to the QBF Frege system operating with lines from C . Such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC 0 [ p ] circuits. In contrast, any non-trivial lower bound for propositional AC 0 [ p ]-Frege constitutes a major open problem. Improving these lower bounds to unrestricted QBF Frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity. In particular, proving a lower bound for QBF Frege systems operating with arbitrary P/poly circuits is equivalent to either showing a lower bound for P/poly or for propositional extended Frege (which operates with P/poly circuits). We also compare our new QBF Frege systems to standard sequent calculi for QBF and establish a correspondence to intuitionistic bounded arithmetic.